Turbulent boundary layer characteristics of pointed slender bodies of revolution at supersonic speeds

APPENDIX A INTEGRAL- THICKNESS DATA REDUCTION TECHNIQUE Since solutions of the inviscid flow equations for the parabolic-arc bodies used in are not constant in the radial this investigation reveal that the inviscid flow properties direction, it was decided not to u s e the conventionally defined boundary-layer integral equations which, in effect, assume that inviscid flow properties a r e constant through the boundary-layer thickness and equal to those at the edge of the boundary layer. Instead, equations similar to those of references 11, 12, and 23 were used as the general expres- sions for the integral parameters in compressible flow over slender axisymmetric bodies where the inviscid flow properties are nonuniform. These equations a r e as follows: Momentum thickness Displacement thickness Velocity thickness Note that for constant inviscid conditions, these equations reduce to their more familiar forms. These equations differ from those of the aforementioned references in the fol- lowing respects: (1) Reference 11 omits the radius term in the foregoing equations and thereby assumes two-dimensional flow. (2) Reference 12 uses wall conditions as a refer- ence instead of the boundary-layer edge conditions used in equations (Al) to (A3).

(3) Reference 23 derives the equations for a viscous wake instead of a boundary layer.

Approximations to the true inviscid flow conditions were made experimentally by measuring pitot pressure profiles external to the boundary layer, where viscous forces a r e negligible, and linearly extrapolating these profiles through the boundary layer to the model surface. The profiles thus determined a r e termed "inviscid" in this report. In addition, linear approximations to the static pressure profiles were obtained by experi- mentally measuring static pressures in the radial direction and fitting the best straight line through the data. Typical static, pitot, and inviscid pitot pressure profiles are shown in sketch 1.

APPENDIX A o Experimental data

- Linear approximation

Y pressure extrapolation P Sketch 1 The static and pitot pressure profiles were combined by using the Rayleigh pitot formula, equation (22), to form the Mach number profiles. Similarly, the static and invis- cid pitot pressure profiles were combined to form the inviscid Mach number profiles.

These two profiles a r e illustrated in sketch 2.

Mach number Sketch 2 The difference between these two profiles is the measure of the viscous effect of the boundary layer and is taken into account by using the integral parameters in the form of equations (Al), (A2), and (A3). These equations, when written in terms of pressure and Mach number, become

APPENDIX A and

6 , = s,"

The integrands of these equations were evaluated at each point sampled in the boundary-layer pitot pressure survey. The integrations were performed numerically by the trapezoidal rule.

The integral thicknesses calculated by equations (A4), (A5), and (A6) a r e a more accurate representation of the boundary-layer effects than a r e the conventionally defined thicknesses, since the conventional definitions make no attempt t o exclude the effects of changes in the inviscid flow field through the boundary-layer thickness.

APPENDIX B AVERAGE-SKIN-FRICTION EQUATION An approximate expression relating the average skin friction on a body of revolution in supersonic flow to the unconventionally defined momentum thickness 0 defined by equation (Al) is derived in this appendix. The flow field is shown in the following sketch: From the theorem of conservation of momentum, the net flux of momentum of the fluid moving through the axisymmetric control volume defined by surface ABCD is equal t o the total force acting on the control volume; this is illustrated in the following sketches: Momentum flux in x-direction A 0 B , I - - -- - - - -k - - - I . - / C c

I / - -

D Total forces in x-direction Equating the forces and momentum flux results in

APPENDIX B where h is the shock height. The total drag in an inviscid flow field can be seen from equation (Bl) t o be -I Assuming that the radial pressure distribution and the shape of the shock wave a r e not changed by the presence of the boundary layer (i.e,, p = pinv and h = hinv) resuits in But, continuity requires that or If the radial inviscid velocity gradient is assumed to be small, Inserting equation (B6) into equation (€33) yields which can also be written But, at y 2 6 , u = uinv so that the second t e r m in equation (B8) is zero. Hence,

DF = 2n S, Pu(uinv - u)(r + y)dy

(B9)

APPENDIX B In coefficient form, equation (B9) is The integral in equation (B10) is the general expression for momentum thickness as defined by equation (Al). Hence, It should be noted that the conventional method of calculating average skin friction (relating wall shearing s t r e s s to the flow parameters by an elemental control-volume analysis and then performing an integration over the body surface), although more exact, is more difficult to use in pressure gradient flows because i t requires an integration of the pressure gradient over the body surface. The approximate method described in this appendix is much simpler to use since it requires measurements only at the body stations where CF is required.

REFERENCES 1. Gazley, C., Jr.: Theoretical Evaluation of the Turbulent Skin-Friction and Heat Transfer on a Cone in Supersonic Flight. Rept. No. R49A0524, Gen. Elec. C o . , Nov. 1949.

2. Bradfield, Walter S.: An Experimental Investigation of the Turbulent Boundary Layer in Supersonic Flow Around Unyawed Cones With Small Heat Transfer and Correla- tions With Two Dimensional Data. Res. Rept. No. 1, Convair Sci. Res. Lab., Mar. 15, 1958.

3. Van Driest, E. R.: Turbulent Boundary Layer on a Cone i n a Supersonic Flow at Zero Angle of Attack. J. Aeron. Sci., vol. 19, no. 1, Jan. 1952, pp. 55-57, 72.

4 . Bertram, Mitchel H.: Calculations o f Compressible Average Turbulent Skin Friction.

NASA TR R-123, 1962.

5. Brown, Clinton E.: Aerodynamics of Bodies at High Speeds. Aerodynamic Components of Aircraft a t High Speeds. Vol. VII of High Speed Aerodynamics and Jet Propul- sion, sec. B, A. F. Donovan and H. R. Lawrence, eds., Princeton Univ. Press, 1957, pp. 244-280.

6. Reshotko, Eli; and Tucker, Maurice: Approximate Calculation of the Compressible Turbulent Boundary Layer With Heat Transfer and Arbitrary Pressure Gradient.

NACA TN 4154, 1957.

7. Wazzan, Ahmed R.; and Ball, W. H.: Body Shape Effects on Skin Friction in Supersonic Flow. AIAA J. (Tech. Notes), vol. 3, no. 9, Sept. 1965, pp. 1770-1772.

8. Sasman, Philip K.; and Cresci, Robert J.: Compressible Turbulent Boundary Layer With Pressure Gradient and Heat Transfer. AIAA J., vol. 4, no. 1, Jan. 1966, pp. 19-25.

Over Insu- 9 . Englert, Gerald W.: Estimation of Compressible Boundary-Layer Growth lated Surfaces With Pressure Gradient. NACA T N 4022, 1957.

10. Winter, K. G.; Smith, K. G.; and Rotta, J. C.: Turbulent Boundary-Layer Studies on a Waisted Body of Revolution in Subsonic and Supersonic Flow. Recent Developments in Boundary Layer Research, Pt. I I , AGARDograph 97, May 1965, pp. 933-961.

11. Clutter, Darwin W.; and Kaups, Kalle: Wind- Tunnel Investigation of Turbulent Bound- a r y Layers on Axially Symmetric Bodies at Supersonic Speeds. Rept. No. LB31425, (Contract NOW 61-0404-T), Douglas Aircraft Co., Inc., Feb. 6, 1964.

12. McLafferty, George H.; and Barber, Robert E.: The Effect of Adverse Pressure Gradients on the Characteristics of Turbulent Boundary Layers in Supersonic Streams. J. Aerospace Sci., vol. 29, no. 1, Jan. 1962, pp. 1-10, 18.

13. Schaefer, William T., Jr.: Characteristics of Major Active Wind Tunnels at the Langley Research Center. NASA TM X-1130, 1965.

14. Truckenbrodt, E.: A Method of Quadrature for Calculation of the Laminar and Turbu- lent Boundary Layer in Case of Plane and Rotationally Symmetrical Flow.

NACA TM 1379, 1955.

15. Sommer, Simon C.; and Short, Barbara J.: Free-Flight Measurements of Turbulent- Boundary-Layer Skin Friction in the Presence of Severe Aerodynamic Heating at Mach Numbers From 2.8 to 7.0. NACA TN 3391, 1955.

16. Peterson, John B., Jr.: A Comparison of Experimental and Theoretical Results for the Compressible Turbulent-Boundary-Layer Skin Friction With Zero Pressure Gradient. NASA TN D-1795, 1963.

17. Tucker, Maurice: Approximate Calculation of Turbulent Boundary- Layer Development in Compressible Flow. NACA TN 2337, 1951.

18. Adcock, J e r r y B.; Peterson, John B., Jr.; and McRee, Donald I.: Experimental Inves- tigation of a Turbulent Boundary Layer at Mach 6, High Reynolds Numbers, and Zero Heat Transfer. NASA TN D-2907, 1965.

19. Ludwieg, H.; and Tillmann, W.: Investigations of the Wall-Shearing Stress in Turbu- lent Boundary Layers. NACA TM 1285, 1950.

20. Ames Research S t a f f : Equations, Tables, and Charts for Compressible Flow. NACA Rept. 1135, 1953. (Supersedes NACA TN 1428.)

Section 20 - Wind Tunnel

21. Volluz, R. J.: Handbook of Supersonic Aerodynamics.

Instrumentation and Operation. NAVORD Rept. 1488 (Vol. 6), Bur. Naval Weapons, Jan. 1961.

22. Kulfan, Robert M.: Turbulent Boundary Layer Flow Past a Smooth Adiabatic Flat Plate. Doc. No. D6-7161, Boeing Airplane Co., May 16, 1961.

23. Sorrells, Russell B., III; Jackson, Mary W.; and Czarnecki, K. R.: Measurement by Wake Momentum Surveys at Mach 1.61 and 2.01 of Turbulent Boundary-Layer Skin Friction on Five Swept Wings. NASA TN D-3764, 1966.

TABLE 1.- SPECIFICATIONS OF MODELS Body profile equation ineness .ength, vlodel Description ratio (r and x incm) cm 0 < x < 63.6 5 Cone 63.6 12.2 r = 0.041~ 2 Cone 12.4 r = 0.0402~ 0 < x < 127 r = 0.0768~ 0 < x < 11.83 8 Cpne- 63.6 12.2 r = -0.1244 + 0.0976~ - 0.000875~2 11.83 < x < 63.6 mrabolic-arc 4 Cone- 127 12.2 0 < x < 23.7 r = 0.0769~ 23.7 < x < 127 ?arabolic-arc r = -0.231 + 0.0974~ - 0.000436~2 6 . 1 r = 0.128~ 0 c x < 17.78 3 Cone- 6 3 . 6 r = -0.442 + 0.178~ - 0.0014~2 17.78 < x < 63.6 parabolic-arc 12.2 r = 0.082~ - 0.000646~~ 0 < x < 63.6 6 Parabolic arc 63.6 r = 0.082~ - 0.000323~~ 0 < x < 127 7 Parabolic arc 127 12.2 r = 0.1333~ - 0.000854~2 0 < x < 127 1 Parabolic arc 127 12.2 NACA RM-10: TABLE I I . - SUMMARY OF TEST CONDITIONS . - Profile number __ M , = 1.61 at I atm, of - M, = 2.20 at pt, atm, of - Station x, CIT vlodel Xtr, cm - ~. ~ 0 . 3 ' 1.01 1 . 7 1 0.61 1 . 0 1 1.36 1 . 7 c 0.6E 0.34 - 46 45 44 2 1 1 1.27 1 70.4 49 48 41 2 5 4 3 100.5 6 54 53 52 50 10 9 8 7 3 116.1 11 58 57 55 4 1 5 14 13 12 121.4 ~- 61 60 59 2 1 74.5 2.54 17 16 64 63 2 115.3 18 .- - 66 65 20 19 3 1 . 2 7 1 35.7 23 22 21 2 54.5 69 68 67 2 58.7 - 24 71 70 4 1 71.4 25 1.90 27 26 74 73 72 2 117.3 28 - 29 77 76 5 2.54 1 35.7 79 78 32 31 30 80 2 5 8 . 7 - 83 82 81 1.90 1 6 3 5 . 7 86 85 84 2 58.7 35 34 33 - 88 87 7 1.90 1 7 1 . 4 36 91 90 89 2 117.3 38 37 - - 8 1.90 1 35.7 40 95 93 43 42 41 94 2 5 8 . 7 - - M, = 1.61 ~~ . - Profile ' Model Station] pt, atm

8 , mm 6*, mm 1 6 , , mm 6, mml

R, - Cf 1 1 1 1.01 9.614 X lo6 0.597 1.532 0.971 8.500 0.002245 0.002643 6.1 2 1 1 .34 3.220 .705 1.802 1.149 9.150 .002698 .003141 6.1 3 1 2 1.36 17.800 .943 2.432 1.519 12.170 .001907 .002324 7.1 4 1 2 1.01 13.760 .939 2.383 1.478 12.120 .002016 .002305 7.1 5 1 2 .34 4.592 1.184 3.017 1.892 14.180 .002376 .002909 6.9 6 1 3 1.70 26.179 1.214 3.055 1.852 15.030 .001710 .002108 6.4 7 1 3 1.36 21.171 1.269 3.215 1.964 15.630 .001764 .002206 6.4 8 1 3 1.01 15.897 1.336 3.404 2.105 16.120 .001848 .002322 6.5 9 1 3 .68 10.623 1.382 3.509 2.168 17.190 .001987 .002414 6.4 10 1 3 .34 5.312 1.470 3.712 2.304 16.250 .002254 .002556 6.3 11 1 4 1.70 27.286 1.400 3.544 2.153 17.580 .001653 .002123 7.0 12 1 4 1.36 22.130 1.436 3.631 2.226 17.610 .001713 .002168 6.8 13 1 4 1.01 16.578 1.466 3.732 2.311 17.640 .001811 .002217 6.7 14 1 4 .68 11.105 1.567 3.986 2.463 17.750 .001935 .002369 6.6 1 5 1 4 .34 5.552 1.734 4.409 2.740 18.820 .002172 .002641 6.5 16 2 2 1.70 25.705 .680 1.630 1.012 10.980 .002057 .002310 7.3 17 2 2 1.01 15.754 .750 1.796 1.126 10.900 .002227 .002530 6.8 18 2 2 .34 5.277 .869 2.105 1.343 10.710 .002673 .002917 6.6

19 3 1 1.01 4.890 .314 .771 .515 I 4.390 .002647 .003027 5.8

20 3 1 .34 1.634 .381 .939 .639 5.440 .003197 .003648 5.8 21 3 2 1.70 12.278 .470 1.176 .762 7.110 .002169 .002573 5.7 22 3 2 1.01

7.467 ' .505 1.251 ' .826 6.950

.002375 .002674 5.7 23 3 2 .34 2.495 .622 1.578 1.069 7.560 .002839 .003259 5.2 24 4 1 1.70 15.911 .559 1.573 1.016 8.960 .002113 .002969 6.1

25 4 1 .34 3.266 .720 1.766 1 1.178 8.650

.002884 .003279 6.0 26 4 26.489 2 1.70 .942 2.224 1.412 12.870 .001897 .002135 6.8 27 4 2 1.01 16.060 .961 2.305 1.446 13.380 .002080 .002274, 6.8 28 4 2 .34 5.366 1.177 2.916 1.876 13.670 .002456 .002788 5.7 29 5 1 .34 1.669 .317 .781 .495 4.510 .003255 .003546 6.6 30 5 2 1.70 13.285 .339 .833 .521 5.900 .002358 .002296 6.4 31 5 2 1.01 8.013 .388 .956 .604 6.020 .002525 .002619 6.7 32 5 2 .34 2.741 .444 1.091 .690 5.820 .003040 .002976 6.7 33 6 2 1.70 13.131 .462 1.145 .703 7.470 .002154 .002421 7.3 34 6 2 1.01 8.013 .467 1.144 .701 7.160 .002377 .002448 7.2 35 6 2 .34 2.665 .573 1.413 .867 7.230 .002817 .003038 7.1 36 7 2 : 1.70 26.103 .837 2.067 1.262 12.990 .001920 .002219 7.6 37 7 2 1.01 16.060 .906 2.253 1.389 13.710 .002077 .002403 7.1 38 7 2 .34 5.366 1.068 2.660 1.655 13.350 .002471 .002832 6.8

39 8 1 1.70 7.987 .295 .734 .471 4.920 .002386 .002850 1 6.1

40 8 1.621 .385 .963 .639 4.370 .003151 .003697 5.9 1 .34 41 8 2 1.70 13.170 .507 1.261 .794 7.500 .002115 .002468 6.3 42 8 2 1.01 8.013 .526 1.292 .820 7.470 .002334 .002515 6.1 43 8 2 .34 2.684 .646 1.601 1.037 7.570 .002775 .003097 6.0 -.

TABLE m.- SUMMARY OF RESULTS - Concluded M , = 2.20 - - rofile Model Station 6*, mm i, mm n 8, mm Pt, a b I , , mm Cf Rx C F - - 44 1 1 1.70 12.052 X l o 6 0.500 1.156 0.861 8.590 l.001891 1.002192 1.1 .002043 .002201 45 1 1 1.01 1.636 .502 1.148 .e49 1.810 1.2 46 1 1 2.553 .IO3 2.460 1.213 .1.480 .002466 .003084 6.1 .34 1 .001642 .002052 4 1 2 1.10 11.301 .851 3.014 1.459 .3.460 1 . 0 48 1 2 1.01 10.903 .882 3.119 1.530 .2.110 .001803 .002119 6.8 49 1 2 .34 3.618 1.146 3.969 2.029 5.660 .002190 .002168 6.5 50 1 3 1.10 19.956 1.069 3.836 1.814 5.450 .001520 .001802 1.3 5 1 1 3 16.466 1.094 .001834 I. 1 1.36 3.899 1.860 .5.320 .001585 52 2.042 !9.310 .001630 I. 0 1 3 1.01 12.596 1.185 4.252 .002009 53 1 3 8.385 1.289 4.632 2.214 18.660 .001138 .002115 6.9 .68 1 3 .34 4.249 1.611 5.580 2.180 !1.350 .002000 .002136 6.8 55 1 4 1.10 21.218 1.201 4,251 1.993 !2.110 .001484 .001131 1.1 56 1 4 1.36 11.113 !1.350 1.251 4.449 2.111 .001542 .001838 1.4 5 1 1 4 1.01 13.127 1.353 4.841 2.291 !9.110 .001584 .002008 1 . 1 58 1 4 .68 8.165 1.453 2.510 .9.250 .001131 .002120 5.115 6.9 59 2 1 1.70 9.610 13.052 .455 1.531 .I49 .002006 .002461 1.5 60 1 1.01 8.060 .416 1.595 .I84 .0.230 .002199 .002586 1.5 61 2 1 2.800 .34 .568 1.903 .912 L0.380 .002691 . 0 03 0 12 1.5 2 2 62 1.10 20.315 .611 2.061 .996 .2.180 .001908 .002168 1.8 63 2 2 1.01 12.438 .655 2.112 1.055 .3.660 .002088 .002296 1.5 64 2 .34 4.291 .I58 2.490 1.245 12.620 .002550 .002621 1.2 65 3 1 1.10 6.138 .435 5.320 2 4 5 .831 .002331 .002650 1.2 3 1 66 .34 1.319 .298 .982 .515 5.240 .003249 .003205 8 . 0 61 3 2 1.10 10.160 .410 1.616 3364 1.690 .002042 .002516 6.2 3 10.050 68 2 1.01 6.326 .515 1.820 .952 .001949 .002803 6.3 3 2 .34 2.166 .562 1.934 1.023 8.330 .002616 69 .003000 6.4 4 1 1.10 12.365 10 .464 1.591 .I88 9.410 .001915 .002328 1.5 11 4 1 .34 2.636 .591 2.004 1.036 8.500 .002658 .002910 1.4 4 2 13.140 12 1.10 20.430 3 0 5 2.801 1.355 .001101 .001991 1 . 6 13 4 2 1.01 12.649 .862 3.008 1.411 13.860 .001844 .002153 1.4 4 2 1 4 .34 4.310 .919 3.304 1.695 13.130 .002355 .002401 6.8 1 5 5 1 1.1a .409 6.313 ,234 .e22 4.100 .002223 .002643 I. 5 1 6 1 1.01 3.851 ,253 ,865 .431 4.850 .002413 .002895 1.4 11 5 1 .34 1.284 1.000 .508 4.840 .003054 .288 .003332 1.4 5 2 1.10 8.480 18 10.409 .311 1.064 .513 .002161 .002161 I. 5 I 9 5 2 1.01 6.326 1.266 .630 1.960 .002302 .002545 .311 1.5 5 2 .34 80 2.109 .430 1.414 .I38 6.900 . 00 2 8 0 I .002915 8.0 6 1.10 8 1 1 6.232 .243 3 4 2 .418 4.450 .002233 .002450 7.4 6 1 1.01 82 3.146 .286 .999 .501 4.980 .002391 .002890 1.2 83 1 .34 1.284 .318 1.114 .560 5.840 .002991 .003233 8.1 84 6 2 1.10 10.231 .409 1.423 .694 1,540 .001910 .002135 6.8 6 2 85 1.01 6.326 .471 1.653 3 1 6 9.460 .002089 .002492 6.6 86 6 2 .34 2.109 .525 1.840 .e91 8.620 .002552 .002187 I. 1 I 1.10 8 1 1 12.435 .419 1.668 .826 8.350 .001933 .002430 6 . 1 I 1 .34 2.660 .945 .002682 88 .519 1.896 10.410 .003033 1 . 6 I 2 1.10 89 20.392 .I46 2.608 1.238 13.940 .001125 .001916 1.3 I 2 1.01 12.649 311 2.846 1.367 13.330 .001864 .002187 9 0 1.4 1 2 .34 4.310 9 1 .945 3.148 1.532 14.410 .002348 .002610 1.5 8 1 . ' .34 1.283 .308 1.082 .550 5.390 .003045 .003110 1.2 2 1.10 8.180 93 8 10.461 .461 1.643 3 2 2 .001886 .002215 6.6 94 8 2 1.01 6.326 .502 1.780 .892 9.010 ,002056 .002499 6.6 2 .34 8.260 95 8 2.109 .569 2.041 1.020 .002491 .002847 1.1 __ O d L- 65- 6987.1 (a) 127-cm-long models.

Figure 1.- Photograph of models.

L-65-6988.1 ( b ) 63.3-cm-long models.

Figure 1.- Concluded.

..o

r

.5 -

rmax Body prof i l e -

---

M o d e l 5-7

/' /

.5 /

Y

'Model 5,' ' % M o d e l 2 /-

1 Station 2 Station

- - 6 3 0 1 I 0

I

0 .1 .2 . 3 .4 .5 .6 . 7 . 8 .9 1.0' (b) Models 2 and 5 (slender cones).

Figure 2.- Continued.

w u1 - ~ - '-2- -2-

---

Body p r o f i l e

. 0 6 ---

.o

----

---

M , = 1.61- r2.20

. 0 8 ---

-- .5

,' S t a t i o n 1 S t a t i o n 2

0 .1 .2 . 3 .4 .5 .6 .7 .8 .9 1 . 0 X

-

( c ) Model 3 (cone-parabolic arc).

Figure 2.- Continued.

cP

.02 ---

A

--

Body p r o f I la-

-L-- 1,o

-------

I r

.5 -

rmax Station 1 .4 .5 . 6 .7 .8 .9 1 . 0 0 .1 .2 . 3 X

-

(d) Models 4 and 8 (cone-parabolic arc) Figure 2.- Continued.

., (e) Models 6 and 7 (parabolic arc).

Figure 2.- Concluded.

S t a t i c p r e s

.

P i t o t p r e s s u r e p r o b e

L-65-6986.1 Figure 3.- Boundary-layer probes.

... . . . . ..

I

- 1

I

p t , a t m P r o f i l e 0 0.34 2 A 1.01 1 I B I

1 . 4

0 .1 . 2 .3

.5 .b .7 . a

.9 U

-

Ub (a) Model 1 , station 1.

Figure 4.- Velocity profiles at a free-stream Mach number of 1.61, 2; 2c 1 E 1 k , p r o f i : ?

p t 9 a t m y 12 0 0.34 5 A 1.01 4 0 1.36 3

e

i C

.1 . 2 .3 .8 . 9 1 .o

. 4 .5 .b .7 U L_ " b (b) Model 1, station 2.

Figure 4.- Continued.

. . ._

I

I

I d

I

I I

I I

I 1

Q 0.34 0 O.b8 A 1.01 0 1 . 3 6

I

I

I

I

I

0 .1 . 2 .3

. 4 .5 . 6 .7 .8 .9 1 .o

U

-

Ub (c) Model 1, station 3.

Figure 4.- Continued.

It ?

14 P r o f i 0 15 0 . 6 8 14 - 1; A 1 . 0 1

e

1 2 0 1.36 C 1 1.

I -

I

m

. 9 1 .o

.a

. I . 4 . 3 .1 U

-

“ b (d) Model 1, station 4.

Figure 4.- Continued.

I O

I l l

I 1

l l

I l l

I l l

I l l

I l l

I l l

a t m P r o f i l e P t 0 0 .

34 18 A 1.

01 17 n 1.

.5 . b . 7 .8 .9 U

-

Ub (e) Model 2, station 2.

Figure 4.- Continued.

a t m P r o f i l f

e

f'

.1 . 2 . 3 .5 .7 .a .9 1 .o

. 4 .b U

-

U b ( f ) Model 3, station 1 .

Figure 4.- Continued.

J w ~- ... . . .... .

. . . . . . . ... . , .

.. .....

- , .,, .. ... ....

..,,.,.,,, , ,,

I

L

I

I

I

I

I

0 . 3 4

i

I

*3 1 A 1.01 1 . 7 0

I 21

I I

I l j

' I

I

I

I

I

I

I

!

I

I

.3 . 4 .5 .b .7

.9 1 .o

U

-

U b (g) Model 3, station 2.

Figure 4.- Continued.

i a

l b P r o f i l e Y - 1; 0 0.34 1 . 7 0 IC E t 1 B p , I ) B n I

.9 1 .o

. a

.7 .b . 4 .5 .2 .3 .1 U

-

" b (h) Model 4, station 1.

Figure 4.- Continued.

p t , atm P r o f i l e Y I

e

0 0.34 A 1.01 D 1 . 7 0

I

p 4

c

0 .1 . 2 .3 . 4 .9

1 .o

(i) Model 4, station 2.

Figure 4.- Continued.

i

p t , a t m P r o f i l e

0 0.34 29

1 .o

.4 . 5 . 6 .1 . 2 .3 U _I_ " 8 Cj) Model 5, station 1.

Figure 4.- Continued.

p t a t m P r o f i l e ] 0 0.34 A 1 . 0 1

3”; 1

1 . 7 0

I

I

I

t ~l

- 1

b f

0 .1 . 2 .3 . 4 . 5

.Q

. 7

. a .9 1 .o

(k) Model 5, station 2.

Figure 4.- Continued.

I

P r o f i l , a t m P 3 0.34 A 1.01 b 1 . 7 0

I

I

I

I

.9 1 .o

.b .7 .5 .3 . 4 . 2 .1 U

-

Ub ( I ) Model 6, station 2 .

Figure 4.- Continued.

I l l 1 l l

I l l 1 1

I I O I

I I I I

1 8

I I O I I

I l l 1

1 6

I l l 1

I l l 1 1

I I

I I I I

P t , a t m . P r o f i l e 1 2 0 0.34

I

37 A 1 . 0 1

I i i i i i i i

L ! ! I ! ! I

l l l l l l l

I I I I I ! !

.5 OO .1 .2 .3 . 4 .b .7 . I U

-

Ub 7, station 2. (m) Model Continued. Figure 4.- Pt, a t m 0 0.34 b 1.70

I

I

.9 1 .o

.8 .b .5 - 4 .1 U

-

Ub (n) Model 8, station 1.

Figure 4.- Continued.

P

P

i

i

d

D J

{I

I

I

.3 .4 .5 . 6 . 7 .a

1 . o

(01 Model 8, station 2.

Figure 4.- Concluded.

1k a t m e - Y 1 : 0 0.34

e

A 1.01 45 c\ 1.70 44 1 c i L i4 D I &

.1 . 2 . 7 .a

.3 . 4 .b .9 1 .o

(a) Model 1, station 1.

Figure 5.- Velocity profiles a t a free-stream Mach number of 2.20.

P r o f i l e 0 0.34 4 9 A 1 . 0 1 4a b 1 . 7 0

I

l l

I

I I

I I

I

l l

I

I

l l A !

I 1

d *A I@

c

I I

..

. L .5 . b . 7

.a .9

U

-

Ub (b) Model 1 , station 2.

Figure 5.- Continued.

0.34 54 0.68

Y - 12

-+.

A 9 1.01 52 1 .36 51 1 70 CI &W . 5 . 4 .1 U

-

" b (cl Model 1, station 3.

Figure 5.- Continued.

P t , a t m P r o f i l e 0 0 . 6 8 A 1 . 0 1 5 7 0 1 . 3 6 n 1 . 7 0

P

.3 . 7 . 8 .9

1 .o

(d) Model 1, station 4.

Figure 5.- Continued.

. .. . . ... . . .

I

I

I

e P r o f i P t 1 o 0.34 61 6 0 A 1 . 0 1 5 9 n 1.70

f

D ‘1 a C .9 1 .7 . b .3 . 4 . 2 (e) Model 2, station 1 .

Figure 5 . - Continued.

p t , a t m Pr-F i l e 0 0.34 64 A 1.01 63 n 1 . 7 0 6 2

U I I I I

_ L l l I I l

I H I I

UI I I

I L I I I

1 l I I I I

UI I I

I I I I I I

I I I L

k k A a L Q 1" fnTy

0 .1 . 2 . 3 . 4 . 5 . b . 7 U

-

" b (f) Model 2, station 2.

Figure 5.- Continued.

P r o f i l e

Y - 12

e

6 6 0 0.34 6 5 n 1.70 C D n 3 - O C D.

Q C

.8 . 9 1 .o

.5 .b .7 .1 .3 . 4 .2 U (g) Model 3, station 1.

Figure 5.- Continued.

I

I

f

ik

i

f

d

k?

"I

I

.3 . 4 .5 . 6 . 7

1 .o

U

-

Ub (h) Model 3, station 2.

Figure 5.- Continued.

I 1( 1 1 1 P r o f i l P t , a Y - 1 3 0.34

e

1.70 c

e

4 I

1 I

i o

. 9 1 .o

I .8

I .7 . 3 . 2 (i) Model 4, station 1.

Figure 5.- Continued.

I 1

l l

l l

l l

I I

p t , a t m P r o f il

0 0.34

* 1.01

1.70 m .4 .7

.8 .9 1 .o

Cj) Model 4, station 2.

Figure 5.- Continued.

I I I I

I

I I I

I I I

I

I I

I

I I

I l l 1

0 0.34 A 1.01 LS 1.70

I

.9 1.

.7 . 4 .5 . 2 .3 U L_ " b (k) Model 5, station 1.

Figure 5.- Continued.

p t , a t m P r o f 3 0.34 3 1.01 1 1 . 7 0 c a' P

' i

c . 2 .3 -4 . 5

. 6 . 7 . 8

.9 1 .o

U

-

Ub (1) Model 5, station 2.

Figure 5.- Continued.

1 8 1 4 a t m P r o f i l e Y

- 1 2

8 3 8 2 8 1

e

e

2 $

L & n I C . 8 .9 1 . 2 . 3 . 4 . 5 . b - 7 .1 U

-

U b (m) Model 6, station 1 .

Figure 5.- Continued.

p t , a t m P r o f i l t 0 0.34 86 A 1.01 85

I I I I I

/ / / I

I I I I

2 1 7

. 8 .9 OO .1 .2 . 3 U

-

U b (n) Model 6, station 2.

Figure 5.- Continued.

P r o f i l 8 8

I

n L

I

+ 0

D b r

I

.8 . 7 . 6 .5 . 4 .3 .1 (0) Model 7, station 1.

Figure 5.- Continued.

/ I l l

I l l 1

I l l

I l l

I I I I

I I I I

I l l 1

I I I I

I I I I I

a t m P r o f i l e 1 P t ’ 0 0.34 A 1.01 1.70 89 .3 . 4 . 5 .b . 7 (p) Model 7, station 2.

Figure 5.- Continued.

2; 2c 1 E It: Y

- 1 2

e

a t m P r o f i l e 0 0.34 1 c € i C

.1 .3 . 4 .7 .8 . 9 1 .o

(q) Model 8, station 1 .

Figure 5.- Continued.

p t , a t m P r o f i l e

i

0 0 . 3 4

e

P L3 1 . 0 1 94 b 1 . 7 0

B

B 4 r --c .8 .7 (r) Model 8, station 2.

Figure 5.- Concluded.

--- u - (#n

ut)

Y

- Profile 20 P r o f i l e P r o f

.8 .6 I .4

n = 6 . 5

.2

= 5*8<

I

I. -- I I L I 1

.2 .4 .6 .81.0 .2 .4 .6 -81.0 .2 .4 .6 . 8 1 . 0 U

u B

(a) M, = 1.61; pt = 0.34 atm.

Figure 6 . - Sample velocity profiles in logarithmic form.

1 7

a

b

t

Profile 24

Profile 16

Profile 11

I n = 6.1 .

.n = 7.0 4

-4

I

, , i V , ,

I

02 .4 .6 .81.0 .2 .4 .6 .81 .2 .4 .6 ,81.0 U

Ut

(b) = 1.61; pt = 1.70 atm.

Figure 6.- Continued.

--- u -

u B U

e

U - Y Profile 55 Profile 65 Prof i le

-

. 8 .6

P

.4

n = 7.8

7*2-4

.2

I

. 1 .2 .4 .6 .81.0 02 .4 .6 .81.0 U (c) M, = 2.20; pt = 1.70 atm.

Figure 6.- Concluded.

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

M o d e l 0 1 0 2 A 3 0 4 0 5 0 6 0 7 F l a t - p l a t e e x p e r i m e n t ( r e f . 22) 0 0 0 ""0 n 6 A A o a A A

I

I 2

01 I I 1 I I 1 1 1 2 4 6 8 1 0 20 40 103 Re (a) M, = 1.61.

I I 1 I I 1 1 1 2 4 6 8 1 0 20 40 103 R e (b) /'&= 2.20.

Figure 7.- Variation of velocity profile index with Reynolds number based on momentum thickness.

1.3- F l a t p l a t e - - - -- - - - - Slender cone (models 2 and 5) 1.2-

I

- - Model 1 I

- --

Model 3

- --- Models 4 and 8

--- Models 6 and 7 1.1-

I

I 1.0

/

e

-

.9-

e

.8-

/’

/

0 .1 .2 . 3 .4 .5 .6 .7 . a .9 1 . 0

Figure 8.- Theoretical momentum thickness ratios calculated by equation (3).

I I M o d e l 0 2 0 5 .0002 I I I I I I 2 4 6 8 1 0 20 O1 40 x lo6

I R X

(a) Mm = 1.61.

.0014- - S l e n d e r - c o n e t h e o r y ? e q . ( 6 ) ) - 0 0 1 2 - .OOlO- '.

.

.0008- [7----. .

. u

--.. D

1 .

- a--J.JJ - -

.0006 b--q--- 0

---

--

---

--

.0004- .0002

i- I

I I I l l I 01 2 40 x 106 4 6 8 1 0 20 r R X (b) M , = 2.20.

Figure 9.- Momentum thickness distributions on models 2 and 5 (slender cones).

.0004 . _ _ F l a t - p l a t e t h e o r y ( e ( 5 ) ) - _ - _ - - - S l e n d e r - c o n e t h e o r y ? e q (6)) .0002 ~ - M o d e l 1 t h e o r y ( e q . ( 3 ) j I I - J I '

4 A 4 2 0 20 40 x lo6

R X (a) M, = 1.61.

p t = 0 . 3 4 atm 0 . 6 8 .0014 \. 01

I

i

.0012- . O O l O -

-"-

.oooa - X - .0006 - .0004 - .0002 I I I I I A

I I

2 4 6 8 1 0 20 4 0 x 106 O1 R X (b) P&= 2.20.

Figure 10.- Momentum thickness distribution on model 1 (RM-10). Ticks differentiate between the stagnation pressure levels.

E -0014 A A P t 0.34 atm

\

/

LT 1.01 .

K L1 ‘.

1 . 7 0 -.

.

-.,,-/-

-.

’/

-.- - -

--. X

-.

---

---

.0006

---

--

-.

---

.0004

t

. 0 0 0 2 t I I O1 I 1 2 I 1 1 I 4 6 8 1 0 20 4 0 x 106 R X (a) M, = 1.61.

F l a t - p l a t e t h e o r y ( e q . ( 5 ) - - - - - .

S l e n d e r - c o n e t h e o r y ( e q . ( . 0 0 1 4 r \ M o d e l 3 t h e o r y ( e q . ( 3 ) ) p t = 0 . 3 4 atm .0010 / 1.70 - X .0004 .0002

i

I I I I I I L I O1 2 4 6 8 1 0 20 4 0 x 106 R X (b) b= 2.M.

Figure 11.- Momentum thickness distribution on model 3 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

. 0 0 1 4 . 0 0 1 2 . 0 0 0 4 k . 0 0 0 2

I I I I 1 I I ’

01 4 6 8 10 2 20 4 0 x lo6 R X (a) M, = 1.61.

- - F l a t - p l a t e t h e o r y ( e ( 5 ) ) _ _ - _ - S l e n d e r - c o n e t h e o r y Teq (6)) . 0 0 1 4 - - M o d e l 4 t h e o r y ( e q . ( 3 ) j

\

‘\ . 0 0 1 2

\

.0006 . 0 0 0 4 (b) M, = 2.20.

Figure 12.- Momentum thickness distribution on model 4 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

.0014- .0012-

'---- p t = a 0.34 a t m 1

/ .0010, \

. 1 . 0 1

'.

.

.

1.70

.--. . /

c r - -.

.0008 W --_

.._ ------. 1

.

-

---_

X -\ -.

--

--

.0006-

- ~

--

--

---

0002 O o o 4 1 I I O1 i L 2 4 6 8 1 0 20 40 x lo6 R X (a) M, = 1.61.

F l a t - p l a t e t h e o r y ( e q . (5)) S l e n d e r - c o n e t h e o r y ( e q . (6)) .0014

\ =-= M o d e l 6 t h e o r y ( e q . ( 3 ) )

.0012 . O O l O .0006 .0004 . o o o z

I

I I b.- I I I 01 L L - - 2

20 40 x lo6 6 8 10 R X (b) M, = 2.20.

Figure 13.- Momentum thickness distribution on model 6 (parabolic arc). Ticks differentiate between t h e stagnation pressure levels,

-Ool4r .0014-

\

.0012 .0012-

p t = \ 0.34 a t m p t 1 = (/ 0.34 a t m

.0010, .0010,

t

-.

'\

'-.> - -1

-. 1 . 0 1

.. . . \__ /- '

e -

w

--.

X -- .- - . . /--

---

--

--

.0006-

--

--- ---

--- ---

-- --

I .0004

I I I I I -

2 4 6 8 1 0 20 40 x l o 6 R x (a) M, = 1.61.

F l a t - p l a t e t h e o r y ( e ( 5 ) ) _ _ _ _ _ _ S l e n d e r - c o n e t h e o r y 7 ; s ( 6 ) ) .0014 M o d e l 7 t h e o r y ( e q . ( 3 ) j .0012 . O O l O ' .0008 I ',- .

.0004 .0002 I ____I 1 I I I 1 20 40 x 106 2 4 6 8 1 0 R X (b) rY, = 2.M.

Figure 14.- Momentum thickness distribution on model 7 (parabolic arc). Ticks differentiate between the stagnation pressure levels.

a3 .0004 . 0 0 0 2 I I I I 1 1

L

O1 2 4 6 8 1 0 2 0 40 x 106 R X (a) M, = 1.61.

F l a t - p l a t e t h e o r y ( e ( 5 ) ) y o d e l - - S l e n d e r - c o n e 8 t h e o r y t h e o r y ( e q . 7;s (3)j ( 6 ) ) -Oo14[ . 0 0 1 2 e - X .0004 .0002 6 . I 1 I I I,

2 A - 8 10 20 40 x 106

R X (b) M, = 2.20.

Figure 15.- Momentum thickness distribution on model 8 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

'1 -

Model 0 2 u 5 .005-

-_

---

77--- - _

.002- - - < _ D -------a_-

-_ -F------ 0

v --------u _ _ _ _ _ _

- . O O l

I I I -u

(a) M, = 1.61.

.006, 1 . I I I I I

1 2 4 6 8 1 0 20 40 x l o 6

* X (b) f & , = 2.M.

Figure 16.- Displacement thickness distributions on models 2 and 5 (slender cones).

F l a t - p l a t e t h e o r y ( e q . ( 7 ) ) - -_ - - - - - - S l e n d e r - c o n e t h e o r y ( e q . (11)) .005 M o d e l 1 t h e o r y ( e q . ( 1 0 ) )

--

---- ---- - - - - ----- -

-Ool t

01 I I I 1 - I 1 I 1 2 4 6 8 1 0 20 40 x l o 6 R X (a1 M, = 1.61.

.006,

t

I I 01 I I I I I ~ . I..

40 x l o 6 1 2 4 6 8 1 0 20 R X

-

(b) b= 2.20.

Figure 17.- Displacement thickness distribution on model 1 (RM-10). Ticks differentiate between the stagnation pressure levels.

F l a t - p l a t e t h e o r y ( e q . ( 7 ) ) .005- - - - - - - - - S l e n d e r - c o n e t h e o r y ( e q . (11)

-- - -_ _ _ _

--__ - -- -___ - .OOl- I 1 - 1 1 1 1 2 (a) Mm= 1.61.

.006 ,-

R X (b) M, = 2.M.

Figure 18.- Displacement thickness distribution on model 3 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

- - - - _ ------ - _ _ _ _ _ _ . O O l - (a) M, = 1.61.

0 0 6 r 0 ' I I 1 I 1 1 - .~ I

1 2 4 6 8 1 0 20 40 x l o 6

R X (b) M, = 2.20.

Figure 19.- Displacement thickness distribution on model 4 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

.~ F l a t - p l a t e t h e o r y ( e q . ( 7 ) ) .005 . - - - - - - - - Slender-cone t h e o r y ( e q . ( 1 1 ) ) -- Model 6 t h e o r y ( e q . ( 1 0 ) )

! 1 I ILL._ - 1 - 0- I

2 4 6 a 1 0 20 40 x l o 6 R X (a) M, = 1.61.

! I 1- 1 I 12

2 4 6 8 1 0 20 40 x l o 6

R X (b) M, = 2.20.

Figure 20.- Displacement thickness distribution on model 6 (parabolic arc). Ticks differentiate between the stagnation pressure levels.

F l a t - p l a t e t h e o r y ( e q . ( 7 ) ) - - - _ _ - - - - - S l e n d e r - c o n e t h e o r y ( e q . (11)) -- M o d e l 7 t h e o r y ( e q . (10)) .oo+ X I I I I I I I 1 2 4 6 8 1 0 20 40 x lo6 R X (a) M, = 1.61.

-----____

-- -

.OOl-

I 1 I I l l 1 I

20 40 x 1 0 ' O1 2 4 6 8 1 0 r7 R X (b) M, = 2.20.

Figure 21.- Displacement thickness distribution on model 7 (parabolic arc). Ticks differentiate between the stagnation pressure levels.

.006 F l a t - p l a t e t h e o r y ( e q . ( 7 ) ) - - - - - - - - - S l e n d e r - c o n e t h e o r y ( e q . (11)) M o d e l 8 t h e o r y ( e q . (10)) .004

.Oo1 t

I I I I I I 1 ’

40 x lo6 4 6 8 1 0 20 1 2 R X la) M, = 1.61.

.006 .005 .004

- 6 * .003

X .oos .001 I I I I I I - I C

2 4 6 8 1 0 20 40 x l o 6

R X (b) M, = 2 . 2 0 .

Figure 22.- Displacement thickness distribution on model 8 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

.025r Model

0 1 0 2 A 3 0 4

.020 L ' , 0 5

F l a t - p l a t e t h e o r y ( e ( 1 2 ) ) -,Slender-cone t h e o r y 9 ; s . ( 1 4 ) ) . . I . I . I I . I I 4 G 8 1 0 20 40 x l o 6 R X (a) M , = 1.61.

I .. J I I I 1 I. I 20 40 x l o 6 4 6 8 1 0 O1 2 R X (b) f & , = 2.20.

Figure 23.- Boundary-layer thickness distribution.

3.7 3.5 3.3 3.1 2 . 9 2.7 2.5 2.3 Figure 24.- Variation of b”/e with Mb.

M o d e l 0 1 0 2 Figure 25.- Variation of 8,/8 with M6.

-__ F l a t p l a t e - - - - - - - M o d e l s 2 a n d 5 ( s l e n d e r c o n e s ) M o d e l 1 M o d e l 3 M o d e l s 4 a n d 8 - -. M o d e l s 6 a n d 7 (a) Ma= 1.61.

(b) M , = 2 . 2 0 : Figure 26.- Theoretical local skin-friction-coefficient ratios calculated by equation 118).

0 8 Flat-plate theory ( r e f 15) -004- --- _ _ - - - Slender-cone theory (eq. ( 2 0 ) )

-- \

.001 I- I

! ! _ I . - _ _ 1 - 1

6 8 10 20 40 x lo6 1 2 4

RX ( a ) M , = 1.61.

.004- (b) M, = 2.x).

Figure 27.- S u m m a r y of local skin-friction-coefficient data.

Model 0 2 0 5 F l a t - p l a t e t h e o r y ( r e f 15) - - _ _ - - - Slender-done t h e o r y (eq. (20)) "f .002

moo1 I I I I -I 1 '

1 2 4 6 8 10 20 40 x l o 6

Rx (a) M , = 1.61.

- .

I-.- . I I LA I d

*006-

--- Model 1 t h e o r y ( e q . (18))

- (J 7.5' c o n e t h e o r y ( e q . ( 1 9 ) ) - .004 ' \ -\-

--

---------. \---

cf - 0 0 2 -

! I ! ! I

. 0 0 1 (a) M , = 1 . 6 1 .

C f 0002- - . w V I

. 0 0 1 u- - I - - 1 - I 1 - - 1 I - I

1 2 4 6 8 1 0 20 40 x l o 6

R X (b) M, = 2.20.

Figure 29.- Local skin-friction-coefficient distribution on model 1 (RM-10). Ticks differentiate between the stagnation pressure levels.

Flat-plate theory ( r e f . 15) - ---- Slender-cone theory (eq. ( 2 0 ) )

=Oo6r

-___ Model 3 theory (eq. (18))

- 0 = 7 . 5 ' cone theory (eq. (19))

.004 c

c , I

- T

= 0.34 atm

'1 n i = O O 2 t Pt ' (a) M, = 1.61.

= O o 6 1 .004

I 1 I I 1- -I I . -_J

,001 40 x lo6 1 2 4 6 8 10 20 RX (b) M,= 2 . 2 0 .

Figure 3 0 . - Local skin-friction-coefficient distribution on model 3 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

F l a t - p l a t e t h e o r y ( r e f . 15) ----- S l e n d e r - c o n e t h e o r y ( e q . ( 2 0 ) ) e o o 6 r

---

M o d e l 4 t h e o r y (eq. (18)) - 7.5O cone t h e o r y ( e q . ( 1 9 ) )

.004 c

~- - 1 - - I I I 1 I

M O O 1 L

6 8 1 0 20 40 x l o 6

1 2 4 RX (a) M, = 1.61.

.004 = O O L 1 - I

1 2 4 6 8 1 0 20 40 x lo6

(b) M, = 2.20.

Figure 31.- Local skin-friction-coefficient distribution on model 4 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

!

Flat-plate t h e o r y ( r e f . 15)

-006 - - - _ _ Slender-cone t h e o r y (eq. ( 2 0 ) )

Model 6 t h e o r y (eq. (18)) - 0 7.5O cone t h e o r y (eq. ( 1 9 ) )

- 0 0 4 L---

+

--

---

1.70 ------L

(a) M , = 1.61.

.004

= 0.34 atm -1-

I I I . _ _ . I . . I 1 _ _ .. .- . - , I - -_I

,001-

1 2 4 6 8 10 20 40 x l o 6

R X (b) b= 2.20.

Figure 32.- Local skin-friction-coeff icient distribution on model 6. (parabolic arc). Ticks differentiate between the stagnation pressure levels.

F l a t - p l a t e theory ( r e f . 15)

- - _ _ _ Slender-cone theory (eq. ( 2 0 ) )

--- Model 7 theory (eq, (18))

- 0 = 7.5O cone theory (eq. (19))

,004

- --\

I I .. 1 _J

1 2 4 6 8 1 0 20 40 x lo6

RX (a) M, = 1.61.

. * O O L 0 0 4

c f .002

1.70

I

I - I 1 I I - 1 . -2

. O O l l

1 2 4 6 8 1 0 20 40 x l o 6

(b) M, = 2.20.

Figure 33.- Local skin-friction-coefficient distribution on model 7 (parabolic arc). Ticks differentiate between the stagnation pressure levels.

Flat-plate theory ( r e f . 15) - - _ _ - Slender-cone theory (eq. (20)) --- Model 8 theory (eq. (18))

,004 - (I = 7.5O cone theory (eq. (19))

1 I I I I I 2 ,001 2 4 1 6 8 10 20 40 x lo6 R X (a) M, = 1.61.

‘ O O T .004

1 - - I I L I I -1

.001- I

2 4 6 8 10 20 40 x lo6 R X (b) M, = 2 . 2 0 .

Figure 3 4 . - Local skin-friction-coefficient distribution on model 8 (cone-parabolic arc). Ticks differentiate between the stagnation pressure levels.

M o d e l 0 1 0 2 A 3 0 4 v 5 0 6 0 7 < ' 8 - F l a t - p l a t e t h e o r y ( r e f . 1 5 ) --_._.

Slender-cone t h e o r y (eq. (21) ) I I I ._ 1 I . L I I l l ! - I I !

2 4 6 8 10 20 40 x lo6 Rx (a) M , = 1.61.

1 2 4 6 8 10 20 40 x l o 6 Rx (b) M , = 2.20.

Figure 35.- Variation of average skin-friction coefficient with free-stream Reynolds number.

L i n e o f 5 p e r c e n t d i s a g

I

.0034-

e

co

!

N I u S I

.E .003OC

Model

)I o 1 (RM-10.)

II .0026-

2 (127 cm cone)

'u Q,

17 5 (63.6 cm cone)

i-' cd L

.0018k

,,y-Line o f ,100 p e r c e n t agreement

Figure 36.- Comparison of methods of calculating average skin-friction coefficient.

.0035 o Equation (B11) A o First t e r m in equation (26) A B o t h terms in equation (26) Q A .0030 A c F

i $ B

.0025

c

.0020 (a) M , = 1.61.

.0015'-

14 -

r - r e e - s t r e a m c o n d i t i o n s c

12 -

a

(0 CI

10 -

8- Y, cm M o d e l 3 2 . 2 0

p t = 1.70 a t m

I X

- 0.56

0 1

0 .01 .02 .03 .04 cP \ Figure 38.- Typical pressure-coefficient and inviscid Mach number profiles.

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