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NACrrlYiA 21
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1229 HEAT TRANSMISSION IN THE BOUNDARY LAYER* By L. E. Kalikhman Up to the present time for the heat transfer along a curved wall in a gas flow only such problems 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 Mach numbers (the ratio of the velocity of the gas to that of sound) and small temperature drops between the flow and the wall, the velocity field does not depend on the temperature field. In 1941 A. A. Dorodnitsyn (reference 1) 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 temperature drops and large velocities use is made of the abovementioned 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 numbers and temperature drops. The method proposed is applicable both for Prandtl number P = 1 and for P 1 1. "Gazodinamicheckaya Teoriya Teploperedachi." Prikladnaya Matematika i Mekhanika, TomX, 1946, pp. 449474. NACA TM No. 1229 I. FUNDAMENTAL RELATIONS FOR THE LAMINAR AND TURBULENT BOUNDARY 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 twodimensional gas flow (fig. i). 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, y, where x is the distance along the arc of the profile from the leading edge, y is the distance along the normal, T is the absolute temperature, p the pressure, p the density, p the coefficient of viscosity, X the coefficient of heat transfer, e the coefficient of turbulence exchange, Mt the coefficient of turbulent heat conductivity, Cp the specific heat, and J the mechanical equivalent of heat. U2 T* = T +  2JCp is the stagnation temperature r p is the velocity of sound K = P cv is the adiabatic coefficient gcp 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 m, 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 fT No. 1229 (au Pu ( V+ ) a] S(pu) + (p~) = o V  ( CT* are^ a+v + v / p = pKT, p = oTn (1.5) where R is the gas constant, and C and n are constants. In the solution of the system equations (1.1) to (1.5) we start out from considerations on the dynamic and thermal boundary layer of a finite (but variable) thickness. The flow outside the dynamic boundary layer approaches the ideal (nonviscous) flow, nonvortical in front of the shock wave and, in general, vortical behind the wave. Equation (1.4) shows that the pressure is transmitted across the boundary layer without change, that is, the pressure po(I) 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 boundary the total energy 10 is constant: u22 u2 U= U2 JcpT + = JcpT., + 2 = JcpT + From this it follows that the stagnation temperature T* thermal boundary layer has a constant value T= TOO =  outside the Thus the flow outside the thermal boundary layer for small velocities Is assumed nearly isothermal while for large subsonic velocities isentropic. (1.6) + v al= C +  CV) a1 \ a ST) TY V FY) (1.7) NACA TM No. 1229 For supersonic velocities the entropy is constant up to the shock wave while after the wave the entropy is constant along each flow line of the external flow about the profile but variable from one flow line to the next. The boundary conditions of the problem are: u = v = O, T* = Tw for y = 0 (1.8) where Tw = Tw(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 IJ = 0. Further u = U(x) for y = By(x) (1.9) T = TOO for y = Y(x) where 8y and Ay 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 (1.3) gives the "trivial integral" T* = Constant. Taking into account the second condition of equation (1.9), we obtain T = To00( 12) = (2.1) The temperature of the wall is equal to the temperature of the adiabatic stagnation: Tk = Too = Tl + (K 1)2 (M= (2.2) The integral (2.1) corresponding to the case of the absence of external and internal heat transfer in the boundary layer, ( = = 0 for P = , was obtained on the basis of the solution of A. A. Dorodnitsyn (reference 1). NACA TM No. 1229 For P = 1, U = Constant, and T = Constant, equation (1.3) is likewise integrated independent of the solution of the remaining equations of the system and gives the qocalled StodolaOrocco integral T* = au + b (2.3) Imposing the boundary conditions we obtain T = TOO [l + ( ) (l : (2.4) From equation (2.3) we also obtain t*  (t* = T, to = TO Tw) (2.5) to The integral (2.4) corresponds to the case where there is similarity of the velocity 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 V Constant [for ( Y) 1 0 or T j Constant or P 9 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 integral of the system (1.1) to (1.5) satisfied by the boundary conditions (equations (1.8) and (1.9)). The temperature at an arbitrary point can then be represented in the form T1 T = T OO u0 for (2.6) T = O l2 + (T, 1) 1 NACA TM No. 1229 As is easily seen, the integrals (2.1) and (2.4) are particular cases of t* t U the second form of the relation (2.6) for q = 1 and = U 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 K P = O = 02(1 Uj2) (3.1) where pO is the pressure on the boundary of the layer (thermal or dynamic depending on which of them is thicker), p02 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 (3.1) K Ki P p1 (1 2) (3.2) S 02 2 + (w 1) where p02 is the density on adiabatic reduction of the velocity to zero. The viscosity is determined by the equation S= O 2 + l) n (n 0.75) (3.3) where 400 corresponds to the temperature TOO, that is, is obtained on the adiabatic reduction of the velocity to zero. NACA TM No. 1229 For any point ahead of the shock we have p = PO(1 p = p0ol( K K1  02) 1  u2 1 (3.4) where p01 and p01 are, respectively, the pressure and density on adiabatically reducing the velocity to zero up to the intersection of the streamline with the shock. As a scale of the velocities it is possible instead of the assumed magnitude j/2i to take the critical velocity a* and the local sound a. As is known a* = (Ki 1)/(K + 1) /2 and = M we obtain a   SU=+ 1 , v + 1 Substituting S(K 1)M/2 1 + ( K 1)M2/2 M= l 1 ( + 1)/2 ( 1 1)12/2 4. Integral Relations of the Momentum and Energy in New Variables Fram equations (3.1) and (3.2) we obtain dp dU OUU S=poU L = POTU (4.1) velocity U 1 * (2.5) In order to distinguish various uses of the symbol X herein, sabscripts 1 and 2 have been added by the NACA reviewer in the translated version. NACA TI No. 1229 where p0 Is the density on the boundary of the layer (thermal or dynamic depending on which is the thicker). We represent equations (1.1) and (1.2) in the form (pu2) + (PU) = p0UU' + + 6) Lu 6 (puU) + (pvU) puU' = o Subtracting the previous equation from the above we obtain + UU'p ) + UUt (   + y [(U 4] = [( + ') ' (4.2) From equation (3.2) we find 1 O= 02(1~1 P0 = P02(i 19) P0 1 = p 2 2 + ( 1)(1 t*/to*) i V2 (4.3) Integrating equation (4.2) term by term from y = 0 to y = Ay, if Ay > By and to y = 6y if 8y > Ay (for definiteness we assume that Ay > By; the same result is obtained if we assume By > Ay), making use of the relation (4.3), 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 u2 L 1 + 2UU'I P 1  Ctc1 a (li@ NACA TM No. 1229 fUy 3x J + UU' I U dy + UU' 0y 0 +1 (4.4) where + ) Jy=0 is the frictional stress at the wall. We now represent equations (1.2) and (1.3) in the form  (puto*) Ox + (pvto*)  u dt0*= 0 dx (put*) + (pvt*)  y dto pu  dx = ( + e) Subtracting the second equation from the first, integrating the result term by term from y = 0 to y = AY (assuming as above for definiteness that Ay > By), and remembering that for y > By the heat transfer oT q = X r and the friction T = of the thermal layer we obtain d Uto* p ai 0 d are equal to zero on the boundary  )o/ dy = (4.6) 1  10y P 1 UU'(T 1) + 1 2 j0A  t) dy = Tw o / (4.5) UP p(1 H dy U) V /w  LOT^ 6 (, 1) Ty = [( NACA TM No. 1229 where 1 Y= C = X (47) =c y=0 w 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 2K Td = 1 a ( (4.8) 0 1 u Noting that the function under the integral sign in equation (4.8) agrees with the expression p/p02 for vw = 1, we introduce a new independent variable of the analogous equation containing in the function under the integral sign the expression P/pO2 for the general case according to equation (3.2) S= 1 (1 ) dy (4.9) 0 1 U2 + (T 1)(1 t*/to0) For Tw = 1 the relation between the coordinates n and y depends on t* the unknown velocity profile. For  = equation (4.9) gives the to U change in variables applied to the problem of the heat interchange of the plate with the gas flow. In this case the relation between n and y likewise depends only on the velocity profile. As is seen from equation (4.9) in the general case the relation between the coordinates n and y depends not only on the velocity profile u(x, y: but also on the temperaturedrop profile t (x, y) which likewise is not initially known but is determined from the solution of the problem. NACA TM No. 1229 Replacing the deniaty p in equations (4.4) and expression (3.2) and passing to the variables x, Tj i) dT +UU(2 (4.6) by its we obtain + 1 22)fR (1 ) d + ,I U~ + UUI'k i+ 1 d 0x t0 t 0) ut+'+,U 1) dTi + (I  * 0) dT = T P02 =PC P02op (4.10) (4.11) 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 lose in momentum and the thickness of the displacement in the plane xz, respectively, by * = H = I d (4.12) we introduce the concept of the thickness of the energy loss (in the plane xTr) e = to0 dj (4.13) This magnitude has a clear physical meaning; namely, the magnitude e 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 U d1 U2 1L (l A = i0 1 u d NACA TM No. 1229 boundary layer would have if its stagnation temperature were equal to the stagnation temperature of the external flow and the true total energy of this mass. The magnitude e thus represents in length units referred to the temperature 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 heatcontent loss (reference 4). The magnitude A te m* \ A* =ear / i T A Jo \ to (4.14) may be called the thickness of the thermal mixing. With the aid of the magnitudes defined by equations (4.12), (4.13), and (4.14) we represent the obtained integral relations of the moment and energy (equations (4.10) and (4..1)) in the final form as U'U d + H + 2 (H + 1) d U 1 de U' to* + e+  ai U to* U' (T 1 ) 7T  +   U(1U2) p U2 02 dt dx Po2Ucpt We note that the integration with respect to y (or n) may be taken from 0 to a so that the relations (4.15) and (4.16) 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 (4.15) and (4.16) assume the form d U' U' T+ S+ (H + 2) + T 1 dx U U T HTe = T p0U2 (4.17) de U' +  dx U to0 to 0 Up0 Po'Vo (4.15) (4.16) dTz dx/ NACA TM Ho. 1229 where to = TO T, To and pO are, respectively, the temperature and density of the isothermal flow outside the thermal boundary layer (it follows from equations (2.6) and (3.2) if we set approximately U = 0, that is, according to equation (3.5) M = 0). The pressure distribution over the profile is determined by the equation of Bernoulli for an incompressible fluid.1 As is seen from equations (4.15) and (4.16) and also from what follows, in the variables x, q the equations of the system (1.1) to (1.3) are simplified 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 Kfrm n for the case of the turbulent layer. The proposed method of the solution of the problems connected with heat interchange permits, of course, generalization of certain other problems in the theory of the boundary layer in an incompressible fluid. II. LAMINAR BOUND LAYER WITH HEAT INTERCHANGE BETWEEN TEE GAS AND TEE WALL 5. Transformation of the Differential Equations Assuming in equations (1.1) to (1.3) 6 = 0 and t* = T* TV, substituting the values p, p, and p according to equations (3.1) to (3.3), we transform these equations to the new independent vari ables x = x and ni, determined according to equation (4.9). The equations of transformation of the derivatives will be K K x + (lx ' )y 1  2 + ( K 1)(1 t*/to*) _ IFram equations (3.1) and (3.5) we have 2i 0 = + K1)M 1 pO 1 + t + 2 + whence setting M = 0, we obtain pO + (po 92) = Constant 2an what follows we restrict ourselves to the case P = 1. NACA TM No. 1229 We obtain (introducing the + v = u x 00 notation 02 = P02 1 2 + (t 1)(1 t*/to*) + v02 (  a 52)1 a d) civ 01 + I = 0 OT 2 + (+ 1)  (v = v p 02 dt u dx K2( v02 (1 2) 1 t* n1 to 1 + (, 1) If it is assumed approximately that still further simplified. n = 1, the obtained system can be 6. Generalization of the Method of Pohlhausen We represent the velocity profile and the stagnation temperature drop profile by the polynomials S= A ) + + A ) 3 v2 + A (6.1) u (5.1) (5.2) 6a OX 3t* u ax +t* + V  on t J9 (5.3)  a2 NACA TM No. 1229 = t BI + BB + B (6.2) To determine the coefficients of the polynomials we set up the conditions 2 a t 62 u t6 t* + u (/8)2 u ai(nl/A)to* t (n/s) U for = 0 8 (6.3) where S2n B2U'T X2 = v02(1 ) .1 Condition (6.3) follows from the u = = 0 for = = 0. Further, a = (1 n) equation of motion (5.3) since = 1i u 0 U a(,T/) U ' 62 ( 2 1 = 0 (n/B)2 0 for = 1 B From equation (53) for n = 0, v = v = 0 we obtain =0 for = 0 (6.5) Similarly to conditions (6.4) for the profile t*/t0* the conditions a t W=A) 0. b(n/A) to* a2 J profile u/U we take for the t* = 0 for 1=1 (6.6) to* O (//Ar)' Wo /a (6.4) 62 6(n/A)2 I " Tw TOO Tw f a t* a o lo6iA to* 16 NACA TM No. 1229 By differentiating equations (5.1) and (5.3) with respect to T with the subsequent equating of 9 to zero, it is easy to obtain the conditions for the third derivatives of u/U and t*/t0* at the wall. These conditions may be useful for various aspects of the method of Pohlhausen. Using conditions (6.5) and (6.6) we obtain 3 /9 12a B1 =  B2 = 6 3B1, B3 = 8 + 3B1, B4 = 3 B1 From conditions (6.3) and (6.4) we now find 12 + X2 6 (3 /9 12o)8/A A2 = 6 3A1, A3 = 8 + 3A1, A4 = 3 Al (6.8) In relations (4.15) and (4.16) there enter, besides 3(x) and e(x), the four unknown functions Tw, H, q HT. 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 (4.5), (4.7), and (4.12) to (4.14). We obtain S= 5 =6 , 20 T7 = T0 nl U w IooTw 5 K ) (1 Vf~ 5AI2 + 12A1 + 144 4 = 6  1260 A1, qw = coCp0 'n1 t (1 A 8 B1 A* = e = 20 e = A(M1 + NiA1) BI  8b()3 + 9 , N1 = b1 3b22 3b 4)+ b ) 24 5B1 4 = 2520 Al = (6.7) (6.9) K Ki1 B1 (6.10) where for M1= 6b2( ) (6.11) 6 B1 b =60 16 3Bi b2 = 420 5 B1 3 = 280' NACA TM No. 1229 for > 1 8 8 B1 M1 20 + B [0.4 + 0.2290('7 2K + B,(1 0.1 0.1143 N1 = 0.0500 0.0429(  0.143c3 + 0.0290( \A I \A, A+ 0.0536/ 2 + 0.0286( 3 3 6X01  0.00961' I o.oo006/) V,,'J + B1( 2 0.0167 + 0.0214 )0.0107 () + 0.o02() 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 by the condition Xk = 12. 7. Determination of the Initial Conditions For subsonic flow and also supersonic in those cases where there is a head wave in front of the profile a critical point U(0) = 0 is formed at the leading edge (x = 0). The latter is a singular point of equations (4.15) and (4.16) in which the derivatives do/dx, de/dx, and so forth, increase to infinity if the initial b, 8 are not subjected to certain special conditions. Substituting the expressions for Tv, q, A* from equations (6.10) and (6.11) into equations (4.15) and (4.16), multiplying the latter by 6 and A, respectively, and equating to zero the coefficients of 1/U these conditions are obtained in the form (7.1) 8B 0,(A)2 = 0 X,(H + 2) + X2(Tw, l) 20 81 = O, r = o 18 NACA TM No. 1229 In the absence of heat transfer (Tw = 1) the first relation (7.1) is a cubical equation in X2. Of its three roots (7.052, 17.75, and 70) only the root (X2)x=0 = 7.052 satisfies the physical conditions. This value of X2 is the one generally assumed initial condition in the theory of the laminar layer for the equation of KarmtnPohlhausen. The equations (7.1) may be conveniently regarded as a system for determining the initial values of (X2)x=0 and (A/B)x=0. We present the results of the computation of these values for 0 < (Tw)xz= < 5 (for n = 0.75). Tw = 0 X2 = 0 0.05 0.10 0.50 1.o0 1.50 2.00 3.00 4.oo 5.00 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 2.74 3.70 4.86 8 The graphs of these results are shown in figure 2. 8. Method of Successive Approximations Simple computation formulas can be 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 2T and have Ut 2(H + 2 U2) 2 U T 1  + + 2 U Y2 U 1 U2 U 1g2 K 2 n1K OPT(CU) A ROv (8.1) where = A =L L L Sx dU x U'= Ar' R02 "oLo2 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 Tw; that is, we set k = c1 + (r cl), where c1 = c(i() 2 d di2 NACA TM No. 1229 After simple transformations we obtain dY 2T1 UUF a1 U9 2 2 +  + 2(, LJ2 + 1 + 2(T 1)  K n2 1 K1 1 2 = Twn2 (1 )1 2A1 2 (k c) (8.3) UR2 5 ) Assuming over a certain part of the boundary layer X1 < x < TXl that A* the ratio = h is constant with respect to i and equal to the mean value for the given segment, we set3 1260(8 BI) c = cl + 2(T 1)h = cl + 2(Tw 1) 2 (8.4) L 20(5A12 + 12A1 + 144) Multiplying equation (8.3) by UJ we obtain d (UiUc) + (c 2) 42 K n2 c1 1  = T (1 U [2A Considering this equation as linear in b2Jc we write its solution In the form S02 c + Tn2 c1 (1 f2)  d (85) i f) (8.5) For TJ Constant there is taken in the exponent c a constant mean value of Tw for each section. NACA TM No. 1229 where 0 = 2A1 g Tw (k cl), 1  C  02To (1 _ 2) x=x0 where C = 0 for XO = O. Substituting under the integral (in the function C and exponent c) certain initial values of (X2)0 and (A/6)0 we obtain a first approximate function A(x). The arbitrariness in the choice of the magnitude cl may be utilized for improving the convergence. For this purpose cl must be chosen such that for each value of Tw the error due to the assumed (inexact) values of X2 and A/6 is a minimum. Since ) itself depends little on A/6 it is the condition of little variation of $ with X2. small dependence of the functions 6/6 and k on in the argument Al on which they depend, X2 = 0, approximate expression S12 + X=(A0) 6 X2=0o52=Ac sufficient to set up Neglecting the relatively X, that is, setting we obtain the 2 tQ X2=0[k) c1] TV W/ XoL =0 11X= In order that C = Constant the coefficient of X2 must be equal to zero, whence we obtain (A1)X2=0Tw 01c = ( k) _ 2=0 6(9/6) S 6X2=0 This dependence of cl on Tw in a wide interval of change of the argument (and practically independent of A/5) is close to a linear one. For Tw = 0.1 we obtain cl = 9.35 for = 1 (c, = 9.12 for =2, c = 9.7 for = 0.5). For T= = 1 we obtain cl = 6.26. Rounding off the last value to cl = 6 we apply the linear relation cl = 9.5 3.5Tw (8.7) Finally multiplying the equation of energy (equation (4.17)) in nondimensional form term by term by 202 ( = and integrating as linear we obtain (8.6) NACA TM No. 1229 =R2 = ( ' 1)2 K + 2B1(TV I2 ) (l 2)1! 1 e XO C' = [2RO2b2( _ 1)2] and C' = 0 for i = 0. The computation of the dynamic and thermal and (8.9) can be carried out in this sequence. with the parameter X2, the analogous composite layers by equations (8.5) We consider, together parameters 2 20 (1 2 f2 S ) 2n (l (1J2) W '2R (m2n 2 e= 2U'T210 By equations (69) and (6.) represented in the form By equations (6.9) and (6.11) represented in the form X5A12 + 12A, + 14 2 = 2 1260 2?e X2A(M1 + NiA1)2 V X2 there are constructed for the given value of T. auxiliary graphs of the dependence of X2 on X2, for various values of X2A and for the dependence of X2G on X2e for various values of X2. The magnitudes taken as initial in the computation by formula (8.5) are determined from where (8.9) NACA TM No. 1229 the initial data. If U/ 0 for 30 = 0, there are taken the values (2)o = 0 and (A/B)0 = 1. If f = 0 for XO = 0, there can be taken as the initial values the values of (X2)0 and (X2A)0 = (A/)02(2)0 according to figure 2. By the values of the functions i(i), that is, X20(1) of the first approximation using the initial value (A/8)0 (for the succeeding approximations it is convenient to use the initial values of X2.), X2(x) of the first approximation is found with the aid of the graph. Further, by equation (8.9) there is computed e(x), that is, X2e(i) of the first approximation, making use of X2(i) of the first approximation and (A/B)0 (in the succeeding approximations there is used the function X2(i) following and X2,(x) preceding). From the values of 2g(1) and x2(i) with the aid of the graph there is obtained X2(i). In those cases where there is a considerable change in the ratio A*/8 (or the function Tw(E)) 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 qw/to* reduced to nondimensional form) and the coefficient of friction are found from the equations N=  = (K M U 1 (8.10) S= + (' 1)4 (8.11) Pum R. UL J "In particular for 1T = 1 formula (8.5) with (2)0 = 0 gives (M. < Mcr 9 ~1+ PC poo00 = o . 7 (u 2 p l d (c = 6) 2 0 1c(l U2) 2 which agrees with the equation of L. G. Loitsiansky and A. A. Dorodnitayn for the computation of the laminar layer without heat transfer (reference 6). In the absence of heat transfer and for small Mach numbers we again obtain from this the quadrature R = o.47 f dx (:R earlier derived by us on the basis of bhe method of KarmanPohlhausen for the laminar layer in an incompressible fluid. (See Tekhnika Vozdushnogo Flota, No. 56, 1942.) NACA TM No. 1229 For small Mach numbers equations (8.5) and (8.9) assume the form n2 ( c1 '0~ (8.12) ( 'o oLp * di (= U0Ij \ 0 /n1 Tar; 2B1 2 (T/To 1)2(U/U.)2 J U 0 (8.13) 9. Dependence of the Reynolds Number R02 on the Parameters of the Flow Taking account of the fact that according to the equation of state p02 =  we represent the parameter R02 in the form P01 p02 R02 = R01 , P01 where R1 = 00 (9.1) The parameter R01 is expressed directly in terms of the Reynolds U.Lo U, number Rm = and the Mach number M = 2 of the approaching flow. From equation (33) to (3) we fin a flow. From equations (3.3) to (3.5) we find S +1 01 = (C1 I .2) 1 + [l ( 1)M_. 2(Kl 01 U ( 1()M92 (9.2) In the case of subsonic subcriticall) velocities we have R02 = R01 = R00. For supersonic velocities the ratio p02/P01 is 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 %0 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 cp 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 2R = U1 (U/U^ )c ( TA P02 P01 1 2 NACA TM No. 1229 K 1 P02 / 1/2(K + 1)M. 0os2 :l 2 K : ( =   M.co (9.3) Pol \1 + 1/2( 1)M cos2 K + 1 + 1/2(K + 1)M2sin 29 tan (c + Po) = (9.4) 2 1 + 1/2( 1)M2cos2( 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 with the direction of the velocity before the discontinuity; in equation (9.3) there is in this case to be substituted p = 0. 10. Boundary Layer in the Flow of a Gas with Axial Symmetry For any axial flow about a body of revolution the integral relations of the impulse and energy have the following form: a 6 d dp S ypu2r dy U pur dy = d 6yr Twr (10.1) dx '0 0 dx d cpu(t* t0*)r dy = qr (10.2) dx 0 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 layer 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 4 as in sections 2,and 3 and introducing the new independent variable I by formula (4.9) we obtain the integral relations of the moment and energy in thb variables x, T: NACA TM No. 1229 H + 2 + (H + 1) .2T + 1 t 2 U'(TM 1) U(l .U2) r' w_ eHT +  =  r POOU2 toI* e+  + r to* Qe oo8= PooUc to* (= L, = ,and so forth) (0.4) SL L (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), (10.4), (6.9), and (6.11) the solution of which we write (for the case where there is no shock wave) in the form 2R00 = 112 6201 Tc( ~T2) 22 e2ROO = C UQ(T _ 1)2F2 _ JI r + x0 ) c 1+I Tnl (i1 2) 2 KI 2 Twn2Bl (1 (10.5) K 2) fl2) (t, _1)2j;2 Bdi] (10.6) where 1C I =c C = I[2ROCci(1 2 t) 22J X=1O C' = [2ROo2(w )2 x=x0 The method of computation does not differ from the case of the twodimensional flow. For the coefficients of the heat transfer and friction the equations (8.10) ani (8.11) remain valid. In the case of the internal problem (flow in nozzles) the Nusselt number and the friction coefficient are determined by the equations K q L nl K1 Noo = Tw( U 2) 0oot0* K 2rw 2 n1 o 1 A1 Too = Tw (1 )  SPOO o00 U5 di U' d9 U' dix U (10.3) NACA TM No. 1229 where X00 is the coefficient of heat conductivity corresponding to the temperature TOO. III. TURBULENT BOUNDARY LAYER IN TEE PRESENCE OF HEAT TRANSFER BETWEEN THE GAS AND WALL 11. Fundamental Assumptions The functions H, HT, Tw, and q. entering equations (4.15) E.an (4.16) are determined by equations (4.12), (4.14), (4.5), and (4.7) which express them as functions of ) and e through the medium of the velocity profile and the stagnation temperaturedrop profile. The present state of the problem of turbulence does not permit representing th velocity profile (and also the temperature profile) by a single equation which holds true from the wall to the boundarylayer limit. The fundamental dynamic and thermal characteristics of the turbulent layer can nevertheless be computed with an accuracy which is sufficient for practical purposes. A fortunate property of equations (4.15) and (4.16) which can be predicted on the basis of the results with respect to noncompressible fluids is that the functions H and HT change very little over the length of the turbulent layer and the functions Tw and qw connected with and 8 by the equations are little sensitive to the actual condition which prevail at a given section of the boundary layer. Hence H and HT (and also magnitudes analogous 'to them) can be taken as constant over x and the relations between Tw and a (the resistance law) and between qw and e (the heattransfer 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 Karmnn according to which the section of the boundary layer is divided into a purely turbulent "nucleus of the flow" and a laminarr 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: 2 T = p (11.1) where I is the length of the mixing path. In other words, in equations (1.1) and (1.3) we set = p22 dl. It follows directly from dy NACA TM No. 1229 this in view of the fact that the turbulence assumption of Prandtl gives Xt = Cpe that the expression for the heat transfer is dT 2 du dT q = n = cppl dy d pP y dy (11.2) The thickness 86y of the laminar sublayer of the dynamic boundary layer, equal for P = 1 to the thickness Ay6 of the thermal sublayer, is determined by the critical Reynolds number (the Karman criterion) (a S 11.5) (11.3) where u7 is the velocity on the boundary of the lminar sublayer, pw and pw are 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 laminar sublayer is permissible on account of the small thickness, we have (12.1) From equations (11.3) and (12.1) we obtain F B = y (12.2) In equation (12.2) we pass to the variable TI. Near the wall on account of the smallness of the terms u2 and t*/t0* we have T0 "0 (1 12)1 1 l2 + (T 1)  K dyi %b w(1 2)1" (12.3) hence K B~aT~lU2)K'1 u15:I4 = <,2 Tw = ,W Y t* to W NACA TM No. 1229 where 65 = At is the thickness of the laminar sublayer for the variable T. Further, K P1 (1 U)1 Pw = 002 (1w S= W, = POOT4 Substituting these expressions in equation (12.2) we obtain = , e RB For the fundamental parameters of the dynamic layer there is here introduced the notation RE = UR02o, Equation (12.1) is into the form T = p027L 2 U 1 2(K1) 11 U TP) (12 V TV/Po2 TV with the aid of equations (3.2) and (4.9) transformed 3K 12 2 (1 U2) (12 [1 2 + (T, 1)(1 t*/t0o] .5) .6) Since for small i the terms U2 and t*/t0* are small and the mixing path I = ky (k = 0.391) where the coordinate y is expressed according to equation (12.3), the "generalized" mixing path 1 near the wall is a linear function of t: S 1 )2(1) ' = kii 172 (1 ) TV1/2 (12.7) In deriving the resistance law in an incompressible fluid a linear mixingpath 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 law and the friction drops to S (12.4) (61 = NACA TM No. 1229 zero as the outer boundary of the layer is approached. Ths assumptions made act in opposite directions and lead to a satisfactory relation between the parameters RE and t. Carrying over this fundamental idea of the logarithmic method into our present theory we set T = T in equation (12.7) 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 2. Integrating equation (12.6) between I and B we obtain the approximate velocity profile: S1 + n 1 (12.8) U kl 8 From equations (12.1) and (12.4) we obtain the velocity at the boundary of the laminar sublayer U r ul The condition of the equality of the velocities of the turbulent and laminar flows on the boundary of the sublayer gives Pr = Cl w nekk ( = RO, C=a eka = 0.326) (12.10) Making use of the velocity profile (12.8) in equations (4.12) we obtain B1 ,1 1=  (12.11) 6 Ft k 6 5 kc Eliminating from equation (12.10) and the first of relations (12.11) the auxiliary variable 8, we obtain the resistance law: BR = ClTwnek (l ) (12.12) We obtain incidentally also the approximate expression for the parameter H: 6* 1 H= (12.13) 0 1 2/kt NACA TM No. 1229 13. Derivation of the HeatTransfer Law In this section we shall give a generalization to the case of a gas moving with large velocities of the heattransfer law earlier derived by us (reference 4) for an incompressible fluid. We construct the function q* = = q + u (13.1) dy J For the turbulent nucleus of the flow we have dt* dt* 2 du dt* q* = At  = c ;= cpp2 (13.2) Sdy dy dy dy(13.2) Transforming this equation to the variable j we obtain 3K ~2 du dt* f2 2 q* = c P02op 2 = d1 )K d= dT [1 U2 + (T 1)(1 t*/to*]3 (13.3) Near the wall the function q* behaves like q, ,that is, differs little from the constant value qw, and the mixing path I depends linearly on I according to equation (12.7). The common mechanism of the transfer of heat and the transfer of the momentum in the flows along solid walls 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 expression for du/dij obtained from equation (12.8) and integrating equation (13.3) from r to A we obtain the approximate profile for the stagnation temperatures t* q U = 1 + In (13.) t Cpto0*T k NACA TM No. 1229 From equation (13.1), assuming a linear distribution of the stagna tion temperatures in the laminar sublayer, we find t I* qw = Cp', 7 Ayz (13.5) where t2* 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: Re = URO2, RO2 U 1 K u2) 1 qoL xooto (13.6) From equations (13.4) and (13.5) we obtain the stagnation temperature on the boundary of the laminar sublayer to CT (13.7) Equating the stagnation temperatures on the boundary of the laminar sublayer and making use of condition (12.4) we obtain R = ClT nexp(kCT)k (RA = T0RO2) (13.8) Substituting the expressions for u/U and t*/to* according to equations (12.8) and (13.4) in equations (4.13) and (4.14) we obtain S1+ In e (. i, kCT 2 k2 CT, 1* 1 A kCT (13.9) From equations (12.10) and (13.8) it follows that = exp(k kt) so that we have S= 1 ( k2T (13.10) NACA TM No. 1229 Eliminating from equations (13.8) and (13.10) the auxiliary parameter A we obtain the heattransfer law Re = C01 n ) exp ktT We obtain incidentally also the approximate expression for the parameter HT: (13.11) (13.12) l/kt 1/kC(l 2/kT) 14. Solution of the Equation of the Turbulent Dynamic Boundary Layer We represent equation (4.15) in the form t' (H + 1) +  U(1 t2) SU'(Ty1)A* 1 1i + =2 = I(l V2) ( TV K (i U U 02 (14.1) We make the change in variables (reference 7): z = ekC(l 2/ki)k2t2 Differentiating this relation with respect to we obtain dz S= Kz dx 1 R0 dz n  A1 (14.2) x and using equation (12.12) S1 2/k k22 (K 1 2/kt + 2/k2t da (14.3) NACA TM No. 1229 From equations (14.1) to (14.3) we obtain U'K(H + 1) + U(1 2) + 'K(Ty 1) U(1 U2) 6* Tw' z + nK ' TV 02 K00o~k2 Z  z n+l SCin+l ClTw K (1 j2)K1 (14.4) The magnitudes H and K layer are assumed constant with as a known.function of 1 then respect to z. which change little along the boundary respect to i. If 6*/a is considered equation (14.4) is a linear equation with Assuming a constant mean value of the ratio = h interval Xo < x < 11 (in the first approximation we may A* 6* turbulent layer assume h = = = H, which holds for obtain the solution of equation (14.4) in the form5 (1 )c M nKc KRO2k2 px TwnKnlc+l K (1 )K1 (/ TO over a certain for the entire the plate),we dxJ (14.5) c + where C = [zT.n (1 u) , /duerj X=X Q~ The constants must be taken equal to K = 1.20, H = 1.4, h = 1.4, c = K(H + 1) + K(T, 1)h (Cl = 0.326; k = 0.391) See footnote (3) on page 19. NACA TM No. 1229 After computation of the dynamic and thermal layer new mean values HTRg of the variables K, H, h = H can be found and, if the deviations "b from the assumed values are considerable,the computation is repeated. If, for greater accuracy, the computation is conducted over segments the values of the constants for each succeeding segment are determined by the values of RE, C, Re, and tT obtained at the end of the preceding segment. In integrating from the point of the transition of the laminar into the turbulent state, the magnitude (z)= is determined from the condition of equality of the initial value of Re to the value of at the end of the laminar segment. In integrating from the leading edge, C = 0. By equations (12.2) and (14.2) the auxiliary graphs of the functions log (RE C1 ) and log z as functions of kt can be constructed once for all. The local friction coefficient is found from the equation 1 + ( 1.) 2+ l l 02 \2 1 i o \Uoo Cl For small Mach numbers equation (14.5) assumes the form KRk2 T 'nKn1 c+l +  TO R UoLp 0 1.0 Re = U R, U, 15. Solution of the Equation of the Turbulent Thermal Layer We represent equation (4.16) in the form dRe 1 d + ;1 dt * Rg =  fUR(1 2)Kl di T 02Tw (15.1) 2Tw Cf =  PU2 2_ Tw K U2) K1 (14.6) where (14.7) C= = (U0 0 Z (T,/(= U1 c (TW/To),j(U/Ujj)0 To0 oT NACA TM No. 1229 Carrying out the change in variables (reference 4) zT = xep (kT) k kT and making use of the heattransfer law (13.1) we find 1 dTv + T di T,* dx 1 dtO* ZT + KT * d 0 ci k2 ZT = KT kC +l ktCiT4 (KT 1 l/kSt 1 2/kT + 2/k2tT The solution has the former Krp 2k2. r( C+ i " '0 1)K U kt C'= [TTT(t 1)KT X=Xo 6The equation of between Re and tT energy (15.1) in connection with the in the form (13.11) in the case T = an equation with separable variables so that together with in the form (15.4) we can use its accurate solution exp(ktT)(kT 3) + 2Ei(kT) = C + ROl Where where F, 1(kCT) = IkT e relation Constant is the solution K (l tp)141 di  du, C = [exp(kT)(kT 3) + 291(kST) u X=X0 (for X0 = 0, C = 0) dzT di (15.2) UR02(1 UL2)"1 (15.3) ZT = ( 1)KT w (1 (1_b2 d.1 (15.4) NACA TM No. 1229 The constant K' must be taken equal to 1.15 (with subsequent check by the results of the computation of the thermal layer). The magnitude ((T)i= is determined by equation (13.11) from the condition of the equality of the value of Re 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 auxiliary graph of the function log zT against kCT can be constructed once for all. The local heat transfer is found from the equation qWL N  __0 n S1i + 1 (K 1) M21 = T T 2 K (1 Uj2) 11 For small values of the Mach number equation (15.4) assumes the form 1 T = T(T/T9T(T;/T KTEC T (Ty/TO^nT(T /TO 1KT 16. Determination of the Profile Drag for Subsonic Velocities The drag of a wing of infinite span (over unit length of span) is obtained from the momentum theorem in the form COQ = ) = Pouc1 00U20 wnere g denotes the sum of the momentumloss thicknesses referred to the upper and lower surfaces and computed at a great distance from rh wing where 8 > and UU.. For the drag coefficient we have 1 ep = 2Q 2 m 1 + 4 1)M2 K (16.2) p L DoiLTo L  nKT On1 W0 (0 +1 vK U 1 U00 kt (15.6) (16.1) (15.5) NACA TM No. 1229 The problem consists in expressing thermal characteristics of the boundary Since in the wake behind the body T7 = and (4.17) for the wake assume the form H + tf 1 u di U d TT , in terms of the dynamic and layer at the trailing edge. 0 and q = 0, equations (4.16) Tw  fj2u (16.3) (16.4) 1 dt0 + = 0 S0 dx 0 We introduce the notation G(i) = 1 U2 Tw 1 + 1 T 1 2 a and represent equation (16.3) in the form 1 di l di = (2 + G) In dx (16.6) Integrating this equation with respect to I from (denoted by the subscript 1) to i = a, we obtain in = (2 + 01) In 1 *1 For U, 0, + O 91 the trailing edge In dG UM T, = 1 the function 0G() goes over into H(i). an incompressible fluid however the hypothesis of Squire and Young (reference 8) on the linear character of the dependence In(U/Ua) on H holds: In(U/U.) Hw H In(Ul/U.) E H1 (16.5) (16.7) For *E2 + T 2 ; = 0 NACA TM No. 1229 Making the analogous assumption ln(U/U ) GG ln(Ul/uI,) G. G (16.8) we obtain7 In .1 i2 l 2 Uf. (16.9) We write down the expression for a1: 2 HI + U2 01 2 1 Ul (16.10) 1 _ U 1 +  1 9T 1U2 "1 It is easily seen that Ho = 1 and BH~ = 1, hence 2 G = S1 2 1 Tm m 1 e, + 12 , i U2 .0 (16.11) 'The same result can be arrived at from the following elementary considerations. Equation (16.7) may be represented in the form ln = (2 + 01) In m + (GC G1) In U 1l Um U. where Um is a certain mean value of the velocity U that lies between U1 and U,. For the usual profile shapes however the ratio Ul/ is, in general, near unity and since the magnitude 2 + 01 exceeds the magnitude Go _1 by several times,therefore for any choice of the mean value of Um the relative error in the determination of S, is not large. Taking the geometric mean m = UVU we again arrive at equation (16.9). NACA TM No. 1229 39 Replacing in equation (16.9) G1 and G, by their values, we obtain finally = 1 (16.12) u./ H1 +5 H1 + 1 U12 2 2 1 12 2 U 2 a v   i 2 1 )1 1 2 (TW i)oa/aCO 1v From equation (10.4) it follows that 0U(Tw 1) = Constant, hence (TV" 1) 9 = (if 1)91 U (16.13) (9 U1 (16.13) For small Mach numbers equations (16.2), (16.12), and (16.13) assume the form cxp = 2 ., H + 5 2 (Tw s = 9u1 00 1 \U it 17. Boundary Layer in a Gas Flow with Axial Symmetry For the turbulent flow about a body of rotation the equations (10.1) and (10.2) in the variables x, y and the transformed equations (10.3) and (10.4) remain valid. Restricting ourselves to the case of the Prandtl number P = 1 and introducing the parameters Rb, R, RE, and we represent the integral relations in the form elHT1 'i (TITW / 0 ) 5. = (Tow NACA TM No. 1229 dE UI(H + 1) 1) 1 1_1  + + + % + + fi T noo 1 W d U(1 ) U( U) r C2 T (17.1) R r  +  R dx r d to* + R= 0 y(1 y t0* T Making use of the drag law equation (12.12),the heattransfer law equation (13.11), effecting the change in variables z = k22' ZT = exp (kT) (l k assuming the little changing magnitudes H, K, XT constant with respect to x, and also a constant mean value for the ratio = h, we obtain a system of linear equations the solution of which has the form S/(I :52) S= 0 XRook2 zT nKi +, + C VcI c (l~ di Cl Ji0 1 ) (17.3) S/(1 _ )0 _ ZT = 1 )T TVnWTiT(TL T l 41ook2 fr2)I c+ 0 m f C' = z (rw, 1)Kr (17.2) NACA TM No. 1229 For the frictional stress and the Nusselt number, equations (14.6) and (15.5) remain in force. In the case of the internal problem ,L K 2 1 N00 URooU(1 2) Of = = (ol  X oo oT* tT T 00 o (17.5) The theory presented, in particular the integral relations of the moment 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 external and internal heat interchange. The computation of the boundary layer by the equations derived in parts II and III on the assumption of the Prandtl number P = 1 permits finding directly for arbitrary Mach numbers (excluding the interval from M = Mcr to M = i): (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 frictional stress along the wall and the profile drag of the wing (in the case M, < M r) for arbitrary 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 computed by the method of Donov (reference 9). In figure 4 are given the curves for the Nusselt number N which assure the uniform cooling of the surface up to the temperature Tw = 0.25Too for RE = 15 x 106, MN = 2, and Mw = 6 for the laminar (lower curves) and turbulent (upper curves) regimes. Translated by S. Reiss National Advisory Committee for Aeronautics NACA TM No. 1229 REFERENCES 1. Dorodnitsyn, A. A.: Boundary Layer in a Compressible Gas. Prikladnaya Matematika i Mekhanika, vol. VI., 1942. 2. Frankl, F. I., Christianovich, S. A., and Alexseyev, R. P.: Fundamentals of Cas Dynamics. CAHI Report No. 364, 1938. 3. Kalikhman, L. E.: Drag and Heat Transfer of a Flat Plate in a Gas Flow at Large Speeds. Prikladnaya Matematika 1 Mekhanika, vol. IX, 1945. 4. Kalikhman, L. E.: Computation of the Heat Transfer in a Turbulent Flow of an Incompressible Fluid. NIII Report No. 4, 1945. 5. Lyon, H.: The Drag of Streamline Bodies. Aircraft Engineering, No. 67, 1934. 6. Dorodnitsyn, A. A., and Loitsiansky, L. G.: Boundary layer of a Wing Profile at Large Velocities. CAHI Report No. 551, 1944. 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. 56, 1943. 8. Squire, H. B., and Young, A. D.: The Calculation of the Profile Drag of Aerofoils. ARC, R & M No. 1838, 1938. 9. Donov, A.: Plane Wing with Sharp Edges in Supersonic Flight. Izvestia Akademii Nauk SSSR, Ser. Matematichoskaya, 1939, pp. 603626. WACA TM No. 1229 Compression shock T0 Figure 1. MG6 a 00 V.2 D.* Da6 08 / Figure 3. # tj / J Ta, Figure 2. Figure 4. ii F1 4 4 f < 0 M tO i k E' 141 E a 0 m 43 4 4~F z , ii y ~F9R H Cu 0 < O ol' t3 0 .1 CU H Hi 14 z 0 Q 4 0 0 cei , 014 A 9I H E4 o\ W d 9 0 ri. fl 0 ( 0 +3 1 fr I C 5.4 1 L4 0 a) 0'  I D0 u > r nd O $ 0 P C mH 60 0 0 0 00 0 414 0 0 0 9 D41 I , D0 00 0M + wr4 P40 :3P 1.1 0M 0 a4 * H tO c0c 5 00 F0 3 4 34 'Q 0a C D t O 0 ra Q a 0) 2 a . 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