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wt i "' I T i;;i q 15.. " S ... .., :" ". "" ." a:.i / il i,: ,, :: ,:, ,, 1. M. LA ?5 25a RM A51l25a :i" t"h .,' ."". ."..: :* : { i;:' ":; :!.. " i i: i':' .. : " J, : ': ":': "t I:..r. :.'" 1 ' ili rC.i; ;.; R~ CH ' ?; .I" ,;: ,' " ,. ; aii i*. .,) < ;.. ., MEMORANDUM iJMlOARY OF AVAILABLE KNOWLEDGE CONCERNING SKIN FReITION AND HEAT TRANSFER AND ITS APPLICATION II, '.'TO THE DESIGN OF HIGHSPEED MISSILES I. By Morris W. Rubesin i i;i:.. Ames Aeronautical Laboratory ; :lf;:i'.: :: : Charles B. Rumsey ,.; 'Langley Aeronautical Laboratory i:~.~..a : and Steven A. Varga .~~' Ames Aeronautical Laboratory i ". i i :: = *.' UNIVERNOSfYS FLORIDA SCM;E'l:.ENTS DEPARTMENT : 20 MARSTON SCIENCE UBRARY 3P PO. BOX 117011 GAtt$VL FL 326117011 USA AL ADVISORY COMMI 1... 4, FOR AERONAUTICS S. i WASHINGTON ; ;November 9% 1951 i" C "", '.; ,.' % ;, ..] .!.,. ., ,= : :" , TTEE .1 t.i ... ... ...: ..".. .... :." ',".. ,::: !." :.' .. ,,4 .. ., ::. :;i::'~ ..' ..: .:,:!.".2 .:6i~ . . ".. :.. .; : ':. /.* :*:; I ~1 ___ ___ _~ ;r ~: . .... .. .. NACA RM A51J25a A SUMMARY OF AVAILABLE KNOWLEDGE CONCERNING SKIN FRICTION AND HEAT TRANSFER AND ITS APPLICATION TO THE DESIGN OF HIGHSPEED MISSILES1 By Morris W. Rubesin Ames Aeronautical Laboratory Charles B. Rumsey Langley Aeronautical Laboratory and Steven A. Varga Ames Aeronautical Laboratory To determine the skin friction and heat transfer on the surfaces of highspeed missiles, it is necessary to know certain characteristics of the boundary layers. These characteristics are: the temperature recovery, the skinfriction coefficients and the heattransfer coeffi cients of both the laminar and turbulent boundary layers and, also, the position of the transition from laminar to turbulent flow. In this paper a review is made of the existing information concerning these characteristics. In addition, comparison is made between existing flight data and results computed by the boundarylayer momentumintegral method in a preliminary attempt to establish some rational way of approaching the design of a missile whose Mach number range and body geometry are markedly different than those of existing data. The problem of determining the position of transition from a laminar to a turbulent boundary layer is very important. For flight conditions in which transition occurs in regions other than very near the nose of the missile, the average skin friction and heat transfer may be influenced more by the location of transition than by the absolute values of the skin friction or heat transfer corresponding to either the laminar or turbulent flow. At present there is no accurate method for determining the location of transition. A compilation of data showing the beginning and end of boundary layer transition is shown in figure 1. The ordinate is the Reynolds number based on the length along the body. The abscissa is the free stream Mach number. Open symbols designate the beginning of transition, whereas the filledin symbols designate the end of transition. Most of these data, compiled by project Hermes, were obtained in the early stages of V2 flight and are, in effect, for a cooled surface. 1This is substantially a reprint of the paper by the same authors which was presented at the NACA Conference on Aerodynamic Design Problems of Super sonic Guided Missiles at the Ames Aeronautical Laboratory on Oct. 23, 1951. NACA HM A51J25a Also shown are data obtained on unheated flat plates at Princeton (refer ence 1) and in the Ames 6inch heat transfer tunnel (reference 2), on an unheated RM10 test body in the Langley 4 by 4foot supersonic tunnel, and on an unheated body of revolution at the Lewis Laboratory. It can be seen that the scatter in the data is enormous; however, the general trend of the data indicates that the Reynolds numbers of the beginning and end of transition increase at the higher Mach numbers. That the data shown do not correlate any better is expected. In these data no control was made of such important quantities as surface roughness, body shape, freestream turbulence, and surface temperature. One of these variables, the surface temperature, was isolated for study in tests performed at the Ames Laboratory on a heated flat plate at M = 2.4 (reference 2). The results of these tests are shown in figure 2. The ordinate used is the Reynolds number based on the momentum thickness whereas the abscissa is the ratio of the surface temperature to the freestream temperature. This form of Reynolds number was chosen to localize conditions, thereby making the results applicable to bodies of revolution with surfacepressure variations. The points at the extreme left are for the unheated case. It can be seen that the Reynolds numbers of the beginning and end of transition are reduced by about 50 percent from the unheated condition for a surface to freestream temperature ratio of 2.8. It is interesting to note that a determination of the momentum thickness Reynolds number at the beginning of transition on an unheated body of revolution tested at the Lewis Laboratory resulted in a value identical to that shown for the unheated plate in this figure. This agreement may have been fortuitous because the Mach number of the body was 3.12 whereas that for the plate was 2.4. Obviously, much more work needs to be done concerning transition before an accurate means is available for predicting its position on a missile. Because there is no alternative, it is recommended that until more information is available the results of figure 1 be used as a guide in design. Before it is possible to determine the heat transfer, and often the skin friction, it is necessary to know the recovery temperature. The recovery temperature can be determined from the usual equation for recovery factor shown in figure 3. In the equation at the left r is the recovery factor, Tr is the recovery temperature of an unheated body, To is the freestream temperature, and Mo is the freestream Mach number. This figure shows a compilation of temperature recovery factor as a function of Mach number obtained in wind tunnels at Mach numbers below 4, and for two flight tests at Mach numbers approximately equal to 2. These data apply to flat plates and bodies of revolution. The length of the vertical bars, which represent windtunnel data, shows the range of variation of the recovery factor with Reynolds number at a NACA EM A51J25a fixed Mach number. In general, the data points lie on two levels, around r = 0.85 and r = 0.90. These levels agree with the usual theoretical values of Pr1/2 for laminar flow and Prl/3 for turbulent flow. The set of data for the laminar boundary layer on a flat plate is not in agreement with the other laminarboundarylayer data, or the theoretical prediction. In addition, two theoretical results showing the effect of Mach number on the recovery factor are also indicated. The turbulent boundarylayer recovery factor determined by Tucker and Maslen (refer ence 3) by extending the approximate Squire analysis to include compres sibility shows a reduction with Mach number. Apparently the variation of the recoveryfactor data does not exhibit this change. It can be concluded, therefore, that the usual theoretical value Pr1/3 be used for the recovery factor in design work through the Mach number range, neglecting the theoretical variation indicated. For the laminar boundary layer, Klunker and McLean (reference 4) have shown that, under flight conditions where extremely high air temperatures occur, the recovery factor decreases with Mach number. These results were obtained from the same basic boundarylayer theory which yields a recovery factor of Prl/2 for the temperature levels occurring in wind tunnels. Thus, the agreement of windtunnel data with Pr1/2 checks the basic theory. The flight datum point shown is at too low a Mach number to indicate any marked reduction. Since the experimental data agree with the basic theory, the work of Klunker and McLean for flight conditions should yield satis factory results for design purposes at high Mach numbers. Several theories exist for determining the magnitude of the skin friction and heattransfer coefficients. For the case of flight condi tions where extremely high temperatures occur, the previously mentioned theory of Klunker and McLean also provides a means to calculate the laminarboundarylayer skinfriction and heattransfer coefficients. In addition, Van Driest (reference 5) has obtained similar results by extending the work of Crocco to include flight conditions with the resulting high air temperatures. Although the Crocco method is restricted in that the Prandtl number is assumed constant and the viscosity is expressed in Sutherland's equation in terms of enthalpy rather than temperature, figure 4 indicates that the results of average skin friction for the laminar boundary layer at Mach numbers below 10 are within 1 per cent of the more exact method of Klunker and McLean. As good agreement is also obtained for the recovery temperature and the local heat transfer, it can be concluded that for practical purposes the two theories give equal results. NACA RM A51J25a The data with which these theories can be compared are relatively meager. Published skinfriction data on unheated flat plates (refer ences 2, 6, and 7) represent the average skin friction from the leading edge to the point of measurement of boundarylayer surveys. These average skinfriction coefficients obtained at Mach numbers around 2 are about 30 percent higher than those given by Crocco's theory made to apply to windtunnel conditions. Similar results were obtained at Lewis from unpublished data on a hollow cylinder placed parallel to the air stream. This discrepancy between theory and experiment has been attri buted to the momentum loss in the boundary layer caused by the bluntness of the sharp leading edge. Unpublished data of average skin friction obtained at the Langley Laboratory on a 60 wedge in a flow at a Mach number of 6.9 exceeds by about 14 percent the estimated theoretical value based on the Crocco method when the wedge is at a zero angle of attack. Further unpublished tests at the Lewis Laboratory have indicated that the laminarboundarylayer theories compare favorably with the experimental average skinfriction coefficients determined experimentally on a conecylinder body at M = 3.85. Although no local skinfriction data have as yet been correlated with the theory, local heattransfer data shown in figure 5 have been determined on a cone having approximately a constant surface temperature (reference 8). The data are, on the average, about 12 percent lower than those given by the Crocco theory, corrected to a cone. In general, it can be concluded that the Crocco theory predicted the skin friction and heat transfer within engineering accuracy up to a Mach number of 7, for the windtunnel tests. It then would be expected that the theories for the laminar boundary layer for flight conditions are adequate for design. For the turbulent boundary layer there are several theories from which the skin friction and the heat transfer on flat plates can be calculated (reference 9). Each of these theories indicates a marked reduction in the skinfriction and heattransfer coefficients with an increase in Mach number or surface temperature. Because of the large effects indicated by the theories and because they are of a semiempirical nature it is important to compare them with existing data. This com parison is made in figure 6 for the case of an unheated flat plate in a wind tunnel at a Mach number of 2.4 (reference 9). It is observed that the average skinfriction coefficient is reduced from the values of the incompressible case; however, the reduction estimated by Von Karman was not realized. The compressible theories for turbulent flow on a flat plate give good agreement with the data over the range of Reynolds numbers below 6,000,000. In figure 7 are shown unpublished local skinfriction data obtained on unheated cylinders with their axes placed parallel to the air flow. The Mach number of these tests was 3.1. The abscissa used in this figure is the Reynolds number based on the momentum thickness. This characteristic dimension was used to avoid the necessity of knowing the exact location of transition. In the lower figure there are shown NACA RM A51J25a the data with natural transition. These data agree approximately with the compressible flatplate theories at the lower Reynolds numbers. Beyond a Reynolds number of 6000 the data drop off toward the Von Karmxn estimation. The data with artificial transition shown in the upper figure exhibit a reduction from the incompressible case; however, the data have a different slope than any of the theories and give no insight into which of the theories agree best with the physical phenomena. Figure 8 is intended to show that a modified Reynolds analogy exists at a Mach number of 2.4. This unpublished datum point was obtained on a cooled flat plate in the Ames 6inch heat transfer tunnel. The ordinate is written in a fashion which permits comparing heattransfer data with theoretical skinfriction computations through a modified Reynolds analogy. The abscissa used in this figure is the Reynolds number based on the momentum thickness to avoid the necessity of knowing the location of transition. The single datum point of heat transfer compares favor ably with the theories of Frankl and Voishel and of Van Driest. In general, it can be concluded from the last three figures of windtunnel data that the compressibleturbulentboundarylayer theories represent the available data of skin friction and heat transfer on flat plates with an accuracy sufficient for design. The same cannot be said from the data obtained on cylinders with their axes parallel to the air stream, except for the data obtained with natural transition at Reynolds numbers below 6000 when based on momentum thickness which did agree fairly well with the theories. Data of skin friction and heat transfer have been measured in flight on the RM10 missile, the earliest of which are included in references 10 and 11. Figure 9 shows time histories of the flight characteristics for a typical boosted PJ~l0 flight during which average skinfriction coef ficients were obtained from boundarylayer rake measurements to a maximum Mach number of 3.7. The characteristics shown are a surface temperature parameter, the Reynolds number based on the length to the rake location just ahead of the fins, the Mach number, and the average skinfriction coefficient. The surface temperature parameter shown was used since its numerical value indicates the magnitude of the heating regardless of Mach number and indicates cooling and heating of the boundary layer by negative and positive values, respectively. The experimental skinfriction coefficients are from 20 to 30 percent higher than Van Driest's theoretical prediction for a flat plate at the test conditions, except near peak Mach number. During the first part of the test which is after booster separation but prior to firing of the sustainer rocket, transition would be well forward on the pointed nose of the missile so that close to 100 percent of the skin area would have turbulent boundary layer and the measured values would be NACA RM A51J25a average turbulent coefficients. It is expected, however, that at the high Mach numbers during sustainer firing, the strong cooling of the boundary layer indicated by the surface temperature parameter would stabilize the laminar boundary layer and cause transition to move back on the body. The measurements during this part of the flight would thus be lower than average turbulent coefficients. During the period after sustainer firing, the heating parameter became less stabilizing, and transition would be expected to move forward causing a relative rise in the average coefficient. These trends are shown by the data. At a time of about 23 seconds, the heating parameter became positive, or destabilizing, and nearly all of the skin area would again be covered by turbulent flow. The 20 to 30 percent difference shown between turbulent flatplate theory and the data for times of nearly complete turbulent boundary layer is attributed to the missile geometry and to the pressure distribution at the flight Mach numbers. Figure 10 shows unpublished flight conditions and results from a cylindrical body with an ogive nose. This configuration more closely approximates a flat plate. The measured values of average skinfriction coefficient are relatively lower than the RM10 results and are in close agreement with Van Driest's flatplate theory. The extent of laminar flow on this model is believed to have been small because of the values of Mach number and Reynolds number, at least during the first half of the test. Presented in figure 11 are values of average skinfriction coef ficient at the condition of zero heat transfer which have been obtained at four points in the skinfriction tests, all occurring at a Reynolds number of about 60 x 10 but at different Mach numbers from 1.1 to 3. Also shown is a value at zero Mach number and 60 x 106 Reynolds number which was recently obtained from rake measurements in underwater tests performed on an RM10 body in the Langley tank no. 1. A flatplate theory is also included to show its variation with Mach number. Below Mach number 1 no reduction is shown by the data. From Mach number 1 to 3, the data show a reduction of about 30 percent whereas the flat plate theory shows a reduction of 35 percent. Local heattransfer coefficients measured in flight on the RM10 missile are shown in figure 12. The data are plotted as NuPr/3 against Reynolds number with the velocity and air properties based on freestream conditions. Above a Reynolds number of about 6 x 106 the heattransfer coefficients are for turbulent flow. Below approximately 2 x 106, the coefficients measured on the nose of model C show a decrease from the turbulent correlation indicative of laminar flow. The values are, however, considerably higher than the laminar theory for a cone. Plotted in the present manner the data lie midway between the laminarboundarylayer theory for a cone and the measured turbulent data. NACA RM A51J25a The heattransfer data for turbulent boundary layer can be repre sented to t7 percent by a line having the equation indicated. These data were obtained over a Mach number range from 1 to 2.8 and at several stations along the body as indicated in the legend. It is interesting to note that, for three models, almost all of the data for all stations along the length of the body and over the complete Mach number range agree to within 7 percent. It is concluded from the flighttest data that for missiles not greatly different in shape from the RM10, and for conditions similar to the test range, heattransfer characteristics for turbulent flow can be obtained from the RM10 equation for design purposes. It should be emphasized that the heattransfer data do not explicitly show a Mach number effect in the range of Mach numbers tested. The test data further indicate that the skinfriction coefficients can be obtained by reference to the flatplate theory in the following manner. For ogive cylinder bodies practically no modification to the theory is necessary. For bodies of higher fineness ratio than the RM10 it would be expected that the values of skin friction are between those of the RM10 and the flat plate. In view of the conclusions drawn from the flighttest data, it is apparent that some rational method is necessary for extrapolating the known data to blunter bodies or to bodies flying at flight conditions much different than those of the available tests. As the flatplate theories including compressibility agreed well with the skinfriction data obtained with the ogive cylinder, it was believed that some method accounting for body shape might bring the theories in line with the data obtained on the RMlO, thereby extending the scope of the data. There fore, computations were made of heat transfer and skin friction for the RM10 shape and flight conditions by means of the wellknown momentum integral method using the Frankl and Voishel flatplate theory. The momentumintegral method consisted of solving the equation shown in figure 13. This equation relates the rate of growth of the boundary layer with the compressibility effect, the acceleration of the air outside the boundary layer, the geometry of the body, and the local skinfriction coefficient. The solution of this equation is obtained through the use of the flatplate relationships of the skinfriction coefficient and the Reynolds number based on the momentum thickness. The solution yields the distribution of the momentum thickness, from which the skinfriction coefficient can be determined. The local heat transfer coefficient is obtained from the local skinfriction coefficient through a modified Reynolds analogy. In figure 14 there is shown a comparison of some preliminary results of the momentumintegral method with the local heattransfer coefficients NACA RM A51J25a measured on the RM10. For the theoretical computations the local skin friction coefficient was expressed in terms of the Reynolds numbers based on momentum thickness according to the flatplate theory of Franki and Voishel. The ordinate shown is the local heattransfer coefficient. The abscissa is the dimensionless length along the body. Two sets of data are shown; the upper set is for a Mach number of 2.3, whereas the lower is for a Mach number of 1.02. It should be noted that the Reynolds numbers of these data are roughly in proportion to the Mach numbers. The solid lines represent the distributions given by the equation representing the bulk of the RM10 data. The dashed line represents the results obtained from the momentumintegral method. At the lower Mach number, and consequently the lower Reynolds number, the momentumintegral method agrees well with the data and the RM10 equation. The results of the Van Driest flatplate theory for these conditions were about 10 percent lower than the data along the entire body. At the higher Mach number the momentumintegral method gave results which are about 15 percent higher than the data on the front of the missile and about 3 percent higher than the data towards the rear of the missile. The data apparently do not show the geometry effect expected from the momentumintegral method. In fact, the Van Driest flatplate theory gives results which pass through the data near the front of the missile and then drop to values about 3 percent low in the rear portions of the missile. From the latter results it can be concluded that, for slender bodies such as the RM10, the RM10 equation or flatplate theory represents the data as well as does the more tedious momentumintegral method at a Mach number of 2.3. The momentumintegral method may become necessary for blunter bodies. The momentumintegral method is evaluated further in figure 15. The ordinate shown is the average skinfriction coefficient and the analogous heattransfer parameter. The abscissa is the Mach number. The average skinfriction data shown are for the RM10 at recovery temperature. These data were shown previously in figure 11. The average heattransfer parameter shown was evaluated from the RM10 equation and it is noted there is no Mach number effect in this equation representing the bulk of the RM10 data. For comparison, two flatplate theories corresponding to the recovery temperature are included. These theories are for condi tions comparable with the curve of the RM10 heattransfer parameter because the same value of the parameter was obtained under conditions of both cooling and heating. It is noted that the momentumintegral method does not reconcile the flatplate theories with the characteristics of the RM10 data in that first, the momentumintegral method does not sufficiently increase the values obtained using flatplate theory to agree with the RM10 skin friction data, and second, the Mach number effect remains in the theories even when based on the momentumintegral method thereby resulting NACA RM A51J25a in a lack of agreement with the RM10 heattransfer equation. It is apparent that no conclusive method for extrapolating existing data to greatly different conditions can be given at present. REFERENCES 1. Ladenburg, R., and Bershader, D.: Interferometric Studies on Laminar and Turbulent Boundary Layers along a Plane Surface at Supersonic Velocities. Symposium on Experimental Compressible Flow. U. S. Naval Ordnance Lab. (White Oak, Md.), June 29, 1949, pp. 6786. 2. Higgins, Robert W., and Pappas, Constantine C.: An Experimental Inves tigation of the Effect of Surface Heating on BoundaryLayer Tran sition on a Flat Plate in Supersonic Flow. NACA TN 2351, 1951. 3. Tucker, Maurice, and Maslen, Stephen H.: Turbulent BoundaryLayer Temperature Recovery Factors in TwoDimensional Supersonic Flow. NACA TN 2296, 1951. 4. Klunker, E. B., and McLean, F. Edward: Laminar Friction and Heat Transfer at Mach Numbers from 1 to 10. NACA TN 2499, 1951. 5. Van Driest, E. R.: Investigation of the Laminar Boundary Layer in Compressible Fluids Using the Crocco Method. Rep. AL1183, North American Aviation, Inc., Jan. 9, 1951. 6. Blue, Robert E.: Interferometer Corrections and Measurements of Laminar Boundary Layers in Supersonic Stream. NACA TN 2110, 1950. 7. Low, G. M., and Blue, R. E.: Measurements to Evaluate Leading Edge Effects on Laminar Boundary Layer on Flat Plate in Supersonic Stream. (Prospective NACA paper) 8. Scherrer, Richard, Wimbrow, William R., and Gowen, Forrest E.: Heat Transfer and BoundaryLayer Transition on a Heated 200 Cone at a Mach Number of 1.53. NACA RM A8L28, 1949. 9. Rubesin, Morris W., Maydew, Randall C., and Varga, Steven A.: An Analytical and Experimental Investigation of the Skin Friction of the Turbulent Boundary Layer on a Flat Plate at Supersonic Speeds. NACA TN 2305, 1951. 10. Rumsey, Charles B., and Loposer, J. Dan: Average SkinFriction Coef ficients from BoundaryLayer Measurements in Flight on a Parabolic Body of Revolution (NACA RM10) at Supersonic Speeds and at Large Reynolds Numbers. NACA RM L51B12, 1951. 11. Chauvin, Leo T., and deMoraes, Carlos A.: Correlation of Supersonic Convective HeatTransfer Coefficients from Measurements of the Skin Temperature of a Parabolic Body of Revolution (NACA RM10). NACA RM L51A18, 1951. NACA RM A51J25a TREND OF END OF TRANSITION TREND OF BEGINNING OF TRANSITION TRANSITION BEGINS ENDS o a HERMES (V2) A4 PRINCETON U. (FLAT PLATE) h L AMES (") a LEWIS (BODY OF REV.) 0 LANGLEY(RMIO) I I I I I I 2 3 4 5 MACH NUMBER, M. Figure 1. Location of boundarylayer transition on different flight and in wind tunnels. bodies in FLAT PLATE Me = 2.4  END OF TRANSITION HEATING START OF TRANSITION 22 24 2.6 2.8 3.0 Figure 2. Effect of heating on boundarylayer transition. 5x106  XUwPc Rx . 2x106  5x1O5 O 2800 r EATING 2400 2000 1200 800 400 2.C I I I f I :) NACA RM A51J25a RECOVERY FACTOR .90  .88 'lI TII(I(rFR MBA~IIP F3 (TURBULENT) 2 S = /PR i TURBULENT LAMINAR == FLIGHT O S WIND TUNNEL ' KLUNKER 8 McLEAN '.., LAMINARR) N.> MACH NUMBER, M, Figure 3. Compilation of temperature recovery factors and comparison with theories. PREDICTED SKIN FRICTION IN FLIGHT T. a KLUNKER McLEAN /ISULAED I iKcn a McLEA 1 IN, VAN DRIEST INSULATED1 (CROCCO'S METHOD) NO DISSOCIATION MA 2 3 4 5 6 MACH NUMBER, M, 7 8 9 Figure 4. Comparison of the theories of Van Driest (Crocco) and of Klunker and McLean for flight conditions. 11 a. I 1. 1.4 1.3 1.2 1.1 1.0 .9 r\LUn S I I I ,, NACA RM A51J25a HEATTRANSFER PARAMETER Nux Rx!. CROCCO THEORY (WITH CONE CORRECTION) CONE DATA I I I I J .2 .4 .6 .8 1.( X DIMENSIONLESS LENGTH, L L Figure 5. Comparison of Scherrer's cone data with Crocco's theory. UNHEATED PLATE M,=2.4 INCOMPRESSIBLE VAN DREIST '""'7 .. VON KARMAN'. 3 4 5 6 7 8 9 10I Figure 6. Comparison with theories of experimental average skinfriction coefficients on a flat plate. .006 .005 CF .004 .003 .002 nnmrI FRANKLY AND VOISHEL 1.5x 10 2 a I I I I I NACA RM A51J25a .0015r .001 Cf 000 2 .0005 .00151 DATA OF BRINICH AND DIACONIS ARTIFICIAL TRANSITION  a V IA I reT FRANKL AND VOISHEL WILSON VON I I , I I I I I I I I I NATURAL TRANSITION Figure 7. Comparison with theories of experimental average skinfriction coefficients on a cylinder in axial flow. M, 2.42 = 2.00 (COOLING) T. Cf =2Nu Rx PrI FRANKL .002 FLAT PLATE EXPERIMENTAL POINT .001J 0013 R 104 Figure 8. Comparison with theories of experimental local heattransfer coefficient on a flat plate. NACA RM A51J25a %5 TwTr 0 TrTs 5 1.0 150x0 6 RX BASED 01 100 RAKE STATIC 50 0 1I 3.5 3.0 M. 2.5 2.0 1.51 1.0 S EXPERIMENTAL .002  VAN DRIEST CF .001 FLAT PLATE THEORY' 0  1 0 4 8 12 16 20 24 28 ~ TIME, SEC Figure 9. Flight data from the RM10 missile and mental average skinfriction coefficients comparison of experi with theory. TIME HISTORIES OF OGIVECYLINDER BODY TEST Tw T OT f T,T 0 Tr T8 1.0 150 x 106 R 100 50 01 2.5 2.0 Me 1.5 1.0 .002 CF .001 0 0 Rx BASED ON LENGTH TO RAKE STATION 10.42 FT I I I I _ __ EXPERIMENT AN DRIEST FLAT PLATE THEORY I l I I I I 2 3 4 5 6 TIME, SEC Figure 10. Flight data from ogivecylinder body and comparison of average skinfriction coefficients with theory. N LENGTH TO N. 10.42 FT 1 I z NACA RM A51J25a R~ 60x106 RMIO RESULTS VAN DRIEST FLATPLATE THEORY VAN DRIEST FLATPLATE THEORY Figure 11. Comparison with theory of experimental average skinfriction coefficients on the RM10 in flight. 106 MODEL STATION < M<2.8 A, B, C, + o 8.9 + a 9.2 O 14.3 1 . X 17.7 NuPr 3.0296 R 17.8 NuPr 3=.571R2 o 18.3 104 o 36.2 CROCCO'S LAMINAR D 49.9 THEORY CORRECTED o 85.3 FOR A CONE A 86.1 + 87.3 Nu Pr 3 123.5 / y REYNOLDS NUMBER 10 Figure 12. Heattransfer results from RM10 flight tests. 003 002 CF .001 F NACA RM A51J25a d8+ H+2M2 dMI dr Cf dx IMI (I+ Mij) dx r dx 2 Figure 13. Integralmomentum equation for bodies of revolution in compressible flow. FLIGHT DATA o M = 2.3 o M,= 1.02 .04 F h .03 BTU FT2 SEC "F .02 RMIO0 EQUATION o Z MOMENTUM INTEGRAL BASED /m ON FRANKL a VOISHEL THEORY EL E U L  0 .1 .2 .3 .4 .5 .6 .7 .8 XL NAC Figure 14. Comparison of experimental and theoretical heattransfer coefficients on RM10. NACA RM A51J25a FRANKLVOISHEL AVERAGE HEATTRANSFER (FLAT PLATE) PARAMETER EVALUATED FROM RMIO EQUATION .002  CF VAN DRIEST S(FLAT PLATE) or 2Nuav Rx Pr3 MOMENTUMINTEGRAL METHOD .001 USING FRANKLVOISHEL THEORY oAVERAGE SKIN FRICTION ON RMIO IN FLIGHT 0 I 0 I 2 3 4 M, Figure 15. Comparison with theories of average skinfriction coeffi cients on RM10 in flight. NACALangley 13155 150 to TO,5 4 0 S ar4C6 I~~ 3 i5 6r.S e g g" ^litr 7r i4~ir 0C. a 4)0. x >M 4 ;5wI> v 7 P 896=05wwo Mil ll. 0 43) :b iiMOOMN 02 m 5d iiSis lil^Bp l l a m t ,S LO F. 0~ 74 119 1 I 11E! 1 Illdl 50 0: tol iE;i 9:o? ECoo 2 M 0 S 0 v a 0 ..O` 0o c a 6', 0j W O Ell 0 'u] ) bg) E09 IOU tl 0 o 40M Q g? >s ~ e sa I r. i g Q ? 0 C) Oc 0 0) 0 ew1U 1 ES 1Fo CO CC a 'a 0 "i 1 o E) Cl o I T 3 wo CC E, 0 to L tka bll 1 0 cd 0. Z kl0l4~Ui~ I1 Si MQ C ^1 S^ k ze o wE <0 LO k.' < 0 W m ~ D ~ ~,. 00 Ual 14J 0 v. 2 6 d P4 :8 .^6 Sa: 5 g j1" *<; / e ^'SS a< V2 mcm Bn ioa g i i U g 3 S . Z0 0 0: E0 E i. .. 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