fO^Cf\'T^-n2^ ^^» . dtt- tot i-k-ADON 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 No. 1229 
 
 HEAT TRANSMISSION IN THE BOUNDARY LAYER 
 By L. E, Kalikhman 
 
 Translation of "Gazodinamicheckaya Teoriya Teploperedachi" 
 Prikladnaya Matematika i Mekhanika, Tom X, 1946 
 
 Washington 
 
 ADrill9i#^''^^^^'^°^^^°^'^^ 
 Aprii -^y^yQcuMENTS DEPARTMENT 
 
 120 MARSTON SCIENCE UBRARY 
 
 P.O. BOX 117011 
 
 GAINESVILLE, FL 32611-7011 USA 
 
^3^ "^^0 >tP J75? 
 
 NATIONAL ADVTSOEY COMMITTEE FOE AERONAUTICS 
 
 TECHNICAL MEMORANDUM No. 1229 
 
 EEAT TRANSMISSION IN TKE BOUNDAEY LATER* 
 
 By L. E. Kallldiman 
 
 Up to the present time for the heat transfer along a curved wall in 
 a gas flow only such prohlema have been solved for which the heat trans- 
 fer between the wall and the incompressible fluid was considered with 
 physical constants that were independent of the temperature (the hydro- 
 dynamic theory of heat transfer). On this assumption, valid for gases 
 only, for the case of small Madi numbers (the ratio of the velocity of 
 the gas to that of sound) and small temperatiire drops between the flow 
 and the wall, the velocity field does not depend on the temperature 
 field. 
 
 In 19^1 A. A. Dorodnitsyn (reference l) solved the problem on the 
 effect of the compressibility of the gas on the boundary layer in the 
 absence of heat transfer. In this case the relation between the tempera- 
 ture field and the velocity field is given by the conditions of the 
 problem (constancy of the total energy). 
 
 In the present paper which deals with the heat transfer between the 
 gas and the wall for large temperatiire drops and large velocities use is 
 made of the above-mentioned method of Dorodnitsyn of the introduction of 
 a new independent variable, with this difference, however, that the 
 relation between the temperature field (that is, density) and the velocity 
 field in the general case considered is not assumed given but is deter- 
 mined from the solution of the problem. The effect of the compressibility 
 arising from the heat transfer is thus taken into account (at the same 
 time as the effect of the compressibility at the large velocities). A 
 method is given for determining the coefficients of heat transfer and the 
 friction coefficients required in many technical problems for a curved 
 wall in a gas flow at large Mach nimbers and temperature drops. The 
 method proposed is applicable both for Prandtl number P = 1 and 
 for P ^ 1. 
 
 "Gazodinamlcheckaya Teoriya Teploperedachi . " Prikladnaya Matematika 
 i Mekhanika, Tom X, 19^6, pp. h^^^-k^k. 
 
NACA TM No. 1229 
 
 I. FUNDAMENTAL REIATIONS FOE THE lAMIKAE AND TURBULENT BOUNDAE?Y 
 LAYER IN A GAS IN THE PRESENCE OF HEAT TRANSFER 
 1. Statement of the Problem 
 
 We consider the flow over an arbitrary contour of the type of a 
 wing profile in a steady two-dimensional gas flow (fig. l). For 
 supersonic velocities we take into account the existence of an oblique 
 density discontinuity (compression shock) starting at the sharp leading 
 edge or a curvilinear head wave occurring ahead of the profile. For 
 subsonic velocities we assume there are no shock waves (value of the Mach 
 number of the approaching flow is less than the critical). 
 
 We denote by u, v the components of the velocity along the axes 
 X, J, where x is the distance along the arc of the profile from the 
 leading edge, y is the distance along the normal, T ia the absolute 
 temperature, p the pressure, p the density, \i the coefficient of 
 viscosity, X the coefficient of heat transfer, e the coefficient of 
 turbulence exchange, X^ the coefficient of turbulent heat conductivity, 
 Cp the specific heat, and J the mechanical equivalent of heat. 
 
 T = T + 
 
 2JCp 
 
 is the stagnation temperature 
 
 a = 
 
 Kp 
 
 is the velocity of sound 
 
 is the adiabatic coefficient 
 
 
 P - ^ 
 
 is the Prandtl number. The remaining notation is explained in the text. 
 The values of the magnitudes in the undisturbed flow are denoted by the 
 subscript CO, the values of the magnitudes at the wall by the subscript w. 
 
 The problem consists in the solution of the system of equations 
 (reference 2) 
 
NACA TM No. 1229 
 
 
 (|i + e) 
 
 ^ 
 ^ 
 
 ^ (pu) + ^ (pv) = 
 
 (1.1) 
 (1.2) 
 
 (■ 
 
 
 + V 
 
 B 
 ^ 
 
 [,. 
 
 O 
 
 (S-) 
 
 57 
 
 ^1) 
 
 (1.3) 
 
 S7 
 
 = 
 
 (i.i^) 
 
 p = pRT, ^l = CT^ 
 
 (1.5) 
 
 where E is the gas constant, and C and n are constants. 
 
 In the solution of the Bystam equations (l.l) to (1.5) we start 
 out from, considerations on the dynamic and thermal houndary layer of a 
 finite (hut variable) thickness. The flow outside the dynamic houndary 
 layer approaches the ideal (nonviscous) flow, nonvortical in front of 
 the shock wave and, in general, vortical behind the wave. Equation (l.^) 
 shows that the pressure is transmitted across the boundary layer without 
 change, that is, the pressure Po(2:) and the velocity on the boundary 
 of the layer U(x) may be considered as given functions of x. The 
 flow outside the thermal boundary layer we assume to occur without heat 
 transfer, that is, outside the thermal layer and on its boundeory the 
 total energy Iq 1b constant: 
 
 JCpT + 
 
 u^ 
 
 U. 
 
 JCpToo + 
 
 = JCpTo 
 
 
 (1.6) 
 
 From this it follows that the stagnation temperature 
 thermal boundary layer has a constant value 
 
 i. 
 
 outside the 
 
 T* = 
 
 _2. 
 
 Jc, 
 
 (1.7) 
 
 Thus the flow outside the thermal boundary layer for small velocities 
 is assumsd nearly Isothermal while for large subsonic velocities isentropic. 
 
NACA TM No. 1229 
 
 For supersonic velocities the entropy is constant" up to the shock wave 
 while after the wave the entropy is constant alon^ each flow line of the 
 external flow atout the profile but variable from one flow line to the 
 nex±. 
 
 The boundary conditions of the problem are: 
 
 u = V = 0, T* = T^ f or y = (1.8) 
 
 where T^ = T^(x) is a given function. In the absence of external heat 
 transfer (across the wall) we have instead of the second condition of 
 
 equation (1.8) the condition I j = 0. Further 
 
 \Sy /y=:0 
 
 u = U(x) for y = 6^(x) 
 
 7' 
 
 T* = T, 
 
 00 
 
 for y = Ay.(x) 
 
 (1.9) 
 
 where 5„ and A^. are the values of y referring, respectively, to the 
 boundary of the dynamic and the boundary of the thermal layer. 
 
 2. Fundamental Expressions for the Temperatures 
 
 For P = 1 equation (l.3) gives the "trivial integral" T* = Constant. 
 Taking into account the second condition of equation (l.9)j we obtain 
 
 T = Too(l-u2) 4 = -^\ (2.1) 
 
 The temperature of the wall is equal to the temperature of the adiabatic 
 stagnation: 
 
 \ = Too = '^"f + 1^^ - 1)^^] P^o = ^) (2.2) 
 
 The integral (2.1 ) corresponding to the case of the absence of external 
 
 and internal heat transfer in the boundary layer, ( — ) = for P = 1, 
 
 \Vy=0 
 was obtained on the basis of the solution of A. A. Dorodnltsyn (reference l), 
 
NACA TM No. 1229 ! 
 
 For P = 1, U = Constant, emd T^ = Constant, equation (l.3) Is 
 likewise integrated independent of tlie solution of the remaining 
 eq^uations of the system and gives the so— called Stodola— Crocco integral 
 
 T = au + b 
 Imposing the "boundary conditions we ottain 
 
 T = T, 
 
 00 
 
 l--^-(T,-l) (l-»)] {\ = l 
 
 From equation (2.3) we also obtain 
 
 w 
 ^00/ 
 
 (2.3) 
 
 (2.U) 
 
 u 
 
 U 
 
 (t = T - T^, to = Tqo - Tv) 
 
 (2.5) 
 
 The integral (2.J+) correspondB to the case -where there is similarity of 
 the Telocity field with the field of the stagnation temperature drop 
 (equation (2.5)) and was used in the solution of the problem of the 
 flow about a flat plate (reference 3)« 
 
 In any more general case U ^ Constant 
 
 for 
 
 (-lo ^ ° 
 
 or 
 
 T^^ / Constant or P ^ 1 the integral of (1.3) Is not known in advance. 
 
 The existence of the trivial integral is not, however, the required 
 condition for the solution of the problem and this fact is fundamental 
 for what follows. 
 
 Let the function T*(x, y) or u(x, y) be integrals of the system 
 (l.l) to (1.5) satisfied by the boundary conditions (equations (I.8) 
 and (1.9) )• The temperature at an arbitrary point can then be represented 
 in the form 
 
 ^00 
 
 Joo 
 
 for 
 
 '■00 
 
 -u2 
 
 (2.6) 
 
 1 - u^ + (T^ - 1) 1 
 
NACA TM No. 1229 
 
 As Is easily seen, the integrals (2.l) and (2.U) are particular cases of 
 
 - t* 
 
 the second form of the relation (2.6) for T = 1 and — - = ^, 
 
 ■to 
 
 respectively. This relation permits expressing together with the 
 temperature also the density and viscosity as a function of the velocity 
 and stagnation temperature drop. 
 
 3. Expressions for the Pressure, Density, and Viscosity 
 
 The pressure p at any point within the boundary layer is determined 
 by the equation of Bernoulli 
 
 P = Po = Po2(i-u2r r = 7= (3.1) 
 
 where Pq is the pressure on the boundary of the layer (thermal or 
 dynamic depending on which of them is thicker), Pqq is the pressure 
 on adiabatically reducing the velocity to zero in the tube of flow 
 passing through the shock wave. The density p at an arbitrary point 
 within the boundary layer is determined by the equation of state (first 
 equation of (1.5 )), equations (2.6) and (3.l) 
 
 K 
 
 P = Po2 li ;i-^ 77 pr^ (1 - ^^)'" (3.2) 
 
 l-u2 + (T^-l)J^l-^^ 
 
 where p^ is the density on adiabatic reduction of the velocity to zero. 
 The viscosity p. is determined by the equation 
 
 H = Hoo 1 - u^ + (Tw - 1)(1 ^) (n;^0.75) (3.3) 
 
 where Hqq corresponds to the temperature Tqq, that Is, is obtained on 
 the adiabatic reduction of the velocity ^o zero. 
 
NACA TM No. 1229 
 
 For any point ahead of the shock we have 
 
 P = PoiCl - U2) 
 
 K. 
 K-1 
 
 J. 
 
 P = P, 
 
 01 
 
 (i - ^y-"- (3.U) 
 
 vhere Pq-, and Pq-, are, respectively, the pressure and density on 
 
 adiahatlcally reducing the velocity to zero up to the intersection of the 
 streamline with the shock. 
 
 As a sc ale of the velocities it is possilple instead of the assumed 
 magnitude |/2iQ to take the critical velocity a* and the local sound 
 
 velocity a. As is known a* = [/(it - 1)/(k + l) J2±q. Substituting 
 -^7 = X,-|_ and — = M we obtain 
 
 a 
 
 a 
 
 K _ 1 _ / (k - 1)M^/2 
 U=i/ X-L, U= ' ' ' ' 
 
 ■^ + 1 l/l + (k- 1)m2/2 
 
 M= / ^1 (2.5) 
 
 i/(k+ 1)/2 - (k^ 1)Xi^/2 
 
 h. Integral Relations of the Momentum and Energy in New Variables 
 From equations (3.l) and (3.2) we obtain 
 
 M-Y- 
 
 'In order to distinguish various uses of the symbol X herein 
 SLibscripts 1 and 2 have been added by the NACA reviewer in the 
 translated version. ■ 
 
8 
 
 NACA TM No. 1229 
 
 where p is the density on the hoimdary of the layer (thermal or 
 dynamic depending on which is the thicker). We represent equations (l.l) 
 and (1.2) In the form 
 
 |_(pua).|-(puv) = p„uu. .|[(...)|] 
 
 I- (puU) + I- (pvU) - puU« = 
 dx ay 
 
 Suhtracting the previous equation from the ahove we obtain 
 
 ^h 
 
 pg(i_|)l ,2jn;.pu(i-|).|^[p,(u-u)| 
 
 + uu» 
 
 p(^^ _ x). xk:.p(i- I). -1^ [(...) 1; 
 
 From equation (3*2) we find 
 
 P0= P02(l-U^) 
 
 1_ 
 
 K-1 
 
 (U.2) 
 
 ^_1 tjg-ug -H (T^-l)(l-t7to*) 
 
 (i^.3) 
 
 Integrating equation ('+.2) term hy term from y = to y = Ay, if A_ > 
 
 and to y = B_. if 6 > Ay. (for def initeness we assume that Ay > &„; 
 
 the same result is obtained if we assume bj > Ay)j making use of the 
 
 relation (^.3), and taking into account the fact that starting from the 
 boundary of the dynamic layer the velocity u is constant along x and 
 the friction T is equal to zero, we obtain 
 
NACA TM No. 1229 
 
 "^^f ^H-D-^-hr^r^K-s)^ 
 
 + uu' /I + 
 
 u^ 
 
 - uy -'o 
 
 
 uu»(Tv-i) r^ 
 
 u^ -^0 
 
 P 1- 
 
 dy = T 
 
 w 
 
 where 
 
 (l+.U) 
 
 V = 
 
 (^+ €) ^ 
 
 S7-'y=0 W/w 
 
 ('+.5) 
 
 Is the frictlonal stress at the wall. 
 
 We now represent equations (l.2) and (l.3J In the form 
 
 2- (puto*) . I (Pvto') - PU ^ 
 
 = 
 
 -— (put*) + T- (pvt*) - pu = .5- 
 
 &x oy di oy 
 
 (n + 
 
 m<i-r^k'H 
 
 Subtracting the second equation from the first, integrating the resuit 
 
 term by term from y = to y = Ay (assuming as above for definiteness 
 
 that /Sy > 5y), and remembering that for Ay > S the heat transfer 
 
 q = X -r— and the friction t = n g:- are equal to zero on the boundary 
 of the thermal layer we obtain 
 
 f-Ut * 
 dx 
 
 ° C^H' 
 
 
 (4.6) 
 
10 WACA TM No. 1229 
 
 where 
 
 1^- Cp 
 
 (■ f L • (J - .) •. (. % 
 
 ■-■.(■i„-'-(sf). 
 
 Is the intensity of the heat transfer at the wall. 
 
 In solving the problem of the "boundary layer without heat transfer 
 between the gas and the wall for large velocities and P = 1, 
 A. A. Dorodnitsyn introduced the change in variables 
 
 K-1 
 
 (1-U2) dy (U.8) 
 
 l-u2 
 
 Noting that the function under th£ integral sign In equation (^.8) agrees 
 with the expression p/Po2 ^°^ '^ ~ ^' ^® introduce a new independent 
 variable of the analogous equation containing in the function luider the 
 integral sign the expression p/po2 ^°^ "^^^ general case according to 
 equation (3.2) 
 
 D 1 -u2 + (T„-l)(l-tVto ) 
 
 For T^ = 1 the relation between the coordinates t) and y depends on 
 
 t* 
 the unknown velocity profile. For — - = ^ equation (^.9) gives the 
 
 change in variables applied to the problem of the heat interchange of 
 
 the plate with the gas flow. In this case the relation between t] and y 
 
 likewise depends only on the velocity profile. 
 
 As is seen from equation (U.9) in the general case the relation between 
 the coordinates t\ and y depends not only on the velocity profile u(x, y] 
 but also on the temperature -drop profile t (x, y) which likewise is 
 not initially known but Is determined from the solution of the problem. 
 
NACA TM No. 1229 
 
 11 
 
 Replacing the density p in equations {h,k) and (4.6) by its 
 expression (3.2) and passing to the varlahles x, t^ we obtain 
 
 + uu» 
 
 ^ u^ \r^/ u\ uu'(T„-i)PA/ t*\ T^ ^, ; 
 
 1 + / (1 - ii) dTi + / 1 dTi = -2- U.IO 
 
 d_ 
 dx 
 
 Ut 
 
 J„ tl 
 
 Sli- 
 
 i 
 
 dTl 
 
 P02°i 
 
 (U.ll) 
 
 where 6 and A are the values of the variable r, referring, 
 respectively, to the boundary of the dynamic and the thermal layers. 
 
 We denote the thickness of the loss in momentum and the thickness 
 of the displacement in the plane it\, respectively, by 
 
 '-I'i{^-^)'^ ^* = ^H=/„'(l-S)i'' (^-12) 
 
 we introduce the concept of the thickness of the energy loss (in the 
 plane xt) ) 
 
 e = 
 
 u 
 
 
 
 5 1 - -^ ^1 
 
 (lt.l3) 
 
 This magnitude has a clear physical meaning; namely, the magnitude 9 
 characterizes the difference between that total energy which the mass of 
 the fluid that flows in unit time through a given section of the thermal 
 
12 
 
 NACA TM No. 1229 
 
 boimdary layer would have If its stagnation temperature were egual to the 
 stagnation temperature of the external flow and the true total energy of 
 this mass. The magnitude 6 thus represents In length units referred 
 to the temperatiire to the "loss" In total energy due to the heat 
 transfer. For small velocities the concept of the thickness of energy 
 loss agrees with the previously Introduced concept of the thickness of 
 heat-content loss (reference k). The magnitude 
 
 .A 
 
 A = eap = 
 
 1 - 
 
 dTl 
 
 ih.lh) 
 
 may he called the thickness of the thermal mixing. 
 
 With the aid of the magnitudes defined by equations (4.12), (U.I3), 
 and (4.1U) we represent the obtained Integral relations of the momenta 
 and energy (equations (1<-.10) and (l+.ll)) In the final form 
 
 d0 ^ U^ 
 dx U 
 
 H + 2 (H + 1) 
 
 U2 
 
 1 -U2 
 
 d + 
 
 U»(Tv - 1) 
 U(l -U2) 
 
 Hm0 = 
 
 W 
 
 02 
 
 (I+.I5) 
 
 d£ 
 djc 
 
 u 
 
 to* 
 
 e = 
 
 %r 
 
 P02^°pto' 
 
 (4.16) 
 
 We note that the integration with respect to y (or t]) may be taken 
 from to 09 so that the relations (4.15) and (4.l6) are general for 
 the theory of the boundary layer of finite thickness and the theory of 
 the asymptotic layer. 
 
 For small Mach numbers (the effect of the compressibility due to 
 the temperature drop) the relations (U.I5) and (4.l6) assume the form 
 
 «.^(H..,..^(^-.)K,e. 
 
 ' W 
 
 "" u \To 7 ^" - 
 
 PoU2 
 
 f . u' ^ ^ to' ^ ^ 
 
 dx u to 
 
 
 > 
 
 (4.17) 
 
NACA TM No. 1229 13 
 
 ■vAiere tg = Tq — T^, Tq and Pq are, respectively, the temperature and 
 
 density of the Isothermal flow outside the thennal boundary layer (it 
 follows from equations (2.6) and (3«2) if we set approilmately U = 0, 
 that is, according to equation (3«5) M = O), The pressure distribution 
 over the profile is determined by the equation of Bernoulli for an 
 Incompressible fluid. ^ 
 
 As is seen from equations (i<-.15) and (U.l6) and also from what follows, 
 in the variables x, t\ the equations of the system (1.1) to (1.3) Bxe 
 Bimplifled and approach in principle the corresponding equations for the 
 incompressible fluids. For this reason the fundamental methods of the 
 theory of the boundary layer In an incompressible fluid may be generalized 
 to the case of a body in a gas flow with heat Interchange. 
 
 We give below the generalization of the method of Pohlhausen for 
 the case of the laminar layer and the logarithmic method of Prandtl— 
 Kann^ for the case of the turbulent layer. The proposed method of the 
 solution of the problems connected with heat Interchange pennlts, of 
 course, generalization of certain other problems in the theory of the 
 boundaiT- layer in an Incompressible fluid. 
 
 II. LAMHIAE BOUNDAEY lAYER WITH HEAT INTERCHANGEE 
 
 BETWEEN THE GAS AND THE WALL^ 
 
 5. Transformation of the Differential Equations 
 
 Assuming in equations (l.l) to (1.3) e = and t* = T - T^, 
 Bubstitutlng the values p, p, and |i according to equations (3«l) 
 to (3.3)i we transform these equations to the new independent vari- 
 ables 1 = 1 and J], deteimined according to equation (^.9). The 
 equations of transformation of the derivatives will be 
 
 
 ^Prcan. equationa (S.l) and (3.5) we have 
 
 Poo 
 
 -Po= [i^|(--i)M^]'"^-i 
 
 PqI 
 
 Jfi 
 
 :.,li^,%=JL^, ,] 
 
 Po«-2-r ^-"^-^-alT 
 
 whence setting M = 0, we obtain 
 
 Pq + 5- (pqU^) = Constant 
 *En what follows we restrict ourselves to the case P = 1. 
 
Ik 
 
 NACA TM No. 1229 
 
 We obtain (introducing the notation v 
 
 02 
 
 ^^oo 
 
 ^02 
 
 . . 1 - u^ + T^ - 1)1 - t7to*) 
 
 dx oil 
 
 1-U2 
 
 a 
 
 + ^1 
 
 
 1 _u2 + (T^_l) h.__ 
 
 n-a 
 
 l> 
 
 (5.1) 
 
 ^ ^ V P02 ^7 
 
 (5.2) 
 
 * dt^ 
 
 St* St 
 
 U TT + V u 
 
 ox Sti 
 
 iic 
 
 = y(l-^2f-ll_. 
 
 02 
 
 Sti 
 
 1 -u2 
 
 .(5.-i)(i-^^ 
 
 n-1 
 
 ^ 
 ^ 
 
 (5.3) 
 
 If it is assumed approximately that n = Ij the obtained system can he 
 still further simplified. 
 
 6. Generalization of the Method of Pohlhausen 
 
 We represent the velocity profile and the stagnation temperature- 
 drop profile hy the polynomials 
 
 s-i(^H^)^-<f/-<^: 
 
 
 (6.1) 
 
NACA TM No. 1229 
 
 15 
 
 t 
 to 
 
 7 = <l) ^ ^^1 ^ <I) ^ <d' 
 
 (6.2) 
 
 To determine the coefficients of the polynomials we set up the conditions 
 
 ^^ u 6 S t* ^ u 
 
 - + a ^ — — + ?^ for ^ = 
 
 a - 
 
 S(Ti/6r U ^Mri/A) V S(ti/6)" 
 
 (6.3) 
 
 where 
 
 A,o "" 
 
 
 V^^(l-u2)^-l 
 
 fl 
 
 , a = (l - n) 
 
 ^w ~ ^00 
 
 '■w 
 
 02 
 
 Condition (6.3) follows from the equation of motion (5*3) since 
 u = V = for T| = 0. Further^ 
 
 ^ = 1, — ^— ^ = 0, i-^ li = for a = 1 
 
 (6.4) 
 
 From equation (5.3) for t) = 0, v = v = we obtain 
 
 .2 t* 
 
 2 +n* 
 
 S(ti/a)^ to- 
 
 + a 
 
 a t^ 
 
 t2 
 
 S(ti/a) to_ 
 
 = for T = 
 
 (6.5) 
 
 Similarly to conditions (6.U) for the profile u/U we take for the 
 profile t /to* the conditions 
 
 *^ = i _i_i_= 
 
 S(n/A) t 
 
 
 
 p.2 t* 
 
 _£ ^ =0 for ^=1 (6.6) 
 
 S(Ti/A)2 to* A 
 
l6 NACA TM No. 1229 
 
 By differentiating eguatione (5.I) and (5.3) vith respect to t) 
 with the subsequent equating of r\ to zero, it is easy to ohtain the 
 conditions for the third deriTatiTes of u/U and t*/to* at the wall. 
 These conditions may he useful for various aspects of the method of 
 Pohlhausen. Using conditions (6.5) and {6.6) we obtain 
 
 3 - /9 - 12a 
 Bi = , B2 = 6 - 3Bi, B3 = ^ + 3Bi, Bi^ = 3 - \ {6.1) 
 
 From conditions (6.3) and (6.i)-) we now find 
 
 12 + Xp 
 
 A-, = , , Ag = 6 - 3Ai, A^ = -8 + 3Ai, Al^ = 3 - Ai 
 
 6 - (3 - 1/9 - 12a )& /A 
 
 (6.8) 
 
 In relations (U.15) and (U.I6) there enter, besides t3(x) and 0(x), 
 the four unknown functions t^, H, q^, Bp. In constructing the profiles 
 
 there were also introduced the auxiliary functions 5 and A. The 
 required six additional equations are obtained by substituting equa- 
 tions (6.1) and (6.2) in equations (U.5), (4.7), and {^,12.) to (U.l4). 
 
 We obtain 
 
 8 _ fi -5Ai^ + 12A-], + li+Ij- 
 
 6* = i3H = 5 -^, t3 = 5 (6.9) 
 
 20 1260 
 
 \ = ^^oo^w"-^ f (1 - ^)' ^ ^1^ <iw = Voo^""^ ^ (1 - u^)' ^ % (6.10) 
 
 A 
 
 8 - B-L 
 A* = eBp = A , e = A(Mi + NiAi) (6.II) 
 
 20 
 
 where for — < 1 
 o 
 
 M, . 6.jg - 8.3(4)3 . 3.u(t)\ % = ^x(f)- 3.,(0^ . ..{ff . .,{f 
 
 _6 -B-^ _ 16 - 3Bi 5 - Bi 2I+ - 5B-L 
 
 ^1 = ~6o~' ^2 = ^20~' ^3 = -^80"' ^^ = 2520 
 
NACA TM No. 1229 
 
 17 
 
 for ^ > 1 
 6 
 
 8 - B 
 
 % = 
 
 20 
 
 ^^1 
 
 -O.U + 0.229c/|j - 0.lU3c/^\ + 0.0290/'- 
 
 ^ ^<l) 
 
 0.1 - 0.11U3 I + 0.0536/^^ - o.O096f''^V 
 
 N-, 
 
 6 
 A 
 
 2 "^ 1 
 
 0.0500 - o.oU29('-") + o.0286('-^) - 0.006/'^) 
 
 -47 
 
 - 0.0167 + O.O211/-J- 0.0107/'-') + 0.002/^^\ 
 
 From equations (6.10 ) and (6.8) it is seen that the point of separation 
 of the laminar layer for the given boundary conditions is determined 
 hy the condition Xp ■= -12. 
 
 7. Determination of the Initial Conditions 
 
 For suhsonic flow and also supersonic in those cases where there is 
 a head wave in front of the profile a crltlcEil point U(0) = is 
 formed at the leading edge (x = O). The latter is a singular point of 
 equations (4.15) and {h.l6) in which the derivatives diS/dx, dQ/dJC, 
 ajid so forth, increase to infinity if the initial -8, 9 are not 
 Buhjected to certain special conditions. 
 
 Substituting the expressions for t^, q^, A from equations (6.10 ) 
 and (6.11) into equations (^^-.15) and {h.l6), multiplying the latter 
 by S and A, respectively, and equating to zero the coefficients of l/U 
 these conditions are obtained in the form 
 
 X,(H .2)1. X,(f„ - 1)| ^ - T^l = 0, ,^(ff I - T^l - (7.1) 
 
l8 NACA TM No. 1229 
 
 In the abaence of heat transfer (T^ = l) the first relation (7»l) 
 is a cubical equation in Xg- Of its three roots (7. 052, 17.75, and — 70) 
 only the root (-^2)^=0 ^ '^•^52 satisfies the physical conditions. This 
 value of X,2 is the one generally assumed initial condition in the theory 
 of the laminar layer for the equation of Karm^in— Pohlhausen. The equations 
 (7.1) may he conveniently regarded as a system for determining the 
 initial values of (X2) _q and {A/5)yr-Q. We present the results of the 
 
 computation of these values for < (T^)x=o "^ 5 (for n = 0.75). 
 
 T^ = 0.05 0.10 0.50 1.00 1.50 2.00 3.00 4.00 5.00 
 
 X.2 = 0.3 0.7 4.05 7.05 7.5 6.0 3.9 2.6 1.9 
 
 -=1.12 1.16 1.18 1.23 1.31 1.50 1.87 2.7^^ 3.70 U.86 
 5 
 
 The graphs of these results are shown in figure 2. 
 
 8. Method of Successive Approximations 
 
 Simple computation formulas can he obtained by generalizing the 
 method of H. Lyon (reference 5). At the same time we modify the method 
 of Lyon with the object of improving the convergence. 
 
 We multiply (4.15) by 2-3 and have 
 
 _ (O Tjt 2(H + 2 - U^) -2 u' T - 1 2 - r,-1 / -On"*-! i3 
 d5c U 1 _ u2 U 1 _ tj2 E_U "^ ^5 
 
 02 
 
 (8.1) 
 
 where 
 
 
 ^^00 
 
 and L is a characteristic dimension. Setting k = 2(H +2) we try 
 in the function _k to separate a certain principal part constant for a 
 given value of T • that is, we set 
 
 k = cj + (k — cj), where c-, = c^Ct^) 
 
NACA TM No. 1229 
 
 19 
 
 After simple transformations we ottain 
 
 dx U L -^ 
 
 \ A_ 
 
 ^3 
 
 _2 UU 
 
 ^ + 
 
 1 - tJ' 
 
 :_[c,.2(T,- 1)^-2 
 
 ^2 
 
 T n-2 _1. 
 w 
 
 (1 
 
 u^r^ 
 
 UE 
 
 02 
 
 ^Hi\-^2(if i^-^i) 
 
 (8.3) 
 
 Assuming over a certain part of the "boundary layer xq < x < x-^ that 
 the ratio — = h is constant with respect to x and equal to the 
 
 m- 
 
 aan value for the given segment, we eet^ 
 
 1260(8 - Bi) 
 
 c = ci + 2(T„ - l)h = CjL + 2(T - 1) ^ — — 
 
 ^ ^ J. w B 20(-5Ai2 + -j^2Ai + Ihh) 
 
 (8.1+) 
 
 Miatiplying equation (80 3) "by U^ we obtain 
 
 dx 1 - U^ 
 
 ;-l 
 
 K 
 
 , T„->^ 5!Z: (1 . u2,«-i 
 
 •-w 
 
 R 
 
 02 
 
 »i|*w-Ki-)'(^-n) 
 
 Considering this equation as linear in ^U^ we write its solution in 
 the form 
 
 I B02 
 
 1-^ 
 U^(l-U2) 2 
 
 1-1+;^ 
 C + r T^^-2 JP-^ (1 -U2) 
 ^^0 
 
 <b dx 
 
 (8.5) 
 
 ^For 11. ^ Constant there is taken in the exponent c a constant 
 mean value of T^ for each section. 
 
20 
 
 NACA TM No. 1229 
 
 where 
 
 2A 
 
 ■1^''w-M|)^^-"1^^ 
 
 C = 
 
 ^^E^U^ (1 - U^) 
 
 1-- 
 
 '02 
 
 (8.6) 
 
 x=x, 
 
 
 
 where C = for Xq = Oo Substituting under the Integral (in the 
 function and exponent c) certain initial values of (^2^n ^^'^ ('^/^ )o 
 we ohtain a first approximate function •a(x). The arbitrariness in the 
 choice of the magnitude c-[_ may be utilized for improving the convergence. 
 
 For this purpose c^ must be chosen such that for each value of T,^ the 
 error due to the assumed (inexact) values of X2 ^nd A/6 is a minimum. 
 
 Since $ itself depends little on A/6 it is sufficient to set up 
 the condition of little variation of with X^ . Neglecting the relatively- 
 small dependence of the functions t3 /5 and k. on X, that is, setting 
 in the argument A]_ on which they depend, X2 = 0, we obtain the 
 approximate expression 
 
 $ a 
 
 ^12+X2 
 
 (^1) (I) ^w 
 
 
 In order that = Constant the coefficient of X2 must be eq.ual 
 to zero, whence we obtain 
 
 ci = (k). 
 
 (^l^g^O^w 
 
 ^2=0 6(Vs) 
 
 \2=0 
 
 This dependence of c^ on T^ in a wide Interval of change of 
 the argument (and practically independent of A/6) is close to a linear 
 
 one. For T^ = 0.1 we obtain c-, = 9-35 for ^=1 (c. = 9.12 for 
 1=2, c^ = 9.7 for I = 0.5). For T^ = 1 we obtain c-^ = 6.^6. 
 Bounding off the last value to c^ = 6 we apply the linear relation 
 
 ci = 9.5 - 3.5T^ (8.7) 
 
 Finally multiplying the equation of energy (equation (4.17)) in nondlmensional 
 form term by term by 20u2 (§ = j-) and integrating as linear we obtain 
 
NACA TM No. 1229 
 
 21 
 
 ^ ^2 = — 
 
 U'^CT^ - 1)^ 
 
 C» + \ T^,""^ 2Bi(Tv - 1)2 U(l - U^)'' I die 
 
 '^0 
 
 (8.9) 
 
 where 
 
 C« = [e^E 2u2(Tv - 1)^- - 
 L ^ x=xo 
 
 and C* = for x = 0. 
 
 The computation of the dynamic and thermal layers ty equations (8.5) 
 and (8.9) can "be carried out in this sequence. We consider, together 
 with the parameter Xq^, the analogous composite parameters 
 
 ^2eq2U't/^ 
 
 ^2,,= 
 
 
 (l_u2f' 2^ (1-U2) 
 
 (1-U2)^ 
 
 CO = 
 
 + 1 
 
 K - 1 
 
 By equations (6.9) and (6.11 ) represented in the form 
 
 f-^k-^ + 12A-L + Ih^ 
 ^2.0 = ^pV 
 
 1260 
 ^2q = ^2^(Mi + NiAi)2 
 
 A 
 
 there are constructed for the given value of T^ auxiliary graphs of 
 the dependence of X^ °^ ^2 ,3 for various values of X^. and for the 
 
 dependence of Xg/, °^ '**-2q f O-f various values of Xg • '^h® magnitudes 
 taken as initial in the computation hy formula (8.5) are determined from 
 
22 
 
 NACA TM No. 1229 
 
 the initial data. If U / for Xq = 0, there are taken the values (X2)o = 
 and (a/5)o = !• If U = for x^ = 0, there can be taken as the 
 initial values the values of (X2)o ancL (X2^)o = (V5)o (^2^0 according 
 to figure 2. By the values of the functions •5'(x), that is, X2q(2) of 
 the first approximation using the initial value (a/5)q (for the succeeding 
 approximations it is ccmvenient to use the initial, values of ^2a'» ^2^^) 
 
 of the first approximation is found with the aid of the graph. Further, 
 "by equation (8.9) there is computed 0(x), that is, >^20(^) °^ "^^® first 
 approximation, making use of \2(x) of the first approximation and (a/6)q 
 (in the succeeding approximations there is used the function X2(x) following 
 and X,2. (x) preceding) . From the values of ^29^^) ^"^^ "^2^^^ ^^*^ *^® ^^^ 
 of the graph there is obtained X2a(^)' ^ those cases where there is a 
 
 considerable change in the ratio A*/i3 (or the function T^(x) ) the 
 computation must be conducted over segments. The required data for each 
 succeeding segment are taken equal to the corresponding values obtained 
 at the end of the preceding segment. The local Nusselt number (that is, 
 the coefficient of heat transfer q^Ao* reduced to nondimensional form) 
 and the coefficient of friction are found from the equations 
 
 n K. 
 
 N = 
 
 ^w^ 
 
 K^o* 
 
 = %r^ 
 
 2t 
 
 Cf = 
 
 w 
 
 _ 2 s n-1 
 
 PaPo 
 
 E„ 
 
 ■•w 
 
 1 + I (k - l)Meo^ 
 
 1 + 1 (k -1)M„' 
 
 (1 
 
 -u2)^-^ 
 
 Bl 
 A 
 
 
 n 
 (1 
 
 K 
 
 U 
 
 s 
 
 (8.10) 
 
 (8.11) 
 
 *In particular for T^ = 1 formula (8.5) with (^2)0 "^ ^ gives 
 
 8^ 
 
 00 
 
 l-£ 
 
 0.U7 
 
 r,c-i 
 
 1-^+ -^ 
 
 U°(l-U2) 2 
 
 
 
 U^-l(l -U2) 
 
 2 K-1 ,_ 
 dx 
 
 (c = 6) 
 
 which agrees with the equation of L. G. Loitslansky and A. A. Dorodnitsyn 
 for the computation of the laminar layer without heat trajisfar (reference 6). 
 In the absence of heat transfer and for small Mach numbers we again 
 obtain from this the quadrature 
 
 (uAj«)6 Jo V^"/ \ n / 
 
 earlier derived by us on the basis of the method of Karman— Pohlhausen for 
 the laminar layer in an incompressible fluid. (See Tekhnlka Vozdushnogo 
 Flota, No. 5-6, 19h2.) 
 
NACA TM No. 1229 23 
 
 For grnnll Mach numbers equations (8.5) ajid (8.9) assume the form 
 
 
 e^E = ^ f (^)" 2Bi & - 1) ^ X ^ (8.13) 
 
 (TjTo - 1)^(uAJ„)2 ^0 W V^O / U^ A 
 
 ■^00 
 
 9. Dependence of the Reynolds Kumher R02 on 
 the Parameters of the Flow 
 
 TaJclng account of the fact that according to the equation of state 
 
 P02 P02 
 
 = we represent the parameter Rno in the form 
 
 POI Poi ^ 
 
 n— *r r- ^^-, r-n 
 
 2 2(k-1) 
 
 . P02 ^ /2ioLpoi , , 
 
 ^02 = ^01 :; — ' where Eqi = (9-1) 
 
 The parameter Eq^ is expressed directly in tenna of the Reynolds 
 
 UooLPoo U„ 
 
 number 'R„ = and the Mach number M^ = — = of the approaching 
 
 1^00 aoo 
 
 flow. From equations (3. 3) to (3.5) ve find 
 
 K+1 _1 
 
 2 
 
 (9.2) 
 
 In the case of subsonic (subcritical) velocities we have 
 Eq2 = Rqj = Eqq. For supersonic velocities the ratio P02/P0I ^^ 
 
 found from the condition of a line of a flow passing through an oblique 
 shock wave (or a head wave) at the leading edge of the body. Considering 
 each surface of the profile separately we denote by Pq the angle which 
 the tangent to the surface of the airfoil at any point makes with the 
 direction of the velocity of the undisturbed flow and, by 9 the angle 
 which the normal to the surface of discontinuity makes with the same 
 direction. From the equations of the oblique shock wave we obtain 
 
 Eqi = R„ =^ (1 - U„2) ""-^ ^ R„ 1 + 1 (k - l)Moo' 
 
 Uqo L ^ 
 
21^ NACA TM No. 1229 
 
 p / 1/2(k + 1)M„co32cp V-VSK 2 2 .-^r / > 
 
 -^ = 5 ( M«,^co82cp i] (9.3) 
 
 Pqi \1 + 1/2(k - 1)M<„ cos2cp/ U + 1 
 
 1/2(k + l)M„,^sin 2cp 
 
 tan (cp + Po) = i — p 5" (9.^) 
 
 "^ 1 + 1/2(k - 1)M<„ cos'^q) 
 
 In the case of a head wave in front of the "body the direction of the 
 velocity after the discontinuity (for the flow line at the profile) may 
 "be considered to coincide vith the direction of the velocity "before the 
 discontinuity; in equation (9.3) there is in this case to "be substituted 
 9=0. 
 
 10. Boundary Layer in the Flow of a Gas with Axial Symmetry 
 
 For any axial flow a'bout a body of revolution the integral relations 
 of the impulse and energy have the following form: 
 
 — / ^ pu^r dy - U ^ / pur dy = - ^ 5yr - T^ (lO.l) 
 dx Jq ^0 dx "^ 
 
 ^J Cppu(t* - to*)r dy = -^^ (10.2) 
 
 In these equations the usual simplifications were made; x is the distance 
 along the arc of the meridional section, d the distance along the 
 normal to the surface, and r(x) the radius of the cross section of the 
 "body of rotation (the change of the radius vector within the boundary 
 layei is neglected). The boundary conditions of the problem and also the 
 assumptions with regard to the external flow are taken to be the same 
 as in section 1. Setting up expressions for T, p, p, and [i as in 
 sections 2_ and 3 and introducing the new independent variable t] by 
 formula (^.9) we obtain the integral relations of the momenta and energy 
 in the variables x. ti: 
 
NACA TM No. 1229 
 
 25 
 
 dx "^ U 
 
 H + 2 + (H + 1) 
 
 U2 
 
 1 - U^ 
 
 U»(Tv - 1) 
 U(l -U2) 
 
 0Hqi + 
 
 yit - 
 
 d = 
 
 W 
 
 POO^ 
 
 ,U2 
 
 (10.3) 
 
 d9 U' - r^ *0 
 
 + -=r + r=r^ + r 
 
 dx u ^ "'^O* 
 
 I* 
 
 %, 
 
 PQO^S^O 
 
 ^ X 
 
 _ d 
 
 = ^, i3 = -.and so forth) (lO.U) 
 
 (L Is a characteristic dimension, for example, the length of the body 
 of rotation. ) Restricting ourselves to the case P = 1 and taking the 
 expressions (6.1 ) and (6.2) we obtain a closed system of equations 
 (10.3), (10.1;), (6.9), and (6.11) the solution of which we write (for the 
 case where there is no shock wave) In the fona 
 
 ^^E 
 
 00 
 
 1-^ 
 
 uC(i_u2) 2-2 L 
 
 1-^ + 
 
 2. , K 
 
 C + r T^-^tr^-l (1 - U2) ^ ^-1 r^o d5E 
 
 e^E, 
 
 00 
 
 U2(T^ - 1)V 
 
 c» + 
 
 where 
 
 (10.5) 
 
 =s n-1 
 
 _o.K-l 
 
 T^^-"2BiU(l - U2) (T^ - D^r^ ^ dx 
 
 2^2 e .,. 
 
 ^ 
 
 A 
 
 (10.6) 
 
 j_FEooU^(1 
 
 1-^ 
 
 U2) 2r2 
 
 x=xo 
 
 C = [02rooU^(Tw-1)^?^]_ _ 
 
 x=xo 
 
 The method of computation does not differ from the case of the 
 two-dimensional flow. For the coefficients of the heat transfer and 
 friction the equations (8.IO) and (8.II) remain valid. In the case of 
 the internal problem (flow in nozzles) the Nusselt number and the 
 friction coefficient are determined by the equations 
 
 NOO = 
 
 ^oo'fco* 
 
 = T. 
 
 ^ n-1 
 
 w 
 
 (1 . u2)K-l |, Cfoo 
 
 POO^ 1^00 U5 
 
26 NACA TM No. 1229 
 
 vhsre Xqq is the coefficient of heat conductivity corresponding to 
 the tesiperature Tqq. 
 
 III. TURBULENT BOUKDAEY LAYER IN TEE PRESENCE OF HEAT 
 TRANSFER BETWEEN TEE GAS AND WALL 
 11. Fundaniental Aeauaiptlons 
 
 The functions H, Bsjt, t^, and g^ entering equations (4.15) 
 
 and {h.l6) are determined by equations (4.12), (k.lk), (U.5), and (4.7) 
 which express them as functions of S and 6 through the medium of the 
 velocity profile and the stagnation temperature— drop profile. The 
 present state of the problem of turbulence does not permit representing 
 the velocity profile (and also the temperatujre profile) by a single 
 equation v/hich holds true from the wall to the boundary—layer limit. 
 The fundamental dynamic and thenaal characteristics of the turbulent 
 layer can nevertheless be computed with an accuracy which is sufficient 
 for practical purposes. A fortiuiate property of equations (4.15) 
 and (4.l6) v/-hich can be predicted on the basis of the results with 
 respect to noncompresslble fluids is that the functions H and Hrp 
 change very little oyer the length of the turbulent layer and the 
 functions t^ and q^ connected with -8 and 9 by the equations are 
 little sensitive to the actual conditiona which prevail at a given section 
 of the boundary layer. Hence H and H^ji (and also magnitudes analogous 
 'to them) can be taken as constant over x and the relations between t^ 
 and -a (the resistance law) and between q_y and 9 (the heat— transfer 
 law) can be set up starting from the assumption that the conditions at 
 the given section of the boundary layer do not differ from the conditions 
 on the flat plate. On the basis of the derivation of these supplementary 
 equations we assume the simple scheme of Karman according to which the 
 section of the boundary layer is divided into a purely turbulent 
 "nucleus of the flow" and a "laminar sublayer" in Immediate contact 
 with the wall. In the latter the turbulent friction and the temperature 
 drop are small by comparison with the molecular. We assume that in the 
 turbulent "nucleus" the frictional stress is expressed by the formula 
 of Prandtl: 
 
 ^ - "''O <--' 
 
 where I is the length of the mixing path. In other words, in 
 equations (l.l) and (l.S) we set € = pZ^ ^. It follows directly from 
 
NACA TM No. 1229 
 
 27 
 
 this In view of the fact that the turbiilence ass^xnption of Prandtl gives 
 X^ = c € that the expression for the heat transfer is 
 
 , dT ,2 d-u dT 
 
 (11.2) 
 
 The thickness 5^^ of the laminar sublayer of the dynamic boimdary 
 layer, ecju'al for P = 1 to the thickness Ayj of the thermal sublayer, 
 is determined by the critical Eeynolds number (the Karman criterion) 
 
 Ut5 
 
 ^I^=a^ (a8Sll.5) 
 
 (11.3) 
 
 vhere u^ is the velocity on the boundary of the l&minar sublayer, p^ 
 and Hy aj*e the density and viscosity at the wall. 
 
 12. Derivation of the Resistance Law 
 
 Assuming that as in the case of the noncompressible fluid a linear 
 variation of the velocity in the laminEir sublayer is permissible on 
 accoiint of the flmfl.11 thickness, we have 
 
 ■^w = ^^^ 
 
 5 
 
 (12.1) 
 
 7l 
 
 From equations (U.3) and (12. l) we obtain 
 
 
 (12.2) 
 
 In equation (l2.2) we pass to the vEiriable r\. Near the wall on account 
 of the smallness of the terms u^ and t*/tQ* we have 
 
 7= (l-u2) 
 
 ^0 
 
 hence 
 
 K 
 1^ 
 
 1_U2.(T„-1) ^-Q 
 
 6y,»i„(i-n2r-i5j 
 
 1-K 
 
 dTi » T„(i -u'^r T) 
 
 (12.3) 
 
28 NACA TM No. 1229 
 
 vhere 5^ = A^ is the thickness of the laminar sublayer for the varlahle t\, 
 Further, 
 
 Substituting these expressions in equation (l2.2) we obtain 
 
 ^=^1,- (-5^ = $) (12.1.) 
 
 For the fundamental parameters of the dynamic layer there is here 
 introduced the notation 
 
 E^ = U^02. ^ - T-^^— rTTP (1-^)^^' ^^ (12.5) 
 
 /Vpo2 Tw ^ 
 
 Equation (l2.l) is with the aid of equations (3'2) and (U.9) transformed 
 into the form 
 
 3k 
 
 .2 „ ^2 
 
 °^ V^^-^ [l - u2 + (T^ - 1)(1 - tVto^P 
 
 Since for small t\ the terms u^ and t*/tQ* are small and the mixing 
 path 2 = ky (k = 0.391) vhere the coordinate y ^s expressed according 
 to equation (l2.3), the "generalized" mixing path I near the vail is 
 a linear function of t): 
 
 ^ = kTi-^(l-U^) (12.7) 
 
 In deriving the resistance lav in an incompressible fluid a linear 
 mixing— path distribution and a constant frictional stress are assumed 
 for the entire section of the boundary layer, from the wall to the 
 outer boundary. Actually the mixing path increases at a considerably 
 slower rate than according to the linear lav and the friction drops to 
 
NACA TM No. 1229 • 29 
 
 zero as the outer boundary of the layer Is approached. The assumptions 
 made act In opposite directions and lead to a satisfactory relation 
 "between the parameters E „ and ^ . 
 
 Carrying over this fundamental idea of the logarithmic method into 
 our present theory we set t = t in equation (l2.7) and assume 
 
 the linear law (equation (12.7)) for the entire section of the boundary 
 layer. We thus assume that as in the case of the noncompressible fluid 
 there will be a mutual compensation of the errors committed in the distri- 
 bution of T and I. Integrating equation (12.6) between t\ and 6 
 we obtain the approximate velocity profile: 
 
 ii = 1 + - m II (12.8) 
 
 U k^ 5 
 
 From equations (l2.l) and (12. U) we obtain the velocity at the boundary 
 of the laminar sublayer 
 
 2 _ g 
 U " 5 
 
 The condition of the equality of the velocities of the turbulent and 
 laminar flows on the boundary of the sublayer gives 
 
 Eg = C^T^^e^^k^ MRg = U5Eq2, C^ = f ^~^°' = 0.326J (l2.10) 
 Making use of the velocity profile (12.8) in equations (1+.12) we obtain 
 
 Eliminating from equation (12. lO) and the first of relations (12. ll) the 
 auxiliary variable 6, we obtain the resistance law: 
 
 E, = CiT/e^^ C"ft) ^^^'^^^ 
 
 We obtain incidentally also the approximate expression for the 
 parameter H : 
 
 5 1 / » 
 
 H = — = (12.13) 
 
 ^ 1 - 2/k^ 
 
30 NACA TM No. 1229 
 
 13. DerlTation of the Heat-ITransfer Law 
 
 In this section we shall give a generalization to the case of a 
 gas moving with large yelocities of the heat-transfer law earlier derived 
 ty us (reference h) for an inconrpressihle fluid. 
 
 We construct the function 
 
 dt* 
 
 q*= X^= q + iru (I3.I) 
 
 dy J 
 
 For the turbulent nucleus of the flow we have 
 
 dt* dt* „ ,2 du dt* 
 
 ^^ ^ U.U an, ,kf au ax (to r,\ 
 
 1 = H TT" = Cp€ = CppZ ;- (13.2; 
 
 '' dy i' dy ^ dy dy 
 
 Transforming this equation to the variable t] we obtain 
 
 du dt* ~2 ^ jf (^ _ ^2f-'i- 
 
 ^^ <^^ ' [l-u2 + (T^-l)(l-t*/to*J^ 
 
 (13.3) 
 
 Near the wall the function q* behaves like q, that is, differs little 
 from the constant value q^., and the mixing path I depends linearly 
 
 on f] according to equation (12.7). The common mechanism of the transfer 
 of heat and the transfer of the momentum in the flows along solid waJJLs 
 provides a basis in the derivation of the law of heat transfer for 
 assuming as before a constant value q* = q„. and the linear law (equa- 
 tion (12-7)) for the entire thickness of the thermal layer. Substituting the 
 sxpression for du/dT) obtained from equation (12.8) and integrating 
 equation (13- 3) from ti to A we obtain the approximate profile for the 
 stagnation temperatures 
 
 = 1 f —^ In J (13.4) 
 
 r Is 
 w 
 
 *0* c_,to*T„kC 
 
NACA TM No. 1229 3I 
 
 From equation (13.I), assuming a linear distribution of the stagna- 
 tion temperatures in the laminar sublayer, we find 
 
 <lw = CpU„ 7— (13.5) 
 
 where t^* is the stagnation temperature at the boundary of the laminar 
 sublayer . 
 
 As the fundamental thermal characteristics of the boundary layer 
 we introduce the following parameters: 
 
 K 
 
 % = U0EO2, C^ = _ - _ (1 - U2) Nqo = —— ] (13.6) 
 
 ^^ ^ w \ ^00^0 / 
 
 From equations (l^'^) and (13.5) we obtain the stagnation temperature 
 on the boundary of the laminar sublayer 
 
 t* 
 
 r^ = T (13.7) 
 
 Equating the stagnation temperatures on the boundary of the laminai- 
 sublayer and making use of condition (12. U) we obtain 
 
 % = CiT^^exp{H^)kt, (E^ = UAR02) (13.8) 
 
 Substituting the expressions for u/U and t /tQ according to 
 equations (12.8) and (13.U) in equations (U.I3) and (k.lk) we obtain 
 
 e /, 1 , Al 1 2 A* 1 /-.^ «\ 
 
 A = (i + n 1^1 rj -: ;; — > — = — (13.9) 
 
 ^ V n VkC^ ^2^^^' A k^^ 
 
 A 
 
 From equations (l2.10) and (13.8) it follows that | = exp(k^^ - kt; ) 
 so that we have 
 
 ^i \ hJ 
 
 i-til^-—^ (13.10) 
 
32 NACA TM No. 1229 
 
 Eliminating from equations (13.8) and (13.IO) the auxiliary parameter A 
 we obtain the heat— transfer law 
 
 •■(' 
 
 Kq = CiT^^ fl - -^ exp k^^, (13.11) 
 
 We obtain incidentally alao the approximate expression for the 
 parameter Hrp: 
 
 iMt 
 ■ % = (13.12) 
 
 l/k^d - 2/kCT) 
 
 l4. Solution of the Equation of the Turbulent 
 Dynamic Boundary Layer 
 We represent equation (k.l^) In the form 
 
 We make the change in variables (reference T)^ 
 
 z = e^^d -2M)k2^^ (11^.2) 
 
 Differentiating this relation with respect to x and using equation (12,12) 
 we obtain 
 
 dx \K^ dx T^ dxy \^ 1 _ 2/k5 + 2/k^^y 
 
NACA TM No. 1229 
 
 33 
 
 From equations (1^4-. l) to (1^4-. 3) we obtain 
 
 dz U»K(H + 1) U«E(Ty - 1) A* ^ T^» ^^™Qg^ ,, r^\K-l /., , n 
 
 <i^ U(l-U2) U(l-U2) ^ 
 
 w 
 
 ClT, 
 
 n+1 
 
 w 
 
 The ma^itudes H and E which change little along the "boimdary 
 layer are aesiimed constant with respect to 5. If A /a is considered 
 as a known. function of x then equation (ll|.4) is a linear equation with 
 respect to z. 
 
 A* 
 Assuming a constant mean value of the ratio — = h over a certain 
 
 interval "xq < ^ < ^1 ^^^ ^^^ first approximation we may for the entire 
 
 A* fi* 
 turbulent layer assume h = -r- = -5- = H, which holds for the plate ), we 
 
 obtain the solution of equation (lU.V) in the form' 
 
 -^vC 
 
 z = 
 
 l/ (i-ug) 
 
 ^nKijC 
 
 C .+ -^i- / (1 - U2)«-1 d3c 
 
 Cl 
 
 ^ 
 
 /(1-U2) 
 
 =;0\C 
 
 where 
 
 (li^.5) 
 
 c = 
 
 zT^"^ /(I -U2)-^] , c = K(H + 1) + K(T^ - l)h 
 
 x=xo 
 
 The constants must be taken equal to 
 
 K = 1.20, H = l.U, h = l.k, (Cl = 0.326; k = 0.391) 
 
 'See footnote (3) on page 19 . 
 
34 
 
 NACA TM No. 1229 
 
 After computation of the dynamic and thermal layer new mean values 
 
 HmRo 
 
 of the variahlee K, H, h = -^ — can he found and, if the deviations 
 
 ■Ht3 
 
 from the assumed values are considerahle ,the computation is repeated. 
 
 If J for greater accuracy, the computation is conducted over segments, the 
 
 values of the constants for each succeeding segment are determined "by 
 
 the values of E^, ^, Kq, and t^m ohtained at the end of the preceding 
 
 segment. In integrating from the point of the transition of the laminar 
 
 into the turhulent state, the magnitude (z):n._Y^ is determined from the 
 
 condition of equality of the initial value of E^ to the value of E^ 
 at the end of the laminar segment. In integrating from the leading 
 edge, C = 0. By equations (l2.2) and (lij-.2) the auxiliary graphs of the 
 
 functions log (r^%'^^1~'^) and log z as functions of k^ can be 
 constmcted once for all. 
 
 The local friction coefficient is found from the equation 
 
 2t, 
 
 w 
 
 PooU« 
 
 1 + - 
 
 T L 2 
 
 w 
 
 J PoiW 52 
 
 For small Mach numbers equation {ik.^) assumes the form 
 
 nK/ 
 
 (VTo)^(uAj„) 
 
 C + 
 
 (li+.T) 
 
 where 
 
 E 
 
 UooLp 
 
 E. = ^ ^E, 
 
 00 
 
 c = 
 
 \w W 
 
 X=Xr 
 
 15. Solution of the Equation of the Turbulent Thermal Layer 
 We represent equation (U.I6) In the form 
 
 dEe ^ 1 ^^0* 
 
 dx 
 
 to'' dx ® ^^tTw °2 
 
 ■'O 
 
 Too/ 
 
 (15.1) 
 
NACA TM No. 1229 
 
 35 
 
 Carrying out the change in variables (reference k) 
 and making use of the heat— transfer law (13.I) we find 
 
 (15.2) 
 
 dZn 
 
 1 ^\ 
 
 1 dto' 
 
 i"' "^^ T; ^ ^T + ^ ^ -^ -T = Kt 
 
 k2 
 
 die 
 
 k^CiV 
 
 m n+1 
 
 URood -U2) 
 
 K 
 
 K-1 
 
 (15.3) 
 
 K,p = 
 
 1 - l/k^T 
 
 2, 2; 
 
 1 - 2/k^rp + 2/k'^^rp' 
 
 The solution has the form 
 
 Zm = 
 
 T - - 
 
 T^"^(T^-1)^ L ^1 
 
 2 t^ 
 
 C« + / T ^ (Tv-l) — (1-ir) djc 
 
 Cl JxQ kt 
 
 il^.h) 
 
 C = 
 
 
 =^ 
 
 x= 
 
 ^The equation of energy (I5.I) in connection with the relation 
 "between E@ and ^m in the form (13.II) in the case T = Constant is 
 
 an equation with separable veLriahles so that together with the solution 
 in the form (15.^) we can use its accurate solution 
 
 K 
 
 k^Roa 
 
 ^ u 
 
 exp(k^rr)(k^ - 3) + 2gi(kCT) = C + „^T , , 
 
 JL (1-U2)'^-I<ii 
 
 where 
 
 Kt ^u 
 
 ^ i(k^T) = / ^ du, C = expCk^T^^^^T " 3) + 2'bim^)\ 
 
 x=Xo 
 
 (for 5cq = Oj C = 0) 
 
36 
 
 NACA TM No. 1229 
 
 'Ihe constant Km must be taken equal to 1.15 (with subsequent check 
 by the results of the computation of the thermal layer). The magnitude 
 (tm)_ _ is determined by equation (l3.ll) from the condition of the 
 
 equality of the value of Eg at the start of the turbulent region to its 
 value at the end of the laminar section. In integrating from the leading 
 edge, C* = 0. The aiixiliary graph of the function log zij against k^j 
 
 can be constructed once for all. 
 
 The local heat transfer is found from the equation 
 
 N = 
 
 Aoot/^ 
 
 02^ 1 r 1 / ^ 2 
 JJ—^ 1 + i (k -1) M«. 
 
 ^^T T^ L ^ 
 
 n 
 
 (i-u2) 
 
 K 
 
 K-1 
 
 (15.5) 
 
 For small values of the Mach number equation (l5.^) assumes the form 
 
 ^T = 
 
 (T^/To)^T(T^/To - 1)^ 
 
 C» + 
 
 KpRk^ 
 
 Cl 
 
 ^0 
 
 x/T \ 
 
 4) 
 
 nErjT-n— 1 
 
 .\^ U 1 
 
 & - iP ^ 
 
 H 
 
 d5c 
 
 (15.6) 
 
 l6. Determination of the Profile Erag for Subsonic Velocities 
 
 The drag of a ving of infinite span (over unit length of span) is 
 obtained from the momentum theorem in the form 
 
 Q = / Po<^cx>(Uoo - ^J 
 
 ^^=^oo-~Z^l(-^j^. 
 
 = Poo^» 
 
 (16.1) 
 
 viiere \ ^ denotes the sum of the momentum-loss thicknesses referred 
 
 to the upper and lower surfaces and computed at a great distance from 
 the wing where 6 — ^oo and U — >Uoo. For the drag coefficient we have 
 
 2Q 
 
 ■'xp - 2 
 
 PoJJoo L 
 
 = 2y v^cc [l + I (k - l)M, 
 
 K-1 
 
 (16.2) 
 
NACA TM No. 1229 37 
 
 The problem confllsts In expressing iSoo i^i terms of the dynamic and 
 thermal characteristics of the houndary layer at the trailing edge. 
 Since in the wake behind the body j^ = and q^ = 0, equations (U.l6) 
 
 and {h.l'j) for the wake assume the form 
 
 ^5-)'- 
 
 d-d -n* f B. + Jfi „ D I- , ^ ^ 
 
 di xj t^* djc 
 
 We introduce the notation 
 
 ^.Ela.A^e^o (16.1.) 
 
 ,_, H + U^ Tv - 1 / ^ ^ 
 
 G(x) = ^ + -^^7% (16.5) 
 
 1 - IF 1 - IT ^ 
 
 and represent equation (16.3) in the form 
 
 l^ = _(2 + G)i-lz.^ (16.6) 
 
 ^ dx dx Uo. 
 
 Integrating this equation with respect to x from the trailing edge 
 (denoted by the subscript l) to i = 00, we obtain 
 
 ln=S2.= (2 + Gn)ln=i+/ ln=-dG (16.7) 
 
 For U, s; 0, ^ = 1 the function G^x) goes over into H(x). For 
 
 ein incompressible flxiid however the hypothesis of Squire and Young 
 (reference 8) on the linear character of the dependence ln(U/croo) on H 
 holds : 
 
 ln(uAjJ _ In(UiAJoD) 
 H« - H ^ - H, 
 
38 NACA TM No. 1229 
 
 Making the analogous assumption 
 
 ln(u/Uoo) ln(Ui/aoo) 
 
 G«,-G 
 
 G-»o — Gr]_ 
 
 (16.8) 
 
 we obtain"^ 
 
 (16.9) 
 
 We write down the expression for G-|_: 
 
 % + Ui2 T - 1 
 
 Gi = ^p- + --h^ -^ \ (16.10) 
 
 1 - U^^ 1 - U^ ^1 
 
 It is easily seen that Hoo = 1 and Hm^ = 1, hence 
 
 G^= ^ + " . ^ (16.11) 
 
 * 1 - U„^ 1 - U„2 ^„ 
 
 ^The same result can "be arrived at from the following elementary 
 considerations. Equation (16.7) may he represented in the form 
 
 In ^ = (2 + Gi) In rA + (Goo - Gi) In =» 
 ■^1 Uc Ugg 
 
 where Uj^ is a certain mean value of the velocity IJ that lies hetween 
 U2_ and IJ„. For the usual profile shapes however the ratio Ui/Uoo Is^ 
 in general, near unity and since the magnitude 2 + Gri exceeds the 
 magnitude Gw — _Gi hy several times, therefore for any choice of the 
 mean value of U^^ the relative error in the determination of i3^ Is 
 not large. Taking the geometric mean Uj^ = j/u^Uoo we again arrive at 
 equation (16.9). 
 
NACA TM No. 1229 39 
 
 Eeplacing In equation (16.9) Grj_ and G„ by their values, we obtain 
 finally 
 
 ?„ = ^t( =ri) (16.12) 
 
 '.(» 
 
 K + 5 Hi + 1 ^^2 u«2 ^ / _ i)ht 0i/^\ (t^^ - l)e„/^% 
 
 Tt = -^ + -— + TT + — T + # 
 
 ^ nTT^T_TT'-2 -, XT '- '- TTT^ 
 
 1-U-," 1-U„ "^ 1-U-," " 1-Uc 
 
 -"1 -^ - ^00 X - u^ 
 
 '00 
 
 From equation (lO.il) it follows that 6U(Ty — l) = Constant, hence 
 
 (^ - 1) e„ = (T^ - i)ei -A (16.13) 
 
 ' ^ ' 1 u, 
 
 CO 
 
 For small Mach numbers equations (l6.2), (16.I2), and (16.I3) assume 
 the form 
 
 
 
 00 
 
 --'©■ &-^ 
 
 ^- = ^Un"J' i^-v ^- = (v-^'^^ 
 
 17. Boundary Layer in a Gas Flow with Axial Symmetry 
 
 For the turbulent flow about a body of rotation the equations (lO.l) 
 and (10.2) in the variables x, y and the transformed equations (IO.3) 
 and (10. U) remain valid. Eestricting ourselves to the case of the 
 Prandtl number P = 1 and introducing the parameters E^, ^, Eq, and tj^ 
 
 we represent the integral relations in the form 
 
ko 
 
 NACA TM No. 1229 
 
 dx U(l-U2) 
 
 _ — ^^ T Rs + — ^ 
 
 ^^^o(i 
 
 U2f-1 
 
 (17.1) 
 
 dE, 
 
 iz 
 
 cLt^* 
 
 ."iB,._-i--^E, = ^i-i5Hoo(l--^)«- 
 
 to* da 
 
 (17.2) 
 
 T w 
 
 Making use of the drag law equation (12. 12), the heat-tranefer law 
 equation (13.II), effecting the change in variahlea 
 
 assuming the little changing magnitudes H, K, K^ constant with respect 
 
 to X and also a constant mean value for the ratio — = h, we obtain 
 5 -a 
 
 a system of linear equations the solution of which has the form 
 
 z = 
 
 /(1-ug)' 
 
 ^nEyC^ 
 
 C + 
 
 -"o^^^^'-^-^^^a-i^F^ 
 
 ^1 Jxo /(1-U2) 
 
 ^9\c 
 
 (17.3) 
 
 C = 
 
 z^v^ 
 
 nK 
 
 /(1-U2)° 
 
 x=xo 
 
 Zrp = 
 
 ^^nKT^T(T^ _ 1)% 
 
 C + 
 
 ^1 ^io 
 
 _^n&i.^l(^ - 1)^ -1(1 - Vi^)^-h^ d5 
 
 K 
 
 z^ff. 
 
 5„'^(T„-l)^r^' 
 
 -'X=Xr 
 
NACA TM No. 1229 1+1 
 
 For the frictlonal stress and the Nusselt number, equations (lk.6) 
 and (15.5) remain In force. In the case of the internal problem 
 
 ^00*0 "T ^^ POO^ 
 
 K 
 
 NOO = ^^ = 77- r ^00^(1 -^r\ C,oo = -^ = ^ :^ (1 - T^^ f-^ 
 
 (17.5) 
 
 The theory presented, in particular the integral relations of the 
 momenta and energy established in part I, permits determining the 
 thermal and dynamic characteristics of the boundary layer at a curved 
 wall in the most general cases, that is, in the presence of ertemal 
 and internal heat interchange. The computation of the boundary layer 
 by the equations derived in parts II and TTT on the assumption of the 
 Prandtl number P = 1 permits finding directly for arbitrary Mach numbers 
 (excluding the interval from *^ = ^^ to M^ = l): 
 
 (1) The coefficients of the heat transfer from the wall to the gas 
 for a given maintained temperature of the wall through heat supplied 
 outside the body and the coefficients of heat transfer from the gas to 
 the wall, that is, required for maintaining the heat conduction within 
 the body at the given temperature of the wall. 
 
 (2) The distribution of the frictlonal stress eO-ong the wall and 
 
 the profile drag of the wing (in the case M„ < M^o ) for arbitrary 
 
 cr 
 
 ratio of the stagnation temperatures and those of the wall. 
 
 For small velocities the obtained equations express the dependence 
 of the heat transfer and the drag on the ratio of the absolute tempera- 
 tures of the flow and the wall (the effect of the compressibility and 
 the change of the physical constants due to the heat interchange). 
 
 In conclusion we give the results of computation of a single example. 
 In figure 3 is given the distribution of the velocities of the external 
 flow for the supersonic flow about a body with two sharp edges. The 
 contour of the body and the position of the discontinuity are also shown. 
 The flow was confuted by the method of Donov (reference 9). In figure h 
 are given the curves for the Nusselt number N which assure the uniform 
 cooling of the surface up to the temperature T^. = 0.25Tqq for 
 
 E„ = 15 X 10°, Meo = 2, and Mo, = 6 for the laminar (lower curves) and 
 turbulent (upper curves) regimes. 
 
 Translated by S. Eeiss 
 National Advisory Committee 
 for Aeronautics 
 
42 NACA TM No. 1229 
 
 REFERENCES 
 
 1. Dorodnitsyn, A. A.: Boundary Layer in a Compreasi'ble Gas. Priklalnaya 
 
 Matematika i Meldiajiika,. vol. VI., 19^2. 
 
 2. Frankl, F. 1., Chrlatianovich, S. A., and Alexseyev, E. P.: Fundamental a 
 
 of Gas Dynamics. CAHI Report No. 36k, I938. 
 
 3. Kalikhman, L. E. : Drag and Heat Transfer of a Flat Plate in a Gas 
 
 Flow at Large Speeds. Prikladnaya Matematika i Mekhanika, vol. IX, 
 19^5. 
 
 k. Kalikhman, L. E. : Computation of the Heat Transfer in a Turtulent 
 Flow of an Incompressible Fluid. NII-I Report No. h, 19^5. 
 
 5. Lyon, H. : The Drag of Streamline Bodies. Aircraft Engineering ^ 
 
 No. 67, 193^. 
 
 6. Dorodnitsyn, A. A., and Loitsiansky, L. G. : Boundary Layer of a Wing 
 
 Profile at Large Velocities. CAHI Report No. 551^ 19hh. 
 
 7. Kalikhman, L. E. : New Method of Computation of the Turbulent Boundary 
 
 Layer and Determination of the Point of Separation. Doklady 
 Akademii Nauk SSSR, vol. 38, No. 5-6, 19^3. 
 
 8. Squire, H. Bo, and Young, A. D. : The Calculation of the Profile 
 
 Drag of Aerofoils. ARC, R & M No. I838, 1938. 
 
 9. Donov, A. : Plane Wing with Sharp Edges in Supersonic Flight. 
 
 Izvestia Akademii Nauk SSSR, Ser. Matematichoskaya, 1939, 
 pp. 603-626. 
 
NACA TM No. 1229 
 
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