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R.M T..5iW2H :: .: :h .. K i: i;: y.. ,, .:" .. . I.: ; '. '. ,: *" .. . SEARCH MEMORANDUM S r TRANSFER AND SKIN FRICTION FOR TURBULENT i!i .. . BOUNDARY LAYERS ON HEATED OR COOLED SURFACES AT HIGH SPEEDS . j : ." By Coleman duP. Donaldson :. Langley Field, Va. UNIVERSITY OF FLORIDA .. DOCUMENTS DEPARTMENT ::120 MARS.TON SCIENCE BRARY P INESVILLE, F 326117011 USA piON AL ADVISORY COMMITTEE OR AE RONAUTICS S WASHINGTON "' .October 2, 1952 'N: ...... *y 1&,4.i UiOi :K ,.! iq .;%g A '. F ."; ... . wd 'iii.L 'z J '. ?, % : ',T :" ', d"", :. . rli; [,. .. '., ', .',. . N k AIL Aibt ar "" .H i' ii '.. : ... .:. : .. :i.,. , : ' .. .: / S: . ii "!J ' I I 3 :,: if 1. *:i .. ' :* i: 1. .. : :j: ,'; : :' . .:: i :i...:.,::.:.i::, : .W::.^ ^ NACA RM L52B04 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM HEAT TRANSFER AND SKIN FRICTION FOR TURBULENT BOUNDARY LAYERS ON HEATED OR COOLED SURFACES AT HIGH SPEEDS By Coleman duP. Donaldson SUMMARY The method presented in NACA TN 2692 for evaluating the skin fric tion of a turbulent boundary layer in compressible flow on an insulated surface is extended to evaluate the turbulent skin friction and heat transfer in compressible flow on a surface which is heated or cooled. The results of this analysis are in good agreement with the heat transfers measured in flight on the NACA RM10 missile up to Mach num bers of 3.7. INTRODUCTION Within the last decade the tremendous increase in flight Mach num bers that has been achieved has placed the problem of aerodynamic heating among the most important of the present time. In particular the problem of the heat transfer through turbulent boundary layers at high speeds has received considerable attention. It has long been known that the heat transfer through turbulent boundary layers is so intimately connected with the skinfriction force exerted by the boundary layer that the two problems are almost one. The analysis of the turbulent boundary layer presented in NACA TN 2692 (ref. 1) permits an easy extension of the turbulent skinfriction law from incompressible to compressible flow. The analysis led to expres sions for the extent of the effective laminar sublayer 6L and for the velocity uL at the point 5L. These results permitted the skinfriction law to be derived. If use is made of the quadratic dependence of temper ature on velocity derived by Crocco in reference 2, the results of ref erence 1 may be extended to obtain the heat transfer through the turbulent 2 NACA RM L52H04 boundary layer in both incompressible and compressible flows. It is the purpose of the present short analysis to make this extension. SYMBOLS parameter, 1 [n(r l) 0 1 k2 uo6 2r, skinfriction coefficient, 2Tw pouo r i .f.(7 1)M2 uL Tw Tadw F = 1 + + + S 2 u\ o To Sr.f.( 1)M L Tw T L0.56 U 0.448 (i_) K constant (0.045) k,m,n,r constants M Mach number qx Nx Nusselt's number,  K AT N5 Nusselt's number, qb K LT q local rate of heat transfer r.f. recovery factor Rx Reynolds number, u Vo RE Reynolds number, u Vo T absolute temperature NACA RM L52HO4 u velocity in xdirection x distance along surface y distance normal to surface 7 ratio of specific heats b boundarylayer thickness e boundarylayer momentum thickness K thermal conductivity p viscosity v kinematic viscosity 0 density T shear stress Subscripts: av average value adw adiabatic wall conditions L conditions at edge of laminar sublayer o freestream conditions w wall conditions DERIVATION OF THE HEATTRANSFER LAW FOR TURBULENT BOUNDARY LAYERS Before the present analysis is presented, a summary of the findings of reference 1 is in order. It was found that for boundary layers of the type o n U( () NACA RM L52H04 the dimensionless effective extent of the laminar sublayer given by Sn L n(r 1) L n+l & k2 ubt and the velocity at the point 5L was given by LL [ln(r 1) Ln"1 uD k2 uo, 1 With these results, it was reasoned that the skin friction was = "lL 5L which, when the necessary substitutions had been made, resulted in following skinfriction formula ln 2 2 Cf r(r n+1 o n1 L ( n1l k2 O ) Sie Since PL TL 0o TL and if it is assumed that was the (5) (6) PL TL 0 (To)L m Tw w w NACA RM L52H04 then 1n cf = 2jn(r 1) n+l k2 ] 2 n2m1 'n+ T n+l RE 6 T For a velocity profile having n = 7, it was found in reference 1 (from skinfriction measurements in incompressible flows) that n(r 1) = 158 k2 so that the local skinfriction coefficient for the compressible case became cf = 0.045 R5 1/4 /(T)0.56 L With these results it is now possible to pass on to the derivation of the heat transfer through the turbulent boundary layer. If a temperature distribution in the boundary layer similar to that used by Crocco (ref. 2) for Prandtl number = 1 and zero pressure gradient is adopted; namely: T =A + B + C 2 the temperature in the bodary layer may then be written the temperature in the boundary layer may then be written T = (Tw Tad) (Tadw To) 2 (10) It might be noted that, although the quadratic form of the tempera ture dependence on velocity was derived under conditions of zero pressure gradient, this form appears also to be a very useful relationship for most supersonicmissile shapes where the pressure gradients are generally quite small. NACA RM L52HOh Differentiating equation (10) and evaluating it at the wall where u = 0 gives uo Tw TUo Ld (11) Hence, the heat transfer at the wall is q= Y/= w(Tw Tadw (12) Ww Uo w From equation (4) it will be seen that S= L (13) so that q, L(T Tadw)L (14) P w6L uo If the values of 6L and UL that are obtained from equations (2) uo and (3) are substituted into equation (14) and the heat transfer is expressed in terms of a Nusselt number based on the dimension 6, the result is ln n1 N q (r 1) n+lUo l n+ w Ng ; 1)] (15) T5 2 J/ o w If it is assumed that K  (i6) K0 4o NACA RM L52HO0 equation (15) may be written 1n n(r 1)n+ k2 Substituting the value n1 of (v)n+l (TL0) and IL Io obtained from the use of equations (6) and (7) results in 1n S[n(r 1 n+l N [ k2 n1 n2m1 n+l n+1 R5 iT  (18) Writing the equations for dimensionless skin friction and heat transfer together where 1n and expressing 2 nr n+ L k2 J 2 cfF = Constant X Rg n+l n1 N8F = Constant R n+l 2 n2m1 F T n+l F T Vo/ as a constant yields (19) (20) (21) Since the righthand sides of equations (19) and (20) are the incompressible expressions for cf and N5, respectively, the effect of Mach number and temperature potential must be contained in the nl n1 5 n+ O n+1 R6 n+ I (17) NACA RM L52H04 factor F. This factor may be evaluated in the following manner. equation (10) From TL = Tw_ (Tw Tadw) To To To (Tad U Lo 2 To V which may be written L 1+ r.f. (_ 1)M2 1 L2 1+ (y 1wl ii To 2 u ou The factor F therefore becomes F = 1 + r. (7 1)M2 (u) 2 \uo) T, Tadw 1 To w o j Tw Tadw 1 10 It may be seen that the second term within the brace represents the effect of Mach number, and the third term the effect of temperature potential on the skinfriction and heattransfer coefficients. The value of is usually of the magnitude 0.4 and although it depends uo Tw Tadw upon M, Rb, and it generally does not have a firstorder To effect upon the factor F. The value of LL, however, must be found uo in the following way: Since 1 n+1 uL (r 1, o k2 u6(3) (22) (23) n2m1  n+ uL Ou (24) NAC ~ 1) h2 1 1 n+l Vn+1 R96 vo 1 where the value of V 7O 1 has been expressed in terms of temperatures by means of equations (6) and (7) and where A n(r 1) 1 k2 R5 Now, if the value of TL given by equation (23) is substituted in To equation (25) there results 1 UL = A n+1 + r M2  uL This equation for may be expressed as uo n+1 /u \m+l I l m+1 1/ A 1 1 Am+l r.f.(y 1) 2 2 +Tw Tadw To 1 m+l Tw Tadw +A To M2UL 2 M  \ uo (26) Sm+ u\ n+l (27) UL uo + r.f.(2 1) M2 + w Ta = 0 2 To / UL and solved graphically for for given values of Rg, (28) M, and Tw Tadw To 1 n+1 =A m+1 (TLn+1 LTO! (25) A RM L52104 NACA RM L52H04 For the particular case of a turbulent boundary layer with a one seventh power velocity profile in air when = 1.4 and m may be taken as 0.76 (an approximation which is usually adequate for calcula tions such as these), the equations necessary for the calculation of heat transfer become NF = 0.02 0.75 NKF = 0.0225R& T Tadw To 1 0.56 UL  o L"or (29) (30) uL and is found from the following equation Uo ( 4.55 18 0.568 w, / (R r r.f.M2 2UL 5 VUQI 0.568 Ty Tadw L (& r T \uT  158\0.568 RS r.f.M2 + Tw Tadw o0 5 To uL Tables I and II give values of UL uo S\2 and (z} \U found by using equation (31) for Mach numbers up to 5 for a range of values of Rg T T ri and Tw Tadw that are useful in heattransfer calculations. To COMPARISON WITH EXPERIMENT In order to compare the results of this analysis with experiment, it is necessary to have measurements of local rates of heat transfer for conditions of turbulent flow when the local values of Rg, M, and where (31) r.f.M2 (uL F= 1 +1 U NACA RM L52HOh 11 T, Tadw are known. This type of information for a range of Mach To numbers, Reynolds number, and temperature potentials can be obtained from the heattransfer and skinfriction measurements made on the NACA RM10 missile in flight and reported in part in references 3 and 4. For two of the body stations at which local heat transfers were measured in reference 3 (stations 85 and 122 inches from the missile nose) the bound ary layer had been surveyed under similar conditions for the skinfriction study reported in reference 4, so that the boundarylayer thickness 5 and hence RE were known. The pertinent measured quantities at these stations for several flight conditions are given in table III. The first six points are taken from data published in references 3 and 4. The last four points are computed from heattransfer data not yet published. The correlation of these data by the present method is shown in figure 1 wherein the measured and theoretical results are plotted in the form NBF versus RS. A reasonably good correlation of the data is obtained for Tw Tadw Mach numbers ranging from 1.6 to 3.7 and temperature potentials T ranging from 0.15 to 1.8. In comparing these experiments with the theory, it was assumed that the boundarylayer profile had a oneseventh power profile even though the surveys show that the power of the boundary layer profile varied somewhat from test to test. This is not considered to be a serious matter in making a comparison between the theory and experiment, as experience has shown that small variations of profile power from the value of seven do not materially affect the magnitude of the heat transfers involved. In general, heattransfer data are not plotted as has been done in figure 1, in terms of local correlations, but in terms of Nx and Rx. This usual practice is generally permissible in the case of a flat plate having no pressure gradient, but it may be of interest to see how the results of the present analysis appear when integrated along such a flat plate so as to be presented in more conventional form. The integration necessary (carried out in detail in the appendix) result in the following relations for incompressible flow with n = 7 1/5 Cf = 0.0578Rx Nx = 0.0289Rx Thus the normal minus onefifth and fourfifth power variations of cf and Nx with Rx are found. NACA RM L52HO For the case of compressible flow, the solution is not quite so Tw Tadw simple and only in the case of a flat plate where is con To stant can the following useful approximations be made. For n = 7, cfG = 0.0906Rx 1 NxG = 0.0453Rx where G = / + r.f.M2 2 5 Vo/av It is, however, desirable for the sake of accuracy and generality to retain the relations given in equations (29), (30), and (31) and correlate turbulentboundarylayer heat transfer and skin friction on the basis of the length 5 rather than x. CONCLUSIONS The method presented in NACA TN 2692 for evaluating the skin friction of a turbulent boundary layer in compressible flow on an insulated surface is extended to evaluate the turbulent skin friction and heat transfer in compressible flow on a surface which is heated or cooled. The results of this analysis are in good agreement with the heat transfers measured in flight on the NACA RM10 missile up to Mach numbers of 3.8. Langley Aeronautical Laboratory, National Advisory Committee for Aeron3atics, Langley Field, Va. Tw Tadw To v L 0.f8 \"% j NACA RM L52H04 13 APPENDIX DERIVATION OF DEPENDENCE OF SKIN FRICTION AND HEAT TRANSFER ON Rx FOR A FLAT PLATE The momentum equation for the boundary layer on a flat plate may be written de Cf dx 2 e Since b is a constant on a flat plate at a given Mach number (Al) (A2) dx c dx 2 f Substituting the expression 2 K n+l cf = into (A2) there results 2 2 K (F.Vn 2e F OF (A4) When M = 0 T T and w Tadw To O, F = 1 so that equation (A4) may be integrated to obtain (A3) NACA RM L52H04 n+l n+1 2 n+l 8 = Kn+3 + 3 n n+ n+3 Uo (A5) The local skin friction and Nusselt M = 0 and Tw = Tadw as n+l Cf = K n+3 n+l Kn+3_ + 1 llxinJ number may then be expressed for 2 2 2 n+3 ,,)n3 2 n+1 n+3 n+3 2) (uox B (vor (A6) (A7) For n = 7, K = 0.045, and I = . so that 5 72 cf = 0.0578Rx Nx = 0.0289Rx / (A8) (A9) These formulas are the more conventional expressions for local skin friction coefficient and Nusselt numbers as a function of Reynolds number. It may be seen from the differential equation (A3) that the extremely simple expressions just derived are not valid for the case when M or Tw Tadw T is other than zero. However, F is not a very sensitive T1 function of R5 and, if an average value of R5 for a given problem is used to evaluate an F = Fay, the resulting approximate equations are extremely useful and fairly accurate. Thus, for a flat plate with a constant surface temperature NACA RM L52H04 n+l n+l 2 S K n+3n n+3 V n+3 Fav n+ \n+ 3 jj n+l 2 2 S= K \n+3/n+1 2en+3vo n+3 cf iav n + 36" uox n+1 N = K n+3 1 2\av/ \n + 3 For n = 7 these expressions become 2 n+l 2\n+3 uon+3 T/ \,,g 1/5 = 0.0906Rx = 0.o453Rx These equations may be written cfG = 0.0906Rx1/5 NxG = 0.0453R,/5 where 5 o2 To r..M(A2 ) (A17) In equation (A17) o) av is an average value of L Uo along the plate. and n+1 n+3 (A10) (All) (A12) (A13) (A14) (A15) (Al6) /1/5 )4/5 Cfx) (Fav Nx 1/5 4/5 Nx (w F a NACA RM L52H04 REFEREE NCES 1. Donaldson, Coleman duP.: On the Form of the Turbulent SkinFriction Law and Its Extension to Compressible Flows. NACA TN 2692, 1952. 2. Crocco, Luigi: Transmission of Heat From a Flat Plate to a Fluid Flowing at a High Velocity. NACA TM 690, 1932. 3. Chauvin, Leo T., and deMoraes, Carlos A.: Correlation of Supersonic HeatTransfer Coefficients From Measurements of the Skin Temperature of a Parabolic Body of Revolution (NACA RM10). NACA RM L51A18, 1951. 4. Rumsey, Charles B., and Loposer, J. Dan: Average Skin Friction Coefficients 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. NACA RM L52H04 TABLE I VALUES OF uL FOR n = 7 uo ULIUo for n = 7 and Rg =  2 x 104 6 x 10 1 x 105 5 x 105 1 x 106 1.5 x 106 Tw Tadw 0 To 0 0.5465 0.4900 0.4463 0.3655 0.3350 0.3192 1 .5603 .5038 .4593 .3778 .3457 .3295 2 .5943 5365 .4911 .4060 .3752 .3555 3 .6318 .5725 .5285 .4392 .4140 .3852 4 .6702 .6108 .5660 .4734 .4352 .4171 5 .7085 .6493 .6032 .5076 .4663 .4463 Tu Tadw = 0.5 To 0 0.5704 0.5140 0.4679 0.3875 0.3560 0.33985 1 .5817 .5250 .4805 .3970 .3650 .3482 2 .6100 .5524 .5067 .4200 .3870 .3691 3 .6432 .5868 .5400 .4492 .4150 .39627 4 .6787 .6220 .5742 .48oo .4430 .42418 5 .7150 .6573 .6089 .5110 .4720 .4525 Tw Tadw =  To 0 0.5140 0.4544 0.4135 0.3375 0.3055 0.2896 1 .5334 .4760 .4320 .3515 .3205 .3052 2 .5759 .5180 .4725 .3885 .3532 .3383 3 .6185 .5606 .5130 .4275 .3900 .3731 4 .6630 .6045 .5554 .4650 .4283 .4095 5 .7070 .6492 .5983 .5058 .4660 .4460 TW Tadw = 1.0 To 0 0.5890 0.5330 0.4885 0.4045 0.3720 0.3561 1 .5990 .5422 .4972 .4120 .3783 .3629 2 .6230 .5651 .5200 .4330 .3970 .3808 3 .6539 .5963 .5499 .4590 .4240 .4052 4 .6860 .6281 .5815 .4870 .4500 .4308 5 .7185 .6610 .6135 .5160 .4775 .4565 Tw Tad 1.0 V0 To 0 0.461 0.401 0.356 0.2755 0.2465 0.232 1 .494 .435 .3905 .310 .2814 .266 2 .553 .496 .449 .3645 .3335 .3165 3 .6085 .5495 .530 .4138 .3794 .362 4 .6561 .5965 .5495 .4558 .4202 .4015 5 .6968 .6385 .5878 .4925 .4555 .435 18 NACA RM L52H04 TABLE II VALUES OF ( 2 FOR n = 7 (,L /ui 2 cfor n = 7 and R.  M 2 x 10 6 10 1 x 10 5 x 10" 1 10 1.5 x 106 T, Tadw ,.3 To 0 0.2'87 0.2401 0.1992 0.1343 0.1122 0.1019 1 .319 .2.38 .21O .1427 .1191 .1086 2 .3 2 .2878 .2412 .1681 .1408 .L264 3 .4C0, 3310 .2793 .1955 .16i50 .192 4 .4492 768 .3204 .2241 .1903 .1740 .50X20 .42'1 .3022 .2528 .2174 .1992 T% Ta, i To 0 O. i325 0.2642. 0.2189 0.1502 0.1267 0.1155 1 .3384 .2757 .2309 .1576 .1332 .1222 2 .3721 .30.1 .2567 .176. .1498 .1362 3 .1i59 .3443 .291' .2034 .1722 .1570 4 .462. .380'. 3297 .2314 .1971 .1799 5 .5084 .8290 .368 .2611 .2228 .2048 Tw Tad = . To 0 0.26442 0.2005 0.1710 0.1139 0.0933 0.0839 1 .285 .2266 .1866 .1236 .1027 .0931 2 .3317 .2683 .2233 .1502 .1262 .1144 3 .3851 .3160 .2669 .1828 .152. .1407 4 .4421 .?652 .311j .2151 .1834 .1677 5 .4965 .4i 3544 .2490 .2125 .1945 T, a 1.0 To 0 0.3469 0.2841 0.2386 0.1636 0.1399 0.1268 S .3588 .2940 .2472 .1697 .1441 .1317 2 .3831 .3193 .2704 .1875 .1576 .1450 3 .4280 .3556 .3024 .2107 .1798 .1624 4 .4701 .39' .3 01 .2372 .2025 .1856 5 .fi.2 .4 369 .3838 .2663 .2280 .2084 Tw Tadw = 1.0 '1.0 0 0.2125 O.1608 0.1267 0.0759 0.06076 0.05382 i .24L4 .1892 .1525 .0961 .07919 .07076 2 .3058 .2'ta .2016 .1329 .1113 .1002 3 .3703 .3020 .2515 .1712 .1439 .1310 4 .4305 .3558 .3020 .2078 .1766 .1612 S .4855 .4077 .3455 .2426 .2075 .1892 NACA RM L52H04 TABLE III MEASURED QUANTITIES USED IN COMPARISON Point M T Tadw, Tw Tadw Rg N5 (See fig. 1) OF To 1 1.59 67 0.145 0.385 x 106 270 2 1.61 58 .126 .697 422 3 2.15 153 .320 .585 388 4 2.19 171 .365 1.39 799 5 2.52 376 .776 .7 450 6 2.58 394 .837 1.973 1130 7 2.60 49 .135 .208 180 8 3.12 90 .258 .457 331 9 3.60 509 1.312 .921 561 10 3.69 745 1.842 1.07 634 . / ,,, %fL)tlA > NACA RM L52H04 Mtm ~t~ia~I:u~n Li;r;iLLLItm.llllllu1rtruimmnrIuJLunLuimmrIwmI21TnJnuuwLILau~ i 'v lIhl4 Iit + # 1 +f1". 1 11 111 10,000 8,000 6.000 4,000 2,000 1,000 800 600 LIiS1 *ii I i. t n i ff iilx Iij:q: ilill:1ll~ii;: [^ti f:^ *^ = I 14tit 1J.. 4 H'U4:iI_ IJI.' i..UJ II RI[ tlih 2 x 10 4 6 8 105 rII __rL Ui. ~U ^l U 1l U1JJ. ....,<. ..~tt~HH illl .... ,.., .*,( ,~...  ...l 4T ^  I^ ll~CI t~lp [:hll..,,;H:;tt~l;l::l I ; iP~t~ill!l^ !h ::,!!;)t  :;,:ts! ! l 4 li:S i;' li~ f 1 ^, :: : a F 0.0225 R r jI j ,:Iij , H M, I t 4 I H t f. 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