Analytic Methods in Aircraft Aerodynamics

r

APPENDIX CALCULATION OF QUADRATURE COEfFICIENTS (dfn/dn)sin e de

II = lim [~ f (n ) - f+1

E n v cos e - cos e E-+O v -I (AI) Introducing equation (23) and differentiating with respect to n yields: (A2) Th .e integral can be solved analytically:

'IT __ ---::---'-- __ ~ sin jJ e "

cos jJe de v (A3) cos e - cos e = 'IT sin 8 v v

i

The relationships of equations (24) transform II into 2 M M-2

f '(n. 2) b . + ~ f (n.)b . (A4)

II = """' f '(n. 2 )b . +

L...J n J+ VJ n J- vJ L...J n J VJ

L

j=1 j =M-1 j =3 where G.b.

J vJ (j = I, 2, M - 1, M and S, 6, . . . , M - 4)

b . =

sin e vJ v (ASa) G.b. - G. c ./sin 8.

J vJ J-2 vJ J (ASb) (j 3, 4) b . = sin e VJ v G.b . - G. 2c . / sin 8.

J vJ J+ vJ J (j = M - 3, M - 2) b . = CASc) sin e VJ v The coeff i cients b . and c. ar e sums over ~ and can be evaluated in closed form: V J VJ M ~ s i n ~e sin ~e .

b.

= 2 L: \! J \! J ~ =l v = i { M[ M + _ 2 _(_S_i_n_M_ e_ v_s_i_n_ 8_ V _-_s...,.:_:_s-;:; 8 :- : _ 8_ v _C_O_S_ 8_ v _)_s_i_n_ M _ 8 _ ]. (:~: ~: v Y) Me cos M8. e . - cos M e Me . sin 6 (sin sin sin \!

v J J J Me = s i n M8 . +

M [ Si n

V J

v cos 8 .

- cos e J v J sin M8 . - sin sin 8 . (1 - cos M 8 cos Me. ) (1 - cos 8 cos 8.) s i n M 8 e v v J v J v J J + (cos 8. - cos e ) \!

J if 8 8 .

-f \!

J M = 2 c sin ~e cos ~8 .

L ~2

\!

\! j J ~ = l sin M e cos M e (2 sin M e - 1) cos

a v} M \! \! \!

M8 cos Me

M2 [S in = +

\! \!

2 sin 2 e sin 8 \!

\!

sin M8 cos 8 \! \!

8 = e .

if \!

J sin 2 8 \!

r

M 2 (COS Me cos Me. sin e + sin Me sin Me. sin e.)

\! J \! V J J c . M2 sin MS cos MS.

VJ v J cos e . - cos e J \!

2M[(1 - cos 8 cos S. )sin MS cos MS. - sin MS. cos MS sin S. sin S ] \! J \I J J \I J \!

+ (cos 8 . - cos S )2 J \I (cos M S cos MS. - l)sin S (2 - cos S . cos S \I J \! J \!

( + sin M S sin M S . sin S .(2 - cos S. cos S + \! .J J J v if 8 "f e.

\! J (cos 8 . - cos e )3 J \I If equations (24) are put into equation (23), the summations over ~ can be evaluated in closed form: . M 5 ( S) sin 118 sin 118

= 2~

m m ~=l sin M8 cos M8 cos 8 m m m = M + sin MS if 8 8 m m sin 8 m (sin M8 cos Me sin 8 - sin M8 cos M8 sin 8 ) m m m

= sin M8 sin M8 +

m cos 8 - cos 8 m

if e "f 8

m r (8) cos ~8 sin ~8 m m M(sin2 M8 - O.S)cos 8 m m

= sin M 8 cos M 8 (M + O.5/sin 8 ) +

m m m sin 8 m if (cos MS cos MS sin e + sin MS sin M8 m m cos M e + m cos e - cos 8 m (1 - cos 8 cos 8 )sin M e cos M 8 - cos M8 sin M8 sin 8 sin 8 m m m m + -------------------------------------------------------------- (cos e - cos 8 ) 2 m where 8 is equal to arc cos n. or arc cos n .. Thus, the coefficients 1 J sm(n ) or sm(nj) are obtained from: i

s ( n ) = 1:. G 5 ( 8 ) (m = 1, 2, M - 1, M, and 5, 6, . . . , M - 4)

m 1f m m (A6a) Gm_ 2 rmC a J] s (n) (m = 3, 4) (A6b) [ G 5 caJ mm m 1f sin 8 m 1 [ G r C a J]

G 5 ( 8 ) _ m+ 2 m

s (n) - - (m = M - 3, M - 2) (A6c) m 1f m m sin 8 m Hence, M M- 2

f (n) = f (arc cos n) = ~ s Cn)f' + +~s (n)f

n

n L..J m nm+ 2

L..J m nm

L: m=l m=M - l m=3 (A7)

l

-- '1

I

!

REFERENCES 1. Landahl, M. T.; and Stark, V. J. E.: Numerical Lifting-Surface Theory - Problems and Progress. Preprint 68-72, AIAA, 1968.

2. Watkins, C. E.; Woolston, D. S.; and Cunningham, H. G.: A Systematic Kernel Function Procedure for Determining Aerodynamic Forces on Oscillat- ing or Steady Finite Wings at Subsonic Speeds. NASA TR R-48, 1959.

3. Multhopp, H.: Methods for Calculating the Lift Distribution of Wings (Subsonic Lifting-Surface Theory). A.R.C . Rep. and Mem. 2884, 1955.

4. Truckenbrodt, E.: Tragflaechentheorie bei inkompressibler Stroemung.

WGL Jahrbuch, 1953, pp. 40-65.

5. Multhopp, H.: Die Berechnung der Auftriebsverteilung von Tragfluegeln.

Luft.-Forschung, vol. IS, 1938, pp. 153-169.

6. Wagner, S.: On the Singularity Method of Subsonic Lifting-Surface Theory.

Preprint 69-37, AIAA, 1969.

7. Hildebrand, F. B.: Introduction to Numerical Analysis. McGraw-Hill Book Co., Inc., New York, 1956.

8. Graham, D.: Chordwise and Spanwise Loadings Measured at Low Speeds on a Large Triangular Wing Having an Aspect Ratio of 2 and a Thin, Subsonic- Type Airfoil Section. NACA RM A50A04a, 1950.

9. Kelly, M. W.; and Tolhurst, W. H.: The Use of Area Suction to Increase the Effectiveness of a Trailing-Edge Flap on a Triangular Wing of Aspect Ratio 2. NACA RM A54A25, 1954.

10. Landahl, M.: Pressure-Loading Functions for Oscillating Wings With Control Surfaces. AIAA J., vol. 6, no. 2, 1968, pp. 345-348.

11. Crespo, A. N.; and Cunningham, H. J.: On the Calculation of the Three- Dimensional Pressure Distribution on Wings With Control Surfaces, Based on the Integral Equation for Subsonic Flow. NASA TN D-54l9, 1969.

EXAMPLES FOR SPANWISE ANGLE-OF-ATTACK DISTRIBUTIONS WITH DISCONTINUITIES WING WITH FL APS WING-BODY COMBINATION :.~'" '. '" ~ ~ ... ~j,'- .~~., ~ "" ..

~

Cb~ "

WING WITH TIP TANKS "' " Figure 1 WING GEOMETRY AND COORDINATE SYSTEM --,--'--' - y, y'

I

' Ate('I7v-O)

! x, x' Ate('I7v +0) I

~ ---- b . , Figure 2 -- -- ------------- CHORDWISE PRESSURE DISTRIBUTION FUNCTIONS ho -I .2 .4 .6 .8 1.0 X F igu re 3 SPANWISE CONTROL-POINT LOCATIONS

r V? r ~ ~

I II , I II

, II I , ,\ II I I I

1\ I i I PRESENT METHOD (E XTENSION OF REF. 6)

. II I, I I I • PREASSIGNED ABSCISSAE

I i 0 GAUSSIAN NODES

, <» ADDITIONAL CONTROL-POINT LOCATIONS I

~ §4l,~*i FLAP

o MULTHOPP'S (REF 3) , TRUCKENBRODT'S (REF 4), AND ORIGINAL METHOD (REF 6) Fi gure 4 SPANWISE PRESSURE MODES, TOTAL LIFT AND LOCATION OF AERODYNAMIC CENTER WITHOUT FLAP AR = 2 .0 M«)=0 .1 3 a=4 .2 ° . 06 .05 I .04 ~ fO .0 3 l I

.02 r

f n I .01 C loc (IOC)Ih L1h C I-- - - 1--- L C L"p Cr (IOC) .. p - ORICINAL THEORY 0.1618 -1.1 % 0. 5921 -0.6% - -- EXTENOEO THEORY 0.1621 -1.3% 0.5991 -1.1% o EXPERINENHREf.8) 0.1600 0.5890 - . 03 . 25 .50 .75 1.00 o ,.., = 2y / b Figure 5 SKETCH OF THEORETICALLY PREDICTED PRESSURE DISTRIBUTION ll Cp Figure 6

L

THREE -DIMENSIONAL PRESSURE DISTRIBUTION y AR = 2 MIX) • .I3 a '" 4 . 2- -- ORIGINAL TliEORY - . - EXTENDED TliEDRY o EXPERIMENT (REF . 8) l>Cp 1.0 o Figure 7 SPANWISE PRESSURE MODES, TOTAL LIFT AND LOCATION OF AERODYNAMIC CENTER W ITH DEFLECTED FLAPS f 2 AR = 2 a = O· 8 = 59· (XOcl'h 1--- ~oc) .. P -.4 - ORIGINAL THEORY 0.8453 -31.5% 0.8111 -2.0'10 - - - EXTENDED THEORY 0. 6415 0.2'10 0.8213 -3.2'10 o EXPERINEHl1REF. 9) 0.643 0 .8 015 ( 'f (y). WITH BOUNOARY-lAYER CON TROll o I I : I I .25 .5 0 : .75 1 . 00 "7=2 Y /b !

'-- --- FLAP --~~ Figure 8 DISCUSSION WILLIAM P. RODDEN, Consulting Engineer: I'd like to ask Dr. Wagner and John Lamar, if he is here from Langley, why we need two more modifications of Multhopp's method. The vortex lattice method and the doublet lattice method for the oscillatory case have been demonstrated in the last few years to be tremendously simple and versatile in terms of the arbitrary types of configura- tions that can be handled. They also seem to have as much accuracy as is available from any other method, and they provide solutions when you are not in a position to guess at the pressure functions, the basic functions needed by Multhopp and Truckenbrodt.

WAGNER: Well, this question is not very easy to answer, because it is true that the vortex lattice methods accurately predict in many cases most of the data a design engineer wants to know. But there are always some cases where we need a better approximation of the physical flow by a lifting surface theory. For instance, if you are going to calculate the leading-edge suction force, it is very difficult to do this with a vortex lattice method because the pressure distribution is not continuous.

I haven't been working so much with vortex lattice methods, so I can only answer this question as far as I have gotten answers from communication with other people. I have learned that these vortex lattice methods are quite sensitive to the selection of the control-point locations and box sizes.

There are some cases where, if you have not selected these control points or box sizes properly, the answers might not be very good.

JOSEPH P. GIESING, Douglas Aircraft Company: You mentioned difficulty in calculating the induced drag distribution by the vortex lattice method. We

used the basic approach where you just use Prandtl's basic F = pvxr. We

get very excellent agreement with, say, Garner's results, so we don't need really to calculate the leading-edge suction.

Also, as far as the optimum control point problem (for the lattice method) is concerned, we believe we have this one worked out. We use one specific set of control point rules, obtained from two-dimensional analysis, and this has given us accurate results in all cases that we have tried for very complicated configurations.

WAGNER: Well, if you have gotten good results, I don't have anyobjec- tions. Maybe it is really better to use those vortex lattice methods. If this is true, why not? Then you can leave all these methods, which are very complicated, such as this lifting-surface method. I don't have any objections against that.

ATLEE M. CUNNINGHAM, JR., General Dynamics, Fort Worth Division: I'd like to say something in support of your method versus the vortex lattice and doublet lattice methods. Although these collocation type methods are more complex, once you get the associated problems worked out, in the end I feel that these types of methods will give some solutions more efficiently.

--- -- - - -- -- - - -- -- As an example, you could take a configuration where you might require 64 doublet lattice panels to obtain nearly the same solution that you would get with six control points through the use of a collocation method. So my con- tention is that although the doublet lattice and vortex panel methods now allow us to treat more complex cases, in the end I believe we will be able to arrive at more efficient solutions once we understand more about these assumed pressure modes and so forth that are required.

I would like to know if you looked at the effect of choosing these off Gaussian chords, that is, what the effect is on trying to account for the spanwise singularity?

WAGNER: I didn't quite understand what you mean. What the effect of the spanwise singularity is, when we are doing what?

CUNNINGHAM: When you are taking these slightly off Gaussian control points.

WAGNER: Qh, I see. I have done thi s before, going off of these Gaussian control points" and what you usually get is very violent oscillation. So if you go away from those Gaussian nodes, you cannot guarantee results, because they depend on the control-point location, and of course this result is not good to use because you don't know whether you get results from the wing configuration or from the location of the control points.

CUNNINGHAM: I have been working with this control surface problem and I have derived a pressure function which also accounts for the chordwise singularity as well as the spanwise variation and I find it works quite well using the standard control points (the control points of Hsu, which spanwise are the same as your spanwise points).

WAGNER: Now, you are talking about chordwise.

CUNNINGHAM: No, I was talking about the spanwise points, the Gaussian points as you had derived. I found that the chordwise points were more critical, and that the solution is very sensitive to where these are located.

WAGNER: Well, in my case, I am using as chordwise control points one at the leading edge, one at the trailing edge; then I am distributing the points between the leading edge and trailing edge equidistantly.

But I have heard from other people - I was talking to some gentlemen at Lockheed - and they indicated it would probably be better to distribute the chordwise location of the control points by the same procedure as we are using for the spanwise locations. But it is advisable to have a control point at the leading edge and at the trailing edge, because there is a paper which was presented by Dr. Jordanl and it indicates that if you are taking more and more chordwise control points, then the Multhopp procedure IJordan, P. F.: Remarks on Applied Subsonic Lifting Surface Theory.

Wissenschaftliche Gesellschaft fur Luft-und Raumfahrt, Jahrbuch 1967, pp. 192-210.

might diverge if you don't take at the same time more spanwise control points and if you are not using the correction for the logarithmic singularities, whereas if you are using a control point at the leading edge and at the trailing edge you can prevent this divergence.

GEORGE R. BARTE, JR., General Electric Company: I'd like to add just a brief historical footnote. In earlier conversations - for the benefit of those attendees here, I had spent a little time with Dr. Wagner, and at the time the question of who came up with what method first was discussed briefly.

It turns out that both in England and in Germany, Truckenbrodt and Multhopp arrived independently at very similar approaches to the question of calculating wing lift distributions.

In Germany at that time, Multhopp, then with Fockewulf and now with the General Electric Company, prefers to reserve some of his advanced work for later publication on the presumption it gives certain advantages in a competitive industry.

J

EDITOR'S COMMENT Following the first paper of the Symposium by Siegfried N. Wagner, Joseph P. Giesing of Douglas Aircraft Company asked for time to present some prepared comments (with figures). He was given ten minutes in which to present a short progress report on the doublet lattice approach in subsonic lifting surface theory. The doublet lattice method is an unsteady extension to the vortex lattice method and is very simple, since it does not require loading functions. However, it does have important limitations and should be applied with special care and knowledge, as emphasized by the comments which follow the prepared slides and written remarks of Giesing.

PREPARED FIGURES AND REMARKS OF JOSEPH P. GIESING, Douglas Aircraft Company FIGURE 1 Figure 1 compares the spanwise lift-curve-slope distribution across a wing-fuselage combination as calculated by Douglas (nonplanar) (ref. 1), Douglas (planar) (ref. 2), and Woodward (ref. 3). Lifting surface panels were placed both on the wing and fuselage for the Douglas (nonplanar) and Woodward methods. The control point on each box in the Woodward method was selected as the 8S-percent point on the basis of two-dimensional calculations.

Woodward, in reference 3, incorrectly suggests the 9S-percent point. The control point for the Douglas methods, both planar and nonplanar, is at the 7S-percent point of each box. The Douglas nonplanar method and the Woodward method agree. Slight disagreement between these and the Douglas planar method is observed however. The Douglas planar method uses an image system within the fuselage instead of lifting surface elements on the fuselage surface. The image system eliminates the wing root singularity (the flow is infinite at the terminus of a lifting surface) and thus is probably more accurate than the Douglas nonplanar or the Woodward methods for this case.

FIGURE 2 Figure 2 compares the static stability derivatives as calculated by Douglas (nonplanar) (ref. 1) and Belotserkovskii (ref. 4), for annular wlngs of various diameter to length ratios.

FIGURE 3 Dynamic stability derivatives for annular wings of various diameter to length ratios are given in figure 3. The oscillatory aerodynamic method was evaluated at low values of reduced frequency for these calculations (see ref. S).

FIGURE 4 Figure 4 presents calculation for a wing with and without pylons. In steady flow a comparison is made between the Douglas nonplanar method and the method of Blackwell, reference 6. Also shown are calculations done for the wing oscillating in pitch (about its apex) at a reduced frequency of 0.5.

FIGURE 5 The pressure distributions as calculated by the Douglas nonplanar method and the method of Zwaan, reference 7, are compared with Experiment. The T-tail is oscillating in yaw at a frequency of 0.55.

FIGURE 6 The induced drag distribution is not calculated in the usual way by the Douglas method. Instead of determining the leading-edge suction along with the normal force drag, the Douglas vortex-doublet lattice method uses the fundamental Kutta-Joukowski law. The upwash at each bound vortex is deter- mined and multiplied by the vortex strength to obtain the local contribution to the induced drag.

Figure 6 compares the spanwise induced drag distribution as calculated by Douglas and by Garner, reference 8. The agreement is very good except at the wing tips where Garner's results seem to be ambiguous. The numbers N in Garner's calculation represent the number of chordwise loading functions used.

It may be remarked that the spanwise distribution of induced drag is very sensitive to the spanwise loading. Small changes in span load cause large variations in the induced drag distribution. The total induced drag is, however, not as sensitive.

FIGURE 7 A comparison of induced drag distribution for a variable sweep configu- ration as calculated by Wagner, reference 9, and Douglas is given in figure 7.

The distributions do not agree nearly as well as in the hyperbolic wing case.

REFERENCES 1. Kalman, T. P.; Rodden, W. P.; Giesing, J. P.: Aerodynamic Influence Coefficients by the Doublet Lattice Method for Interfering Nonplanar Lifting Surfaces Oscillating in a Subsonic Flow. Rep. DAC-67977, McDonnell Douglas Aircraft Co., October 1969.

2. Giesing, J. P.: Lifting Surface Theory for Wing-Fuselage Combinations.

Rep. DAC-67212, McDonnell Douglas Aircraft Co., Aug. 1, 1968.

3. Woodward, F. A.: A Unified Approach to the Analysis and Design of Wing- Body Combinations at Subsonic and Supersonic Speeds. J. Aircraft, vol. 5, no. 6, November-December 1968.

4. Belotserkovskii, S. M.: The Theory of Thin Wings in Subsonic Flow.

Plenum Press, N. Y., 1967.

5. Rodden, W. P.; and Giesing, J. P.: Application of Oscillatory Aerodynamic Theory for Estimation of Dynamic Stability Derivatives. Paper 5630, McDonnell Douglas Aircraft Co.

6. Blackwell, J. A., Jr.: A Finite Step Method for Calculation of Theoreti- cal Load Distributions for Arbitrary Lifting-Surface Arrangements at Subsonic Speeds. NASA TN 0-5335, 1969.

7. Zwaan, R. J.: Application of a Method for Estimating Experimental Pressure Distributions to an Oscillating T-Tail. Rep. TR 68048L, National Aerospace Laboratory NLR, The Netherlands, Jan. 1968.

8. Garner, H. C.; Hewitt, B. L.; Labrujere, T. E.: Comparison of Three Methods for the Evaluation of Subsonic Lifting-Surface Theory. NLR TN G. 65, June 1968.

9. Wagner, S.: On the Singularity Method of Subsonic Lifting Surface Theory.

Paper 69-37, AIAA, Jan. 1969.

LIFT DISTRIBUTION ON A WING FUSELAGE COMBINATION -0- . -0- ' --0- DOUGLAS (NONPLANAR) ________ DOUGLAS (PLANAR) ~WOODWARD °0~~-0 ~ . 2~L-~O L .4~~0~.6~L-~~~ y/ b/ 2 Figure 1 STATIC STABILITY DERIVATIVES FOR ANNULAR WINGS

r- L -j

o DOUGLAS - BELOTSERKOVSKII 2.0 .-- --r~_.__~__, 0.20 1.5 f-- -4- - #-- ---!

III

0 ;>- 0 C 0.15 La CMq 1.0 f----!l '--+----!

~

!

CO~:O IYf1

0.10

II

0. 05 0.25 UIJ 00 1.0 2.0 3.0 00 1.0 2.0 3.0 1.0 2.0 3.0 Oi l Oi l Oi l Fi g ure 2 DYNAMIC STABILITY DERIVATIVES FOR ANNULAR WINGS 1.2 1.2 --0- DOUGLAS h 1.0 1.0

\

!

0.8 0.8 C .;.

M

I \

0.6 v-o--.

~

\

I I

0.4

II I

0.2

II

o 1.0 2.0 3.0 o 1.0 2.0 3.0 D/ l D/ l Figure 3 LIFT DISTRIBUTION ON A WING WITH PYLON PY LON BLACKWELL D OU GLA S K = 0 K = 0.5 o _ . -I:::r- . - OF F o _ .. 0-. - ON C l~ a c 0.2 0.4 0.6 0.8 1.0 y/ b /2 Figure 4 PRESSURES ON A T-TAIL OSCILLATING IN YAW REAL o 0 o [] EXPERIMENT O ~~-------------~-~--~O~o~ ---- ZWAAN ---- DOUGLAS STATION C -6 -8 '- --'-----'--'---'---' 4 .- --~ I M ~ A ~ GI ~ NA ~ R ~ Y ----' I'V'" <>

-2 wee

cG

YAWING AXIS 4 '-~ --~~~ ~-7 o 0.2 0.4 0.6 0.8 1.0 X/ C F igure 5 INDUCED DRAG DISTRIBUTION HYPERBOLIC WING o . 20 .---.--.----,----.---, o . 15 i--=-'~--+_-t---+----; 0 . 10 C dj 0 . 05 C L - GARNER - 0 .0 5 ~_t_--+_-::-+_:_--t7''--i o DO UGL AS - 0 . 10 '---'-----'-------'------'---' 0 . 2 0 .4 0 .6 0.8 1.0 o y/ b/ 2 Figure 6 INDUCED DRAG DISTRIBUTION I~ ~ ~ b-'" "">q; \ ~ --WAGNER -2 , I --0-- DOUGLAS I Cd; -4 -6 -8 I , -10 , I ~ -12 o 0.2 0.4 0.6 0.8 1.0 y/ b/2 Figure 7 -- -- --~ - DISCUSSION JACK N. NIELSEN, Nielsen Engineering and Research Inc.: From what I understood you to say earlier, you indicated that the induced drag you calcu- late was just as accurate as the induced drag that Siegfried Wagner gets using his method. I don't know Garner's results with which you compared.

Well, do they treat the leading-edge singularity accurately?

GIESING: Will you show the second to the last slide? Garner showed a conversion study with various numbers of terms. As you see, n, n2, n3, n4, that's his conversion study. Garner was putting more chordwise variables to determine his spanwise -- NIELSEN: The number of chordwise stations is the thing that is important.

You see, the problem is you have to get the strength of the leading-edge singularity accurately. Depending on how far your nearest control point is from the leading edge, you get more or less leading-edge suction.

GIESING: Right.

NIELSEN: Therefore, I think that the accuracy of your method basically depends on how many chordwise panels you put on the wing, but that's not the case in Wagner's work because he uses the Munk stagger theorem to get the strength of the leading-edge singularity. So it seems to me there is a basic advantage to his method.

GIESING: Well, Wagner uses the coefficients of the terms in the chord- wise series to get the leading-edge suction. That's right.

We do not pretend to get the leading-edge suction at all. We do not get it. We use the basic Kutta-Joukowski law.

NIELSEN: The strength depends on how close you get your control point to the leading edge.

GIESING: So does the upwash.

ROBERT T. STANCIL, LTV Aerospace Corp.: That also means that your answer requires full leading-edge suction.

GIESING: Full leading-edge suction? Yes.

STANCIL: It means that you cannot correlate some percentage of leading- edge suction or assume no leading-edge suction, is that right?

GIESING: Well, we haven't even thought about it, but on the face of it, no, you can't.

STANCIL: I think in the real world you never do achieve full leading~ edge suction, but it is significant to know approximately how much or I whether there is some peak in it that you would not expect to get in the real world, GIESING: I agree.

SIEGFRIED WAGNER: First of all, I would like to say that both of these methods, the vortex lattice methods and the lifting-surface type methods, have their significance and are very important. For instance, I have no doubt that your methods are very well applicable to wing design, and they predict the lift and the pitching moment reasonably well. Also, these methods are applic- able to more complicated configurations to which the lifting-surface method, the continuous distribution of vorticity, is not yet applicable. But the basic approach of the lifting-surface theory will probably ultimately give you the best answer because your method depends on the panel size or the posi- tion of the control points, and I have no doubt that you can arrange these variables so that you always get fine answers. But there might be questions in a very small region of the flow field where we have to know the flow field very accurately, for instance, near the leading edge or near slots of deflected flaps, and so on. Therefore, we need a better answer, and for this reason I am working on these lifting-surface theories to develop them to a point where they can be used as easily as the vortex lattice methods, but we are far away from this point .

Therefore, I am not going to say that I am replacing all your methods.

I see the significance and importance of your methods. I think the lifting- surface methods as such are also very important to be studied, and it is important to continue their development to where they can be as easily handled as your methods.

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THE USE OF FINITE ELEMENT METHODS FOR PREDICTING THE AERODYNAMICS OF WING-BODY COMBINATIONS By Ralph L. Carmichael and Charles R. Castellano Ames Research Center and Chuan F. Chen Rutgers University SUMMARY The method of finite elements is a procedure for solving linear partial differential equations by the superposition of a large number of elementary solutions in such a way as to approximate the exact boundary conditions.

Computer programs based upon this method are able to predict the subsonic and supersonic aerodynamic characteristics of wing-body combinations with very few restrictions on geometry. Comparisons between finite element solutions and established benchmark solutions indicate that the approximations involved introduce negligible amounts of error.

INTRODUCTION The principal objective of this paper is to summarize the principles of the method of finite elements for solving problems in the external aero- dynamics of complex configurations in subsonic and supersonic flow. The method of finite elements is a procedure for solving linear partial differen- tial equations with complicated boundary conditions by superposing a large number of elementary solutions in such a way as to approximate the exact boundary conditions.

The secondary objective is to describe the computer programs developed by the NASA which enable one to estimate the aerodynamic characteristics of wing-body combinations. The ultimate objective of this work is to prepare a collection of procedures of sufficient generality to handle the actual geo- metrical description of flight vehicles, although the current programs require a somewhat idealized configuration.

SYMBOLS

AR aspect ratio

b span local chord C area average chord of wing, span induced drag lift coefficient dCL da section lift coefficient

Cz

pitching-moment coefficient pressure on lower surface - pressure on upper dynamic pressure root chord Mach number q dynamic pressure y distance coordinate in span direction a angle of attack sweep angle THEORY Concept of the Method of Finite Elements - The Airship Problem One of the first problems in aerodynamics to be studied by theoretical methods was that of the pressure distribution on an airship (fig. 1). Once the configuration was idealized to a body of revolution at zero angle of attack, it was found that the flow about the airship could be represented by a distribution of sources along the axis and that an integral equation could be written which related this source distribution to the geometry of the air- ship. For certain simple mathematical shapes, this integral equation could be solved, and from the resulting source distribution one could compute the

L

--- pressure loading on the airship. A typical solution (for a quadratic varia- tion of radius) is shown in the lower left of figure 2. The general case remained intractable, however, because of the mathematical difficulties in treating complicated expressions and because airship dimensions were not specified by mathematical equations but by tables of coordinates from engi- neering drawings. Since the exact closed-form solution could not be obtained, various attempts at approximating the solution were trie~ and the method of finite elements was one of the most successful of these approximations.

The essential concept of the method of finite elements is illustrated in the lower right of figure 2. Since the smooth curve representing the exact solution cannot, in general, be obtained, it is assumed that a simple function (dashed lines) closedly approximates the smooth function. (Note: a simple function is defined as one that assumes only a finite set of values.)

This means that we are now representing the physical boundary of the airship by a finite number of sources of constant strength instead of a continuous source distribution.

The mathematical problem of the source distribution of constant strength can be solved in closed form in terms of elementary functions; hence the problem is to find the best choice of these finite values of the simple function in order to approximate the exact solution. This is done by estab- lishing a set of control points on the body and requiring that the flow be tangent to the surface at each point. If there are N sources, then there are N control points and the problem can be formulated as the simultaneous solution of N linear algebraic equations in N unknowns. With computers, it is feasible for N to assume quite large values, although it has been our experience that values of 25 to 50 provide adequate accuracy.

Let us emphasize that the velocities induced by each of the individual constant source elements are exact solutions of the linearized equation of flow and hence the superposition of all of them is also an exact solution to the partial differential equation. The approximation that is made is in the boundary conditions; the flow is required to be tangent to the body only at the N control points and not at every point on the body surface.

Finite Element Concepts for Configuration Synthesis The distribution of sources along a line simulates the flow about a body of revolution at zero angle of attack. For the representation of the flow about other shapes, there are other basic finite element solutions (fig. 3). In each case, there is a closed-form solution to the fundamental linearized equation of flow, either subsonic or supersonic. The line doub- lets, in conjunction with the line sources, simulate a body of revolution at angle of attack. The surface elements represent both thickness and lifting effects of wings. The surface thickness elements are wedges with swept lead- ing and trailing edges. The surface lifting elements are thin surfaces which support a constant pressure differential across the surface. Such a surface is, by its nature, highly warped near the root and tip regions. The combina- tion of surface thickness and lifting elements simulates the flow about a thin lifting wing of finite thickness.

Combination of Finite Elements to Represent a Wing-Body Combination By use of the finite element solutions described above, we can describe components of airplane configurations. As stated previously, the line sources and doublets represent an isolated lifting body and the surface thick- ness and lifting elements represent an isolated lifting wing. If the elements representing these isolated components are placed in close proximity, as in a wing-body combination, the resultant flow about the body will pene- trate the wing and vice versa; hence the boundary conditions will be violated.

In order to represent a wing-body combination, it is necessary to use addi- tional elements to satisfy the boundary conditions in the presence of inter- fering fields. One method for doing this is to locate surface lifting elements on the surface of the body (fig. 4) and adjust the pressure across the wing and body panels simultaneously to insure that there is no mass flow through the surface of the wing-body combination. If there are N finite elements, then we establish N control points on the configuration and require that the net velocity satisfy the boundary condition at each of these points. In general, the more elements used to represent a configuration, the greater the accuracy of the solution, and the practical upper limit on the number of elements is the maximum number of simultaneous linear algebraic equations that can be solved.

APPLICATION Comparisons Between Finite Element Method and Exact Linear Theory Since this procedure is an approximate method, it is desirable to study the magnitude of error introduced by the assumption of finite elements.

To make such estimates, some comparisons are made between well-known estab- lished solutions and solutions obtained by finite elements. In supersonic flow, there are exact solutions to the linear theory for conical wings. In figure 5, the results of the finite element computing program are compared to exact linear theory (ref. 1) for both supersonic and subsonic leading-edged delta wings. The solid lines are the exact solution and the symbols are the program output. In figure 6, a comparison is made for a swept, constant- chord wing with subsonic leading and trailing edges. Here, the exact solu- tion may be obtained by superposition of several conical fields (ref. 2).

As in the previous case, the finite element procedure provides quite adequate approximations to the exact solutions.

In subsonic flow, in contrast to supersonic, there are no exact soltuions to which we may compare program results. For the purposes of making similar comparisons at subsonic speeds,we use the highly perfected lifting surface theories based on Multhopp's original idea (refs. 3 and 4).

Figure 7 shows the results of one such comparison. The solutions labelled "modified Multhopp" are from Lamar (ref. 4), and the finite element solutions are from our computing program. As can be seen the results are very close.

These and other similar results indicate that the method of finite elements is very accurate for solving problems in subsonic and supersonic wing theory.

For wing-body combinations, there are yet fewer benchmark results with which to make studies of the accuracy of approximate methods. Some of these results and comparisons to finite element solutions may be found in refer- ence 5. It is concluded in reference 5 that the finite element (or panel) method does not introduce significant errors into the calculation of wing- body interference at supersonic speeds.

Application to Complex Configuration The method of finite elements is, therefore, as accurate as any existing linear theory procedure for calculating the aerodynamic character- istics of these simple shapes. It has the added virtue of being programmed for rapid solution on a digital computer. The great value of the method lies in its ability to generate solutions for complex configurations which can be solved in no other way, for example, the lifting body under the lifting wing, biplanes, ring-wings, and wing-body-tail configurations (fig. 8). Studies are now in progress to explore the application of linear theory to such configura- tions that rely on interference for the generation of desirable aerodynamic characteristics. The predicted results for the supersonic biplane are shown in figure 9. The infinite aspect ratio case was treated by Licher (ref. 6), and the results are duplicated by finite element theory. The finite aspect ratio solutions are not obtainable by other theories. Figure 10 illustrates the use of the program in predicting the characteristics of a configuration with multiple lifting surfaces. The experimental data are from the supersonic transport concept known as SCAT-17. In figure 11, the predicted and measured results for a ring-wing are compared (ref. 7), The small struts used to attach the ring-wing to the body must be considered to obtain proper correlation.

Computer Programs The Ames Research Center of the NASA has undertaken the development and distribution of computer programs for predicting pressure distributions on wing-body combinations at subsonic and supersonic speeds. Some early versions of these programs are described in references 8 and 9, The latest version of these programs, commonly known at the "Ames Wing-Body Program" is now available for distribution. This version contains significant improvements over the program of reference 9. The execution time has been reduced by a factor of 10 to 2~ and the input requirements have been greatly streamlined. Also, many of the restrictions on geometry have been elimina- ted and the subsonic surface elements derived by Woodward (ref. 10) have been incorporated, so that the complete wing-body interference problem can now be solved at subsonic as well as supersonic speeds. The development of these programs is continuing at Ames, and the ultimate goal is a program that will handle the actual geometry of flight vehicles, including propulsion units, internal flow, exhaust plumes, separated flows, shock waves, and so forth.

CONCLUDING REMARKS The method of finite elements has been shown to be an accurate and straightforward approach to the approximate solution of linear partial differential equations with complicated boundary conditions. Computer programs based on this method are now able to handle wing-body combinations with very few restrictions of geometry. With a reasonable amount of progress in both aerodynamic theory and computer technology, it will be possible to create computer programs that will predict the aerodynamic characteristics of complete flight vehicles. The development of these programs is a major element in the goal of aeronautical research and development, namely "pro- viding all information required for a designer to go from paper designs to production with complete confidence of success" (ref. 11).

I

L REFERENCES 1. Jones, Robert T.; and Cohen, Doris: High-Speed Wing Theory. Princeton University Press, 1960.

2. Cohen, Doris: Formulas for the Supersonic Loading, Lift, and Drag of Flat Swept-Back Wings With Leading Edges Behind the Mach Lines. NACA Rep. 1050, 1951.

3. Wagner, Siegfried: Some Recent Developments in Subsonic Lifting-Surface Theory. Paper presented to the NASA Symposium on Analytic Methods in Aircraft Aerodynamics, Oct. 28-30, 1969.

4. Lamar, John E.: A Modified Multhopp Approach for Predicting Lifting Pressures and Camber Slopes for Composite Planforms in Subsonic Flow.

NASA TN 0-4427, 1968.

5. Carmichael, Ralph L.: A Critical Evaluation of Methods for Computing Wing-Body Interference at Supersonic Speeds. Paper 68-08, 6th Congress International Council of the Aeronautical Sciences, Munich, West Germany, Sept. 9-13, 1968.

6. Licher, R. M.: Optimum Two-Dimensional Multiplanes in Supersonic Flow.

Rep. SM18688, Douglas Aircraft Co., Jan. 1955.

7. Morris, Odell: Aerodynamic Characteristics in Pitch of Several Ring- Wing--Body Configurations at a Mach Number of 2.2. NASA TN 0-1272, 1962.

8. Carmichael, Ralph L.; and Woodward, Frank A.: An Integrated Approach to the Analysis and Design of Wings and Wing-Body Combinations in Super- sonic Flow. NASA TN 0-3685, 1966.

9. Woodward, F. A.; Tinoco, E. N.; and Larsen, J. W.: Analysis and Design of Supersonic Wing-Body Combinations, Including Flow Properties in the Near Field. Part 1 - Theory and Application. NASA CR-73l06, 1967.

10. Woodward, Frank A.: Analysis and Design of Wing-Body Combinations at Subsonic and Supersonic Speeds. J. Aircraft, vol. 5, no. 6, Nov.-Dec.

1968, pp. 528-534.

11. Harper, Charles W.: Prospects in Aeronautical Research and Development.

J. Aircraft, vol. 5, no. 5, Sept.-Oct. 1968, pp. 417 - 426.

J Fi gure 1 SIMULATION OF AIRSHIP FLOW FIELD BY AXIAL SOURCE DISTRIBUTIONS AXIAL SOURCE DISTRIBUTION FIN I TE EL EMENT EXACT APPROX IMATION Fi gure 2

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FIN ITE ELEMENT COMPONENTS FOR CONFIGURATION SYNTHESIS • POINT ELEMENTS • SOURCE (SINK) • DOUBLET • LINE ELEMENTS • SOURCE • DOUBLET • VORTEX • SURFACE ELEMENTS • THICKNESS • LIFTING Fi gure3 COMBINATION OF FIN ITE ELEMENTS TO REPRESENT WING-BODY COMBINATION • BOD Y THICKNESS BY LINE SOURCES • BODY LIFT BY LINE DOUBLETS • WING THICKNESS BY CONSTANT SOURCE PANELS • WING LIFT BY CONSTANT PRESSURE PANELS • WING-BODY INTERFERENCE BY CONSTANT PRESSURE PANELS Figur e 4 COMPARISON BETWEEN EXACT LINEAR THEORY AND PROGRAM RESULTS FOR DELTA WINGS

/- ~ / -- CONICAL THEORY

, , 006V' FINITE ELEMENTS

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"- f3 cot A = 0 .8 f3 cot A= I . 2 f3!::. Cp 8 a 6 · o .2 .4 .6 .8 1. 0 0 .2 .4 .6 .8 1.0 FRACTION OF LOCAL CHORD Figu re 5 COMPARISON BETWEEN EXACT LINEAR THEORY AND PROGRAM RESULTS FOR A SWEPT WING f3 IR = I. 92 f3COS A =0 .6 2y/b=0 . 75 f3!::.Cp 2 a -I -- NACA REPORT 1050 2y/b =0.95 O V'06 FINITE ELEMENTS -2 -3 I I I I o . 2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 FRACTION OF LOCAL CHORD Figure 6 FINITE ELEMENT APPROACH TO LIFTING - SURFACE THEORY M::::O SPAN LOADING -- MODIFIED MULTHOPP

~~ --I T

o FINITE ELEMENT 1.4 1R=1.7 b 1.2

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RESULTS FINITE .4 MODIFIED ELEMENT MULTHOPP (120 PANELS) .2 C 0.0364 0.0359 La .187 .188 CD/C L o .5 1.0 C .0229 .0233 ma FRACTION OF SEMISPAN Fi gure 7 • NON PLANAR INTERFERENCE CONFIGURATIONS F igure 8 DRAG DUE TO LIFT OF A SUPERSONIC BIPLANE GAP/CHORD = I .6 --- ~ = co (LiCHER) o ~=CO) FINITE ---{]- ~ = 6 ELEMENTS ----fr- ~ = 3 .4 .2 o .5 1.0 1.5 Figure 9 • LIFT AND PITCHING MOMENT CHARACTERISTICS OF A WING-BODY- TAIL COMBINATION M=0.7 .02 - --- THEORY .15 -- EXPERIMENT .01 - C m .10 ,.,,~'" ~AILO FF -..:'-' .05 -...:: .....

-.01 - " .....

, ' .....

TAIL ON " ..........

, , I I L -.02 .5 1.0 1.5 I 2 3 -.5 0 o C a, deg L Figure 10 PREDICTION OF RING-WING CHARACTERISTICS BY FINITE ELEMENT METHOD o EXPERIMENT (NASA TN D-1272)

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-- THEORY -!===~- (WITH STRUTS) -"- --T-

4[p

, , --- THEORY (STRUTS NEGLECTED) 8- .4 - 6- / .2 - / / / / 4- .0 ---"(;).;:::------- / / C . / m V 2- / -.2 - O----r!f------ -.4 - -2 L I I - .6 L I -4 0 4 8 12 -2 O . 2 4 6 8 ANGLE OF ATTACK , deg C L Fi gure 11 ---~ ---------- DISCUSSION PETER B. S. LISSAMAN, Northrop Corporate Laboratory: The author kept on talking about subsonic. Does he mean subsonic or does he mean incompressible?

If he means subsonic, how would he take into account compressibility?

CARMICHAEL: Well, let's say compressible within the limits of the Prandtl-Meyer transformation. There is no nonlinear treatment of the com- pressible flow equations, so this would be a small perturbation theory for compressible speeds.

THEQDORE R. GOODMAN, Oceanics, Inc.: I have the distinct impression that a graduate student at Cornell University in the early '50s under Bill Sears solved the supersonic biplane with a finite aspect ratio rectangular planform, and if you would contact him I am sure he would tell you exactly how to get hold of that information.

WILLIAM J. EV~S, Grumman Aircraft: Ralph, is your limitation still one hundred wing panels?

CARMICHAEL: I haven't modified the program that I am running here, even though we now do have a larger machine. I see no reason why not, and I under- stand that some people who have gotten our program and modified it run up to 300 finite elements.

EVANS: Is there any indication of computer running time for those cases?

CARMICHAEL: The time is proportional to n , where n is the number of panels. We did say 150 panels in 4 minutes. If you used 600 panels, that is, 4 times as many, the time required would be 16 times as long. These are all on a 360, model 67, FORTRAN-H.

EVANS: Have you replaced the matrix inversion procedures?

CARMICHAEL: No, they are still single precision. That would be a limit if you used a very large number of panels. In fact, I am ~ot aware of how the people who have run this up to 300 panels do the inversion.

It may still work. You know, these aerodynamic matrices are very well conditioned in the sense that the big, strong elements are on the diagonal, so you may be able to do single precision inversion up to 200 or 300.

JOSEPH P. GIESING, Douglas Aircraft: I don't want you to give all your secrets away at one time, but I would like to know, since you are using a small-perturbation compressible flow analysis, how do you handle pointed type bodies with your lifting surface elements?

How do you handle the nonplanar aspects of the elements that you place on the fuselage?

CARMICHAEL: Oh, yes, there is a real problem, that's right, if you try to put up to the nose.

GIESING: Or back by the tail.

CARMICHAEL: We solve this by avoiding that problem.

I know that is a real problem. We ran into that in the early days, trying to put the panels as you saw in that picture (fig. 4), right up to the nose, making a cone solution with panels. It violates the assumption that the panels are nominally parallel to the free stream.

GIESING: I just have one other comment. Using the recommended 0.95 does not give you the right answer in subsonic flow for this method.

We came up with another magic number, 0.85. We found that this worked exactly right in two dimensions for wings without camber.

CARMICHAEL: Okay. I hate to get into the nitty gritty of this thing, but the question of where you put that control point in the individual panel has been the subject of great grief and hand-wringing and all that stuff.

Seventy-five percent was supposedly a magic number at one time, too.

SIEGFRIED WAGNER: I just wanted to say maybe the Multhopp-Truckenbrodt methods aren't bad.

CARMICHAEL: When we agree with it, we say well, that must be the right answer.

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CALCULATIVE TECHNIQUES FOR TRANSONIC FLOWS By John R. Spreiter Stanford University and Stephen S. Stahara and William H. Frey Nielsen Engineering and Research, Inc.

SUMMARY A summary of old and new ideas and results is presented to show that a theory already exists that is capable of accounting for many of the properties of transonic flows, that the fundamental equations, although nonlinear, are amenable to solution by a number of methods, and that the full potential for developing calculative techniques for three-dimensional flows has not been explored. Further progress is definitely possible and some examples of new developments are provided.

INTRODUCTION The desire to design aircraft capable of crulslng efficiently at speeds as close to Mach 1 as possible without incurring excessive drag penalties and the development of military aircraft able to maneuver at transonic speeds are leading to a renewal of interest in transonic aerodynamics. Although nearly half a century has elapsed since NACA, predecessor to NASA, initiated its long line of studies of transonic flow by sponsoring a series of experi- mental investigations by Briggs, Hull, and Dryden (ref. 1), the development of techniques for transonic analysis has continued to be "a real challenge to technology," as noted in a recent survey of emerging technologies by Lamar (ref. 2). He went on to say: "The dominant nonlinear characteristics of mixed transonic flows have been far too complex for analysis, especially in the three dimensions of real vehicles."

In spite of, or perhaps because of, this state of affairs, it is the purpose of this paper to report on an effort to develop calculative techniques for predicting transonic aerodynamic characteristics of wing-body combina- tions. This investigation is currently in progress at Nielsen Engineering and Research, Inc., under the sponsorship of Ames Research Center. While this goal may seem unduly optimistic in view of Lamar's assessment of the field and although the contract has been under way for only six months so that our specific advances to date are modest, we are confident that much progress can be made and will present a summary of existing knowledge and some new results to support this view.

DISCUSSION First of all, it should be recognized that considerable progress was achieved in the analysis of transonic flow in the decade and a half following World War II (see Spreiter, refs. 3 and 4; Guderley, ref. 5; and Ferrari and Tricomi, ref. 6, for significant summaries). Most important was the estab- lishment of the basic concept that there is a major body of transonic flow problems that can be analyzed adequately within the framework of steady inviscid flow theory rather than the more general, and far more difficult, framework of unsteady viscous compressible flow theory. For these problems the basic equations are therefore those associated with Euler's equation of motion rather than with the Navier-Stokes' equations. Furthermore, the general concern with efficient flight of streamlined objects permits the assumptions of small disturbances and irrotational flow for transonic speeds, just as in most aerodynamic analyses of subsonic and supersonic flows.

Although the familiar Prandtl-Glauert equation of linearized compressible flow theory shown on the bottom left of figure 1 is also based on similar assump- tions, its well-known degeneracy at free-stream Mach number M equal to I led Oswatitsch and Wieghardt (ref. 7), Busemann and Guderley (ref. 8), Guderley (refs. 9 and 10), and von Karman (refs. 11 and 12) to develop a new nonlinear theory of small disturbance flow with M near or equal to unity. Although the basic differential equation for the perturbation velocity potential ~ has been written in several slightly different forms, the form presented on the bottom right of figure 1 was shown (Spreiter, ref. 13) to be generally advantageous because it (a) provides greater accuracy, (b) is no more diffi- cult to solve, and (c) is applicable not only to transonic flows, but also to subsonic and supersonic flows as well. This equation thus provides a basis for a unified flow theory for all Mach numbers from 0 to that supersonic Mach number (at least 2 or 3) at which linearized supersonic flow theory must be supplanted by the nonlinear theory of hypersonic flow.

In spite of the seemingly simple form of the differential equation for transonic flow, the presence of the term involving ~ x~xx introduces great complications. These difficulties arise not primarily from the nonlinearity that this term introduces, but rather from the change of the basic character of the differential equation from elliptic to hyperbolic type as the sign of 1 - M 2 - M 2(y + l)(~x/U ) changes from positive to negative . Since no 00 00 00 general mathematical theory exists for such equations, aerodynamicists and applied mathematicians working in transonic flow theory have had to develop their own methods. Several of the more successful and generally applicable of these are listed on figure 2 together with pertinent comments.

Listed first is the hodograph method, since it is the most firmly founded mathematically and has provided most of the exact solutions with which the results of approximate theories can be compared . The key step in this method , summaries of which are given by Guderley (ref. 5) and Ferrari and Tricomi (ref. 6), is the linearization without approximation of the transonic flow equation by interchange of the dependent and independent variables through use of either of two classical transformations of compressible flow theory, the Molenbroek transformation or the Legendre transformation. The differential equation that results in either case can then be transformed simply, by introducing new normalizing variables, to a linear equation of mixed elliptic-hyperbolic type named after Tricomi for his pioneering mathematical studies of its properties begun approximately 50 years ago. Solutions for a number of problems of aerodynamic interest have been determined in this way by superposition of more elementary solutions found by application of either analytical or numerical techniques. In all except certain sp e cial applica- tions, however, the analysis must proceed in an indirect manner because the shape of the body cannot be prescribed in advance but must be found as part of the solution. Although many valuable results have been obtained in this way, the fundamental restriction of the hodograph method to two-dimensional planar flows precludes its further consideration in the present study directed toward wing-body combinations.

The integral equation method is an approximate method based on considera- tion of a nonlinear integral equation derived from the differential equation for transonic flow through application of Green's theorem. It originates from a series of papers of Oswatitsch (refs. 14 and 15), Gullstrand (refs. 16-19), Spreiter and Alksne (ref. 20), and Spreiter, Alksne, and Hyett (ref. 21). More recently, reviews of this method have been given by Zierep (ref. 22) and by Ferrari and Tricomi (ref. 6). Although the initial steps of the integral equation method are analytic, the calculations proceed directly toward the solution for a body of specified shape through a combination of numerical and iterative procedures. At present, the method has been deve loped for planar flow past nonlifting airfoils for M oo < 1, and the results for such cases appear to be the most satisfactory of those provided by any of the methods listed on figure 2. The method is potentially more versatile than these results indicate, and discussions of preliminary aspects of extensions to other cases may be found in the references cited above. Precise details for such cases remain to be developed, however.

The parabolic method stems from the daring proposal by Oswatitsch and Keune (ref. 23) that the nonlinear term in the transonic flow equation could be approximated satisfactorily for flows with M oo = 1 past slender bodies of revolution by replacing ¢x¢xx by K ¢ x' where K is a constant. The method derives its name from the parabolic type of the resulting approximate differ- ential equation for ¢ which has the form of the equation of heat conduction.

This procedure was subsequently applied to planar flows past thin airfoils, and extended to other Mach numbers by the simple expedient of retaining the term (1 - M oo 2)¢xx' Reviews of these developments have been given by Maeder (ref. 24) and Hosokawa (ref. 25), the two leading contributors. Although the procedures are direct and simple, the results lack accuracy generally and must be judged to be the least satisfactory of those provided by any of the methods listed on figure 2.

The local linearization method which grew out of the parabolic method of Oswatitsch and Keune is yet another approximate method for solving the tran- sonic flow equation. This method has been applied to planar flow past thin airfoils and to axisymmetric flow past slender bodies (Spreiter and Alksne, refs. 26 and 27) for the Mach number ranges M oo ~ 1, M oo ~ Mcr,l and Moo ~ Mcr,u' where Mcr,l and Mcr,u refer to the lower and upper critical Mach numbers that bound the transonic range. It has also been applied by Alksne and Spreiter (ref. 28) to flows with M ~ I past nonlifting wings of finite span having simple planform and airfoil oo shapes. In addition to being the most versatile of all the methods presently available for predicting aerodynamic properties of thin wings and slender bodies at transonic speeds, the local linearization method has consistently displaced accuracy comparable with the best theoretical and experimental results. For these reasons, this method has been selected for further study as holding the greatest promise for successful extension to wing-body combinations. Briefly, the basic idea underlying the method is that of linearizing the transonic flow equation by replacing either ~ x or ~ xx in the nonlinear term by a constant A, solving the simplified equation, and then introducing different values for A for different points in the flow. This procedure might be considered equivalent, in some sense, to replacing the original nonlinear partial differential equa - tion by a different linear partial differential equation at each point.

Results obtained by such a procedure depend, of course, on the choice of A and must be assembled to determine the final results. This step is accom- plished by putting the results into such a form that a first-order nonlinear ordinary differential equation is obtained for the streamwise perturbation velocity component u = ~ x after A is replaced by the quantity it origin- ally represented. In many cases, this equation is of sufficiently simple form that it can be integrated analytically and the solution expressed in closed form. In other cases, the integration must be performed numerically, but the equation is of such a form that standard methods can be applied.

The parametric differentiation method of Rubbert and Landahl (ref. 29) is a recent addition to the list of procedures for obtaining satisfactory approximate solutions of the transonic flow equation. The difficulties associated with the nonlinearity of the basic equ ation are avoided by con- sidering the linear problem governing the rate of change of the flow velocity with respect to the airfoil thickness ratio parameter and integrating solu- tions of this problem over the thickness ratio. This procedure is equiva- lent to summing a series of perturbations in airfoil thickness and has the advantage, shared with the local linearization method, of moving all the non- linearity to a first-order ordinary differential equation where it causes little difficulty. The method has been applied to planar flow past nonlift- ing airfoils, and the results are either identical to or very nearly the same as those of the local linearization method . There is obviously considerable merit to this method, and further study to extend the range of cases for which it is explicitly applicable is clearly warranted. The close relation- ship between the results of the parametric differentiation and local lineariza- tion methods, in spite of the seemingly different nature of the approximati ons involved, is also provocative; further investigation of the reasons for this may be anticipated to lead to a better understanding of the mathematical foundations of both methods. For the present, however, we will continue to apply the method of local linearization, knowing that it is likely that many of the results so obtained might also be effectively reproduced by suitable extensions of the parametric differentiation method.

Solution of the transonic flow equation by completely numerical procedures is included in the list of methods not so much because of the

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demonstrated ability of such procedures to yield useful results, but because of the promise that this approach holds with the continued improvement of electronic computing capability. Although a number of groups are developing these methods, progress has been slow, and very few details or results for transonic flows have been published to date. It is clear, however, that the completely numerical solution of transonic flow problems is not only expen- sive but probably limited to two-dimensional flows with the present genera- tion of computers. For these reasons, we have not considered it profitable at this time to seek solutions for transonic flow about wing-body combinations by application of completely numerical techniques.

The first and simplest transonic flow problem to which the local linearization method was applied was flow with M oo 1 past thin nonlifting airfoils having a finite angle at the leading edge. The variation of the pressure coefficient C = (p - p )/(p U /2), or its transonic similarity counterpart (p, along ~he surfac~ of ~uch an airfoil is found to be given by the expression at the top of figure 3, in which Z represents the ordinates of the airfoil surface and x* represents the x-coordinate of the point on the airfoil surface at which the flow accelerates through the speed of sound.

Although x* is known a priori to be at the shoulder of a single ~ or double- wedge airfoil (see, e.g., ref. 5), the location of the sonic point on a con- tinuously curved profile must be found as part of the solution. It is, a~cording to the method of local linearization, at the point x at which the second equation shown on figure 3 is satisfied. In the original presentation of these results, Spreiter and Alksne (ref. 26) applied the theory to a family of airfoils tested by Michel, Marchaud, and Le Gallo (refs. 30 and 31) having thickness ratios T from 6 to 12 percent ' and positions of maximum thickness from 30 to 70 percent of the chord. The ordinates of the airfoils with maximum thickness at 30-, 40-, and 50-percent chord are proportional to 1 - (x/c) - [1 - (x/c)]n where n= 6.05, 3.38, and 2 , respectively. In the original comparisons I the integrations were performed analytically, but it was necessary to approximate the exponent 6.05 by 6, and 3.38 by either 3 or 3.5, in order to carry out the indicated integrations. Even with this simplification, the final expres- sions are so lengthy that even substituting numbers into them to obtain spe- cific results is a laborious task. Nevertheless, the comparisons, such as that shown on figure 3 for airfoils with n = 3.38, served to demonstrate that the theoretical results are indeed both versatile and in good accord with experiment.

As the first step in the application of electronic computers to the method of local linearization, we have programmed the general equations shown on figure 3, and applied them to calculate the pressure distribution on the airfoil defined by the above expression with n = 3.38. The new results have been added to the plot in figure 3, and we note that they fall between the previous analytic results for n = 3 and 3.5 as they should. All three sets of theoretical results are also in satisfactory agreement with the experi- mental pressure distributions for all four airfoils tested except near the trailing edge, where interactions between the boundary layer and the trailing shock waves disregarded in the theory are undoubtedly responsible for the discrepancies.

The pressure distributions shown on figure 3 are di~layed in transonic similarity form so that the results appear in terms of C instead of Cpo p This is done to facilitate comparison of a single theoretical curve with data for four affinely related airfoils having different thickness ratios. The corresponding results for Cp are presented in the left-hand part of figure 4 for a specific airfoil of the same family having a thickness ratio of 1/12.

In addition to the results for M oo 1, the theoretical pressure distributions indicated by the local linearization method for the lower and upper critical Mach numbers are also displayed. In the right - hand part of figure 4 are shown the variation with M oo of the minimum value of C~ for subsonic flow and the maximum value of Cp for supersonic flow. The lntersections of these curves with the curve representing the variation of the critical pressure coefficient C with M defines the boundaries of the transonic range.

p cr 00 Of particular interest is the remarkable width of the transonic range, which for this particular airfoil extends from M = 0.75 to 1.66. Moreover, this range would be even greater were the airfoil lifting.

Results for axisymmetric flow past a body of revolution having the same ordinates as the airfoil considered in figure 4 are presented in figure 5.

In the left-hand part are shown the theoretical pressure distributions on the body surface for M = 1 and for the lower and upper critical Mach numbers.

Also included for comparison are the experimental results for Moo = 1 for this body, as measured in the Ames 14-Foot Transonic Wind Tunnel by McDevitt and Taylor (ref. 32) and abridged for clarity of representation. As with previous comparisons with other bodies of revolution having simpler expres- sions for the ordinates (Spreiter and Alksne, ref. 27; and Spreiter, Smith, and Hyett, ref. 33), the results calculated using the local linearization method agree well with the experimental results over most of the body but disagree noticeably over the rear of the body. Comparison with the results presented in figure 4 shows that the magnitudes of the peak pressure coeffi- cients are much smaller for axisymmetric flow past a body of revolution than for planar flow past an airfoil having the same profile. As a result, the transonic range is also much smaller and, as can be seen from the right-hand part of figure 5, extends from M = 0.94 to 1.29 for the present example.

With respect to the discrepancies between the theoretical and experimental results near the rear of the body, many would be inclined to dismiss further discussion by attributing the differences to shock-wave-- boundary-layer interaction effects not included in the theory. While there is little doubt that such effects are important, we believe that the experi- mental results for the rear of this body are subject, in addition, to signi- ficant interference effects of the wind-tunnel walls. Although there is no explicit experimental evidence to support or refute this statement for the specific body considered in figure 5, the data on figure 6 for a closely related pair of bodies are definitely relevant. The plots on the left show pressure distributions measured at M oo = 1 on two bodies of revolution having ordinates proportional to 1 - (x/Z) - [1 - (x/Z)]n and diameter-length ratio D/l of 1/12. The results shown in the upper plot are for a parabolic-arc body, for which n = 2, and those in the lower plot are for a body with maximum thickness at 30 percent of its length, for which

n = 6. Both wind-tunnel models were 6 inches in diameter, the same as that

for which data are shown in figure 5. The data for these bodies obtained by Taylor and McDevitt (ref. 34) and McDevitt and Taylor (ref. 32) in the Ames 14-Foot Transonic Wind Tunnel with square test section are indicated by the open circles. The closed circles indicate the data obtained when the same actual models were subsequently tested under choking conditions in the Ames 12-Foot Pressure Wind Tunnel which has solid walls and a circular test section (Spreiter, ref. ' 4; and Spreiter, Smith, and Hyett, ref. 33). The principal points to observe are that both sets of experimental data are in essential agreement with each other and with theory over most of the body but that all three sets of results are widely divergent over the rear of the body.

The source of these differences has been discussed previously (Spreiter, ref. 4; Spreiter, Smith, and Hyett, ref. 33; and Berndt, ref. 35), but it is worth reviewing here because similar considerations are likely to be of impor- tance in most comparisons of theoretical and experimental results for tran- sonic flow past slender bodies or wing-body combinations. This can best be done by considering the diagram of figure 6 which shows the characteristic lines for an unbounded flow with M = I past a parabolic-arc body of revolu- <Xl tion having a diameter - length ratio of 1/12. These results have been calcu- lated by application of the transonic similarity rule for axisymmetric flow (Oswatitsch and Berndt, ref. 36) to a related diagram given by Oswatitsch (ref. 37) for a parabolic-arc body of revolution with a diameterrlength ratio of 1/6. The position of the wall with respect to the model in the tests in the 12-foot pressure wind tunnel is as indicated, and the nearest part of the wall in the tests in the 14-foot transonic wind tunnel is 7/6 as far away.

Although it was thought at the time of the tests in the l4-foot transonic wind tunnel that the 6-inch diameter of the models was sufficiently small to avoid significant effects of wind-tunnel-wall interference, figure 6 shows that this ' may not be the case because characteristics, or Mach w . aves, origi- nating from the forepart of the body are indicated to be reflected from the walls onto the aft part of the body. It can be seen, moreover, that the most upstream reflected characteristic strikes the body at about x/l = 0.6 in the l2-foot wind tunnel test, and only slightly aft of that location in the l4-foot wind tunnel. The effect of the reflected waves striking the body is t .o make the pressure coefficients more negative in the l2-foot wind tunnel, because the outgoing characteristics represent expansion waves that reflect from the solid wall of the tunnel as rarefaction waves. The effects are amplified, moreover, because of the focusing characteristics of the reflected axisymmetric waves as they collapse down onto a part of the body that has a smaller circumference than that from which they originated. The sign of the corresponding effects in the l4-foot transonic wind tunnel is not so simple to ascertain, since the reflections from the partly open wall of that wind tunnel are very nearly equal in magnitude, but opposite in sign, to that of the reflectlons trom the solid wall of the l2~foot wind tunnel. In addition to the direct effects of the reflected waves impinging on the rear of the body, there exists the distinct possibility of significant augmentation arising from the interaction of the boundary layer with a shock wave that may form adjacent to the body. The latter may form either because of coalescence of compression waves reflecting from the body or because of boundary-layer separation resulting from the wall-induced steepening of the adverse pressure gradients.

In either case, it is clear that considerable additional study will have to be made before it is possible to properly evaluate the significance of discrep- ancies between theoretical and experimental pressure distributions on the aft parts of bodies of revolution such as those i llustrated in figures 5 and 6.

Perhaps one of the best-known properties of transonic flows is the transonic area rule of Whitcomb (ref. 38). It states that, near the speed of sound, the zero-lift drag rise of a slender wing-body combination (thin low- aspect-ratio wing, slender body) is primarily dependent on the axial distribu- tion of cross-sectional area normal to the air stream. This rule, which was proposed on the basis of certain fundamental, yet elementary, statements regarding the nature of transonic flow fields and demonstrated experimentally, is closely related to the transonic equivalence rule which relates the flow around a slender body of arbitrary cross section to the flow around an "equiv- alent" nonlifting body of revolution having the same longitudinal distribution of cross-section area Sex). The latter rule was first proposed for transonic flow past thin nonlifting wings by Oswatitsch (ref. 39) and later extended initially to lifting wings by Spreiter (ref. 3) and subsequently to slender bodies of arbitrary cross section, including wing-body combinations, by Heaslet and Spreiter (ref . 40). Figure 7, which is an extension to wipg~body combinations of a rather similar figure presented by Spreiter (ref . 3) for thin wings, summarizes the theoretical essentials of both the equivalence and area rul~s. Most important is that the expression for the perturbation veloc- ity potential ¢ in the vicinity of a slender body of arbitrary cross section is approximately of the form ¢ = ¢2 + g(x) where ¢2 is the solution of Laplace's equation ¢yy + ¢zz = 0 for the given boundary conditions in the yz plane at each x station, and g(x) is an additional contribution depen- dent upon M and Sex) but not on the shape of the cross section. It is thus possible to determine g(x) from the solution of the simpler problem of axisymmetric flow past the equivalent body. The equivalence rule, which is described in mathematical terms by

¢ = ¢2 , a + ¢ 2,t - ¢2,B + ¢B

in which each component of ¢ has the meaning indicated in figure 7, follows immediately by writing ¢ = ¢2 + g(x) for the body of arbitrary cross section and subtracting the corresponding expression for the equivalent body.

The order of error in the transonic equivalence ' rule has been established by Heaslet and Spreiter (ref. 40) for thin wings of aspect ratio A, chord c, and thickness ratio T. It was shown that the magnitude of the quantity ¢/U c retained in the equivalence rule is O(AT Zn A), whereas that 4 2 of the quantities discarded in the derivation for M = I is O(A T Zn A).

Since the magnitude of the quantities discarded in the derivation of the corresponding result in linearized subsonic and supersonic flow past slender bodies is O(A T Zn A), it follows that the equivalence rule ought to be applicable to wings of greater aspect ratio at M oo = 1 than at any other Mach number.

-- --- " -- -- -- -- -- -- ------ . -- - -- Once the appropriate expression has been constructed for ~ ,the pressure distribution on or near the surface of slender bodies may be determined by use of the expression for C shown on figure 7. The results may, in turn, p be integrated to obtain expressions or values for the total forces, including lift and drag, and moments on slender bodies or wing-body combinations of arbitrary cross section. Since the aerodynamic loading, lift, and all lateral forces and moments depend on differences in pressure between pairs of points at the same longitudinal station, these quantities depend solely on ¢2 and are therefore independent of M oo ' In particular we may note, as discussed by Spreiter (ref. 41) and Heaslet, Lomax, and Spreiter (ref. 42) even before the discovery of the transonic equivalence rule, that these quantities may be calculated quite adequately by linearized slender-body theory even though M may be unity.

It is evident that the transonic · area rule of Whitcomb is closely related to the transonic equivalence rule. In a detailed examination of the relation between these two rules, Heaslet and Spreiter (ref. 40) derived the expression shown on the bottom of figure 7 between the drag D of a slender body of arbitrary cross section and the drag DB of the equivalent body of revolu- tion. Each of the integrals is a line integral along a curve that is situated in a plane perpendicular to the x - axis and that traverses the base of the body and any vortex wake which may be present. The difference D - DB is thus independent of M oo and is the same as given by linearized slender~body theory. If the arbitrary body is inclined at angle of attack a , the first integral provides a contrlbution to the drag that is proportional to a .

This quantity is exactly the vortex drag associated with the production of lift. If attention is confined to nonlifting cases, several classes of shapes exist for which the contribution of the two remaining integrals cancel, and D = DB as proposed by Whitcomb. One important class includes shapes that taper to a point at the rear, since then both integrals vanish as the contour shrinks to a point. Another includes shapes that are cylindrical at the base, since then 3¢ 2 t/ 3n = 3¢2 B/ 3n = O. Still another includes bodies for which the equivalent body and the original body have the same shape and surface slopes at the base, since then both integrals are carried out over the same contour, along which ¢2 ,t = ¢2 B and 3¢2 ,t/ 3n = 3¢ 2B/ 3 n, and the integrals again cancel. These and many other cases for which the integrals cancel constitute the class of shapes for which the transonic area rule applies.

We now turn to two new applications of the transonic equivalence rule that are significant to the further development of calculative techniques for three-dimensional transonic flows . The first of these, summarized on fig- ure 8, exploits the result that ¢ = ¢2 + g(x) = (U /2n) (dS/dx)Zn r + g(x) in order to calculate the properties of the flow field at points removed from the body surface. The plot in the lower left shows the theoretical pressure distribution for M oo 1 on the surface of a parabolic-arc body of revolution with a diameter-length ratio of 1/12, as indicated by the local linearization method and data from the experiments of Taylor and McDevitt (ref. 34) in the Ames 14-Foot Transonic Wind Tunnel. The two plots in the upper left show the corresponding comparisons of calculated and measured pressure distributions along lines parallel to the body axis but removed from it by distances of 4 and 8 times the maximum diameter D of the body. The plot on the right shows the variation of C with distance measured laterally from the mid- p point of the body axis. Except for the discrepancies near the rear of the body that have already been discussed in connection with the presentation of the same results for the surface pressure distribution in figure 5, the agreement between theory and experiment is generally satisfactory. This is particularly so when considered with respect to possible applications of the transonic equivalence rule to configurations having wings or related extrem- ities of such size that they, rather than the body, provide the major contribu- tion to Cp at lateral distances of the order of those for which results are shown in figure 8.

The second application, summarized in figure 9, is to flows with M = 1 past two different slender bodies with elliptical cross section having ratios alb of major to minor axes of 1.5 and 3. The experimental data are those of McDevitt and Taylor (ref. 43) for the longitudinal variation of C at the p extremities of the major and minor axes. The theoretical results have been determined by using the transonic equivalence rule to account for the differ- ences between the elliptic bodies and the equivalent body of revolution and the local linearization method to provide the results for the equivalent body.

The expression for ~2 ,t required to describe the two-dimensional flow asso- ciated with the growing and shrinking of the elliptic cross sections is known (see for instance Nielsen, ref. 44) to be given by the relation shown on fig- ure 9, where R.P. stands for the real part of the complex function that follows. As in the previous comparisons, the theoretical and experimental results are in good agreement except near the rear of the bodies, where at least part of the discrepancies must be attributed to the extraneous effects of the wind-tunnel walls.

CONCLUDING REMARKS It is hoped that this summary of old and new ideas and results has served to show that a theory already exists that is capable of accounting for many of the properties of transonic flows, that the fundamental equations, although nonlinear, are amenable to solution by a number of methods, and that the possibilities for the development of calculative techniques for three- dimensional transonic flows are not nearly so bleak as indicated by the quota- tion in the opening paragraph. Further progress is definitely possible but will come only with determination and the expenditure of effort, and these have largely been lacking in this field during the past decade.

REFERENCES 1 . Briggs, L. J.; Hull, J. F.; and Dryden, H. L.: Aerodynamic Character- istics of Airfoils at High Speeds. NACA TR 207, 1925.

2. Lamar, W. E.: Military Aircraft: Technology for the Next Generation.

Astronautics and Aeronautics, vol. 7, 1969, pp. 68-78.

3. Spreiter, J. R.: Theoretical and Experimental Analysis of Transonic Flow Fields. NACA-University Conference on Aerodynamics, Construction, and Propulsion, vol. II, "Aerodynamics," 1954, pp. 18/1-18/17 .

4. Spreiter, J. R.: Aerodynamics of Wings and Bodies at Transonic Speeds.

J. Aero/Space Sci., vol. 26, no. 8, Aug. 1959, pp. 465-487.

5. Guder1ey, K. G.: Theory of Transonic Flow. Pergamon Press, Oxford, England, 1962.

6 . Ferrari, C.; and Tricomi, F.: Transonic Aerodynamics. Academic Press, N. Y., 1968 .

7. Oswatitsch, K.; and Wieghardt, K.: Theoretische Untersuchungen Uber stationare Potentialstromungen und Grenzschichten bei hohen Geschwindigkeiten. Lilienthal-Gesellschaft fur Luftfahrt-forschung, Ber. 13/1, 1942, pp. 7-24 (also available as NACA TM 1189).

8 . Bus em ann , A.; and Guderley, K. G.: The Problem of Drag at High Subsonic Speeds. Rep. and Trans. No. 184, British M.A.P., March 1947.

9. Guderley, K. G.: On the Transition from a Transonic Potential Flow to a Flow with Shocks. Tech. Rep. F-TR-2160-ND, AAF, Air Materiel Command, Wright Field, Aug. 1947.

10. Guderley, K. G.: Considerations of Structure of Mixed Subsonic- Supersonic Flow Patterns. Tech. Rep. F-TR-2168-ND, AAF, Air Materiel Command, Wright Field, Oct. 1947.

11. von Karman, T. : Supersonic Aerodynamics - Principles and Applications.

J. Aero. Sci., vol. 14, no. 7, July 1947, pp. 373-402, Discussion, pp. 403-409.

12. von Karman, T.: The Similarity Law of Transonic Flow. J . Math. and Phys., vol. XXVI, no. 3, Oct. 1947, pp. 182-190.

13 . Spreiter, J. R.: On the Application of Transonic Similarity Rules to Wings of Finite Span. NACA Rep. 1153, 1953. (Supersedes NACA TN 2726.)

14 . Oswatitsch, K.: Die Geschwindigkeitsverteilung bei lokalen Vberschallgebieten an flachen Profilen. ZAMM, Bd. 30, Nr. 1/2, Jan./Feb . 1950, pp. 17-24.

-- -~-~-~ 15. Oswatitsch, K.: Die Geschwindigkeitsverteilung an symmetrischen Profilen beim Auftreten lokaler Vberschallgebiete. Acta Physica Austriaca, Bd. 4, Nr. 2/3, Dec. 1950, pp. 228-271.

16. Gullstrand, T. R.: The Flow Over Symmetrical Aerofoils without Incidence in the Lower Transonic Range. KTH Aero TN 20, Royal Inst.

Tech., Stockholm, Sweden, 1951.

17. Gullstrand, T. R.: The Flow Over Symmetrical Aerofoils without Incidence at Sonic Speed. KTH Aero TN 24, Royal Inst. Tech., Stockholm, Sweden, 1952.

18. Gullstrand, T. R.: A Theoretical Discussion of Some Properties of Transonic Flow Over Two-Dimensional Symmetrical Aerofoils at Zero Lift with a Simple Method to Estimate the Flow Properties. KTH Aero TN 25, Royal Inst. Tech., Stockholm, Sweden, 1952.

19. Gullstrand, T. R.: The Flow Over Two-Dimensional Aerofoils at Incidence in the Transonic Speed Range. KTH Aero TN 27, Royal Inst.

Tech., Stockholm, Sweden, 1952.

20. Spreiter, J. R.; and Alksne, A.: Theoretical Prediction of Pressure Distributions on Nonlifting Airfoils at High Subsonic Speeds. NACA Rep. 1217, 1955. (Supersedes NACA TN 3096.)

21. Spreiter, J. R.; Alksne, A. Y.; and Hyett, B. J.: Theoretical Pressure Distributions for Several Related Nonlifting Airfoils at High Subsonic Speeds. NACA TN 4148, 1958.

22. Zierep, Jurgen: Die Integralgleichungsmethode zur Berechnung schallnaher Stromungen. In Symposium Transsonicum, K. Oswatitsch, ed., Springer-Verlag, Berlin/Gottingen/Heidelberg, 1964, pp. 92~l09.

23. Oswatitsch, K.; and Keune, F.: The Flow Around Bodies of Revolution at Mach Number 1. Proc. Conf. on High-Speed Aeronautics, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., Jan. 20-22, 1955, pp. l13r13l.

24. Maeder, P. F.: The Linear Approximation to the Transonic Small Disturbance Equation. In Symposium Transsonicum, K. Oswatitsch, ed., Springer-Verlag, Berlin/Gottingen/Heidelberg, 1964, pp. 112-125.

25. Hosokawa, Iwao: A Simplified Analysis for Transonic Flows Around Thin Bodies. In Symposium Transsonicum, K. Oswatitsch, ed., Springer-Verlag, Berlin/Gottingen/Heidelberg, 1964, pp. 184-199.

26. Spreiter, J. R.; and Alksne, A. Y.: Thin Airfoil Theory Based on Approximate Solution of the Transonic Flow Equation. NACA Rep. 1359, 1958. (Supersedes NACA TN 3970.)

27. Spreiter, J. R.; and Alksne, A. Y.: Slender Body Theory Based on Approximate Solution of the Transonic Flow Equation. NASA Rep. 2, 1959.

28 . Alksne, A. Y.; and Spreiter J. R.: Theoretical Pressure Distributions l on Wings of Finite Span at Zero Incidence for Mach Numbers Near 1.

NASA TR R-88, 1961.

29. Rubbert, P. E.; and Landahl, M . T.: Solution of the Transonic Airfoil Problem Through Parametric Differentiation. AIAA J., vol. 5, 1967, pp. 470-479.

30 . Michel, R.; Marchaud, F.; and Le Gallo, J.: Etude des ecoulements transsoniques autour des profils lenticulaires, a incidence nulle.

O.N.E.R.A. Pub . No. 65, 1953.

31. Michel, R. ; Marchaud, F.; and Le Gallo, J.: Influence de la position du maitre-couple sur les ecoulements transsoniques autour de profils a pointes. O.N.E.R.A. Pub. No . 72, 1954.

32. McDevitt, J . B. ; and Taylor, R. A.: Pressure Distributions at Transonic Speeds for Slender Bodies Having Various Axial Locations of Maximum Diameter. NACA TN 4280, 1958.

33. Spreiter, J. R.; Smith, D. W.; and Hyett, B. J . : A Study of the Si mulation of Flow with Free-Stream Mach Number 1 in a Choked Wind Tu nnel . NASA TR R-73, 1960.

34. Taylor, R. A. ; and McDevitt, J. B.: Pressure Distributions at Transonic Speeds for Parabolic-Arc Bodies of Revolution Having Fineness Ratios of 10, 12, and 14. NACA TN 4234, 1958 .

35 . Berndt, S. B.: Theory of Wall Interference in Transonic Wind Tunnels.

In Symposium Transsonicum, K. Oswatitsch, ed., Springer-Verlag, Berlin/Gottingen/Heidelberg, 1964, pp . 288-309.

36. Oswatitsch, K.; and Berndt, S. B .: Aerodynamic Similarity at Axisymmetric Transonic Flow Around Slender Bodies. KTH Aero TN 15, Royal Inst. Tech., Stockholm, Sweden, 1950.

37. Oswatitsch, K.: Die Berechnung wirbelfreier achsensymmetrischer Uberschallfel~er. Osterreichisches Ingenieur-Archiv, Band X, Heft 4, 1956, pp. 359-382.

38 . Whitcomb, R. T.: A Study of the Zero-Lift Drag-Rise Characteristics of Wing-Body Combinations Near the Speed of Sound. NACA Rep. 1273, 1956.

(Supersedes NACA TM L52H08.)

39 . Oswatitsch, K.: Die Theoretischen Arbeiten Ober Schallnahe Stromungen am Flugtechnischen Instut der Kungl . Tekniska Hokskolan, Stockholm.

Proc. Eighth Int. Congo on Theo. and Appl . Mech., 1953 .

40. Heaslet, M. A.; and Spreiter, J. R.: Three-Dimensional Transonic Flow Theory Applied to Slender Wings and Bodies. NACA Rep. 1318, 1957.

(Supersedes NACA TN 3717.)

41. Spreiter, J. R.: The Aerodynamic Forces on Slender Plane- and Cruciform-Wing and Body Combinations. NACA Rep. 962, 1950. (Super- sedes NACA TN's 1662 and 1897.)

42. Heaslet, M. A.; Lomax, H.; and Spreiter, J. R.: Linearized Compressible- Flow Theory for Sonic Flight Speeds. NACA Rep. 956, 1950. (Super- sedes NACA TN 1824.)

43. McDevitt, J. B.; and Taylor, R. A.: Force and Pressure Measurements at Transonic Speeds for Several Bodies Having Elliptical Cross Sections.

NACA TN 4362, 1958.

44. Nielsen, J. N.: Missile Aerodynamics. McGraw-Hill Book Co., New York, 1960, p. 30.

BASIC CONCEPTS GOAL: TO CALCULATE PRESSURE DISTRIBUTION, AERODY NAMIC CHARACTERISTICS, AND FLOW FIELD OF WING-BODY COMBINATIONS AT TRANSONIC SPEEDS EQUATIONS, EXACT AND APPROXIMATE NAVIER-STOKES EO. FOR UNSTEADY COMPRESSIBLE VISCOUS FLOW ASSUME STEADY INVISCID FLOW EULER EQ. OF STEADY COMPRESSIBLE FLOW ASSUME SMALL DISTURBANCES AND IRROTATIONAL FLOW, 0 S M<I) S - 3 \ LINEARIZED THEORY TRANSONIC THEORY (i-M<I)) CPXX +CPyy +CPZZ = 0 2 M<I)(Y+I)

I

(i-M<I))cpxx+C/>Yy+CPzZ = U<I) CPr, CPxx TRANSONIC RANGE EXCLUDED UNIFIED THEORY FOR SUBSONIC, \ TRANSONIC, AND SUPERSONIC FLOW

I

Figure 1 PRINCIPAL METHODS OF SOLUTION HODOGRAPH EXACT, LIMITED TO PLANAR FLOW, INDIRECT, ANALYTIC OR NUMERICAL INTEGRAL EQUATION APPROXIMATE, DEVELOPED ONLY FOR M<I) <I PLANAR FLOW BUT POTENTIALLY VERSATILE, DIRECT, NUMERICAL INTEGRATION PARABOLIC METHOD FOR Moo = I, AND EXTENSION FOR ALL M<I) APPROXIMATE, PLANAR OR AXISYMMETRIC FLOW, DIRECT, SIMPLE, BUT NOT ACCURATE GENERALLY .

LOCAL LINEARIZATION APPROXIMATE, DEVELOPED FOR PLANAR, AXISYMMETRIC, OR NONI,.JFTING FINITE SPAN WING FLOWS WITH Moo" I, Moo:S Mer, l' OR M 00 ~ Mer u AND POTENTIALLY MORE VERSATILE, DIRECT, RELATIVELY SIMP'LE, ACCURATE PARAMETRIC DIFFERENTIATION NEW, PLANAR RESULTS SIMILAR TO THOSE OF LOCAL LINEARIZATION, DIRECT, PROMISING COMPLETELY NUMERICAL DEPENDS ON INCREASING CAPABILITIES OF ELECTRONIC COMPUTERS / EXPENSIVE, UNDER DEVELOPMENT FOR TWO DIMENSIONS AT PRESENT Figure 2 -4 o -2 o C p EXPERIMENT T o .06 n=3. 38.(gh =.40 max 2 o .08 Mco= I [MICHEL et 01. 1954] "" .10 "V .12 0 .2 .4 .6 .8 1.0 x/C Figure 3 APPLICATION TO PLANAR FLOW -1.2,- -------, 8 Cp .

-. M =. 75j min Mco= 1 co \ ,i' .... " -.4 / ' , M =I . 66 co I ' C ,' ", ~ p o L... / ••• _~"'-e'~ ["7 " , Mer l = . 75 --j • I I , I I C pmax I Mer u =I. 66 1 I • , .8 t_[(I_~)_(I_g)3.38] I , SUBSO N IC' 'SUPER IC - t-- T RAN SO N IC --\- T= 1/12 1.0 .5 o 2 x/C Figure 4 APPLICATION TO AXISYMMETRIC FLOW o EXPERIMENT, 14' TRANSONIC WIND TUNNEL, Mexl = I [MCDEVITT AND TAYLOR, 1958] - .3,..-- ---------, .3,..------,.-.---------,

r(1!. ) =(I-M~)

-.2 exl .2 \ U er M~(y+l)

\

-.1 u .1 C p U exl 0~~===3~~$ O~-----*~----~ ,. _ 0 ', ... I k 94 I• 29 I }I, - M -I.29 0" Mer, l = • 1 \ : Mcr,u· exl I . 0/ " _.I : \ ' • . M =·94 ' : exl

f=~ -.2 i; \ (U:)

: : "

: TiAI' : "- -.3 ,---,S""UBS"",OI""IC--':----LLSO_IIC,-,,- ' -,-,su",-p£"",m~lC--, .3 '------'---------' o .5 1.0 o 2 xn Figure 5 EFFECT OF WIND TUNNEL WALLS AND BOUNDAR Y LAYER ON PRESSURE DISTRIBUTION ON SLENDER BODIES OF REVOLUTION, Mexl = I - .2 CHARACTERISTIC DIAGRAM OF BOD Y IN WIND TUNNEL -.1 C p WALL " 12' WIND TUNNEL .1 \ PARABOLIC-ARC BODY, 0 = 6 " -- EXPER IM ENT, Dn = 1/12, D = 6" o 14' TRANSONIC WIND TUNNEL, Mexl = I .1 [MCDEVITT AND TAYLOR, 1958) • 12' SOLID WALL, CHOKED .2 [SPREITER et 01.,1960) .3 LL----'-----.J - THEORY, LOCAL LINEARIZATION 0 .5 1.0 x/Z Figure 6 (I-M~) cP + cP + cf> xx yy zz ,Z um. ~ .•• , .---------~-------, EQUIVALENT BODY WITH SAME S(x)

- 6 ~

AS WING-BODY I ~CP2B + CPB , + 9 (x) FOR SMALL r AREA RULE FOR DRAG VORTEX DRAG

D = D - Pro (i<k acp2,a ds ' + fcp aCP2,l _,( cp aCP2B )

ds dS B 2 JC 2,a an c c 2,1 an I]B 2B an B Figure 7 PRESSURE DISTRIBUTION IN FLOW FIELD OF A SLENDER BODY OF REVOLUTION, MID = I C ........... .

- .l ~ PO .... . ... . . SA" .1 _.1 r-4D ~ r

T: .50

CPO~~ D .1 o o , o , , o -.05 -.10 - .15 , , C , p EXPERIMENT, 14' TRANSONIC WIND TUNNEL 0:6" BODY [TAYLOR AND McDEVITT, 1958] .3 - THEORY SURFACE : LOCAL LINEARIZATION FLOWFIELD: 4>' ~~ (~) lr . r+g(x) PRESSURE: Cp • _.1- 4> - ~ l UID X UID r Figure 8 PRESSURE DISTRIBUTION ON PARABOLIC-ARC BODY WITH ELLIPTIC CROSS SECTION, Moo = I THEORY: - 8 = 0°, --- 8 = 90° LOCAL LINEARIZATION AND EQUIVALENCE RULE

_ rU(X)S'(X) CT+(CT~a2 + b2)~]

<P2 ,t - R·p·l~ Ln 2 S = 17" ab, CT = Y + iz o. EXPERIMENT 14 ' WIND TUNNEL [MCDEVITT AND TAYLOR, 1958] -.15,-- ---- -----, .5 o 1.0 xlL Figure 9 DISCUSSION NORMAN MALMUTH, North American R9ckwell Science Center: I was wondering, Dr. Spreiter, whether you cared to comment on the relative advantages of unsteady and steady difference methods compared to the integral equation method, which you briefly mentioned .

SPREITER: No, I don't think I could make any very good comments on that.

M ALMUTH: I have another question about the applicability of your technique of local linearization to flows with a shock. Have you given any attention to this recently?

SPREITER: Not yet. So far our concern has been to develop programs which will let us do quickly what we already know; and we are using them to explore more configurations to get a firmer feeling of where we can go and where we cannot.

MALMUTH: My last question is on the equivalence rule. Can you extend it to larger incidence angles beyond its original limitation of small angle of attack compared to thickness ratio?

SPREITER: I don't know that the restriction you quote applies all of the time, since you can actually apply the equivalence rule to a wing with no thickness, ideally, a lifting flat plate, and it works perfectly well. I think in some of these instances one lacks a full definition of the boundaries of applicability. There are various ways in which these theories are appli- cable; and one must be a little careful when in possession of a demonstration or proof that it is true under certain circumstances not to regard that as a necessary condition, but more as a sufficient condition.

JAN RAAT, General Dynamics/Convair: I would like to draw attention to still another approximate method to attack the transonic potential equation.

One can rewrite the equation in the form (1 - M 2 ) ~ xx + ~ yy = 0 and then assume M to be a function of x alone. This simplification has also been made by Rubbert and Landahl. However, at this stage one does not have to go into parametric differentiation. The simplified equation can be handled by operational techniques so that the problem reduces to a nonlinear ordinary differential equation, which can then be treated by the WKBJ method for slowly varying functions. We have done some work along these lines at Convair and we have obtained encouraging results for symmetric bodies at zero incidence.

SPREITER: Actually I could have extended my list by several more schemes .

What you are describing sounds quite similar to the method that Julian Cole and Royce developed at one time in which they replaced ~ x ~ xx term with x( ¢ xx). They then applied the method to a body of revolution having, I believe, a parabolic arc profile.

Mrs. Alksne and I inspected that method at the time; and what we found was that it would indeed give a good result for a parabolic arc body of revolution and also, as I recall, a parabolic arc airfoil. But as we tested it by going to other bodies, with maximum thickness fore and aft of the midpoint, its success deteriorated very badly.

I think these are all closely related concepts of approximation, and don't regard them so much as competing with each other as supplementing.

Ultimately, we may have a better theory than any of them.

RAAT: As far as accuracy is concerned, I think operational methods and Rubbert and Landahl's technique produce about the same results. In either case, one starts out with the crucial assumption of the local Mach number M being a function of x alone. From then on it essentially is a matter of applying different tools to the same problem.

I agree with you that the various approximation schemes that have been employed, usually lead to surprisingly similar results.

SPREITER: Yes, and perhaps this would be even more clear if one had a better insight into similar ideas. Heaslet, for example, in the Japan National Congress of Applied Mechanics in 1959, which probably is very unknown since it. was only published in the proceedings volume in Japan, has a very provocative development which leads to the same results as the local linearization method. In his development, he starts with a reciprocity theorem, a nonlinear one, and relates the transonic flow to another flow in the reverse direction about the same object. He then makes a plausible approximation, and 10 and behold, the familiar results emerge. I think it is a very interesting situ a tion where you have several approaches that seem rather different in the beginning, but end up with either identically or very closely the same results.

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PROCEDURES FOR DESIGNING SUPERSONIC BODIES OF REVOLUTION FROM PRESCRIBED SURF ACE PRESSURE DISTRIBUTIONS By Raymond L. Barger NASA Langley Research Center SUMMARY An iterative procedure for designing a body from a prescribed surface pressure coefficient distribution is described. The method of successive approximation was pre- ferred over a possible analytic inversion procedure because any slight deviation from exactness in computing the body shape, made in order to obtain an analytic solution, would cause a greatly amplified error in the final pressure distribution. The present method also facilitates comparing design and off-design performance. Results calculated for a static pressure probe designed by this method indicated that it could be effectively used over a range of Mach numbers including the deSign Mach number.

INTRODUCTION There are a number of problems involving bodies of revolution at supersonic speeds for which it would be advantageous to relate the body design directly to a desired surface pressure distribution. Such problems arise, for example, in the design of fuselages for experimental wing- body combinations, in pressure- sensor design, and in the design of certain specialized types of proj ectiles.

It does not appear to be possible to obtain a direct analytic solution for the body shape in terms of the pressure distribution , except by using a theory that is so approxi- mate that the results are essentially useless. However, an iterative method that appears to work well (provided, of course, that the prescribed pressure distribution is a reasona- ble one) is described herein.

It is apparent that in prescribing a surface pressure distribution there must be some limitation on the initial positive portion in order that the condition of shock attach- ment be satisfied. It is also necessary to place some constraints on both the area and the slope of the negative portion in order to obtain a body with a reasonable base area and in order to prevent a sudden necking down or "wasping" of the body shape. Any such sudden or large reduction in body radius would cause premature flow separation on the resultant body and might result in the calculation of a negative radius, in which case the procedure would fail to converge. In other cases the procedure would converge, so these constraints are not critical with regard to convergence ; however, if one prescribes a completely absurd pressure distribution , the resulting shape will be unsatisfactory in some way.

SYMBOLS C body surface pressure coefficient p l body length M Mach number R body radius S body cross-sectional area t,x axial coordinates U tabulated function u,v dimensionless axial and radial disturbance velocities , respectively denotes increment in indicated quantity Subscripts: denotes nth iteration n free stream Superscript: denotes derivative in axial direction GENERAL PROCEDURE The general procedure is summarized first , and some of the steps are described subsequently in more detail. The general procedure includes the following six steps: (1) An initial body shape is assumed. (The convergence of the procedure is not at all sensitive to this initial guess; the program as written simply uses a parabolic shape.)

(2) Its pressure coefficient distribution is calculated. (This calculation is described subsequently.)

(3) The error is computed - that is, the values of Cp for the body are subtracted from the desired values of Cpo (4) The body slope distribution is changed in such a way that the error in step (3) is reduced. (This step is also discussed subsequently.)

(5) The new slope function is integrated to obtain an improved body shape - that is, a body that has a pressure distribution nearer to the desired distribution.

(6) The improved shape is introduced into the program as the input for the next iteration.

This procedure is repeated until the error is reduced to a negligible value.

PRESSURE CALCULATION Step (2) of the general procedure requires some further discussion. The basic expression used in this pressure calculation is Lighthill's approximate solution of the linear equation for the nondimensional axial velocity (ref. 1, p. 455) ()1 rxurx-tJdS'(t)

u x = - 27T J lPR(t)J ,8 R(t)

O in which ,8 has the constant free-stream value. This expression is modified in the analysis of reference 2, which makes use of an approach somewhat similar to local lin- earization to obtain the following equation (given here in the present notation):

1 rx [ x - t J[1 + u(tDdS' (t)

u(x) = - 27T J U lf3(t)R(t~ ,8 (t)R(t)

O This equation contains the local ,8 instead of the free-stream value, and a factor of 1 + U has been inserted. This new factor is a result of using the exact boundary condi- tion in Lighthill's original derivation instead of the slender-body approximation.

It is seen that, whereas the basic Lighthill expression is simply a formula for u, the modified expression is an integral equation for u. Furthermore, it is highly non- linear because u is involved not only on the left-hand side of the equation and explicitly under the integral but also in each ,8. An approximate solution to this equation was obtained in reference 2 by the method of parametric differ entiation , with the body thick- ness ratio used as the parameter.

Inasmuch as parametric differentiation is a complicated procedure, both theoreti- cally and computationally, that method was not used in the present analysis. Instead , the integral equation was solved by successive appro x imation , as indicated by where the functions u , f3 , and R have the local values , as in the preceding equation.

n For convenience, U is taken to be identically zero. This iterative procedure has the o advantages that it converges rapidly, it is a natural method for machine calculation, and it yields essentially the exact solution of the integral equation (whereas the parametric differentiation solution is approximate) . The solution of this equation for u is the basis of the pressure calculation, because after u has been calculated, v is obtained from the exact boundary condition: Then Cp can be computed in terms of u and v by using the exact expression for Cp(u,v} (ref. 3, eq. (9.9}).

Since the design procedure relies on performing a pressure calculation and then reducing the error in it , the accuracy of this calculation is a crucial consideration for the method. Figures 1 to 3 show some comparisons of computations on known shapes for which accurate solutions are available just to give an indication of the accuracy of this pressure calculation method.

It is convenient to use cone calculations to compare theories because exact solutions are available and because, for such comparisons, it is easy to observe the variation of the pressure coefficient with Mach number. In figure 1, results obtained by the present theory are compared with some results presented in reference 1 (p. 468) for a cone with a half-angle of 15 • The data for the exact solution are from reference 4. The Broderick second-order theory is an expansion in powers of the square of some parameter such as the tangent of the cone semivertex angle. Thus the second-order theory includes terms up to the fourth power in this parameter, whereas the first-order theory includes terms no higher than second degree. For the curve labeled "Linear Equation Solved Exactly," the linear solution for u was used together with the exact boundary condition and the exact pressure coefficient formula. It is seen that even the second-order theory is appli- cable only over a small Mach number range , whereas the present theory is accurate over an extensive range , the only exception being the re~ion within a few tenths of the minimum Mach number - that is , the Mach number for which the shock detaches .

Figure 2 shows a comparison of results obtained by the present theory with some data given in reference 5 for a cone with a half-angle of 20 . The Van Dyke second-order theory is somewhat more accurate than the Broderick second-order theory in figure 1; however , it begins to diverge Significantly for the higher Mach numbers. Again the pres- ent theory is inaccurate only within a few tenths of the detachment Mach number. The Significance of the weakness of the theory in this region as far as the overall design prob- lem is concerned is that if one is designing for a very low supersonic Mach number, it is expedient to check in a table of pressure coefficient versus cone half-angle and Mach number to make sure that the value of Cp prescribed at the nose corresponds to a cone half-angle well within the attachment capability for the design Mach number.

In order to illustrate the application of the direct pressure calculation to nonconical bodies, figure 3 shows some results for a parabolic body at two free-stream Mach num- bers. The Lighthill first- order theory is somewhat more accurate than a strict linear theory. The agreement of the present theory with the method of characteristics is very good at M oo = 2, with just small deviations at M oo = 4.

These comparisons in figures 1 to 3 indicate that the accuracy of the pressure cal- culation is good .

BODY SLOPE CORRECTION Step (4) of the general procedure corrects the body slope so as to reduce the error in the distribution of Cpo The change in Cp at a pOint x due to a change in R' at point t (for x;; t) is given by the following expression (modification of equation given in ref. 1, p. 453):

2AR'(t)[1+u(tU U J

x - t AC ~)= U P (3 (t) (3 (t)R(t) Here U is the same tabulated function used in step (2), the pressure calculation. In general, changes are to be made in Cp and in R' at all points. Consequently this calculation can become quite involved . However, it can be shown by an argument uti- lizing the fact that U = 1 when t = x that the following Simple equation is an approxi- mate expression for ACp(X): 2AR' (x) ~ + u(x)] () AC p x = (3 (x) This simplification does represent a rough approximation , but it is used here to relate incremental quantities; furthermore, the error does not have to be eliminated on anyone iteration but just reduced significantly. In this equation the function aCp(x) is simply the error in Cp(x) obtained from step (3) of the general procedure. The equation is easily solved for aR ' to obtain the new slope distribution for the new body shape: SAMPLE APPLICATION The procedure converges rapidly. Four iterations appear to be adequate even for the worst cases. Machine time on the Control Data 6600 computer system is about 2 min- utes for the entire program.

Figure 4 shows the results for one sample application, the design of a static pres- sure probe. The problem is to design a sensor tha t will read the free-stream pressure over a range of Mach numbers.

The sequence of calculations is as follows: The pro g ram input was represented by the solid curve - the design distribution of Cp for a free-stream Mach number of 2.5.

With this input the body shape shown in the figure was computed. Then direct pressure calculations were made for the shape at the off- design Mach numbers, with the results indicated by the dashed lines.

It is seen that , if an orifice is located at about the 55-percent station , the probe should read the free-stream pressure over a considerable range of Mach numbers. It is interesting that although the detailed body shape is determined by the single design Mach number, in certain applications like this one it is possible to obtain a result that is useful over a range of Mach numbers.

CONCLUDING REMARKS A rapidly converging iterative procedure for designing a supersonic body of revolu- tion from a prescribed surface pressure distribution has been developed. The method facilitates comparison of design and off-design performance. Results calculated for a static pressure probe des .igned by this method indicated that it could be effectively used over a range of Mach numbers including the design Mach number.

REFERENCES 1. Lighthill, M. J.: Higher Approximations. General Theory of High Speed Aerodynamics.

Vol. VI of High Speed Aerodynamics and Jet Propulsion , sec. E, W. R. Sears , ed., Princeton Univ. Press, 1954, pp. 345-489.

2. Kacprzynski, J. J.; and Landahl, M. T.: Recent Developments in the Supersonic Flow Over Axisymmetric Bodies With Continuous or Discontinuous Slope. AIAA Paper No. 67-5, Jan. 1967.

3. Liepmann, H. W.; and Roshko, A.: Elements of Gasdynamics. John Wiley & Sons, Inc., c.1957.

4. Sims, Joseph L.: Tables for Supersonic Flow Around Right Circular Cones at Zero Angle of Attack. NASA SP-3004, 1964.

5. Van Dyke, Milton D.: First- and Second-Order Theory of Supersonic Flow Past Bodies of Revolution. J. Aeron. Sci., vol. 18, no. 3, Mar. 1951, pp. 161-178.

COMPARISON OF SOLUTIONS FOR PRESSURE ON A CONE

--

BRODER I CK SECOND - ORDER THEORY ------ FI RST - ORDER THEORY

~

--- LINEAR EQUATION SOLVED EXACTLY 15° .4 EXACT SOLUTION PRESENT THEORY .3 p C \ ---- ....-/

~,' . ---

.2 "- '~ v- __ ~ ______ -.~ ______ ~

.......... ---

.1 .... -...... -----.

3 4 5 o 1 2 M.x, Fi gure 1 COMPARISON OF SOLUTIONS FOR PRESSURE ON A CONE -- EXACT SOLUTI ON -- VAN DYKE SECOND-ORDER THEORY

~

FI RST -ORDER THEORY 20° .6 o PRESENT THEORY

\

.4

,~

, .....

..... ,-

---

--

---

2 3 4 5 o 1 Moo Figure 2 COMPUTED C DISTRIBUTION FOR PARABOLIC BODY p FINENESS RATIO, 7 -- CHARACTERISTICS ----- LIGHTHILL FIRST-ORDER THEORY Moo =4 -.1

----

o PRESENT THEORY C O~---------=r~~--------------~ p FOR -.1 Moo- 4 .I C p .2 '---------------,-,.........,----------------'----1 0 FOR M =2 .1 00 .2 o .2 .4 .6 .8 1.0 X/I Figure 3 EXAMPLE CASE OF APPLICATION OF METHOD PRESSURE-SENSOR DESIGN -- DESIGN Moo=2.5 ---- OFF-DESIGN M =1.5 oo ---OFF -DES IGN Moo=3 .5 -.1 X/1 Figur e 4 DISCUSSION THEODORE R. GOODMAN, Oceanics, Inc.: Last spring I attended the Canadian Aeronautical Institute in a joint meeting with the AIAA in Ottawa, and a paper was presented which used exactly the same six-step procedure but in designing incompressible bodies, and I think that the two papers together cover the complete Mach number range. You might be interested to look it up.

BARGER: I am glad you mentioned the subsonic calculation . I have been working on a similar problem for the subsonic case in an effort to include the compressibility.

If you use the von Karman source distribution formula, similar to this Lighthill formula in the supersonic case, · again using the local value of 13 , and solve the resultant integral equation by iteration or successive approxi- mation, then you obtain a pressure calculation which is considerably more accurate than a similar strictly linear type of computation, especially for blunt bodies where, as you know, near the nose you have real problems in the compressible ca~e. But you get a very accurate calculation for the direct pressure calculation. I haven't been able to perform the inversion as yet.

WALLACE D. HAYES, Princeton University: In the supersonic case the problem is hyperbolic in the sense that there is no upstream influence, and you do not seem to take advantage of this property in your method. Your method is the same method that would be appropriate for the subsonic case .

If you were worried, which I guess you aren't, about computational efficiency, it seems to me you should take this into account . Perhaps yours uses so little machine time it's not worthwhile.

BARGER: To some extent, this is taken into account in the calculation, and it has a lot to do with the rapid conversion of the method, and it seems to be the " reason that as yet I haven't been ab Ie to do it for the subs'onic case; but that is really a matter of time, a matter of debugging the program .

But it is taken into account in the machine program . I just failed to mention it.

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EMPIRICAL METHOD FOR ESTIMATING PRESSURES ON ELLIPTIC CONES By George E. Kaattari Ames Research Center SUMMARY A method is presented for estimating the pressure distribution over elliptical cones at supersonic Mach numbers at angle of attack. The method is based on an empirical correlation between experimental pressures in the symmetry planes of elliptic cones and the pressures given by two-dimensional shock theory. The method is applicable for cones whose ellipticity ratios b/a range from I to 6, and whose maximum semiapex angles are less than 30° at Mach numbers greater than 2. Results given by the method are shown to agree well with experimental values.

INTRODUCTION There is currently considerable interest in flow solutions for cones at supersonic speeds. Exact solutions are difficult to obtain, particularly for elliptic cones at angle of attack. Engineering estimates of pressure distri- butions can be made by the tangent-cone method or by Newtonian theory. How- ever, these methods are reliable only for large angle cones of low ellipticity.

The following analysis will describe the development of a method for predicting pressure distributions on elliptic cones at supersonic Mach num- bers. The method is based primarily on unique correlations between pressures in the symmetry planes of elliptic cones and pressures given by two- dimensional shock-expansion theory.

NOTATION A* normalized base area coordinate (fig. 3) a cone base vertical semiaxis cone base horizontal semiaxis b pressure coefficient c pressure coefficient at A* PA* c circular-cone pressure coefficient, a = 0 Pee c two-dimensional or wedge pressure coefficient p w cone pressure coefficient at A* 0, a = 0

cone pressure coefficient at A* I, a = 0

=

differential operator interpolation constants cone length free-stream Mach number

exponent of A* , a = 0

exponent of A* , O

a. t-

y vertical coordinate from cone axis z horizontal coordinate from cone axis angle of attack, deg

incremental cone pressure coefficient at A* = o due to

angle of attack incremental cone pressure coefficient at A* = I due to angle of attack 8* correlation constant (fig. 5) wedge angle, zero angle of attack wedge surface angle at angle of attack cone semiapex angle in vertical plane cone semiapex angle in horizontal plane ANALYSIS Zero Angle of Attack T ypical correlations developed between experimental pressures (refs.

1-4) in the symmetry planes of elliptic cones and the corresponding two- dimensional or wedge pressures when the wedge angles are equal to the cone semiapex angles are shown in figure 1. Linear correlations result at a given Mach number for a family of cones that have the same ratio of base area to length but differ in cross-section ellipticity. Specifically, the slope is only a function of Mach number and the ratio !ab/l. Circuiar cones are included because the pressures at this coordinate point may be readily deter- mined from supersonic handbooks (e.g., ref. 5). If, in addition, the value of the slope at this point were also known, the correlation line would be completely defined. The pressures in the symmetry planes of any elliptic cone in the family could then be determined from the correlation line with the two-dimensional (wedge) pressure coefficient corresponding to the cone angle in the plane of symmetry and the free-stream Mach number.

Experimentally determined slopes, including those of figure 1 and certain theoretical values (ref. 6), were plotted on logarithmic scales as a function of the ratio of circular-cone pressure to wedge pressure. The general, linear correlation shown in figure 2 was found to result for the indicated wide range of cone geometries and Mach numbers. The slope of this correlation is closely

represented by (c kp )1.6 The pressure correlation lines of figure 1 are

p

eel w

now fully defined and the pressures in the symmetry planes of any elliptic cone in the family become determinable. Next, we attack the problem of deter- mining the circumferential pressure distribution between the planes of symmetry.

By apparent mass theory, the chord force of a cone depends only on its maximum cross section or base area. Experimental indications (ref. 1) are that the chord forces of elliptic cones of a given length and Mach number also depend on the base area and are almost independent of ellipticity ratio. New- tonian theory predicts the same result for cones in a restricted ellipticity ratio range. It is therefore assumed, with some confidence, that the chord- force coefficient of an elliptic cone can be considered equal to . that of a circular cone of the same base area and length.

Figure 3 outlines the manner in which the correlations of figures 1 and 2 and the above assumption with regard to chord-force coefficient may be used to estimate the circumferential pressure distribution on an elliptic cone at zero angle of attack. A quadrant of an elliptic cross section is shown. The variable A* represents the ratio of the area of the indicated sector to that of the whole quadrant. The merit of this variable is that when the pressure distribution is integrated with respect to A* over the quadrant, the chord force or average pressure coefficient with respect to the base area results.

The pressure distribution is now characterized by the known pressures cpo and c in the planes of symmetry and the area under the pressure distribu- P1 tion curve which, by the previously stated assumption, is numerically the chord-force coefficient of an equivalent area circular cone or the mean pressure c .

p cc It is now assumed that the pressure variation is a monotonic function of the argument A* modified by a distorting exponent no which remains to be determined. The simple derivation indicated in figure 3 relates the known

I

J

pressures, cp , cp , and cp to the integrated, assumed pressure distribu- o 1 cc tion which, in normalized form, is a function only of no. Numerical values for the pressure coefficients fix the value of f(no) and in turn, nO (fig. 4). The pressure-distribution function may then be evaluated as a function of A* from equation (2).

Effect of Angle of Attack It was again possible to correlate the pressures in the vertical plane of symmetry of cones at angle of attack with two-dimensional wedge pressures by ·an extension to the procedure described for zero angle of attack. On the left-hand portion of figure 5 is a schematic representation of the correla- tion technique used. The wedge angle 0 wo is determined so that the cor- responding two-dimensional pressure is equal to the known pressure on the cone in the vertical plane of symmetry at zero angle of attack . The ratio of the wedge angle 0wo to the vertical semiapex angle O YO is assumed to be a constant 0* insensitive to angle of attack. Thus, if the cone is at angle of attack a , the corresponding wedge angle ow a to give the same pressure should be equal to o* ( O y + a ). The validity of this assumption was borne out by the good correlation between predicted and experimental pressures shown on the right-hand portion of figure 5 for the range of variables ind i cated.

Predicting the effect of angle of attack on the pressures on cones in the horizontal plane of symmetry is somewhat more involved than in the verti- cal plane. Figure 6 shows the Newtonian theory result that the change in the hori z ontal plane pressure 6cp with angle of attack is equal to the nega- tive value of t~e zero angle of attack pressure c times sine squared of P1 the angle of attack. Also shown on the figure are values from a sophisti- cated theory of Babenko (ref. 7) for circular cones. A linear correlation fit to these data in the low angle-of-attack range (0° to S°) is represented by the top line for a wide range of Mach numbers and cone apex angles. It was discovered, however, that the ordinate value 6c /sin a had to be P1 multiplied by a hypersonic similarity parameter M oo sin Oz in order to bring into correlation the data of those cones whose value for M oo sin Oz was less than unity. The Newtonian or bottom line was considered to represent data at a ; 90° since the horizontal plane pressure will then be close to zero by reason of the resulting large flow expansion, that is, 6C 1 ; -c 1 .

p p A technique for interpolating between the low-angle correlation (0°) expressed in the form -6c Ic ; (0.500 + 1.33C )/c sin a and the high- P1 P1 p I, PI angle correlation (90°) expressed as . - lI C ' 'P/cPl ;; sin a was devised as fol- lows : In figure 7 the line 0-1 has the slope (0.500 + 1.33C )/c and rep- P1 P1 resents the low-angle correlation curve. The line 0-2 has the slope of unity (1) and represents the Newtonian andlor high-angle-of-atte.ck correlation curve. The solid curve represents a monotonic interpolation function of sin a which is tangent to the low- and high-angle correlation curves in the neighborhoods of their respective validity (sin a = 0 and 1.0). The simplest monotonic interpolation function was found to be a series expression of three terms with the ordinate ~Cp /c as the independent variable; that is, p I I In order to satisfy the known slopes and ordinates required of the interpola- tion function, the values of the coefficients k were found to be sin <5 z and 0.500 + 1. 33c PI The term M oo sin 0 z is included only when its value is less than unity '. The above equation for sin a was evaluated for various numerical values of kl to tabulate sin a as a function of ~Cpl/cPl' This tabulation was used to construct curves of -~cPl/cPl as a function of angle of attack for var- ious values of k • These curves are presented in figure 8. Although fig- ure 8 is based on circular-cone data, the results are assumed to be valid for elliptic cones as well.

The pressure in the horizontal and vertical planes of symmetry is shown now for a cone at angle of attack. Through symmetry, the spanwise derivative of the pressure distribution at the vertical plane of symmetry is necessarily zero. These end conditions are not sufficient constraints to define the pressure distribution reliably, so additional pressure information is necessary.

The pressure derivatives given by Babenko's data for circular cones in the horizontal plane of symmetry at angle of attack compared with the simple result of Newtonian theory are presented in figure 9. Surprisingly, a good, one-to-one correlation results. Again, while this correlation is based on circular-cone data, it is assumed to be valid for elliptic cones as well.

With this correlation, an additional constraint to the pressure distribution curve is in hand.

It is convenient to treat the angle-of-attack pressures as increments to be added to the zero angle-of-attack pressures. Figure 10 is a normalized plot of circular-cone incremental pressures against A~. It is again assumed that the distribution is a simple monotonic cosine function of the argument A* raised to some power na' The relationship of na to the assumed pres- sure distribution is easily found by the differentiation indicated in the figure. The Newtonian value of TI sin Oz sin 2a /tan 0y is substituted for the end-point slope (dCp/dA*)1, allowing na and thus the incremental pressure distribution to be evaluated .

In the case of elliptic cones, a modification to the Newtonian pressure derivative given is required. This correction takes into account the fact that an elliptic cone does not have a constant zero angle-of-attack reference pressure distribution and although the pressure derivative is zero at A* = I, it rapidly assumes a value other than zero at A* < 1. It was found for a wide range of cone ellipticity ratios that a good approximation to the zero angle-of-attack pressure distribution was characterized by a curve whose pressure derivative in the vicinity of A* = 1 is 2(c - cp)' This value p 1 0 was added to the Newtonian value to give the effective derivative of the incremental pressure distribution at A* = 1 or COMPARISON OF PREDICTED AND EXPERIMENTAL RESULTS Z ero Angle of Attack The present method is compared in figure 11 with the tangent-cone method, Newtonian theory, and experiment (refs. 1-2). The pressures are plotted as a function of the normalized spanwise coordinate z/b. For cones of large ellipticity at Mach number 2, the present method agrees better with experi- ment than does the tangent-cone method or Newtonian theory. The superiority of the method is less pronounced with respect to the tangent-cone method for cones with b/a ratios close to unity as is to be expected.

Angle of Attack The comparison of experimental (ref. 2) and predicted results for pressur~ distributions for two elliptic cones at Mach number 6 presented in figure 12 indicates good agreement. For cones of low ellipticity such as those shown, the tangent-cone method will give good results at small angles of attack. For highly elliptic cones at large angles of attack, the present method is expected to give better results than the Newtonian or tangent-cone methods.

In figure 13 the present method is compared with a recent line method of Bazzhin (ref. 8) and modified Newtonian theory for a bla = 3, elliptic cone at Moo = 7 and angle of attack of 30°, The present method is in good agree- ment with the method of Bazzhin. The Newtonian theory predicts the maximum pressure well in this case, but the pressure distribution differs considerably from the other predictions.

___ _

j

CONCLUDING REMARKS It has been shown by comparison with experiment that good estimates of pressure in the symmetry planes of elliptic cones can be made by utilizing a simple empirical correlation with two-dimensional or wedge pressures. These pressures, in conjunction with experimentally verified constant chord-force coefficients for a given cone family and Mach number, are sufficient infor- mation to enable good estimates to be made of pressure distributions for cones at zero angle of attack.

The effect of angle of attack on pressures in the symmetry planes of cones was also shown to correlate in a simple manner with wedge theory and with Newtonian theory.

REFERENCES 1. Jorgensen, Leland H.: Elliptic Cones Alone and With Wings. NACA Rep.

1376, 1958.

2. Zakkay, Victor; and Visich, Marian, Jr.: Experimental Pressure Distribu- tions on Conical Elliptical Bodies at M = 3.09 and 6.0. Pibal Rep.

467, AFOSR TN 59-10, AD 208591, March 1959.

3. Chapkis, Robert L.: Hypersonic Flow Over an Elliptic Cone: Theory and Experiment. Guggenheim Aeron. Lab., Calif. Inst . Tech. Hypersonic Res. Proj. Memo. No. 49, May 1, 1959.

4. Ortloff, Charles R.: Hypersonic Approximation for the Inviscid Flow Over Conical Bodies Without Axial Symmetry and Comparison With Test at Mach 8. Pibal Rep. 728, AFOSR 1634, Jan. 1961.

5. Ames Research Staff: Equations, Tables, and Charts for Compressible Flow. NACA Rep. 1135, 1953.

6. Van Dyke, Milton D.: The Slender Elliptic Cone as a Model for Non-Linear Supersonic Flow Theory. J. Fluid Mech., vol . 1, no. 1, May 1956, pp. 1-15.

7. Babenko, K. I.; Voskresenskiy, G. P.; Lyubimov, A. N.; and Rusanov, V. V.: Three-Dimensional Flow of Ideal Gas Past Smooth Bodies. TT F-380, 1966, NASA, from Prostranstvennoye Obtekaniye Gladkikh tel Ideal'nym Gazon Izdatel'stvo Nauka, Moscow, 1964.

8. Bazzhin, A. P.; Trusova, O. N.; and Chelysheva, I. F.: Analysis of Perfect Gas Flow Around Elliptic Cones at High Angles of Attack.

Izv. Akad, SSSR. Mekh. i Gaza (Moscow), no. 4, 1968, pp. 45-51.

EXAMPLE CORRELATION OF CONE AND WEDGE PRESSURES a = 0°. Z PLANE Moo = 6 NEWTONIAN .4 / ~z / / .3 / / / CPeone / .2 / / Moo = 2 .1 Jab / I =0 .1 4 / .1 o .2 .3 .4 .5 .6 CPwedge F ig ure 1 SLOPES OF CONE-WEDGE PRESSURE CORRELATION LINES a = 0° 1.0 - o EXPER I MENT .8 - o t:. VAN DYKE-SECOND ORDER THEORY .6 - o NEWTONIAN AND .4- HYPERSONIC ESTIMATED 6CP e eon 6CPwedge .2 -

c )1.6

6Cpeone Pee ( 6CPwedge = Cpw 1.4 < Moo < 10 . 10 - 0 . 06 < ..JCib/l < 0.30 .08 - I < b/a < 8 .06 - I I I I .2 .4 . 6.8 1.0

C IC

Pee Pw Fi gur e 2 METHOD OF DETERMINING PRESSURE DISTRIBUTION a = 0° ASSUMPTIONS: I (I) fcp dA* = Cpee o 1+ COS7T(A*)no (2) I I ( * )nO = f[ + COS 7T A ] dA* o a = f (no) (3) b Figure 3 THE FUNCTION f( no) 1.0 SCALE CHANGE .8 .6 .4 . 2 o .2 .4 .6 .8 1.0 2 3 4 5 6 7 8 9 10 Figure 4 CALCULATED AND EXPERI MENTAL PRESSURES IN VERTICAL PLANE OF SYMMETRY 0° < a < 20° 2 < Mco < 10 1.4 < b/o < 6 .8 .6 .4 Cpo .2 ~wo 8 = a*(8 + a) y wa o Figure 5 EFFECT OF ANGLE OF ATTACK ON PRESSURES IN HOR I ZONTAL PLANE OF SYMMETRY FOR CI RCULAR CONES 2.0 o 0 ° < a < 5° o } BABENKO o 1. 6 2 < Mco < 7 o Figur e 6 BASIS OF INTERPOLATION TECHNIQUE 1.0 INTERPOLATION FUNCTION o 1.0 sin a Figure 7 PRESSURE DECREMENT IN HORIZONTAL SYMMETRY PLANE .5 .4 k = . 500 + 1.3~ CPI = co I C Moo Sin 8z p, .3 -~CPI C P1 .2 .1 1.5 0 5 10 15 20 25 30 a, deg Figure 8 PRESSURE DERIVATIVES IN THE HORIZONTAL PLANE OF SYMMETRY FOR CIRCULAR CONES 1.0 .8 dC .6 p ) ( - dA* I BABENKO .4 2 < Moo < 7 0 0 2.5 < 8 < 30 y 0 0 .2 2 .5 < a < 30 o .2 .4 .6 .8 1.0

(~~~)

INEWTONIAN Figure 9 ESTIMATION OF PRESSURE DISTRIBUTION DUE TO ANGLE OF ATTACK ilC - ilC p P1 ilC - ilC Po P1 1.01---: :::::-- - ~ ~ u U <l <l o a. a.

U U <l <l o Figure 10 , I I

l

-- - ~ - - - - ------ - -- ~~----- --= --------- - --- i-- I PREDICTED AND EXPERIMENTAL PRESSURES a = 0° .3 Moo = 2, b/o = 3 Moo = 2, b/o = 6 .2 - PRESENT METHOD / -- TANGENT CONE .

C p I ---- NEWTONIAN , .1

. /

. . . . ?"

--- '

../ '

--------=---= =--:=----

--= =- -=---== =-- -== =- --::: -:: - -' ",' .3 M", = 3, b/o = 1.4 Moo = 3, b/o = 1.8 .2 C r.,... --0-=:: p --- .1 -:;

-

---

-------- ... ----- o .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 lib lib Figure 11 PREDICTED AND EXPERIMENTAL PRESSURE DISTRIBUTIONS M", = 6, a #- 0° PREDICTED t:. 0 EXPERIMENT b/o = 1.39 b/o = 1.79 .4 a = 100 .2 .1 o .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 z/b lib Figure 12 COMPARISON OF PREDICTION METHODS .6 -~-"':";;--------- ......

~~ ......

'\ .5 , \ MIX) = 7 ~ \ .4 ./Cib I l =0.176 ~ \.

b/o = 3 ~ \ a = 30° ~ \ I .3 I I Cp I , -- PRESENT METHOD : .2 -- BAZZHIN,et. a!. ~ I ------ MODIFIED NEWTONIAN . 1 O~--------------~ ----=--" I I I o .2 .4 .6 .8 1.0 z/b Figure 13

L_ ~ _~

DISCUSSION ERNEST O. MARCHAND, ARO, Inc.: This is not in the form of a question but a comment that may be useful to some people here. There is a program, available upon request, by D. J. Jones (Aeronautical Report LR-507) of the National Aeronautical Establishment, Ottawa, Canada, that is capable of calcu- lating elliptic cones at angle of attack. To my knowledge, this is the only readily available program. It runs very quickly for all the sharp cones. I think the 360-50 running time is only a minute and half.

This report seems to have gotten a rather limited distribution, and many people don't know about it. I thought it appropriate that I should bring it up.

LORNE C. DUNSWORTH, CCI Marquardt Corp.: I wonder if you would comment on the possibility of extending this technique to elliptic ogives at angle of attack. Is there a way to do it?

KAATTARI: .1 don't know. I have thought of it, but I don't know if it is possible. I couldn't say at this time. I'd be glad to try it.

It may be that one could use Prandtl-Meyer flow and start from the cone, but I don't know, it might be quite difficult. For the angle-of-attack case equal to zero it might be feasible, but for the angle-of-attack case it would be quite difficult.

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IMPROVEMENT OF WING-BODY INTERFERENCE THEORY FOR HYPERSONIC SPEEDS By William C. Pitts Ames Research Center SUMMARY Calculated results from two theories are found to agree equally well with experimental data for the lift-curve slope and for the stability derivative for the Mach number range 5.3 to 10.7. The results agree despite the fact that one theory predicts a large effect of wing-body inter- ference and the other neglects interference effects . Theoretical examination of load distributions on the body and wing, with and without interference effects, verifies that, for the example considered, the total force and moment is the same by both theories, but the load distributions are very different .

INTRODUCTION The subject of wing-body interference has been extensively explored both theoretically and experimentally for the subsonic and supersonic speed ranges, but not very much has been done in the hypersonic speed range. In reference 1 an extensive comparison was made between theory and experiment for about 70 sets of data of widely different configurations for subsonic, transonic, and supersonic speeds up to Mach number 2. It was found that the lift - curve slope was predicted within 10 percent for all sets of data and that wing-body interference accounted for between 20 to 30 percent of the lift of the wing-body combinations . The purpose of the present program was to see if this large and favorable wing-body interference effect per- sisted to the hypersonic speed range.

TOTAL FORCE AND MOMENT In lieu of an adequately established hypersonic Mach number theor y , the supersonic theory of reference I was used as the first tool of the investigation. This theory was used to estimate the pitching moment and the lift-curve slope for a model that was tested at Ames (ref. 2). The model is shown in figure 1. The body has a fineness ratio 12 and a Sears- Haack nose. The wing has a delta planform with a 70° leading-edge sweep angle . The wing is 4 percent thick with flat surfaces; the bottom surface would pass through the body axis if extended.

In figure 2, experimental data for C and a ~/ a cL are compared L a with calculated results from two theories. Both of these quantities are predicted reasonably well by the theory of reference 1 (NACA Rep. 1307) within the Mach number range 5.3 to 10.7. This good agreement plus the fact that 25 percent of the lift is due to wing-body interference effects, accord- ing to the theory, suggests that wing-body interference is important at hypersonic speeds as well as at supersonic speeds. However, the tangent- cone, tangent-wedge method also predicts the experimental data well (except for acm/ a cL at M = 10.7) and it includes no interference effects.

Obviously, then, some compensating effects are present in one or both of these theories, and the question of the importance of wing-body inter- ference is not resolved by this comparison.

THEORETICAL LOAD DISTRIBUTION Correction Factors To study this problem further the program was extended to look at the load distribution on the wing-body combination. The wing-body interference theory of reference I could not be used for this extension because it gives only the total force and moment. Instead, the linearized, quasi- cylindrical theory of Nielsen (ref. 3) was chosen for studying the load distribution. This theory has been demonstrated to be applicable at super- sonic speeds by comparison with experiment (ref. 4) but several assumptions made in the theory must be examined before it can be applied to the hyper- sonic speed range. The first assumption is that the slender-body theory of Beskin properly predicts the body upwash. Other assumptions are that in the region of the wing the local dynamic pressure and the local Mach number are the same as in the free stream. These assumptions will now be examined in turn beginning with the body-upwash assumption.

The mechanics of body upwash are summarized in figure 3. Any body inclined at an angle a to a flowing gas will have a normal component of velocity V which streams around the body in the cross-flow plane. Then, N according to the slender-body upwash theory of Beskin, the local angle of attack in the wing plane varies along the span as (1) where a is the body radius, and r is the radial coordinate. At the

wing root, where air = 1, the wing is effectively flying at twice the angle

of attack of the body. This then is the reason wing-body interference is favorable.

This upwash model has been found to be adequate at low supersonic Mach numbers but until recently its use in the hypersonic speed range has not been justified. Now the method-of-characteristics calculations by Rakich (ref. 5) have shown that upwash varies in a manner similar to the Beskin equation except for an empirical factor o .

(2) a = [1 + o {a/r) ] a L This was found to be true for Mach numbers up to 10.7. The term in the brackets is then a factor that corrects the free-stream angle of attack to the local angle of attack.

The correction factors for Mach number and dynamic pressure will now be developed with the aid of figure 4. The sketch at the top of the figure shows a wing mounted on a cylindrical-body section. The theory of ref- erence 3 will give the pressure distribution aft of the wing Mach wave. The method of reference 5 will give body-alone pressures forward of the wing.

F or this figure the subscript 00 refers to free-stream conditions and the subscript L refers to conditions near the cylindrical part of the body.

Calculations by the method of characteristics show that although flow parameters vary rapidly near the body nose they become virtually independent of axial distance in the subscript L region. For Mach numbers less than 2 the shock is relatively weak and the subscript L values are essentially equal to the subscript 00 values. At hypersonic speeds they can differ significantly, and the pressure, P 2 , cannot be calculated directly in terms of the free-stream conditions by simple theories. However, it can be cal- culated in terms of subscript L conditions because they are essentially free-stream conditions to the wing. The pressure coefficient according to linear theory is then (3) The pressure coefficient can then be expressed in terms of free-stream conditions by simple algebraic manipulation of equation (3) (4) The second term of equation (4) is generally large in the body-nose region, and must be retained there, but calculations by the method of character- istics show that even for M oo = 10.7, this term approaches zero aft of the nose and can be neglected in the subscript L region. Thus the pressures on the winged region of the combination can be expressed in terms of free- stream conditions and three correction factors . The first is the upwash correction factor a Li a given by equation (2). The other two factors, qL/q oo and Soo / SL correct q and S to local values. The product of these three factors is called A. Then the pressures on the winged region of the wing-body combination can be expressed in terms of the product of free- stream conditions and the total correction factor.

p = (5) Although A and its components are virtually independent of axial distance they do depend on r. Calculations by the method of characteristics (ref. 5) show that the radial variation can be very closely approximated by the a (a/r) form of equation (2). The pressure distributions that will be presented later have had this correction factor incorporated into the calculations.

Figure 5 shows how these factors vary with Mach number at the wing- body juncture. Note that at Mach number 2 both SL and qL are essentially equal to free-stream values so that the assumption of this fact (for M ~ 2) in reference 3 was valid. Also the use of slender-body theory to estimate body upwash was valid in this M ach number range. As the Mach number is increased from 2, each of the individual correction factors diverges from the slender-body theory value. However, the effects of these divergences are compensating and, fortuitously, the net effect is that A, as calculated by the method of characteristics, remains very nearly equal to the slender-body theory value for A throughout the Mach range.

Calculated Loading The effect of these correction factors is shown in figure 6. The figure shows the chordwise pressure distribution at the wing-body juncture for the model shown in figure 1. The pressure coefficient P and the axial distance x from the wing leading edge are presented in normalized forms S P/ a and x/ S a, where a is the free-stream angle of attack. All three curves were calculated using the quasi-cylindrical theory of reference 3. The only difference between the curves is the manner in which the correction factor is calculated. For the upper curve the method of characteristics was used to calculate A. For the middle curve slender-body theory was used, and for the lower curve no correction was used for upwash, SL or qL' As would be expected from the results of figure 5, it makes little difference whether the method of characteristics or slender-body theory is used to calculate the correction factor. However, there is a large difference if no correction is used for local flow conditions.

Figure 7 shows the same thing for the entire wing. The ria = 1, or wing-body juncture plane, shows the same pressure distribution as fig- ure 6. The pressure distributions in the ria = 2 and ria = 5 planes are also shown. In all planes, the upper curve is the pressure coefficient with correction factor and the lower curve is the pressure coefficient for no correction factor. All along the wing leading edge the pressure is the same as that of an isolated flat plate inclined at the local angle of

attack. At the wing root (ria = 1) the pressure starts decreasing

immediately in the chordwise direction. At ria = 2 the pressure rises in front of the Mach wave from the wing root leading edge and then decreases behind the Mach wave. The rise is due to this wing element experiencing the influence of inboard elements which are flying at higher effective angles of attack. At ria = 5 the wing is entirely in front of the Mach wave and the body upwash effect is small so that the pressure is nearly constant over the entire chord and the two curves are very nearly the same.

In contrast to the varying pressure level predicted by this theory, the tangent-wedge method predicts a uniform pressure over the entire wing, indicated by the three hash marks at the leading edges of the three radial stations. This is also the pressure level obtained if the interference theory pressure distribution is averaged over the entire wing. Thus the lift predicted on the wing is the same for both theories. The interference theory, however, predicts a larger nose-up moment on the wing than does the tangent-wedge method. This nose-up moment is compensated for by a nose- down moment contributed by interference pressures from the wing acting on the afterbody. This is shown by the pressure distribution on the body in figure 8. To simplify the isometric drawing the body is shown as a flat surface in the wing plane. In this view the Mach wave appears as a straight line rather than a helix around the cylindrical body. Because the cylindrical-body section generates no lifting pressure according to the theory of reference 3, this pressure is due entirely to interference on the body from the wing.

CONCLUDING REMARKS In summary, two theories have been compared . One of them includes wing-body interference effects and the other does not . The two theories agree with each other and with experimental measurements of the total lift and pitching moment for the wing-body combination considered. They also agree on the amount of lift on the wing and on the body. However, they differ on the distribution of lift and hence on the moment acting on the wing and on the body. Fortuitously these differences cancel. The question still remains as to which, if either, of the two theories is correct. The answer to this question awaits experimental evidence from programs now in progress.

REFERENCES 1. Pitts, William C.; Nielsen, Jack N.; and Kaattari, George E: Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds. NACA Rep. 1307, 1957.

2. Nelms, Walter P., Jr.; and Carmichael, Ralph L.: An Experimental and Theoretical Investigation of a Symmetrical and a Cambered Delta Wing Configuration at Mach Numbers From 2.0 to 10.6. NASA TN 0-5272, 1969.

3 . Nielsen, Jack N.: Quasi-Cylindrical Theory of Wing-Body Interference at Supersonic Speeds and Comparison With Experiment. NACA Rep. 1252, 1955.

4. Pitts, William C.; Nielsen, Jack N.; and Gionfriddo, Maurice P.: Comparison Between Theory and Experiment for Interference Pressure Field Between Wing and Body at Supersonic Speeds. NACA TN 3128, 1954.

5. Rakich, John V.; and Cleary, Joseph W.: Theoretical and Experimental Study of Supersonic Steady Flow Around Inclined Bodies of Revolution.

Preprint 69-187, AIAA, 1969.

MODEL

Figure 1 COMPARISON OF THEORY AND EXPERIMENT . 05 TANGENT-CONE TANGENT-WEDGE TR 1307 (PITTS, NIELSEN, KAATTARI) o EXPERIMENT g' .04 l:J Qj Q. .03

-

w a..

g 02 en · w > gj .01 ., -.2 u ~ ~ 0 I--+---+---'\ob-lc---:-+---+----l o ~ 0 :::-- __ !;,; f---- ..................

.........

.........

I I -- ...J .2 -.01 2 4 6 8 10 12 Moo Figure 2 CORRECTION FACTOR FOR BODY UPWASH

alia ~t ~I

o 2 ria a =[I+(a/r) Ja (SLENDER BODY) L 2J (CY FROM METHOD OF [ a L = 1+ CY (air) a CHARACTERISTICS) Figure 3 CORRECTION FACTORS FOR MACH NUMBER AND DYNAMIC PRESSURE Figure 4 VARIATION OF CORRECTION FACTORS WITH MACH NUMBER 3.0

METHOD OF ~ ~L

2.5 CHARACTERISTICS~ ~- "' ~-->.

2.0 ~ SLENDER-BO;;--- THEORY 1.5 1.0 METHODOF ~qL .5 CHARACTERISTICS qco o 2 468 10 12 MACH NUMBER Figure 5 CHORDWISE PRESSURE DISTRIBUTION AT WING- BODY JUNCTURE Mco=7.4 , A . . = 70 L E

WITH >. BY METHOD OF

CHARACTERISTICS, >'(0, q ,13) WITH>' BY SLENDER-BODY THEORY, >'(0) /3p o CORRECTION FACTOR, >'=1

(%=~= :c= I)

o .4 .8 1.2 1.6 2.0 x//3a Figure 6 - ------ _.-- - - - --- - PRESSURE DISTRIBUTION ON WING Moo = 7.4 ,A .=70° LE WITH A BY METHOD OF CHARACTERISTICS NO CORRECTION 5 FACTOR, A=I ria {3P 3 a o .4 .8 1.2 1.6 2.0 2.4 xl{3o Figure 7 PRESSURE DISTRIBUTION ON BODY Moo = 7.4 , ALE.= 70 WITH A BY METHOD OF CHARACTERISTICS {3p BODY 0 -...l...- -== --r ---- -- 45 ~------------~=-~~~--90° ------ - MACH WAVE FigureB DISCUSSION A. R. GEORGE, Cornell University: If these sigmas and lambdas were determined for this one shape that you showed, do you expect that you would get this close an equivalence between the slender body and the characteristics calculations in general?

PITTS: We are talking now about the cylindrical section of bodies. I don't think that the nose shape is going to have too much of an effect if you get far enough aft of the body shoulder.

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THEORETICAL STUDIES OF VORTEX FLOW ON SLENDER WING-BODY COMBINATIONS By E. S. Levinsky and M. H. Y. Wei Air Vehicle Corporation and Ralph L. Maki Ames Research Center SUMMARY An analytical procedure is prese nt ed f or determining the nonlinear lift and pressure distribution on slender win g- b od i es with leading-edge vortices.

The theory is restricted to flows which sati sfy the usual constraints imposed by slender-body theory.

Both conical and nonconical developments are presented. The theoretical results are evaluated by comparing with wind-tunnel force and pressure data.

Both theory and test data show large increases in lift due to leading-edge vortices for even the smaller size strakes. Good correlation was obtained over the complete angle-of-attack range for cones with strakes of 50 percent of the body radius or greater. For these configurations the lift was approxi- mately two times the linear theory value at angles of attack a equal to

twice the strake semiapex angle a. For cones with 25-percent and la-percent

strakes, good agreement was found except in the intermediate range 2 < a /a < 4.

Calculations for several nonconical configurations also agreed well with test data. Among the nonconical geometries considered were a cone with a fil- leted double-delta wing, a wing-body w ith a curved nose and strakes, and a conical body with variable incidence strakes.

I NTRODUCTION The present paper deals with the calculation of the nonlinear lift and pressure distribution on wing-bodies with leading-edge vortices. Such vorti- ces are known to increase the lift at large angles of attack to several times that predicted by the usual linear analyses, thus invalidating the linear approach at large angles of attack. Both conical and nonconical theories are presented; however, the analysis is limited to configurations for which the usual slenderness assumptions apply.

A slender wing-body configuration with leading-edge vortices is shown in figure 1. The model, photographed in the Ames 7- by la-Foot Wind Tunnel, is a blunted cone with small-span wings (strakes) at an angle of attack of approximately 30°. The vortices are known to have a pronounced effect on lift and pressure distribution, hence the motivation for the present analysis. The well-known model for the leading-edge vortex consists of a feeding spiral sheet originating from the wing tip and winding inward into a central vortex core (see fig. 2). This model will be used in the subsequent analysis.

The general type of configuration geometry to be studied is also shown in figure 2. Effects of body cross section, relative strake-to-body size, strake dihedral, nose blunting, variable strake sweep angle, and variable strake incidence angle are included in the theory. Thus, the theory should " be of use in evaluating the low-speed approach, landing, and control capabil- ities of recoverable booster or spacecraft configurations of high volumetric efficiency.

Several types of vortex flows have been observed for slender bodies and wing-bodies. For a body alone at a large angle of attack, two strong vortices are known to form on the leeward side (refs. I and 2) (see upper sketch in fig. 3). The separation point is determined by viscous effects for this case, and no complete theory which includes determination of the separation point has as yet been obtained. The present analysis deals with wing-body combina- tions for which the separation point is fixed at the wing leading edge (middle sketch in fig. 3). A priori knowledge of the separation point greatly simpli- fies the theory and enables an inviscid formulation to be used. It is conceiv- able that with very small span strakes the leading-edge vortex may influence the flow only locally in the region of the leading edge. In this case, body vortices could still occur as sketched in figure 3. Although the simultane- quS occurrence of both body and strake vortices has not been confirmed visu- ally, anomalous data obtained with small-span strakes can be explained by this hypothesis, as will be shown later.

Considerable research has been carried out on the leading-edge vortex for low aspect ratio wings and bodies as summarized in figure 4. The first successful analysis was carried out by Brown and Michael (ref. 3), who treated the conical flow past a flat delta wing. They used a simplified model for the vortex in which all the vorticity was lumped into the vortex cores. The vor- tex cores were joined to the wing tip by a cut. No feeding sheets were included. Their calculations gave the correct overall trend for the variation of lift with angle of attack. However, the vortex location was calculated to be too far outboard, and the overall lift was significantly overpredicted.

Mangler and Smith (ref. 4) improved the formulation by also including a feeding v9rtex sheet of a specified shape. This restriction was later removed by Smith (ref. 5) who considered a segmented vortex sheet and deter- mined both its strength and position. The calculations by Smith (ref. 5) agreed remarkably well with test data. The present authors have generalized the conical Smith procedure to apply to bodies with small span strakes (ref. 6), and have also extended the formulation to nonconical flows

-

SYMBOLS vertical half-dimension of elliptical body; also, radius of a circular body horizontal half-dimension of elliptical body b lift coefficient based on planform area lift coefficient based on force and area ahead of station x pitching-moment coefficient about apex based on body length and C m planform area C (x) pitching-moment coefficient about apex based on moment, length, m and planform area ahead of station x i N number of vortex sheet segments n inward normal to vortex sheet

static pressure ( ~ p = p - p oo )

p free-stream dynamic pressure q R Reynolds number r polar radius in physical plane u component of free-stream velocity along x complex velocity potential w x,y,z body-centered Cartesian coordinates Z complex variable (y + iz) Z* complex variable in transformed crossflow plane angle of attack separation angle r vortex strength <5 strake semiapex angle

no parameter in transformation Z* = 2*(Z)

e polar angle ~ parameter in transformation Z* = Z*(Z) ~o 6 ~ velocity potential jump across vortex sheet ~ velocity potential angle between r and tangent to sheet a distance along vortex sheet Subscripts m mean value v isolated vortex core Superscript complex conjugate DISCUSSION Conical Analysis The usual slender-body approach is assumed in the analysis. The major steps in the theory have been outlined in figure 5. The governing differen- tial equation is the Laplace equation in the crossflow plane Z = Y + iz. The linear theory requires that the tangency condition be satisfied on the surface of the wing-body. A simple means for accomplishing this is to conformally map the wing-body into a vertical slit in the transformed Z* plane. A general- ized transformation for carrying this out is shown in figure 5, and applies to wing-bodies of elliptical cross section (b/a is the ratio of the major-to- minor axis) and of arbitrary dihedral angle. The quantities no and t;o are defined in reference 6.

In the nonlinear theory, additional boundary conditions, as listed in figure 5, must be satisfied to determine the strength and position of the vor- tex core and sheet. A typical six-segment sheet and vortex core are shown in r figure 6. The core is joined to the end of the sheet by a cut. The Kutta condition, which states that the velocity is finite at the wing tip, provides a single algebraic expression for determining the unknown vortex strengths and positions. Two additional algebraic relations are provided by the zero force I condition, which requires that the resultant force on the vortex and cut must vanish. The pressure and normal velocity boundary conditions, which state

I

that both pressure and normal velocity must be continuous across the vortex sheet, are satisfied at the midpoint of each sheet segment. These provide 2N r

L

additional algebraic relations making a total system of 2N + 3 equations.

The unknowns to be determined are the vortex core strength, the two vortex core position coordinates, the N vortex sheet strengths, and the N vortex sheet radius vectors.

The equations are nonlinear and must be solved by i terat i on. T his i s I accomplished with a computer program based upon the iteration procedu~e origi- nally developed by Smith (ref. 5). Sample results are presented in figures 7 through 9. Additional results may be found in reference 6.

The results in figure 7 are typical of calculations for bodies with strakes of exposed semispan equal to 50 percent of the body radius (50-percent strakes) or greater. The corresponding test data in this and subsequent fig- ures were obtained in the Ames 7- by IO-Foot Wind TUnnel for models of differ-

ent semiapex angle e. Test conditions varied between a dynamic pressure of

50 psf and 75 psf. Good agreement between the nonlinear theory and test data was found over the entire angle-of-attack range shown (except when stall or vortex breakdown occurred, viz., at values of angle of attack a > 30 ).

Figure 8 shows a similar comparison with 10-percent strakes. Good agreement with test data is found up to a/ e ~ 2 or 2.5. In the range 2.5 < a/ e < 4, the theory was consistently below the test data. Beyond

a/e = 4, the theory becomes multivalued, and three possible theoretical solu-

tions exist for the same a/ e . The lower nonlinear solution gives a weak leading-edge vortex lying close to the wing tip, whereas the upper solution contains a strong vortex located well above the wing. The middle nonlinear solution results in a vortex of intermediate strength. The test data, on the other hand, form a smooth transition between the theory at low a/ e and the upper theoretical solution at large a/ a . No experimental evidence of multiple lift values was found.

The failure of the theory in the range 2 < a/ a < 4 may be due to the existence of body as well as leading-edge vortices, as postulated in the mixed case of figure 3. The estimated lift increment from body vortices has been ~dded to the calculated nonlinear lift in figure 8 in order to check the magnitide of this correction.

It is of interest to examine the nonlinear analysis in the limit of vanishingly small strake size and to compare with test data for a cone. The separation angle 8 must now be varied parametrically, since as mentioned in the Introduction, 8 is no longer fixed by the strakes, but is determined by viscous effects. The resulting calculations are shown in figure 9 and indicate that the lift increases with decreasing separation angle. No nonlinear solu - tions were found for a/ e < 2, irrespective of the value 8. This is in agree- ment with previous work by Bryson (ref. 2) , who used the Brown and Michael model for a cone.

Nonconical Analysis The nonlinear theory has been extended to nonconical configurations.

In the nonconical theory, the tangential flow and Kutta conditions are essentially unaltered from the conical theory. The remalnlng boundary conditions, however, are no longer algebraic as was the case for conical flow.

The zero force condition on the vortex and cut now becomes a first-order ordinary complex differential equation (see fig. 10), and is essentially the same as originally used by Bryson (ref. 2). The pressure continuity condition for a general nonconical vortex sheet becomes a quasilinear first-order partial differential equation. The general solution to this equation was obtained and is shown in figure 10. The continuity of normal velocity condition for a gen- eral vortex sheet is given by the third equation in figure 10. (Geometric definitions for some of the symbols in figure 10 may be found in figure 4.)

The system of algebraic and differential-integral equations describes an initial value problem starting from a given axial station. The system has been solved numerically by assuming the flow to be conical ahead of the ini- tial station. Thus, the conical theory is used to generate initial data for the nonconical theory. Some sample results are presented in the following figures.

Calculated values of CL(x), Cm(X)!CL(X), and of a function proportional to vortex core strength up to station x are shown in figure 11 for a cone with a double-delta wing and fillet at a = 24°. The theoretical results appear to approach the conical asymptotic values for 0 = 18°.

A comparison of the calculated C and C values with wind-tunnel data L

is shown in figure 12. A single theoretical point at a = 24° and the known

linear theory slope at a = 0° were used in establishing the faired curve

shown in figure 12.

The calculated upper surface spanwise pressure distribution for the same configuration is compared with test data in figure 13 at two axial sta- tions. The theory agrees reasonably well with the test data for the filleted wing at both stations. Removing the wing fillet significantly affected the

pressure at x = 0.60, directly behind the break in sweep, but had only a

minor effect at the downstream station.

Additional comparisons of the nonconical theory with wind-tunnel test data are included in figures 14 through 16. Thus, figure 14 shows a compar- ison of the calculated values of C and C versus a with test data for L a configuration consisting of a body with ~n ogive nose and curved 50- percent strakes. A comparison of upper surface pressure data at a = 24° is shown in figure 15 for the same configuration.

The nonconical theory was also used to calculate the effect of changing the strake incidence angle £ over the rear 45 percent of a conical wing-body (control effectiveness). The calculations for CL versus a are compared with test data on figure 16.

CONCLUSION AND RECOMMENDATIONS Analytical procedures have been developed for calculating the nonlinear lift and pressure distribution on conical and nonconical wing bodies with leading-edge vortices. Except for rel~tively small strakes in the range 2 < a/ a < 4, agreement between both the conical and nonconical theories and subsonic test data was highly encouraging.

The nonlinear theory is at present limited to flows with lateral symmetry and to configurations which satisfy the geometric and Mach number constraints of slender-body theory. Removal of some of these restrictions would increase the usefulness and applicability of the theory. Thus, removal of the require- ment for lateral symmetry would allow analysis of yawed flight and the evalu- ation of nonlinear lateral stability. The formulation of a companion theory for supersonic flows also appears to be highly desirable, since leading-edge vortices have been observed at Mach numbers well above one (ref. 7). Exten- sion of the theory to include nonslender effects arising from the apex, trailing edge, step control deflections, etc., although needed from a practical standpoint, is considered very difficult to accomplish.

REFERENCES Allen, H. J.; and Perkins, E. S . : A Study of Effects of Viscosity on l.

Flow Over Slender Inclined Bodies of Revolution. N ACA TR 1048, 1951.

2. Bryson, A. E.: Symmetric Vortex Separation on Circular Cylinders and Cones. J. Appl. Mech., vol. 26, no. 4, 1959.

3. Brown, C. E. ; and Michael, W . H., Jr.: On Slender Delta Wings With Leading-Edge Separation. NACA TN 3430, 1955.

4. Mangler, K. W.; and Smith, J. H. B.: A Theory of the Flow Past a Slender Delta Wing With Leading-Edge Separation. Proc. Roy . Soc., A., vol.

251, 1959, pp. 200-217.

5. Smith, J . H. B.: Improved Calculations of Leading-Edge Separation From Slender Delta Wings. RAE Tech. Rep. 66070, March 1966.

Levinsky, E. S.; and Wei, M. H. Y.: Nonlinear Lift and Pressure I 6.

Distribution of Slender Conical Bodies With Strakes at Low Speeds .

NASA CR 1202, 1968.

7. Jorgensen, L. H.: Elliptic Cones Alone and With Wings at Supersonic Speeds. NACA TR 1376, 1958.

VORTEX FLOW VISUALIZATION Figure 1 TYPICAL NONCONICAL SLENDER-BODY CONFIGURATION WITH VORTEX SHEET v~ a VORTEX SHEET VORTEX CORE VARIABLE INCIDENCE FOR CONTROL Figure 2 L I SOME VORTEX FLOWS • BODY ALONE • SEPARATION POINT VARIABLE • DEPENDENT ON VISCOUS EFFECTS • STRAKE I BODY (TREATED IN CURRENT ANALYSIS) • STRAKE VORTICES DOMINANT • SEPARATION FIXED BY SHARP L. E.

• MIXED Figure 3 RELATION TO PREVIOUS WORK INCOMPRESSI BLE FLOW BROWN a MICHAEL ISOLATED VORTEX CORE

\

/ f:' , 1955 NO FEEDING SHEET ( \ MANGLER a SMITH ADDED FEEDING SHEET 1959 OF SPECIFIED SHAPE

o

SMITH SEGMENTED FEEDING 1966 SHEET

eCJ

GENERALIZED TO CURRENT ANALYSIS STRAKE I BODY CONFIGURATIONS GENERALIZED TO NON CONICAL FLOW Figure 4 CONICAL ANALYSIS SLENDER-BODY THEORY a2~ a2~ - + - =0 a y 2 az 2 • TANGENCY CONDITION (LINEAR THEORY) TRANSFORM STRAKE IBODIES (Z PLANE) INTO VERTICAL SLIT (z* PLANE)

Z*2 = __ I _ {b [zL(b2-a2~ 1/2 -aZ-i(b-a)7] }2_e

(b-a)2 !J 0 0 • ADDITIONAL CONDITIONS (NONLINEAR THEORY) KUTTA CONDITION ZERO FORCE CONDITION PRESSURE CONTINUITY CONDITION NORMAL VELOCITY CONTINUITY CONDITION Figure 5 NONCONICAL ANALYSIS SYMBOLS

t

va w=g;-iljt ~<t> = ~ JUMP ACROSS SHEET m = MEAN VALUE ON SHEET Figure 6

I

_J

--------- COMPARISON OF THEORY WITH SUBSONIC EXPERIMENT CIRCULAR CONE , 50% STRAKES SYMBOLS DENOTE DATA FOR 6°5 85 12° 50 5 q 5 75 pst 1.4 5 R 5 1.6 x 10 (per tt) NONLINEAR tj THEORY C\J

EXPERIMENT 'b

'" u ()() <:) C\J c .E "- ..J LINEAR 40 (J THEORY 2 4 6 8 o sin alton 8 Figure 7 COMPARISON OF THEORY WITH SUBSONIC EXPERIMENT CIRCULAR CONE, 10% STRAKES SYMBOLS DENOTE DATA FOR 4.4° 5 85 11° 50 5 q 5 I 00 pst 1.4 5 R 5 1.8 x 10 (per tt) <:) tj C\J '" 0 u GO BODY VORTEX INCREMENT C\J c ADDED +- "- ..J (J LINEAR THEORY 0 2 4 6 8 sin a/tan 8 Figure 8 COMPARISON OF THEORY WITH SUBSONIC EXPERIMENT CIRCULAR CONE WITHOUT STRAKES SYMBOLS DENOTE DATA FOR 4°:S 8:s 10° ASSUMED 50 :s q :s 100 pst f3 1.4 :s R:S 1.8 x 10 (per tt) 400 eJ C\J if) u C() C\J c LINEAR , THEORY ...J C Rl27ro8 u L 2 4 6 8 sin a/ton 8 Figure 9 NONCONICAL ANALYSIS • ZERO FORCE CONDITION (SAME AS BRYSON, 1959) v dry ( ) [dW rv ( I ) dZ ] v -d zv-zm = rv -d- + -, =-=- -v- d x Z 27T1 z-zv X Z = Zv FORCE ON CUT FORCE ON VORTEX • PRESSURE CONTINUITY CONDITION 6<:> = (M<:>/ocr) [cr-l(op~ocr)m dX] • NORMAL VELOCITY CONTINUITY CONDITION 8 I JX ocp/on r(x, 8) = r cotcp d8- - , - dx J v o 0 Sin cp • INITIAL VALUE PROBLEM START WITH CONICAL SOLUTION Figure 10 NONCONICAL THEORY CONICAL BODY WITH DOUBLE -DELTA WING a= 24° 1.6 C (X) -- - L ~ __ _ 1.2 / Cm(~~ ___ _ .8 ~-=~-ar~ - 7" CONICAL .4 ASYMPTOTES FOR 8=18° o .2 .4 .6 .8 1.0 AXIAL DISTANCE, X Figure 11 COMPARISON OF THEORY WITH SUBSONIC EXPERIMENT CONICAL BODY WITH DOUBLE-DELTA WING ~ THEORY EXPERIMENT 1 .2 THEORY .8 THEORY

/

C L .4 o 20 40 o -.4 -.8 a, deg C , ABOUT APEX m Figure 12 SPANWISE PRESSURE DISTRIBUTIONS CONICAL BODY WITH DOUBLE-DELTA WING -- THEORY EXPERIMENT o NO FILLET

----~F~I~LL~~ ~ T -

o WITH FILLET 0.6 0.8 -3 X=0.6 X=0.8 -2 t,p q -I o .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0 FRACTION OF LOCAL SEMISPAN Figure 13 LIFT AND MOMENT COEFFICIENTS OGIVE NOSE BODY WITH CURVED 50% STRAKES 0 EXPERIMENT

~+E?21-

-x- THEORY 1.0 NONCONICAL .8 THEORY /J

Y

.6 x x C L .4

//

.2 I x ......

. ...... ...... ~ LlNEAR THEORY ......

.6 .8 0 10 20 30 40 0 .2 .4 a -Cm Figure 14 PRESSURE DATA OGIVE NOSE BODY WITH CURVED 50% STRAKES UPPER SURFACE ONLY o EXPERIMENT

--THEORY -E - a$ - +~

0.43 0.7\ xo/Xf =0.43 xo/Xf =0.71 -3 -3 f-- - BOOY --~' I -' STRAKE--j ~ -- BOOY -- ~' I ----' STRAKE --I -2 -2 6P q -\ -I o o 20 40 60 80 100 0 20 40 60 80 100 LOCAL SEMISPAN, percent Figure 15 LIFT COEFFICIENT CONE WITH 50 % VARIABLE INCIDENCE STRAKES 1.0 0.55 THEORY ~0 .8 - x- E =0 [J . ~0 .6

rg

C L .4 EXPERIMENT o E = 0 -b- E = -2° .2 o € = -4° o € = -6° 0 10 20 30 a, deg Figu re 16 DISCUSSION CAPT. DAVID FINKLEMAN, Air Force Academy: I'd like to hear your comments on the procedure due to Sacks, which introduces discrete vortices. Perhaps you classified this with empirical methods, but it need . not necessarily be so.

By increasing the number of vortices you arrive at a completely force-free vortex sheet, rather than one that is only force free in the mean, such as the Brown-Michael approach.

LEVINSKY: Well, I know there have been several methods. One I am familiar with is by Nielsen Engineering, where they had a large number of free vortices representing the sheet.

FINKLEMAN: I say this in particular because I know the Sacks approach seems to indicate good agreement for the same configurations you have examined. This was a NASA Contractor Report a couple of years ago. CA. H.

Sacks, R. E. Lundberg, and C. W. Hanson, "A Theoretical Investigation of the Aerodynamics' of Slender Wing-Body Combinations Exhibiting Leading-Edge Separa- tion," NASA CR-719, March 1967) LEVINSKY: If you take more turns of the vortex sheet than we have shown, eventually the force-free condition of the core and the cut should become less and less significant.

FINKLEMAN: Problems may arise with this method when one attempts to trace the wake behind, for instance, an arrow-shaped wing. If you have dis- crete vortices they are easy to follow, but with a sheet it is not quite clear how the wake could be followed.

LEVINSKY: We have used a Slender-body approach. We are constrained to very small changes in the x direction of the configuration. If you go beyond the base of the body the vortices are no longer growing in strength ., as they are in these cases. They become free. We are not talking about that kind of a case. The sheet would tear. You are not dealing with free vortices.

This is a feeding sheet model.

FINKLEMAN: One other short question. You say you have handled cases at incidence. Am I correct in assuming that you are using the solutions to Laplace's equation for a shoulder or high-wing configuration as against one with only a mid-wing?

LEVINSKY: We have carried out a calculation with two degrees of incidence, and it is in the written version. As for mid-wing versus high wing, this is a wing displacement effect whereas the incidence effect was considered due to a change in wing slope only. The displacement effect was not included in our incidence calculations, just the slope effect.

We have compared our results with test data, and the comparison is reasonable.

FINKLEMAN: And then you always took the wing, you said, along the diameter?

LEVINSKY: Yes, we used a mid-wing configuration.

RAYMOND SEDNEY, Martin Company: Can you tell us the origin of this multiple-valued nature of the solution?

LEVINSKY: Yes, this occurred in the conical cases. With small strakes it turns out that you can get three different solutions which satisfy all of the boundary conditions. We have taken the Brown and Michael approach without any vortex sheet and found the same results. In other words, there appear to be three different strengths and three different positions for the vortex core and sheet which satisfy the force-free condition, the Kutta condition, etc. This is a consequence of the fact that there may be more than a single solution to a set of nonlinear equations.

ROBERT L. TRIMPI, NASA Langley Research Center: I may have missed it, but I don't think you told us what computer you were using.

LEVINSKY: We used a CDC-3600.

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METHODS FOR CALCULATING NONLINEAR CONICAL FLOWS By Jerry C. South, Jr., and E. B. Klunker Langley Research Center INTRODUC TION In 1935 Busemann (ref. 1) introduced the concept of a general conical flow field as one in which the velocity vector is constant along any ray emanating from a common point in the flow. Solutions for such self-similar conical flows are of interest to the aerody- namicist for a number of reasons, among which two are: first, significant regions of the flow about many practical configurations are conical or nearly so (moreover, the super- sonic flow past any pointed body with a shock wave attached at the apex is conical in the immediate neighborhood of the point) and, second, conical bodies and wings are the sim- plest class of three-dimensional shapes and thereby provide "bench-mark" cases for both experimental and theoretical studies in inviscid supersonic flow.

Two basic approaches are available for the numerical development of "exact" 1 nonlinear conical solutions: (1) Distance-asymptotic methods, where approximate distributions of the flow vari- ables and shock-wave shape are used near the apex as initial values for continuing the calculation downstream by some three-dimensional computation scheme. The calculation proceeds until conical similarity conditions are sufficiently satisfied.

(2) Methods which invoke the conical self-similarity and thereby reduce to two the number of independent variables are referred to simply as "conical" methods for brevity.

Both general approaches have their merits. The distance-asymptotic techniques develop the solution as a well-posed initial-boundary problem for equations of hyperbolic type, and convergence is "almost" guaranteed from both physical and theoretical consid- erations. However, to achieve a satisfactory solution in many problems where a fine mesh is needed, these methods require a large amount of computer storage and time.

The conical methods reduce the problem to two dimensions but in the more difficult form of a free-boundary problem for equations of elliptic or mi~ed type. In fact, many of the conical methods are similar to methods used for solving the blunt-body problem.

The methods of references 2 to 7 are examples of the distance-asymptotic method.

References 2, 3, and 4 consider circular cones at angle of attack, references 5 and 6 1Exact in the sense that the only approximation made is the reduction of the gov- erning partial differential equations to ordinary differential equations or algebraic equa- tions by using finite-difference expreSSions for the derivatives with respect to one or more of the coordinates.

~ - - - -- -- - - - - -

l

include cones of elliptic cross section, and reference 7 presents calculations for the com- pression side of conical delta wings with shock wave attached not only at the apex but also along the swept leading edges.

Examples of conical methods appeared as early as 1929, when Busemann (ref. 8) constructed the axisymmetric conical flow by numerical-graphical construction in the hodograph. Reference 9 cites many of the approximate and exact conical methods docu- mented up to about 1964; therefore, they are not all discussed in the present paper .

Most recent conical methods are of the "inverse" type, in which an analytic function of two or more parameters is assumed for the conical shock wave and the governing partial differential equations are solved by marching inward until some body shape is obtained.

Various inverse methods are reported in references 10 to 14. Unfortunately, these methods have not been successful for constructing solutions for body shapes which pro- duce a shock wave requiring many parameters for an adequate description; only circular cross-section cones at incidence have been amenable, whereas solutions for elliptic cones have been obtained only painstakingly. It is apparently unfeasible to use the aforemen- tioned inverse methods for elliptic cones at incidence. other conical methods that have been used incorporate the method of relaxation in regions where the cross flow (velocity component normal to a conical ray) is subsonic and the governing differential equations are of the elliptic type; a two-dimensional method of characteristics is used in super- sonic cross-flow regions where the equations are hyperbolic. Such approaches are developed in references 15 to 17 in which flat delta wings with attached leading-edge shocks are considered. It is not known if these methods can be easily coded for effi- cient machine calculation for nonflat conical wings, but it seems doubtful. Moreover, the . results in reference 17 appear to be erroneous, as shown later in this paper.

I The present paper concentrates attention on a method that is " direct " in the sense

that the body shape is given and is one of the bounding coordinate surfaces; yet the shock wave is another bounding coordinate surface, and the governing differential equations are i solved by integrating inward from the shock. Thus, in that respect, this method is like the I

inverse methods. The method has two distinguishing features: (a) the coordinate trans- I

formation which maps the region between the shock and body onto a rectangle and (b) the solution of the transformed problem by the method of "lines" or "straight lines," so-called

I

by various authors. Neither feature is new, yet together they prove to be an efficient means of solving free-boundary problems such as the supersonic blunt-body problem or

I

conical flows. The convenient device of transforming the shock and body surfaces into bounding coordinate surfaces is widely used in computational aerodynamics, for example, J references 2 to 7 and 18 to 21. The basic idea of the method of lines is to discretize all I but one of the independent variables in the partial differential equations so that a system of appr oximate, simultaneous, ordinary, differ ential-diff er ence equations is obtained. The I

_

---- -- -- -- -- approximate system can be solved numerically by an efficient routine such as a fourth- order Runge-Kutta method, and very good accuracy can be obtained with large steps in the direction of the continuous variable. The same purpose is achieved by the better- known (in the United States) method of integral relations (ref. 22). The difference between the two methods is that the method of integral relations, in its usual formula- tion (refs. 19,23, and 24), requires new algebraic development for each higher approxi- mation (Le., more lines) and the equations grow more complex; whereas the method of lines system is written recursively with arbitrary number and spacing of lines.

The present method, including the transformation of the shock and body surfaces, is hereafter called the method of lines. In reference 25, the work of Telenin and his coworkers is Cited, in which they used the method of lines for numerical solutions to the axisymmetric, supersonic blunt-body problem. That work was extended to the three- dimensional blunt-body problem and reported in reference 26. More details of the pre- ceding work are given in reference 27.

Makhin and Syagayev (ref. 28) recognized the difficulties associated with Syagayev's earlier inverse method (ref. 14) and applied the transformation discussed previously in which the body and shock become bounding coordinate surfaces. Their procedure was essentially the method of lines, although they used a first-order Euler method to integrate toward the body; to maintain accuracy, a small step size was used in the inward-marching direction (l/64th of the local shock-layer thickness). The modification allowed them to solve the direct elliptic-cone problem, which was intractable by the earlier inverse approach.

Bazzhin and Chelisheva (ref. 29) applied the method to conical bodies at large angles of attack where supersonic cross flow always occurs. Their procedure was quite similar to approaches to the blunt-body problem; they used the method of lines on the windward, high-pressure side of the conical body up to and beyond the region where the cross flow becomes supersonic. A conical, two-dimensional method of char- acteristics was used to continue the solutions in the supersonic cross-flow region. More results, with emphasis on the characteristics solUtions, are given in reference 30. An interesting result in reference 30 was that in some instances it was possible to continue the characteristics solution through the leeward plane of symmetry of elliptic cones at large incidence; thus, symmetry conditions were violated and this indicated the impossi- bility of flow without embedded shocks in the general case. Neither reference 29 nor 30 suggests using the method of lines for the small-to-moderate incidences in which the cross flow is everywhere subsonic, nor was any mention given to the application in reference 28.

2The word "Attached" in the title of reference 26 is an obvious errOr in translation and should read "Detached."

_~_J

-- -- - - - - - -- - - -- -- - - --- -I During the progress of the present work, Jones (ref. 31) reported a method similar in many ways to the present procedure. He was apparently unaware of the work in the Soviet Union cited herein and did not refer to his method as a method of lines, although it is the same. Jones obtained solutions for circular and elliptic cones and a conical body with a four-parameter, smooth, cross-section contour having both concave and convex portions. The method was shown to be accurate and efficient, and he was able to experi- ment numerically with some of the more theoretical questions concerning conical flows such as the "lift-off" of the vortical singularity (refs. 32 and 33). He stated that the method was restricted to conical flows with subsonic cross flow, which is not found to be true in the present work.

The present paper outlines a method of lines for automatic computation of conical flows, including not only elliptic cross-section cones but also the compression side of conical delta wings with shock attached along the leading edge. In the latter problem, large regions of supersonic cross flow occur, yet the method is applicable there without modific ation.

SYMBOLS a semimajor axis of ellipse semiminor axis of ellipse b pressure coefficient i,j indices indicating line number free-stream Mach number N number of lines p pressure r distance along ray u,v,w component of velocity in r-, 1/-, and ~-direction, respectively free- stream velocity

component of velocity in plane 1> = Constant

component of velocity normal to plane cf> = Constant nondimensional spanwise coordinate cylindrical coordinate in downstream direction z angle of attack shock angle y ratio of specific heats convergence criterion on normal velocity at conical surface € T/ angle measured in plane normal to body from rayon surface to ray in field T/s value of T/ at shock

eo conical apex angle of body in vertical plane of symmetry

A sweep angle ~,T arc length along intersection of unit sphere with conical body p denSity cf> polar angle in cylindrical coordinates (see fig. 2) METHOD Conical Coordinates The equations governing supersonic, inviscid flow of an ideal gas are written in a body-oriented, orthogonal, conical coordinate system (r ,T/,~) as suggested in reference 34 where r is the distance along a conical ray, T/ is the angle measured from the body surface to the ray in a plane ~ = Constant, and ~ is a measure of the arc length along the intersection of the body surface with a sphere of radius r centered at the body apex.

It should be noted that the body is the conical surface T/ = 0, and the contour along which

I

~ is measured is a plane curve only in the special cases of a circular cone or a flat delta wing. This coordinate system has the advantage that the associated velocity components are the natural conical components (u,v ,w); the v-component is zero at the surface TJ = 0 and the magnitude of the velocity component normal to a conical ray (hereafter called the "cross-flow" component) vv + w2 governs the type (elliptic or hyperbolic) of the par- tial differential equations for the conical flow. Furthermore, certain singularities appear

where the cross-flow component vanishes Uv2 + w2 = 0), and it is extremely important

that such points be recognized and interpreted correctly. In reference 35, for example, a cylindrical coordinate system was used, and the authors apparently assumed that a nodal singularity occurred where the two cylindrical components vc,wc vanished; they actually manufactured a singular node at that point and obtained an erroneous discontinu- ous solution for an elliptic cone at zero incidence. This is not to say that the use of cylindrical or other nonconical coordinates is wrong but rather that more care must be taken with them. Jones (ref. 31) used cylindrical coordinates with the method of lines, and he computed the same case as in reference 35 and obtained a valid continuous solution.

Transformation to a Rectangular Region

With the conical similarity (8~ = 0) the partial differential equations involve two

independent variables, TJ and ~,and the problem is solved on a spherical surface r = 1. To facilitate the formation of the boundary values at the shock wave, whose posi- tion must be determined as part of the solution, it is convenient to introduce the following change of variable s: (1) and

T = ~

where TJ = TJs(~) is the equation of the shock-wave contour on the spherical surface

r = 1. The chain rule gives 8 1 8 (2a) 8TJ = TJ ~ s 8 ~ dTJ 8 8 s -=----+- (2b) a~ TJ dT a~ aT s so that the partial differential equations take the form (3) Similar forms hold for the three velocity components u, v, and w. The density is elim- !

inated from the system as a differential variable by using the Bernoulli equation and its derivatives. It should be noted that the function TJs defining the unknown shock contour and its derivative dr}s/d7 appear expliCitly in the transformed equations. Likewise, because a body-oriented system is used, terms involving the body surface curvature also appear in the right-hand side of the equations, although they are not indicated in equation (3).

The problem is thus transformed to a rectangular region, where the cross-sectional contours of the shock wave and body surface are mapped onto the lines ~ = 1 and ~ = 0, respectively.

The Method of Lines The region of interest in the ~,7 plane is now divided by N lines parallel to the ~-axis (N + 1 lines for delta wings with attached leading-edge shocks); it is not neces- sary that the strips be of equal width. For the elliptic cone, figure 1 illustrates the division in the physical ~,r} plane and the transformed ~ ,7 plane for N = 9. At each strip boundary or line, the system of equations is reduced to ordinary differential- difference equations by replacing the derivatives 8/8 7 by finite differences or, say, by derivatives of polynomials in T. One of the aims in the present work was generality with simpliCity; hence, it was decided to use the derivative of a Lagrange interpolation polynomial to approximate 8/8 7. This use allowed experimentation with unequal line spaCing, noncentral differencing, and arbitrary (within reason) number of lines to be included in the interpolation-difference formula. The current program uses an equal number of lines on either side of the line at which 8/ 8 7 is computed; therefore, cen- tral differencing is obtained when the line spacing is equal. The difference, or poly- nomial, approximations for 8/ 8 7 cause the differential equations along any line to be coupled to those along the other lines. There results a system of 4N simultaneous ordi- nary differential equations which are integrated by a fourth-order Runge-Kutta method.

Boundary Conditions Shock-wave conditions. - The shock wave shape is unknown and must be determined in the solution. The flow variables are to satisfy the usual three-dimensional shock jump conditions at ~ = 1.

Flow tangency at surface. - The condition of flow tangency at the surface must be satisfied, that is at ~ = 0, the normal velocity component must be zero: (4)

V(0,7) = °

Symmetry conditions.- So far, only problems with at least one plane of symmetry have been solved. Symmetry is accounted for in the formula for the 7 derivatives by properly reflecting points (lines) about the symmetry plane. For the elliptic cone both the windward (i = 1) and leeward (i = N) lines correspond to symmetry planes. For the compression side of delta wings, the line i = 1 is a symmetry plane.

Leading-edge shock conditions.- For the delta wings, the line i = N + 1 corre-

sponds to the sharp leading edge, and the condition of shock attachment is enforced there.

At that line the slope of the shock wave and the values of the flow variables can be deter- mined directly from the shock conditions; hence, no differential equations need to be inte- grated along i = N + 1. In forming T derivatives at lines near the leading edge, the number of points used in the derivative formulas is automatically reduced, if necessary, to retain an equal number of lines on either side of the line at which a/aT is computed.

For example, if a five-point formula is being used generally, with the derivative evaluated at the middle point, then the number of points in the formula is reduced to three at line i = N to maintain an equal number of points on either side of this line.

Determination of the Shock Shape The form of the shock-wave cross section is given by the unknown function 11 = 11S(T). If 11s and d11s/dT were known, then all the information concerning the shock-wave geometry would be known, and the values of the functions p, p, u, v, and w at the shock wave (~ = 1) could be evaluated from the shock jump conditions.

These values could be used to start the numerical integration at ~ = 1 and proceed down to the body surface at ~ = 0. Only the correct shock function 11s will cause the flow-tangency condition (eq. (4)) to be satisfied; thus, there must be a relation between the function 11S(T) and the normal component at the surface V(O,T). This is the baSiS, then, for determining the correct shock shape.

Newton iteration for shock shape.- The number of unknown values is equal to the number of normal components vi(O) = V(O,Ti). A Newton-type iteration procedure has been devised for adjusting 11s,i to achieve m~lvi(O)1 ~ E, where E is a prescribed accuracy criterion. The steps in the procedure are straightforward and easily automated and are given as follows: (1) Assume an initial set of values 11s i (i=l, ... ,N) based on experience, approxi- , mate solutions, or previously computed cases with conditions close to those desired.

(2) Numerically differentiate the 11s i values with the formulas coded for obtaining , T derivatives.

(3) With 11s,i and d11s,i/dT from steps (1) and (2), solve the shock jump condi- tions for Pi' Pi' ub Vb and Wi (i=l, ... ,N).

(4) Use the results of step (3) for initial values to start a numerical integration of the system of 4N equations from ~ = 1 to ~ = 0, and evaluate the normal components at the surface Vi(O).

--~ - - - --

(5) Test m~lvi(O)I. If m~lvi(O)1 ~ €, then the shock shape is good and the prob- 1 1 lem is considered solved. Otherwise continue to the following perturbation cycle .

. (6) Perturb each parameter 11 ,j independently and in order. That is, change s 11s,j to 11 ,/l + 0), where 0 is a small number (say, 10- ), and repeat steps (2) to (4).

s This results in small changes in each vi(O) due to the perturbation in 11 ,j' Hence the s jth column of an N by N matrix of influence coefficients or partial derivatives is gener- ated with each perturbation.

(7) Solve the usual first-order linear system (5)

~ tV'(O~ (i=l , ... ,N)

L T--:- A11s,j = -vi(O)

j=l 11s,J to obtain the increments A11 ,j required to correct the shock shape and drive all s vi(O) - o.

(8) Use the new shock parameters 11 ,j + A11 ,j to start a new cycle at step (2).

s s Note that a complete cycle requires N + 1 integration runs: one "pivotal" run and N perturbation runs.

The shock-wave determination procedure of reference 31 differs somewhat from the present one in that the shock function is given by a trigonometric polynomial and the coefficients are chosen to minimize the sum of squares of the normal components at the surface. The computational scheme used herein appears to require more computation, but in the authors' experience it was the only one that ~as generally satisfactory.

ApprOximate starting shock shapes.- Usually a very good estimate of the shock shape is required for a successful calculation and convergence. The exceptions are

circular cones at moderate relative incidences (e~ ~ 0.5) and delta wings at large super-

sonic Mach numbers (Moo ~ 3.0). For most other cases, however, considerable care

must be exercised in chOOSing the initial shock shape to start the iterations previously described. A poor initial estimate can result in any of several program failures, such as negative pressures or vanishing denominators caused by excessive supersonic cross flow, which introduce characteristic-type singularities in the differential equations (Le., when a ~ = Constant line is tangent to a conical characteristic). Moreover, as the num- ber of lines is increased, the apprOXimation to the elliptic-type partial differential equa- tions improves, and the solutions tend to become sensitive to a small degree of "rough- ness" in the shock shape. A maximum number of lines (or a minimum AT) is reached beyond which the numerical integration from the shock results in a singularity between the shock and body, even though the shock shape is a "good" one for a calculation with fewer lines. In the cases studied with this method so far, the maximum number of lines -- ---- - has been about 19 or 20, but such a large number is seldom necessary except in severe cases of very flat elliptic cones (axiS ratio ~ ~ 0.25).

In order to obtain a good initial estimate of the shock shape for such cases as elliptic cones at incidence, a "simpler" case was computed first and then the input parameters were changed gradually toward the desired configuration, a new converged shock shape being obtained with each change of input parameters (e .g., y, Moo, eo, CI', and b/a). The history of the set of TJ i as a function of the set of input parameters s , was incorporated in an extrapolation procedure to predict the new shock shape for the new set of input parameters. Such a procedure was used in reference 31 and was also successful in the present work. The procedure is completely automated in the present program for circular and elliptic cones, and the variation of any of five different input parameters is allowed.

The initial approximation for the delta-wing shock shape could be estimated directly for any angle of attack CI' (up to that causing detachment from the leading edge), sweep angle A, and Moo' The estimate used was an even function in T which requires only an estimate of TJs,l (the value of TJ in the plane of symmetry). The s function is contrived to give TJ N+1 = 0 (the condition of attachment at the leading edge) s and the correct value for (dTJs/~h)N+1 which is calculated from the shock conditions.

The function is (6) The value of TJ 1 used in equation (6) is a tangent-cone approximation, increased by a s , factor 1.2 to avoid the Mach wave condition for very thin wings at small CI', as follows: (7a) where (ref. 36) (7b) In a few instances, the value of TJs 1 was so far off that the required corrections , ~TJs i were sizable, resulting in too much "roughness" in the shock shape; subsequent , iterations would fail for reasons already described. However, it happens that in this event the first correction for TJ 1 is quite good so that restarting the iteration with s , this value in equation (6) has always been successful.

Extrapolation to Surface At the surface, ~ = 0, the derivatives du/d~ and dw/d~ are infinite because of the well-known vorticallayer adjacent to the surface in conical flows (ref. 32). The derivatives dP/d~ and dv/d~ are finite, however, and this fact allows extrapolation of the functions p and v to the surface from ~ > O. ill fact, it is only Vi that is extrapolated during the iteration runs in order to evaluate the magnitudes of Vi (0) (i=1, ... ,N). When the convergence criterion is met, then the pressure p is also extrap- olated to ~ = O. The surface entropy is a constant on the surface, and if the value is known, then the isentropic surface density can be calculated as a function of the extrap- olated pressure and the surface entropy. The Bernoulli equation relates p, p, u, v, and w; since v = 0 at the surface and p and p are known, this gives one equation with two unknowns, u and w. The other equation needed for determining u and w is the differential equation (8) that is valid at the surface. Substitution of equation (8) into the Bernoulli equation gives a single nonlinear differential equation as follows:

2')1 p. 2 (8U.)2

(i =1 , ... ,N) (9)

.y-=-T p~ + ui + a: = Constant

The usual formula is used for evaluating T derivatives, and equation (9) is solved by Newton iteration. When convergence is achieved, equation (8) gives Wi; hence, the isentropic surface values are completely determined. This is essentially the same pro- cedure as that of reference 31.

Computation of Surface Entropy For the circular cone, the surface entropy is the value that occurs in the windward symmetry plane, line i = 1. For the elliptic cone with the free-stream velocity vector lying in the plane of the minor axis, the surface entropy is assumed to be the maximum value at the shock wave. When the free-stream vector is in the plane of the major axis, the surface entropy is piecewise constant; that is, it has the value of the windward sym- metry plane on the surface segment where W(O,T) < 0 and the value of the leeward sym- metry plane on the rest of the surface. The surface entropy for the delta wings with con- vex surfaces is the same as the leading-edge value. Other possible configurations are not covered in this paper.

RESULTS Elliptic Cone The present method was applied to the calculation of supersonic flow past an elliptic

cone with y = 1.4, M oo = 5.8, ~ = 0.5, and e := 6.0 • The calculated shock shape and

o pressure distribution are shown in figure 2 (a). The calculation was performed with N:= 17 lines with unequal spaCing; the density of lines in the region of large surface cur- vature was about double that elsewhere as shown by the dashed lines in the section view.

The same problem was calculated with N = 17 and equal spaCing and the results were of

comparable accuracy; it is not yet clear if unequal spaCing offers any advantage.

The calculated shock shape is shown for a:= 6 , where the leeward side of the cone is alined with the free-stream vector. This angle of attack is difficult because at M oo = 5.8 the free-stream pressure is small, and pressures lower than the free-stream value are predicted on the leeward side as can be seen in the Cp plot. The abscissa of the Cp plot is the angle cp as shown in the section view. (The windward symmetry plane corresponds to cp = -90 .) The present results for the pressures are compared with the experiments of Chapkis (ref. 37), and the agreement is excellent except on the leeward side (cp > 0) where a viscous buildup occurs to raise the level of the measured pressures. Typical of the inviscid calculations for both circular and elliptic cones, a pressure minimum occurs away from the leeward plane of symmetry when the relative incidence a/eo approaches unity.

Computation history.- The elliptic cone at a:= 0 was solved by starting from a circular cone (~ = 1) with semiangle equal to the desired semiangle (11.8 ) in the plane of the major axis of the ellipse (~= 0.5). The axis ratio b/a was changed in increments of -0.1 until the desired configuration was obtained. The solutions for various angles of attack were then computed beginning with zero angle of attack. Table 1 gives the history of the automatic increase in a with the number of cycles (one cycle consists of N + 1 integration runs of the coupled 4N differential equations) required to converge the shock shape and print for each a. The convergence criterion for the normal velocity compo- nents was set at E:= 10- , and the final value of m~lvi(O)1 is also shown. Each inte- gration from the shock to the body was made with 14 integration steps of variable size.

The total central processing time required for these 16 cycles was about 14.5 minutes on a Control Data 6600 computer system. Thus each cycle required an average of about 56 seconds of central processing time.

Supersonic cross flow.- For a:= 6 , a small region of supersonic cross flow occurs next to the surface (0 ~ ~ ~ 0.12) on the leeward side where the pressure drop is stron- gest (10 < cp < 25 ). Whenever supersonic cross flow occurs around circular or elliptic cones, convergence of the normal components vi(O) is more difficult at the lines in that

l

region. If the supersonic cross-flow region is not extensive, convergence is possible, as demonstrated by the present example.

Conical Delta Wings Perhaps more appropriate to this compilation of papers are the wing-like bodies.

The present method has been applied to the problem of conical delta wings with the shock wave attached at the sharp leading edges. In this problem the lower (compression) sur- face is independent of the upper (expansion) surface, and the two can be treated separately.

The method of lines, as formulated herein, is only applicable to the compression side where the shock wave forms the outer boundary. Nevertheless, the compression-side results are valuable from both theoretical and practical standpoints. As mentioned earlier, the results from the present method provide a check for more complex three- dimensional calculation methods, and it will be seen that they corroborate results from other conical methods and experiment, too. Practically speaking, at large supersonic Mach numbers nearly all the aerodynamic-force contribution comes from the compres- sion side.

In these problems the cross-flow component {v2 + w2 is supersonic at the leading edge and remains supersonic for some distance toward the wing center line. As stated before, the partial differential equations governing the conical flow are of the hyperboliC type in regions where the cross flow is supersonic, and a two-dimensional conical method of characteristics can be used to construct the exact flow in such regions. Maslen (ref. 15) was the first to describe this approach for conical delta wings, and it has been exploited extensively in reference 30 for the flow about elliptiC cones at large angles of attack. A computer program using the conical method of characteristics has been devel- oped at the Langley Research Center by C. W. Chiang3 and Richard D. Wagner, Jr.; they supplied the results presented in this paper although the method has not yet been published.

Vincenti (ref. 38) proposed an approximate method which can also be used in regions of supersonic cross flow; it is analagous to the familiar shock-expansion method for two- dimensional and axisymmetric flows. The approximation reduces to a nonlinear ordinary differential equation that is numerically integrated along the surface in the spanwise direc- I tion from the leading edge inward to the cross-flow sonic point. Results obtained by Richard D. Wagner, Jr., by that method are presented and are referred to as a "conical I shock expansion" method.

I

The supersonic cross flow in the delta-wing problems presents no difficulties for the method of lines; this is contrary to the case of the circular or elliptic cone at

I

3 NRC -NASA Resident Research Associate on leave from the University of Denver.

_1

incidence where convergence becomes more difficult when a region of supersonic cross flow occurs, as already mentioned.

Parabolic-arc cross section.- Reference 7 presents calculated results for a number of conical delta wings, with a three-dimensional implicit finite-difference scheme being used which is sometimes referred to as the "BVLR" 4 method or the "half-step" method.

Delta wings with both flat and parabolic-arc cross sections were presented in the refer- ence, and the results from reference 7 are compared herein with results from the pres- ent method and other methods. Planform and section views of a thin parabolic cross-

section wing are shown in figure 2(b) with the calculated shock shape for a = 10 . The

0 0

conditions for this problem are Moo = 4, A = 50 , and eo = 3 . The cross-flow sonic

line is shown as a heavy dashed line in the section view, and the layout of the computa-

tional lines for N = 9 is depicted by the finer dashed lines which appear normal to the

surface.

In the three wing problems of this paper, equal line spacing was used. Recall that this means equal ll.T tncrements where T traces around the wing contour in the sur- face r = 1; thus, when projected onto the plane z = 1 in which the body cross section is prescribed, the lines appear to spread as the leading edge is approached. The solid curve for the shock represents the present results, whereas the squares represent Voskresenskii's results. The squares are not meant to coincide with the dashed lines; they were replotted from the continuous curves shown in reference 7 by using constant intervals in the spanwise (X) direction. The replotting of those curves for the shock shape is probably not as accurate as the pressures because although Voskresenskii's shock appears to lie slightly inside the present result near the center line, shock pres- sures by both methods (not shown here) agree very well. The conical method of charac- teristics also gives a shock shape from the leading edge to the sonic line, and the results (not shown) coincide with the solid curve drawn through the present results.

On the right side of figure 2(b) the surface pressure coefficient is plotted against x,

a nondimensional spanwise coordinate; x = 0 at the wing center line and x = 1 at the

leading edge. The vertical dashed lines indicate the spanwise location of the cross-flow sonic point5 at the surface and, hence, the limit of applicability of both the conical method of characteristics (MOe) and the conical shock-expansion method. The method of lines, conical characteristics, and conical shock expansion agree very well in the supersonic cross-flow region. Voskresenskii's results are generally quite close to the present results with some small discrepancies. At the leading edge, where the pressure and other variables can be calculated accurately from the algebraiC shock conditions, results 4After Babenko, Voskresenskii, Lyubimov, and Rusanov, the co-authors of reference 3.

5As the angle of attack is increased, the cross-flow sonic point moves toward the leading edge and reaches it just before leading-edge detachment occurs.

-

from all theories should agree exactly; yet Voskresenskii's result for the pressure is noticeably low. His pressure calculation just inboard of the leading edge rises slightly above that of the three conical methods, but then appears to agree well with the present results in the central subsonic cross-flow region. It appears, then, that a small error in the leading-edge boundary condition has little effect on the computation over the rest of the wing by Voskresenskii's three-dimensional, distance-asymptotic method.

Circular-arc cross section. - Reference 39 presents experimental data for various conical delta wings with attached leading-edge shocks; one of the models had a circular- arc cross section, a good test problem for the different computational methods. The con-

ditions for this problem are M oo = 8.1, A = 50 , and 8 = 6.54 • Figure 2(c) shows the

shock shape calculated by three methods: the methods of lines and conical characteris- tics, and a three-dimensional method of characteristics (3D MOC) described in refer- ence 40. In reference 40, results are shown only for delta wings with a region of con- stant flow at the leading edge; that is, the outboard portion of the wing cross section is either flat or a wedge. Results were not shown for delta wings with curvature at the leading edge, such as the parabolic-arc or circular-arc cross-section wings, because difficulties were experienced with such wings. The source of these difficulties has not yet been located, but some preliminary results are shown in figure 2(c). The methods of lines and conical characteristics agree very closely for the portion of the shock wave bounding the supersonic cross flow, but the 3D MOC of reference 40 predicts a shock layer that is more than 20 percent too thin over most of the span.

I For the problem of figure 2(c), the wing thickness and Mach number are both about twice the values used in the problem of figure 2(b); it can be seen that the cross-flow

I sonic line leans closer to the wing center line. This behavior is caused mainly by the

I increase in M oo and is similar to that of the sonic line in the blunt-body problem , (ref. 9).

I The spanwise pressure distribution is also shown in figure 2(c), where the various I

I theoretical results are compared with the data of reference 39. The hatched bands are

used to represent the measurements for several different Reynolds numbers and stations , downstream from the wing apex. The measured pressures are higher than the inviscid

I

I predictions over the entire surface; this is the expected hypersonic boundary-layer dis-

'I placement effect, while boundary-Iayer-shock-wave interaction is most severe near the leading edge (x = 1). A comparison (not shown) was made between results from the pres- ,

r ent method and the experimental data for Moo = 5.08, and the agreement was better

I because the viscous displacement and interaction effects were not as severe.

1 --------------------------------------------------------------------

'I 6Work supported by the NASA Langley Research Center under Contract NASl-7850.

7Calculations were done with N = 9 and N = 18, and the results coincided. The

: N = 18 results were used to plot the cross-flow sonic line.

I

I

- - ~- - - - - - - -_.

-- -- -- - - - - - - - - - - Two different results by the 3D MOe are shown. The long-dash-short-dash curve in figure 2(c) shows an earlier unpublished result that is clearly in error over the entire span. The authors of reference 40 modified the method and produced the more recent (but still preliminary) results represented by the triangles. These recent results are close to the other theories in the supersonic cross-flow region (although the corre- sponding shock shape is not in agreement, as already mentioned), but they depart from the method of lines results in the central region. When attempts were made to calculate farther downstream in hopes of improving the results, the calculation became unstable and diverged. Because of the failure of the method for this conical problem, the validity of any other results obtained by this method should be questioned. The point to be made is the necessity for rigorous checks on three-dimensional computational schemes; conical-flow problems provide excellent check cases because they are three dimensional but can be solved by two-dimensional methods.

Flat delta wing.- The classical problem of the flat delta wing is shown in figure 2(d).

This problem has the interesting feature that flow properties are constant between the cross-flow sonic line and the leading edge; the sonic line is thus a singular line connecting a constant-property region to one of varying properties. The exact, inviscid pressure \ distribution should exhibit a "corner" or discontinuous slope at the cross-flow sonic point, \ and the shock wave should be planar from the leading edge to the sonic line. This weak singularity is not treated in any special way in the present method and the same is

I

assumed true of reference 7.

I

Both methods produce nearly identical results as seen in figure 2(d). The open circles are used to show the actual location of the computational lines in the present

I

method for N = 12; the results are surprisingly smooth around the sonic line. Although

I the assumed starting shock shape (eq. (6)) is analytic, the final converged shock points lie very close to a straight line between the leading edge and the cross-flow sonic line,

I

as they should.

I In the methods of references 17 and 40, the flow properties are set constant in the

supersonic cross flow, and in reference 17, an effort was made to account for the weak I

Singularities. Even so, the results of reference 17 (solid circles) appear to be in con-

I

sider able disagreement with the other methods, particularly at the larger values of 11; it seems reasonable to assume that the other methods are quite accurate. It is also note-

I

worthy that although Babaev (ref. 17) attempted to account for the singularities, his I results for the surface pressure distribution appear to be very smooth at the sonic point, without a corner. Results by the method of reference 40 (3D MOC) are shown on the

I

Cp plot for 11:::: 15 , and they are in fair agreement with the present results and those of reference 7.

-- - - -- -- -- -- - - Convergence history.- Figure 3 illustrates the convergence history for the flat delta wing problem used for figure 2(d). The surface pressure distribution obtained with each new pivotal shock shape, including the initial apprOximation given by equa- tion (6), is shown. The key of the figure gives the maximum normal velocity magnitude at the end of each cycle and the approximate elapsed central-processing time for the Control Data 6600 computer system.

Convergence with increasing N.- Figure 4 is an illustration of the accuracy that can be achieved with smaller values of N. With only one line placed between the leading

edge and the wing center line (N = 2), the accuracy is surprisingly good. More lines are

used, however, to give better resolution of the spanwise distribution. The approximate central-processing time and number of cycles required to converge each case are shown in the key.

CONCLUDING REMARKS The method of lines has proved to be an efficient, versatile procedure for con- structing numerical solutions to conical-flow problems. The method has heen applied to circular and elliptic cones at incidence and to the compression side of conical delta wings with attached leading-edge shocks. The results for surface pressures are in good agreement with experiment except where viscous effects become important, such

I

as on the leeward side of the elliptic cone at relatively large incidence and near the leading edges of delta wings.

I

The method can be used for circular or elliptic cones at such incidences that small embedded regions of supersonic cross flow occur, but the convergence becomes more difficult. In the delta-wing problems, however, Significant regions of supersonic cross flow occur, and the method is applicable with no difficulties.

Several other approaches for calculating conical flows were compared with the method of lines and were found to be in close agreement. One approach is a three- dimensional, implicit, finite-difference scheme that attains the conical solution asymp- totically, whereas two others are strictly two-dimensional conical methods. These last two are applicable only in regions of supersonic cross flow, but they provide valuable corroboration for more general methods.

A three-dimensional method of characteristics was shown to fail for the problem of a delta wing with curvature at the leading edge. It appears that such conical-flow problems are excellent check cases for more general three-dimensional computational schemes because they are three-dimensional but can be solved by two-dimensional coni- cal methods.

l

REFERENCES 1. Busemann, A.: Aerodynamic Lift at Supersonic Speeds. Ae. Tech!. 1201, Rep.

No. 2844, Brit. A.R.C., Feb. 3, 1937. (From Luftfahrtforschung, Bd. 12, Nr. 6, Oct. 3, 1935, pp. 210-220.

2. Moretti, Gino: Inviscid Flowfield About a Pointed Cone at an Angle of Attack.

AIAA J., vol. 5, no. 4, Apr. 1967, pp. 789-791.

3. Babenko, K. 1.; Voskresenskiy, G. P.; Lyubimov, A. N.; and Rusanov, V. V.: Three- Dimensional Flow of Ideal Gas Past Smooth Bodies. NASA TT F-380, 1966.

4. Rakich, John V.: A Method of Characteristics for Steady Three-Dimensional Super- sonic Flow With Application to Inclined Bodies of Revolution. NASA TN D-5341, 1969.

5. Bebenko, K. 1.; and Rusanov, V. V.: Difference Methods of Solving Three-Dimensional Problems in Gas Dynamics. NASA TT F-10,827, 1967.

6. Gonidou, Rene: Supersonic Flows Around Cones at Incidence. NASA TT F-ll,473, 1968.

7. Voskresenskii, G. P.: Chislennoe Reshenie Zadachi Obtekaniia Proizvol'noi Poverkhnosti Treugul1nogo Kryla v Oblasti Szhatiia Sverkhzvukovym Potokom Gaza (Numerical Solution of the Problem of a Supersonic Gas Flow Past an Arbitrary Surface of a Delta Wing in the Compression Region). Izv. Akad.

Nauk SSSR, Mekh. Zhidk. Gaza, no. 4, 1968, pp. 134-142.

8. Busemann, A.: Drucke auf kegelformige Spitzen bei Bewegung mit Uberschallgesch- windigkeit, Z. Angew. Math. Mech., Bd. 9, Heft 6, Dec. 1929, ,Pp. 496-498.

9. Hayes, Wallace D.; and Probstein, Ronald F.: Hypersonic Flow Theory. Vol. I - Inviscid Flows. Second ed., Academic Press, 1966, pp. 526-536.

10. Briggs, Benjamin R.: The Numerical Calculation of Flow Past Conical Bodies Supporting Elliptic Conical Shock Waves at Finite Angles of Incidence. NASA TN D-340, 1960.

11. Mauger, F. E.: Steady Supersonic Flow Past Conical Bodies. A.R.D.E. Rep. (B)3/60, Brit. War Office, May 1960.

12. Stocker, P. M.; and Mauger, F. E.: Supersonic Flow Past Cones of General Cross- Section. J. Fluid Mech., vol. 13, pt. 3, July 1962, pp. 383-399.

13. Eastman, D. W.; and Omar, M. E.: Flow Fields About Highly Yawed Cones by the Inverse Method. AIAA J., vol. 3, no. 9, Sept. 1965, pp. 1782-1784.

--- --- -- I~

I

, 14. Makhin, N. A.; and Syagaev, V. F.: Numerical Solution of the Problem of Supersonic Flow Past Conical Bodies at an Angle of Attack. Fluid Dyn., vol. 1, no. 1, Jan. -Feb.

1966, pp. 100-101.

I

15. Maslen, Stephen H.: Supersonic Conical Flow. NACA TN 2651, 1952.

16. Babayev, D. A.: Numerical Solution of the Problem of Flow Round the Upper Surface I of a Triangular Wing by a Supersonic Stream. U.S.S.R. Comput. Math. Math. Phys., no. 2, 1963, pp. 296-308.

I

17. Babaev, D. A.: Numerical Solution of the Problem of Supersonic Flow Past the Lower Surface of a Delta Wing. AIAA J., vol. 1, no. 9, Sept. 1963, pp. 2224-2231.

18. Vaglio-Laurin, Roberto; and Ferri, Antonio: Theoretical Investigation of the Flow Field About Blunt-Nosed Bodies in Supersonic Flight. J. Aero/Space ScL, vol. 25, no. 12, Dec. 1958, pp. 761-770.

19. Belotserkovskiy, O. M.: Supersonic Gas Flow Around Blunt Bodies - Theoretical and Experimental Investigations. NASA TT F-453, 1967.

20. South, Jerry C., Jr.: Application of the Method of Integral Relations to Supersonic Nonequilibrium Flow Past Wedges and Cones. NASA TR R-205, 1964.

21. Sedney, Raymond; and Gerber, Nathan: Shock Curvature and Gradients at the Tip of Pointed Axisymmetric Bodies in Non-Equilibrium Flow. J. Fluid Mech., vol. 29, pt. 4, Sept. 27, 1967, pp. 765-779.

22. Dorodnitsyn, A. A.: General Method of Integral Relations and Its Application to Boundary Layer Theory. Vol. 3 of Advances in Aeronautical Sciences, Macmillan Co., 1962, pp. 207-219.

23. Chushkin, P. I.; and Shchennikov, V. V. (B. A. Woods, transl.): Calculation of Certain Conical Flows Without Axial Symmetry. Lib. Transl. No. 926, Brit.

R.A.E., Dec. 1960.

24. Brook, John W.: The Method of Integral Relations for Conical Flow - Theoretical Analysis. Res. Mem. RM-193 (Contract No. AF 33(616)-6400), Grumman Aircraft Eng. Corp., Oct. 1961. (Available from DDC as AD 266 501.)

25. Roslyakov, G. S.; and Telenin, G. F.: Survey of Computational Work on Steady Axisymmetric Gas Flow Carried out at the Computational Center of Moscow State University. Numerical Methods in Gas DynamiCS, G. S. Roslyakov and L. A. Chudov, eds., NASA TT F-360, Israel Program Sci. Trans!., 1966, pp. 1-11.

(Available from CFST!, U.S. Dep. Com.)

, 26. Telenin, G. F.; and Tinyakov, G. P.: A Method of Calculating the Three - Dimensional Flow Past a Body With an Attached Shock Wave. Sov. Phys. - Doklady, vol. 9, . no. 2, Aug. 1964, pp. 132-133.

27. Tiniakov, G. P.: Investigation of the Three-Dimensional Supersonic Flow Around Ellipsoids of Revolution. Lockheed Missiles & Space Co. Transl. (From Izv. Akad.

Nauk SSSR, Otd. Tekhn. Nauk, Mekhan. i Mashinostr., no. 6, 1965, pp. 10-19 .)

28. Makhin , N. A.; and Syagayev, V. F.: Numerical Solution of the Problem of Supersonic Flow at an Angle of Attack Past Conical Bodies. NASA TT F-10,481, 1966.

29. Bazzhin, A. P.; and Chelysheva, 1. F.: Primenenie Metoda Priamykh k Raschetu Obtekaniia Konicheskikh Tel pri Bol' shikh Uglakh Ataki (Application of the Straight- Line Method in the Calculation of Flows Past Conical Bodies at Large Angles of Attack). Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, no. 3, 1967, pp. 119-123.

30. Bazzhin, A. P.; Trusova , O. N.; and Chelysheva, 1. F.: Raschet Techenii Sovershen- nogo Gaza Okolo Ellipticheskikh Konusov pri Bol'shikh Uglakh Ataki (Calculation of Ideal-Gas Flows Around Elliptical Cones at Large Angles of Attack). Izv. Akad.

Nauk SSSR, Mekh. Zhidk. Gaza, no. 4, 1968, pp. 45-51.

31. Jones, D. J.: Numerical Solutions of the Flow Field for Conical Bodies in a Super- sonic Stream. Aeronaut. Rep. LR-507 (NRC No. 10361), Nat. Res. Counc. Can.

(Ottawa), July 1968.

32. Ferri, Antonio: Supersonic Flow Around Circular Cones at Angles of Attack. NACA Rep. 1045, 1951. (Supersedes NACA TN 2236.)

33. Melnik, R. E.: Vortical Layers in Supersonic Conical Flow. Hypersonic Boundary Layers and Flow Fields, AGARD CP No. 30, 1968, pp. 26-1 - 26-20.

34. Scheuing, Richard A.; Brook, John W.; Mead, Harold R.; Melnik, Robert E.; Hayes , Wallace D.; Donaldson, Coleman duP.; Gray, K. Evan; and Sullivan , Roger D.: Theoretical Prediction of Pressures in Hypersonic Flow With Special Reference to Configurations Having Attached Leading-Edge Shock - pt. 1. Theoretical Investigation. ASD TR 61-60, pt. I, U.S. Air Force , May 1962.

35. Babenko, K. 1.: Investigation of a Three-Dimensional Supersonic Gas Flow Around Conic Bodies. Applied Mechanics, Henry GOrtler and Peter Sorger, eds., Springer-Verlag, 1966, pp. 749-755.

36. Hord, Richard A.: An Approximate Solution for Axially Symmetric Flow Over a Cone With an Attached Shock Wave. NACA TN 3485, 1955.

37. Chapkis, Robert L.: Hypersonic Flow Over an Elliptic Cone: Theory and Experi- ment. J. Aerosp. ScL, vol. 28, no. 11, Nov. 1961, pp. 844-854.

38. Vincenti, Walter G.; and Fisher, Newman H., Jr.: Calculation of the Supersonic Pressure Distribution on a Single-Curved Tapered Wing in Regions Not Influenced by the Root or Tip. NACA TN 3499, 1955.

39. Mead, Harold R.; and Koch, Frank: Theoretical Prediction of Pressures in Hyper- sonic Flow With Special Reference to Configurations Having Attached Leading-Edge Shock - Pt. II. Experimental Pressure Measurements at Mach 5 and 8. ASD TR 61-60, Pt. II, U.S. Air Force, May 1962.

40. Beeman, E. R.; and Powers, S. A.: A Method for Determining the Complete Flow Field Around Conical Wings at Supersonic/Hypersonic Speeds. AIAA Pap.

No. 69-646, June 1969.

___ J

TABLE 1 COMPUTATION HISTORY FOR AN ELLIPTIC CONE

~ = 1.4 ; Moo = 5.8; b/ a = 0.5; eo = 6.0 ; N = 17J

m~xlvi(O)1 a Cycles I 2.2 x 10- 0.05 2 1.0 2 9.9 x 10- 2.0 4.7 x 10- 3.0 3.7 x 10- 2.5 x 10- 4.0 2 1.6 x 10- 5.0 2 8.1 x 10- 6.0 3 LAYOUT OF COMPUTATIONAL LINES SHOCK WAVE CROSS SECTION ~ It . __ ,-~ ~ ,-"S_H~OC_K-.~_AV,E __ .-, 7 I I I I I I I I I I I I I I I I I I I I I I I I I I I I. T 0 1 i= I 2 345 678 \ 9 CONICAL BODY SURFACE ~ i = I PHYSICAL (~,7]) PLANE TRANSFORMED (TI~) PLANE Figur e I

- -- --- - - - - - ~- - - ---

-- SHOCK SHAPE AND PRESSURE DISTRIBUTION ELLIPTIC CONE; Moo = 5.8 0, deg • 0 o 4 EXPERIMENT (REF. 37) o 6 -METHOD OF LI NES (N = I 7) C p .12 o 0 0

O~ ______ ~~ __ D===

-90 0 90 q, F igu re 2( a) SHOCK SHAPE AND PRESSURE DISTRIBUTION CONICAL DELTA WING; PARABOLIC CROSS SECTION; Moo = 4 - METHOD OF LINES (N e9) o VOSKRESENSKII (REF. ?)

.3 0 CONICAL MOC • CONICAL SHOCK EXPANSION SUBSONIC ~ CROSS FLOW .2 CROSS - FLOW . II>-<J--C..-- SONIC LINE x o .5 1.0 x Fi g ur e 2( b) SHOCK SHAPE AND PRESSURE DISTRI BUTION CONICAL DELTA WING; CIRCULAR-ARC CROSS SECTION; MCO=B .I - METHOD OF LINES (N = 9) A RECENT} 3D MOC --- OLDER o CON ICAl MOC • CONICAL SHOCK EXPANSION ,\\\\\W EXPERIMENT (REF. 39) .4 =IOO ~'\~ . 3 SUBSONIC I .,' " " CROSS FLOW., ,. 4° \" \\\\\, \~ \ .(:>.. \ ~ ~\\\\\\ _..".......'" I \\\\\\\,\\~ - --- ,\\,\\\\\\\\ 0 \\~ I \\\\\\\\\ 0 ~\\\\ \\\\\\\\\,,\\\\\\\ "\\ \\\ ~~c.:.:-r- \\\\\\\ a SHOCK FOR a = 10° _ I L- ____ L- ___ ---,-J . 0 .5 1.0 x Figure Z(c) SHOCK SHAPE AND PRESSURE DISTRIBUTION FLAT DELTA WING; M(X) = 4 -- VOSKRESENSKII (REF. 7) o METHOD OF LINES (N = 12)

t .3 6, 3D MaC

500 • BABAEV (REF . 17) .!.c~>--<)----(:r--<> .2 Cp CROSS-FLOW SONIC LINE 1.0 .5 a x Figure Z(d) I 154

L __ _

SURFACE -PRESSURE CONVERGENCE HISTORY FLAT DELTA WING; N=12; Moo=4.0; A=50o; a=15° SUBSONIC .26 CROSS FLOW / / / / / / ./ .22 INITIAL ",,/ APPROXIMATION ~ ./ ./ __ -- -- CYCLE ELAPSED

---

TIME, sec .1 8

---

0 7

---

t::.

I 90 0 2 ---0-- 3 .5 x Figure 3 CONVERGENCE WITH INCREASING N FLAT DELTA WING; Moo = 4.0; A:: 50 ; ex :: 15° SUBSONIC : .26 CROSS FLOW ---:l .24

•

- VOSKRESENSKII (REF . 7) METHOD OF LINES N NO. CYCLES TIME, sec .22 • 2 2 5 o 4 2 15 t::. 8 3 104 . 200~ ------------------.~ 5 ----------~------~ 1.0 X Figure 4 DISCUSSION HARRY A. DWYER, University of California, Davis: I was just wondering how you located the shock position in the method of lines. How do you determine the shock location?

SOUTH: I must have done that pretty poorly on slide 3. Let us look at slide 3. [Information from slide 3 was incorporated in written version; there- fore, this slide is reproduced at the end of this discussion.]

We specify the ordinates of the shock, and then we numerically differ- entiate the ordinates. Differentiating the ordinates we get the cross slopes of the shock wave, and this gives us all we need to satisfy the Rankine- Hugoniot jump conditions. This gives us the pressures and velocities behind the shock, so we can integrate the ordinary differential equations to the surface.

Now, when we get to the surface, the normal components are not zero, and we want to drive them to zero by selecting the right shock ordinates. In other words, the first guess for the shock location was wrong. I explained how we make a small perturbation on each ordinate and generate a matrix of influence coefficients. We invert it to obtain corrections for each one of the ordinates, and hence a new estimate for the shock location. We keep repeating the cycles until we have driven all the normal components at each line to zero.

WILBUR L. HANKEY, USAF Aerospace Research Laboratory: Have you looked at the lees ide of a delta wing or a cone at high angle of att . ack, in which the vortical singularity was of any sufficient strength to lift off the surface? It would seem that with the method of lines you could get into trouble with the singularity.

SOUTH: Well, the vortical singularity ... There was a comment earlier about a piece of work by D. J. Jones, of the National Research Council in Canada, and the method that Jones uses is almost identical with this. His work was published while we were still wrapping up the elliptic cone part and working on the wings, and he presented some results indicating the lift- off of the vortical singularity.

As far as I know, this is the only method that has actually given some results indicating the actual lift-off of the vortical singularity other than some work done by Gonidou with the BVLR method. This method, strangely, although it is very simple and very unsophisticated, seems to have little trouble with the nodal singularity on the lees ide, at least before this "lift-off" occurs. I won't get into too much discussion of this. The method of lines seems to work better and easier than any other method as far as that feature is concerned.

L __

We will never use this method for the leeside of wings. For one thing, we might work on the inviscid problem for years and finally succeed, but people wouldn't believe it anyway, because I think it is mainly viscous- dominated with vortical eddies at the leading edge. For another reason, there is no well-defined outer boundary on the leeside for the application of this method in its current formulation.

ROBERT FELDHUNE, U. S. Naval Ordnance Laboratory: Did you say that with your technique you do actually see the vortical singularity lift off the surface of the elliptical cones?

SOUTH: We have not done it, but Jones has, for circular cones, and you can look in his report and see it. He computed a low Mach number case, which was about a 1.8 Mach number, and a 12.5° half-angle cone, up to a relative incidence of 1.2 and beyond; that is, up to about 15° angle of attack and beyond.

The shock wave on the leeside was still not a Mach wave, and the normal component of velocity in the leeward plane of symmetry actually passed through zero above the surface and then went positive and then back to zero again. So there were two zeros in the normal component of velocity in that leeward plane. He states that his results indicate lift-off. We played with that briefly and we would always get into trouble. That seemed to be a limitation with our program. With his, it apparently wasn't a limitation.

SIDNEY A. POWERS, Northrop Corporation: Since we are doing the conical three-dimensional method of characteristics, I'll have to put in a commercial here.

In answer to Mr. Hankey, this method has been applied and does work on the expansion side. We run 4°, 8°, and 12° leeside solution for a 45° swept wing at Mach 3.

Now, our method follows streamlines, so we in essence bypass the vortical singularity problem that way.

We agree with Ralph Carmichael's comment this morning that one should have a good inviscid solution to give you something to base further judgments on. So we think that the leeside solutions are important. We have done some solutions with and without imbedded shock on the inside. [Mr. Carmichael's comment was made in paper no. 2 of this compilation.]

-- - -- -----

--- - --- - - - -

- - ~ METHOD OF "LINESII (I) WRITE GOVERNING EOS.IN FORM: SHOCK-WAVE

:~ = p{p,p,U,V,w,~, it'1t'1r}

CROSS SECTION [ ~=U d..'L=V ~=W 9 8 iF) '0") '0") I \ 7

(2) N LINES t=t NORMAL TO BODY

I l I I I I (3) FINITE DIFFERENCES FOR o'{ I (4) ASSUME SHOCK SHAPE (5) INTEGRATE FROM SHOCK TO BODY (6) EVALUATE NORMAL COMPONENTS AT BODY (7) ITERATE SHOCK SHAPE TO ACHIEVE Vj=O, j=I, .. . ,N . S lid e 3.

I

i

I

I

-- --- - - - - - -------------------------------------------------------------------- APPLICATION OF THE METHOD OF CHARACTERISTICS TO NONCIRCULAR BODIES AT ANGLE OF ATTACK By John V. Rakich Ames Research Center SUMMARY The generalized boundary conditions for bodies with noncircular cross sections are described, and numerical stability problems arising from these boundary conditions are discussed. Surface-pressure coefficients for a pointed elliptic cone at angle of attack are compared with experiment and with results of other theoretical methods. Flow-field results are presented for a nonconical wing-like body consisting of an elliptic cone forebody and an elliptic-wedge afterbody.

INTRODUCTION A method of characteristics for three-dimensional flow was described in references I and 2. Although the method was developed in a general way, applications in those papers were limited to bodies of revolution. The main difference for bodies with noncircular cross sections is in the application of the boundary condition at the body surface. In the present paper the gen- eralized boundary condition is described and tested on a wing-like body with elliptical cross sections.

Applications of the method of characteristics to noncircular bodies have been made previously by Moretti in reference 3. That method is one of first- order accuracy and was developed specifically for bodies with simple sections (flat and cylindrical surfaces in combination). The flow over a delta wing with supersonic leading edges was recently computed by Beeman and Powers (ref. 4) by a method of characteristics employing the distance-asymptotic technique. (Conical solutions may be considered analogous to the steady- state motion produced by a piston with an impulsive start.) A noncharacter- istic method, developed in the Soviet Union, has been applied to elliptic cones in references 5 and 6. These also use the distance-asymptotic techni- que to get the conical solution. Numerous inverse methods have also been employed in the past for elliptic cones (e.g., ref. 7). In two recently reported methods (refs. 8 and 9), inverse methods have been successfully auto- mated to find the shock shape consistent with the given body.

In the present study, flow over the nonconical wing-like body shown in figure 1 is calculated to test the proposed method. The elliptic-cone fore- body solution is obtained by the distance-asymptotic technique. The forebody and afterbody solutions are obtained with the same computer program that main- tains second-order accuracy in terms of mesh spacing.

PRINCIPAL SYMBOLS speed of sound; l ocal span a characteristic directions in ~ plane pressure coefficient Mach number + body surface outer normal N pressure p s* projection of streamline on ~ plane u,v,w cylindrical velocity components 2 2 2 V velocity magnitude,/u + v + w x)r) ~ cylindrical coordinates A A A ex,er,e ~ unit vectors along x, r, ~ a angle of attack I \ £ surface upwash angle S flow angle in ~ plane

I

~ crossflow angle normal to ~ plane

l

THEORY \

I

Differential Equations The present description will begin directly with the compatibility I equations derived and used in references I and 2. These are

l

i3 ap as

cos sin sin ~ cos ~ ----- + cos 4> sin --= + + (1)

e)

e )]

S (F 2

"* [(PI

r r

pv ac * ac *

I

1 1

I S ap

ae

sin 4> cos ~

sin --= e) _ S (F 2 + sin (2)

-- --- - cos 4>

"* [(PI _ cos

e )]

\

p V2 aC 2* aC 2 *-

r r

l

l _____ _

cp sin sin e ~= (3) + F 3 * * as rE 11 1 ap ~= (4) ---+ F4 d S* a d S* where C1* and C2* are projections of the Mach cone on the plane ~ = constant, s* is the projection of the streamline on that plane, and E11* is the direction cosine relating unit vectors 5 and 5* (see ref. 2). The Fi (i = I, . .. , 4) on the right-hand side of these equations contain the so- called cross derivatives which vanish for two-dimensional or axisymmetric flow but are always present in three-dimensional flow. The circumferential derivatives are evaluated by means of Fourier analysis, as described in ref- erence 2. However. for the present calcula ti ons, the Fourier coefficients are multiplied by Lanczos' a factors to damp high-frequency oscillations (see ref. 10).

In deriving equations (1) and (4), use was made of a cylindrical x, r, ~ coordinate frame (see fig. 1). The angles 8 and cp are defined relative to the planes ~ = constant; 8 is the flow direction measured in the plane; and ~ is angle by which the velocity vector is turned out of the plane. In terms of velocity components, u, v, w, along the x, r, ~ directions, these angles are: e = tan- y (5) u w tan- (6) For general body shapes the surface boundary condition will involve all three velocity components. This will introduce a coupling between flow angles 8 and ¢ which does not occur for bodies with circular cross sections.

While it is possible to simplify the boundary condition by using reference planes normal to the body surface, the basic equations would be more complex and the class of body shapes restricted. Therefore the basic cylindrical coordinate system was retained in spite of complications in the boundary con- dition. The general boundary condition is derived next.

Surface Boundary Condition The boundary condition at the body surface may be written generally as + + V • N = 0 (7) Let the body be given by (8) g(x, r, ¢) = r - f(x, ¢) Then the surface normal may be written -fxe + e - (l/r) f ¢e¢ x r N = (9) /1 + f / + (1/r 2 )f ¢2 where the subscripts indicate partial derivatives. The unit vector along the

streamline direction may be written in terms of flow angles e and ~ as

follows: A A A A ( 10) S cos ~ cos 8 ex + cos ~ sin 8 e + sin ~ e¢ r -+ substituting equations (9) and (10) into equation (7), noting that V = Vs, one can obtain the following expression for tan e (see ref. 2),

f x + (t an <p / r ) f ¢ fi + f / - [(tan ~ / r) f ¢ ] 2

( 11) tan 8 = 1 - [(tan <p /r)f ¢ ]2 COMPUTATIONAL PROCEDURE The first effect of the generalized boundary condition is to change the procedure for computing the conditions at a typical body point (see fig. 2).

Since tan e is not equal to the body slope a f/ ax , the streamline projec-

tion s* is not tangent to the body. Consequently, the integration of equa- tion (3) requires interpolation for data at point D on the initial data line.

This is the main change in program logic for noncircular bodies. For circu- lar bodies, conditions at point A are employed without need for interpolation.

To complete the calculation for point C on the body, equation (2) is applied on C * using interpolated data at point B in the standard manner described in reference 2, equation (2) is replaced by the boundary condition equation (11), and equation (4) is replaced by a body entropy condition.

The body entropy condition mentioned here for the first time, requires additional comment. It is possible to employ equation (4) directly without explicitly introducing the entropy function. However, since the body entropy is usually known, the computation is simpler and more accurate if a surface

_J

entropy condition is employed. This is especially true near conical singular points where density gradients are multivalued. For general conical flows it is always possible, in principle, to trace the stagnation streamlines from the body to the shock in order to determine the body entropy. However, detailed streamline tracing is a complication of questionable value in the light of experience with analogous blunt-body flows. Numerous studies have found that the stagnation entropy is very nearly equal to the maximum entropy on the shock. Therefore, as a practical expedient, the body entropy has been set equal to the maximum shock entropy in the present conical-flow calculations.

Numerical Stability A stability analysis of the general nonlinear equations is too difficult to perform so one usually relies on a linear analysis. The stability condi- tion for linear hyperbolic equations is the well-known Courant-Friedrichs-Lewy or C.F.L. condition. It states that the difference equations must include the domain of dependence of the differential equations. This means that the Mach line in figure 3 must lie above point A as shown. Although the C.F.L.

condition is derived on the basis of linearized equations, and is rigorously shown to be only a necessary condition, it has been found to be sufficient in most practical applications. This was true for previous applications of the present method to bodies of revolution. However, the C.F.L. condition was not sufficient in present applications because of the coupling of the cross- flow angle through the boundary condition, equation (11).

Previous practice was to use a step slightly less than the C.F.L. condi- tion. A value of 0.8 (C.F.L.) worked well for most applications. When this condition was applied to an elliptic-cone calculation, the computation became I unstable as illustrated in figure 3. This figure shows the variation of ¢ with distance starting with flow conditions that depart only slightly from the conical solutions. For conical flow ¢ should be constant, but with a step size of O.S (C.F.L.), ¢ oscillates with increasing amplitude as x increases. When the step was reduced to 0.6 (C.F.L.), the amplitude was decreased but the computation still slightly unstable. Finally, with 0.4 I (C.F.L.), it is seen that the oscillations are quickly damped, and the computation is stable.

RESULTS Elliptic Cone The present method-of-characteristics program has been used to obtain conical solutions by means of the distance-asymptotic technique. Starting with an approximate solution, the computation is carried downstream until variations of flow properties along conical rays decrease to a specified error. The maximum shock entropy is applied at the body as a boundary condi- tion. The initial solution need not be accurate, but the relaxation process

I

tends to be slow if the initial solution is poor. Specialized conical methods have been developed (e.g' refs. 8 and 9) that are more efficient but l more li~ited. These methods start with an assumed conical shock and march to the body . The process is repeated until the surface boundary condition is matched.

To test the present method, a 2:1 elliptic cone at M = 5.8 and 4° angle of attack was selected. Experimental results were available from refer- ence 11 as well as theory from reference 9, The surface pressure distribu- tion is shown in figure 4. The results of reference 9 agree better with experiment; the present method appears slightly low with the biggest differ- ence occurring just to the leeward side of the leading edge. It is believed that the difference in theories is due to the coarser mesh employed in the present calculations. In reference 9, 17 planes were used and they were more densely spaced near the leading edge. The present calculations initially employed 13 planes and 6 points between the body and shock. The field was later refined to 11 points and 15 planes. Computing time was about 35 min- utes on an IBM 7094, mod I computer.

A Wing-Like Body To demonstrate the generality of the present method, a nonconical body, which has been studied for hypersonic transport missions (ref. 12), was selected. Flow computations have been performed for the delta-wing-like body shown in figure 5. This body is described as an elliptic-cone forebody with an elliptic wedge afterbody. This test body presents an especially difficult test because of the expansion corner at the junction between the cone and the wedge. This corner region was approximated with a small radius arc, so that the program could be applied directly without special treatment of the sur- face discontinuity. The approximating arc was made 0.01 of the nose length.

About 35 steps were used in calculating across this arc segment.

Results of the circumferential pressure distribution at the start , mid- point, and the end of the arc are shown in figure 6. Note that the pressure on the leading edge ~ = 90° is unchanged, as it should be. Pressure on the windward and leeward sides, on the other hand, changes rapidly in this region.

The symbols on the elliptic-cone curve (x = 1.0) show results of Kaattari's

empirical method (ref. 13) and numerical results of South et al. (ref. 9).

Agreement is good except for the region just leeward of the leading edge where the present predictions are slightly lower than those derived by the referenced methods.

The axial variation of the surface pressure is shown in figure 7. The pressure is constant on the elliptic cone, x < 1, and remains essentially

constant for the leading edge ~ = 90°. On the other planes, the pressure

drops rapidly on the transition arc, 1 < x < 1.01, and then rises for a short distance behind the corner. This type of pressure variation is typical of those observed for sphere-cones . An overexpansion usually occurs near regions where the body slope or curvature is discontinuous. Experiments with

L _

blunted cones indicate that the boundary layer tends to smooth the discontin- uous pressure gradients predicted by inviscid theory.

The t w o-dimensional pressure for a sharp expansion corner at ~ = 180° is also shown in figure 7. It is somewhat below the value obtained with the approximating arc segment. The present method should yield the two- dimensional value in the limit as ~x of the arc tends to zero.

The radial variation of pressure is shown in figure 8 for several x stations in the windward plane, ~ = 180°. Moving from body to shock, n = 0 to 1, the pressure first overshoots and then relaxes to the elliptic- cone value. The overshoot results from numerical interpolation with a rela- tively coarse finite difference mesh (11 points were used between the body and shock). The pressure should be equal to the elliptic-cone value outside the initial Mach wave from the corner, and should be less than that value inside the Mach wave. The approximate location of the initial Mach wave is indicated by the small lines on the pressure curves. Previous experience with similar flows has shown that the pressure overshoot is decreased when the mesh is refined (see ref . 2).

Figures 9 and 10 show the axial and radial variations of flow angle e.

Figures 11 and 12 show similar curves for the crossflow angle ~. On the 90°

and 180° planes, the flow angle e is constant behind the corner as specified

by the body shape. However, on intermediate planes, e varies with distance

in accordance with equation (11). The radial variations of the flow angles are qualitatively similar to those for the pressure described above. These angles approach the elliptic-cone value some distance off the body surface, but with some overshoot, as described in the discussion of pressure variations.

The sign reversal of the crossflow angle in figure 11 raises a question about the shape of streamlines on the wedge afterbody. To visualize better the streamline direction, an upwash angle measured on the body surface is introduced here. Let E be the actual surface stream angle measured with respect to the meridional plane, ~ = constant, and defined as posi tive for an upwash. Then E may be expressed as follows in terms of flow angles e and ~ , and body slopes fx and f ~ : (12) For a circular body, or, more generally, for f ~ = 0, equation (12) gives E = -~. It is also noted here, for reference, that the conical crossflow velocity is given by Wc = V sin E (13) The axial variation of E is shown in figure 13 for the elliptic-cone elliptic-wedge body. The variation is similar to that shown for the cross- flow angle, ~, in figure 11. On the elliptic-cone forebody, E is positive 0 0 on the 75 plane and negative on the 105 plane. This indicates a flow away from the leading edge in the direction of decreasing pressure. A reversal of sign occurs at about x = 1.01 which seems unusual. A closer examination shows that this peculiar reversal is caused by the curvature of the reference line from which E is measured. The reference line is the intersection of

the plane ~ = constant with the body surface and is curved on the elliptic-

wedge afterbody. The sketch in figure 13 shows the reference line for ~ = 105 (to the windward side of the leading edge). All the reference lines, except the ~ = 90 line, will approach the x axis at the trailing edge where the body closes.

CONCLUDING REMARKS A three-dimensional method-of-characteristics program has been applied to noncircular bodies at angle of attack. Results for an elliptic cone were verified by experiments and by other available conical-flow solutions. Sta- bility problems were encountered which were overcome by using a step size less than that allowed by the Courant-Friedrichs-Lewy condition. The C.F.L.

condition gives the maximum allowable step for linear hyperbolic equations; it is a necessary but not a sufficient condition. Present results indicate that in nonlinear problems the maximum step can be appreciably less than the C.F.L. step. This illustrates a need for a more restrictive stability condi- tion for complex, nonlinear flow-field calculations.

The present method was applied to a nonconical body with elliptic cross sections to demonstrate its generality. While the results appear reasonable and internally consistent, additional verification with experiment is essential.

While the present examples were all pointed bodies, the method is not so restricted. Flows over blunt-nosed bodies have also been calculated using a locally supersonic starting solution obtained from an inverse blunt-body method. Other problems which can be treated in a straightforward manner by the present method of characteristics include sharp wings with supersonic leading edges, and simple wing-body combinations. However, additional work is required to include the special leading-edge boundary condition for "super- sonic" wings. For wings with subsonic leading edges, the leading-edge bound- ary condition is more difficult. At the present time, it is not clear how the problem of a subsonic leading edge is best solved numerically.

REFERENCES 1. Rakich, John Vi and Cleary, Joseph W.: Theoretical and Experimental Study of Supersonic Steady Flow Around Inclined Bodies of Revolution.

Preprint 69-187, AIAA, Jan. 1969.

2. Rakich, John V.: A Method of Characteristics for Three-Dimensional Supersonic Flow With Applications to Inclined Bodies of Revolution.

NASA TN D-534l, 1969.

3. Moretti, G.: Three-Dimensional Supersonic Flow Computations. AIAA J., vol. 1, Sept. 1963, p. 2192.

4. Beeman, E. R.; and Powers, S. A.: A Method for Determining the Complete Flow Field Around Conical Wings at Supersonic/Hypersonic Speeds. Pre- print 69-646, AIAA, June 1969.

5. Babenko, K. I.; and Rusanov, V. V.: Difference Methods of Solving Three-Dimensional Problems in Gas Dynamics. NASA TT F-lO,827, 1967.

6. Gonidou, Rene: Supersonic Flow Around Cones at Incidence. NASA TT F-ll,473, 1968.

7. Briggs, Benjamin R.: The Numerical Calculation of Flow Past Conical Bodies Supporting Elliptic Conical Shock Waves at Finite Angles of Incidence. NASA TN D-340, 1960.

8. Jones, D. J.: Numerical Solutions of the Flow Field for Conical Bodies in a Supersonic Stream. Natl. Res. Council Canada Aeron. Rep. LR507, July 1968.

9. South, J. C.; Klunker, E. B.; Wagner, R. D.; and Chiang, C. W.: Flow- Field Calculation for Conical Wings and Bodies. Methods for Calculat- ing Conical Flows. NASA Symposium on Analytical Methods in Aircraft Aerodynamics, Oct. 28-30, 1969.

10. Hamming, Richard W.: Numerical Methods for Scientists and Engineers.

McGraw-Hill Book Co., Inc., 1962.

11. Chapkis, Robert L.: Hypersonic Flow Over an Elliptic Cone: Theory and Experiment. Calif. Inst. Tech., Guggenheim Aeron. Lab. Memo. 49, May 1959.

12. Gregory, T. J.; WilCOX, D. E.; and Williams, L. J.: The Effects of Propulsion Systems-Airframe Interactions on Performance of Hypersonic Aircraft. Preprint 67-493, AlAA, July 1967.

13. Kaattari, George E.: Empirical Method for Estimating Pressures on Elliptic Cones. NASA Symposium on Analytical Methods in Aircraft Aerodynamics, Oct. 28-30, 1969.

Figure 1 SURFACE BOUNDARY CONDITION

at ton¢ af (El)2 _(ton¢ .El.)2

- + -- ael> I + ax r ael> ax r t 8 on = ------------~---------------- I_(ton¢ ~)2 r ael> 8 = ton-I Y...

u v

t Lu

r B~S *

t ~ /'" ~C*

~x 2 PLANE el> = CONST PLANE x = CONST Fi gu re 2 NUMERICAL STABILITY ci -- ~ s * _- ')lI --...

• /' C2 orp I ( I op . . )

-- - - - - - + sin rp sin e

Os * r pv oct>

----

- /

---- / / /

>/1

/ / STEP CFL o 0.4

A &t~

STEP o .6 • .8 CFL .08 -

•

[J cf> .04 .

•

O ~I --~----~--~----~--~ .S3 .S4 . S5 . SO .SI . S2 x Fi gure 3 CON I CAL TEST CASE AXIS RATIO = 2 SWEEP = 7S.2° M = 5.S a = 4° . 10 -- PRESENT METHOD --- SOUTH et. 01. Ref. 9 h .OS o EXPERIMENT Ref. " .06 . 04 .02 o 30 60 90 150 120 180 ct>, deg Fi gure 4 NONCONICAL TEST CASE O M=7 a=I MESH {II POINTS BODY TO SHOCK 13 PLANES

-- _. ~ 1 , --c 2/3-l

a/b=2

@II~

a CON~ ARC ~EDGE ~ 0.01 -:r::::=== - ~

~b

CONE SECTION Figure 5 SURFACE PRESSURE CIRCUMFERENTIAL VARIATION o SOUTH et al. Ref. 9 o KAATTARI Ref. 13 X= 1.0 CONE ..-r-~~ CONE ARC WEDGE 1.01 .1 I .03 150 180 30 60 90 120 o <%>, deg Figure 6 (

l __

SURFACE PRESSURE AXIAL VARIATION ~ 1 0 WINDWARD . 1 - / SHARP CORNER _L _ <t> = 180 I .03 L I .98 1.02 1.06 1.10 1.14 X Figure 7 PRESSURE RADIAL VARIATION 5 .0 x = 1.0 <:> = 180· IN I TIAL MACH WAVE . 1 I I I I I .05 o .2 .4 .6 .8 1.0 r- r b 'T] = r s - rb Figure 8 FLOW ANGLE AXIAL VARIATION .3 - / LEADING EDGE .2- __ ---, .1 - 8, rod 01-----++---+---+---+---+--+---1-----1 -.1 -

CONE ARC WEDGE +

1 80 -.2 - " I ' \"". _________ --""- __ 1_ _ _° __ _ ............. WINDWARD I I I -.3 L .98 1.02 1.06 1.10 1.14 x Figure 9 FLOW ANGLE RADIAL VARIATION 4>=180° .2 Ol--r--,~-,L-~r------- 8, rod -.1 -.2 I -.3 o .2 .8 1.0 Figure 10 '-- - CROSSFLOW ANGLE AXIAL VARIATION . 08 - .04 - O-----+~~+---~----~--~ LEADING EDGE 4>, rod -.04 - - .08 - CONE ARC WEDGE .. I .... I.

- .12 L 1 1 I .98 1.02 1.06 1.10 1.14 x Figure 11 CROSSFLOW ANGLE RADIAL VARIATION ¢ = 105 .08 x = 1.0 O~-----;L--------- __ ~---- cpo rod -.04 1.2 -.08 1 1 1 -.12 o .2 .4 .6 .8 1.0 Figure 12 SURFACE UPWASH ANGLE AXIAL VARIATION ¢> = 105 Vro

- - =-- -

PLAN VIEW-WINDWARD ¢>=105° 10 - E, deg -10 - CONE ARC WEDGE · 1· · 1· I I I -20 L . 98 1.02 1.06 1.10 1. 14 X Figure 13 DISCUSSION RAYMOND SEDNEY, Martin Company: I don't want to quibble at all with your excellent presentation, especially the part about the stability and the CFL condition, but I can't help but generalize from some experience in a com- pletely different problem, and I just wonder if the fact that the CFL condi- tion is violated so drastically might indicate it would be worthwhile to look into some more efficient grid scheme. Have you considered that at all?

RAKICH: Well, I'm not sure what you would mean by a more efficient grid scheme. I think a grid which employed body normal reference planes would probably be better from the stability criterion, because this is what we used before for bodies of revolution, and we didn't run into this type of instabil- ity problem. We tried to be general in using radial reference planes, and this coupled the crossflow angle and pressure too strongly and was really the cause of the instability. I don't know what other kind of a grid you might use, however.

GINO MORETTI, Polytechnic Institute of Brooklyn: Apropos of CFL rule, are you sure that the rule has - been applied correctly in your work? If you work with finite differences instead of working with characteristics, then the elements which enter into the denominator of the expression which gives the allowed 6x depend essentially on crossflow values. It seems to me that you are using other velocity components, so probably you are using here a 6x which does not satisfy the CFL rule as it was originally written for multi- dimensional flow. I cannot make a precise comment, but I would like to put a question mark on the way you applied the CFL rule.

RAKICH: As I understand the rule, it just states that the domain of dependence of the difference equations must include that of the differential equations.

MORETTI: For the entire crossflow?

RAKICH: Yes, so it means you have to include points from reference planes at least beyond where the Mach lines would intersect the initial data surface.

MORETTI: I think that, on occasions, ~x may still be smaller than the one you use, and it may well depend on the grid.

So, no precise comment can be made, unless one provides a detailed analysis of the scheme which has been used.

RAKICH: Well, let me restate, then, what I said earlier. Can I have slide 3, please?

As I understand the CFL rule, it just says that we have to include points from our difference mesh just beyond the intersection of the actual Mach line with the initial data line. Of course, as you point out, this is just one plane, and the Mach lines we show here are the projections of the Mach cones onto that reference plane. We imagine all the Mach cones lined ~p, and these make a Mach surface that crosses this reference plane.

Now, in the crossflow direction consider a similar p~ojection of Mach cones onto the surface r = constant. All we have to do to satisfy the CFL condition is to use data from reference planes outside the intersection of these crossflow Mach lines with the initial data surface. Since the present Fourier method uses data from all the planes, ~ = 0 to TI , this has been satisfied.

MORETTI: How big is the speed in the crossflow? Is it supersonic?

RAKICH: In this case, no.

FRED R. DEJARNETTE, Virginia Polytechnic Institute: I believe you add some sort of smoothing function to your characteristics method, that is some- thing like a secoRd derivative term. Was that used in this characteristics program?

RAKICH: Yes, it was. This is not always used, but in certain cases it is. When the entropy layer on a blunted body is very thin, and also for conical flow,we find it helps out if we do employ that type of difference scheme for the streamwise calculation.

DEJARNETTE: I would like to add that possibly this has an effect on the stability criterion, because other people, using something like a Lax-Wendroff scheme, have found that they have to use something somewhat less than the CFL stability criterion in order to get stable solutions.

RAKICH: This is something that could be investigated, and I suppose I l should.

SIDNEY A. POWERS, Northrop Corp.: John, your slide 8 (fig. 11), where you show a crossflow angle of an axial variation, it looks a little bit strange.

This angle ~ is essentially the upwash angle, due normal to the leading edge.

Is that correct? And yet you don't show any axial variation as you go down- stream. Would you expect this cross flow angle or this upwash angle along the leading edge to change in this wedge region somewhat drastically actually, due to the influence of this fairly sharp corner up there?

RAKICH: No, I didn't expect a drastic change. We are talking about this line here on a 90° plane, right?

I didn't have any very strong feeling on whether it should or should not vary. It just turned out that it dian't. I guess it is because the pressure gradient is not changing much on the leading edge since the pressure is nearly a maximum there. If you recall slide 5 (fig. 6), the pressure was practically constant on that leading edge and dropped sharply to each side of the leading edge, but it was pretty much symmetrical.

lThis question was studied after the conference and the smoothing function used does not affect the stability problem discussed in this paper.

TIME-DEPENDENT NUMERICAL METHOD FOR TREATING COMPLICATED BLUNT-BODY FLOW FIELDS By Richard W. Barnwell Langley Research Center INTRODUCTION Time-dependent methods have been used for several years to calculate numerical solutions for the flow fields about blunt bodies traveling at supersonic speeds in inviscid, compressible gases. The time-dependent method of characteristics has been used by Sauerwein (ref. 1) to treat this problem. However, time-dependent finite-difference methods provide a much more efficient means of making these calculations.

Several of the early finite-difference methods were based on the "artificial vis- cosity" technique, which treats shock waves as continuous compressions with steep gra- dients where the flow would otherwise be discontinuous. Artificial viscosity techniques of first-order accuracy in the mesh spacings have been developed by Bohachevsky and Rubin (ref. 2), Harlow (ref. 3), Gentry et al. (ref. 4), and others.

A finite-difference technique of second-order accuracy which treats embedded shock waves as continuous compressions has been developed by Lax and Wendroff (ref. 5).

Burstein (ref. 6) and others have used this technique to solve the blunt-body problem.

Another approach to the blunt-body problem is to treat the bow shock wave as a discontinuity. Godunov et al. (ref. 7) have developed such a method which produces solu- tions of first-order accuracy. Masson et al. (ref. 8) have used this method to calculate flow fields about several complicated shapes. A method of second-order accuracy which treats the bow shock wave as a discontinuity has been developed by Moretti and Abbett (ref. 9).

The purpose of this paper is to describe a time-dependent numerical method simi- lar to that of reference 9 which has been used to study some fairly complicated blunt-body flow fields. Examples of the types of complexities which have been treated are shown in figure 1. These complexities include embedded shock waves, sharp sonic corners on the body profile, and nonuniform free-stream properties upstream of the bow shock wave.

All the results which are presented in this paper are for axisymmetric bodies at zero angle of attack. However, it should be noted that the techniques presented herein can be applied to more complicated flow fields.

SYMBOLS

functions of flow properties, i = 1, 2, 3, 4

a speed of sound H total enthalpy M Mach number p pressure Pt stagnation pressure r perpendicular distance from axis of symmetry rb base radius r c corner radius rn nose radius s distance along surface from axis Sc distance along surface from axis to corner t time to initial time tl specific time u,v velocity components normal to and tangent to body normal, respectively V magnitude of velocity

-

V velocity vector X distance along reference line from axis smallest and largest values of X associated with corner Y nondimensional distance along body normal Yb distance from body surface to reference line a constant appearing in equation (4) y ratio of specific heats B distance from body to shock along body normal percent deviation of free-stream velocity at distance r = rb from axis E from center -line value p density angle between body normal and free -stream direction Subscripts: av average center line I,II first and second bicharacteristics 00 free stream METHOD In general, all time-dependent methods treat the blunt-body problem as an initial- value problem. This approach is possible because the governing equations for time- dependent flow, which are written as 8p - (-)

at + V· pV = 0

(1)

are hyperbolic everywhere in the flow fie~d. The quantities p, V, p, and H in equa-

tions (1) are the density, velocity vector, pressure, and total enthalpy, respectively. It should be noted that for steady flow, the governing equations are hyperbolic only where the flow is supersonic. Where the flow is subsonic, the governing equations are elliptic.

In general, the region of computation in the physical plane must enclose all of the region of subsonic flow. This requirement is necessary for the calculations to be stable, and it means that the sonic line must lie inside the downstream boundary of the region of computation. The upstream boundary of the region must lie on or beyond the bow shock wave. For axisymmetric flow fields, the region of computation can be terminated at the axis of symmetry.

Solutions for steady flow are obtained with time-dependent methods in the following manner. First, an approximate initial solution is assumed at t = to in order to start

the calculation. Next, the time -dependent equations are integrated from t = to to

t = to + At in order to get the solution at t = to + At for all the points in the region of computation. The time-dependent method of characteristics or one of several finite- difference methods can be used to integrate the time-dependent equations. This integra-

tion process is repeated to get the solutions at t = to + 2At, to + 3At, . . . . At each

time interval, the solution just obtained is treated as the initial solution and the integra- tion process is repeated to get the new solution. Steady results are obtained when the solutions at successive time intervals have converged sufficiently. Thus a solution is obtained for the steady blunt-body problem, which is a boundary-value problem with mixed governing equations, by treating a related initial-value problem, the unsteady blunt-body problem, with hyperbolic governing equations.

The region of computation and the coordinate system which are used in the present

method are shown in figure 2. Due to the symmetry of the flow field, the lower boundary I

of the region of computation is located at the axis. The left boundary is located at the bow ; ::::u::: ::::sn:::::s ~: ~~:d~~:: s~r~:::~:::y ::~:~t;o:f' 1 :~:heYsh~~~r:~~a~e~t ~~~c:::_ I I face. The X coordinate is measured along a reference line which is located at a con- stant distance Yb from the body surface. This choice of the line along which X is measured provides a convenient means for rotating the coordinate system about sharp corners. The quantity 0 is the distance from the body to the shock along the local body normal. The velocity components u and v are normal to and tangent to the body normal, respectively.

The present method is similar to that of reference 9 in that the bow shock wave is treated as a discontinuity, the time-dependent method of characteristics is used at the shock wave and body surface, and a time-dependent Lax-Wendroff finite-difference method is used between the shock and body. However, in the present treatment the characteristic compatibility relations are integrated along bicharacteristics rather than along l one-dimensional characteristics as in reference 9. Also, the governing equations for the finite-difference calculations are used in conservation form in this treatment, whereas in reference 9 they are used in expanded form.

When the governing equations are written in conservation form, the coefficients of the partial derivatives are not functions of the dependent variables. The equations of the present method are written as

(i = 1, 2, 3, 4) (2)

where {) is a function of the independent variables X and t and the quantities Ai, Bi, Ci, and Di are functions of the independent variables X, Y, and t and the dependent variables p, p, u, and v.

The present treatment employs an explicit two-step finite-difference method. The first step leads to a preliminary solution of first-order accuracy, and the second step provides a corrected solution of second-order accuracy. The equations for the prelimi- nary and the corrected solutions are written as (i = 1, 2, 3, 4) (3)

and ~ t ( - ~t+~J [(a4A~t (a4A~t J

i

A.)t+~t= (A.\t + ~t (aAi~ + aA + a (~X)4 -T + (~y)4 -T

(

1 X, Y 1) X,Y 2 at X Y at X Y ax X Y ay X Y " , , (i=1,2,3,4) (4)

respectively. Numerical values for the partial derivatives aAil at and aAil at in equa-

tions (3) and (4) are determined from the finite-difference equations which are analogous to equations (2). These finite-difference equations are obtained by replacing the partial derivatives with respect to X and Y in equations (2) with central-finite-difference expressions. The quantities (Ai)~v in equations (3) are the averages of the values of the quantities Ai at the points (X + ~X,Y), (X - ~X,Y), (X,Y + ~ Y), and (X,Y - ~ Y) at time t. The fourth-order terms in equations (4) are added to insure stability. A value of a = 1/32 was used to determine the present results.

There are two advantages associated with the use of the conservation form of the governing equations. First, the use of this form permits stable calculations of flow fields containing embedded shock waves with no further consideration being given to the presence of these shocks. These embedded waves are represented by continuous compressions which extend over several mesh spacings. The flow-property profiles across these compressions have steep gradients where the discontinuity would otherwise be. Second, the use of the conservation form results in improved solutions at grid points near a sharp corner where the flow is singular. Let Xinin and x1'nax be the values of X associated with the normals to the upstream and downstream surfaces at the corner, respectively, as shown in figure 2. The second advantage is realized because the quan- tities Ci in equations (2) are proportional to Y for X~in;2 X ;2 X~ax and can be written as (i = 1, 2, 3, 4) where the quantities C are functions of the dependent and independent variables.

i Therefore, the partial derivatives aCi/ay are well-behaved near the corner, although the individual flow properties and the quantities Ci are singular there. It has been found that the quantities Bi are well-behaved near the ' corner and that the only compli- cation which arises during the computation of the partial derivatives aBi/ aX occurs for values of X where the curvature of the reference line changes discontinuously.

As stated previously, the time-dependent method of characteristics is used to deter- mine the flow properties at grid points on the body surface. In general, three character- istic compatibility relations are used at each body point. The procedure which is used at a sharp corner is somewhat different. It should be noted that the present coordinate system maps the point at the corner onto the line Y = 0 and Xinin;2 X ;2 xinax. If the flow upstream of the corner is supersonic, the standard procedure is used for the

point Y = 0 and X = Xinin, but if the flow upstream of the corner is subsonic, the solu-

tion at this point is obtained by solving two characteristic compatibility relations and the condition M = 1 simultaneously. It has been found that the flow properties at the grid

locations Y = 0 and Xinin < X ;2 xinax can be related to those at the point Y = 0 and

X = Xinin by the transient analog to the Prandtl-Meyer solution.

Let s be the distance along the surface from the axis, and let s = sc at the sharp corner. The two compatability relations which are used to determine the sonic solution

at the grid location Y = 0 and X = Xinin at some time t = t 1 are integrated along

bicharacteristics which lie in the s-t surface. This surface is shown in figure 3.

There are two alternate procedures which can be used. The first is to integrate the com- patibility relations

dP~ (dU~ 2 (1 av u ~

(5)

- + pa - = -pa - - + - cos cp

(

dt I dt I 0 ay r and dP\ a2(~\ - 0 (6) ( dt)n - dt)n -

I

along t~e bicharacteristic with slope

dS\ = u + a

(7) ( dth and the bicharacteristic with slope (8) respectively. The quantity a in equations (5), (6), and (7) is the speed of sound, and the quantities rand 1> in equation (5) are the perpendicular distance from the axis and the angle between the body normal and the free-stream direction, respectively. The second procedure consists of integrating equation (6) and the transient Bernoulli equa- tion along the bicharacteristic with the slope defined in equation (8). It has been found that results obtained with the two procedures are essentially the same.

RESULTS AND DISCUSSION The present results for the flow past a flat-face cylinder for a ratio of speCific heats of 1.4 and a f~ee-stream Mach number of 2.81 are compared with the experimental

I

results of Kendall (ref. 10) in figure 4. Kendall's results are shown as solid lines in the

figure and the present results are shown as Circles. The shock-wave and sonic-line I

shapes are shown on the left. The open circles are the present results for the shock- wave location, and the solid circles are the results for the sonic-line location. The pres- sure distribution is given at the right.

I It should be noted that the grid used is fairly coarse. There are four grid spaces across the shock layer and nine along the shock (five of these sUbtend the face and four

I

subtend the corner).

I

In figure 5, results are presented for the effect of nonuniform flow on the shock- wave and sonic-line shapes for a 60 blunted cone with a ratio of nose radius to base

I

radius of 0.25. The ratio of specific heats is 1.4 and the free-stream Mach number at the center line is 10. The nonuniformity of the flow consists of a parabolic dependence

I

of the free-stream velocity on the distance from the axis. The quantity € is the per-

I

cent deviation of the velocity at a distance r = rb from the axis from the center-line value. The free-stream pressure and total enthalpy are constant. This type of nonuni- I formity may be similar to the disturbed flow in some wind tunnels with strong viscous effects at the walls.

It is seen in figure 5 that the nonuniformity has a marked influence on the shock- wave and sonic-line shapes. The general trend is for the shock wave to move closer to the body in the stagnation region and farther from the body in the transonic and super- sonic regions as E is increased. The sonic point at the shock moves outward away from the axis as E is increased. The present results indicate that the sonic point is posi- tioned at the corner for E = 0 and 0.01 but that it is located upstream of the corner on the cone surface for E = 0.03 . This displacement of the sonic point will be discussed subsequently.

The effect of this nonuniform flow on the pressure distribution for the 60 cone is shown in figure 6. The pressure p is nondimensionalized with the center-line value of P 00 V ~, and it is plotted as a function of the distance from the axis along the surface.

It can be seen that for E = 0.01, the pressure in the vicinity of the corner is

reduced by about 20 percent, and for E = 0.03, the pressure near the corner is reduced

by about 50 percent. It should be noted that these severe pressure reductions are accompanied by equally severe reductions in the free-stream Mach number. At a dis-

tance r = rb from the axis the free-stream Mach number for E = 0.01 is reduced

16 percent from 10 to 8.4, and for E = 0.03, it is reduced 34 percent to 6.6.

It can be seen that the present results are irregular in the region just upstream of the corner. It has been found that this irregularity decreases as the grid is refined.

This irregularity may exist because an average value of the curvature of the reference line is used at points where this quantity changes discontinuously .

The calculations for E = 0.01 are compared with the results of the method of

integral relations of South in figure 6. It should be noted that these results have not been published previously. The basic method which South used is given in reference 11. The close agreement of the present results and those of the method of integral relations is encouraging in the absence of experimental data on the subject.

The Newtonian pressure distributions are also presented in figure 6. The standard Newtonian equation was used, but the radial dependence of the quantity P oo V~ was included. For E = 0 and E = 0.01, the flow remains subsonic all along the face, and the present results differ from the Newtonian results as expected. As stated previously, the present results indicate that the sonic paint is located on the cone surface upstream of the corner for E = 0.03. The present results for the pressure for E = 0.03 differ from the Newtonian results in the region where the flow is subsonic. However, in the region between the sonic point and the corner, the results of the two methods are in close agreement.

In general, there is a reluctance to accept solutions which indicate that the sonic paint on the surface of a body in a symmetric inviscid flow field is located on a straight segment of the surface. In fact, Shifrin (ref. 12) has proved that for symmetric, non- isentropic, inviscid plane flow such a solution cannot exist. However, it should be noted that this proof does not apply directly to the present cases for axisymmetric flow.

It is possible that the truncation error of the present method is responsible for the displacement of the sonic point upstream of the corner for E = 0.03 since this error is viscouslike in nature. However, it is also possible that the present results represent the inviscid flow correctly and that the displacement occurs because the dynamic pressure at points off the axis for E = 0.03 becomes too small to support subsonic flow at the sur- face all the way to the corner.

In figure 7 results are presented for the effect of Mach number on the shock-wave and sonic-line shapes for a 45 truncated cone with rounded corners. The cone is fol- lowed by a cylindrical afterbody.

At a Mach number of 2.8, the flow in the nose region is independent of the flow on the conical portion of the body since the sonic line at the first shoulder extends all the way from the body to the bow shock. The flow expands around the first shoulder and becomes supersonic. It then compresses on the conical portion of the body and becomes subsonic. At the second shoulder, it again expands to the supersonic state.

The regions of subsonic flow are connected for a Mach number of 2.6. However, a bubble of supersonic flow remains at the first shoulder. For a free-stream Mach num- ber of 2.4, the subsonic zone in the vicinity of the conical portion of the body enlarges so that it extends all the way to the bow shock wave. There are two isolated regions of supersonic flow at this Mach number: one at the first shoulder and one at the shock wave.

For a Mach number of 2, the flow field upstream of the second shoulder is subsonic with the exception of the supersonic bubble at the first shoulder. At this Mach number, the entire subsonic region is affected by the presence of the conical surface.

The compression which occurs over the conical portion of the body starts as a con- tinuous compression at the first shoulder. For higher Mach numbers, the compression fan probably merges to become a weak embedded shock wave which intersects the bow shock wave near the first inflection point. Experimental data (refs. 10 and 13) indicate that these embedded shock waves do form. On the basis of the present results, it cannot be determined whether or not this effect occurs since the present method smears embedded shock waves so that weak shocks cannot be distinguished from continuous com- pressions. It is clear that the continuous compression extends all the way to the bow shock wave for Moo = 2 because the flow is subsonic throughout the compreSSion region.

Results which show the effect of Mach number on the pressure distributions for the truncated cone are presented in figure 8. The free-stream Mach numbers which are treated include the four treated in figure 7 and infinity. The pressure is nondimensional- ized with its stagnation-point value and is plotted as a function of distance along the sur- face from the axis.

It is seen in the figure that the results correlate quite well on the face and on the upstream portion of the first shoulder. On the rearward part of the shoulder, a Mach number effect appears which becomes very pronounced on the conical portion of the body.

This effect is monotonic with the higher nondimensional pressures corresponding to the lower Mach numbers. Note that the curves representing the results for the lower Mach numbers converge as the second shoulder is approached. The curves between the junc- tion of the first shoulder and the cone and the first point on the cone where calculations are made are dashed because the present results do not indicate what is happening there.

Mach number effects are observed on the downstream portion of the second shoulder and the cylindrical afterbody, but they are small.

Some further applications of the present method are shown in figure 9. One of the problems to be treated is concerned with the blunt body at angle of attack. The bodies under consideration include those with sharp sonic corners since a means has been found to compute the cross flow at these corners. In principle, blunt bodies with no symmetry at all can be treated.

The second application involves the calculation of flow about delta wings with the shock attached only at the tip. It is hoped that the present techniques for treating sharp corners on blunt bodies can be extended to treat the flow at the sharp leading edges of these wings. It should be noted that for the cases of interest, the flow over these wings is always supersonic. Therefore, the governing equations for steady flow are hyperbolic everywhere in the flow field, and results for steady flow can be calculated directly using the distance from the tip rather than time as the marChing variable. This procedure has been used by Moretti (ref. 14), Babenko et al. (ref. 15), and others to calculate supersonic flow past pointed bodies.

REFERENCES 1. Sauerwein, Harry: A General Numerical Method of Characteristics. AIAA Paper No. 65-25, Jan. 1965.

2. Bohachevsky, Ihor 0.; and Rubin, Ephraim L.: A Direct Method for Computation of N onequilibrium Flows With Detached Shock Waves. AIAA J ., vol. 4, no. 4, Apr.

1966, pp. 600-607.

3. Harlow, Francis H.: The Particle-in-Cell Method for Numerical Solution of Problems in Fluid DynamiCS. Vol. XV of Proceedings of Symposia in Applied Mathematics, Amer. Math. Soc., 1963, pp. 269-288.

4. Gentry, Richard A.; Martin, Robert E.; and Daly, Bart J.: An Eulerian Differencing Method for Unsteady Compressible Flow Problems. J. Comput. Phys., vol. 1, no. 1, Aug. 1966, pp. 87-118.

5. Lax, Peter D.; and Wendroff, Burton: Difference Schemes for Hyperbolic Equations With High Order of Accuracy. Commun. Pure Appl. Math., vol. XVII, no. 3, Aug.

1964, pp. 381-398.

6. Burstein, Samuel Z.: Finite-Difference Calculations for Hydrodynamic Flows Con- taining Discontinuities. J. Comput. Phys., vol. 1, no. 2, Nov. 1966, pp. 198-222.

7. Godunov, S. K.; Prokopov, G. P.; and Zabrodin, A. V.: A Difference Scheme for Two- Dimensional Non-Steady Problems of Gas Dynamics and Calculation of Stream Flow With a Detached Shock Wave. UCRL-Trans-1060/L/, Lawrence Radiat. Lab., Univ.

of California, Mar. 26, 1965.

8. Masson, B. S.; Taylor, T. D.; and Foster, R. M.: Application of Godunov's Method to Blunt-Body Calculations. AIAA J., vol. 7, no. 4, Apr. 1969, pp. 694-698.

9. Moretti, Gino; and Abbett, Michael: A Time-Dependent Computational Method for Blunt Body Flows. AIAA J., vol. 4, no. 12, Dec. 1966, pp. 2136-2141.

10. Kendall, James M., Jr.: Experiments on SupersoniC Blunt-Body Flows. Progr.

Rep. No. 20-372 (Contract No. DA-04-495-0rd 18), Jet Propulsion Lab., California Inst. Technol., Feb. 27, 1959.

11. South, Jerry C., Jr.: Calculation of Axisymmetric Supersonic Flow Past Blunt Bodies With Sonic Corners, Including a Program Description and Listing. NASA TN D-4563, 1968.

12. Shifrin, E. G.: The Direct Problem of Plane Symmetrical Flow With Detached Shock Wave About a Smooth Convex Profile. Sov. Phys. - Doklady, vol. 12, no. 1, July 1967, pp. 4-6.

13. Hastings, S. M.; Persh, J.; and Redman, E. J.: Experimental Investigation of the Pressure Distribution on Axi-Symmetric Flat-Face Cone-Type Bodies at Super- sonic and Hypersonic Speeds. NAVORD Rep. 5659, U.S. Navy, Oct. 1, 1957.

14. Moretti, Gino: Inviscid Flowfield About a Pointed Cone at an Angle of Attack.

AIAA J., vol. 5, no. 4, Apr. 1967, pp. 789-791.

15. Babenko, K. 1.; Voskresenskiy, G. P.; Lyubimov, A. N.; and Rusanov, V. V.: Three- Dimensional Flow of Ideal Gas Past Smooth Bodies. NASA TT F-380, 1966.

--- - -- - - -

BLUNT-BODY FLOW -FI ELD STUDY TYPES OF COMPLICATIONS TREATED NONUNIFORM FREE- EMBEDDED SHOCK SONIC CORNERS STREAM PROPERTIES WAVES Figure 1 REGION OF COMPUTATION AND COORDINATE SYSTEM REFERENCE x*

LlNE0 /-r;~ ' ----

X~in).'" V SHOCK mini Figure 2 BICHARACTERISTICS AT CORNER dS\ = u+a ( dtlr Time Sc DISTANCE ALONG SURFACE Figure 3 FLOW PAST FLAT -FACE CYLI NDER Y = 1.4; Mro= 2 . 81 -- KENDALL (EXPER I MENT) 0, • PRESENT RESULTS SHOCK WAVE AND SONIC LINE PRESSURE DISTRIBUTION l.0o---or--O--O p Pro V2ro .5

o

S/Sc Figure 4 EFFECT OF NONUNIFORM FLOW ON SHOCK-WAVE AND SONIC-LINE SHAPES FOR BLUNTED CONE y=I.4; (Moo) =10 i E=0 . 03 .01 PARABOLIC VELOCITY DISTRIBUTION o Voo(rl = (Vco)i ~ - E (r;r b)2) t---~ poo(r) = CONSTANT

T

I--_~ Hoo(r) = CONSTANT

--L.---I.,I;~_~l

Figu re 5 EFFECT OF NONUNIFORM FLOW ON PRESSURE DISTRIBUTION FOR BLUNTED CONE y = 1.4 ; (Moo)ct.. ::: 10 --PRESENT RESULTS 1.0 o METHOD OF INTEGRAL RELATIONS -----NEWTON I AN .8 p .4 SONIC POINT .2 I

o .2 .4 .6 .8 1.0

atSc F igure 6 EFFECT OF MACH NUMBER ON SHOCK-WAVE AND SONIC-LINE SHAPES FOR TRUNCATED CONE y=IA 2 A. .

'< 45 ~ rc

1-. .- 1 '-- rc=0.25 Mc:o= 2.0 Figure 7 EFFECT OF MACH NUMBER ON PRESSURE DISTRIBUTION FOR TRUNCATED CONE y=1.4 --M =2 .0 CD 1.0r----_ 2.4 2 .6 2 .8 / .8 ro I I ......-- - - :: _-:-=:s--- I, _- ______ _ 11 /':--- .6 SON 1 C I /: 1/" ___ ----.

PRESSURE '-- ~ -1 11t -- - LEVEL \ ift' __ ....------ ~ / - .2 --1 I-SHOULDER -I \ f--SHOULDER \\ ::-- ~- .2 .4 o .6 Figure 8 FUTURE EXTENSIONS BLUNT BODY DELTA WI NG WITH SHOCK AT INCIDENCE ATTACHED ONLY AT TIP " .. '. .' . 0.' ~ .•. _ . ..

~

F igure 9 _J DISCUSSION FERNANDO L. FERNANDEZ, The Aerospace Corp.: I had a question, just out of curiosity. You didn't show many results for high Mach number, free-stream flow. Is there any difficulty that might occur with these methods as one gets to high Mach numbers?

BARNWELL: The answer is no, there are no difficulties. In general, the results presented are for Mach numbers of interest in airplane work.

LEONARD WALITT, Applied Theory, Inc.: This, I assume, is an explicit method. Do you have any stability criterion you have to satisfy when you integrate your finite difference equations?

BARNWELL: Yes, I do. The stability criterion used is the von Neumann condition. A linearized stability analysis would tell you that the time step is 0.707 times that determined from the CFL condition. Our time step is a little bit smaller to be safe.

WALITT: Because of this, how close can you put your zones to define imbedded shock surfaces?

BARNWELL: I don't understand.

WALITT: Well, in the area of imbedded shock waves you would have to have zones pretty close together. You indicated the flow field was smeared over a sh ock .

BARNWELL: Yes, the imbedded shock is smeared. At the bow shock waves, I use the method of characteristics so there is no smearing there.

WALITT: I just have one other comment. Do you really believe that methods like this are practical for viscous three-dimensional, time- dependent flows? You indicated in your last slide that you did. I have to disagree with you.

BARNWELL: I would use techniques like this to calculate the subsonic region.

WALITT: I think you may have to develop a new kind of computer for calculating viscous, time-dependent flow fields in three spatial dimensions.

BARNWELL: Well, I don't agree.

GINO MORETTI, Polytechnic Institute of Brooklyn: I am not answering you, of course, but Dr. Walitt. I would like to mention a blunt-body time- dependent program which I did about 3 years ago for Sandia Corporation, and which has been used since then by Sandia and a large number of other indus- tries. Nobody has complained that they need to buy a new computer. The ones who use the CDC-6600 can perform the computation in 6 minutes.

Now, since I have the microphone let me tell you that I like your presentation, but I do not believe that the equations which you used take care of imbedded shocks.

ROBERT CHARLES GUNNESS, JR., Boeing Company: I understand that some of the Lax-Wendroff type methods experience difficulties when the sonic line is crossed. Was this a problem?

BARNWELL: Yes, I did have trouble with stability. My problem did not arise at the sonic line but rather at the stagnation streamline. At both places the eigenvalues of the amplification matrix go to zero and neutral stability can occur. I had to put in a fourth order damping term to insure stability in the vicinity of the stagnation streamline.

GUNNESS: Will this appear in the paper?

BARNWELL: Yes.

EARLL MURMAN. Boeing Scientific Research Laboratories: Could you explain again why you chose the time~dependent method of characteristics for handling the boundary condition at the body, and do you think it is better than other methods for handling the boundary condition? I haven't seen the method before.

BARNWELL: I used it because it worked. Dr. Moretti has used this procedure before and has published several papers in which it is used.

MURMAN: Did Moretti use it for a solid body or did he just use it for shock waves?

BARNWELL: No, he used it at the body surface also.

One difference in his technique and mine is that he uses a one- dimensional characteristic and I use a bicharacteristic for the line along which the compatibility relation is integrated.

MORETTI: Yes, and no. In 4 years I changed my mind many times.

However, as a general rule, if the numerical scheme reflects the physical nature of the problem, then it is bound to work.

J

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AN ANALYSIS OF COANDA JET FLOWS By M. H. Y. Wei Air Vehicle Corporation and Victor R. Corsiglia Ames Research Center SUMMARY This work concerns the second-order theory of a static laminar Coanda jet flow around a circular cylinder with external flow entrainment. It is assumed that the jet is very thin in comparison with the cylinder radius; hence the matched asymptotic theory for high Reynolds number proposed by Van Dyke is used. Accordingly, the second-order theory is being formulated including the effects due to curvature and entrainment. The objective of this research is to predict the laminar separation point. Preliminary results indicate that the curvature and entrainment effects tend to promote separation.

INTRODUCTION It is well-known that jets, particularly two-dimensional jets, show a strong tendency to become attached to nearby solid surfaces. This phenomenon is usually associated with Coanda (fig. 1) who discovered the effect in his numerous experiments (refs. I and 2). An outline of the computational scheme appears in figure 2. Some applications are noted in figure 3 (e.g., cyclic circulation control of rotors, and jet flap wings). For laminar flow, the effect is also used with fluidic amplifiers.

For a better understanding of the effect, a comparison between the free jet and wall jet is made in figure 4 . Here a free jet emerging from a slot of a finite width h is shown. Downstream of the slot, the flow at the edge of the jet mixes with the surrounding air so that the width of the jet increases and the jet velocity decreases. If we replace the x-axis with a solid surface, the free jet becomes a wall jet. The velocity profile of the wall jet consists of an inner boundary layer for y < Ym and an outer free jet layer for y > Ym' Furthermore, if the pressure in the jet is slightly lower than the ambient pressure, the free jet will not deflect due to symmetry. However, for a wall jet, the pressure differences will cause The transition length from a parabolic velocity profile for the nozzle flow to the jet flow velocity profile is assumed to be small (of the order of 2 3 Re- / ).

-- 1

- -- ----- -- ------~-~ ---- --- --

the jet to deflect toward the wall. Particularly for a wall jet along a curved surface, the centrifugal force tends to make the jet pressure slightly lower than the ambient pressure and the jet will become strongly attached to the walL It is realized that the complexity of the problem makes analysis of this effect in a free stream very difficult. To gain fundamental understanding of the problem, the wall jet around a simple curved surface without any external flow should first be examined. The objective of this work is to develop a theory for Coanda jet flow around a circular cylinder for predicting the vari- ation of wall shear, the approximate separation point, and the external mass entrainment. The authors are grateful to Prof. Libby for his stimulating discussions concerning this objective.

SYMBOLS See also figure 5.

f dimensionless similarity function defined in Glauert's solution normalization constant in G1auert's solution f100 h slot width, ft pressure, Ib/ft P R cylinder radius, ft Re Reynolds number, dimensionless \ u, v tangential and radial velocities, ft/sec (u ' Uc defined in fig. 10) D

I

x, y rectangular coordinates, ft (Ym defined in fig. 4) y

I normalized y-coordinates, Y = y~

R \ 0 jet thickness, ft

I

Y --, defined as dimensionless n 6 3/ \ 6 angular coordinate, rad

I

\) kinematic viscosity, ft /sec \ p density, slugs/ft EQUATIONS AND ASSUMPTIONS As shown in figure 6, the equations to be considered are the incom~ pressible Navier-Stokes equations in cylindrical coordinates. According to - l_~ __ _ _ _ the Prandtl boundary-layer theory, the terms in the equations can be grouped into different orders (i.e., the first, second, and higher orders). It is noted that the equations are invariant by translation in the e coordinate.

Assumptions made for this analysis are: 1. Thin jet - the thickness of jet is small compared with cylinder radius.

2. Laminar flow - the Reynolds number of the flow based upon the cylinder radius is in the order of 3x l0 or less.

3. Incompressible flow.

METHOD OF SOLUTION For viscous flow at high Reynolds number, Professor Van Dyke at Stanford University has developed a matched asymptotic expansion theory (refs. 3 and 4). The problem is, then, to apply his method systematically to the Navier- Stokes equations. Accordingly, as shown in figure 7, we define an inner and an outer expansion for the tangential velocity. The outer equations take account of the external entrainment flow. The equations are Laplace equations subject to matching boundary conditions calculated from the inner equations.

The inner equations are boundary-layer equations. The normal coordinate y is stretched by a factor equal to the square root of the Reynolds number.

FIRST-ORDER INNER SOLUTION The first-order inner equations are determined to be Glauertts wall-jet equations (ref. 5) in which the jet is assumed to flow along a flat wall with~ out any normal and tangential pressure gradient. The velocity distribution of Glauert's solution across the jet is shown in figure 8. Because of the homo- geneous boundary conditions, Glauert's solution contains an undetermined normalization constant. This constant, as shown in reference 6, can be deter- mined by the product of jet momentum and jet volume flow. After the constant is determined, a unique relation between the jet thickness and the 8 coordinate is obtained: Re 8 (1) The streamlines of Glauert's solution, as shown in figure 9, indicate that the entrainment flow is perpendicular to the main jet flow. This imposes a boundary condition for the determination of the first-order outer solution, which must properly describe the upstream entrainment flow. The boundary

condition at y = a for the outer first-order stream function is

1 4 4f 8 / 1jJ(1)(~, = for 8 > 0 (2) e \=0 0 8 < 0 = for which yields a sink distribution of

V (l )(L 3

> for 8 0 (3) -f 1 00 \ 8 1 8 )y=O = R' for 8 < 0 along the cylinder surface.

SECOND-ORDER CORRECTIONS There is no separation from the first-order inner solution. Second~order inner solutions are needed to predict separation. It has been shown in ref- erences 3 and 4 that the second-order inner correction may be split into two parts, namely, curvature and displacement as noted in figure 10. Due to curva - ture, a normal pressure gradient is generated by the centrifugal force of the flow. This introduces a tangential pressure gradient in the tangential momen- tum equation. However, this term cancels with the other second~order terms in the tangential momentum equation. In addition, the changing of the arc length between the y coordinates also introduces a viscous shear term vy( a u/ay2) into the momentum equation.

To find the second-order inner correction due to entrainment, one must solve the first-order outer equation. The outer equation is given by the Laplace equation, and the solution must satisfy the boundary conditions, equa- tion (2) or (3); a sink distribution is thus required along the cylinder surface.

The solution can be found by the use of Poisson's formula for a circle (ref. 7). The outer stream function is seen to be bounded at large values of y. From the Poisson's integral, it i1 found that the external entrainment l flow induces a tangential velocity u at the outer edge of the boundary D layer.

0.504 0.25 +-+ 0.109 - 0.043 8 - 0.00078 ••• (4) 8 3 4 8 / This series expansion is an approximation for Poisson's solution at y = R for 8 < 2n .

The equations to account for the curvature effect are shown in fig- ure 11. The boundary conditions are homogeneous, but the equations are non- homogeneous. The solution to the second-order curvature equations is shown in figure 12 and indicates that the curvature effect tends to decrease the velocity near the wall and increase the velocity in the outer part of the layer.

To obtain the curvature effect, a resultant velocity profile can be formed by adding the second-order curvature correction to Glauert~s wall-jet solution. It is found that the velocity profile (fig. 13) becomes flatter with increasing e coordinate, indicating decrease in wall shear as e increases.

The equations to take account of the displacement effect are shown in figure 14. The tangential velocity at the outer edge of the boundary layer is given by equation (4). The solutions of the second-order displacement equa- tions corresponding to the first two terms in the series expansion (eq. (4)) are shown in figure 15. The displacement effect decreases the wall shear and tends to promote separation. A plot of the resultant velocity profile includ- ing the first six terms in the series expansion is shown in figure 16 indicating a significant reduction in wall shear.

UNIQUENESS OF THE SECOND-ORDER THEORY The above solutions of the problem are not unique. There may exist a set of infinite discrete eigensolutions, each of which satisfies the homo- geneous boundary conditions and exhibits exponential decay at infinity (refs. 8 and 9). However, in the second-order inner equations, there is an eigenfunction with eigenvalue equal to zero. This solution decays with the

e coordinate at the same rate as Glauert's solution; hence, one cannot simply

dismiss it by invoking the principle of minimum singularity (ref. 10). Thus in the formulation 9f the second-order theory, the need for deriving a unique condition to include the eigensolution becomes apparent. When the constant associated with the eigensolution is determined, there is a possibility that the wall shear predicted by the theory will be significantly changed. Pre~ sently we are trying to find a condition for determining the constant. Lindow and Greber (ref. 11) noted similar results in their study of the similarity solution of the second-order wall-jet equation.

CONCLUSIONS Based upon the work completed to date, the preliminary conclusions of this brief progress report may be summarized as follows (fig. 17): 1. The first-order solution is similar and gives no separation (this is the Glauert wall jet).

2. The curvature effect allows separation. However, the separation angle seems unreasonably large if the unknown constant is taken to be equal to zero.

3. The displacement effect decreases wall shear significantly, promoting separation.

4. The theory needs experimental verification. Tests in oil to observe the laminar separation angle are planned.

REFERENCES 1. Metral, A.: Sur un Phenomene de Deviation des Veines Fluides et ses Applications. Coanda Effect, Cabinet Technique due Ministere de l'Air (1938) .

2. Metral, A.; and Zerner, F.: L'Effect Coanda, Publication Scientifiques et Techniques du Minister de l'Air, No. 218 (1948).

3. Van Dyke, M. D.: Higher Approximations in Boundary Layer Theory. Part 1.

General Analysis. J'. Fluid Mech., vol. 14, 1962, pp. 161",177.

4. Van Dyke, M. D .: Higher Approximation in Boundary Layer Theory. Part 2.

Application to Leading Edges. J. Fluid Mech., vol. 14, 1962, pp. 48l~ 495.

5. Glauert, M. B.: The Wall Jet. J. Fluid Mech., Aug. 1956, pp. l6l~177.

6. Parks, E. K.; and Petersen, R. E.: Analysis of a Coanda Type Plow.

AlAA, vol. 6, no. 1, Jan. 1968.

7. Morse, Philip M.; and Feshbeah, Herman: Methods of Theoretical Physics.

McGraw-Hill, 1953, pp. 370-374.

8. Libby, P. A.; and Fox, H.: Some Perturbation Solutions in Laminar Boundary-Layer Theory. Part 1. The Momentum Equation. J. Pluid Mech., vol. 17, 1963, pp. 433-449.

9. Stewartson, K.: On Asymptotic Expansions in the Theory of Boundary Layers. J. Math. and Phys., vol. 36, 1957, pp. 173-191.

10. Van Dyke, M. D.: Perturbation Method in Fluid Mechanics. Academic, New York & London (1964), pp. 131-132.

11. Lindow, B. G.; and Greber, I.: Similarity Solution of a Laminar, Incompressible Jet Flowing Along a Curved Surface. AlAA, vol. 6, no. 7, July 1968.

COANDA JET FLOW Uro = 0 START OF JET LAMINAR ,." , >,/>,///" ",,- ' ....---- BOUNDAR Y LA Y ER '\ \ \ \ \

--t- SEPARAT ION

~~~~~~~~~~ ~ I \

\ \\

\ \ \ \ \ "- \ "- \ "- \ "- \ Figur e 1 TO PREDICT COANDA JET FLOW INNER REGION OUTER REGION (NEAR SURFACE) (AWAY FROM SURFAC E) ,----- -- ---" - -- - -----, ORDERED NAVIER - STOKES II II ( LAPLACE EQUATION ) II EQUATIONS . .

I I I 1st ORDER SOLUTION I I 1st ORDER I I BOUNDARY LAYER THEORY I I 1st ORDER~ SOLUTION GLAUERT SOLUT IO N I I /1 NO FREE STREAM, :/ 2nd ORDER SOLUTION I I INFLOW ALONG SURFACE I I 2nd ORDER I

I

I BOUNDARY LA Y ER THEORY I I I I

TOTAL 1 SOLUTION

I I I st ORDER SOLUTION I I + CURVATURE EF F ECT I I + ENTRAINMENT EFFECT I I L ______ _ ___ ~L __ ___ _ ___ ~ Figure 2 COANDA JET THEORY • APPLICATIONS • CIRCULATION CONTROL, JET DEFLECTION - TURBULENT FLOW • FLUIDIC CONTROLS - LAMINAR FLOW • OBJECTIVE • DEVELOP A THEORY TO PREDICT COANDA JET FLOW SEPARATION • APPROACH • FIRST PHASE - CONSIDER ONLY LAMINAR FLOW WITHOUT FREE STREAM ON A CIRCULAR CYLINDER • LATER PHASES - EXTEND THEORY TO INCLUDE TURBULENT FLOW, FREE STREAM, AND OTHER GEOMETRIES Fi gure 3 TWO-DIMENSIONAL FREE JET y

-----

HYPOTHETICAL ORIGIN OF THE JET

---

TWO-DIMENSIONAL WALL JET y

-------

HYPOTHETICAL ORIGIN OF THE JET Fi gure 4

l

NOTATION OF JET .... TANGENTIAL VELOCITY u....... .

. .. RADIAL VELOCITY v ..

. .. TANGENTIAL AND RADIAL COORDINATES e, Y ...

.. ... AMBIENT PRESSURE Pro ·· .. SURFACE PRESSURE AT THE CYLINDER Ps ·· ·· .

Figure 5 NAVIER - STOKES EQUATIONS • CONTINUITY +(V+Y~)'O ( B!+1Y.)

aB ay ay I sl ORDER 2nd ORDER • TANGENTIAL MOMENTUM V au u au a u) ( I a (Pip) vu 1/ au) ( dY + ~ as -1/ dY2 + R;y dB + ~ - Rry ay + 1st ORDER 2nd ORDER u I a u 2 av ) ( 1/ ~ - (~2 ~ + CR:Y)2;; ·0 HIGHER ORDER • NORMAL MOMENTUM 1/' KINEMATIC VISCOSITY P • DENSITY Figure 6 • ASSUMPTIONS

• TH IN JET (* < I) - BOUNDARY LAYER ANALYSIS IS APPLICABLE

• LAMINAR FLOW (Re = R"U < 3 x 10 ) • INCOMPRESSIBLE FLOW • METHOD OF SOLUTIONS • OUTER REGION (VALID FOR Y LARGE) LAPLACE EQUATION • INNER REGION (VALID FOR Y SMALL) NAVIER - STOKES EQUATIONS • EXPANSIONS • OUTER EXPANSION u = J... (J{I) (1.. 8) + J... 'U(2) ('1 8) + - __ .IRe R' Re R' • INNER EXPANSION u· u{O)('1..1Re 8) + J... u{l)(1 ..IRe 8) + - - - R ' ..IRe R ' Figu re 7 FIRST-ORDER INNER SOLUTION GLAUERT WALL JET EQUATIONS : au{O) av{O) ap{O) ---ae+----ay=0; ay=O u{O) autO) +V{O) au{O) = " a u{O) a8 ay ay2 y Ymax .2 o Fi gureS FIRST-ORDER INNER STREAMLINES FOR WALL JET ENTRAINED STREAMLINES y./Re R DIVIDING STREAMLINE h _ SLOT [ 2 WIDTH LL~-L ____ L- __ -L __ ~ 2 3 4 o 8 ~ RADIANS F ig u re 9 SECOND-ORDER CORRECTIONS U{/) = U(~) + U(~ • CURVATURE EFFECTS: (U(~)) • CHANGING ARC LENGTH BETWEEN Y COORDINATES • NORMAL MOMENTUM EQUATION IS INCLUDED • INDUCED TANGENTIAL PRESSURE GRADIENT TERMS DROP OUT • DISPLACEMENT EFFECT: (U\~)) • UPSTREAM ENTRAINMENT: \ NONZERO TANGENTIAL VELOCITY AT OUTER EDGE OF JET Fi gu re 10 SECOND-ORDER CURVATURE EQUATIONS U(I) c • CONTINUITY elu(l) elv(1) elv(O) _c + _c + V(O) + Y - = 0 el8 elY elY • TANGENTIAL MOMENTUM elU(!) elu(O) elU(~} elU(O) el ug) U(O) _c + u(l)- + V(O) - + vO) --- -- = cl8 c cl8 elY elY cly2 elp(1) el clU(O) (0) el (0)

- ---ae + elY Y 1fT -V elY Yu

• NORMAL MOMENTUM elpO) (0) 2

-w- =(u )

Y =..,.IRe y/R Figure 11 VELOCITY CORRECTIONS DUE TO CURVATURE U (I) c 204 - PROFILE INDEPENDENT 2.0 - OF 8 1.6 - Y Ymax 1.2 - .8 - 04- -.4 -.2 o .2 .4 .6 .8 1.0 u(l)/u(l) c / c max F igure 12

I

I

L

- - - - - - - - - -- - - TO OBTAIN CURVATURE EFFECT U=[U(Ol(7)l+ fl<Xl/Re / U~1(7)]/8112 3 4 (7) = Y /8 / , fl<Xl= NORMALIZATION CONSTANT) U(7), 8>8h) u(7),8 h) .5 o .1 .2 .3 .4 U Figure 13 SECOND-ORDER DISPLACEMENT EFFECT u(/) D • OBTAIN INFLOW VELOCITY DISTRIBUT/.ON AT OUTER EDGE OF JET BY SOLVING OUTER FLOW EQUATION 1 3/4 U(ci (<Xl, 8) = 0.504/8 + 0 . 25/8 + 0.109 - 0.043 8 ......

• SOLVE aut I) av(/) _0 +_0 =0 a8 ay (I) (0) 2 (I) ~ ((I) (01) vIOl oUo V(I) ~ _ 0 Uo 08 u 0 u + oY + 0 ay - oy2 • BOUNDARY CONDITIONS u~) (0, 8) = 0 3 / 4 (I) (81- 8 8 8 u 0 CD, - 0.504/ +0.25/ +0 . 109- 0.043 ......

Figure 14 VELOCITY CORRECTIONS DUE TO DISPLACEMENT 3/4 U~) = (f,(7])18 + f2 (7])18 + f3(7]) + ••• ) 8 - 6- 7J - 4- .8 1.6 Figure 15 RESULTANT VELOCITY PROFILE (u) 1/2 U = 4 {f(O) (7J) + -'-f(l) (7J. )}/8 .jRe fico fO(7J) ONLY (GLAUERT WALL JET) .2 .4 .6 o Figure 16 SEPARATION POINT • FIRST-ORDER SOLUTION IS SIMILAR, i e NO SEPARATION • CURVATURE EFFECT ALLOWS SEPARATION, HOWEVER, SEPARATION ANGLE SEEMS UNREASONABLY LARGE • DISPLACEMENT EFFECT DECREASES WALL SHEAR SIGNIFICANTLY, PROMOTING SEPARATION • EXPERIMENTAL VERIFICATION TESTS IN OIL Figure 17 DISCUSSION STANLEY G. RUBIN, Polytechnic Institute of Brooklyn: I just want to ask you a question about the entrained streamline that you show on Slide 9 and a little bit on Slide 10. It indicates that the streamline flow has negative slope and all streamlines go away from the origin of the jet.

Now, a while ago I had a short note on a free jet expansion.

WEI: Yes.

RUBIN: In that particular case a turnaround occurs indicating a sink- like action of the potential flow. I'm just curious as to this effect in your case.

WEI: Yes, that's right. I tried to find the solution first for the plane wall jet. It did turn around, very similar to Rubin's work on the free jet. However, on a cylinder, you will find that the stream function for entrainment flow is bonded at infinity. Now, Rubin's solution for the plane free jet is not bonded, and it is over here that we are going to get a dif- ference between the slopes of the entrainment streamlines.

Furthermore, on the cylinder surface you will find that the inflow veloc- ity distribution has a reversal of slope. (Drawing on board.) The inflow velocity is very large near the jet origin, directing from left to right.

This inflow velocity diminishes to zero tentatively assumed at here and grows in reverse direction over the rest of surface. Of course, at the reversal point, the tangential velocity is equal to zero and the radial velocity is not equal to zero. It seems to me that this kind of entrainment flow is very reasonable because you would expect that some place near the point of reversal the jet peels off from the cylinder surface.

RUBIN: Down there, but I am really concerned with where the flow approaches the origin.

WEI: Yes, what you are concerned with, kind of going like that. (Drawing on board) RUBIN: That's only on a free jet?

WEI: This is correct. When you get into wall jet on cylinder, you will find that the slope of the entrainment flow is different from the plane free jet.

RUBIN: Let me just ask one additional question. Since your initial solution and all your subsequent solutions are based on similarity, have you investigated what the effect of the initial conditions would be, and the pos- sibility of having eigensolutions dependent on the initial conditions1 L WEI: Yes, there are eigensolutions. If the eigenvalues for the eigen- solutions are not equal to zero, you can invoke Van Dyke's minimum singularity principle that the eigensolutions belong to higher orders. How- This even, as I mentioned, there is an eigensolution with zero eigenvalue.

I solution decays at the same rate as the wall-jet solution. That is why said that I found a solution which ended with an undetermined constant. I still don't know how to determine the constant associated with the eigensolution.

RUBIN: That's the solution you were referring to?

WEI: Yes, that's right, a solution corresponding to zero eigenvalue.

For the plane free jet, there is no eigensolution with zero eigenvalue.

RUBIN: I don't know if you are familiar with this, but do you know of Professor Libby's work on this subject?

W. J. McCROSKEY, Army Aeronautical Laboratory: I am also hung up on some of the details of the entrainment. Is it not correct that you have assumed that all of the entrainment in the outer flow is radially inward?

WEI: Yes.

McCROSKEY: This is .an assumption which, in the first place, I wonder how good an assumption this is, on what basis you can make that, and secondly, if you make this assumption it sort of looks to me like you would get into trouble. There must be some kind of singularity or very funny behavior just upstream of 8 = Bh.

WEI: Yes, that's correct. The method assumes a radially inward inflow on the cylinder surface. Professor Van Dyke has worked out several other cases for the entrainment effect. One is the round jet. For the round jet, the zero-order solution shows that the entrainment flow is perpendicular to the jet flow. He used this as a boundary condition to calculate the outer solution. From the inner and outer solutions he constructed a composite stream function which is valid for all regions. The round jet ' has an exact solution. The exact solution was then used to check the composite solution.

It turned out very close. So this is one indication that the method of matched asymptotic expansion seems to be very fruitful in treating this kind of problem. Of course, there are other illustrative examples. All I know is that they check very well with the exact solutions.

Referring to your second question, the velocities are indeed all singular at the jet origin. The entrainment effect introduces singular solutions of higher order than the wall-jet solution.

McCROSKEY: You're just throwing away all the regions upstream of this, in terms of the entrainment source.

WEI: Oh, you mean what is the upstream boundary condition on the stream function?

McCROSKEY: You have some entrainment from upstream.

WEI: The boundary condition is that the stream function is equal to ze r o for the 6 -coordinate less than zero. This is for a plane wall jet, allowing some entrainment from upstream. For the cylinder, we only have to conside r the boundary condition on the 6 -coordinate between zero and 2n .

--- - --- ------ --- ~ -- - -- - -- - -- - -- --~~---- ~ A NEW METHOD FOR CALCULATING NEAR AND FAR FIELD PRESSURE ABOUT ARBITRARY CONFIGURATIONS By Frank A. Woodward Aerophysics Research Corporation Lynn W. Hunton and Anthony R. Gross Ames Research Center SUMMARY A new method of calculating pressure signatures and associated shock wave patterns in the near and far field about arbitrary wing-body combinations is described herein and illustrated with several examples.

The principal difference between this method and the currently accepted theory of G. B. Whitham is shown to be in the aerodynamic representation used.

Both methods use a linear distribution of singularities to represent a body of revolution, but the new method includes additional planar singularity distri- butions on the surfaces of wing-body combinations to improve the aerodynamic representation in the near field. In the far field, this spatial distribution of singularities gives the same result as Whitham's equivalent body of revolution in each azimuthal plane.

Examples of pressure signatures are presented only for bodies of revolution pending completion of the wing-body analysis computer program.

Good correlation between theory and experiment is shown by these examples.

The new method is expected to overcome some of the present limitations of the Whitham theory and to provide improved near field pressure signature estimates for arbitrary wing-body configurations.

INTRODUCTION Calculations of the pressure signature and shock wave pattern in the field surrounding arbitrary airplane configurations at supersonic speed have been based for a number of years almost exclusively on a theory presented by G. B. Whitham in 1952 (ref. 1). Whitham's theory depends on the smooth slender-body assumption in which the airplane volume and lift are represented by separate equivalent line source singularity distributions along the body axis. The predictions of this theory have been verified by extensive wind- tunnel and flight-test measurements and are generally accepted as a basis for estimating sonic boom overpressures.

Recent studies (ref. 2) have indicated a number of limitations to the Whitham theory in areas of current interest, for example, in the analysis of (a) Nonslender configurations (b) Vertical displacement of wing (high wing, dihedral) (c) Flow field definition very near the vehicle Cd) High Mach numbers In an attempt to alleviate these deficiences, the method described here has been formulated for calculating pressures and velocities to first order throughout the entire flow field about arbitrary wing-body configurations.

The method w ill be illustrated by several examples.

NOMENCLATURE pressure coefficient first-order correction function (Whitham) (y + l)M 4/1:2 S 3/2 k 00 00 total number of singularities K Mach number M first-order correction function (present method) N r radial coordinate

s singularity strength

u dimensionless perturbation velocity in x direction v dimensionless perturbation velocity in y direction x,y,z Cartesian coordinates origin of first-order singularity a flow inclination angle y ratio of specific heat for air cone half-angle Mach angle

e cylindrical coordinate

f I

L

Subscripts free-stream condition singularity type j singularity number k r radial direction ANALYSIS The procedure presented herein is based on a recently developed distributed singularity method described by R. L. Carmichael (ref. 3). The configuration is represented by a large number of aerodynamic singularities distributed along the body axis and on panels located on the wing and body surfaces. The panel subdivision of a typical wing-body combination is illus- trated on figure 1. The six different types of singularities used in the analysis to simulate volume, camber, incidence, and interference effects are listed on the figure. Since the assumptions underlying this method and the procedures used in determining the potential functions and velocity components are described in reference 3, these details will not be repeated here. Com- pared with the Whitham theory, the method introduces two important analytical advantages. First, the spatial distribution of singularities gives an improved aerodynamic representation in the near field while retaining the correct asymptotic form in the far field. Second, the surface pressures, forces, and moments acting on the configuration are obtained automatically in addition to the pressure signatures in the field, thus greatly facilitating the overall aerodynamic analysis.

The first step in the pressure signature calculations is the determina- tion of the characteristic lines in the field. Linearized theory gives the result that the disturbance field from each singularity is contained within the Mach cone from the singularity origin since the disturbances are assumed to have zero strength and to propagate into the field at the speed of sound of the fluid at rest. The slope of the characteristic lines of an outgoing wave in this "zero order" flow field is given by dx dr - cot ~oo = Boo where ~oo is the Mach angle of the undisturbed flow.

The theory of characteristics gives the exact slope of the outgoing characteristic lines as dx cot ~ - tan a dr - cot(ll + a) = 1 + cot ~ tan a 21 7

j

where ~ is the local Mach angle, and a is the local inclination of the flow measured from the free-stream direction. Following Whitham, a first-order approximation to the slope of the characteristics may be obtained by expanding the exact equation for small values of B v , where B = cot ~ r and vr = tan a (ratio of the radial to free-stream veloc i ties). If, in addi- tion, a linear relationship is assumed between the axial velocity ratio u and the local Mach number M, such that and then, This approximation to the slope is accurate only if S Vr is small. An alternate form of this equation is obtained, valid for higher Mach numbers, if the exact equation is inverted and expanded for small va lues of v r/ B.

Then, M 4 M 2 dr I y + I 00 00 ::; -- u + (v + Boo u) r dx 2 B 3 2 Boo B oo 00 The first equation is referred to as the low Mach number expansion; the second, as the high Mach number expansion. The Whitham theory always uses the low Mach number expansion. This may introduce s i gn i ficant errors if the theory is applied at high Mach numbers. The present theory, on the other hand, may use either the low or the high Mach number expansion.

The equations of the first-order characteristic lines are obtained by integrating the slope equations outward from the body surface along the zero order characteristics using the linear theory expressions for u and v .

r The integration is performed in planes of constant e for each of the singularities used in the representation, as illustrated on figure 2.

The low Mach number expansion gives an axial shift ~x to the zero order characteristic line, while the high Mach number expansion gives a corresponding radial shift ~r. Because of the nature of the line integrals being evaluated, it is found that ~x = -B ~r. For convenience in interpret- ing the results, an equivalent axial shift ~Xl can be defined in the high Mach number case in terms of ~r and the local slope of the characteristic line. In compression regions, ~x is always greater than ~Xl, as indicated on the figure.

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_ ._ - - - - - - - - - - -- - - - -- - - - - - - - - -- - - - -- - - - - -- - --

In the Whitham theory, the axial shift is given by the simple functional relationship where the F function depends only on the equivalent area distribution in azimuthal planes. The present theory does not permit this convenient separa- tion of variables. Instead, the axial shift is expressed in terms of six near field N-functions, one for each of the six types of singularities. For example, where N· is the axial shift calculated for the kth singularity of type . S . Jk h h f h k h . 1· d K . h 1 J, k 1S t e strengt 0 t e t slngu ar1ty, an j 1S t e tota number of singularities of type j used in the representation.

The first term of the asymptotic expansion of the N functions for large radial distance can be expressed exactly in the form of the Whitham F function . Thus, either theory will give the same results in the far field for a given distr i bution of singularities.

The pressure signatures in the field are now constructed by relocating the linear theory pressures along the first-order characteristic lines rather than the zero order Mach lines. The procedure is illustrated on figure 3 for a 15° half-angle cone at Mach 3. The zero order pressure signature for r = 0.5 rises rapidly from zero at the Mach line and reaches its surface value on the cone. The first-order pressure signature is obtained by apply- ing the appropriate axial shift 6x(x ,r) to the zero order signature for o each point on the curve. The forward distortion of the signature indicates a compression region containing overlapping characteristic lines. The envelope of the characteristics defines a limit cone corresponding to the most forward extent of the pressure signature.

The forward curvature of the signature between the Mach line and the limit line introduces a physically unrealistic solution. Instead, a shock wave will occur in this region which will provide a discontinuous pressure rise from zero to a value on the upper limb of the pressure curve. The shock wave is located by applying the principle that the shock wave bisects the angle formed between the upstream and downstream characteristic lines at that point. Within the approximations of the theory, this is equivalent to locating the shock wave so that the area under the distorted pressure signature is preserved, as indicated on the figure.

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EXAMPLES The shock wave angle for a 150 cone is presented as a function of Mach number in figure 4. The results of the present first-order near field theory are compared with Whitham's first-order far field theory and a recently published second-order far field theory given by Caughey (ref. 4). The present theory is seen to agree well with the Ames Research Center Cone Tables (ref. 5) over the entire Mach number range. It should be noted that the present theory is limited to Mach numbers below Mach tangency, or 3.86 in this example.

The effect of distance on the form of the pressure signature produced by a 150 half-angle cone-cylinder at Mach 1.7 is shown on figure 5. The com- pression waves from the cone move forward to form the front shock, as pre- viously described. The expansion fan from the shoulder extends rapidly out into the field, modifying the pressure signature significantly as it over- takes the compression region. The near field signature represents an inter- mediate stage in this process resulting in a relatively flat-topped initial pressure distribution. The far field signature is first formed when the first expansion wave from the shoulder reaches the shock. This results in a typical N- wave pressure distribution which persists for all greater dis- tances away from the body. The expansion wave is terminated by a weak rear shock wave in this example, followed by a trailing region of negative pres- sure. The far field pressure signature is not complete until all the charac- teristic lines emanating behind the shoulder are absorbed into the rear shock.

At this point, the front and rear shocks become equal in strength, and the pressure behind the rear shock returns to the free-stream value.

A comparison of the theory with experiment for a 12.75 half-angle cone- cylinder at a distance of 80 cone lengths is given on figure 6 for M = 1.69.

The present method is seen to give an accurate representation of the signature, especially in the expansion region between the shocks.

In figure 7, results are shown for a blunt-nosed body of revolution.

Although the present method of analysis is restricted to bodies with nose angles that lie within the Mach cone, the pressure signatures of blunt bodies can be approximated by adding a conical extension to the nose that lies just inside the Mach cone and is tangent to the body surface. In this example the pressure signature is calculated at a distance of 10 nose lengths at M = 2.01.

The pressures in the expansion region agree well with experiment, but the theory overestimates the pressure jump across the front shock. Because of the extreme pressure gradient behind the nose in this example, a small error in shock wave location can cause a large error in the pressure jump.

CONCLUDING REMARKS A new method for calculating the pressure signatures around a wing-body combination in the near and far field has been developed and programmed for the digital computer. When complete the new method is expected to -overcome some of the present limitations of the Whitham theory and provide improved near field pressure signature estimates for arbitrary configurations.

Examples of pressure signatures calculated for bodies of revolution show good agreement between theory and experiment.

REFERENCES 1. Whitham, G. B.: The Flow Pattern of a Supersonic Projectile. Commun.

Pure App1. Math., vol. 5, 1952, pp. 301-348.

2. Hunton, Lynn W.: Current Research in Sonic Boom. Second Conference on Sonic Boom Research. NASA SP-180, 1968, pp. 57-66.

3. Carmichael, Ralph L.; Castellano, Charles R.; and Chen, Chuan F.: The Use of Finite Element Methods for Predicting the Aerodynamics of Wing-Body Combinations. NASA Symposium on Analytic Methods in Aircraft Aerodynamics, October 28-30, 1969.

4. Caughey, David A.: Second Order Wave Structure in Supersonic Flows.

NASA CR-1438, September 1969.

5. Ames Research Staff: Equations, Tables, and Charts for Compressible Flow. NACA Rep. 1135, 1953.

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.------1 AERODYNAMIC REPRESENTATION x BODY VOLUME • LINEAR SOURCES • QUADRATIC SOURCES BODY CAMBER AND INCIDENCE • LINEAR DOUBLETS • QUADRATIC DOUBLETS WING THICKNESS • CONSTANT SOURCE PANELS WING CAMBER, INCIDENCE AND WING-BODY INTERFERENCE • CONSTANT PRESSURE PANELS Figure 1 I

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LOCATION OF APPROXIMATE CHARACTERISTICS

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LOW MACH - __ .. / P (x, r,8) P (x, r,8) I I /

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8 r HIGH MACH / / / X = Xo + {3a:J r

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i I ~----~------------~---X ~~---------- - y Xo J LOW MACH x - Xo I r=--+ LH HIGH MA CH /3a:J I r Figure 2 I

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CONSTRUCTION OF PRESSURE SIGNATURE M=3 15° CONE r = 0.5 SURFACE PRESSURE

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ZERO ORDER C p PRESSURE PRESSURE SIGNATURE SIGNATURE DISPLACED BY 6x 1.00 I I SHOCK WAVE LOCATED BY .75 AREA BALANCE .50 .25 \ 0 .5 1.0 1.5 2.0 X Figure 3 SHOCK WAVE ANGLE ON 15 CONE 55 - SHOCK WAVE~~ WHITHAM ~ (I st ORDER FAR FIELD)

~

45\~ "

\~~ ./

8 • deg - ,,~ - _____ / s ""'~ CONE TABLES 35 - ',~:::-....

"O, ~ PRESENT METHOD

'~ ~:::::-..- (I st ORDER NEAR FIELD)

/ ... '- ~ :::--------

25 - CAUGHEY .......... - --- (2nd ORDER FAR FIELD) - - - - _ - MACH ANGLE / ------_ I I I I 15 L 3.8 3.0 3.4 2.6 1.8 2.2 1.4 Moo Figure 4 EFFECT OF DISTANCE ON THE SIGNATURE M = 1 .7 15° CONE FAR FI ELD SIGNATURE NEAR FIELD SIGNATURE x Figu re 5

COMPARISON OF THEORY a EXPERIMENT - POINTED BODY

M = 1.69 . 012 - r / l = 80 . 008 - - PRESENT THEORY o EXPERIMENT . 004 - o t:.p P - . 004 - I I I I - . 008 L I I 4 6 o 2 -6 - 4 -2 t:.x/Z Figure 6

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COMPARISON OF THEORY a EXPERIMENT - BLUNT BODY

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F z

·1 .06 - M = 2.01 kXl/4 o = r . 04 - T = 10 --- PRESENT THEOR Y .02 -

6: (~t4 0 EXPERIMENT

I I I - . 02 L - .8 -.4 .4 ~ X (~r / 4 Figu re 7 DISCUSSION A. R. GEORGE, Cornell University: I'm not sure I understand. Does your theory use an expansion of the exact characteristics for large SVr?

WOODWARD: Well, we have this option and have found in the studies done so far that above about a Mach number of 2 that the high Mach number expansion seems to give an improvement in the location of the characteristic lines.

GEORGE: This would only be valid in the near field because certainly v r decreases as you move out.

WOODWARD: This is right, but we feel that the far-field shock location is dependent on its path through the near field. We are trying to improve the approximation in the near field to better understand the far-field signature.

GEORGE: How, do you switch from one to the other?

WOODWARD: No, the method is valid all the way out into the far field.

In fact, it will give you the Whitham theory in the far field as the first term of the expansion in the normal way. It is a continuous solution all the way out.

GEORGE: One other question. For large SVr I wouldn't expect the area balance-shock relation to hold. Have you done any work on that? As far as I know, it has never been worked out.

WOODWARD: That's right. We have had some difficulties with it, but at the moment we are basing our shock location on the area balance technique.

This is one area in which we probably will be doing more work.

HARRY W. CARLSON, NASA Langley Research Center: Frank, as you know, the Whitham approach has been quite useful for airplane configuration work because you can take a quite complex airplane and reduce it to an equivalent body of revolution, get the F function and then calculate the signature from this.

We have observed that it gives good results for moderate supersonic speeds, but there is some deterioration for higher supersonic speeds, say at Mach 3 and above, and particularly in the near field.

Now, have you given any thought to using the approach you have for finding the slope of these characteristic lines, to use these as a correction to the Whitham method, employing the same technique for describing the F function, but using this as a way of finding a new aging factor or a new way of calculating the advance of one characteristic relative to another?

WOODWARD: Yes, we have. As a matter of fact in the development of our displacement functions we have taken the far field limit of the disturbance from each one of the singularities, and this, in a sense, is the Whitham F function. As a matter of fact, they all show the vr behavior that is char- acteristic of the far field solution. So with that in mind, I think that it --~- - ----~~~ will be possible to tie this together and come up with a simple program for calculating the far field F functions, perhaps correcting them slightly, too, as you mentioned.

CARLSON: Yes, I think that this is a good approach because I believe the development of a system such as you have outlined here for handling complete airplane configurations is going to be very complex, and perhaps a long time off, but provisions for correction to the method of Whitham would be one that we could start employing almost immediately.

ROBERT L. TRIMPI, NASA Langley Research Center: I missed it, but where do you switch over from your high Mach number to your low Mach number approxima ti on?

WOODWARD: Well, it depends on the magnitude of the product of B times the radial velocity. At the present stage of development it is really a choice that you make at the start of your calculations whether you use one or the other, but it can be checked by evaluating the magnitude of this product.

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PROPAGATION OF SONIC BOOM THROUGH A STRATIFIED ATMOSPHERE By Wallace D. Hayes Princeton University and Harry L. Runyan, Jr.

Langley Research Center SUMMARY A method for predicting the sonic -boom signature due to a maneuvering aircr aft has been presented. This method includes the effects of a variable, stratified atmosphere.

The method is based on geometric acoustic principles and accounts for the use of a non- linearity to account for the weak shock field. The analytical results have been compared with experimental data and excellent agreement has been shown, particularly with regard to signature length.

INTRODUCTION Sonic booms have become of prime importance in the design and operation of super- sonic aircraft. A need has been felt for a comprehensive analysis and algorithm, real- ized in a practicable computer program, which would provide more realistic calculations for sonic-boom signatures in the atmosphere. This paper will present a broad outline of such an analysis; the details of a full analysis, including the computer program, are given in reference 1.

Earlier algorithms for sonic boom have used various simplifying assumptions. A basic aim of the present algorithm has been to avoid these assumptions as far as possible and thus provide a more general treatment of the problem. Thus, the present algorithm includes the following features: (1) The inclusion of maneuvering aircraft in calculation of a sonic-boom pressure (2) An appropriate ray-tube-area calculation based on linear geometric acoustics (3) Results in the form of complete signatures, without far-field assumptions, obtained through the use of an "age" variable in the calculation of nonlinear effects The present procedure assumes a horizontally stratified atmosphere with horizontal winds. This limiting assumption is of great practical interest and considerably simpli- fies the calculation. However, the effects of wind turbulence have not been considered.

The analysis is largely a rational synthesis of existing theories described in the literature and includes some new theoretical development. A principal new theoretical development is in the calculation of ray-tube area. The analysis is also new in the care- ful piecing together of a number of calculations, principally in the relation of the wave system and rays issuing from the aircraft stratified atmosphere with winds. This rela- tion requires the consideration of a Galilean transformation connecting a local coordinate system with the fixed coordinate system.

Figure 1 illustrates the geometry of the problem. At any given instant, t = t , and

aircraft is in one position, and associated with the aircraft is a shock-wave system, which near the aircraft is approximately conical and which extends down to the ground; an observer senses the shock wave as a loud noise. The shock wave, however, was created

at an earlier time, t = to, and the disturbances travel approximately along rays. The

key to the present method is the correct linear analysis of propagation in the atmosphere by using the method of geometric acoustics, analogous to geometric optics. Therefore, the following analysis can be divided into a number of topics, including (1) ray tracing, (2) Blokhintsev invariant, (3) determination of the ray-tube areas, (4) flow near the air- craft, (5) definition of an age variable, (6) determination of shock shape and magnitude, and, finally (7) a comparison of theory and experiment.

SYMBOLS A horizontal component of area of ray tube A area vector of ray tube normal to rays

An area of ray tube normal to rays

a speed of sound a initial speed of sound o

c ray velocity

c velocity of propagation of wave normal to wave front (eq. (1» n I j I

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L _________ _

-------- ------_ - _____ J integral defined by equation (9) F integral defined by equation (9) F· integrals defined in equation (8) i horizontal unit vector Mach number M n unit vector normal to wave front ii' horizontal component of n p pressure q perturbation velocity r radius radius vector horizontal component of r

s area distribution of equivalent body of revolution (primes refer to derivatives)

backward facing distance from normal to wave front temperature t time time along aircraft trajectory to time of initiation of disturbance t1 time at which disturbances form a shock u(z) horizontal wind velocity as a function of altitude u wind vector velocity ~ horizontal wind component in x ,y 1 coordinate system Ut initial horizontal wind velocity component in x,y coordinate system o ~, ~ horizontal wind velocity component V aircraft velocity V E parameter defined by equation (7) x ,y,z fixed coordinate system; east, north, and above ground, respectively x ,y l' -z1 coordinate system alined with wave normal y ratio of specific heats aircraft climb angle

e angle from horizon of ray

initial angle from horizon of ray Mach angle, sin- ~ heading angles linear phase

l_

~ 1 actual phase P air density Po initial air density <P velocity potential <Px derivative of velocity potential with respect to x cP azimuth angle (see fig. 2) CP r azimuth angle of wave normal relative to aircraft T age variable defined by equation (11) DISCUSSION Ray Tracing The first three topics are concerned with linear geometric acoustics. Ray tracing is tracing of the waves as they propagate through the stratified atmosphere. In geometric acoustics, the signal essentially follows wave fronts as given by

c = a + ii . ii (1)

n and as illustrated in the following sketch:

where a is the speed of sound, e is the angle from the horizon of the ray, ii. is the

unit vector normal to the wave front, and IT is the wind vector velocity .

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The ray velocity may then be given by df - - - (2) -==c==an+u dt Equation (2) could be integrated for the rays if the value of ii was known, but this value, in general, is not known. An expression relating ii to the changes in the speed of sound and the wind velocity is dii ( -) - - r::- - ( -) .;1 (3) - == Va + Vu . n - nLn . Va + n· Vu • ~ dt The simultaneous solution of equations (2) and (3) is required, and these equations consti- tute a fifth-order system.

In the present method, if a stratified atmosphere is assumed, so that the wind is horizontal and the temperatures and density are also known in stratified layers, then equation (3) can be reduced to a form of Snell's law of geometrical optics. For use herein, Snell's law then appears as

ii' == 1 cos e == Constant along each ray

(4) C C n n where ii' is the horizontal component of ii.

The ray equations, obtained from the simultaneous solution of equations (2) and (4), then appe ar as dx _ a cos e sin !J - U x d(-z)- asine ~ == a cos e cos !J - uy (5)

d( -z) a sin e

dt _ 1

d(-z) - a sin e

where lix and Uy are the horizontal wind components and !J is the heading angle.

The vertical distance z has been chosen as the independent parameter in place of t, since most of the input data are known as a function of altitude. These equations may be solved by quadrature, and in this quadrature Snell's law can be applied.

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Now, one remark on the minus sign in front of the wind velocity components, U x and lly. It has been said that "the north wind doth blow and we shall have snow." There is a meteorological convention that a wind that blows in the direction of south is called a north wind, so uy is positive for a north wind and that is the only reason a minus sign is used on these components.

Blokhintsev Invariant In the preceding section, the path of the acoustic waves was traced, but none of the important parameters, such as the pressure variation down the rays, have been related.

For a stationary atmosphere, Rayleigh derived an invariant as given below or

where p is the air denSity, q is the perturbation velocity, An is the area of the ray

tube normal to the rays, and a is the speed of sound. For a moving atmosphere, Blokhintsev derived an analogous invariant expression pq cn - 2

-- c . A = pq AC sin e (6)

n a where A is the horizontal component of the area of the ray tube. Using the horizontal component of the area is more natural, since the problem is defined in terms of a hori- zontally stratified atmosphere.

Finally, for later use, a new invariant parameter V is introduced and defined as E or (7) 1/2

V = i(pa A sin e cos e )

E Ray-Tube Area In equation (7), the ray-tube area is of fundamental importance, and an expression for the area must be determined. In figure 2 is depicted a two-parameter family of rays required to define a ray-tube area. One parameter is the azimuth angle cf> which is measured with respect to the aircraft path in the direction the ray is considered to go.

The other is the time t measured by a clock on the aircraft at the instant the ray is emitted. The precise definition of this ray-tube area may be given in terms of a Jacobian taken at constant altitude, so that it is a Jacobian of the horizontal coordinates x and y, taken with respect to the two ray-tube parameters, t and <p . The final expression for ray-tube area for a maneuvering aircraft is given as

sin Yo ) V sin <p cos tJ. cos Yo J Sco

A = 1 + (12 - Ut 11) - 11 - sin tJ. sin 8 0 cos 8 S cf> 0 0

~

V sin cf> cos tJ. cos Yo ~ Sv

- cos 8 12 S <p o sco S Sco S ) ( 2) (8) + - ~ - - -X... 1113 - 12 ( s ta S <p S <p S ta where

I (-z) = S cos 8 d(-z)

1 -z a2sin3 8 o u cos 8 d( -z) t a sin 8 c n and where Yo represents the aircraft climb angle (y in ref. 1),

Co = -- 8' and

cos If the aircraft were in straight and level flight, then locally, at least, the ray-tube area would simply be proportional to the distance away from the flight path, so the ray-tube area would be expected to have terms that are proportional to a quadrature, an integral taken from the aircraft downward. But now, beeause of maneuvering in which the state of the aircraft changes as ta changes, the orientation of these two rays at a later time may be different from the orientation of the two rays at an earlier time.

After quite a bit of algebra, a ray-tube area in the form of equation (8) is derived in terms of the three quadratures 1 , 1 , and 13' These are taken from the aircraft 1 2 position downward toward the ground. The terms in the first two lines of equation (8) are linear in the quadrature, and no terms which depend upon the maneuver are present. In the third line, there is a term with derivatives with respect to ta and a factor which is quadratic in the quadrature; these third-line terms are required for the maneuver calculations.

The path of a ray as it propagates through the variable atmosphere and how its pertinent quantities will change during the travel of the acoustic wave have been discussed.

The next section is concerned with connecting this system of rays to the aircraft.

Flow Near the Aircraft The initial conditions must be obtained from a study of the flow near the aircraft which is a well-established theory. The velocity potential for a body of revolution may be given as where x ' yo' and zo denote the source points and x, y, and z denote the field O points. The denominator may be factored into two terms, one term proportional to the distance from the flight path and a second term, containing a parameter proportional to the shape of the body of revolution, which has historically been termed the "F" function.

The velocity is obtained from (9) where which is analogous to the Whitham "F" function. In this expression, S is the cross- sectional area for a body of revolution, but for a body with lifting surfaces, for instance, it is an effective cross-sectional area, as discussed in reference 1.

Through the perturbation velocity q/a, V E and F can now be related. In equa- tion (7), V is given in terms of q/a, and F is related to q/a in equation (9).

E After the necessary algebra, the relation between V and F is given as E (10) Age Variable The analysis thus far has been linear; now the nonlinear aspects must be introduced so that a realistic shock-wave structure can be obtained. For this an age variable, given in equation (11), will be introduced.

The propagation velocity, equal to c = an + ii in the undisturbed fluid, is changed

by (~a + q)ii, where ~a is the perturbation in the speed of sound. This quantity is defined by

~a = r...:....!. ~ = r...:....!. q

2 pa 2 so that the change in propagation velocity is simply (y + 2 )qn • If the phase were expressed as distance s measured backwards normal to the wave fronts, the signal would experience a phase shift arising from the change in propa- gation velocity given by where t is time along the ray defined in equation (5).

In treating the phase, distinction must be made between the actual phase variable and the linear-theory phase variable. The nonlinear effect is basically the difference between the two. The actual phase is termed ~1 and the linear phase is ~. The local distance phase s is related with ~1 by ds = c d~1' The expression for the change n in propagation velocity in terms of ~1 becomes

_ r..±..l ~ = _ r..±..l q

dt 2 c 2 Co cos e

n This equation describes the phase shift for a particular point on the signal. The point on the signal found at ~ according to the linear theory is actually found at ;1' which is a function of ; and t.

The perturbation velocity q can be expressed in terms of V E by using equa- tion (7) to obtain

d;1 = _ 2::...±.-1 V E

dt 2co cos 8(pA sin 8 cos 8)1/2 The variable t may be replaced by -z by using equation (5). To transform the equa- tion for the phase shift to a canonical form, the age variable T is introduced; this age variable is defined by

T = L.!:.l S d( -z)

(11) 2co -z a sin 8 cos 8(pA sin 8 cos 8)1/2 The phase shift is then governed by

(12) I

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in the canonical form desired.

With the basic assumption that the linear results for the ray-tube area and

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Blokhintsev invariance still hold with only the phase shifted, the following relationship

holds: I

I (13)

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where V E (;) is the linear solution (independent of t or T because of the invariance).

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Here the actual phase ;1 satisfies equation (12) at constant ; and also the initial con- dition ~1 = ~ at T = O. The solution of equation (12) is then

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(14)

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The linear phase ~ is a function of ~1 and T, which may become multivalued in ~1.

When T is multivalued, equations (13) and (14) must be modified to take into account the presence of shock waves.

In a uniform atmosphere, T is proportional to the distance to the one-half power, whereas in an actual layered atmosphere, T is given by a convergent instead of divergent integral. (See eq. (11).) Now, the fact that T is given by a convergent integral means that it has a definite upper bound, and thus the signal never gets older than a certain age.

This means that the signal freezes and approaches an asymptotic shape. No matter how far _ the Signal progresses, it never gets any more distorted than this asymptotic distortion.

Sonic-Boom Signatures The analysis is now complete and the resultant shock wave or sonic-boom signature can be defined. In figure 3 is shown a progression of curves needed in developing the signature. The top curve, VE(O, is obtained by use of equation (10) from a knowledge of the F function and is plotted against ~. Utilizing the transformation of equation (14), the curve appears as indicated in the second curve, which is essentially a shift in phase and is multivalued. Because the compression parts of the signal move faster than the speed of sound relative to the medium, the resultant Signal is distorted. The high- pressure parts of the signal move toward the head of the wave and the low-pressure parts move toward the rear of the wave.

r Eventually this curve will fold over. This folding over will indicate that at a given point in the signal three different values can be obtained, and thiS, of course, is physical nonsense. Actually, what happens is that a shock wave appears, and this shock wave may be shown to appear where equal areas occur; that is, by using the equal-area rule, the location and strength of the shock can be obtained. A convenient method of locating the equal areas is shown in the third curve of figure 3, which is essentially the integration of V E' By using the crossing of the upper branch of the curve, the location of the shock wave is obtained, and the final signature is shown on the bottom curve.

RESULTS The foregoing analysis has been applied to a specific aircraft, and the results are shown in figure 4. Two examples are given, one for the aircraft flying at M = 1.4 at 35 000 feet, and the second for M = 3.0 at 70 000+ feet. The shaded area represents the experimental results and the dashed curve represents the analytical results. Two different atmospheric inputs were used in this analysis, one was a uniform atmosphere to which was applied a standard correction factor which approximated the effects of a stand- ard atmosphere; the second input was the standard atmosphere. For both inputs, good correlation with experimental data, with regard to the magnitude of the pressure, was shown; however, the standard-atmosphere input gave much better agreement with regard to signature length. This method has also been used for maneuvering aircraft by Haefeli, as reported in reference 2.

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I CONCLUDING REMARKS A method for predicting the sonic-Loom signature due to a maneuvering aircraft has been presented. This method includes the effects of a variable, stratified atmosphere.

The method is based on geometric acoustic principles and accounts for the use of a non- linearity to account for the weak shock field. The analytical results have been compared with experimental data and excellent agreement has been shown, particularly with regard to signature length.

REFERENCES 1. Hayes, Wallace D.; Haefeli, Rudolph C.; and Kulsrud, H. E.: Sonic Boom Propagation in a Stratified Atmosphere, With Computer Program. NASA CR-1299, 1969.

2. Haefeli, Rudolph C.: Sonic Boom Propagation From Maneuvering Aircraft. AIAA Pap. No. 69-1134, Oct. 1969.

--------- PROBLEM DEFINITION FLIGHT PATH ~ ___________ / -"~!I.k--------- ~ -------- RAY-TUBE AREAS

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STRATiFiED ATMOSPHERE u{z) p{z) T{z) ------- ~-=....,...:'r_-------- GROUND SiGNATURE Figure 1 RAY-TUBE AREA AIRCRAFT FLIGHT PATH 01 FFERENTI AL RAY-TUBE AREA z = CONSTANT \.. NEIGHBORING RAY '" (to + 8to' cf> + 8cf» NEIGHBORING RAY (to + 8to' cf» chi 0YI

oto Fa

A(t cf>,z)= +

o ' o chi 0YI dfdf Figure 2 -- --- ------ -

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SONIC-BOOM SIGNATURE Figure 3 CALCULATED AND MEASURED BOOMS LOCKHEED SR-71

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-----CALCULATED -EXPERIMENT I STANDARD ATMOSPHERE "CORRECTED" UNIFORM ATMOSPHERE

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M = 1.4; 35 000 ft

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Figur e 4

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--~ DISCUSSION ALFRED GESSOW, NASA, Washington: Applying this method to a practical case, to what extent is the maximum overpressure affected by a nonuniform atmosphere as compared with a uniform atmosphere?

HAYES: Well, if you take a uniform atmosphere you get completely the wrong results, so these have to be corrected, and they are usu . ally corrected by putting in the corrected acoustic impedance. In the atmosphere which is at a constant temperature, which is the simplest case, the correction factor is simply the square root of the pressure ratio.

Now, if you compare the uniform atmosphere method with this method for a constant-temperature atmosphere, what you would come up with is that the area under the lobe would be the same with both methods, and so, since the uniform atmosphere method would give you too large a signal range to the N wave, it would give you too Iowa value for the overpressure by a small amount. The differences are not very great.

GESSOW: I guess what was in back of my question was someone's suggestion that since there is quite a difference in temperature distribution in the atmosphere, one might take advantage of this and fly a path, in the practical case, so as to mlnlmlze the boom signature. However, if these effects are not that great, then such a scheme may not be too practical.

HAYES: Yes, I don't think we can get away from the boom that easily.

LORNE C. DUNSWORTH, CCl Marquardt Corp.: Is the ground reflection factor essentially 2?

HAYES: Well, the theory will give you 2. In practice various people use 1.9 or something like this. As far as I know there is no theoretical reason to use 1.9 instead of 2 so the justification for using 1.9 instead of 2 is purely empirical, I don't like it, but this is the way I am trained. I don't like things empirical.

L_ AN ANALYSIS OF FINITE-DIFFERENCE TECHNIQUES APPLIED TO EQUATIONS GOVERNING CONVECTIVE TRANSFER By Harvard Lomax Ames Research Center SUMMARY A study is presented of fixed-mesh, finite-difference techniques as they apply to the numerical calculation of hyperbolic partial differential equa- tions. Effects on diffusion or dissipation and dispersion are considered.

INTRODUCTION A survey of the presentations made at this conference points clearly to the fact that the development of numerical wind tunnels for practical airplane shapes is emerging as a reality. One aspect of this development is the possi- bility of constructing finite-difference techniques that can, by means of a computer, proceed directly from the basic partial differential equations and boundary conditions to the solution for the entire flow field. The technical journals contain many papers giving results of such solutions for a variety of techniques. We have investigated some of these methods to see if there was in some sense a "best" one to use for a given aerodynamic problem posed in the Eulerian system. The purpose here is to review briefly some of the results of this investigation.

PROBLEM DEFINITION One can ask: What is the best way to numerically analyze a set of differ- ential equations? If we seek to attach any generality to the answer, some testing procedures must be constructed that are representative enough to serve as a common denominator for the various techniques that can be posed. Tests for ordinary differential equations are comparatively easy to construct. For

example, the numerical integration of u ' = du/dt = ou + e~t by any given dif-

ferencing technique enable " s one to classify that technique with respect to all others when applied to the coupled set -+ -+-+ (1) u' = [A]u + f(t) where the elements of {A] are constants. Here, by classify, we mean specifi- cally to predict in a solution: (1) the accuracy given to the driving eigen- values of [A]; and (2) the bound on the parasitic eigenvalues.

The situation is much more complicated in the case of partial differential equations. In such cases it appears to be necessary to prescribe more about the physical nature of the problem being solved before attempting to analyze, let alone optimize, a numerical method that might be applied to it.

The following is an attempt to describe some broad classes of problems that occur in the study of the partial differential equations governing fluid dynamics, each class having peculiarities which make the "best" choice of numerical procedure peculiar to it.

Initial and Boundary Conditions One categorization of fluid flow problems applies to the type of restraints imposed on the various surfaces of contact. Three quite different sets of conditions arise. One leads to the study of transient phenomena in which, by definition, the initial and boundary conditions play about equally important roles. These are probably the hardest problems of all from the numerical viewpoint, and they are not discussed directly in this report.

Another leads to the study of periodic phenomena in which the initial condi- tions prevail throughout the complete time history of the solution. A certain class of these problems leads to the least numerical difficulties and is given the most attention in the following discussion. Finally, another set leads to the study of steady-state problems in which, by definition, the final solution is independent of the initial conditions or any transient behavior, and depends, therefore, only on the boundary condition. These problems lead to interesting numerical concepts that are often discussed under the name "relaxation," but, for want of space, will not be considered here.

Each of the above categories is most efficiently analyzed by a different numerical procedure; the reason being most effectively illustrated by examin- ing equation (1). Its solution can be written in the form "" o- t (2) u = ~Ce J + P.S.

"""' 0 - t where ~ce J is the complementary solution, which is independent of f(t), and P.S. refers to the particular solution. In the transient case both solu- tions must be resolved with equal accuracy; hence, its relative difficulty.

The periodic cases considered in the following are those in which f(t) is zero so that the particular solution disappears. In actual practice this rep- resents a great simplification in the numerical adaptation of the boundary conditions. Finally, the steady-state cases (as defined in the above) are those for which f is independent of t and the final solution is identical to the particular solution which is (3) In this case the complementary solution disappears and all the eigenvalues o- J are parasitic.

I

.----- --- ------------------ Genuine and Weak Solutions Most of the analysis of numerical methods applies to linear equations.

Since the actual partial differential equations of interest in fluid flow are generally not linear, the qualification is generally made that they can be considered to be locally linear; and, in that sense, they can be relied upon to respond in a predictable way to a given differencing scheme. The usual pro- cedure is to choose a numerical method that has a given order of accuracy on the basis of a local Taylor series expansion and, furthermore, satisfies some stability requirement in the linearized framework. With the added realization that one must also satisfy a dispersion requirement (discussed in the next section) this approach is usually practicable and is widely accepted when the dependent variables describing the flow field are everywhere continuous.

In most regions of a fluid flow the variables are continuous and the mathematical solution in those regions is referred to by Lax (ref. 1) as genu- ine. However, if supersonic flows occur, shocks appear and surfaces along which the dependent variables are discontinuous must be admitted. Lax refers to the mathematical description of such phenomena as weak solutions. Quite different numerical methods are recommended depending upon whether or not the desired solution is purely genuine. Both cases are considered but, at this point, the problem of calculating flows with shocks is considered in more detail.

If a weak solution exists in the region of numerical calculation t wo possibilities arise: in one, the location of the discontinuity is in some way determined and the differencing formulas are never permitted to use values on both sides at once; in the other, the differencing formulas are allowed to cross the discontinuities (thereby locating them automatically) but in such a way as to limit the resulting errors. Only the latter is considered below.

Figure I shows what can be expected if one tries to represent a discon- tinuity by a set of continuous functions. One period of a periodic step wave is shown together with its representation by 3 to 11 harmonics from a Fourier series. The gross representation of the step wave is clear, but the local errors, caused by the omission of high-frequency terms, can be a sizable per- centage of the jump at the discontinuity. Furthermore, the local error is not limited to the location of the jump, but is spread over the entire step, about equally for the number of harmonics shown. Examples of how particular differencing schemes calculate a moving periodic step wave are given later.

Dissipation and Dispersion , Consider the equation 2 3

au a u a u

-+ (4) a 2 -- 2 + a 3 -- 3

ax

ax ax

lThe term "weak", rather than "shock", is used because the formulas given by Lax provide the strength and location of any discontinuous solution of which the Rankine-Hugoniot is only one.

247 I

I ___ ~ __ ~ ____ ~_~ ______________ ~--1 -, )

If at t = 0 the initial condition u = eiwx is imposed, the oscillatory

solution (5) results. If a 3 is zero, the term al is the speed of convection of u or the wave speed. It is also, in such a case, the slope dx/dt of the characteristic in the xt plane. Clearly, from a glance at equation (5), a2 controls the damping of the solution. In heat conduction problems a2 is referred to as the diffusion coefficient. In the Navier-Stokes equations, when multiplying terms in which u represents a velocity, it is referred to as the coefficient of viscosity. The term a 3 is a measure of dispersion or phase shift. When a3 is not zero, the wave speed is (al - a 3w 2) which differs for each w, that is to say, for each initial frequency.

If a2 in equation (4) is zero, the equation is said to be conservative for u. Let the real part of equation (5) be the solution and let u be representative of the "energy" of the system. If a2 is zero, the "energy" is conserved and 2Tf / W 2 Tf u dx = - (6)

S

o w regardless of t.

Of course conservation laws are extremely important to the physicist. In fact the partial differential equations for fluid dynamics represent the con- servation of mass, momentum, and energy. Because of this, most modern numer- ical techniques used to solve conservative systems are constructed so that the d iff er en ce equations are also conservative. That is to say, some summation (i.e., numerical integration) of certain components over the space mesh is forced to be invariant with the timewise calculation. The point to be made here is that such techniques control the gross effects of terms such as that containing a2 in equation (4) -- that is, they are numerically stable and nondissipative; but they have no such control on terms such as that containing a 3' This we see at once because the integral in equation (6) is not only invariant with t but also with a 3' In short, conservative differencing methods tend to force all of their numerical error into dispersion.

THE SIMPLEST REPRESENTATIVE CONVECTION EQUATION The simplest partial differential equation containing convective phenomena is

au au

(7)

at - -c ax

It has been used as a representative equation by many authors and is I I I I 248 I I I

I I

l ____________________________________ ~ ___ J

sufficiently profound to display many of the difficulties involved in the numerical calculation of fluid flow. If c is a constant and periodic boundary conditions are imposed, any initial variation of u is transmitted in time as a wave with velocity c in the x - direction (see eq. (S)). In the following we consider two kinds of such waves: one, simple harmonic waves moving from left to right as time proceeds (the direction is given to fix the phase shift); and the other, a step wave whose harmonic analysis is shown in figure 1.

If c in equation (7) is set equal to u, a nonlinear equation results which is often used (ref. 1) as a model that is representative of the non- linear, shock-like phenomena contained in fluid flow problems. In the following it is used to analyze a single, nonperiodic, weak solution moving from l e ft to right as time proceeds.

TIME-CONTINUOUS METHODS There are two distinct approaches to the numerical calculation of equation (7) in conventional fixed-step meshes. The choice of one or the other depends mostly on whether or not the conditions are such that a weak or "shock-like" solution is contained in the answer. In this part we survey methods that are appropriate for problems whose solutions are entirely genuine.

If the right side (space derivatives) of equation (7) is replaced by some discrete differencing scheme but the left side (time derivative) is allowed to remain continuous, the equation reduces to a set of coupled ordi- nary differential equations identical to equation (1). If c is a constant, the solution to these equations is given by equation (2) in which P.S. is zero if the boundary conditions are periodic. We note that the solution is critically dependent upon the eigenvalues O J.

Consider the following simple differencing scheme for the right side of equation (7) u) ~ _1_ [-(1 + S )u· + 2Su , + (1 - S )U '+l] (8) ( a ax j 2 t::. x J- 1 J J If the constant 13 is taken to be -1, 0, or +1, equation (8) represents a forward, central, or backward differencing scheme, respectively. The or der of the error term is the same for all nonzero 13 , but the eigenvalue structure of [A] is drastically altered by various choices of 13 in the region indi- cated, In fact, the real parts of the OJ can be positive, negative, or zero, depending on the value chosen for S. All of this takes place without any regard for the physical nature of the problem.

However, the situation is clarified if we ask not what is the error introduced by replacing a u/ ax in equation (7) with equation (8), but what is the exact equation solved by the difference-differential combination? This latter equation turns out to be (9) It is easy to show that all even derivatives in this equation are multiplied by S and all odd ones are independent of S. Comparing with our previous discussion of equation (4), we see that any nonzero choice of S introduces a dissipation or diffusion term into the equation actually being solved, a term that is completely alien to the nature of the physical situation modeled by the original equation (7).

We have now arrived at an important point. If S is set equal to zero and equation (8) replaces the space derivative in equation (7), a numeriGal technique has been constructed that guarantees the conservation of "energy" in the sense discussed in equations (4) through (6). Combined with an appro- priate timewise numerical integration, the method is stable and nondiffusive.

It is apparent, however, that a wave of frequency w which travels in the x- direction with the speed c in an exact solution, will travel with a speed approximately equal to c[l - ( w2~x2)/ 6] in the numerical computations. The entire error in the space differencing appears in dispersion. In the simple problem being considered all central differencing schemes for au/ax are conservative, the higher the order, the less the error in dispersion.

In order to illustrate the above discussion it is necessary to discuss the error brought about by the numerical integration of the time variable in the intermediate, time-continuous form that arises after the spare deriva- tives are made discrete. One of the principal reasons for approaching the problem from this point of view is that this kind of integration (i.e., the numerical solution of coupled ordinary differential equations) has been so thoroughly analyzed. Application of methods such as ~he Runge-Kutta (or any other explicit, one-root method) to equation (1) reduces that expression to a set of difference equations the solution of which, after n time steps of ~t, can be written (10) where each Aj is related to a corresponding OJ in equation (2) by a polynomial of the form Aj = I + b (OjM) + b 2 (ojM)2 + ..• + bk( Oj'~~ t/ (11) The depend on the particular method, and for maximum accuracy are made to bk be l/k!

in which case Aj is equal to a truncated Taylor series expansion o·M e J of Here we need only consider the time-continuous form of equation (7) in which some central differencing scheme has been used for a u/ax and periodic boundary conditions applied. In such a case all the OJ are imaginary, and a L ___ _ ---------------- -~----.-.

fourth-order Runge-Kutta method provides the relationship between Aj and aj~t

shown in figure 2 (note that OJ = iaj)' It is clear from inspecting equation

(10) that the time integration 1ntroduces no dissipation for IAjl = 1, and is unstable for IAjl > 1. If IAjl < 1, dissipation is introduced by an amount depending upon how much less than one it is. If we study figure 2 with these remarks in mind, we arrive at the following conclusions regarding the effect of applying the fourth-order Runge-Kutta method to the problem posed: 1. For small enough. ~t there is practically no dissipation « 1 per- cent after 700 steps if a~t ~ 0.3).

2. Any frequency such that a~t ~ 2 . 6 is heavily damped.

3. For a spectrum of frequencies such that amax~t ~ 2.6, the method acts as a low-pass filter, heavily damping the high frequencies without affecting the low ones.

Finally, before the illustrations are presented, something should be said about the ratio of step sizes ~ t/~x. In classical studies of the numerical solutiori of equations similar to equation (7), considerable importance is attached to the term \I :: c~t/~x, which is often referred to as the Courant number. The condition is often imposed that \I must be less than or equal to one. Physically we must recognize that a "domain of dependence" exists in hyperbolic partial differential equations and the above condition is based on that concept. In the following, \I will be presented as high as 2.5 with quite acceptable results. This comes about , however, because the Runge-Kutta type methods used for the time integration are sequences of predictor- corrector formulas and the value of \I is not connected to the characteristic cones in the usua.l way. Actually, in no case presented did the numerical method violate the principle involved in the "domain of dependence."

Figure 3 shows the effect of dispersion caused by a three-point central differencing of a u/ ax i n equation (7), c being a constant. The exact solution u = sin 6n(x - ct) is shown by the solid line between x = 0 and 1.0 after 200 and 500 time steps had been integrated using a fourth-order Runge-Kutta method ,' In the x differencing 50 x points (excluding the last point) were used so ~x = 1/50.

Since ~t was chosen such that c~t/~x = 1, 6nc~t = 6n/50 ~ 0.38. This is

representative of a~t = 0.38 in figure 2, so practically no damping can be expected over several hundred time steps, and this is in fact verified. The dispersion, however, is quite evident.

Figure 4 shows how the dispersion can be minimized by increasing the order of the x-wise central differencing scheme, and figure 5 shows how dis- sipation can be brought about by increasing the time step so that a~t in figure 2 falls in an area for which IAj I is significantly less than one.

J

Figures 6 through 8 show the effect of analyzing equation (7) by the Runge-Kutta method when the boundary conditions are periodic and the initial condition is a step wave. Figure 6 shows the results after t has been increased from 0 to 40 using, again, 50 equally spaced x points. In this case equation (8) was used for the x derivative and 8 was taken to be 0.0, 0.1, and 0.2 for the three cases shown. The effect of adding "artificial viscosity" in this fashion is evident and typical. Figure 7 shows the first 5 time steps in a method which puts no damping in the au/ ax approximation (by choosing 8 in eq. (8) to be zero), but does put damping on the higher fre- quencies in the composition of the time dependency, by fixing 6t at a value such that a 6t in figure 2 is around 2.5. Figure 1 shows what happens max when all of the high-frequency terms above the limits 3 through 11 are removed.

Figure 7 illustrates the initial phase of their gradual removal. Figure 8 shows how such low-pass filtering methods resolve a periodic step wave after the same elapsed time as was used for the results presented in figure 6 for methods that damped all frequencies.

FIXED-MESH APPROXIMATIONS TO THE METHOD OF CHARACTERISTICS If a problem contains both a genuine and a weak solution, the "best" numerical procedure would compute both solutions with a given accuracy as if the other were . not present. A popular procedure for doing just that in two- dimensional and axisymmetric flows is the well-known method of characteristics.

In two-dimensional, linearized flow-field analysis, the equispaced, fixed-mesh, second-order differencing scheme of Lax-Wendroff is identical to the method of characteristics when a parameter similar to the Courant number is set equal to one. In higher dimensional and nonlinear cases, such a correspondence does not exist. Nevertheless, the published results of a large number of investi- gators have shown that this method (or a variant of it) gives acceptable results under the more stringent conditions.

Almost all second-order fixed-mesh difference schemes in popular use (e.g., refs. 2 through 8) for calculating flows with embedded shocks are identical to the Lax-Wendroff method when applied to the very simple linear form given by equation (7). When applied to higher order or nonlinear equations, however, they have individual differences. Some of these methods were tested under common conditions, and the one recommended here, both from the point of view of accuracy in resolving a genuine as well as a weak solution, and simplicity in programming, is that presented by MacCormack (ref. 8). As applied to equation (7) it can be written in several predictor- corrector forms, one of which is I -n+l n c6t ( n = u.

u - 6x u + j j

- u~) ) 1

J J r (12)

c.t (ii~+l _ ii~+l)] ul!+l 1 [ -n+l

I = ul!

"2 u + j 6.x J J -1 J J I I I

I

I

I

l __

-- -- -- - -- - - - -~ When applied to the solution of the periodic wave shown in figure 3, equations (12) gave the results shown in figure 9. Again 50 x points were used. Various values of the ratio c ~ tj ~ x were chosen and all five cases were run for the same total time, which was identical to that for the 200 time step case in figure 3. The low Courant numbers show nearly the same dis- persion as in figure 3 and considerably more dissipation. The higher Courant numbers show very little dispersion or dissipation because here the method is approaching the method of characteristics which is exact.

The results of applying equations (12) to the periodic step wave are shown in figure 10 and are typical of the Lax-Wendroff methods for linear problems when the mesh points do not lie along the characteristics as time proceeds.

The important nonlinear case in which equation (7) is replaced by

au a (u j2)

(13)

at = - ax

is illustrated in figure 11. The figures show how a discontinuity (or "shock") moves from left to right started by setting u = 1.0 on the upper level and u = 0 . 0 on the lower one . The upper two figures illustrate the behavior for time-continuous methods with damping entered through both the x and the t derivative as explained previously. The lower figure shows the results for MacCormack's method.

On the basis of these (and similar) tests MacCormack's version of the Lax-Wendroff method was used to analyze the conical flow field about a tri- angular wing mounted on a cone. A Mach number was chosen such that the wing edges were sonic, and the combination was placed at angle of attack. Both the compression and expansion sides (which, of course, are independent of one another) were computed. A typical result for the pressure distribution along the body and ~ing surface and vertically through the shock layer is shown in figure 12. Further pr~liminary details are presented in reference 9.

REFERENCES 1. Lax, P. D.: Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation . Commun. Pure Appl. Math., vol. 7, 1954, pp. 159- 193.

2. Richtmyer, Robert D.: A Survey of Difference Methods for Non-Steady Fluid Dynamics. National Center for Atmospheric Research Technical Note 63-2, Aug. 1962.

3. Lax, P. D.; and Wendroff, B.: Difference Schemes for Hyperbolic Equations With High Order of Accuracy. Commun. Pure Appl. Math., vol. 17, 1964, pp. 381-398.

4. Burstein, S. Z.: Numerical Methods in Multidimensional Shocked Flows.

AIAA J., vol. 2, 1964, pp. 2111-2117.

5. Emery, A. F.: An Evaluation of Several Differencing Methods for Inviscid Fluid Flow Problems. J. Compo Phys., vol. 2, 1968, pp. 306-331.

6. Fromm, J. E.: A Method for Reducing Dispersion in Convective Difference Schemes. J. Compo Phys., vol. 3, 1968, pp. 176-189.

7. Lapidus, A.: A Detached Shock Calculation by Second-Order Finite Differences. J. Compo Phys., vol. 2, 1967, pp. 154-177.

8. MacCormack, R. W.: The Effect of Viscosity in Hypervelocity Impact Cratering. Preprint 69-354, AIAA, Jan. 1969.

9. Kutler, P.: Application of Selected Finite Difference Techniques to the Solution of Conical Flow Problems. Ph.D. thesis, Iowa State University, 1969.

I - - - - -- -- --- I =l- t:: ""'-0

~=~

Figure 1.- Harmonic representation of a step.

1.0 r-----t--<::::-- ---t- 1>..1 0 2 3 4 O=h 1.000 . 998 1>..1 .996 . 994 0 .5 1.0 O=h Figure 2.- The absolute value of the numerical root A versus the product of the ste~ size, h, and the absolute value of the time continuous root, cr = jial, for the fourth-order Runge-Kutta method.

- - ------- ___________________________ J

200 TIME STEPS '000 TI ME S TEPS 3 POIW T CEwT DI FF . ~TH RU W GE ~ U TT ~ SO X POIWTS . COUR~WT = , . 00 Figure 3.- Pure dispersion in a time-continuous method; solid line, exact method; dots, numerical.

Figure 4.- Fourth-order Runge-Kutta time integration of a periodic harmonic wave defined by 50 x paints. Top: v ~ c ~ t/ ~ x = 2.5, 3-point central difference. Center: \) = 2.0, 5-point central difference. Bottom: \) = 1.6, 7-point central difference.

Figure 5.- Fourth-order Runge-Kutta time integration of periodic harmonic wave defined by 50 x pOints. All cases represent 3-point central differ- ence. Top: v = 2.5. Center: v = 1.6. Bottom: v = 0.8.

Figure 6.- Fourth-order Runge-Kutta time integration of periodic step wave defined by 50 x points. Results given for v = 1.0 after t = 40 c~x.

Beta in equation (8) is 0.0 for top, 0.1 for center, and 0.2 for bottom.

I

I

I

___ __ _______ _ ___ J

Figure 7.- Fourth-order Runge-Kutta time integration of periodic step wave defined by 50 x points using 3-point central differencing scheme and v = 2.5. Results shown after 1, 2, 3, 4, and 5 steps, top through bottom, respectively.

Figure 8.- Fourth-order Runge-Kutta time integration of periodic step wave defined by 50 x points. Top: v = 2 .5, 3-point central differencing.

Center: v = 2.0, 5-point central differencing. Bottom: v = 1.6, 7-p oint central differencing.

I

I

I I

L -.J

~OOO o SO ~ OO o 10 28b Figure 9.- Lax-Wendroff solution of periodic harmonic wave defined by 50 x points.

/\ COUR4WT o 80

l r

v

r

v o . ''0

\

r

V L4 l( WENDROFF SO l( PO INTS '10 T STEPS 4T COUR4NT 1 0 TOT4L TIMES S4ME 4LL C4SES Figure 10.- Lax-Wendroff solution of periodic step wave defined by 50 x points.

3 CEN T B O . OS COUR~NT 2 . S

1,---

3 CENT . B = 0 . 20 COUR~WT = 2 S

\-----

LAX WENDROFF COUR~NT = 1 0 Figure 11.- Solutions of nonlinear equation (13). Top and central represent fourth-order Runge-Kutta integrations with various forms of damping.

Bottom represents MacCormack's version of the Lax-Wendroff method.

cph

...

ALONG VERTICAL PLANE

CP~

. ~

ALONG BODY ALONG WING CONICAL WING-BODY COMBINATION CROSS SECTION Figure 12.- Development of flow about a conical wing-body combination using MacCormack's method. Solution shown not fully converged.

L

DISCUSSION RAYMOND SEDNEY, Martin Company: I hate to in any sense detract from the \ interesting material you presented, but I want to raise a strong objection to

I the rather provocative opening statement about numerical wind tunnels being

on the verge of appearing. I can find statements like that going back to 1948 - or a little earlier than that - when there was a tremendous break- I through on computing machines, and people really expected this.

i I personally think it is rather dangerous to make statements like that I because I feel the government wastes a great deal of money supporting work I where the sales pitch is exactly that: you know, we are going to replace wind I tunnels with computing machines.

I I . So I would like to ask you a question. Do you visualize this happening I In the next one year, or ten years or a hundred years?

LOMAX: That's too loaded a question, but I think that each year I can I see significant progress in this area, and I think that we can cut back on

I wind-tunnel tests and rely more and more heavily in certain well understood

I areas on the methods that are being developed, and as they get more and more sophisticated they can go hand in hand.

I

I When you bring in viscosity and turbulence and all this sort of thing,

I of course, I can't answer; perhaps that's a hundred years. But when you bring

i in the Eulerian forms of the equations, and other computations such as are

l presented in the other papers, you begin to get a feeling that yes, there is

I something more immediate that can be relied upon.

SIDNEY A. POWERS, Northrop Corp.: This is a tremendously fascinating piece of work, Mr. Lomax. I would just like to ask what is the difference in the Langley or the Ames method that made it so smooth? Could you describe the finite difference method you said you developed here that made it so smooth?

LOMAX: The details of the method are outlined in the written version.

I have picked here a case (I didn't do it deliberately; actually I chose some cards at random when I had the camera working) where the shocks were exceptionally sharp. This is the best you can expect, one reason being that in this case the shock structure is rather uniform. If you' have a problem where you have a rather weak shock in the presence of a strong one and try to use the same differences in every direction for both, you get a smearing of the weaker one. I do say that we did analyze a lot of these numerical methods, and this one was picked because it did give us the sharpest resolu- tion we could find. As I say, the details are in the written version.

PETER B. S. LISSAMAN, Northrop Corp.: I would like to make a few comments which were suggested by a previous comment. First, I'd like to say this was a very entertaining and illuminating paper.

I

I

I

I

_J ------------------- ------------ I am still interested in your remarks about electronic wind tunnels because I think that they are rather a long way away. It seems to me, as somebody who has spent a lot of sleepless nights and even some sleepless days , over trying to design wings, that the only interesting problem in wings are when the flow shows compressible regions, which is the cruise state of the airplane, or when it shows highly viscous separated regions, which is the landing state for the airplane. In fact, we are forced these days, almost, to make all our airplanes into biplanes, only we choose to pack the one wing inside the other one, and get it out with a lot of trouble when we want to land the airplane.

Now, I am very interested in all these incompressible techniques that are being developed, and I think I made some remarks yesterday which didn't sound the way I meant them to sound. I think that is the way we have to go to start, but I do believe that possibly we have to devote more attention to rather rough and crude numerical techniques toward tackling the very difficult problems, the problems of imbedded supersonic regions or imbedded viscous regions. If we could do that to within even 15 percent, then we could use the computer as a very valuable preliminary design tool.

I am thinking about, specifically, for example, a wing, perhaps of a subsonic airplane. You start to notice some shock pattern there in a wind tunnel, and then you ask your design engineer, "If I try to waist the fuselage or do something or maybe droop the leading edge a little bit, how would that help us?" The fact is that none of our computing methods, at least that I know of, can give us any indication about what we can ~o there.

Now, I will admit it is a formidable problem. As you say, it may be a hundred years away, but I think maybe that is all the more reason for us to address ourselves to some of those hard, messy prob lems at which we are forced to make approximations than to spend too much time toward refining known, regular, analytic techniques.

I would like to point out that this is in no way directed toward the talk that you have given. In fact, I just used the comment as an excuse to stand up and say this.

EARLL MURMAN, Boeing Scientific Research Laboratories: There was a research paper published in "Mathematics and Computations" in the July 1969 issue authored by Abarbanel and Zwas, who did a shock wave calculation using the Lax-Wendroff technique, similar to what you have shown today, only they used what they called an iterative technique of guessing Lax-Wendroff solu- tions and iterating. They ended up with a very sharp profile such as the one you showed that MacCormack produced. Would you care to comment, if you are familiar with this paper, how the iterative use of Lax-Wendroff technique damps the oscillation that is usually shown with shock wave?

LOMAX: I am not, right offhand, familiar with the paper. The partic u lar method which you saw here tends to damp the oscillations because it takes first one side to evaluate the space derivative and advance the solution and then, if you like, iterates or, if you like, corrects as in a predictor - corrector sequence, takes the other side for the same derivative to make the second advance. Then it repeats the cycle.

MacCormack's method actually holds true when you have three-dimensional problems in which case you use all four sides in a cycle. In this way, in each individual advance, he is sampling only one side and hopefully not cross- ing the shock. I believe that is one secret of his success.

GINO MORETTI, Polytechnic Institute of Brooklyn: I am a little more familiar than you, probably, with that paper. A friend of mine at Sandia Corporation made some experiments on that technique, and found a very great dispersion. What you saw in the published paper seems to be a squeezed figure. In looking at the figure very closely, it seemed to me that about 20 points were used to cover the shock transition. This is not any better than what one obtains by using the old-fashioned Lax scheme. I didn't spend too much time analyzing this paper, but offhand it seems to me that what they achieved with this iterative Lax-Wendroff technique is to put a lot of addi- tional viscosity which was not present in the original scheme.

I would like to make another comment about the MacCormack technique. I became aware of it a few days ago, and I played a little bit with it. Now, in simple, nonlinear problems where we know the exact solution it came out that not only the order of magnitude but even the numerical value of the errors coincide almost perfectly with the modified version of the Lax-Wendroff technique, which I have used many times.

Now, that is very interesting, because I must confess that my way of using the Lax-Wendroff scheme is much more complicated than the MacCormack scheme as far as programming is concerned, so I am now planning to switch to MacCormack's technique systematically. However, I wouldn't say that MacCormack's scheme allows you to go through a shock, because I think that if \ you go through a shock you do something against physics, and there is no mathematical gimmick which can do that.

LOMAX: You can go through a shock with the method of characteristics.

MacCormack's method tries its best to do that. It fails insofar as it can't do it exactly.

I didn't bring this out, but one of the big advantages of MacCormack's method is its Simplicity in programming and its low storage space.

MORETTI: Yes, I agree very much with you, and I even believe that such a scheme could be adopted using centered differences all the time, which is a trifle simpler than in MacCormack's version. I tried these centered differ- ences on one problem with no deterioration of the results.

One great advantage is that one can handle viscous flOWS, probably, with no coding difficulty, whereas when I tried to use my original scheme for a viscous flow the coding became cumbersome.

And if I can add a word of optimism from the East Coast to what you said, I am sure that wi thin two or three years we will be much b-etter off with this electronic wind tunnel, so, Ray, don't be worried. It took time and money to to develop conventional wind tunnels and the same happens for these numerical techniques, but we are definitively making progress.

~ _________ ~ __________________ ~ ____________________ J

LOMAX: Apparently, the phrase numerical wind tunnel is a good one to use to bring out discussion.

FRED R. DEJARNETTE, Virginia Polytechnic Institute: I notice the shock waves looked very good and sharp, particularly compared to the basic Lax scheme. However, the basic Lax scheme when applied to just a basic Prandtl- Meyer expansion smears out expansions tremendously. I was wondering how the expansions in the MacCormack technique compared, say, to just basic Prandtl-Meyer expansions.

LOMAX: The written version again, I think that will help. The Lax method is very dissipative and dispersive if used off design, and I definitely recommend that you do not use the Lax method unless you really know what you are doing. The Lax-Wendroff is second order, and therefore it is much less dissipative. It does have troubles when you come to discontinuities in the slope, as any second order method will, and some of the wiggles you saw in the film were evidence of this. Aside from this difficulty we have used the method on expansion fans and have gotten very good results.

------ ---- --- ---------- ------------------------------------ -------------------- --

A NUMERICAL METHOD FOR COMPUTING THREE-DIMENSIONAL VISCOUS SUPERSONIC FLOW FIELDS ABOUT SLENDER BODIES L. Walitt and J. G. Trulio Applied Theory, Inc.

and L. S. King Ames Research Center SUMMARY A numerical method has been developed for calculating steady, three- dimensional, viscous, compressible flow fields about slender bodies at angle r of attack. The method is based on the equivalence principle and on a two dimensional, time-dependent numerical technique for solving the Navier-Stokes equations. The equivalence principle relates the axial coordinate of the three-dimensional problem to the time coordinate; thus, the steady three- dimensional problem becomes directly analogous to the two-dimensional problem of a cylinder (not necessarily circular) expanding with time. The basic fea- tures of both the numerical'technique for two-dimensional, time-dependent flow and the equival e nce principle as applied to this problem are presen t ed.

Since the equivalence principle assumes axial velocities everywhere near the axial component of the free-stream velocity, the no-slip condition at the body surface in the axial direction cannot be applied. The flow field, including separation and body vortex development, is then predicted with a model incorporating a viscous crossflow with an inviscid axial flow. An anal- ysis of the errors introduced by this approximation is presented.

Numerical calculations were made and compared with experimental results for an ogive-cylinder and an airplane fuselage configuration. Flow conditions x

were Moo = 1.98, R oo = 4.68 I0 /ft, and a = 10° for the ogive~cylinder and

x M oo = 2.50, R oo = 9.l I0 /ft, and a = 15° for the fuselage configuration.

Results are presented as static pressure distributions on the body surface , velocity vector plots, and contour maps in the flow field at selected axial locations. Good agreement between theory and experiment was obtained; maximum deviations of numerical surface pressures from corresponding experimental values were no more than 6 percent of free - stream dynamic pressure for both problems. However, some differences were noted. Boundary-layer separation and body vortex positions differed from experimental locations on the ogive- cylinder, and the shock induced by the fuselage canopy was predicted at a slightly different location. These differences are considered attributable to neglect of axial viscous effects, exclusion of turbulence phenomena, and th e approximations introduced by the equivalence principle in describing the inviscid axial flow.

INTRODUCTION For highly maneuverable advanced aircraft as exemplified in figure 1, the prediction of flow-field characteristics for nonsimple geometries at high angles of attack and high flight Mach numbers transcends the capabilities of linear theories. The need thus exists for an analytical or numerical method for providing an accurate, detailed description of the vehicle flow field for determining both fuselage configuration and inlet-airframe integration effects.

Prediction of the three-dimensional flow field about a slender body immersed in a supersonic airstream presents a formidable mathematical problem to which no exact solution has been obtained, either analytically or numeri - cally. Successful numerical methods using the method of characteristics and the equivalence principle have been developed for three-dimensional, steady, inviscid flow fields. The work of Gallo and Rakich (ref. 1) is representa t ive of the characteristics approach while the development by Van Dyke (ref . 2) of the theory introduced by Hayes (ref. 3) is representative of an approach that uses the equivalence principle.

In this research program the equivalence principle was applied to convert time-dependent viscous, compressible flow calculations in two space dimensions to steady motion of a viscous, compressible fluid in three space dimensions .

The two-dimensional flow calculations were obtained by numerically solving the time-dependent compressible Navier-Stokes equations. In this paper the numerical technique is described in detail and results of its application to an ogive-cylinder body and to a fuselage forebody are presented.

SYMBOLS a Lagrangian coordinate radius a A surface area abscissa of center of smaller circle of fuselage cross section abscissa of center of canopy circle AS ordinate of center of smaller circle of fuselage cross section ordinate of of the body wi t h r espect t o t he horizo nt al axis line reference slope of axis of the body with respect to ho r izon t a l reference B2CZ') line ordinate of center of canopy circle BS surface of the body B(x,y,z) local sound speed

c

drag force drag coefficient, q oo Ac lift force lift coefficient, p - Poo pressure coefficient, q oo specific heat at constant pressure specific heat at constant volume diameter d d displacement vector E specific internal energy bounded oscillating function used to determine discretization error H stagnation enthalpy direction cosines of the normal to the body's surface in the x, y, and z directions, respectively m mass M Mach number M momentum vector absolute error in Mach number direction cosines of the normal to a shock surface in the x, y, and z directions, respectively cycl e number N mesh point density N p pressure stagnation pressure 26 7

________ ~ __ ~ __________________________ J

absolute error in pressure pi tot pressure dynamic pressure, q (1/2) pU R Reynolds number radius of smaller circle of fuselage cross section radius of larger arc of fuselage cross section radius of canopy circle position vector of a mesh point about a fuselage cross section B.(j ,k) position vector of a mesh point about a circular cross section B.'(j,k) S entropy surface describing system of shock waves about a body S(X,y,t) t time time step temperature T velocity component in x direction u one-dimensiQnal velocity, U(a,t) U U velocity vector v velocity component in the y direction v specific volume

v volume

w velocity component in the z direction

w work rate

x coordinate normal to plane consisting of the body axis and the perpendicular to that axis coordinate normal to the plane consisting of the horizontal x' reference line and the perpendicular to that line Lagrangian coordinate, X = X(a,t)

x

I L- ____________________________________________________ _ y coordinate normal to body axis y' coordinate normal to the horizontal reference line z coordinate along the body axis Zl coordinate along the horizontal reference line a free-stream angle of attack with respect to body axis a local angle of attack with respect to a horizontal reference line, e y' direction) tan -1 (veloc~ ty ~n Z I direction veloclty ln X' direction) . angle, tan -1 (Velocity in (3 sidesllp velocity in Zl direction Cn ratio of specific heats, -L y C v discretization error in a property (eq. (29)) p density T maximum slope of surface of body with respect to the free-stream flow direction stagnation condition property downstream of a shock free-stream condition Lagrangian coordinate at time zero reduced variable actual variable variable associated with body cross section FORMULATION OF EQUIVALENCE PRINCIPLE The two-dimensional time-dependent equations of motion (independent variables x, y, t) can be transformed directly to steady-state three~ dimensional (x, y, z) equations for slender bodies through the Hayes equiva- lence principle. The equivalence principle relates the steady~state flow field over a slender body to an equivalent time-dependent flow field in one less space dimension. In particular, steady three~dimensional flow about a body can be reduced under certain conditions to two-dimensional time-dependent flow in a plane normal to the free-stream flow direction.

A simple example of the application of the equivalence principle is illustrated in figure 2, namely that of steady flow over an axisymmetric body at zero angle of attack. The steady-state two-dimensional axially symmetric flow field is analogous to time-dependent one-dimensional axially symmetric flow in a plane caused by an expanding cylinder. The cross-section plane of unsteady analogy moves downstream with free-stream velocity and the outline of the moving boundary js given by the trace of the original shape in the cross- section plane. At the station z (see fig. 2), the steady~state flow field in this plane is analogous to the flow field about the expanding cylinder at the

time, t = z/U ' To make the equivalence analogy valid, the normal velocity of

oo the expanding cross section Un, is given by the product of the free-stream velocity and the tangent to the surface. Far from the body, free-stream con- ditions are imposed (i.e., P = P , p = p , u = v = 0).

00 00 According to convention, the free-stream flow direction is chosen as the principal axis along which the linearization approximations for equivalence are made. However, for bodies at angle of attack as illustrated in figure 3, it is more convenient to choose the body axis as the principal axis. In planes normal to the body axis, the cross sections can usually be defined in terms of a few simple analytic expressions, which result in simpler, more accurate surface boundary conditions and facilitate the development of finite difference meshes about the cross sections. Because of this choice, free- stream crossflow will exist as an upstream boundary condition, but this choice of axis in no way affects the accuracy of the results.

It is convenient to discuss the equivalence principle in terms of flow about a slender body at angle of attack. Let the z axis coincide with the axis of the body, and let the cross section of the body lie in the x, y plane (see fig. 3). It is also assumed that the z component of the local velocity vector is approximately equal to the z component of the free- stream velocity vector, that is, w = U cos a (1) oo Under this hypothesis, it follows that time-dependent flow in the x, y plane transforms to time-independent motion in x, y, z space according to the equations (2) z y y (3) (4) x = x This result - the equivalence principle - is derived for an inviscid fluid in appendix A.

In order for equations (2), (3), and (4) to represent a valid mapping between a flow field in x, y, t space and a flow field in x, y, z space, the time-dependent solution in the x, y plane must be augmented by a three- dimensional boundary condition at the body surface (5) ~ U cos a + ~ v + ~ u = 0 z 00 y x where ~x' ~Y' ~ z are the direction cosines of th e normal to the body surface in the x, y, and z directions, respectively. Equation (5) states that at the body, the component of the local velocity normal to the body surface is zero. The application of the boundary condition, equation (5), in the x, y plane implies that the body cross section varies in time and that the velocity

normal to the surface is equal to - ~ zU oo cos a/ ~ ~ + ~~. This normal velocity

expression reduces to the product of the axial velocity (U cos a) and the oo local tangent to the surface.

A second boundary condition, a constraint on tang ential flow at the body surface, must also be specified. Since the full time-dependent Navier-Stokes equations are solved in the x, y plane, a no-slip boundary condition is imposed at the surface of the body. We require th at v ~ - u 9- = 0 (6) x y It is seen in appendix A that the boundary condition, equations (5) and (6), are compatible with the assumptions inherent in the equivalence principle and permit a time-dependent viscous calculation to be made in the x) y plane.

Thus, with the inclusion of boundary equations (5) and (6), a solution of the two-dimensional time-dependent Navier-Stokes equations can be applied directly to steady three-dimensional viscous flow.

The three-dimensional boundary layer on the body surface has been replaced by a two-dimensional boundary layer since the no-slip condition is imposed at the body surface only in the crossflow direction. This description of the flow field is poor in the boundary layer, but appears physically reasonable when either inertial forces predominate or crossflow viscous effects predomi- nate in determining the flow field. Solving the Navier-Stokes equations in the crossflow plane provides a mechanism for effecting flow separation and subse- quent development of spiral vortex sheets on the lee side of a body at angle of attack without recourse to empiricism. The extent that this model agrees with the physical situation must be carefully docum en ted by comparisons with other methods and with experiment.

The computational procedure is then as follows. Initially we have a uniform flow field in the x, y plane with a velocity in th e y direction equal in magnitude to U sin a . The slender-body cross section grows with oo time in the x, y plane. At discrete times solutions of the full time- dependent Navier-Stokes equations are obtained such that the cross-section surface boundary condition, equations (5) and (6), are satisfied; also, the flow far from the cross section is uniform with a crossflow velocity in the y direction equal to U sin a . The time-dependent solution in the x, y oo plane is then related to the steady-state three-dimensional flow field about the body through the transformation equations (2), (3), and (4).

-- - ----- - - ----- - ------- ---~ -- ----- - ---.,,------------- - - -- -- DESCRIPTION OF THE NUMERICAL METHOD The numerical method used to solve the time-dependent Navier-Stokes equations in two space dimensions is embodied in a computer code called "AFTON 2PE." The finite difference equations in the AFTON 2PE computer code are based on a physical model of the continuum. By a physical model of the con- tinuum, it is meant that a framework be defined from which finite difference analogs of the full Navier-Stokes equations can be derived. The model utilized in this paper is described in some detail below. Since the Navier-Stokes equations are being used, this continuum model is not related to the particular phenomenon which is under study: once a physical model of the continuum is verified, its range of applicability extends to many flow situations.

In the conventional use of physical models, it is assumed that the partial differential form of the Navier-Stokes equations applies in general.

A physical model is then postulated for the particular phenomenon to be studied and the equations are reduced to ordinary differential equations to obtain either an analytical or numerical solution. The Lees-Reeves (ref. 4) near-wake model is a good example of a conventional physical model. This model is concerned with the base flow and near-wake subregions shown in fig- ure 4. To obtain a solution the boundary-layer approximations are used and pressure and velocity profiles in these regions are assumed. However, this solution is only valid in the base and near-wake regions; whereas, the entire flow field of figure 4 can be predicted with one numerical scheme utilizing a valid model of the continuum.

For illustrative purposes, the space-time continuum model employed in this research is presented in detail for the case of time-dependent one- dimensional flow of a compressible, inviscid fluid. This model was im p lied by a set of finite difference equations developed by von Neumann and Richtmyer (refs. Sand 6). Trulio and Trigger (ref. 7) then derived the con- tinuum model from a careful analysis of the von Neumann-Richtmyer equations.

Finite difference analogs of the continuity equation, momentum equation, the First Law of Thermodynamics will be derived in a Lagrangian coordinate sys- tem. Let a be the Lagrangian coordinate, and X(a, t) be the Eulerian coor- dinate. That is, X(a, t) gives the position at time t, of a fluid element that was originally at position a. Consider the Lagrangian coordinates al, az , and al/ Z (al/ Z == (1/2) (al + az )) shown in figure S(a). Since the system is Lagrangian, the mass between the trajectories labeled al and a2 remains a constant. Let the one-dimensional space continuum be represented by a dis- crete set of zones, designated "thermodynamic" zones. At time zero, let the boundaries of these zones be spaced at a constant interval ~a along the X axis (see fig. Sea)) and denoted by a ~ ( ~ == 0,1,2,3, ... ,L). Let each of the thermodynamic zones be one unit high and one unit wide. Consider another set of zones designated "momentum" zones superimposed on the thermody- namic zones, such that each momentum zone surface always divides the mass of the thermodynamic zone which contains this surface in half. A schematic dia- gram showing a momentum zone and two thermodynamic zones at the time t is r I I I I L--- _______ _ shown in figure Seb). The Eulerian coordinates which define the two thermo- dynamic zones are XCa _ , t), XCa , t), and XCa + , t ) , while th e Eulerian t t t 1 coordinates of the momentum zone are XCa t _Cl/ 2 )' t) and X(a + / ) ' t ) .

t 2 C1 The thermodynamic variables, such as internal energy, specific volume, and pressure, are assumed constant throughout a thermodynamic zone. The velocity is assumed constant throughout a momentum zone. Thus, thermodynamic variables are in effect centered at the momentum zone surfaces; that is, Pi = P(a _(1/ 2 )' t) denotes the pressure in the thermodynamic zone X(a _ , t), i i 1 X(a , t) which effectively acts at X(a _(1/ 2 )' t) (see fig. S(b)). M omentum t t zone variables are in effect centered at the thermodynami c zone surface s ; that is, U ~ = UCa ~ , t) denotes the particle velocity i n the momentum zone X(a _ / )' t), X(a~_Cl/ 2 )' t) which effectively acts at X(a ~ , t).

t 2 C1 Since an explicit formulation is the goal, the variables of motion are not only displaced spatially (as discussed above), but they are displaced in n time as well. Let ~t denote the uniform time interval and t Cn = 0, 1, 2, ... , N) denote the time after n uniform time intervals. The variables associated with thermodynamic zones are defined at integer times; that is, n P~-(1/ 2 ) = p(a -(lf2)' t ) denotes the pressure at the time tn. The momentum t zone variables are defined at half-integer times; that is, n C1 ~-C1f 2 ) -_ UCan, t - / )) d h '1 l' ,n-C1f 2) U N N enotes t e partlc e ve OClty at a tlme t , and the Eulerian coordinate position X(a , tn-Clf 2 )).

t Finite difference analogs of the continuity, momentum, and first law equations follow directly from the physical model of the continuum presented in figure S(b). Since a Lagrangian coordinate system is employed, the zones I of figure SCb) will be displaced continuously from their initial positions; that is, aI, . . . ,aL to coordinate positions X Cal, t), . . . , X CaL' t).

Let us calculate the properties at the time, tn, for the thermodynamic zone 1 having Lagrangian coordinates a i _I' a and the momentum zone having i Lagrangian coordinates at -Clf 2 )' ai +Clf2)' The initial values for this cal- , pn - 1 En - 1 vn - 1 pn -1 En - 1 vn - 1 for culatlon are i -(lf 2 )' i -C1f 2 )' i -(l/ 2 )' i +(l/ 2 )' i +(1I 2 )' i +Cl/ 2) n n the thermodynamic zones and X~ ,X Xn Un-Clf 2) U -(lf 2) un-C1f 2) for N -l i ' t +l' i -I ' i ' £ +1 the momentum zones. The objective is to update these variables by one time step in an explicit manner.

The finite difference analog to the conservation of mass is derived from the expression for the volume change of thermodynamic zone a i _I' a .

i

vn _ n-l

~-(lf 2 ) _ un-Clf2)]A (7)

[U = N ~ -l ut

i- (1/2) V i - (1/2) where n-l n-1 volume of zone at time t V Q, _ (1/22 a Q, _l' a Q, n n volume of zone t at time VQ,_(1/2) a Q, a Q, _l' Based on equation (7), and the fact that the mass of material in the thermody- namic zone a Q,-I ' a Q, ha s the constant value Po 6 a, we find that U n -(1/ 2) _ U n - (1/ 2) v - v n n-1] Q, Q, -1 Q, - (1/2) Q, - (1/2) (8) Po 6t 6a [ where Po is the density at time zero, and v~_(1/ 2) is the specific volume n n n of material in the zone at time t (note that (po 6 a)v .Q, _(1/ 2) ==V .Q, _(l/ 2)) · For an inviscid, adiabatic fluid, the first law equation for a system in equilibrium is applicable: DE _p Dv (9a)

Dt = Dt

The first law finite difference analog to equation (9a) is derived by first writing down a finite difference analog to the term -P(Dv/Dt) and then equat- ing this to the rate of change of internal energy in the zone. Since the zone mass is constant and the pressure is homogeneous in a thermodynamic zone, the term -P(Dv/Dt) becomes

P 6V )

(9b) ( - 6t.Q, _ (1/ 2) where On the basis of equation (9b), the finite difference first law equation (10) n-1 where E.Q, _(1/ 2) is the specific internal energy of thermodynamic zone The equation of state for a perfect gas is: E = Pv (y - 1) (11) Finally, the finite difference equation of motion for momentum zone a Q. _( 1/ 2 )' a Q. +(1/ 2) becomes: \ = [pn-(1/ 2) _ pn-(1/2)] (12) Q. - (1/ 2) Q. + (1/ 2) ,where I \ I I defines the momentum in zone at time tn.

and a Q.

Equations (8), (9), and (10), derived from the physical model presented in figure S(b), are solved in the following manner. First, the specific volume, v~_(1/ 2 )' is calculated from equation (8). Then, the pressure P~-(1/ 2 )' and specific internal energy E~_(1/ 2 ) are calculated from equations (9), (10), and (11). The continuity and first law equations are then solved for specific volume and pressure in all thermodynamic zones. Based on th.e pressure field, equation (12) can be solved for the momentum M~. The particle velocity n+(l/2) . .

U Q. ~s found from the forward extrapolat~on formula (13) where

m = p tJ.a

Q. 0 \ n U +(1/ 2) with time establishes the new i lntegration of the particle velocity Q.

Eulerian position, xn+l This can be done for all momentum zones. Thus, all the initial values c&ted above can be updated by one time step and the process repeated.

The success of this method coupled with the failure of other more recent numerical schemes to improve upon it, led Trulio and Trigger to a careful analysis of these finite difference equations in order to discover the reasons for their success. They found that the finite difference equations possessed the same self-consistency property of form as the original differential equa- tions from which they were derived, that is, the finite difference equations for momentum conservation and the first law implied an exact conservation of total energy (internal and kinetic) finite difference equation (ref. 7). In other words, the continuity, momentum, first law, and conservation of total energy relations are redundant by one. In most other numerical schemes if one tries to derive a conservation of total energy relation from finite dif- ference analog of momentum conservation and the first law, error terms, usually assumed to be "second order," are produced. These numerical error terms are believed to be a primary source of the difficulties encountered in many other numerical schemes. Appendix B demonstrates the self-Gonsistency property of the finite difference equations (8) to (13).

In addition to self-consistency of form, the numerical method has the following properties: (a) Since the method is explicit, stability criteria must be met in order to obtain physically meaningful numerical results.

In gen- eral, the time step must be small enough so that a sound signal cannot cross a zone in a time step (refs. 5 and 6) .

(b) The numerical error has been correlated for this scheme. It has been found that the absolute error in a property is inversely pro- portional to the linear mesh point density to the three-halves power (ref. 8).

The physical model, from which equations (8) to (13) have been derived has been extended to two space dimensions (ref. 9). In that work, specific finite difference equations were formulated and their self-consistency proper- ties demonstrated. Because they provide a base for the present calculation, a brief description of the derivation of these equations will be presented.

Consider two-dimensional flow of a viscous, compressible fluid in the x, y plane. As in the one-dimensional case, let us divide the continuum into two types of zones; namely, quadrilateral zones and momentum zones. See figure 6. The assumptions governing this analysis are as follows: (a) All zones are polygons.

(b) The density, specific internal energy, pressure, stress tensor, and velocity derivatives are homogeneous in a quadrilateral zone.

(c) The velocity vector is homogeneous in a momentum zone.

(d) The zones have unit thickness normal to the plane of motion.

As in the one-dimensional case the continuity and first law equations are solved for each quadrilateral zone to determine the density, specific internal energy, and stress. When the stress tensor depends only on strain, , the first law equation corresponds to that of thermodynamics. However, for a , I

276 I

I 1

l ____ ~ _________________________________________________________________ J

stress-rate of strain dependence, the first law equation is a relationship involving intel~al energy, the specific internal energy being the difference between total and kinetic energy. Based on the stress tensors in each of the four quadrilateral zones which comprise a momentum zone (see fig. 6), the momentum equation is solved for each momentum zone. To accommodate the equiv- , alence principle boundary conditions, the finite difference equations in AFTON 2PE are written in a generalized coordinate system where the four mesh points comprising a quadrilateral zone can move with arbitrary velocity. The numerical method is described in detail for a generalized coordinate system and an Eulerian coordinate system in two places (refs. 9 and 10).

All of the salient properties of the numerical method described in the one-dimensional example are preserved in two-dimensions and will also be pre- served in three spatial dimensions; in that respect the numerical theory is internally consistent. The momentum and first law finite difference equa- tions in two dimensions imply an exact finite difference equation for total energy. The stability criteria are the same as in the one-dimensional case.

The one-dimensional correlation of absolute numerical error in a property extends directly to two dimensions. In two dimensions the absolute numerical error is inversely proportional to the area mesh point density to the three- quarters power (ref. 10).

ORDER OF ERROR OF THE NUMERICAL METHOD In this section the order of error of the numerical method is determined.

The numerical error is defined as the absolute difference between the actual value of a quantity and the value computed numerically. The error in this method stems from three sources, the equivalence principle, discretization errors, and neglect of axial viscous effects. The errors inherent in the equivalence principle lie in the neglect of velocity perturbations along the free-stream flow direction and have been deduced by Van Dyke (ref. 2). The discretization error is defined as the difference between the exact solution to the continuum motion equivalence principle equations for a given system, and the values of the flow variables computed from a finite difference approx- imation to these equations of motion; discretization error results basically from the substitution of a discrete set of points for the space-time continuum.

The errors introduced when axial viscous effects and turbulence phenomena are neglected are manifested in the accuracy with which separation and vortex locations can be predicted. , Since the determination of the accuracy of the method is of primary importance, all sources of error will be discussed at some length.

To deduce the order of error in the equivalence principle assumption, Van Dyke introduced what he termed "reduced" independent and dependent vari- ables, and he redefined the functions describing the body and shock-wave sur- faces. To obtain these reduced variables, consider a coordinate system where the z axis is alined with free-stream flow direction. Let the surface of the body be described by B(x, y, z) = 0, and let the complete system of shock waves be described by Sex, y, z) = O. In this coordinate system the reduced variables are as follows:

I

J

x = xlT I

y = y/, (14) Z = Z U = Uoo,u(x,y,z) v =

UooTvex,y,z) I (15)

w U [1 + ,2w (x , y , z) 1 oo p P M2 2p C - -) = ooY 00' x,y,z (16) p p p(x,y,z) (17) B =

ii ex,)" z) I

(18) S = S(x,y,z) where , is the maximum slope of the body surface with respect to the free- stream flow direction. This transformation of variables was introduced into the continuity, momentum, and first law equations. Reduced parameters were considered of order one or less and terms that contained ,2 explicitly were discarded. A set of reduced equations, linear in w and derivatives with

respect to z, resulted. The boundary conditions for the reduced equations

became: At the body surface aB - aB aB (19) 0 at B 0 -+ v

=

u ax +

az =

ay Far upstream of the body p -+ yM ,2 -+ _00 00 as Z (20) -+ p 1

I

The parameters Moo and, of the full problem enter the reduced problem only in the combination MooT, which appears only in the upstream boundary con-

dition on P (eq. (20)). Since P must be of order one or less for these

equations to be valid, M oo ' must be of order one or greater for this theory to be consistent far upstream of the body, If this consistency condition is satisfied, the maximum error in a reduced variable must be of the order ,2, because terms of order ,2 have bee~ omitted in the reduction. Since Moo, is of order one, an error of order ,2 implies that the error is also of order l/M oo 2, thus the theory becomes more accurate at high Mach numbers. At super- sonic speeds and wIth Moo' = 0(1), this error is not negligible. Maximum errors

L ___ _

-- --------- - \, f expected from the equivalence principle assumption or static pressure and \ local Mach number are derived below.

The absolute difference between the actual reduced pressure, Pa , and the reduced pressure calculated from this theory, P, is (21) Combining equations (16) and (21) yields

Ip - pi

IP - pi ~ _a-- - ~'T 2

a 2q ooT where ~ is the free-stream dynamic pressure ((1/2)p oo U )' Therefore, th e oo absolute error in the pressure is as follows: Since experimental and numerical local Mach numbers are compared in this research effort, it is important to determine the order of error in the M ach number. The Mach number M is defined as 11 2 2 2 2

M = (u + v + w )

(23) C where C is the local sound speed. From Bernoulli's equation, the local sound speed can be expressed in terms of the components of local velocity (24 ) where Coo is the free-stream sound speed. Introducing the reduced veloc i ties u, v, and w from equations (15) into equations (23) and (24) yields the rela- tion for the Mach number in terms of reduced velocities.

M (25) - 2 _ [(y 2 2 2 ] 1/ 2 2 2

- 1)/2]T (w T + 2w + v + w )

[Moo - T4 Since iT, v, and w are of order one, terms in could be neglected in equation (25).

T2(2w + v +

1 + ii )

]"2

M~ (26) 2 2

[ -2

[ (y - 1) /2 ] T (2w + v + u ) Moo -

Equation (26) can be linearized in terms of changes in iT, v, and w by

employing a first-order Taylor's expansion. Let Ma represent the actual

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local Mach number and M represent the value computed by this numerical method. From a first-order Taylor's expansion

dM) - (dM) A- (dM) A-

(27)

M ~ M + - . /:'u + -=- uV + -=- uW

(

a au u=v=w=l av u~v=w=l aw u=v=w=l

The coefficients of equation (27) can be determined by differentiation of equation (26) and evaluation of the derivatives at the values u = v = w 1, since barred quantities are assumed of order one. The final equation is as follows: 2 (y (28) 1 + where

I liM I = 1M - MI

a If the reduced equations are rewritten in terms of the actual unbarred parameters, and the substitution z = Uoot is made, the time-dependent equa- tions of motion in two space dimensions result.

The discretization error in the above time-dependent equations of motion in two space dimensions is based on a general rule that relates the error to the density of mesh points employed in the numerical integration process. The derivation of this error rule can be found in reference 8; the relation is as follows: - 3/ 4 f N E = 2 (29) where E is the absolute discretization error in a given property, N is the two-dimensional mesh point density (i.e., number of zones per unit area of the x, y plane), and f2 is a bounded oscillatory function. If the same problem is run with two different meshes, that is, a medium and fine mesh, the function f2 can be evaluated from the values of a property and mesh point densities of the medium and fine meshes at the same point in space and time. For the case where the static pressure error is required, (30) where P is the fine mesh pressure, Pm is the medium mesh pressure, N is m f the mesh point density of the medium mesh, and Nf is the mesh point density for the fine mesh.

The error introduced by neglecting axial viscous effects and turbulence phenomena is the most difficult of the three sources of error to evaluate. In fact, one of the primary objectives of the study was to determine how well the l __ _ flow field could be predicted with axial viscous effects neglected. No attempt was made to evaluate this error analytically. Instead, numerical results obtained with this method are compared with experimental data. These compari- sons are discussed in some detail in succeeding sections concerned with the numerical results.

In this research effort the effects of mesh point density on the discretization error were not investigated; each problem of the program was run with only one mesh. Therefore, the order of error of the numerical method, which was established by comparing the numerical results with experi- mental ones, included all three sources of errors.

DESCRIPTION OF PROBLEMS SOLVED AND MESHES USED The AFTON-2PE computer code, modified to accommodate the equivalence principle boundary conditions, was applied first to the flow field about an ogive-cylinder configuration and second to a fuselage geometry representative of an advanced tactical fighter plane. In both problems, air, represented as a gamma law gas (y = 1.4), was considered and adiabatic flow was assumed throughout the flow field. For the ogive-cylinder problem the free r stream Mach number was 1.98, the angle of attack was 10° with respect to the axis x of the body, and the free-stream Reynolds number was 4.68 l0 /ft. For the fuselage problem the free-stream Mach number was 2.5, the angle of attack was IS° with respect to the horizontal, and the free-stream Reynolds number x was 9.1 l0 /ft. In this section the cross-sectional shapes for both problems are defined and the finite difference meshes generated about these cross sections are described.

The axisymmetric ogive-cylinder configuration is composed of an ogive which is three maximum cylinder diameters long and a cylinder 7.3 diameters long. This configuration is schematically illustrated in figure 7, where the equations which describe the variation of the radius of the body with axial distance are also indicated. In this problem the axis of the body, which is straight in this case, was chosen as the principal axis for the equivalence analogy between the steady and unsteady flows.

The fuselage had a drooped nose which resulted in a curved central axis of the body. This geometry is schematically illustrated in figure 8. The central axis of the body is composed of a straight portion, inclined 7~1/2° from the horizontal, and a curved portion which begins at a horizontal station of 15 inches (see fig. 8). To avoid introducing curvature effects into the equations of motion, the 7-1/2° reference line was chosen as the principal axis for the equivalence analogy. In this coordinate system, the angle of attack of IS° with respect to the horizontal becomes 7-1/2° with respect to the 7-1/2° reference line; hence a = 7-1/2°. The canopy, included in this problem, is indicated in the cross sections normal to the 7_1/2° reference line shown in figure 8.

The fuselage configuration has cross-sectional shapes normal to the 7-1/2° reference axis whose peripheries can be approximated by circular arcs I and straight lines. In fact the fuselage cross section is circular to a I \

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horizontal station of 9.44 inches from its nose, is asymmetric between horizontal stations 9.44 and 10.88, and includes a circular canopy between horizontal stations 10.88 and 30.68. The fuselage cross section at a horizon- tal station at the canopy location is shown schematically in figure 9. The parameters describing this periphery are also indicated in the figure. These parameters have been curve-fitted as functions of distance along the central axis of the fuselage.

A subroutine of the AFTON 2PE computer code has been developed for generating finite-difference meshes around an asymmetric half-body of a gen- eral fuselage-shaped cross section. This subroutine is based on previous work on finite-difference mesh development for a circular cylinder (ref. 10). The general cross-sectional shape of the half-body is assumed to consist of two circular arcs, two straight-line segments tangent to the circular arcs, and a circular canopy (see fig. 9). The procedure adopted in the calculation was as follows. The area of the half-body was computed and a half-circle of equiva- lent area was located with its center at the coordinate origin. The finite- difference mesh for this half-circle was calculated from a modified stream function and potential function from potential flow theory about a cylinder.

The problem was to transform the cylinder mesh into a new mesh around the asym- metric half-body according to some suitable rule of transformation. In the subroutine developed, each half-circle mesh point (designated hereafter as an "unprimed" mesh point) was transformed into a half-body mesh pOint (designated hereafter as a "primed" mesh point) in the following manner. First, the periph- ery of the half-body shape was divided into as many equal arcs as the half- circle periphery. Then, surface vector displacements were obtained between corresponding unprimed points on the half-circle and primed points on the asym- metric half-body. Based on these surface vector displacements, unprimed mesh points in the flow field were displaced to their primed locations. Consider an unprimed mesh point in the flow field having a position vector R(j, k), where the integer k is associated with a potential-like line, and the integer j is associated with a streamline-like line. Let the surface vector displace- ment corresponding to the same k line be denoted as ~(k). The position vec- tor of the primed point R'(j, k) is determined from the following equation: (31) where a is the radius of the half-circle and R(j, k) is the magnitude of the position vector ~(j, k).

The finite-difference meshes for the ogive-cylinder were composed of 35 j lines and 90 k lines and continuously deformed with the radius of the body. The initial radius was 0.00046875 foot and the maximum radius was 0.046875 foot. Since the ogive-cylinder cross section is circular, only the equations of the mesh generating subroutine which pertained to the circular cylinder mesh points (unprimed mesh points) were used to continually calculate new meshes as the radius of the cross section changed. The finite-difference mesh in a cross-sectional plane normal to the body axis at a radius of 0.01 foot is shown in figure 10. The finite-difference mesh corresponding to the I maximum radius (0.046875 ft) is indicated in figure 11.

r

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The finite-difference meshes for the fuselage configuration were composed of 35 j lines and 99 k lines and continuously deformed as the body cross-sectional shape deformed. The mesh generating subroutine of the AFTON 2PE computer code continually calculated new meshes as the fuselage shape deformed. The finite-difference mesh in the cross-sectional plane nor- mal to the central axis at a horizontal station 7 inches from the nose of the fuselage is shown in figure 12. At this station the fuselage cross section is circular. The finite-difference mesh corresponding to a horizontal sta- tion 25 inches from the fuselage nose is shown in figure 13. The canopy is also indicated in figure 13.

BOUNDARY CONDITIONS AND INITIAL CONDITIONS In the x, y planes, \ the finite-difference meshes are bounded by an upstream boundary, a lateral boundary, and a boundary composed of the symmetry line of the cross section and the body cross section itself. The density and specific internal energy are given their free-stream values at the upstream boundary while the velocity of material normal to this boundary, v ' is oo evaluated from v = U sin ex (32) 00 00 where ex is the angle of attack with respect to the principal axis. Equa- tions (5) and (6) are satisfied at the body's surface, while the fluid is allowed to slide without friction at the system's lateral boundary and sym- metry lines. The downstream boundary condition was based on the method of characteristics. It was used previously in low-speed wake-flow calculations and gave a good approximation to the flow in this region. Reference 11 describes this downstream boundary condition in some detail.

The initial condition consisted of a uniform flow field in the x, y plane with V oo = U sin ex in which the body impulsively appeared at zero time.

oo In both problems, the initial cross-sectional radius was approximately 1 per- cent of the maximum radius.

RESULTS OF OGIVE-CYLINDER PROBLEM The ogive-cylinder problem was the first attempted in this study, This problem was a good test of the method in that vortices had been experimentally observed to occur on the leeward side of the body. The crossflow Mach number was 0.344 and the crossflow Reynolds number was 0.7625 x l0 based on the maxi- mum diameter of the body. The problem was run 3406 cycles (i.e., time steps) on the UNIVAC 1108 computer, requiring approximately 10 hours of computer time. Solutions were obtained from the body's nose to an axial station 8.35 maximum cylinder diameters downstream. At this cross-sectional plane a well- developed pair of vortices were calculated. The problem duplicated the body geometry and free-stream conditions of a wind-tunnel test (ref. 12). In gen- eral, agreement between numerical and experimental data was good, providing evidence that the numerical method is applicable to bodies for which separa- tion and the subsequent development of spiral vortex sheets occur.

________ - _ __ _ _______________________________________ J

---------------------------- - - ---- -- - -- - - ----~-------------------------- To investigate the qualitative behavior of the flow field, it was convenient to exhibit the data in the form of vector plots of the velocities of the fluid particles; the tail of each vector corresponds to a mesh point.

The sequence of events as one moves down the axis of the body is described by figures 14 to 17. In figure 14 the flow field is shown at a station 0.502 maximum cylinder diameters from the nose of the body. Although the finite- difference mesh is relatively coarse with respect to the radius of the body at this station, the bow shock is indicated as well as the expansion which occurs on the leeward side. In figure 15, at a station 2.99 maximum body diameters, where the ogive section ends, the bow shock is better defined. Figure 16 shows the flow field at a station of 4.92 maximum diameters, where separation first appears on the leeward side of the body. The spiral vortex sheets that develop on the leeward side of the body are indicated in figure 17. The for- mation of the bow shock, the leeward expansion, and subsequent development of the spiral vortex sheets are all in qualitative accord with experimental observations . The accuracy of these numerical results is considered below.

Numerical pressure distributions around the body are compared to experimental data at various axial stations in figure 18. As can be seen from the figure, quantitative agreement is achieved from the body nose to an axial station 4.92 diameters. At this station the numerical data indicate that separation had occurred on the leeward side of the body (see the velocity vector plot in fig. 16), whereas the experimental data showed leeward separa- tion at a station approximately 6.00 diameters aft of the body t· s nose. As a result, the numerical pressure data on the leeward side of the body differed slightly from the experimental data between stations 4.92 and 6.00 diameters down the body axis. At stations greater than 6.00 diameters from the cylinder's nose, the experimental data also showed separated flOW, and the numerical and experimental pressure coefficient data were in close agreement 7.63 diameters from the nose of the cylinder.

The circumferential positions of the separation points and vortex centers vary with axial location in a manner determined experimentally by measurements of pitot pressure at various body cross sections. Corresponding separation point and vortex center positions were also found from velocity plots of the numerical flow field. The calculated separation points and vor- tex centers were found to lie about 20° closer to the windward side than the corresponding experimental values as seen in figure 19. For example, in a crossflow plane 8.3 diameters down the body axis, the numerical separation point and vortex location were lOgO and 137°, respectively, while the corre- sponding experimental values were 130° and 160°. The discrepancy between measured and calculated separation point and vortex center positions, while not very large, is perhaps the least satisfactory feature of the calculation.

The difference is due to viscous effects of axial motion being neglected as well as to the exclusion of turbulence phenomena.

The flow field predicted numerically is laminar. However, the actual flow field is turbulent as a result of both axial and crossflow viscous effects. In the numerical method axial motion is treated as if the flow were inviscid. For laminar flow, separation would begin on the leeward side of the body at an axial station upstream of the station at which turbulent sepa- ration begins. Also, the separation point would be closer to the windward 28 4 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - side when the flow is laminar. From the pressure coefficient data for a cir- cular cylinder (ref. 13), it was found that in turbulent flow at a Reynolds S x number of 6.7 lO , separation occurred at 120°; in laminar flow at a Reynolds S x number of 1.85 lO , separation occurred at 90°. Thus, the difference between the numerical and experimental separation positions is in qualitative agree- ment with the difference between laminar and turbulent separation positions.

To obtain more realistic theoretical predictions, it will therefore be neces- sary to generalize the equivalence principle account for turbulent effects due to axial flow; it is felt that the principle, which was generalized for this program by including Newtonian viscosity in the crossflow equations, could be extended satisfactorily by the addition of eddy viscosity terms.

Based on the equivalence principle theory, an error analysis of the surface pressure results of figure 18 was made. The maximum slope of the ogive-cylinder body with respect to the free-stream flow direction was 28.92° which corresponds to a T = 0.552, and occurs on the windward side of the body at the nose. For this value of T the hypersonic similarity parameter M oo T is 1.09, which satisfies the consistency condition of the theory. Since con- sistency is satisfied, the maximum absolute error in the surface static pres- sures, l~pl, that one should expect from this theory is about 19 percent of the free-stream dynamic pressure (see eq. (21)). The maximum absolute error in surface static pressure, or maximum difference between numerical and experimental pressures, for each of the cross sections of figure 18 is pre- sented in table 1. Table 1 shows that the errors are very much less than the maximum error calculated from the equivalence principle theory of Van Dyke.

It is believed that the static pressure errors in the vicinity of the body's nose would be of the order of 19 percent and would decay rapidly as T decreased. For example, at the station 0.502 diameter, the parameter T is 0.481, which results in a predicted absolute maximum error of 11 percent of free-stream dynamic pressure. Therefore, the numerical results are consistent with the equivalence principle theory of Van Dyke.

TABLE 1.- MAXIMUM SURFACE PRESSURE ERROR IN VARIOUS CROSS-SECTIONAL PLANES Axial station, l ~ pl/q, maximum cylinder diameter percent 0.502 5 2.990 1 1.289 3 3.970 5 4.920 4 5.830 4

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7.630 3

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Drag and lift coefficients were determined on the basis of numerical pressure data only; shear stress effects were not included. To compute these

I

coefficients, the pressure coefficients on the ogive-cylinder surface were

I

numerically integrated to determine the lift and drag. Comparison of the

I

numerical and experimental coefficients, as shown in figure 20, was found to be very satisfactory. The coefficients of total drag from the two sources I proved to be almost identical functions of distance from the nose of the

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ogive. Differences in the lift coefficient curves began to be significant at a distance of about 2.75 diameters from the nose of the ogive, increased to a maximum at about 4.5 diameters, and then decreased; the numerical lift coef- ficient was approximately 5.8 percent higher than that measured at about 4.5 diameters from the ogive nose, and about 2 percent higher at 8.35 diameters from the nose.

An overestimated lift coefficient is consistent with the fact that separation occurred earlier in the numerical flow field than in the experimen- J' tal field. Lift coefficients are calculated from pressure forces on the sur- face of the body, as projected in a plane parallel to that of free-stream

flow. Since the area of the appropriate projection is almost entirely derived I

from the body's long cylindrical surface, and separation takes place on that surface, separation that occurs too early increases the lift. I

On the other hand, the drag coefficient should be relatively independent I

of the location of separation. Since the ogive becomes cylindrical at a

station 3.0 diameters from its nose, the projected surface area normal to the I

direction of free-stream flow is very small for the long cylindrical surface aft of this station, and the corresponding pressure forces contribute very little to the total drag. Quantitative agreement ln the pressure drag coefficients is therefore physically reasonable.

RESULTS OF FUSELAGE PROBLEM For the fuselage problem the crossflow Mach number was 0.288 and the x crossflow Reynolds number was 1.15 l0 /ft. Cross-section flows have been calculated in this problem in planes normal to the 7-1/2° reference line to a horizontal station 19.5 inches aft of the nose. At this station the curve bounding the body's surface represents an asymmetric fuselage configuration with a canopy. To reach this station the problem was run 2079 cycles (time steps) on the UNIVAC 1108 computer. This calculation required approximately 6 hours of computer time.

From velocity vector plots of the cross-sectional flow field, the qualitative behavior of the fuselage flow can be examined. The sequence of events as one moves downstream along the central axis of the fuselage is indi- cated in figures 21 to 25. The vectors in these figures correspond to the particle velocities of the flow at each mesh point of the finite difference mesh; the tail of each vector corresponds to the mesh point. Initially, a bow shock forms at the nose and an expansion fan, caused by the interaction between the expanding body and the crossflow, appears on the leeward side.

This is indicated in figure 21 at a station 1.135 inches from the fuselage nose. This flow pattern continues as the fuselage cross section grows until it reaches the canopy which induces a shock wave in the flow field (see fig.

23). As the canopy radius reaches maximum (fig. 24) and starts to decrease, a rarefaction develops in the flow field above the canopy, leading to the formation of vortices (fig. 25).

The quantitative behavior of the predicted flow about the fuselage was ascertained by comparing the numerical results to experimental measurements.

I

L- _ _ The fuselage configuration of figure 6 was tested in the Ames 8- by 7-Foot Wind Tunnel. Static pressures were measured along the surface of the fuselage, and flow-field measurements with conical probes were made in cross-sectional planes normal to the horizontal reference line at a station 19.5 inches aft of the fuselage nose. The static-pressure instrumentation of the fuselage fore- body surface is sketched in figure 26. The pressure taps were located in cross-sectional planes normal to the horizontal reference line. The point of intersection of the 7_1/2° reference line with this cross-sectional plane defined the axis from which pressure tap locations were measured along the periphery of the cross section. Pressure taps were located _90°, _60°, -30°, 0 , and +24° from this axis (see fig. 26). Thus, with respect to stations along the horizontal reference line, the instrumentation was located along the five planes indicated in figure 26.

Numerical and experimental distributions of surface pressure coefficients are compared in figure 27 along the five planes of figure 26. It is seen that excellent agreement has been achieved along the -90°, _60°, and _30° planes.

The numerical and experimental pressure coefficients agree along the 0° plane until station 18. Along the 24° plane there is a discrepancy at station 15.

These discrepancies can be attributed to two sources : (a) a canopy shock wave - axial boundary-layer interaction, which occurs experimentally but not numerically, and (b) a slightly different calculated canopy shock location which will be described in more detail in the discussion of the flow-field results. The canopy shock-axial boundary-layer interaction spreads the shock pressure rise over a greater distance than calculated.

As in the case of the ogive-cylinder problem, an error analysis was made of the surface static pressures. The maximum slope of the fuselage geometry with respect to the free-stream flow occurred on the windward side of the body at its nose (see fig. 8). Its value was 22°, which corresponds to T = 0.400 and M oo T = 0.998. Therefore, the consistency condition of the theory is satis- fied. According to equation (22), the maximum absolute error then becomes 5 percent of the free-stream dynamic pressure. On the basis of the results of figure 27, the maximum error, /6P//q oo ' in each of the planes was determined and the values are tabulated in table 2. It is seen from the table that the errors recorded are consistent with the maximum error calculated from the equivalence principle theory.

TABLE 2.- MAXIMUM SURFACE PRESSURE ERROR ALONG THE FIVE PLANES OF FIGURE 33 Plane, deg !6 P!/q oo ' percent -90 2.0 2.3 -60 3.4 -30 0 5.6 24 5.8 At first glance, the relatlvely small errors of table 2 are surpr~slng since the hypersonic equivalence principle is applied at supersonic speeds.

One would expect only linearized supersonic theory to apply in this Mach num- ber range. However, an analysis of the errors shows that linearized supersonic theory is only slightly more accurate for this fuselage. The error in j

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l I I linearized supersonic theory is of order T which results in a maximilln error in static pressure of 4 percent of free-stream dynamic pressure. Thus, for a body with maximum slope, T, gr e at enough such that M oo T = 0(1), both the hyper- sonic equivalence principle and linearized supersonic theory give satisfactory results at supersonic speeds for M oo > 2.

Contour plots of the local pitot pressure ratio (Pp/P ), local sideslip s oo angle (S ), local Mach number (M), local angle of attack Cae)' and local total pressure ratio (Ps/P ) were generated from the numerical results and compared s oo to corresponding contour plots from experimental data. The comparisons were made in a cross-sectional plane normal to the horizontal reference line at a station 19.5 inches from the fuselage apex. See figures 28 through 32.

The local pitot pressure, P ' is effectively the stagnation pressure p measured by a probe whose axis is parallel to the local flow direction. As in the case of the surface pressures the pi tot pressure is measured directly; therefore, it provides a reliable measurement for comparison. The numerical .

pitot pressure was calculated from the Rankine-Hugoniot equations for a normal shock, where upstream of the shock the local Mach number and stagnation pres- sure were assumed to exist. The numerical and experimental pitot pressure ratios (Pp/P ) of figure 28 indicate good agreement in the lower quadrant of s oo the flow field, where the two sets of Pp/P contours are nearly coincident s in the vicinity of the body. This quant~tative agreement is in accord with the agreement obtained between surface static pressures on the -30 plane, which intersects the body in this region. In the upper quadrant of the flow field there is a discrepancy between the experi m ental and numerical contours.

The numerical results indicate that the canopy shock intersects the fuselage surface above that point indicated by the experimental data. The increased dO\IDward distance of travel by the canopy shock in the experimental case accounts for the discrepancy in contours in the upper quadrant of the flow field. Although axial boundary-layer effects and discretization errors influ- ence the canopy shock location, it is believed the equivalence principle approximation of neglecting the axial perturbation velocity is the primary cause of this discrepancy. The axial perturbation velocity is zero upstream of the body apex; thus the bow shock and windward flow field were calculated correctly. However, these perturbation velocities do exist in the vicinity of the canopy. The canopy shock is thus predicted imbedded in a flow field dif- fering from the actual situation, and consequently, the location and strength of the canopy shock do not agree with experimental data.

The local sideslip angle, S, is measured in the cross-sectional plane normal to the horizontal reference line and is the flow inclination in the x', z' plane (see fig. 9). As in the case of the pitot pressures, the com- parison of the experimental and numerical sideslip angle contours shown in figure 29 indicates that the predicted flow field in the lower quadrant is nearly correct. Furthermore, the predicted location of the canopy shock and the predicted flow-field contours in the upper quadrant deviate from experiment as in the pitot pressure comparisons.

Contours of constant Mach number and constant local angle of attack with respect to the horizontal reference line (see fig. 8) are presented in figures I I L- ____ _ - - - - - - - --- --------------- - - -

- - - - - - - - --

30 and 31, respectively. It is seen from these figures that the numerical Mach number and angle-of-attack contours do not match the experimental contours even in the lower quadrant of the flow field. This discrepancy is not in accord with the surface static pressure comparison, pitot pressure comparison, and sideslip angle comparison discussed previously.

The reasons for this disagreement can be found in discretization errors in the numerical method, errors introduced by the equivalence principle assumption, and errors inherent in the experiment and data reduction process.

The error in Mach number due to the equivalence principle has been determined previously (see eq. (28)). Equation (28), evaluated with T = 0.40 and M oo = 2.5, results in an absolute error of about 0.7 percent of the free~stream Mach number. This corresponds to an absolute Mach number error of about 0.02.

Consider the experimental contour of figure 30 for a constant Mach number of 2.50. It is seen from figure 30 that this contour coincides with the numeri- cal contour for a constant Mach number of 2 .40, which results in an absolute Mach number difference of 0.10. Thus, t he observed absolute difference in Mach number between the numerical and expe ri m ental results is five times the maximum error one would expect from the eq u ivalence principle alone. Also, it is well known that Mach number and angle of attack are much more difficult - to measure accurately than surface static pressures and pitot pressures.

Therefore, one must conclude that the experiment and data reduction process are, at least in part, responsible for these discrepancies.

The calculated local total pressure recovery contours (Ps/Ps oo ) are shown in figure 32. The predicted total pressure recovery varies from 1.0 to 0.88 in the flow field, with recoveries near 1.0 throughout most of the flow region. This parameter is the most difficult to calculate accurately and to determine experimentally. Although the experimental data for this fuselage configuration are still preliminary, an error analysis indicates that the values of Ps/Ps oo can be determined only within ±0.03. For a total pressure ratio bandwidth from 0.88 to 1.0, this error is too large to yield any significant contour data. Consequently, no experimental data are shown.

Future refinement of the set of conical probe calibration data will hopefully reduce this error to a more meaningful level.

Numerical lift and drag coefficients for the fuselage problem are compared to corresponding lift and drag coefficients determined from inviscid, linearized supersonic theory for flow about an axisymmetric slender body at angle of attack (ref. 14). Linearized supersonic slender body theory was used because there were not enough pressure data to determine the lift and drag experimentally. The lift and drag coefficients from the numerical method were determined - by numerical integration of the surface pressure coefficients over the surface and are based on a cross-sectional area corresponding to the maxi- mum equivalent radius of the fuselage, amax = 3 inches. As in the case of the ogive-cylinder problem, shear stress effects were not included. It is seen from figure 33 that the numerical drag coefficient distribution shows greater drag than the distribution from theory, due in part to the slenderness assump- tion of the theory. On the other hand, the numerical lift coefficient distri- I bution nearly corresponds to that of linearized supersonic theory. This I comparison does indicate that the lift and drag coefficients calculated by the I numerical method of this pa~er, if not correct, are at least of the right order.

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-- - ----- - -- - CONCLUSIONS A numerical method has been developed for calculating three-dimensional flow fields about slender bodies at supersonic speeds. The primary advantage of the method is that detailed predictions may be obtained for realistic fuse- lage configurations at angle of attack. Secondly, boundary-layer separation and development of spiral vortex sheets are predicted by the method. Major sources of error in the method are neglect of the axial perturbation velocity, neglect of axial viscous effects, and exclusion of any turbulence phenomena.

The method can adequately predict the surface pressures and flow-field characteristics outside the boundary layer for bodies where the hypersonic similarity parameter, M oo T, is unity or greater and for free-stream Mach numbers of 2 and above. For the ogive-cylinder and fuselage problems, the surface static pressure distributions differed from corresponding experimental values by no more than 6 percent of the free-stream dynamic pressure. These devia- tions are contained within the predicted error bounds of the equivalence prin- ciple. From velocity vector plots and contour plots, the structure of the calculated flow field appeared correct, at least qualitatively. Contour plots of pitot pressures were of the same general shape as the experimental contours for the fuselage configuration, and good quantitative agreement was obtained in the lower quadrant of the flow field. Since the error inherent in the equivalence principle becomes smaller as the Mach number is increased and with the condition M oo T = Oel) maintained, the accuracy of the numerical method will improve at higher Mach numbers. Additional comparisons between numerical results and experiment at higher Mach numbers would be desirable, however, to strengthen this conclusion.

The positions of boundary-layer separation points and vortex centers were not accurately predicted by the numerical method. Neglect of axial viscous effects and exclusion of any turbulence phenomenal which were observed experi- mentally, are considered responsible for these inaccuracies in the computed results. From the two problems which were run, it was not possible to deter- mine the relative effects of viscosity in the axial and crossflow directions or to determine to what extent exclusion of turbulence affected the numerical results. Solution of another problem duplicating the body geometry and flight conditions of an experimental test at hypersonic speed and in the laminar flow regime would provide more understanding of the relationship between axial and crossflow viscous effects. At this flight condition differences between numer- ical and experimental results are attributable predominantly to neglect of axial viscous effects since errors in the equivalence principle are minimized and turbulence is not a consideration.

--------------------------------~------------------ APPENDIX A THE EQUIVALENCE PRINCIPLE In this section the equivalence principle for steady, inviscid three- dimensional flow is derived. An Eulerian coordinate system fixed with respect to the body and having its z axis along the axis of the body is used throughout this section (see fig. 2).

The equations of motion for steady, inviscid three-dimensional flow about a body are: Continuity a a a

ax (pu) + a-y(pv) + az [p (U cos a + w)] = 0 (AI)

oo Momentum au au au 1 ap U- ' +v-+ (w + U cos a) - - (A2a) 00 az - ax ay p ax 1 ap av av av (w + U (A2b)

u-+v-+ cos a) - =

az ax oy p ay ow oW aw 1 ap (A2c)

u ax + v ay + (w + U cos a) az = - p az

oo First law I

' oE oE

u-+v-+

(w + U cos a) ~ = -P ru ~ 1. + v ~ 1. + (w + U cos a) ~ 1:.J

ax oy

00 az L ax P oy p 00 az p

(A3) I Boundary conditions at body I l I (U cos a + w)t + vt + u~ = 0 (A4a) I 00 z y x ! Boundary conditions at shock (A4b) I , , , I 2 2 2

I Poo + pooUoo(cos an + sin any) = P + ps[(U cos a + ws)nz + vsny + usnx]

z s oo

I

(A4c) ---------------~---~----------------------------------------~ oo y P 1 U2( .)2 -- + -2 00 cos an + Sln any z 1 y - P oo (A4d) where E is the internal energy per unit mass, P is the pressure, p is the density, U is the free-stream speed, w is perturbation velocity in the z oo direction, v is the velocity in the y direction, u is the velocity in the x direction, and £x, £y, and £z are the direction cosines of the body nor- mal in the x, y, and z directions, respectively. The subscript s refers to properties downstream of the shock, and n ' ny, n are the direction x z cosines of the shock normal in the x, y, and z airections, respectively.

Boundary conditions far from body z = _ 00 :

w = a (A4e)

v = U sin a , u = 0 oo IntrodUCing the transformation equations z = U (AS)

y = y

x = x

and the slender body assumption U »w into equations (AI) through (A4) yields Continuity (A6) ~ + ~(PU) +~ (pv) = 0 at ax ay Momentum

oW 1 ap 1

aw aw (A7a) -+ u- +

v ay = - pat

u cos a at ax

av + u av + v av = 1 ap

(A7b)

at ax ay - p ay

(A7c) -- ------------ - - --- - ---------- - -- First law aE -+ (AS)

at

Boundary conditions at body (A9a) Boundary conditions at shock (A9b)

P + p U (-cos a sin S + sin a cos S)2 = P + ps(vsny + usnx - U cos a sin S)2

s oo 00 00 00 (A9c) p U (-cos a sin S + sin a cos S)2 _Y~::- _00_ + "" y - 1 Poo p y s ( U sl'n Q) 2 (A9d)

= Y _ 1 P; + vsny + usnx - "" cos a ~

where S is the angle made by the shock surface with respect to the z axis in the y, z plane. Equations (A6), (A7b), (A7c), and (AB) are independent of wand, thus, represent the equations for time-dependent motion in the x, y plane. Therefore, the steady 3-D equations of motion have been trans- formed to a time-dependent 2-D set of equations. The 3-D shock boundary con- I ditions also reduce to a nonsteady shock in the x, y plane moving with I velocity proportional to U"" cos a sin S. In summary, the necessary and suf- ficient conditions for the equivalence principle to be valid are: 1. The z component of the local velocity vector must be approximately equal to the z component of the free-stream velocity vector, and 2. The time-dependent sol . ution in the x, y plane must satisfy the three-dimensional boundary condition equation (A9a) at the body surface.

The calculational procedure for inviscid flow is then as follows: First, equations (A6), (A7b), (A7c), (AS), and (A9a) are solved for v, u, P, and p. The perturbation velocity w may be subsequently obtained from the Bernoulli equation, which is (w + U cos a)2 + v + u y P (AlO)

y - 1 P + ----------~2~----------- = constant

Therefore, the equivalence between a steady three-dimensional flow and a time-dependent two-dimensional flow is proven.

If the conditions (1), (2) are satisfied, viscous effects can be included in the time-dependent calculations in the x, y plane without vio- lating the equivalence principle assumptions. A no-slip boundary condition can be applied at the surface of the cross section in the x, y plane in conjunction with the equivalence principle boundary condition equation CA9a).

I'~ ~-- I I APPENDIX B CONSERVATION OF TOTAL ENERGY In this section the self-consistency property of form that the finite difference equations (8) to (13) possess is demonstrated. This demonstration proceeds in three steps. First, the finite difference kinetic energy equation is derived for the momentum zone a~_(1/2)' a~+(1/2)' Then the finite differ- ence first law equation is derived for this momentum zone. Finally, these relations are added to determine the finite difference equation of total energy for the momentum zone a~_l' a~-(1/2)' The kinetic energy equation for momentum zone a~_(1/2)' a~+(1/2) can be derived from momentum equation (12) which is centered at the time n t -(1/2). Equation (12) can be written in terms of velocities by employing the forward extrapolation relation, equation (13): n n U +(1/2) _ U -(3/2)

~ ~ = [pn-(1/2) _ pn-(1/2)]

(Bl) ~t ~-(1/2) ~+(1/2) Multiplying equation (Bl) by U~-(1/2) yields the kinetic energy equation for momentum zone a~_(1/2)' a~+(1/2)' I un- (1/2) [pn- (1/2) _ pn- (1/2)J I ~ ~- (1/2) H(1/2) (B2) I I The first law equation for momentum zone a~_(1/2)' a~+(1/2) is derived as follows: the internal energy for this zone equals half the sum of the internal energies of thermodynamic zones a~_l' a~ and a~, a~+l (see eqs.

I

I

(9) and (10)). This division of internal energy comes directly from the I relationship between thermodynamic and momentum zones specified when the

I

physical model of figure 6 was postulated. The -P(~v/~t) term for momentum I zone a~_(1/2)' a~+(1/2) comes from half the -P(~v/~t) for thermodynamic I zones a~_l' a~ and a~, a~+l' The first law equation for momentum zone I a~_(1/2)' a~+(1/2) becomes: I

I

I

I I I

295 !

____ ~ __ J

n n ] n-l n-1 (1/2) E~_(1/2) + E~+(1/2) - (1/2) E~_(1/2) + E~+(1/2) m~ ~t n-1

i P~-(1/2)

= { n-1 n } n- n -

+ _P_~+-->-(1....:.../_2~) .,.....+_P_~_+-"-(1~/_2~) [u (1/2) _ U (1/2)]

(B3) 2 ~+l ~ The finite difference equation for conservation of total energy for momentum zone a~_(1/2)' a~+(1/2) results from the addition of equations (B2) and (B3).

n = [wD-(1/2) _ W -(1/2)] (B4) ~- (1/2) ~- (1/2) where n n U -(1/2)U +(1/2) n ~ ~ [ n n ] 1 H = JI,

E~_(1/2) + E~+(1/2) 2 +

n-1 n .. (1/2) pn n + U -(1/2) + p U~ Wn- (1/2) ~-(1/2) ~- (1/2) 5/,-1 ::: ~- (1/2) 2 2 n n-1 n- n U (1/2) + U -(1/2) + P JI,+ (1/2) n PJl,+(1/2) JI,+l JI, W -(1/2) ::: ~+ (1/2) Equation (B4), derived from finite difference analogs of the first law and momentum equations, is a reasonable finite difference expression for total energy conservation. The rate of work done on the surface, having the n-(1/2) Lagrangian coordinate aJl,_(1/2) at time t , is the product of the time-averaged pressure in the thermodynamic zone aJl,_l' aJl, and the space- averaged velocity between surfaces a n_ and an. The internal energy of the No 1 n No momentum zone a~_(1/2)' a~+(1/2) at t1me t, 1S half the internal energy of the two thermodynamic zones that contain it. Finally, the kinetic energy I

I

I

L __ _

r~~ - - - . -~~- -------------- ---~- - ------- ----------- is the product of the of momentum zone a -(1! 2) ' a +(1! 2) at time tn, t t n -(l! 2) and tn+(l! Z ). It is t velocities at the surface a t , at times believed that this self-consistency of form property, which the differential equations possess, and which the finite equations preserve, is the primary reason for their success in numerical calculation of one-dimensional time- dependent flow fields.

REFERENCES 1. Gallo, W . F.; and Rakich, C. V.: Investigation of Methods for Predicting Flow in the Shock Layer Over Bodies at Small Angles of Attack. NASA TN 0-3926, 1967.

2. Van Dyke, M. D.: A Study of Hypersonic Small-Disturbance Theory. NACA Rep. 1194, 1954.

3. Hayes, W. D.: On Hypersonic Similitude. Quarterly Appl. Math., vol. 5, 1947, pp. 105-106.

4. Lees, L.; and Reeves, B. L.: Laminar Near Wake of Blunt Bodies in Hypersonic Flow. AIAA J., vol. 3, no. 11, 1965, pp. 2061-2074.

5. von Neumann, J.; and Richtmyer, R. D.: A Method for the Numerical Calculation of Hydrodynamic Shocks. J. Appl. Phys., Vol. 21, no. 3, March 1950, p. 232.

6. Richtmyer, R. D.: Difference Methods for Initial Value Problems. Wiley (Interscience), New York, 1957.

7. Trulio, J. G.; and Trigger, K.: Numerical Solution of the One- Dimensional Hydrodynamic Equations. UCRL 2667, 1961.

8. Trulio, J. G.: Studies of Finite Difference Equations for Continuum Mechanics. WL-TDR-64-72, 1964.

9. Trulio, J.: Theory and Structure of the AFTON Codes. Tech. Rep. AFWL TR- 66-19, 1966.

10. Wa1itt, L.: Numerical Studies of Supersonic Near-Wakes. Ph.D. Thesis, Univ. of California, Los Angeles, 1969.

11. Trulio, J. G.; Niles, W.; and Carr, W.: Calculations of Two- Dimensional Turbulent Flow Fields. NASA CR-430, 1966.

12. Jorgensen, L. H.; and Perkins, E. W.: Investigation of Some Wake Vortex Characteristics of an Inclined Ogive-Cylinder Body at Mach 1.98. NACA RM A55E31, 1955.

13. Schlichting, H.: Boundary Layer Theory. McGraw-Hill Book Co., Inc., 1955, p. 19.

14. Liepmann, H. W.; and Roshko, A.: Elements of Gas Dynamics. John Wiley and Sons, Inc., 1962, pp. 239-247.

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FLOW FIELD IN FUSELAGE CROSS SECTION PLANE Figure 1 APPLICATION OF THE EQUIVALENCE PRINCIPLE TO AN AXISYMMETRIC BODY SHOCK CROSS SECTION PLANE Of BOOY CROSS SECTION Of RADIUS Q UNSTEAOY ANALOCY SHOC K r- - ---l ---;-f--, "'" ___ ~---T\ : : / -:t: /" :

~~-_J)P ~, Q;H

J

C,----U U .. U..... L .. _ ,j. ___ -.-l

dz t • ZlU cn EQUIVALENT THREE DIMENSIONAL STEADY PROBLEM TWO DIMENSIONAL UNSTEADY PRO B LEM Figure 2 ---------------- ----- -------------------------------- --- -- - ------------ --- - I

I

, I

I

I I I

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FLOW FIELD ABOUT AN AXISYMMETRIC BODY I I AT ANGLE OF ATTACK Figure 3 FLOW REGIONS ABOUT A BODY MOVING AT SUPERSONIC VELOCITY BOW SHOCK TRAILING SHOCK DIFFUSING INNER CORE REAR STAGNATION POINT REGION OF RECIRCULATION BASE REGION Figure 4 -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- ONE-DIMENSIONAL, TIME -DEPENDENT MOTION IN LAGRANGIAN COORDINATES PARTICLE TRAJECTORIES IN A LAGRANGIAN COORDINATE SYSTEM 01 a 02 EULERIAN COORDINATE, X (a, t) THERMODYNAMIC AND MOMENTUM ZONES FOR ONE -D IMENSIONAL TIME -DEPENDENT MOTION THERMObYNAMIC THERMObYNAMIC ZONE ZONE I I MOMENTUM I I ZO I NE

i i

L

x Figur e 5 SCHEMATIC DIAGRAM OF A FINITE DIFFERENCE MESH 3 4 4 5 - I QUADR I LA TERAL QUADRI LA TERAL ZONE ZONE 0 a b 2 r= ----i 5

'- - --- + -- ---

I I I I MOMENTUM ZONE I I I

lQ

Q l

6 2 : : I I I I I I L _ _ ___ , __ __ --.J d c c QUADRILATERAL QUADRILATERAL ZONE ZONE 9 8 7 Fi gu re 6

I

I

I

_J

OGIVE-CYLINDER BODY GEOMETRY L 2 y = -S.75d + jS5.5d (Z-3dl y IS0° 0 0 90 $-90 d=I.125in.

~ O ~

~ e

9.2Sd

'1

Figure 7 SCHEMATIC OF FUSELAGE CONFIGURATION FLOW CONDITIONS

Moo = 2.5 I

Re =9. lxI0 /ft oo

a'l5" D

7.5 REFERENCE LINE

HORIZONTAL REFERENCE LINE ~ ~ ~ ____ ~ ==

1 ~ _ s: ~---- --=-- -------

Uoo~ I I

FUSELAGE 19.5" CENTRAL AXIS STA 0 .0" 9.0" 15.0" Figure 8

- - - - - - - - - - -

DEFINITION OF FUSELAGE CROSS-SECTIONAL PARAMETERS 7 1/2 REFERENCE y' LINE y z' HORIZONTAL x' REFERENCE X LINE Figure 9 FINITE DIFFERENCE MESH FOR OGIVE-CYLINDER RADIUS 0.0 I ft CROSS FLOW - Figure 10

I

_

~

- - - - - ~ - - - - - - - _ .

FINITE DIFFERENCE MESH FOR OGIVE-CYLINDER RADIUS 0 . 046875 ft CROSS FLOW- Figure 11 FINITE DIFFERENCE MESH AT STA 7.0 CROSS FLOW- F ig u re 12

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I

I

--------------- -------------- ------ FINITE DIFFERENCE MESH AT STA 25.0 CROSS FLOW- Fi gure 13 VELOCITY VECTOR PLOT OF FLOW FIELD 0.502 MA X BOD Y DIAMETERS FROM NOSE RADIUS 0.0 1 4 5 ft Figure 14 VELOCITY VECTOR PLOT OF FLOW FIELD 2.99 MAX BODY DIAMETERS FROM NOSE RADIUS 0.046875 ft

" - -- -

----

Figure 15 VELOCITY VECTOR PLOT OF FLOW FIELD 4.92 MAX BODY DIAMETERS FROM NOSE RADIUS 0.046875ft - .. ,," .

- - -

"", '/ / ' .. ~ ~"I"'#~ "/ /' / ... "',/ . / " .

- -

Figure 16 VELOCITY VECTOR PLOT OF FLOW FIELD 8.35 MAX BODY DIAMETERS FROM NOSE - - - - - - - - - - .-.-.- - - - - - - .

. -- ---.--.--.-:.- Figure 17 OGIVE -CYLINDER SURFACE PRESSURE DISTRIBUTION DATA OF JORGENSEN .5 ZlD AND PERKINS (REF 12) .4 0.502 -- NUMERICAL METHOD .3 c.

u .2 r: .1 z w u u..

u.. 0 w 2.99 u w a: 4.92 ::::J en 0 5.83 en w a: 7.63 0..

- . 1 I I I I -.2 0 20 40 60 80 100 120 140 160 CIRCUMFERENTIAL

ANGLE, e, deg

Figure 18

J

TRAJECTORIES OF SEPARATION AND VORTEX CENTER POINTS 180 - EXPERIMENTAL VORTEX

L ~:~E~~:~ECTORY

co 160 -

--

W

---

-..J VORTEX CENTER TRAJECTORY (!)

Z _____ / NUMERICALLY CALCULATED <{ -..J 140- <{ -- i=

-

z W

--- r ---------

0:: W EXPERIMENTAL SEPARATION ~ 120- POINT TRAJECTORY :::J u 0:: U SEPARATION POINT TRAJECTORY NUMERICALLY CALCULATED I I I 100 L 5 678 9 10 AXIAL STATION, z/d Figure 19 COMPARISON OF NUMERICAL AND EXPERIMENTAL LIFT AND DRAG COEFFICIENT DISTRIBUTIONS OGIVE -CYLINDER BODY o .28 A 6 6 6 6 6 6 6 6 '" .24 .20 ...J u .16 0:: o J'.12 .08 .04 7 8 9 o 2 3 4 5 6 STATION, z/d Figure 20

I

~ VELOCITY VECTOR PLOT AT STA 1.135 ---~ .-----~~--- - - - ---~~ --- .. -~-------- - - Fi gure 21 VELOCITY VECTOR PLOT AT STA 10.0 ---- ----- ----- Fi gure 22 VELOCITY VECTOR PLOT AT STA 12.0 --:;.--

-

---- ----- Figure 23 VELOCITY VECTOR PLOT AT STA 17.0

------

...................... ..

Figure 24 VELOCITY VECTOR PLOT AT STA 19 .6 .--=----~ Fig ure 25 STATIC PRESSURE INSTRUMENTATION ON FUSELAGE FOREBODY LANE ' INTERSECTION OF 7. 50 deg REFERENCE LINE WITH PLANE ~ 2~ -30 • STATIC TAP _ 60

~

-90 PLANE, deg HORIZONTAL REFERENCE LINE o -30 _ -90

~2== ~_JUtIt;b~:±l~t::P c=.- -60

I

0 STA 19 . 5" 7.5 REFERENCE LINE Figur e 26 COMPARISON OF NUMERICAL AND EXPERIMENTAL SURFACE PRESSURES EXPER I MENT PLANE - NUMERICAL RESULTS p

C :! ~ ~-L-L~ ~~~ __ t~t~ ~~~~ g ~ =_ ~ J g __ ~~- 3 L g-L ~ -L ~_4~t~

.2 ~

C . ~ p - . 1

.2 ~

p

C . ~~--~--------------=-~~~~~~~~~~

-.I IIIIIIIIII ~~~

C p _. '0 [ t--~------=--~-===~1.!::::::: Ll = ' = Ll ~

[ I I I I I I I I I o 2 4 6 8 10 12 14 16 18 20 FUSELAGE STATION. in .

Figure 27 COMPARISON OF NUMERICAL RESULTS WITH EXPERIMENT CONTOURS OF LOCAL PITOT PRESSURE RATIO FUS STA 19 .5 Moo = 2 . 50 Reoo =9 . 1 xlO /ft a = 1 5°

°v ~ ~G<~

NUMERICAL EXPERIMENT METHOD

.r)! )/

,,~/

0.48 0.52 0.54 0. 50 F igu re 28

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L

COMPARISON OF NUMERICAL RESULTS WITH EXPERIMENT CONTOURS OF LOCAL SIDESLIP ANGLE, f3 FUS STA 19.50 NUMERICAL EXPERIMENT METHOD Figure 29 COMPARISON OF NUMERICAL RESULTS WITH EXPERIMENT CONTOURS OF LOCAL MACH NUMBER, M FUS STA 19.50 2. 50 2.45 2.40

~ ~

SHOCK '\,. .-" /'

NUMERICAL --.-"

EXPERIMENT METHOD

2~

2.55 2.50 2.45 2.40 2.35 2 . 35 Figure 30 COMPARISON OF NUMERICAL RESULTS WITH EXPERIMENT CONTOURS OF LOCAL ANGLE OF ATTACK, Q e FUS ST A 19.50 CANOPY 19 SHOCK ~18 17 /"18

u/ "

NUMERICAL EXPERIMENT METHOD t2d!:: 12 II Figure 31 CONTOURS OF LOCAL TOTAL PRESSURE RATIO, Ps/Psoo FUS STA 19.50 .99 Figure 32 LIFT AND DRAG COEFFICIENT DISTRIBUTIONS FUSELAGE CONFIGURATION .20 -- NUMERICAL METHOD --- INVISCID, LINEAR THEORY .16 FUSELAGE CROSS SECTION IS CIRCULAR UP TO THIS STATION u...J .12 cr o UO .08 . 04 o 2 4 6 8 10 12 14 16 18 20 STATION, in.

Figure 33 DISCUSSION JAN RAAT, General Dynamics/Convair: Assuming that computing costs can be kept within reasonable limits - which is perhaps a bit optimistic at this time - numerical attacks on the hypersonic small-disturbance equations may hold some promise for unsteady flow fields, that is, flow problems in which the given body shape carries out a time-dependent motion. The equivalence principle holds true for unsteady flow. In that case one might consider a series of "planes of unsteady analogy" that follow each other at discrete time intervals.

WALLACE D. HAYES, Princeton University: I find myself in the somewhat peculiar position of trying to put down the equivalence principle, so I think a couple of remarks on what I think the proper role of the equivalence principle is are in order.

The equivalence principle was established primarily to establish a similitude, and as with most similitudes, the principal value is conceptual.

It leads to an intuition and an understanding of the problem.

I believe that this type of approach to the problem of calculating supersonic flows is sound, and the way the equivalence principle fits in is that it will lead you to this particular approach.

The way the equivalence principle then ought to be applied is that it should be used strictly in the equations in the process of testing out numeri- cal schemes to find out what will not work, what will work, what works very well. But when you actually set up a computational scheme for actual computa- tions you should use the exact equations. They are not appreciably more com- plicated than the time-varying equations from the equivalence principle and you are not trying to establish a similitude. In considering x as though it were a time you are already using whatever you have acquired from the equivalence principle as leading to understanding.

WALlTT: I wonder if Dr. Trulio would like to make a comment abo~t that?

JOHN TRULIO, Applied Theory, Inc.: We used the equivalence principle in this way because to calculate three-dimensional flow directly would lead us to a very difficult implicit numerical problem. Here we are approximating (and making some error in the process) the exact statement of the problem that you feel we really should address. I think in the end the justification (for our procedure) has to lie in the comparison between actual data and the computed solution.

I

I would prefer myself to work toward the goal of transient three- r dimensional numerical schemes (and we are) which make no approximation other than discretization assumptions, and which hopefully can be shown to lead to

I

vanishing error to the limit of finer and finer meshes.

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The work reported here is, in a sense, a digression from the main path I we have followed, but I think it may be a useful one.

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- - - - - - - - - - - - - - -- - -

- - - - - - - - - -- LARS E. ERICSSON, Lockheed Missiles and Space Co.: I think that when you try to expand this to the unsteady case you have to be very cautious.

You have to make sure that you include the actual viscous flow effects, the boundary-layer collection on the leeward side with associated separation and free-body-vortex generation. I think some results obtained here at Ames by Tobak, Peterson, and others show, for instance, that at high angles of attack you can have free-body vortices on cone cylinder bodies, where the complete effect induced downstream by those vortices is determined by the relative crosswind at the conical nose section. .

WALITT: I don't know what effect the crossflow at the nose section has on the lee vortices. However, when we start a problem, the calculations in the vicinity of the body nose are quite inaccurate, simply because we cannot afford to make the zoning fine enough at the nose of the body. Due to this

l

coarse zoning, we have recorded oscillations in properties, such as the pres- sure, at the body nose. However, the oscillations dampen out and the answer approaches the correct answer at a station within a half diameter of the body's nose.

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UNIDENTIFIED MEMBER FROM THE FLOOR: You have no way of including the

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effects of axial flow?

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WALITT: Not exactly. However, we could include the turbulence effect.

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For example, in the ogive-cylinder problem, we did calculate flow separation and vortex formation. However, the separation point occurred about a diameter

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upstream from where the experimental separation point occurred. This devia- tion was a result of axial viscous effects and turbulence. The crossflow

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Reynolds number in the plane of calculation was laminar, whereas the actual axial Reynolds number was probably turbulent.

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We believe we could, in a very empirical way, account for the axial

I

turbulence effect. This can be done by adding crossflow eddy terms to the equations of motion. However, the effects of the axial viscous terms in the

I

equations of motion would be more difficult to include.

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JACK N. NIELSEN, Nielsen Engineering and Research, Inc.: I was wondering, were these data on the ogive-cylinder the Jorgensen and Perkins data?

WALITT: Yes.

NIELSEN: I know that they measured the position of the vortex core with the aid of the vapor stream technique and pi tot tubes. Did you compare your vortex position with these experimental ones?

WALITT: Yes.

NIELSEN: Were they in good agreement?

WALITT: No. At each sectional plane, the centers of the numerically calculated vortices were about 20° closer to the windward side of the body

I

than the centers of the experimentally determined vortices. The separation

I

- -- - -- - -- - -- - -- --- -- also occurred earlier in the numerical calculations. The axial viscous terms and turbulence, which were not included in this method, account for this discrepancy. As I said before, both effects can be included in this method.

NIELSEN: Another comment, too. It seems to me that insofar as you are not getting separation, the question on the equivalence, whether the equiva- lence is applicable or not is exactly the same question as of whether the body is slender, in the sense of slender-body theory. Considering the body and the Mach number, it doesn't look like it is too slender, and therefore some of the points you made about that equivalence rule being applicable or not might be ques ti onab 1e .

WALITT: The deviations between the numerical and experimental data did conform to the maximum error predicted by Van Dyke for the equivalence princi- ple at these flight conditions and for this geometry.

NIELSEN: Did he do it for viscous or inviscid?

WALITT: Inviscid, in the NACA paper he presented in 1954. However, it is possible to include crossflow viscous effects without violating any of the conditions of equivalence.

WILLIAM J. EVANS, Grumman Aircraft Engineering Corp.: For one thing, I notice that your pressure data does not go up over the canopy. Did you take data in that region?

WALITT: Yes, we have numerical data allover. The problem was the wind-tunnel test. They obtained data only on the five planes which I referred to in my talk. I would like to see another experiment run in conjunction with another numerical calculation. The experiment should be specifically designed for the numerical calculation, so that we could make a better evaluation of

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the numerical results.

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ROBERT E. MELNIK, Grumman Aircraft Engineering Corp.: This question is directed more to Wally Hayes. I don't think anyone said, but I think the

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equivalence principle is basically a hypersonic thing. You are neglecting r the streamwise perturbation and velocity. You can do this in the inviscid part of the flow field if the disturbances are small and the Mach number is

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high, and you can't do it at the boundary layer, so that for the Mach numbers you are doing it at, I think Wally was saying that it really doesn't apply r to this case except in conceptual terms.

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WALLACE D. HAYES, Princeton University: It applies only inviscidly.

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MELNIK: Well, he made his calculations for Mach 2.5.

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WALITT: The Mach number is important, but so is the hypersonic similarity parameter, MooT, which in this case turned out to be 0.998. Van r Dyke's theory predicts that the maximum error in a property is proportional

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to T2, where T is the maximum slope of the body with respect to the free- stream flow direction. Since M T is of the order one, the error is of the

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r

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order M -2. For this case the maximum static pressure error was 5 percent co of the free-stream dynamic pressure. The static pressure errors we recorded were about 2 percent of free-stream dynamic pressure.

MELNIK: What you are saying is that hypersonic small disturbance theories apply to 15° angle of attack at Mach number 2.5.

WALITT: For this particular body and at this angle of attack it applies.

The only way you can prove that is to measure the maximum slope of the body and determine the errors. For relatively thick bodies in the supersonic flight regime, the theory applies.

ELY S. LEVINSKY, Air Vehicle Corp.: First I would like to know if there is a viscous equivalence principle.

WALITT: No, not that I know of. What we have effectively done is included viscous crossflow effects without violating the equivalence principle theory.

LEVINSKY: If you were to continue this calculation downstream on your ogive-cylinder, would you find a change in the position of the vortex?

WALITT: Yes, it starts and the vortex center grows with distance LEVINSKY: And the boundary layer would be ageing somehow?

WALITT: Yes, the crossflow boundary layer would change.

LEVINSKY: Well, you probably haven't done this, but it would be of interest to look at these boundary-layer calculations that you make, say, for a cone, and compare with available solutions, for instance that by F. Moore for a cone, which is a three-dimensional boundary-layer calculation.

WALITT: We would have to redo the boundary-layer calculation. In other words, this method gives the pressures on the body and the flow field away from the body outside the boundary layer. In the area of the body itself, the calculations are not correct because axial effects are excluded. But you could take the calculated pressures and go back into a three-dimensional boundary-layer program and estimate what the boundary layer would be like as a second step.

LEVINSKY: I am trying to get at how we evaluate the separation point that you predict and the formation of the vortices.

WALITT: The extent to which the viscous crossflow determines separation determines the accuracy with which we can predict separation.

The only case tried was the ogive-cylinder body, and we predicted separa- tion 1 diameter upstream of where it actually occurred. Now, both axial vis- cous effects and turbulence phenomena are the causes of the deviation. At this point we didn't know which effect is more important. However, with more research, we believe we can include these effects in the method.

LEVINSKY: Can you really predict separation if you only have two or three points inside the boundary layer, like you said?

WALITT: Yes. However, I don't think our separation point on the periphery of the body is very accurate. It might be plus or minus a couple of degrees on either side of the correct value, and the axial location might be a quarter of a diameter from the correct station. However, there is no reason to make it three mesh points. We did this becaus e of economic factors. We could have made it ten, fifteen, or twenty points in the boundary layer and have gotten a closer prediction.

LEVINSKY: Just so we understand this, was the calculation made essentially at a cross flow Reynolds number of a million?

WALITT: The fuselage calculation was made at a crossflow Reynolds number of about a million per foot.

LEVINSKY: Because all the other calculations of this type that we have seen in the literature basically are much lower Reynolds numbers, this is a significant achievement.

WALITT: Well, it is a million per foot and the maximum body cross section is a half a foot, so it is 500,000.

LEVINSKY: This is still large compared to the other calculations which are on the order of a thousand.

WALITT: Yes, but in this particular case it was one million per foot.

That is why we were only able to get three points in the boundary layer.

RAAT: It seems to me that the preceding discussions hav.e left unclarified some of the details concerning the applicability of the unsteady analogy. It may be useful to recall that the equivalence principle is based on the hypersonic small-disturbance equations and this implies the double

limit M + 00, T + ° (T being a characteristic inclination of the stream-

lines with respect to the oncoming flow). Under these conditions the stream- wise velocity perturbation is expected to be an order T smaller than the lateral velocity perturbation. This makes it possible to set the streamwise velocity equal to its undisturbed value and to treat the lateral flow problem separately. Now, at the indicated Mach numbers of 1.98 and 2.50, one would not expect hypersonic small-disturbance theory to be very accurate even in the inviscid part of the shock layer. At such low supersonic Mach numbers it seems more realistic to assume that the inviscid streamwise and lateral velocity perturbations are of the same order of magnitude.

JERRY C. SOUTH, JR., Langley Research Center: My question is really out of ignorance; I have never done anything with Navier-Stokes equations, and I haven't ever used a dissipative difference scheme, but somebody showed a paper by McCormick to me last night that I believe was an inviscid calculation of the flow past a sharp - cornered, blunt body, and he had no physical viscosity; it was an inviscid flow model, yet there was an eddy around the corner, and it was apparently caused by the dissipation of his difference scheme.

I

L

My question is, if you have dissipative terms in your difference scheme, could you set your physical viscosity to zero and still get the eddy around the canopy?

WALITT: I don't think so. We tried a problem like that in another case.

We at one time calculated a Karman vortex street about a cylinder at Mach 0.2 and Reynolds number 100, and in that particular problem we were interested in what would happen if we set the viscosity to zero. This was done and we obtained the potential flow solution around the cylinder.

We feel that there is a numerical viscosity as well as the dissipative viscosity; there is no question about it, but we feel that if the mesh is fine enough the numerical viscosity is small.

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----------- - - - - - - - -- - ---- -------- - - - - - - --- - -- ---- EXACT SOLUTION FOR RADIATION OF SOUND FROM A SEMI-INFINITE CIRCULAR DUCT WITH APPIlCATION TO FAN AND COMPRESSOR NOISE By Donald L. Lansing Langley Research Center INTRODUCTION Calculating the rotational noise radiated from the compressor of a modern turbofan engine is currently a problem of great interest. At present there exist two approximate analytical procedures for treating this problem. One of these was developed under con- tract to NASA by M. V. Lowson while at Wyl e Labor atories (ref. 1). The other was sketched out by J. M. Tyler and T. G. Sofrin of Pratt & Whitney Aircraft Division of United Aircraft Corporation in reference 2. This paper presents a third, an exact theory, which is compared with the two Simpler approximate methods.

The three theoretical models used in the present investigation are shown in table I.

The first model, developed by Lowson (ref. 1), consists of an unducted rotor or rotor- stator combination. This model predicts the free-space radiation properties of the noise source and hence may be described as a "source-only" model. This model has a minimum of mathematical and conceptual complications. However, it is not possible to add acoustic liners, to include shear flow through the duct, or to account for the modal structure of the sound within the duct without starting all over again with a new theory.

The second model, which was suggested by Tyler and Sofrin (ref. 2), consists of a hard-wall annular duct which is terminated at the open end by an infinite baffle. Tyler and Sofrin discussed the transmission of individual modes along the duct and their radiation from the open end; thereby, the general mathematical form was obtained for the sound field. However, they did ' not combine the duct modes to represent the noise sources within the compressor. The baffle is an artificial device included only for the purpose of calcu- lating the radiation patterns in front of the duct. Furthermore, this model neglects any reflection or coupling phenomena which occur at the open end. In spite of these simplifi- cations, the model is very attractive because it does incorporate a duct using tractable mathematics and can act as a stepping stone toward analytical studies of ducts containing acoustic treatment and nonuniform axial flow.

The third model, developed in this paper, consists of a semi-infinite circular duct .

with hard walls. One end is open to the free field, and this open end is unbaffled so as to resemble a sawed-off pipe. A uniform free-stream velocity parallel to the duct axis is assumed. This flow may be used to simulate the effect of a steady flight condition on the radiated noise or, if one is concerned only with conditions within the duct, to simulate the effect of the inlet or exhaust flow on the noise transmission along the duct.

The contribution of the present paper is twofold. First, the ideas laid out by Tyler and Sofrin (ref. 2) have been synthesized so that now the duct modes are combined to rep- resent thrust and torque forces on the rotating blades. The fact that there is an unambig- uous mathematical procedure for summing hard-wall duct modes deserves mention since J many questions seem to arise on the matter of how the sound energy should be distributed among the various modes. The second and main contribution of the present paper consists I of generalizing the work of Levine and Schwinger (ref. 3), Carrier (ref. 4), and Vajnshtejn J (ref. 5) on the transmission and radiation of single modes from an unbaffled circular pipe to include flow effects, high frequencies, and high circumferential and radial mode num-

I

bers. Furthermore, these individual mode results are combined together to represent thrust and torque forces on rotating blades. The coupling of the sound field within the

I

duct to the external free field is treated rigorously within linearized acoustic theory so that a realistic assessment of the diffraction and reflection phenomena at the open end is possible. Thus, the model of the present paper provides an exact solution which is use-

I

ful in appraising the validity of approximate methods.

I

SYMBOLS

I

An(x,r;u,:>",w) function defined by equation (9) j a duct radius

I

number of rotor blades B

j

c speed of sound

I

force vector acting on fluid arising from thrust and torque on rotating blades spanwise loading function Fn,j(r) function defined by equation (11) h(a,:>..,M) Bessel functions k frequency parameter, w/c

L __

l it nondimensional frequency parameter, aw/c or nB(an/c) distance of rotating blades from inlet l M Mach number, U/c tip Mach number integers n,j perturbation pressure p Q mass source term associated with blade thickness

s source function

source spectrum t time

I

free-stream velocity, positive into duct U

I

u,A,a transform variables I

v number of stator vanes

I

perturbation velocity vector v I x,r cylindrical coordinates, x positive out of duct

I

Y(A) = VA - (MA - k)2 angle around duct axis

e

mth zero of J ~(x) ambient density

I

azimuth angle I shalt speed w frequency A prime with a symbol denotes differentiation with respect to argument.

THEORY The governing differential equations for the mathematical formulation of the radia- tion problem are the linearized equations for the continuity of mass 1 Dn - -=.!:.+P V·v=Q (1) c2 Dt 0 and the conservation of momentum DV - (2)

Po - = -Vp + F

Dt where D a a -=- - u- Dt at ax These two equations may be combined into an inhomogeneous wave equation for the pres- sure as follows: 1 D2 V p=-:!:C£+S (3) 2 2 c Dt The source function S depends upon the gradient of F and ~ as shown in the fol- lowing equation: (4) The mathematical problem is to find a solution of equations (1) and (2) for p and v which satisfies the following auxiliary conditions: (1) There must be no flow through the duct wall.

(2) The radiation condition which requires the source within the duct to generate I only outgoing waves at infinity must be satisfied.

(3) The perturbation pressure must be continuous everywhere in the free field as well as through the open end into the duct.

i (4) The radiated energy generated by the source must be finite. This condition

places a restriction on the behavior of p and v at the sharp rim at the open end.

This problem constitutes a mixed boundary-value problem which can be solved exactly by using a combination of separation of variables and complex variable methods.

The quantities F and Q can be assumed known from the geometry and operation con-

, ditions of the compressor. Consequently, S is known throughout some region within the

I duct. By separating the variables where possible, the wave equation with its complicated

source term S can be transformed to a more manageable problem which can be solved by using the Wiener-Hopi technique (ref. 6).

By using Fourier series around the duct axis, Fourier integrals for time and in the axial direction, and Hankel integrals in the radial direction, the source term Sex, r, e, t) can be broken down into its spectrum Sn(U,A,W) as shown in the following equation: +00 +00 ine

S(x,r,e,t) = ~ L e SS 1 Sn(U,A,W) In(ur) e-i(Ax+wt)u du dA dw (5)

(27T) n=- 00 _ 00 0 The spectrum can be evaluated directly from the source distribution by using Similarly, the pressure can be represented in terms of its spectrum as indicated in the following equation: +00 +00 ine

p(x,r,e,t) = ~ L e SS Sooo Sn(U,A,W) An(x,r;u,A,w) e-iwtu du dA dw (7)

(21T) n=- 00 - 00 The unknown function An(x,r;u,A,w) must be determined so that p(x,r,e,t) satisfies the wave equation (eq. (3». The governing differential equation for An(x,r;u,A,w) is

a2 1 a n 8 ( 8 ')J -iAx

(8)

-2 + - -a - -2 + - - M - + lk An(x,r;u,A,w) = In(ur) e

8r r r r 8x2 ax

~

This is a Helmholtz type equation which has only two independent variables x,r rather than the four original variables x,r,8,t. The important transformation accomplished is that the forcing function on the right-hand side of the equation is no longer a complicated function of space and time but is a comparatively simple function of just the right form for the rest of the mathematics.

The problem is now reduced to finding An(x,r;u,A,w). Once this has been accom- plished, equations (6) and (7) make it possible to determine the pressure for any S(x,r,8,t) distribution. The method used for finding An(x,r;u,A,w), which satisfies equation (8) and the auxiliary conditions, depends upon a knowledge of the analytical properties of the Fourier transform of An(x,r;u,X,w) in the complex transform plane.

This method is referred to as the Wiener-Hopf technique. The final solution for An(x,r;u,X,w) is given by the following equation: iAx

. _ -In(ur) e- 1 roo ~n(ry(a» K~(ay(a»

(0 ~ r ~ a~ .

1ax e- da An(x,r,u,X,w) - 2 + - An(a) , u + yZ(A) 27T -00 ~(ry(a» In(ay(a» (a ~ r) (9) This solution is the sum of two terms. The first term accounts for the inhomogeneous term in the differential equation for An(x,r;u,A,w) and the second term takes care of the auxiliary conditions. The second term is a contour integral whose integrand has two forms; one within the duct, 0 ~ r ~ a, and another outside the duct, a ~ r. The func- tion An(a), given by the equation An(a) = 2i u J~(ua) y(~ h(a,A,M) (10) 1 - M2 ru2 + y2(X)l(A + _k_) K(n) (A) (a _ A) (a __ k_) K(n)(~ ~ LJ 1-M + 1+M - where

hMa- k

(0 ~ M < 1)

h(<l',A,M) = l~A - k

(-1 < M ~ 0) depends upon the transform variables in the source spectrum U,A,W and contains n functions Kt ) (A) and K~n)(a) which are themselves defined by contour integrals of the following type:

1 K(n)(A) = _1_ S+oo loge t2K~(a'Y(~» I~(a'Y(~)~ d'"

(11) oge + 2 . '" - A ., 1Tl -00 ., Before proceeding to any results, it will probably be helpful to discuss briefly the nature of the source distributions which occur in fan and compressor noise problems and show the basic expression for the far-field pressures. The sketch shows several variables of the problem. For simplicity, the effects of

\--<

mean flow are ignored. The sound generating mechanism is the thrust and torque forces acting on the rotating blades. These forces are periodic in nature and hence may be expanded in Fourier series. The individual terms in this Fourier series have the form i Fn)r) e [(nB-jV)8-nBnB where the function Fn,j(r) depends upon the spanwise varia- tion of the loading.

Each such source term contributes to the radiated noise field. The far-field pres- sure arising from one such source term as predicted by the present theory is an infinite series as shown in the following equation: ~~ ______ ~~~ ______ ~J ~~ ______________ ~~ ______________ ~/~ Loading factor Directivity factor Propagation factor (12) where

v = nB - jV

Each term in the series arises from a single radial duct mode J v(i fJ. ) and consists

m of a loading factor which depends upon the spanwise distribution F n,j (r), a directivity factor which determines how the pressures vary with the azimuth angle l/J measured outward from the duct center line, and an important propagation factor which determines how efficiently a mode carries sound energy along the duct to the inlet. A mode for which

I

J

the square root is imaginary is said to be "below cutoff." The power of e is then a negative real number so that the sound pressures decrease rapidly with l. A mode for which the square root is real is said to be "above cutoff." The exponential function varies sinusoidally with l and represents a wave traveling along the duct. These duct modes are very efficient carriers of sound energy. Because of the large number of rotating blades and high tip speeds, compressors generally operate with many modes above cutoff. On the other hand , ducted propellers usually have far fewer blades so that at subsonic tip speeds they are operating entirely below cutoff.

RESULTS AND DISCUSSION Some of the results obtained from the three radiation models applicable to the fan and compressor noise problem are presented. Recall that these models are the Lowson model based on the source alone, the Tyler and Sofrin model which uses a duct ending in an infinite baffle, and the present model which consists of an unbaffled duct.

Figure 1 shows the variation of the sound power due to thrust and torque forces on a blade element at 0.8 of the duct radius with the nondimensional frequency . parameter k.

The radiated sound power is the total sound energy produced by the source and hence is an indication of its noise generating efficiency. For nB - jV = 5, ducted fans with subsonic tip velocities operate in the range k < 5, whereas compressors are associated with k = 10 to 28. It can be seen that results obtained from a synthesis of Tyler and Sofrin's ideas are in excellent agreement with the present theory over the entire frequency range.

At certain values of k, the present theory appears to have nearly vertical discontinuities.

These spikes are actually very narrow resonances which occur at the natural acoustic frequencies of the duct cross modes. The Tyler and Sofrin model shows very abrupt increases in the radiated sound power at these frequencies but does not peak to indicate a resonance. However, both duct models indicate large sound pressures within the duct near resonance. The difference in the behavior in the free field is believed to arise from the fact that the Tyler and Sofrin model neglects the coupling of the internal and external acoustic fields through the open end. For the length-to-radius ratio used, the Lowson model and the two duct models differ by 10 to 20 dB (re unity) in the low k range. At higher frequencies the two duct models give results which are at least 5 dB below the source-only model. There are broad frequency ranges where the difference is as much as 20 dB. For the conditions shown here the Lowson model appears to greatly over- estimate the radiated noise.

Figure 2 shows the effect of varying the number of stator vanes on radiated sound power. The number of rotor blades B is held fixed at 16. The tip Mach number Mt of the rotating blades is 0.6. For very low and very high numbers of stator vanes the duct is operating below cutoff, hence the steep dropoff in sound power at either end of the graph. When the number of stator vanes is between 16 and 26 blades, both theories give essentially the same results. However, when the number of stator vanes is between 6 and 16, the sound power levels predicted by the present theory are as much as 10 to 15 dB less than those predicted by the Lowson model. The Tyler and Sofrin model gives results (not shown) which are indistinguishable from the present theory.

Figure 3 shows a comparison of sound directivity patterns calculated for the three models. Directivity patterns show how the radiated sound pressures vary as a function of azimuth angle l/J measured outward from the center line of the inlet. These patterns are polar plots of the far-field sound pressure levels in a plane through the duct axis.

As indicated at the upper left of the figure, the origin of the directivity plot is taken at the center of the inlet of the duct. Horizontally to the left is directly ahead of the inlet, and vertically upward is at right angles to the inlet. The directivity pattern as predicted by the present theory consists of one main lobe at about 50 to the duct axis. The direc- tivity pattern computed from the Tyler and Sofrin model is in good agreement with pres- ent theory except near right angles to the inlet where the Tyler and Sofrin model assumes a baffle. As mentioned earlier, radiation patterns behind the inlet cannot be predicted with this model. The results from the Lowson model are obviously all together different in size and number of lobes. In particular, sound pressures at right angles to the inlet appear to be greatly overestimated.

CONCLUDING REMARKS This paper has outlined an analytical method for investigating the radiation of noise generated by rotating blades within a hard-wall circular duct. The method is based upon a rigorous solution of the wave equation plus boundary conditions and hence provides an exact set of calculations against which Simpler theories can be compared. The method accounts for the diffraction effects at the end of the duct and is capable of giving the com- plete radiation pattern. Reflection effects of the inlet are also included. These effects appear to produce resonances in the radiated sound power at the frequencies of the acoustic modes. It has been found that the baffled duct model is in excellent agreement with the present method for both radiated sound power and directivity patterns in front of the inlet. However, it was found that the source-alone model did not agree well with the present method.

REFERENCES 1. Lowson, M. V.: Theoretical Studies of Compressor Noise. NASA CR-1287, 1969.

2. Tyler, J. M.; and Sofrin, T. G.: Axial Flow Compressor Noise Studies. SAE Trans., vol. 70, 1962, pp. 309-332.

3. Levine, Harold; and Schwinger, Julian: On the Radiation of Sound From an Unflanged Circular Pipe. Phys. Rev., vol. 73, no. 4, Second ser., Feb. 15, 1948, pp. 383-406.

4. Carrier, G. F.: Sound Transmission From a Tube With Flow. Quart. Appl. Math., vol. XllI, no. 4, Jan. 1956, pp. 457-561.

5. Vajnshtejn, L. A. (J. Shmoys, transl.): The Theory of Sound Waves in Open Tubes.

Propagation in Semi-Infinite Waveguides, Res. Rep. No. EM-63 (Contract No. AF-19(122)-42), Inst. Math. ScL, New York Univ., Jan. 1954, pp. 87-116.

6. Noble, B.: Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations. Pergamon Press, Inc., c.1958.

TABLE I THEORETICAL MODELS FOR FAN AND COMPRESSOR NOISE RADIATION MODEL INVESTIGATOR REMARKS DESCR I PTI ON SOURCE ONLY - MATHEMATICAL AND CONCEPTUAL SIMPLICITY -NO DUCT LOWSON -ACOUSTIC LINERS AND SHEAR FLOW CANNOT (REF. 1)

iF

BE INCLUDED ANNULAR DUCT - ACOUSTI C LI NERS AND SHEAR FLOW CAN TYLER WITH BAFFLE BE ADDED SYSTEMA TI CALLY AND SOFRIN -USE OF BAFFLE IS QUESTIONABLE

~ (REF. 2)

CIRCULAR DUCT - EXACT TREATMENT OF REFLECTION AND WITHOUT BAFFLE DIFFRACTION EFFECTS OF INLET LANSING

(PRESENT () - D I FFI CULT TO GENERALIZE

»=g

PAPER) RADIATED SOUND POWER DUE TO THRUST AND TORQUE TORQUE/ THRUST = 0.75; nB - jV = 5; l/ a =1 0.----------------------------- -10 -20 I SOUND POWER, I -30 I dB I PRESENT THEORY I I -40 I LOWSON (REF. 1) I I -50 I o SYNTHES I ZED FROM I TYLER AND SOFRIN (REF. 2) I -60 0 4 8 16 20 28 k Figure 1 EFFECT OF STATOR VANE NUMBER ON SOUND POWER TORQUE/THRUST = 0.75; B = 16; Mt = 0.6 0.----------------------------------- / ....

/ ........

-10 \ -20 \ I \ I I \ SOUND POWER , 30 dB - \ \ I I \ -40 I \ I \ I -- PRESENT THEORY \ I -50 I \ - - - - LOWSON (REF. 1) \ \ -60 '-- __ --'--l. __ -'- __ ---'. ____ L- __ ...l...- __ -'-- __ .L.L __ --...J o 4 8 12 16 20 24 28 32 NUMBER OF STATOR VANES Fi gure 2 DIRECTIVITY PATTERNS DUE TO THRUST AND TORQUE TORQUE/THRUST = 0. 75; nB - j V = 5; k=12 LOWSON (REF. 1) SYNTHES I ZED FROM TYLER AND SOFR I N (REF. 2) PRESENT THEORY .....

/ \ / I 40 30 20 10 o 10 20 30 40 SOUND PRESSURE LEVEL,dB Figure 3 A CRITICAL EVALUATION OF PANEL FLUTTER AERODYNAMICS By Peter A. Gaspers, Jr.

Ames Research Center SUMMARY Generalized forces based on Piston theory, quasi-steady theory, unsteady two-dimensional potential flow, and unsteady three-dimensional potential flow are compared. Curves are presented of typical generalized force compo- nents as functions of Mach number and length-to-width ratio. More comprehen- sive comparisons between Piston theory and unsteady three-dimensional theory in the form of the maximum difference of all off-diagonal components of a 6x6 matrix of generalized force components are presented as functions of Mach number, length-to-width ratio, and reduced frequency.

INTRODUCTION In linear supersonic panel flutter analyses there are four aerodynamic theories in common use. These are the linear Piston theory, quasi-steady theory, unsteady two-dimensional potential flow, and unsteady three- dimensional potential flow. Usually the simplest theory is used that will yield acceptable results for the Mach number and length-to-width ratio of interest. However, no detailed comparison of the predictions (i.e., pressures or generalized forces) of the various theories is available, and at present the choice of a particular theory is usually based on order of magnitude con- siderations or on previous flutter results. Within the framework of linear aerodynamic theory the unsteady three-dimensional potential flow may be con- sidered exact and the other theories approximate. In this paper generalized forces based on the approximate theories are compared with the exact three- dimensional results to provide a quantitative measure of their ranges of applicability.

SYMBOLS a panel length Fourier coefficient (see eq. (25)) a· J b panel width Fourier coefficient (see eq. (26)) b· J wind-tunnel width B c .,d. see equation (29) J J Galerkin expansion coefficient

en

chordwise deflection mode spanwise deflection g(y) i imaginary unit, ~ j integer Mk K

f32

wa k V Z.,m,n integers M Mach number dimensionless perturbation pressure, p* = qp p(x,y,t) component of complex perturbation pressure Pn(x,y) pV2 q dynamic pressure, --2- generalized force component (see eq. (4)) see equation (18) t dimensionless time, t* = (V)t u(x,y,t) transverse panel deflection nth mode shape ~(x,y) v free-stream speed w downwash velocity (see eq. (9)) dimensionless rectangular coordinates, x* ax, y* by x,y B Yz.

I

~)2 2J1I2

r· ~2p(3b + K J

~

E. see equation (24) J dimensionless rectangular coordinate 8 see equation (27) dimensionless rectangular coordinate p free-stream density see equation (9) or (20) <j>(x,y,t) dimensionless velocity potential component of complex velocity potential w circular frequency Superscript dimensional quantity * ANALYSIS The coordinate system and panel geometry are shown in figure 1. The analytical formulations of the various theories in terms of the perturbation pressure p(x,y,t) and the transverse deflection u(x,y,t) of the panel and derivations leading to expressions for the generalized forces are given below in dimensionless form. The relationships between dimensional and dimension- less variables are given in the list of symbols.

MODIFIED LINEAR PISTON THEORY (REF. 1) The perturbation pressure is given by

X Y t) = ~ (au + au)

( P

" (3 ax at

The panel deflection is taken in the form N ikt u(x,y,t) (2)

L Cntin (x,y) e

n=l where the ~ are specified functions. Hence, p(x,y,t) has the form: N ikt (3)

p(x,y,t) = L CnPn(x,y)e

n=l The generalized force components Q are defined by: mn 1 1 (4)

Q = j[ j[ ~(x,y)Pn(x,y)dx dy

mn o 0 we consider Um(x,y) = sin mnx sin ny which are eigenfunctions for a plate with simply supported edges. A straight- forward analysis yields: mn

[1 _ C-l)m+nJ}

Q = mn 2 _ n 2

i {m

m , nl

(5 ) k Q = i rnm

2B

QUASI-STEADY THEORY (REF. 2) The perturbation pressure is given by ~ [au + (M2 - = 2) au] p(x,y,t) u(O,y,t) (6) S ax M2 1 at

°

This differs from Piston theory only in the factor (M2 - 2)/(M - 1), and the resulting generalized forces are (7) . (M2 - 2) k nm

Q = ~ M2 _ 1 2B

UNSTEADY TWO-DIMENSIONAL POTENTIAL FLOW (REF. 3) The perturbation pressure is given by

a¢ a¢)

(8)

p(x,y,t) = -2 ax + at

(

where the velocity potential ¢(x,y,t) is given by: (9) Hx,y,t) where W(X,y,t) M(x - 0 M(x - 0 T2 = M - 1 M + 1 Let N ikt (10) u(x,y, t) = 1: Cn~(x,y)e n=l and make the transformation yielding x TI n

au . k ) -iMK (x-O L

Hx,y, t) at.: + 1 un e (

I

o 0 (11) Using (ref. 4) where Jo(Z) is the Bessel function of the first kind of zero order results in N ikt

<t>(x,y,t) = L Cn<pn(x,y)e

n=l (13) (14) (15) integrating by parts and using

llm(O,y) = llm(l,y) = °

results in (16) Following the method of references 5 and 6 the Bessel function is approximated by a sum of trigonometric functions in order to perform the integration for <Pn: i

__ -L ~ e (x-OK cos [(Z + 4) rJ

l (l7)

Jo[K(x - ~)] L...J

Z=Q where L must be chosen to yield the desired accuracy. Let urn = sin mnx sin ny Substituting into equations (12) and (16) and performing the integrations there result m -f n (18) [(_l)m cos YZ - 1] Q rnm (19) where Mk K = S2 The derivation of the generalized force expressions given here closely parallels that of reference 7. The expressions for Q contain removable mn singularities at YZ = mn, and numerical accuracy will be impaired if YZ is sufficiently close to mn. One method of avoiding this problem is to expand Qrnn in a Taylor series in YZ about mn. Another method is to interpolate numerically in the vicinity of a singularity.

THREE-DIMENSIONAL UNSTEADY POTENTIAL FLOW (REF. 3) d <P d <P)

p(x,y,t) = -2 dX + at

( where (20) <P(X,y,t) au au W(X,y,t) -+

at ax

2 2

b S f/2

s)2 R (y - [(X - 0 - a a(x - a(x - s) = Y + = y - ~2 S1 bS bS M2(x M2(x - E,;) - MR - s) + MR = = '1 '2 Let N ikt for 1 u(x,y,t) = Cnun(x,y)e O.s.x.s.

1: 1 1 n=1 -2~y~2 elsewhere (21) = 0 where the un are specified functions. Then <p(x,y,t) N (22) = 1: n=l (23) Now Sl or s2 or both may lie off the panel (outside the unit square) in which case the inner integral of equation (23) cannot be readily evaluated.

Kobett (ref. 8) circumvented this difficulty by expanding the spanwise part of un in a Fourier series valid over the entire range of ~. One can also account for the presence of wind-tunnel walls by constructing an image system for the spanwise deflection and representing it as a Fourier series. Let - -- - - - - -- - g(z;) = 0 and let gpCZ;) be the periodic function obtained by reflecting g(Z;) and its images in ~alls 2p (dimensionless) apart where p = B/2b and B is the dimensional distance between the walls. Then g ( z; ) gp ( z; ) ~p

= I z; I

g ( - z; 2p) -2p < z; < -p

=

g (- z; + 2p) z; < 2p P < g ( z; + 4p) for all z; gp ( z; )

=

p and the Fourier series is cos j n z; + b. sin j nz; ) (24) 2p J 2p j = 0

o j I 0

2P j nz; g ( Z; )d Z; (25) cos

= 1- i

aj 2 2p P p - 2p 2P I i .

(26) b· = - sin ~ g ( Z; )d Z; J 2p 2p P - 2P M aking the transformation (27) in equation (23) results, after some manipulation, in 4>n(x,y) J

= - ~ L (Eja cos ~;Y + b sin ~;Y)Iax [f (0 + ikfn( ~ )]e-ikM(x-O J [ (x-Of j]d~

n ' j j o j=O (28) The integral in equation (28) is in essentially the same form as equation (12) for the two-dimensional case. From the two-dimensional result, the general- ized force expressions are obtained as follows: + mn(m 2 -n 2 )TI 2 (BjZ+k) 2 [1 - (_l)m cos B ] jZ - i(-1)mmn(m 2 -n 2 )TI 2 (BjZ + k) 2 sin B } (29) m f n jZ where B K P - 2b B = tunnel width 1/ 2 j ny 1/ 2 J' ny d· = sin g(y)dy c· cos --- g(y)d y J

J - 1/ 2 zp-

-1/ 2 2p 1

As in the two-dimensional cas~ expressions (29) contain removable singular- ities that may be handled as previously discussed. The summation limits J and L are chosen to give the desired accuracy. Th e derivation given here is similar to that of reference 8. It should also be noted that thr e e- dimensional generalized forces have also been calculated by the numerical inte- gration technique known as th e Box method. See references 9 and 10, for example.

RESULTS AND DISCUSSIO N In figure 2 the absolute value of a typical diagonal generalized force

component IQ111 is plotted as a function of Mach number fo~ a square panel

and a reduced frequency of 1.0 for each of the four theories. It is well

known that the quasi-steady theory breaks down in the vicinity of M = 1:2, and

the figure clearly shows this in a quantitative way . It is also of interest that, for M > 1.3, the simplest theory, Piston t h eory, gives the best results for this combination of parameters.

In figure 3 a typical off-diagonal component Q i s plotted for the same parameters as in figure 2. The agreement is ve ry good for Mach numbers above 1.3 . Note that in this case Piston theory and quasi-steady theory yield identical results. In figure 4 Q is shown as a function of length-to-

width ratio for M = 1.6 and k = 1.0. It is usually stated in the lit e rature

that Piston theory and quasi-steady theory should be restricted to Mach numbers above l.~ but limitations on alb are hardly ever given . It is apparent from the figure that some restriction on length-to-width ratio is also needed since the approximate theories are in error by about a factor of 3 at alb = 10.

For alb > 1.0 the Piston theory predictions are generally comparable to or better than those of either the quasi-steady theory or two-dimensional theory; therefore, if an approximate theory can be used, Piston theory is preferable because of its simplicity. For these reasons, in what follows only Piston theory results are compared to the three-dimensional results. Also, it is known from flutter studies that the off-diagonal components of the gener- alized force matrix dominate the diagonal components in the parameter range where Piston theory is applicable. Since each component of the Piston theory generalized force matrix in general has a different error, in what follows the maximum percent difference between Piston theory and three-dimensional theory for all off-diagonal components of a 6x6 matrix is used as a measure of the error. In figure 5 the percent difference is shown as a function of length-to-width ratio for several Mach numbers and a reduced frequency of 1.0.

For a given Mach number the error rises sharply beyond a certain le ngth-to- width ratio and the alb at which the rise begins increases with increasing Mach number. In figures 6 and 7 contours of constant percent difference are plotted in the M - alb plane for reduced frequencies of 0.2 and 2.0. From these figures the ranges of M and alb for a given error can be conveniently obtained.

CONCLUDING REMARKS On the basis of the results presented it appears that when Piston theory or quasi-steady theory is used, limitations should be placed on length-to- width ratio as well as Mach number, and to a lesser degree, the reduced fre- quency should be considered. The maximum acceptable error in the generalized forces depends on the particular problem in which they are used,but a conveni- ent cutoff is indicated by the point where the rapid rise in error with increasing length-to-width ratio occurs.

L

REFERENCES 1. Ashley, H.; and Zartarian, G.: Piston Theory - A New Aerodynamic Tool for the Aeroelastician. J. Aero. Sci., vol. 23, no. 12, Dec. 1956, pp. 1109-11lB.

2. Fung, Y. C.: The Flutter of a Buckled Plate in a Supersonic Flow.

AFOSR TN 55-237, California Institute of Technology, July 1955.

3. Garrick, I. E.: Nonsteady Wing Characteristics. Vol. VII of High Speed Aerodynamics and Jet Propulsion. A. F. Donovan and H. R. Lawrence, eds., Princeton Univ. Press, Princeton, New Jersey, 1957.

4. Erdelyi, A., ed.: Higher Transcendental Functions. Vol. II, McGraw Hill Book Co., Inc., 1953, Formula 2, p. Bl.

5. Fettis, H. E.: Numerical Calculation of Certain Definite Integrals by Poisson's Summation Formula. Mathematical Tables and Other Aids to Computation 9, July 1955.

6. Luke, Y. L.: Simple Formulas for the Evaluation of Some Higher Transcendental Functions. J. Math. Phys., vol. 34, Jan. 1956, pp. 29B-307.

7. Luke, Y. L.; and St. John, A.: Supersonic Panel Flutter. WADC TR 57-252, July 1957.

8. Kobett, D. R.: Research on Panel Flutter. NASA CR-80, 1964.

9 . Cunningham, Herbert J.: Flutter Analysis of Flat Rectangular Panels Based on Three-Dimensional Supersonic Unsteady Potential Flow. NASA TR R- 256, 1967.

10. Dowell, E. H.; and Voss, H. M.: Theoretical and Experimental Panel Flutter Studies in the Mach Number Range 1.0 to 5.0. AIAA J., vol. 3, no. 12, Dec. 1965, pp. 2292-2304.

COORDINATE SYSTEM AND PANEL GEOMETRY

1 y. ~

b 1--------, v

-

L- ____ ---l __ ... _ x. t a Figure 1 TYPICAL DIAGONAL GENERALIZED FORCE COMPONENT AS FUNCTION OF MACH NUMBER alb = 1.0 .t = 1.0 1.6 - 1.4 - 1.2 - \ -, \ \ \ \ \ \ \ \ \ 1.0 - \ \ IQIII \ \ \ , .8 - \ \ _____ 3-D \ '"\ 2-0 , .6 - PISTON .4- QUASI-STEADY .2 - O'-----'----I>£--'--_....L.....----I 1.0 1.2 1.4 1.6 1.8 2.0 M Figure 2 TYPICAL OFF-DIAGONAL GENERALIZED FORCE COMPONENT AS FUNCTION OF MACH NUMBER 3.0 _ a/b=l.o /; = 1.0 2.5 2.0 - " \ / 2-D " ' <" : \ 3-D , , , I ~ -- , I 1.5 , ,

: "

I " 1.0 - QUASI-STEADY .5 AND PISTON o 1-' _ __'__--'-_.J.-_-'------' 1.0 1.2 1.4 1.6 1.8 2.0 M Figure 3 VARIATION OF TYPICAL OFF-DIAGONAL COMPONENT WITH LENGTH - TO-WIDTH RATIO M = 1.6 /; = 1. 0 1.2 -

------- ,, ----

-- = ~'=--- "'2 -D " 3-D " """ 1.0 ~ QUAS I -STEADY " AND PI STON " , , , .8 - , , " , , , , , , , , , .4 - ' ....

"" .... -

.2 - 0,- 1 __ ~ ___ ~_~__'_~ __ ~ ___ '__~~.J._~ .1 . 2 . 4 .6 .8 1.0 2 4 6 8 10 alb Fig ure 4 PERCENT DIFFERENCE BETWEEN PISTON AND 3-D THEORIES FOR OFF -DIAGONAL GENERALIZED FORCE COMPONENTS J = 1.0 140 - 1 20 - w (6X6 MATRIX) ~ 100 - w 0:: ~ 80 - u..

Ci !z 60- w U 0:: ~ 40- 20 - .2 . 4 . 6.81.0 2 4 6 8 10 alb Figure 5 CONTOURS OF CONSTANT PERCENT DIFFERENCE BETWEEN PISTON THEOR Y AND 3-D THEOR Y FOR OFF- DIAGONAL COMPONENTS

e = 0 .2

9 - 8 - (6X6 MATRIX) 7 - % 6 - 5 - alb 4 - 3 - 2 - I - o I L- __ ---'- ___ --' I 2 3 M Figure 6 CONTOURS OF CONSTANT PERCENT DIFFERENCE BETWEEN PISTON THEORY AND 3-D THEORY FOR OFF-DIAGONAL COMPONENTS /; = 2.0 % 9 - 8 - 7 - 6 - 5 - alb 4 - 3 - (6x6 MATRIX) 2- 1- o ,L- __ ----'- ___ -1 I 2 3 M Figure 7 DISCUSSION HOLT ASHLEY, Stanford University: I guess if Wally Hayes can knock the equivalence principle I can knock piston theory, although it is worth pointing out that the version Mr. Gaspers has been using in his paper was originally developed by Dr. Peter Jordan in Germany during World War II in its linearized form.

I think very wisely you didn't try to state any general conclusions about when that method can be used, but looking at the curves, one might conclude that for the relatively higher aspect ratio panel you can get away with taking piston theory down to a pretty low supersonic Mach number. I don~t think that conclusions is incorrect for the higher aspect ratio panels that you described here. But for the benefit of those aero-elasticians that might not know too much about aerodynamic theory, I would like to issue a warning about the use of the piston theory generally at the lower Mach numbers. In our original paper on the subject (Ashley and Zartarian, J. A. S., 23, 12, Dec. 1956) we suggested that for wing flutter calculation probably a Mach number, normal to the leading edge, in excess of 2.5 is a fairly safe rule on the basis of practical calculations.

That brings in the fact that some panels and also a great many wings for supersonic aircraft are swept, and I think it would be very dangerous to draw a conclusion about even h~gh aspect ratio swept wings and panels on the basis of calculations made essentially for what amounts to unswept wings.

A. R. GEORGE, Cornell University : I am not an aero-elastician at all, but it struck me that the fact that you can use narrower panels at higher Mach numbers is related perhaps to the fact that the Mach angle is narrower and I was wondering whether normalizing the aspect ratio by the tangent of the Mach angle would collapse these curves, such as in figure 11, to more nearly one curve?

GASPERS: I don't think there is any analytical way of doing what you suggest. The three-dimensional generalized forces are complicated functions of Mach number and aspect ratio and in addition the curves of figure 11 are envelopes of all the off-diagonal elements of a 6x6 generalized force matrix.

GEORGE: The theoretical reason is that the aerodynamic influence of a part of a panel is limited by its downstream Mach cone.

GASPERS: It might be possible to normalize the curves in an approximate manner, but I have not investigated this.

HARRY L. RUNYAN, JR., NASA Langley Reserach Center: I just might make one comment about piston theory. I think it is a very valuable tool, and we used it on the X-IS where we were worrying about the flutter of the vertical tail, and we wanted to clear the airplane for flutter for around Mach 6. We ran flutter tests of small wedges, since we had a tail with a cross-section wedge, and three-dimensional linear theory was 400 or 500 percent in error, whereas piston theory agreed with the experiment, and this was because we could include the effect of thickness using piston theory.

GASPERS: You mean by including nonlinear term?

RUNYAN: Yes. We had more terms than you had. In piston theory you can expand the pressure expression to the number of terms you want, which brings in new nonlinear terms. It is really a very valuable theory from Mach 2 and higher.

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RECENT ADVANCES IN RESEARCH ON COMPRESSIBLE TURBULENT BOUNDARY LAYERS By Ivan E. Beckwith Langley Research Center SUMMARY A review is given of significant developments since 1967 in methods for computing compressible turbulent boundary layers . Because of the indeterminate nature of relations between the fluctuating flow correlations and mean flow quantities, the theoretical predic- tions must be evaluated by comparisons with experimental data. Some limited compari- sons are shown for the three main categor i es of t heoretical approaches which are herein designated as (1) integral methods, (2) finite-difference solutions of the partial differential equations, and (3) correlation techniques . Selected methods from each category are described in some detail so that possible reasons for differences in predictions can be identified. New developments in experimental methods and results are also reviewed briefly.

Two principal conclusions are offered. First, the greatest need at the present time is for more detailed and more reliable experimental data for both mean and fluctuating flow properties within turbulent boundary layers for a wide range of Mach numbers, pres- sure gradients, and wall-temperature gradients. Measurements of surface shear stress and heat transfer should also be obtained when possible. Second, while the integral methods are of value for developing and asseSSing correlations of profile parameters and trends or "laws" for skin friction and heat transfer, the finite-difference methods appear to have significant advantages for future requirements.

One of the principal advantages of the finite-difference methods is the conceptual simplicity that can be utilized in the models of the turbulent-flux terms. Since detailed predictions of nonsimilar mean flow profiles, surface shear stress, and heating are obtained, the reasons for poor predictions of experimental results can be identified and appropriate modifications can be easily made. Through the use of this process which has been called "numerical experimentation," basic understanding of turbulent-flow mecha- nisms can be improved. Another advantage of the finite-difference methods is that they can be more easily extended to include detailed effects of normal pressure gradients, to account for unusual boundary conditions such as surface mass transfer and external vor- ticity , and to compute three-dimensional boundary layers.

INTRODUCTION The purpose of this paper is to review briefly the more significant improvements and extenSions, developed since 1967, of methods for predicting turbulent-boundary-layer characteristics. The state-of-the-art for incompressible flows, at least for two- dimensional and zero-mass-transfer applications, has been extensively treated at the 1968 Stanford conference on turbulent boundary layers (see ref. 1). A principal conclu- sion of this conference was that (ref. 1, p. 479) "A considerable number of methods can now predict integral parameters for incompressible 2D flows nearly as well as we have any right to expect in view of the uncertainties indicated by the momentum imbal- ances in the data." This statement emphasizes that reliable experimental data for a broad range of flows and conditions are essential in both the development and evaluation of theoretical approaches. In order to assess the accuracy of the predictions, "standardized" sets of data are desirable such as were tabulated in reference 2 for the particular problem area treated at the 1968 Stanford conference. The objectives and principal results of this conference were reviewed by Morkovin and Kline at the 1968 Langley symposium on turbulent boundary layers (see paper no. 2 of ref. 3).

Several review papers are available that present some excellent discussion and assessments of theoretical developments up through 1967. Spalding reviewed theoretical methods in reference 4 and advocated the use of an effective-eddy-viscosity hypothesis and numerical solutions of the partial differential equations. Also available are three review papers by Bradshaw - a brief but general treatment of boundary-layer problems (ref. 5), a more complete review of turbulent boundary layers (ref. 6), and a review (ref. 7) of extensions to the method developed by Bradshaw, et al. (ref. 8) for utilizing the turbulent kinetic energy equation in numerical methods. Azzouz and Pratt (ref. 9) have reviewed the turbulent-mixing problem. This latter problem has also been treated recently (ref. 10) by a method which utilizes the turbulent-kinetic-energy equation.

Earlier reviews of theoretical methods and experimental data have been given by Thompson (ref. 11), Rotta (ref. 12), and Hornung (ref. 13).

The aforementioned reviews, except references 5, 12, and 13, were concerned pri- marily with incompressible boundary layers. In references 5 and 12 some special prob- lems and results for compressible turbulent boundary layers were considered briefly.

The paper by Hornung (ref. 13) is one of the more complete and general reviews of theory and experiment for compressible boundary layers available up to 1966.

Several numerical methods that were developed originally for the calculation of incompressible turbulent boundary layers have been recently extended to adiabatic com- pressible flows (refs. 14 to 17). The present review is concerned with these and other more recent developments in the prediction of compressible turbulent boundary layers.

Much of the material utilized in this review is available in the proceedings of the 1968 Langley symposium (ref. 3) and in papers presented at the AIAA Fluid and Plasma Dynamics Conference, San FranCiSCO, California, June 16-18, 1969.

SYMBOLS A* parameter in Van Driest's wall damping function (eq. (20d)) Mach number parameter (eq. (22)) a.

parameters used in Pinckney's integral method (pv)w B dimensionless blowing rate, (PU)e skin-friction coefficient, specific heat at constant pressure functions of Me and TwiTe fl,f2 mixing-length functions (eqs. (20a) and (20b)) G any dependent variable H total enthalpy, h + (u /2) H* form factor, 0*/ e

Hi kinematic form factor, oil ei

h static enthalpy K Prandtl's mixing-length constant, approximately 0.4 K* proportionality constant (eq. (36)) A lie dUe K laminarization parameter, -- u2 dx e k molecular thermal conductivity L reference length l mixing length modified mixing length to account for effect of curvature (eq. (40)) total-enthalpy mixing length (eqs. (18)) velocity mixing length (eqs. (17)) M Mach number NSt n exponent in equation (33) C f1.

p molecular Prandtl number, k Npr C € n turbulent Prandtl number based on total enthalpy, ~ K turbulent Prandtl number based on static enthalpy pressure p q flux of total enthalpy turbulent flux of total enthalpy (eq. (6b)) q' magnitude of fluctuation in velocity vector, Vu '2 + v'2 + w'2 2H L PsV e reference Reynolds number, f1.s X Peue Reynolds number based on x length, f1.e

- *

Pll (\ e eddy-viscosity Reynolds number, € u 8 Pe e Reynolds number based on momentum thickness, free-stream Reynolds number per foot, r lateral radius from axis of symmetry longitudinal radius of curvature of body surface haw - he recovery factor, He - he T temperature stagnation temperature u velocity component in x-direction v velocity component in y-direction - p'u'

-

v effective normal velocity, v+-_- P velocity component normal to x- and y-directions w

x modified defect parameter of reference 15 (eq. (28))

boundary-layer coordinates, parallel and normal to surface, respectively x,y a local wall angle with respect to body center line function in modified mixing-length relation, taken herein as the constant 7 0* dp pressure-gradient parameter, -- TW dx y ratio of specific heats boundary-layer thickness

0* displacement thickness, SO (1_ i3ii .\.E- dy

w 0\ peuejr o~

kinematic displacement thickness , SO (1 - iiJ...£. dy

I

o \" u~rw

eddy viscosity (eq. (9a)) E

total-enthalpy thickness, SO (1- e) pii -f- dy

e

o peue w

. u pu r

O( -~--

e momentum thIckness, r 1 - - -- -- dy

J ue Peue rw

O H- Hw e normalized total-enthalpy parameter, He - Hw

kinematic momentum thickness, SO(1 - iiJ ii ...£. dy

e·

I

o u~ue rw

eddy conductivity of total enthalpy (eq. (9b)) K molecular viscosity molecular kinematic viscosity, Il / P

LTJ transformed coordinates (eqs. (33))

1T profile parameter in Coles I law of the wake P density T shear stress TT turbulent shear stress (eq. (6a)) <P eddy-viscosity relations of reference 15 (eqs. (26) and (27)) X modified law-of-the-wall parameter of reference 15 (eq. (24)) Subscripts: a w adiabati c wall e local edge of boundary layer exp experimental L laminar max maximum o free-stream stagnation s reference conditions theo theoretical w wall or surface 0() free stream ahead of bow shock Superscript: body shape index (j = 0 for two-dimensional flow; j = 1 for axisymmetric flow) A bar over a symbol indicates a mean value except where otherwise noted. A prime denotes a fluctuating value.

ABBREVIATIONS C . T. correlation techniques F.D. finite-difference methods I. M. integral methods AREAS WHERE SIGNIFICANT ADVANCES HAVE BEEN REPORTED Experimental Methods and Results The indeterminate nature of the turbulent-flow problem requires that experimental data be relied upon for validation of the prediction methods to a much greater extent than for laminar flows. Therefore, while the emphasis of the present compilation is on ana- lytic methods, it is appropriate to begin with a few remarks on some recent experimental results.

For any given experimental situation, it is clearly mandatory to obtain as many independent measurements as possible of both mean and fluctuating flow properties. The direct measurement of surface skin friction with floating-element force balances has until recently been limited to adiabatic flows with zero pressure gradient. Improvements in the construction and calibration of these balances have now allowed their use under conditions with pressure gradients and large heat transfer. A description of two differ- ent balances is available in references 18 and 19, and results obtained with them are given in references 20 and 21, respectively. (Ref. 21 is essentially the same as paper no. 10 of ref. 3.)

In order to specify the mean flow profiles across a boundary layer, at least two independent measurements are required. Thus, for example, from measurements of pitot pressure and total temperature, all mean flow profile parameters such as Mach number, velocity, and density can be computed if the usual assumption of constant static pressure across the boundary layer is accepted. When the streamwise pressure gradi- ents on a body in supersonic flow are large, the pressure gradients normal to the surface are appreciable and become larger as the Mach number increases. The local static pres- sures within the boundary layer should then be measured as in reference 22, for example.

For flat-plate-type flows where the entire development of the boundary layer occurs with essentially zero pressure gradients, the assumption of negligible normal pressure gradi- ent is usually justified. Recent experimental investigations on flows of this type where two independent sets of measurements were obtained are reported in references 23 to 27.

The turbulent boundary layers on the walls of hypersonic nozzles, while subject to the effects of upstream pressure and temperature gradients, are usually thicker than those on models at comparable unit Reynolds numbers. These boundary layers are there- fore convenient for detailed probing. Recent examples of such detailed profile data, where again more than one profile parameter was measured, are available in refer- ences 20 and 28 to 37. When these data from nozzle-wall boundary layers are plotted in

the form of e as a function of F (sometimes referred to as Crocco plots), a nearly

quadratic variation is obtained (ref. 35). Most flat-plate-type data, on the other hand, show on the average a more nearly linear variation as would be expected for dp/dx ~ 0 and N T "" 1.0 for the entire boundary-layer development. Thus, even though most pr , of the nozzle-wall data were obtained with small local pressure gradients, it seems that the effects of large upstream pressure and temperature gradients that are present in the nozzle flows may persist far downstream. Another possible cause of this difference between flat-plate-type flows and nozzle-wall flows has been identified in paper no. 12 of reference 3. In this paper, Jones has described a total-temperature deficiency found near the settling-chamber wall of a typical hypersonic facility and has indicated that such

a deficiency could account for at least part of the quadratic trends of the variation of e

with F.

Measurements of fluctuating flow quantities in compressible flows are needed in order to extend the present limited knowledge of relations between fluctuating-flow cor- relations and mean flow quantities. The investigation of reference 37 is therefore note- worthy because the fluctuations in density were measured for the first time in a turbulent boundary layer. An electron-beam probe was used to measure both mean and fluctuating denSity in the boundary layer on the wall of a shock tunnel at a free-stream Mach number of about 9. The wall- to total-temperature ratio varied from about 0.06 to 0.16. Mean and fluctuating pitot pressures were also obtained in this investigation. These data on the fluctuating density and pitot pressure were utilized in references 38 and 39 to obtain estimates of the root-mean-square intensity of longitudinal velocity fluctuations for these conditions. By comparison with previous data it was tentatively concluded (ref. 39) that the magnitude and distribution of the root-mean-square values of both the denSity and velocity fluctuations are not greatly affected by Mach number and heat transfer if appro- priate normaliZing parameters are used. One exception to this statement is that the vis- cous sublayer does appear to increase in thickness, relative to the total boundary-layer thickness, as the Mach number increases (see also refs. 33 and 40).

Another technique for measuring fluctuating flow properties is the crossed-beam correlation technique (ref. 41), which yields such data as relative turbulence intensities and integral scales of turbulence (ref. 42).

Another new instrument has recently been developed that is designed to provide direct measurements of all three components of fluctuating velocities (ref. 43). This technique depends on the Doppler principle applied to changes in frequency of scattered light from a laser beam by particles added to the flow. In order for the particles to follow the random motions of the flow with sufficient accuracy, the density of the liquid or gas flow must be high and the particles must be small. To date, the technique has there- fore been used mostly in liquid or high-density gas flows, although preliminary data have been obtained in a supersonic jet at Mach number 3. (See ref. 44.)

A further experimental result of conSiderable significance is the accurate measure- ment of surface heat transfer on a flight vehicle with a turbulent boundary layer at Mach 20 (ref. 45). Also of significance for flight applications is the correlation of tur- bulent heating on the windward and leeward sides of cones, swept plates, and delta wings at angle of attack (paper nos. 17 and 18 of ref. 3). The tests reported in these two papers were conducted at free-stream Mach numbers of 7.4 and 6.0, respectively, and for angles of attack up to 12 and 50, respectively.

Analytic Approaches Since the subject of the present compilation is analytic methods, the remainder of this paper is devoted to a discussion of the three principal approaches to the calcu- lation of compressible turbulent boundary layers and comparisons between experimental data and predictions obtained from selected methods. For the present purposes these three approaches are categorized as (1) integral methods (I.M.), (2) finite-difference solutions of the partial differential equations (F.D.), and (3) correlation techniques (C.T.).

Before proceeding to a description of these three categories and comparisons of typical theoretical results with data, the classes of problems that can now be treated with expec- tation of obtaining reasonable predictions are considered.

Before the development of high-speed computing machines, the methods used by the design engineer who required estimates of skin friction and heat transfer were limited primarily to the C. T. methods (category (3)). Such methods have been reviewed and classified by Spalding and Chi (ref. 46) and are directly applicable only to flat-plate-type flows with zero pressure gradients. These methods provide closed-form expressions (or working charts) for skin friction as a function of Me, Re or Rx, and TwiTe.

Hence, additional assumptions for the Reynolds analogy factor 2NSt/Cf and the recov-

ery factor r are required to obtain the heat transfer. When both large heat transfer

and large pressure gradients are present locally, the more recent I.M. and F.D. methods (categories (1) and (2)) provide more realistic predictions than the C.T. methods.

A somewhat different situation exists for nozzle-wall boundary layers where the boundary layer near the nozzle exit has been subjected to large pressure or wall- temperature gradients during its upstream development while local conditions are nearly uniform. The boundary layer is then in the process of "relaxing" or adjusting to the local uniform conditions, and the question arises as to how fast (or over how long a streamwise distance) the boundary-layer profiles, surface shear stress, and surface heating come into equilibrium with these local conditions. That is, does the boundary layer "remember" its upstream history? Comparisons of limited nozzle-wall and flat- plate data (ref. 47) indicate that the surface shear stress and heating may not be much influenced by upstream history. However, the profiles across the boundary layer, par-

ticularly the relation between e and F mentioned previously, may not relax to

equilibrium forms until downstream distances of many boundary-layer thicknesses have been reached. Experimental evidence of such effects are available in references 35 and 36. Theoretical studies of these effects are possible with the F.D. methods as shown in reference 34. The I.M. approaches that incorporate appropriate profile correlations and nonequilibrium control parameters, such as the method of reference 48, could also be applied to these relaxing flows.

Special problems in turbulent boundary layers that will not be considered in this review because they are treated in other papers of this compilation are as follows: (1) Three-dimensional boundary layer with nonuniform (or nonequilibrium) surface mass transfer, large heat transfer, and large pressure gradients (This problem is treated by an F.D. method in paper no. 19.)

(2) Interaction between turbulent boundary layer and shock wave (A technique for computing the interaction is presented in paper no. 21 and utilizes the concept of an inviscid outer layer and a laminar viscous inner layer; the solution for this inner layer may be obtained by any of the methods reviewed in paper no. 20.)

(3) A separating turbulent boundary layer (The technique of paper no. 21 could presumably be applied to a separating turbulent boundary layer; the method of Fernandez and Lees (ref. 49) could also be applied to this type of problem, providing a suitable expression for the dissipation integral could be formulated.)

DESCRIPTION OF ANALYTIC METHODS Basic Equations Boundary-layer equations.- The boundary-layer equations for turbulent flow may be derived from the conservation equations for a viscous heat-conducting fluid by substituting mean and fluctuating parts for the instantaneous flow variables, applying the Reynolds time-averaging process, and finally neglecting higher-order terms. For two-dimensional or axisymmetric time-steady flow, the resulting equations for continuity, x-momentum, and y-momentum, respectively, are as follows: (1)

-- aii -- aii ap 1 a t - ~- aii --;-;~

J

pu - + pv - = -- + - - r fJ. - - pv u (2)

ax ay ax rj ay ay

,oii ap 1 a (- -2)

(3) ± r c = - ay - r ay \r J pv' j ------- In equation (3) the upper and lower signs should be used for flows on bodies and in noz- zles, respectively. (These sign conventions apply to flows with simple, or nonreflecting, Mach wave systems.)

The equation for total enthalpy is (4)

-- aH -- elH 1 a tJ'~k elH --I -~ 1 ~ - elU~~ pu-+pu-=-- r ---pvH'+J.l.l---u-

ax ely rj By C ely Npr By p All terms of order 0 or smaller have been neglected in equations (1) to (4), and the relative order of magnitude of the coordinates and flow variables is taken as x,r ,r c,ii,P,H: Order of 1 y,v,( )'( )': Order of 0 (5) "P. ,K: Order of 0 where ( )' denotes the fluctuating part of any flow variable except J.I.' and k' which are considered of order 03• In general, the radius to a point in the flow is given by

r = rw ± y cos a

where the upper and lower signs apply to external and internal flows, respectively. When

o « rw, this relation becomes

and d drops from equations (2) to (4).

Turbulent-flux terms. - The terms which include correlations of fluctuating quantities

TT = -pv'u' (6a)

and ---I --- ---- qT = -pv'H = -pv'h' - upv'u' (6b) have the effect of increaSing the flux of momentum or total enthalpy due to the turbulence and are therefore known as the Reynolds shear stress and enthalpy flux. (In the

conventional sense, the term - P v'h' would represent the turbulent flux of heat; how-

c p ever, for the present purpose, it is convenient to retain the total-enthalpy correlation

-pv'H' ~ Since there is no general relation between these correlations of fluctuating

quantities and the mean flow variables, empirical models of the correlations must be used to solve equations (1) to (4).

Static-pressure variation.- It should be noted that the normal-momentum equation (eq. (3)) is generally not included in the system of equations to be solved because of the additional complications. (These complications are associated with the hyperbolic nature of the complete continuity and normal-momentum equations with higher-order terms included (see ref. 50). Presumably, when only the first-order terms for turbulent flows are retained in the normal-momentum equation, as in equation (3), the system of equa- tions (1) to (4) is still parabolic and conventional solution techniques should be applicable.)

When r c is small, the normal pressure gradients are large and are presumably caused mainly by the wall curvature. For this Situation, therefore, the pressure distribution across the boundary layer is nearly the same as for the inviscid flow, as shown in ref- erence 51. The effect of normal pressure gradients can then be easily included in inte- gral methods, as shown, for example, in references 52 and 53.

When r c is large, the normal pressure gradients may still be appreciable. Their magnitude may then be evaluated from equation (3) which for two-dimensional flow and large r c can be integrated to give (7) Then if v,2 z..!. q ,2 (ref. 54) and T T z 0.15 (ref. 8), the following equation is obtained: 4 pq,2

p _ 5 TT

--1--- (8) - 3 - pw pw The variation in static pressure across the boundary layer caused by turbulent velocity fluctuations can then be estimated after formulation of a suitable model for turbulent shear stress.

Eddy-Diffusivity Concepts General relations. - By analogy with the molecular shear stress and heat transfer, it is often assumed that eddy-diffusivity coefficients relate the turbulent correlations to

-I

the corresponding mean flow gradients. From equations (6a) and (6b) the total shear stress and enthalpy flux then become - au aii (9a) T = TL + T T = J.L - + € - oy oy and

- -

k oH K oH (9b) q = qL + qT =--- +--- cp oy cp oy Again by analogy with the molecular Prandtl number cpJJ.

(10)

Npr =-_-

k the turbulent Prandtl number is defined as (l1a) or by the use of equations (6) and the definitions of € and K (eqs. (9a) and (9b)) _ v'u' 8H/8y (l1b)

Npr,T - === 8-/8

v'H' u Y Since this definition of N Pr T is in terms of the total enthalpy rather than the static , enthalpy, it corresponds to the "total" Prandtl number defined in reference 34. An ele- mentary analysis and the results of finite-difference solutions presented in this reference indicated that values of Npr T may be as large as 2.0 for nozzle-wall boundary layers.

, From the definitions of €, K, and Npr T, equations (2) and (4) may be written as ,

-- au -- ou op 1 0 t o( ou~

(12)

pu - + pv - = -- + --:- - rJ J.L + €)-

ox oy ox rJ 8y oy and (13)

-- oH -- oH 1 0 tJ O ~Q II. € ~OH ( 1 J- ou~~ pu-+pv-=--:-- r ---'=--+ -+1l1---u-

ax ay rJ ay N N 'E oy Npr, ay pr pr ,

-

Since the appropriate boundary conditions on u and Hare

L

(y = o~

(14)

(y _ oo~

it can be seen that for ail/ax = 0, Npr = 1.0, and N T = 1.0, a solution to equation (13) pr , is the Crocco integral II - Hw (15) ----=F He - Hw if H , ue, and He are constants. From the definition of Npr T (eq. (llb)), the ratio w , of the streamwise-momentum and total-enthalpy flux correlations for an ideal gas under these conditions is then v'u' (y - l)Me (16) e v'H' u = (. y _ 1 2\ (, T '\ e \1 +-2- M )\l- T:; Hence the assumption of N T ~ 1.0 implies that the turbulent momentum flux and pr , enthalpy flux are caused by the same or closely related mechanisms since their ratio is constant for given values of Me, u , and Tw/To. It follows that when any of the afore- e

mentioned restrictions ~:~ ~ 0, Npr ~ 1.0, and constant TW) are not satisfied, the

assumption of Npr T ~ 1.0 may not be applicable.

, Mixing-length expressions.- One of the more successful models for eddy- diffusivity relations is based on Prandtl's mixing-length hypothesis, which states that

v' ~ u' ~ lu aii and G' ~ lG aG where G is any quantity diffused or transferred by the

ay ay action of turbulence. Application of the hypotheSiS to Reynolds stress or turbulent shear stress then gives (from eqs. (6) and (9)) --I -, - 2 aii aii (17a) TT = -pv u = Pl -- u ay ay or - 2 aii (17b)

€ = Pl -

u ay where the absolute sign is used to insure that the shear stress will have the same sign as the velocity gradient. Similarly, the flux of total enthalpy becomes

-- - laul aH

(l8a) qT = -pv'H' = Plu - lH- ay 8y or (18b) The turbulent Prandtl number (eq. (lla)) is then (19) When Npr T > 1.0 (see ref. 34), equation (19) indicates a smaller mixing length for , enthalpy than for velocity.

For direct application to the calculation of boundary layers, the velocity mixing length lu in the outer part of the boundary layer is usually scaled to the boundary-layer thickness (ref. 4). That is, (20a) where the function f1 is based on experimental data such as those of references 55 and 56 for incompressible flows. Calculations by Maise and McDonald (ref. 57) and Patanker and Spalding (ref. 17) have shown that the same function is applicable to adia- batic compressible flows on flat plates (small values of dp/dx). Results obtained by Bushnell and Beckwith (ref. 34) indicate that for application to nonequilibrium compres- sible flows with pressure gradients, the mixing-length function may also depend on the

incompressible form factor It· hence

1 , (20b) These and other results (for example, ref. 39 and paper no. 15 of ref. 3) indicate that the turbulence properties of compressible turbulent boundary layers can be correlated on the basis of scale factors that depend only on the kinematics of the flow rather than the dynamiCS. Thus, the variations in density apparently have little effect on the empirical models of the turbulent flux terms which retain their basic kinematic character (see also refs. 58 and 59).

In the near-wall region of a boundary layer where the only length scale factor is the distance normal to the surface, the Prandtl mixing-length relation becomes (20c)

lu = Ky

Experimental data show that K ~ 0.4. Again, this value of K is found to be nearly a universal constant. When the boundary-layer equations are integrated all the way to the wall as in some of the F.D. methods (refs. 15, 16, and 34, for example), it is necessary to modify equation (20c) to provide the correct behavior of TT (or €) as the wall is approached. Analysis of the basic equations shows that € should vary as y4 or y3 as y - 0 (see refs. 60 and 61). Van Driest's wall damping function provides this type of variation and also accounts for the effect of Reynolds number (or density level) on the I sublayer thickness (ref. 62). Equation (20c) then becomes (20d) where A* ~ 26. This damping function has also been used in references 16 and 17. A different function was used for the same purpose in reference 15. For application to problems with wall blOwing, A* may be considered a function of the blowing parameter 2B/Cf as shown in reference 34.

Eddy viscosity in the outer part of the boundary layer. - Clauser introduced an alter- nate approach to the mixing-length formulation for eddy viscosity (ref. 63) with a Reynolds number parameter (21) CI~user showed that for equilibrium flows the value of this parameter was nearly con- stant at R€ ~ 60 in the defect or "wake" portion of the boundary layer where the direct influence of the wall is negligible. When this formulatfon is applied to adiabatic com- pressible flows, the same constant value of R€ ~ 60 gives good results (refs. 15 and 16)

if the scale factor oi is retained in the incompressible form. Hence, this is another

example of how the characteristics of turbulence for compressible flow seem to depend only on the kinematics of the flow.

Category 1: Integral Methods (I.M.)

As is well known, the integral methods are based on solutions of various integral forms of the equations of motion. That is, the momentum and enthalpy equations (usually written as eqs. (12) and (13)) are combined with the continuity equation (eq. (1)) and inte- grated across the boundary layer to give two ordinary differential equations for the

momentum thickness e and total-enthalpy thickness 0. These equations cannot be

solved without additional information or assumptions for the form factor H*, the surface

I

___ J

-,

shear stress T and surface heat transfer qw as functions of e, 8, and the boundary

w , conditions of the problem such as wall temperature and pressure gradients. An additional .

difficulty for compressible flows is that H* is a function of not only the velocity profile I shapes but also the density (or temperature) profiles. Since general formulations for these functions are not yet possible, many different procedures based on various assump- tions have been developed. Typical of these procedures is the technique of first multi- plying equation (12) (combined with eq. (1)) by a weighting function and then integrating the new equation across the boundary layer. In this way, a new ordinary differential equation can be obtained for each weighting function used. Thus, for example, if the weighting functions are just y or ii, the resulting integral equations are known as the moment-of- momentum equation and the kinetic-energy equation, respectively. For turbulent bound- ary layers additional unknown integrals involving the turbulent shear stress then arise.

With weighting functions y and ii, these unknown integrals are called the shear-stress integral and the dissipation integral, respectively. A large body of literature is con- cerned primarily with the development of correlations and expressions for these integral quantities.

Since about 1962 an improved method of integral relations has been developed wherein the weighting functions are linearly independent functions of ii (see paper no. 14 of ref. 3; ref. 64; and ref. 1, pp. 16-29 and 46-53). The proponents of this method claim that, in principle, the solution of the resulting system of ordinary differential equa- tions can approach the correct solution of the original partial differential equation if the weighting functions satisfy all boundary conditions and are chosen to avoid Singularities.

In this approach the unknown shear-stress integrals are evaluated by the use of conven- tional eddy-viscosity relations such as equations (17) to (21). Thus, one advantage of the simpler integral methods is lost since the detailed behavior of the turbulent shear stress or heat transfer across the boundary layer has to be specified.

Rather than to attempt a detailed discussion of the many different integral methods and the great variety of assumptions used in their formulation, the following discussion is limited to four methods that utilize ii or y as weighting functions. Predictions from these four methods are compared with data and some of the assumptions used in each method are discussed briefly.

Reshotko-Tucker method (ref. 65).- The method of reference 65 was included herein mainly because it is representa . tive of several of the early approaches. Also, as pointed out by McDonald (paper no. 6 of ref. 3), the method has been widely used and some simple modifications of the basic approach (for example, refs. 66 and 67) have provided much improved predictions. (The more important of these modifications was to the expres- sions for the shear-stress integral.) The method of reference 65 utilizes the momentum and moment-of-momentum integral boundary-layer equations which are solved directly in the transformed plane for the transformed momentum thickness and form factor.

Stewartson's coordinate transformation was used. The skin-friction relation was obtained by application of Eckert's reference-enthalpy expression (ref. 68) to the Ludwieg-Tillmann skin-friction equation for incompressible flow (ref. 69) . In order to express the moment-of-momentum equation in terms of the form factor, power-law velocity profiles were used, a functional relation between the shear-stress integral and form factor was assumed, and integrals of the enthalpy profiles were assumed to have the same functional relation to the form factor as in laminar flows. All these functions and assumptions were applied in the transformed plane. Finally, to compute the heat transfer, the Reynolds analogy factor was modified to account for the effect of pressure gradient by analogy with a laminar-flow result. The results computed by this method and presented herein were taken directly from paper no. 6 of reference 3.

In retrospect, the many assumptions used in the Reshotko-Tucker method seem dif- ficult to justify. For some flow conditions, the much Simpler and straightforward method developed by Charles B. Johnson and described briefly in reference 35 would probably give predictions at least as good as the method of reference 65. This statement is based on the good results obtained from Johnson's method for a wide range of conditions (ref. 35). Also in Johnson's method the velocity and enthalpy profile correlations were based directly on experimental data, the calculations were carried out in the physical plane, and the Spalding-Chi skin-friction relation (ref. 46) was used.

Camarata-McDonald method (ref. 70).- The method of reference 70 has been included in the present review because the momentum and moment-of-momentum integral equations have been used just as in the Reshotko-Tucker method except that two signifi- cant modifications have been introduced which, on conceptual grounds, might be expected to improve the predictions. The procedure is described briefly in paper no. 6 of refer- ence 3 and also in reference 1 (pp. 83-98) as applied to incompressible flows. The two significant modifications of the Reshotko-Tucker method are the procedure for evaluating the shear-stress integral and the use of Coles' transformation (ref. 71) for velocity pro- files and skin friction. In order to evaluate the shear-stress integral, the "extended mixing-length hypothesis" was introduced wherein the value of the mixing length in the outer part of the boundary layer varied in the stream wise direction as governed by an integral form of the turbulence kinetic-energy equation. In this way the effect of the upstream history on the local turbulence properties (that is, on the mixing length) is accounted for. Consequently, this new method presumably represents a conceptual improvement over the previous methods of references 65 to 67 where the shear-stress

in!egral was assumed to be a function of local conditions only (IIi' or a combination of

Hi and Cf).

In order to compute heat transfer, a Reynolds analogy factor would be required.

However, published results from this method have been limited to predictions of skin friction, momentum thickness, and form factor.

Alber-Coats method (ref. 48).- In the method of reference 48 the momentum and mean kinetic-energy integral equations are solved simultaneously in the physical plane for the dependent variables 0* and 7T. The velocity profiles and a differential equa- tion for skin friction were obtained from Coles' incompressible "law-of-the-wall" and "law-of-the-wake" forms (ref. 72) which w~re adapted to compressible flow by replacing the incompressible velocity by Van Driest's generalized velocity

* 1. -1 u

u = u - SIn a- (22)