NACA RESEARCH MEMORANDUM HEAT TRANSFER AND SKIN FRICTION FOR TURBULENT BOUNDARY LAYERS ON HEATED OR COOLED SURFACES AT HIGH SPEEDS By Coleman duP. Donaldson Langley Aeronautical Laboratory Langley Field, Va. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 117011 GAINESVILLE, FL 32611-7011 USA NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON October 2, 1952 [-^^ 1^1 ^ ^ ^ ( ^^^^' *. *-' (J NACA RM L52H0i| 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 compressihle 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 RM-10 missile up to Mach num- bers of 3-T' INTRODUCTION Within the last decade the tremendous increase in flight Mach num- bers that has been achieved lias 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 skin-friction force exerted by the boundary layer that the two problems are almost one. The analysis of the turb\ilent boundary layer presented in NACA TN 2692 (ref. l) permits an easy extension of the turbulent skin- friction law from incompressible to compressible flow. The analysis led to expres- sions for the extent of the effective laminar sublayer 5^ and for the velocity u^ at the point 5^. These results permitted the skin-friction 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 NACA RM ■L52H0ij- ■boioiidary layer in both incompressible and compressible flows. It is the purpose of the present short analysis to make this extension. SYMBOLS parameter, n(r - 1) Vol 1 n+1 ^oy Cf skin-friction coefficient, 2t w PQ^O F = < 1 ^liIAi^lJK ■ - 0" To I " uj ' G = (l) 1 + r.f.(7 - 1)m2 ^y m m (^ - 4 O.kkS K k,m,n,r M Nx q r.f. Rx Rr constant (0.0^^5) constants Mach n\;mber qx Nusselt's number, Nusselt's number. q6_ local rate of heat transfer recovery factor UnX Reynolds number, Reynolds number. Vo Ug6 ^o absolute temperature NACA RM L52H0i+ u velocity in x-direction X distance along surface y distance normal to surface 7 ratio of specific heats 6 boundary- layer thickness 6 boundary- layer momentum thickness K thermal conductivity [1 viscosity V kinematic viscosity p density T shear stress Subscripts: av average value adw adiabatic wall conditions L conditions at edge of laminar sublayer o free-stream conditions w wall conditions DERIVATION OF THE HEAT -TRANSFER 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 u_ ^ /y\ n (1) NACA RM L52H04 the diraensionless effective extent of the laminar sublayer — was 6 given by n(r - 1) ^L k2 u„S n+1 (2) and the velocity at the point 5^ was given by n(r - 1) ^L k2 UqS n+1 (3) With these results, it was reasoned that the skin friction was Ut ^-"IL = -S (M which, when the necessary substitutions had been made, resulted in the following skin-friction formula c^ = 2 1-n P(r - 1)> +1| n+l/vo n+1 Pl VL\n+l l^i PoVol (5) Since PL _^ Po Tl (6) and if it is assumed that ^ (El ^^o \To m (7) NACA RM L52H04 then 1-n p n-2m-l Cf = 2 ~n(r - 1) k2 ^^^. ^^M^)^" (8) For a velocity profile having n = 7> it was found in reference 1 (from skin-friction measurements in incompressible flows) that I}!^JlJ1 = 158 k2 so that the local skin-friction coefficient for the compressible case became cf = 0.045 R6 [^] (9) 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(^^- the temperature in the boundary layer may then be written T = T„ - (t^ - Tadw)^ - (Tadw " ^o) (^ ^ (lO) 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 supersonic-missile shapes where the pressure gradients are generally quite small . NACA RM L52H0i+ Differentiating equation (lO) and evaluating it at the wall where — = gives ^o ■^wjJ^adwA /^\ ^o /W/w (11) Hence, the heat transfer at the wall is ■'^wV'^w ~ '^adwj /dul (12) From equation (4) it will be seen that ^^-u. L^L V^y/w ^'w^L (13) so that %f - c C^w " Tadw, ^^L^ / Un (IM Ut If the values of St and — that are obtained from equations (2) ^o and (3) are substituted into equation (ih) and the heat transfer is expressed in terms of a Nusselt number based on the dimension &, the result is 1-n n-1 N5 = n(r - 1)" n+l/uoS\n+l ^^ HL •^o ^^w (15) If it is assumed that (16) NACA RM L52H04 'quation (l5) may be written N5 n(: 1-n n+1 R& n-1 n+1 /v^\ n-1 n+1 -12 _ (17) n-1 Substituting the value of (— ^| equations (6) and (7) results in and — ^ obtained from the use of ^^o 1-n j^_]_ n-2ni-l N& n(r - 1) .-in+l k2 R6 n+1, n+1 (18) Writing the equations for dimensionless skin friction and heat 1-n transfer together and expressing 2 ^(^ - 1) k2 n+1 as a constant yields Cj.F = Constant X Rg 2 n+1 (19) n-1 Wc,F = Constant p n+1 (20) where n-2m-l , n+1 F = (21) Since the right-hand sides of equations (l9) and (20) are the incompressible expressions for Cf and Ng, respectively, the effect of Mach number and temperature potential must be contained in the 8 NACA RM L52H0i+ factor F. This factor may be evaluated in the following manner. From equation (lO) El - Zk . (^w - Tadw) ^ _ (Tadw_l2olK^ To To To Uq To luo; (22) which may be written T„ 2 '' - 1)M^ V" ^-U" Z^A.M (23) The factor F therefore becomes F = < 1 + £lll (7 - l)M^ 1 - ^^2 ^ w -'■adw |-|_ V " "o^ n-2m-l n+1 (24) 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 skin-friction and heat-transfer coefficients. The Ut value of — = is usually of the magnitude 0.4 and although it depends Ur upon M, Rg, and Tw " Tadw , it generally does not have a first-order effect upon the factor F. The value of -^, however, must be found ^o in the following way: Since ^ ^ Ur - 1) VlI 1 n+1 (3) NACA RM L52H04 1 ^ = / n(r - 1) n ri io I k2 R& Iv, n+1 1 n+1 n+1 I) m+1 \n+l (25) where the value of -^ has been expressed in terms of temperatures by means of equations (6) and (7) and where n(r - 1) 1^ k2 R6 (26) Now, if the value of —^ given by equation (23) is substituted in equation (25) there results ^ = A Uo 1 ^ < l,r^ 1) m2 M"^ 1 - ^ u -*^w ~ 1'ai ditA _ ^' m+l n+1 V- (27) Ut This equation for — may be expressed as n+1 ■■f.(y - 1) „2K\g ,,"^+1 ^v - ^adw u^ M"|-^| + A T u -^o o m+l A 1 + ■f-(7 - 1) y{2 ^ \ " ^ad w\ (28) and solved graphically for — for given values of Rg,, M, and '^w " '^adw 10 NACA EM L52H0i^ For the particular case of a turbulent boundary layer with a one- seventh power velocity profile in air when 7=1.^ and m may be taken as O.76 (an approximation which is usually adequate for calcula- tions such as these) , the equations necessary for the calculation of heat transfer become N5F = O.O225R5 0.75 (29) where F = 1 + r.f .M^ ~~5 -^ -'■w " ■'■adw -^ 0.56 (30) Ut and -^ is found from the following equati on Ur 158\°-^^^ L ^ r.f.M^ ^ T y - T a J ^ R& V 5 Tn / ^R& / \ (31) L / L\ Tables I and II give values of — and — found by using ^o \^/ equation (31) for Mach numbers up to 5 for a range of values of Rg and „ — 2ii!i. that are useful in heat-transfer calculations. 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 R5, M, and NACA RM L52H0U 11 To are known. This type of information for a range of Mach numbers, Reynolds number, and temperature potentials can be obtained from the heat-transfer and skin-friction measurements made on the NACA RM-10 missile in flight and reported in part in references 3 and k. 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 skin-friction study reported in reference h, so that the boundary- layer thickness 6 and hence Rg 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 h. The last four points are computed from heat-transfer 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 N5F versus R5 . A reasonably good correlation of the data is obtained for /- iw ~ i adw Mach n\Jinbers ranging from 1.6 to 3-7 and temperature potentials -^o ranging from 0.15 to -1.8. In comparing these experiments with the theory, it was assijmed that the boundary-layer profile had a one-seventh 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, heat- transfer data are not plotted as has been done in figure 1, in terms of local correlations, but in terms of N^ 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 integrations necessary (carried out in detail in the appendix) result in the following relations for incompressible flow with n = 7 c-f = O.0578RX Nx = 0.0289Rx Thus the normal minus one-fifth and four-fifth power variations of Cf and Nx with B.^ are found. 12 NACA m L52H0^ 1 For the case of compressible flow, the solution is not quite so rp _ "P J 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 = O.O906R. -1/5 N^G = O.Oi+53Rx V5 where 1/5 <1 + r.f.M^ T - T -^w ^ adw av 0.41+6 It is, however, desirable for the sake of accuracy and generality to retain the relations given in equations (29), (30), and (31) and correlate turbulent-boundary-layer heat transfer and skin friction on the basis of the length 6 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 RM-10 missile up to Mach numbers of 3-8. Langley Aeronautical Laboratory, National Advisory Committee for Aeronautics, Langley Field, Va. NACA RM Ly2H0h 13 APPENDIX DERIVATION OF DEPENDENCE OF SKIN FRICTION AND HEAT TRANSFER ON R^^ FOR A FLAT PLATE The raoraentura equation for the boundary layer on a flat plate may be written (Al) d0 ^ Cf ix " 2 Since — is a constant on a flat plate at a given Mach number o ^ = -^ c^ (A2) dx 20 * Substituting the expression Y n+1 , , Cf = ^ — R5 (A3) / T _ T ' w -^adw F|M, -^- , R5 into (A2) there results d8 ^ _&_ k/vo Y^'^ dx 20 fIuoS/ (AM T - T When M = and -^ §^ =0, F = 1 so that equation (a4) may be To integrated to obtain 1^ NACA RM L52H04 n+1 n+1 ^ n+1 'n?^^)""c^r" The local skin friction and Nusselt number may then be expressed for M = and T^ = Tadw as n+1 2 2 --"^^(mfr^r £+1 ^_ n±l "1+3 ,„ , . oM"+3/„„ .n+3 -^Hif) w For n = 7, K = 0.0^+5, and |- = -^ so that 5 72 -1/5 Cf = 0.0578Rx (a8) 4/r Nx = 0.0289Rx (A9) These formulas are the more conventional expressions for local skin- friction coefficient and Nusselt numbers as a function of Reynolds niffliber. 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 - "^adw Tl is other than zero. However, F is not a very sensitive function of R5 and, if an average value of Rg for a given problem is used to evaluate an F = F^y^ "the resulting approximate equations are extremely useful and fairly accurate. Thus, for a flat plate with a constant surface temperature NACA EM L52H0i+ 15 n+l n+1 ^ n+l ■^-^3 n+3 (AlO) and n+l Cf = ,"^3 /n + 1 20V^3/Vo \n+3 I^Favy Vn + 3 S/ \uoXy (All) n+l n+l Nv = 1/kN"^3 /^ ^ 1 20\^-"3/uoxy+3 2\Pav/ \n + 3 5/ VVq / (A12) For n = 7 these expressions "become ■5x1/5/ nV5 ^ -1/5 = 0.0906Rj ir'i^-) :(l)'^'W/'=0.0453H. V5 (A13) (AlU) These equations may he written CfG = 0.0906Rx"-'-/5 (A15) N^G = 0.0453Rx^'^^ where (Al6) G = ||l'^^J 1 + -f-M' In eauation (Al?) (— ) 1-m •*o/av T T ■^w adw -^ 0/ av 0.hh8 (A17) ^L is an average value of — along the plate. av l6 NACA EM L52H0U REFERENCES Donaldson, Coleman diiP.: On the Form of the Turbulent Skin -Friction Law and Its Extension to Compressible Flows. NACA TN 2692, 1952. Crocco, Luigi: Transmission of Heat From a Flat Plate to a Fluid Flowing at a High Velocity. NACA TM 69O, 1932. Chauvin, Leo T., and deMoraes, Carlos A.: Correlation of Supersonic Heat-Transfer Coefficients From Measurements of the Skin Temperature of a Parabolic Body of Revolution (NACA RM-IO) . NACA RM L51A18, 1951. Rumsey, Charles B., and Loposer, J. Dan: Average Skin -Friction Coefficients From Boundary-Layer Measurements in Flight on a Parabolic Body of Revolution (NACA RM-IO) at Supersonic Speeds and at Large Reynolds Numbers. NACA RM L51B12, 1951- NACA RM L52H04 17 TABLE I VALUES OF — FOR n = T "o M uj^/uq for n = 7 ard % = - 2 X lo'+ 6 X 10^ 1 X lo5 5 X lo5 1 X 10^ 1.5 X 10^ Tw - Tadw _ Q To 0.5^+65 0.1+900 0.1+1+63 0.3655 0.3350 0.3192 1 • 5603 .5038 .1+593 .3778 .31*57 .3295 2 .59^*3 -5365 .1+911 .1+060 .3752 .3555 3 .6318 .5725 .5285 .1+392 .1+11+0 .3852 1+ .6702 .6108 .5660 .1+731* .1+352 .1+171 5 .7085 .61+93 .6032 .5076 .1+663 .1+1+63 Tw - Tadw - 5 To O.570U 0.5l'+0 0.1+679 . 3875 0.3560 0.33985 1 .5817 ■ 5250 .1+805 .3970 .3650 .31*82 2 .6100 .5521+ .5067 .1+200 .3870 ■ 3691 3 .61+32 .5868 .51+00 .1+1+92 .1+150 . 39627 k .6787 .6220 .571+2 .1+800 .1+1+30 .1+21+18 5 .7150 ■ 6573 .6089 .5110 .1+720 .1+525 Tw " Tadw Q ^ To 0.5l*tO 0.1+5I+1+ 0.1+135 0.3375 . 3055 0.2896 1 .533'* .1+760 .1+320 .3515 .3205 • 3052 2 -5759 .5180 .1+725 .3885 .3532 ■ 3383 3 .6185 .5606 .5130 .1+275 .3900 .3731 1+ .6630 .601+5 .5551+ .1+650 .1+283 .1*095 5 .7070 .61+92 .5983 .5056 .1+660 .1+1+60 Tw - Tadw ^_g • To 0.5890 0.5330 0.1+885 0.1+0 1+5 0.3720 0.3561 1 .5990 .5^+22 .1+972 .1+120 .3783 .3629 2 .6230 .5651 .5200 .1*330 .3970 .3808 3 • 6539 .5963 .51+99 .1*590 .1+21+0 .1(052 It .6860 .6281 .5815 .1+870 .1+500 .1+308 5 .7185 .6610 .6135 .5160 .1*775 .^565 Tw " T^dw _ 2^ Q Tq 0.1+61 0.1*01 0.356 0.2755 0. 21+65 0.232 1 .1+91* .1+35 .3905 .310 .2811+ .266 2 • 553 .1+96 .1+1+9 .36I+5 .3335 ■ 3165 3 .6085 .51+95 .530 .1+138 .3791+ .362 1+ .6561 .5965 .51*95 .1*558 .1+202 .1*015 5 .6968 .6385 .5878 .1+925 .1+555 .1*35 18 NACA m. L^2R0k TABLE II 2 VALUES OF C^^ FOR n = 7 i M 'u-^lu-^^ for n = 7 and R5 = ■ — 2 X 10** 6 X 10** 1 X lo5 5 X lo5 1 X 10^ 1.5 X 10^ "^w " "^adw Q To 0.2987 0.2401 . 1992 0.131*3 . 1122 0.1019 1 ■ 3139 .2538 .2110 .1427 .1191 .1086 2 ■ 3532 .2878 .2412 .1681 .1408 .1264 3 .1)013 .3310 .2793 .1955 .1650 .1492 1* .1(1*92 .3768 .3204 .2241 .1903 .1740 5 .5020 .4216 .3622 .2528 .2174 .1992 Tw " "^adv „ _ = 0.5 To O.325I+ 0.2642 0.2189 0.1502 . 1267 0.1155 1 ■ 3384 .2757 .2309 .1576 .1332 .1222 2 • 3721 .3051 .2567 .1764 .1498 .1362 3 .4159 . 31*1*3 .2916 .2031* .1722 .1570 k .4629 .3869 .3297 .2311* .1971 .1799 5 .5084 .4290 • 3683 .2611 .2228 .2048 Tw - Tadw _ Q = To 0.2642 0.2065 0.1710 0.1139 0.0933 0.0839 1 .2845 .2266 .1866 .1236 .1027 .0931 2 .3317 .2683 .2233 .1502 .1262 .1144 3 .3851 .3160 .2669 .1828 .1546 .1407 4 .4421 .3652 .311't .2151 .1834 .1677 5 .4965 .4147 .3544 .2490 .2125 .191*5 To 0.3469 0.2841 0.2386 0.1636 0.1399 0.1268 1 .3588 .2940 .2472 .1697 .1441 .1317 2 .3831 .3193 .2704 .1875 .1576 .1450 3 .4280 .3556 .3024 .2107 .1798 .1624 4 .4701 .3945 .31*01 .2372 .2025 .1856 5 .5162 .4369 .3838 .2663 .2280 .2084 To 0.2125 0.1608 0.1267 0.0759 0.06076 0.05382 1 .2440 .1892 .1525 .0961 .07919 .07076 2 .3058 .2460 .2016 .1329 .1113 .1002 3 .3703 .3020 .2515 .1712 .11*39 .1310 1* .4305 .3558 .3020 .2078 .1766 .1612 5 .4855 .4077 ■ 31*55 .2426 .2075 .1892 NACA RM L^2E0k 19 TABLE III MEASURED QUANTITIES USED IN COMPARISON Point M Tw - 1'adw > -■■w -'^adw K5 N6 (See fig. 1) OF To 1 1.59 - 67 O.li+5 0.385 X 10^ 270 2 1.6i - 58 .126 .697 1+22 3 2.15 --153 -.320 .585 388 k 2.19 --171 -.365 1.39 799 5 2.52 --376 -.776 .7 1+50 6 2.58 - -39^ -.837 1.973 1130 7 2.60 - 49 •135 .208 180 8 3.12 - -90 -.258 .h^l 331 9 3.60 - -509 -1.312 .921 561 10 3.69 ~ -7^5 -1.81+2 1.07 63U 20 '^z<,*-*Tt-» 7^ > 7X MAC A RM L52H0^ 10,000 mttttHtttttri^-miiTH+HiHtiHtt ro 2 X 10 Figure 1.- Comparison of experimental results (refs. 3 and k) and present method. NACA-Langley - 5-2-65 - 75 N M M ■o 0) ■^ CO 2k 5 '— ' C ^ < ^ (U (U O CM o •3 3 c u 2 O H " ^ CO Oh o E o ^ s ■s g X T3 o < Q Z C^ NX S Sz 29 •« En O :g <; < rt W Z Z K q5 w H ; CO o 5 w S <-> • g OS -a •a c o Q H 3 ^ S = 2 £1 X O - a: ^ X (11 rt a> c " rt to 3 tc _ CM O 3 C Zf :; H c ^ , (U ^ < -g cii y -3 "3 2 2 c (D -a i " .2 S S >.os g fc ■S ra > Z :; m o ^ 03 H 5! ^ '-' • § z S; o < 5«-§y -s ° 9^ z w ' W J J o D o OP O -a CO a— « x; ■a c 3 t! S3 lis i; — to S to to o< iS S = S o £ OJ ? < ^ u a <; SZ < < z 31 0) (U S i o U 3:' o E eg csi ^ I . ^ CO o ^ c •3 u 3 H o o M I— i ^ il rt go X -o Q CO t— I I— I o: 3 . H ca f-* 3 b. gz 29 « H < U °g " z S g| 3>z «-a H < § H <; rt w z z K § c u o u D a Ui :^ H W Q § T3 , ^■S^ 2 _2 ^H Q rt T3 •a Hi j3 o o bC bf s o CO c: (U ^" I— -r! M n\ ^ K W <^ S K o= ^ K s X c CD 3 ca o Zf H c <-3 <: >-. z 2 5 rt £ s CO _o 0) d . H ■0 a: 3 (K Z o c: ° w rt i « .2 3 C t* £ m 3 C4 ^^ a o a -a 03 M g - Z w ^ 03 O T3 CO 5 .2 S ^ to ■" o = ^ " M - O -^ Cfl r- C -3 aj C j;; 0) ID ^ < < z to o to ji: •a c ^ z: to to « „ « S S f;'3 9- > o o ^ u = < Sz X o CM to 3 > <" Z s ^ w _ O bO < Oi A, U Q (? K a 1^ z " ■s -S -S " 2 ^ E^ <3> C S CS lU o _^ « 8 .£f (V i: to o M !-c bO c o rt o ' U :S < Z Z K CO U u <: « 03 ai fc ° Z W (d .-3 ^8 .5 g 3 „ to 2 cs £ 3 C -a s oi < < z ctJ ^ — I "^ (U to •" o 2 l2 C -3 -a S « 3 ca cj s s m o ^ to c< ■« < < z ^ 0) (U T-1 « -= " H - £ o; o j: ■ »< =5 6£^1 £ = R o 5 rH c C < -; a> v O . •a o < a z CO 1.^ ^H a . f-i Ig 3 fc Sz S2 Is pi <: § H tj s < < Id w z z a oii u H < Ixl Q <§■ S = 2^ o < K-Sy S- ffl CQ W J J o S o _ n o T3 (Tj o Q 0< 5 "g . 1 §it, o