Analysis of two dimensional inviscid model of jet impingement under vertical-takeoff airplane

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ANALYSIS OF TWO DIMENSIONAL

INVISCID MODEL OF JET IMPINGEMENT

UNDER VERTICAL-TAKEOFF AIRPLANE

by Robert SiegeZ und Murvin E. Goldstein

Lewis Reseurch Center

CZeveZund, Ohio

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N W A S H I N G T O N , D . C. J U N E 1 9 6 9 ,

I

i I TECH LIBRARY KAFB, N M ANALYSIS O F TWO DIMENSIONAL INVISCID MODEL O F J E T IMPINGEMENT UNDER VERTICAL-TAKEOFF AIRPLANE By Robert Siege1 and Marvin E. Goldstein Lewis Research Center Cleveland, Ohio NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For s a l e by t h e Clearinghouse for Federal Scientific and T e c h n i c a l Information Springfield, Virginia 22151 - CFSTl price $3.00 . .. . . .. . . . . - . . _. . -~ ABSTRACT Conformal mapping was used to obtain the f r e e streamline pattern and surface p r e s - s u r e distribution for downward jet flow issuing from two parallel slot nozzles. After striking the ground, a portion of the flow from each jet moves along the ground toward the centerline between the jets. When these two portions collide, an upflow results which strikes a plate above the ground simulating the undersurface of an airplane fuse- lage. Typical flow patterns a r e shown to illustrate the effect on the f r e e streamlines of nozzle height, nozzle spacing, plate height, plate width, and plate angle.

ii ANALYSIS OF TWO DIMENSIONAL INVISCID MODEL OF JET IMPINGEMENT UNDER VERTICAL-TAKEOFF AIRPLANE by Robert Siege1 and M a r v i n E. Goldstein Lewis Research Center SUMMARY Conformal mapping was used to obtain the f r e e streamline pattern and surface pres- s u r e distribution for downward jet flow issuing from two parallel slot nozzles. After striking the ground, a portion of the flow from each jet moves along the ground toward the centerline between the jets. When these two portions collide, an upflow results which strikes a plate above the ground simulating the undersurface of an airplane fuselage.

Typical flow patterns are shown to illustrate the effect on the f r e e streamlines of nozzle height, nozzle spacing, plate height, plate width, and plate angle.

INTRODUCTl ON Under a vertical-takeoff (VTOL) airplane, as illustrated by figure 1, there is a com- plicated flow pattern. The downward directed jets in figure l(b), react against the ground, and a portion of the flow is turned outward and flows along the ground. The remaining flow moves inward, and under the fuselage the opposing s t r e a m s collide and .

move upward. Portions of this upward flow will move around the fuselage o r will r e c i r - culate under the wings. The upward flow in some instances provides a useful lifting force. However, it is generally undesirable because of recirculation of hot exhaust This recirculating flow may be partially ingested gases around portions of the airplane.

into the engines, along with dust and debris from the ground that has been entrained (refs. 1 and 2).

The complicated interaction with the ground and the airplane, and the presence in many instances of turbulent mixing, makes the analytical prediction of the three- dimensional flow flow field under a vertical-takeoff airplane extremely difficult. To obtain analytical solutions a simplified incompressible, isothermal flow model is employed herein that retains some of the major features of the actual flow field. The analysis is (a) Plan view o f airplane.

CD-10386-02 (b) Flow pattern under airplane.

Figilre 1 . -Flow configuration f o r fan-pod VTOL aircraft.

an extension of that in reference 3 . As shown in figure 1, a fan-pod type of configuration is being considered. A two-dimensional approximation will be made to study the flow in the region between the engines. The flow is assumed to be inviscid; this could be approxi- mately true when the nozzle exit planes a r e within a few nozzle widths of the ground and within a few widths apart. F o r these conditions there will be only a small entrainment of surrounding fluid into the jet region prior to the flow being turned by the ground and The two-dimensional model is shown in figure 2. A plate, turned under the fuselage.

which can be at an angle to the ground, has been used to simulate each half of the under- surface of the fuselage.

By using a two-dimensional inviscid model, the determination of the flow field ‘L G r o u n d F i g u r e 2. -Two-dimensional model of jet deflection by g r o u n d and fuselage.

becomes a free streamline problem in constant p r e s s u r e surroundings. A solution can be obtained by using the Helmholtz-Kirchhoff method (ref. 4 ) . This conformal mapping procedure is used to find a functional relation between the complex conjugate velocity and the complex potential of the flow. Then the flow configuration is obtained by integrating the complex potential multiplied by the reciprocal of the complex conjugate velocity.

The flow pattern depends on the flow condition leaving the nozzle. The condition assumed herein is that the flow leaves perpendicular to the nozzle exit plane. This is a reasonable condition for nozzles of small width o r when turning vanes are used to guide the flow. The interaction of the flow with the ground produces a nonuniform downward velocity leaving the nozzle; the velocity in the central region of the nozzle exit plane is l e s s than that along the nozzle sides.

The final analytical results were evaluated f o r several combinations of the param- eters, such as plate height, plate width, and spacing between nozzles. Typical f r e e streamline flow patterns a r e given along with the pressure coefficient along the plate, and the velocity distribution a c r o s s the nozzle exit plane. In addition to revealing the nature b of the flow, the inviscid solution provides the upper limit of the pressures achieved on the ground and fuselage. The inviscid solution is also the zeroth order configuration which is needed to compute viscous entrainment by the flow.

SYMBOLS A a constant a constant AO pressure coefficient cP H quantity defined in eq. (A8) I flow region in p p l a n e J flow region in W-plane K complete elliptic integral of first kind K' defined by K'(k) = K(kl) k modulus of elliptic integral k'

defined as d K 2

M quantity defined in eq. (18) p r e s s u r e P ambient pressure outside jets P O functions defined in eqs. (B4) anc ) of r e 3

Q

R functions defined in eqs. (B2) and (B3) of ref. 3 T complex variable in T-plane, [ + i q t complex variable in t -plane U velocity in x-direction V velocity velocity at nozzle exit

-

average velocity at nozzle exit velocity along f r e e streamlines vO V velocity in y-direction W complex potential, @ + i+b coordinates of center of nozzle exit plane nondimensionalized by A

x ~ , y~

coordinates in physical plane X, Y ,i Z complex variable, x + iy tilt angle of plate, fig. 3 P

r flow region in T-plane

angle of deflected jet, fig. 3 Y A width of nozzle 'I widths of s t r e a m s flowing to left and to right 'L' 'R complex conjugate velocity, u - iv t P density 5 , rl coordinates of T-plane 0 quantity defined in eq. (A?)

9 velocity potential

Q stream function

51 function defined in appendix A of ref. 3 Subscripts: refer to stagnation points, fig. 3 A,G,H f fuselage g ground L flow to left R flow to right

ANALY SI S

The analysis presented herein is a generalization of, and quite similar to, that given in reference 3 . F o r this reason only those parts of the analysis differing from reference 3 will be discussed in detail. The configuration of the flow, as shown in figure 2, is sym- metric about a vertical plane between the nozzles, hence only half the flow field need be The boundaries of the jet in the physical plane (z-plane) a r e shown in figure 3 .

considered.

Since the region outside the jet is at constant pressure, Bernoulli's equation shows that

the velocity has constant magnitude along the f r e e streamlines a, f ? , and a. The

n - corresponding direction of the velocity must be along the boundaries on AB, AG, and respectively to the bottom of the fuselage, the vertical line of symmetry, and the ground.

The point H is a stagnation point at which the direction of the velocity along must The dashed line is the dividing streamline of the flows going to the right and reverse.

the velocity is left. The points G and A a r e also stagnation points. Along the line vertically downward .

A s in reference 3, let be the complex conjugate velocity u - iv. From the spec-

ified conditions on the velocity at the boundaries of the flow field, it can be deduced that the flow field in figure 3 maps into the interior of the region I of the hodograph plane

ti’ I -iv

- U I Figure 4. - Flow region i n hodograph (-plane (( = u - iv).

Figure 3. -Flow boundaries i n physical z-plane (z = x + iy).

shown in figure 4. Let W = @ + ilc/ be the complex potential for the flow, where @ is

the velocity potential and lc/ is the stream function. Considerations analogous to those in reference 3 show that the flow field maps into the interior of the region J of the complex potential plane in figure 5.

The solution in the physical plane is obtained from the integration

z = f dW + constant

To carry out this integral, and W must b e expressed in t e r m s of the s a m e vari- able of integration. This is done by appropriately mapping the rectangular region I ? of an intermediate T-plane (described in ref. 3 and depicted in fig. 7) into the regions I and J of the 5 and W planes, respectively. The corresponding positions of the boundaries are indicated by the lettering scheme in figures 4, 5, and 7.

of r, I, and J

I C F Figure 5. - Complex potential plane (W = @ t io).

I I Illli I I i

Mapping Function Between 5 and T-Planes

The function 5 which properly maps the rectangle r into the region I of the

E-plane can b e constructed by using the function S2 defined and studied in appendix A of reference 3. By a procedure completely analogous to that in reference 3, it can be demonstrated that the proper mapping function 5 is given by

provided that the relation between tH, tG, and tA is chosen t o satisfy the following

requirement. Along the plate the argument of [(T) must equal the direction along the plate, so that from figure 4 It follows by the use of equation (A24) of appendix A of reference 3 that the argument of equation (2) is

a r g c ( ~ ) = - + - fl T ~ H - + I ) + - - + I ) + - - + I j 3 p ) ; TEAS

2 2 K 4 K 2 K Equating the previous two expressions for arg <(T) gives the condition relating tH, 5G7 and (A: Mapping Function Between W and T Planes

The function W which properly maps the rectangle r of the T-plane (fig. 7) into

the region J of the W-plane (fig. 5) can be constructed by use of the intermediate t-plane shown in figure 6. An application of the Schwarz-Christoffel transformation shows that the mapping that transforms the upper half t-plane onto the region J of the W-plane in the manner indicated in figures 5 and 6 is defined by

L

Figure 6. - Intermediate t-plane.

t - tH

-- dw - iAo * h t > o (4) dt (1 + kt)(tF - t)

r of the T-plane is mapped onto

Now it is shown in reference 3 that the rectangle the upper half t-plane (fig. 6) by

t = sn(T, k); T E r (5)

This can be combined with equation (4) to eliminate t and find the desired mapping from the T-plane to the W-plane. The relation given in reference 3 is used along with the relation i

Upon carrying out this procedure the required function that maps r onto J is found

to be (ksntHsnT - 1) dnT

-- dW - iA - T E ~

dT (1 + ksnT)[l - dn(qi, k')snT] '

where A is a new constant equal to By using partial fractions equation (6) can be rewritten as iA k(1 + SnSH) dW - ~-

v 1 - sn T (1 + k s n T )

Integrating this expression and neglecting an unimportant integration constant give D

c- c

The flow through the nozzle is equal to the difference between the stream function at

points D and C. Noting that W = Q, + i+ and that @(C) = Q,(D), the flow must equal

+(D) - +(C) = b ( D ) - W(C]/i. When the coordinates of D and C in figure 7 are used,

the flow throughthenozzlebecomes: E ( - K + io) - W(K + iO]/i. Hence, if VI is the

average velocity at the nozzle exit, -

W(-K + io) - W(K + io)

V A = I i p s n T DsnT .

dt . I .

11 - tZ

- dn(qF, k ' ) q

J

iA - - k ( l + snSH)sn(qF, k') sin-' + snT

k ' s n ( q F ~ + &(?7F, "1 k') - k s n t d sin-'

[I + k s n T j

sn(&) = *1 and choosing the appropriate branches for the inverse Using the fact that sine, we find from equation (7)

W(K) = iA {[h(qF7 k') - ksnt;H]

k'sn(qF, k')k dn(qF, 1 2

W(-K) = iA

{ [ & I ( . , k') - k s n g d - k ( l + sntH)sn(qF, k')

k'sn(r7F7k')[k + dn(qF,k'j] 2 Hence, which gives the constant A as - A A = - k'sn(qF, k') An alternate form f o r A will be given by equation (15).

As shown in figure 3, the interaction of the jet with the ground causes it to divide.

The asymptotic width of the flow region to the right is designated by 6R, and that to the left by tjL. The flow to the left must equal the jump in the imaginary part of W at the In order to evaluate this jump, notice that F is at T = K + i% along the F.

point boundary T = K + i q in figure 7 and that

sn(K + iq) =

dn(r1, k') Then, along this boundary, the inverse sine t e r m s in W (eq. (7)) become k') - dn(*,k',] where k +

)/dn2(q7k') - 11

sin-' = -i In i + k'

dn(q, k') + k

1 +

{

where k dn(q, k') + kf2sn(q7 k') + 1 f 2 ( d = ~~

k + dn(q, k')

Since 0 < dn(q, k') I 1 aqd sn(q, k') is positive for 0 I q I K' it is found that

fl(q) > 0 and f2(q) > 0 for 0 5 q 5 K' and, consequently, the logarithms of f l and f 2

a r e real.

I n k') - d n ( w , k ' l t e r m and Hence, choosing the appropriate branch for the substituting in equation (7) gives II I 1 1 1 . 1 1 1 1 I I I I1111111. I I 1111 I . 1111111 1 . 1 1 1 1 1 I II-111111-I.- I I II II I I I AT %W(K + iq) =

1 [Idn(qF7 k') - k P ? I < ~

k'sn(qF, k?)[k + dn(*,k'] 2

Since the velocity at the point F must be Vo, it follows from equations (9)'and (10) that, if GL is the asymptotic perpendicular jet width (fig. 3) and E > 0, Equation (8) is used to eliminate A giving

k + dn(qF7 k') - k ( l + s n t H )

GLV0 = V I A - - ~.

k + dn(qF, k') - k ( l + S n t H ) F - sn(qF, k ' g

Continuity requirements dictate that

GRV0 + GLV0 = ATI

Hence, using equation (ll), the 6L can be eliminated to obtain From equations (12) and (11) the ratio of asymptotic widths is By rearrangement and Using equation (14) in equation (8) gives the constant A as

n L 1 + - 6R

6L -

Integration t o Yield Coordinates in Physical Plane

When the complex conjugate velocity 6 and the complex potential W as functions of the parametric variable T a r e known, the physical variable z can b e found as a function of T by using equation (1). The integration is carried out in the T-plane instead of in the W-plane, so that equation (1) becomes The origin of the coordinate system has been chosen at the point G. Substituting equa- tions (2), (6), and (15) in equation (16) gives

r

$ 1

Average Velocity at Nozzle Exit To find an expression for vI/Vo, note that the nozzle width is given by

A = z(-K + io) - z(K + io)

Then it follows from equation (17) 6R "1 l+q .

The quantity M defined in this equation will be used subsequently.

P r e s s u r e Coefficients The pressure coefficients C and C along the ground and fuselage are defined P, g P,f by

P(X, 0) - Po

c =

Pt g 1 2 - PV0 P(xf,Yf) - Po Cp,f = 1 2 - PV0 where xf,yf are the coordinates of the points along the fuselage. Using Bernoulli's equation gives Cp,f = 1 -

S u m m a r y of Analytical Relations

F o r convenience, the results of this section are now collected in one place.

s n t H = L - 6R k + k Sn(qF, k') 6 , L W(T1 i 1

. I _

-

7~ [&(qF, k')+k] + (1 + sntH)kF(17F'k') -

VI A d n ( w , k ' ) - s n T

k') - k sntH] sin-' [

dn(%,k')snT - 1

-1 k + snT

- k ( l + snCH)sn(W7 k') sin

[l + k snT]} (7' 8, - - k(l + Sn<H)Sn(%7 k') A

Vo k + dn(qF,k') - k ( l + SntH)[l - Sn(qF,k'i]

Cp,f = 1 -

Additional forms of equations (2) and (17) that a r e useful for computer evaluation are given in the appendix.

C O M PUTATl ONA L PROCEDURE If all coordinates a r e nondimensionalized by the nozzle width A, there a r e five independent parameters governing the flow configuration: plate (fuselage) height above the ground, plate width, plate angle, nozzle height, and spacing between nozzles. The plate 0 appears explicitly in the analytical expressions and can be directly specified as angle an input variable in the computer program used to evaluate numerical results.

The other four physical quantities must be found by computation in t e r m s of four convenient input quantities all having values between 0 and 1: k, 6L/6R, vF/K', and tA/K. From a chosen k, the k', K, and K' can be foundfrom

k' = ,'1 - k2

dw K(k) =

1 ' d 1 - k2 sin2 w

K'(k) = K(k')

Then tH can be found from equation (14), and all quantities which a r e necessary to

tG from equation (3) will then be known. The quantities M and vI/Vo are determine then found by carrying out the integration in equation (18), using the S2 functions eval- uated by the method described in the appendixes of reference 3 . The dimensionless width of the stream flowing to the right 6R/A is evaluated from equation (12), and 6,/A is computed from (6R/A)(6L/6R), where 6L/6R is one of the specified input quantities.

With all these quantities evaluated, the height of the plate is found from equation (A6), and its width by integrating equation (A5) to an upper limit of K. These dimensions, along with the specified angle p , fix the position of point B. The streamline can then be plotted by u s e of equation (A12). The distance in equation (A10) is then evaluated The vertical separation of points D and E to fix the horizontal position of the nozzle.

along the right f r e e streamline is found from the second of equations (A9) by integrating to an upper limit of K ' . This distance, along with 6R/A, is used to determine the noz- zle height. The f r e e streamlines originating at the nozzle are then computed from equa- tions (A9) and ( A l l ) and a r e drawn starting, respectively, from points D and C .

The pressure coefficients along the ground and fuselage are found from equations (19) and (20) using the velocities evaluated from the second equations of (A4) and (A5).

RESULTS AND DISCUSSION As revealed by the analysis, there are several independent parameters (all made dimensionless by dividing by the nozzle width) governing the flow: the nozzle height, spacing between nozzles, plate height, plate width, and plate angle. It was not feasible to systematically compute flow patterns for the wide variety of possible combinations of these parameters. To limit the number of computed flow patterns somewhat, the plate simulating the underside of the fuselage was fixed in a horizontal orientation with the exception of the results shown in figure 12 where a comparison is made with a plate tilted at 45'. The plate was generally positioned at the same height as the nozzle, and only two nozzle heights were considered (YN = 1 and 2).

Consider figure 8(a) as a typical set of results. On each set two cases are given, one in solid lines and the other dashed. Each case shows the velocity distribution across the nozzle exit plane, the f r e e streamlines bounding the moving fluid, the pressure coefficient along the ground, and the pressure coefficient along the plate (fuselage).

Figures 8(a) and (b) illustrate the effect of moving the nozzle and plate upward to- gether, as when the airplane is taking off. Part @) is for a longer plate than part (a), but both portions of the figure exhibit the same general features. In the upper position the exit velocity from the nozzle is more uniform; hence, the flow is increased a s the nozzle is raised since there is less reaction from the ground. The increase in flow results in the integral of the ground pressure coefficient being a little l a r g e r for the upper nozzle posi- tion, thus providing a small increase in lift. The pressure coefficient acting on the under- side of the plate is decreased a little as the plate and nozzle a r e raised, thereby reducing the lift on the fuselage. In figure 8@) the stream deflection by the date is so large for the upper nozzle position (dashed) that t h e stream passes back into the region of the nozzle exit. In an actual flow the two s t r e a m s would collide. The analysis used herein does not allow for such interference and permits the two s t r e a m s to pass independently through each other (on different sheets of the complex plane). Actually, there would be interfer- ence and probably a recirculating flow region for this configuration.

Figure 9 demonstrates the effect of changing the horizontal position of the nozzle.

Parts (a) and (b) a r e for a low nozzle (YN = l), while parts (c) and (d) a r e for a high noz- The dominant effect is that the flow going to the left and impinging under zle (YN = 2).

the fuselage becomes quite small as the nozzles approach the plate. Thus having the fan pod adjacent to the fuselage rather than as shown in figure l(a) will reduce the hot gas circulation around the fuselage. Recall that this is a two-dimensional analysis. I f the nozzles were round rather than in a slot o r pod configuration, there would be an additional effect of nozzle spacing resulting from the radial spreading of the flow as it moves out- ward along the ground. This effect along with entrainment would tend to decrease the flow under the fuselage as the spacing between nozzles is increased.

3.2- CL W Lu . 2 - . 4 - .l- \ -'\

-- +

o - 0 - ---_

0 . 4 .8 1 . 2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4 . 4 4. a Dimensionless coordinate, XlA (b) Plate width -0.9.

Figure 8. - Concluded.

I

3 . 2 r I

1.0- I

' 2.8-

.9c I .a- 2 . 4 r

' .-I

\ '.

--

--

0 . 4 . 8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Dimensionless coordinate, xIA (a) Low nozzle (YN = 1); plate width -0.5.

Figure 9. -Effect of horizontal position o f nozzle.

I ,'$),ground

\ \

'"1 . 9 2 . 8 1 \

\ \ \ \ \ \ \ \ \ . 8 - 2.4- \ \ U c \ a , e $ .1-g m \ m 2.0- -E \ 1.0 c d d m \ V 0 \ e- . 6 - - . 5 n .+- L 0 c a , I \ al .- .- V U V \ .- VI 1 . 6 - ~ L c \ \ al al 8 *5-z U c 0 a , a , L ._ VI c VI VI VI . 4 - E 1.2- VI ? !

.- CL a n a , 'c1 c c m - \ a e . 3 - \ .8- \ \ \ . 2 - \ KFree streamlines .__ ', /'\, ----- - - - _ _ _ _ _ _ _ _ _ .4- \ . I - 0 - 0- (b) Low nozzle (YN = 1); plate width -0.8.

Figure 9. -Continued.

I 3.2-

8 . 5 - $

.-

z v) =i c a VI v) 2 . 4 - .E 1.2- n n 'E) c e . 3 - , , , . - F r e e streamlines W .8- . 2 - 0 . 4 .8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Dimensionless coordinate, XlA (c) High nozzle (YN = 2); plate width -0.4.

Figure 9 . -Continued.

N W W c

_ _ _ - _---- -- --

cFree streamlines

d ! ; - - - - -- - - - - - -

. 2 c i

I \ \ \ '\

.lk

'.

0 . 4 . 8 1.2 1.6 2.0 2.4 2 . 8 3 . 2 3.6 4 . 0 4.4 4.8 Dimensionless coordinate, x/A (d) High nozzle (YN = 2); plate width X I . 5.

Figure 9. - Concluded.

3.211

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - - -

1.0- ~ 7 7 - \’ I’ cp, ground .4 . 8 1 . 2 1.6 2.0 2.4 2.8 3 . 2 3.6 4.0 4.4 4.8 Dimensionless coordinate, xlA (a) Low nozzle (YN = 1).

Figure 10. - Effect of plate width for fixed nozzle position.

3 . 2 r I I : : .8- .2- . 4 - .l- 0- 0- 0 . 4 .8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4. a Dimensionless coordinate, xlA (b) High nozzle (YN = 2).

Figure 10. -Concluded.

3.2 - 1.0-

2.8 -

\ .9- .81-

.-

aJ L o .

u . 3 - c 0 .4 . 8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 Dimensionless coordinate, xlA Figure 1 1 . -Effect of plate height for fixed nozzle position.

l : f . 8 r -0 c 3 .3- e c3 .2-

. l-

0- In each of figures lO(a) and @), the plate width is varied while the nozzle is at a fixed position. The (a) and (b) p a r t s a r e for nozzle heights YN = 1 and 2, respectively.

For the wider plate the deflection of the left-moving stream is increased and there is a large stagnation region below the plate, especially when the nozzle is in the lower posi- tion. As the plate width is increased, the space between the nozzle and plate is dimin- ished, thereby reducing the flow t o the left. Raising the plate while keeping the nozzle fixed also increases the jet deflection, as shown in figure 11.

Figure 12 shows the effect of tilting the plate while keeping fixed the horizontal pro- jection of the plate and the nozzle position. The flow pattern is changed only a small amount, the stream flowing to the left being turned more by the tilted plate. The exact shape of the underside of the fuselage is thus of minor importance.

The previous figures have given the reader a quantitative appreciation of the flow

trends resulting from varying the five independent parameters governing the flow config -

The flow patterns can also be used a s the zeroth order solution for evaluating uration.

entrainment into the flow regions.

Lewis Research Center, National Aeronautics and Space Administration, Cleveland, Ohio, March 24, 1969, 129-01 -07 -07 -22.

APPENDIX - WORKING FORMULAS FOR COMPUTING FLOW PATTERN In this appendix equations (2) and (17) will be rewritten in various forms which a r e convenient f o r computing the velocities on and the positions of the various boundaries of the flow. To accomplish this the quantities defined in appendix B of reference 3 will be used. The positions along the free streamlines are found by integrating equation (17) along the sides of the rectangle of figure 7 where T = i q a . By using the various f o r - mulas for the change of argument of elliptic functions, the following relation is obtained in which the elliptic functions depend on real arguments:

L

Similarly, along the top of the rectangle, which corresponds to the solid boundaries and plane of symmetry, 6R

sn(%,k') + -

6L I 6R

I 1 + -

L

Using the results and definitions of appendix B of reference 3 and substituting equa- tions (Al) and (A2) in equations (2) and (17) give the following working formulas (note that M w a s defined in equation (18)): Position and Corresponding Velocity Along Nozzle Exit

J

Position and Corresponding Velocity Along Ground Position and Corresponding Velocity Along Fuselage

J

Height of P o i n t A o n Fuselage Above Ground

Coordinates of Free Streamlines

Define and Hi as follows Then the coordinates along the the right f r e e streamline extending between the points D and E are The last term is the horizontal displacement of the left branch of the jet at infinity.

The coordinates along the free streamline between C and F a r e The coordinates along the f r e e streamline between B and F a r e

REFEREN C E S

1. Spooner, S. H. : The V/STOL Aircraft Environment. Paper No. 68-GT-40, ASME, Mar. 1968.

2. Kemp, E. D. G. : Studies of Exhaust Gas Recirculation f o r VTOL Aircraft. Paper NO. 67-439, AIAA, July 1967.

3. Goldstein, Marvin E. ; and Siegel, Robert: Two Dimensional Inviscid Jet Flow From NASA T N D-5064, 1969.

Two Nozzles at an Angle to a Plane Surface.

4. Birkhoff, Garrett; and Zarantonello, E . H. : Jets, Wakes, and Cavities. Academic P r e s s , 1957.

NASA-Langley, 1969 - 1 E-4903