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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1400 
 
 BOUNDARY LAYER 
 
 By L. G. Loitsianskii 
 
 Translation 
 
 Pogranichnyi sloi." Mechanics in the U. S. S. R. over Thirty Years, 
 
 1917-1947. 
 
 Washington 
 May 1956 
 
16 i[ o y f i ^ so/ 
 
 NACA TM 1400 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1400 
 
 BOUNDARY LAYER* 
 By L. G. Loitsianskii 
 
 1. INTRODUCTION . 
 
 The fundamental, practically the most important branch of the modern 
 mechanics of a viscous fluid or a gas, is that branch which concerns it- 
 self with the study of the boundary layer. The presence of a boundary 
 layer accounts for the origin of the resistance and lift force, the 
 breakdown of the smooth flow about bodies, and other phenomena that are 
 associated with the motion of a body in a real fluid. The concept of 
 boundary layer was clearly formulated by the founder of aerodynamics, 
 N. E. Joukowsky, in his well-known work "On the Form of Ships" 1 published 
 as early as 1890. In his book "Theoretical Foundations of Air Navigation," 
 Joukowsky gave an account of the most important properties of the bound- 
 ary layer and pointed out the part played by it in the production of the 
 resistance of bodies to motion. The fundamental differential equations 
 of the motion of a fluid in a laminar boundary layer were given by Prandtl 
 in 1904; the first solutions of these equations date from 1907 to 1910. 
 As regards the turbulent boundary layer, there does not exist even to 
 this day any rigorous formulation of this problem because there is no 
 closed system of equations for the turbulent motion of a fluid. 
 
 Soviet scientists have done much toward developing a general theory 
 of the boundary layer, and in that branch of the theory which is of greatest 
 practical importance at the present time, namely the study of the boundary 
 layer at large velocities of the body in a compressed gas, the efforts of 
 the scientists of our country have borne fruit in the creation of a new 
 theory which leaves far behind all that has been done previously in this 
 direction. We shall herein enumerate the most important results by Soviet 
 scientists in the development of the theory of the boundary layer. 
 
 *"Pogranichnyi sloi." Mechanics in the U.S.S.R. over Thirty Years, 
 1917-1947, pp. 300-320. 
 
 Joukowsky, N. E.: forme sudov. Trudy Otdeleniya fizicheskikh 
 nauk Obshchestva lyubitelei estestvoznania, 1890. (See also N. E. 
 Joukowsky, Collected Works. Vol. II. Gostekhizdat, 1949, pp. 627-639.) 
 
NACA TM 1400 
 
 2. LAMINAR BOUNDARY LAYER FOR CASE OF PLANE- PARALLEL MOTION 
 OF INCOMPRESSIBLE FLUID 
 
 The solution of the problem of the motion of an incompressible fluid 
 in the stationary laminar boundary layer reduces, as is known, to the 
 obtaining of integrals of a nonlinear system of partial differential 
 equations: 
 
 ox 
 
 + 
 
 V 
 
 Bu 
 Sy 55 
 
 u 
 
 dU 
 dx 
 
 + 
 
 V 
 
 a 2 u 
 Sy 2 
 
 bu 
 ox 
 
 + 
 
 Sy 
 
 
 
 
 
 
 
 (2.1) 
 
 where the unknown functions u(x,y) and v(x,y) are the velocity com- 
 ponents along and normal to the surface of the body at the points of the 
 boundary layer, U(x) is the initially given longitudinal velocity com- 
 ponent on the outer boundary of the boundary layer, x and y are the 
 coordinates along and normal to the surface of the contour, and v = u/p 
 is the kinematic coefficient of viscosity. The boundary conditions of 
 the problem have the form 
 
 u = 0, v = for y = 
 u ->u(x) for y -»■ °° 
 
 (2.2) 
 
 where at times there is the further requirement of satisfying a given 
 distribution of velocities u = U/-j(y) at the initial section of the 
 layer x = . 
 
 The conditions of existence and uniqueness of solutions of equations 
 (2.l) have been considered by N. S. Piskunov (ref. 46). 
 
 The question of an effective method for solving equations (2.l) for 
 an arbitrary given function U(x) has not yet been answered. The existing 
 exact solutions of the system of equations (2.1) for boundary conditions 
 (eq. (2.2)) 'refer only to certain special classes of functions U(x) as, 
 for example, a linear function, a monomial to some power, certain very 
 simple exponential combinations, and so forth. 
 
 The application of purely numerical devices is not of great use be- 
 cause what is of fundamental importance is the possibility of taking into 
 account the effect of the form of the pressure distribution on the motion 
 in the boundary layer and not the accurate determination of the unknown 
 velocity components in a given special case. This is why from about 1921 
 extensive use was made of approximate methods for computing the laminar 
 boundary layer that were based on the application of the general integral 
 
NACA TM 1400 
 
 theorems of the mechanics of a fluid, especially the momentum theorem. 
 The methods of Karman and Pohlhausen are primarily methods belonging to 
 this class. 
 
 By applying the momentum theorem to an element of the boundary 
 layer, bound by the normal sections of the layer at the points x and 
 x + dx and the outer boundary of the layer y = 5(x), where the function 
 S(x) is conventionally assumed finite even though actually the effect of 
 the viscosity extends asymptotically to infinity, there may be obtained 
 the simple integral condition 
 
 *£♦$<» 
 
 -8-H- 
 
 + 6*) = - 
 
 pU< 
 
 (2.3) 
 
 where the prime denotes differentiation with respect to x. (This equa- 
 tion may also be derived strictly from equations (2.l) by employing the 
 accurate boundary conditions (eq. (2.2)). The two conventional boundary- 
 layer thicknesses 5*(x) and &**(x) are defined by the integrals 
 
 ** n u ( i u \ a 
 'So "l 1 -") 47 
 
 (2.4) 
 
 denoted, respectively, as the displacement thickness and loss of momentum 
 thickness, while the magnitude x defined by the equation 
 
 '- ■ w (l)y< 
 
 represents the frictional stress on the surface of the body; the symbols 
 6 and °° in the upper limit of the integral denote the possibility of 
 employing either the theory of the boundary layer of finite thickness or 
 the asymptotic theory. 
 
 Suppose we are given, in a boundary- layer section, the distribution 
 of the velocities expressed in the form of a polynomial of the fourth 
 degree with respect to the nondimensional coordinate tj = y/oc with 
 coefficients which are functions of x. Then, by satisfying the 
 conditions 
 
 u = 0, 
 
 S 2 u 
 
 Sy 
 
 2 - 
 
 u = 
 
 ciu 
 U > By" " °> 
 
 UU 1 
 
 a2 
 
 o u 
 
 oy 2 
 
 for y = 
 for y = 5 
 
 ] 
 
 (2.5) 
 
NACA TM 1400 
 
 the polynomial approximating the velocity distribution may be given the 
 form 
 
 U 
 where 
 
 H = <p(T),\) = 2tj - 2t] 3 + T] 4 + i Xtj(i - T)) 3 (2.6) 
 
 U'6 2 
 
 X=^- (2.7) 
 
 This magnitude, which is a function of x, plays the part of a parameter 
 of the group of curves (eq. (2.6)) determining the form of the velocity 
 profiles in the sections of the boundary layer and is, therefore, often 
 termed the form parameter. 
 
 The momentum equation (2.3) may be expressed in the form of an 
 equation for the determination of X as a function of x: 
 
 ^=4fg(x) +£ k(x) (2.8) 
 
 dx U U 
 
 where we must put (ref. 35) 
 
 l> ± - (2H** + H*)X 
 
 g(x) - 
 
 k(X) = 
 
 
 with the notation 
 
 H * = T = f (1 " ^^ *** = V \f ^ - ^ 
 
 \> (2.9) 
 U \ m /dcp\ 
 
 'd 
 
 b ± = 
 
 3 5/y=0 
 
 For the given form (eq. (2.6)) of the function <p(t],X) the magnitudes 
 (eqs. (2.9)) are functions of the parameter X, and equation (2.8) is a 
 nonlinear ordinary differential equation of the second order for the 
 determination of X as a function of x. By solving this equation for 
 the initial condition x = 0, X = Xq, where Xq is determined from the 
 
NACA TM 1400 
 
 condition that the right-hand side of equation (2.8) is finite at the 
 critical point x = 0, we obtain X.(x) , and hence, by equation (2.7), 
 S(x) also. Then there is no difficulty in computing the magnitudes 
 &*, 5**, and t w whereby the problem is solved. 
 
 The abscissa of the point of separation is determined from the 
 condition 
 
 ^ " ^(1) 
 
 y=0 
 
 This briefly is the Pohlhausen method for approximately solving 
 the equations of the laminar boundary layer. 
 
 Notwithstanding the roughness and small justification of the assump- 
 tions, this method, as numerous computations have shown, has proven 
 itself entirely satisfactory in the region of negative and small positive 
 longitudinal pressure gradients in the boundary layer, but entirely un- 
 suitable in the afterpart of the layer in the presence of a pressure 
 rise sufficiently steep to be accompanied by separation of the boundary 
 layer from the surface of the airfoil. In addition to this deficiency 
 in principle, the method ceased to be of service, also from the point of 
 view of practical application, since for solving the fundamental non- 
 linear differential equation (2.8), it required the use of complicated 
 graphical or analytical computing devices. 
 
 A whole series of Soviet investigations may be cited that were con- 
 cerned with the simplification of the practical application of the above 
 method. Thus, A. P. Melnikov (ref . 4l) worked out a method for the 
 numerical integration of the fundamental equation instead of its graph- 
 ical solution. K. K. Fedyaevskii (ref. 54) showed the possibility of 
 the approximate linearization of this equation and the consequent re- 
 duction of the solution for simple quadratures. A. A. Kosmodemyanskii 
 (ref. 19) substituted for the approximating polynomial (eq. (l.6)) the 
 product of a polynomial of the second degree by a trigonometric function 
 and applied the method of successive approximations to solve the differ- 
 ential equation thus obtained. 
 
 A. N. Alexandrov (ref. 2) [NACA note: Ref. 2 in turn refers to 
 NACA Rep. 527, "Air Flow in a Separating Laminar Boundary Layer" by G. B. 
 Schubauer, 1935. ] worked out a numerical method for integrating equation 
 (2.8), maintaining the velocity profile (eq. (2.6)) in the convergent 
 part of the layer, but for the diffuser part constructing a new polynomial 
 satisfying the boundary conditions obtained from equation (2.5) by adding 
 
 a new exact condition d^u/dy^ = for y = and dropping the old 
 
 condition d u/dy = for y = 6. This device gave good agreement of 
 the computation with experiment for the case of the flow about an 
 
NACA TM 1400 
 
 elliptical cylinder at different angles of attack, whereas the old method 
 led to a result which contradicted experimental findings, namely the 
 absence of separation in the after region of an elliptic cylinder with 
 ratio of axes 2.96 to 1 and zero angle of attack. 
 
 The method of Alexandrov does not, however, rest on a sufficiently 
 well-founded theoretical basis and possesses little accuracy, being, 
 moreover, extremely complicated computationally, the method was not able 
 to satisfy the increasing demands for a suitable computation of the 
 boundary layer. In the early part of 1941 there appeared in the U.S.S.R. 
 new, very much more accurate methods based on simple theoretical consid- 
 erations and, in addition, very suitable for practical application. 
 
 L. G. Loitsianskii (ref . 35) introduced the following two functions 
 of the form parameter (X) : 
 
 f (X) ■ XH** , F(X) - 2H**[b 1 - \(2H** + H*) ] 
 
 (2.10) 
 
 The functions g(X) and k(X) entering equation (2.8) are expressed in 
 terms of them as follows : 
 
 dX 
 
 dX 
 
 Equation (2.8) can then be reduced to the form 
 
 df U' 
 
 U" 
 
 51= u_ F(f) +±L f 
 
 dx U v ' U' 
 
 (2.11) 
 
 that is, to a differential equation determining f as a function of x 
 if the system of equations (2.10) is regarded as a parametric relation 
 between F and f through the parameter X. The parameter X is thus 
 excluded from consideration and replaced by a new form parameter f , 
 according to equations (2.10) and (2.9): 
 
 f = 
 
 U'8 
 
 *-*2 
 
 (2.12) 
 
 The form parameter f has the principal advantage as compared with the 
 parameter X because it does not contain the conventional nonphysical 
 magnitude 5 and is equally applicable to the theory of the layer of 
 finite thickness as well as to the more strict asymptotic theory. As 
 will be explained below, this form parameter has in addition a number of 
 other advantages . 
 
NACA TM 1400 
 
 The problem is thus reduced to that of determining once and for all 
 the functional relations 
 
 F = F(f), 
 
 = H(f) 
 
 ^ 
 
 uU 
 
 = 5(f) 
 
 (2.13) 
 
 after which, by solving equation (2.1l), it is possible to find succes- 
 sively f(x), then by equation (2.12) to find 6-h-*(x), by the third part 
 of equations (2.13) to find T w (x) , and finally, if required, by the 
 
 second part of equation (2.13) to find S*(x) . All these magnitudes are 
 encountered in the study of the flow about bodies and their resistance. 
 
 To establish equation (2.13), it is possible, for example, to make 
 use of the following one-parameter approximation of the velocity profiles 
 in the sections of the boundary layer (ref . 35) : 
 
 5 « 1 + ai (l - ^) n + a 2 (l - T!) n+1 + a 3 (l - rO n+2 (2.14) 
 
 where the coefficients a^, ag, and a^ are determined from the boundary 
 conditions on the surface of the cylindrical body: 
 
 u - 0, ^ . - Sgl, ^ = for y = (2.15) 
 
 oy^ oy° 
 
 and the exponent n, characterizing the degree of contact of the curve 
 of equation (2.14) with the straight line u = U on the outer boundary 
 of the layer, is considered as a function of the parameter X; that is, 
 in contrast to the old methods, it changes in passing from the forward 
 part of the layer to the rear part. To determine the relation between 
 n and X, use was made of a class of exact solutions of the equations 
 of the boundary layer for the case of a prescribed velocity of the ex- 
 ternal flow in the form of a monomial U = cx m . With a high degree of 
 approximation, it was possible to use the simple linear relation 
 n = 4 + 0.15X, which gives good agreement of the magnitude of the non- 
 dimensional friction coefficient £ computed by the present method, and, 
 from the above-mentioned class of exact solutions, for different ex- 
 ponents of degree m corresponding to different flows of a fluid in con- 
 verging and diverging channels. 
 
NACA TM 1400 
 
 The same idea was more consistently carried out in the cooperative 
 work of N. E. Kochin and L. G. Loitsianskii (ref. 2l). Instead of the 
 family of curves of equation (2.14), they made use of tables of values 
 of the velocity u(x,y) that were computed with great accuracy for the 
 class of problems U = cx m . 
 
 A. M. Basin (ref. 3) proposed employing the family of velocity pro- 
 files for the same purpose in place of equation (2.14) 
 
 8- £ ♦ §(* - - f i)~ 
 
 sin 2 
 
 satisfying at the point of separation all the conditions of equation 
 (2.15) and at the anterior critical point the boundary conditions of 
 equation (2.5) . 
 
 An ingenious solution of the same problem was given by E. E. 
 Solodkin who showed that it was possible in equation (2.14) to choose a 
 linear relation between n and X to satisfy approximately, at the 
 same time, both the equation of momentum and the equation of energy. It 
 is then no longer necessary to use a class of exact solutions. According 
 to Solodkin, the relation n = 4 +0.27 X holds. 
 
 All previous examples give approximate the quantitative results for 
 equation (2.13). Omitting in our present review the tables of these 
 functions and the graphs showing their variation, we remark only that 
 the function F(f) deviates little from the simple linear dependence 
 
 F(f) = a - bf (2.16) 
 
 where the constants a and b have definite initially computed values 
 fluctuating within certain limits depending on the device used for deter- 
 mining the approximating velocity profiles in the sections of the bound- 
 ary layer. It is possible, for example, to assume, on the average, the 
 values a = 0.45 and b = 5.7 leading to a deviation of F(f) from the 
 straight line of equation (2.16) by only a few percent. Because of the 
 equality equation (2.8), equation (2.3) maybe integrated in quadratures 
 and the solution has the form 
 
 f= aLM U b " 1 (?)d? (2.17) 
 
 If desired, it is possible to take into account the deviation of 
 the function F(f) from the straight line of equation (2.16) and to intro- 
 duce a correction in the solution of equation (2.17) but, as computations 
 show, there is practically no need for this. From equation (2.17) there 
 are readily obtained f(x), S'^x), and so forth. The condition of 
 
NACA TM 1400 
 
 separation will be f = f s = constant, where this constant likewise has 
 different values depending on the approximation used; there may be 
 assumed, for example, f s = - 0.085 or other values of f s close to it. 
 
 To compute t w and &*, it is necessary to have recourse to tables for 
 £(f) and H(f) given in the previously cited references. 
 
 The one-parameter method described previously is widely applied at 
 the present time for computing the flow of wing profiles and other cylin- 
 drical bodies in a two-dimensional flow. The further increase in accu- 
 racy of the method by passing to a large number of parameters and using 
 the equation of energy (L. S. Leibenzon, ref . 27, and a number of other 
 authors) is associated with extreme complication, and evidently is not 
 dictated by necessity, since the use of the single-parameter method al- 
 ready gives sufficiently good accuracy for smooth wing profiles. 
 
 3. TURBULENT BOUNDARY LAYER IN PLANE- PARALLEL MOTION 
 
 OF INCOMPRESSIBLE FLUID 
 
 Depending on the shape of the cylindrical body in the flow, the 
 condition of its surface, and also the structure of the approach flow, 
 the laminar boundary layer turns into a turbulent boundary layer over a 
 certain small transitional region generally taken to be a point called 
 the transition point. To compute the resistance of the wing and to deter- 
 mine the character of the flow about it and also, of particular impor- 
 tance, to estimate correctly the maximum lift force of the wing, it is 
 essential to be able to predict the position of the transition point. 
 Much had been done in this direction even before the start of the war by 
 Soviet aerodynamicists. Especially to be noted are the numerous experi- 
 mental investigations serving as the basis for devising empirical methods 
 for determining the position of the transition point. Thus, E. M. Minskii 
 (ref. 43) investigated the effect of the turbulence of the approach flow 
 and of the longitudinal pressure drop on the transition point on the 
 upper surface of a wing. 
 
 From the curves presented in the work of Minskii, it may be seen 
 very clearly how the transition point is displaced upstream of the flow 
 with increased turbulence of the flow and also with increase in the angle 
 of attack of the wing. Similar tests were conducted by Minskii for the 
 circular cylinder. On the basis of his investigations and numerous tests 
 of other authors, Minskii proposed a generalized empirical diagram from 
 which it is possible to determine approximately the position of the tran- 
 sition point being given certain averaged characteristics of the test 
 conditions. 
 
10 NACA TM 1400 
 
 Soviet scientists have worked out new experimental devices for deter- 
 mining the position of the transition point under laboratory and natural 
 conditions. N. N. Fomina and E. K. Buchinskaya (ref . 62) have conducted 
 an extensive experimental investigation of the boundary layer on a plate, 
 a wing, and a body of revolution with the aid of total- pressure micro- 
 tubes. The velocity profiles obtained by them permit estimating the 
 position of the transition point. Similar measurements on the surface 
 of biangular profiles were conducted by I . L. Povkh (ref. 47). The 
 investigations carried out in the last 10 years by P. P. Krasilshchikov, 
 K. K. Fedyaevskii, and others have considerably increased our under- 
 standing of the part played by transition phenomena in the development 
 of the interaction force between the body and the fluid (refs. 22, 55, 
 and 60) . 
 
 The effect of the above-mentioned factors on the heat transfer of 
 bodies in a fluid flow was investigated by a number of authors in the 
 aerodynamics and thermal physics laboratories of the Leningrad Poly- 
 technical Institute (L. G. Loitsianskii, P. I. Tretyakov, V. A. Shvab; 
 refs. 40, 68, and 70) . On the basis of these investigations concerned 
 mainly with the intensification of the processes of heat exhange in 
 steam boilers, an original method was proposed for determining the tur- 
 bulence of a fluid based on the measurement of the heat given off by a 
 calibrated body, a sphere, depending on the displacement of the line of 
 transition (thermal scale of turbulence) . 
 
 In 1944, an extremely simple semi -empirical theory of the transition 
 of a laminar layer into the turbulent layer was proposed by A. A. Dorod- 
 nitsyn and L. G. Loitsianskii (ref. 10) . On the basis of the consider- 
 ation that the principal reason for the transition of the laminar layer 
 to the turbulent layer is the occurrence of premature instantaneous 
 local separations of the laminar boundary layer in the region located 
 farther upstream than the point of stationary separation arising in the 
 absence of external disturbances, the authors proposed the following 
 simple formula for determining the abscissa of the transition point: 
 
 (3.1) 
 
 where t is a certain constant, characteristic of the given flow, and 
 is determined by the equation 
 
 r- f s(7ir**L . _ 
 
 (re*>-o 
 
 where f s is the separation value of the form parameter that is given 
 by f s = -0.085. The expression in parentheses on the right side of 
 equation (3.2) is computed, once and for all, for a given aerodynamic 
 
NACA TM 1400 11 
 
 wind, tunnel from tests on a plate or other body for which the point of 
 transition coincides with the point of minimum pressure. This very 
 approximate semi-empirical theory was sufficiently well confirmed by 
 numerous Soviet and foreign tests. 
 
 The more accurate theory, presented at the end of the paper cited 
 above, shows that, in fact, the constant y is a function of the non- 
 dimensional velocity at the transition point. There is also given an 
 explicit relation between the magnitude y and the intensity and scale 
 of the turbulence. It is important to note that the previously mentioned 
 semi-empirical theory can be easily generalized to the case of motion of 
 large velocities where it is no longer permissible to neglect the effect 
 of the compressibility of the air. 
 
 Let us now turn to the question of the turbulent boundary layer on 
 a wing profile. The absence of a rational theory of the turbulent 
 boundary layer has not up to the present permitted devising a theoret- 
 ically justified method for its computation. The first solutions of 
 this problem for the case of the wing profile were based on the utili- 
 zation in the boundary- layer sections of the velocity distributions 
 corresponding to a known power law, for example, the l/l power law, de- 
 rived for the steady motion in a pipe. As is known, power laws have the 
 fundamental defect that laws of such type are applicable only within a 
 certain range of Reynolds numbers. 
 
 The first investigator to overcome this deficiency was G. A. Gurz- 
 hienko (ref . 6) who applied a logarithmic velocity distribution not de- 
 pending on the Reynolds number to the computation of the turbulent 
 boundary layer. By making use of a logarithmic formula for the veloc- 
 ities in the sections of the boundary layer, Gurzhienko reduced the 
 problem to a certain relatively complicated differential equation and 
 gave a method of integrating it by successive approximations. From its 
 very nature, this method cannot take into account the effect of a longi- 
 tudinal pressure gradient on the shape of the velocity profile and it is 
 therefore not applicable to those cases where such a gradient is of 
 importance. 
 
 The first attempt to take into account the effect of the longi- 
 tudinal pressure gradient on the velocity distribution in a turbulent 
 boundary layer is that of K. K. Fedyaevskii (ref. 57) who presented a 
 new theory of the turbulent boundary layer based on the application of 
 the idea of "mixing length". 
 
 The proposed law of variation of the "mixing length" with the dis- 
 tance from the wall is the same for the boundary layer as for the pipe. 
 By approximating the distribution of the friction stress in a cross 
 section of the layer by a method analogous to the previously mentioned 
 device in laminar motion, Fedyaevskii established the form of the one- 
 parameter family of velocity profiles in the sections of tne layer, 
 
12 NACA TM 1400 
 
 choosing for the form parameter a magnitude equal to the ratio of the 
 longitudinal pressure drop over a length equivalent to the thickness of 
 the boundary layer to the friction stress at a given point on the sur- 
 face of the wing. By generalizing the idea of a laminar sublayer for 
 the case of the presence of a longitudinal pressure drop and applying 
 the formula for the velocity to the boundary of the sublayer, Fedyaevskii 
 obtained a formula for the resistance after which the equations of the 
 problem formed a closed system and the solution was carried to the end. 
 
 The method of Fedyaevskii was subsequently developed in the direc- 
 tion of greater convenience of computation by L. E. Kalikhman (ref. 12), 
 who also carried out a large number of computations of the boundary 
 layer for different wing profiles and showed the effect of the shape of 
 the profile, the lift coefficient, and other factors on the flow about 
 the wing. 
 
 A somewhat different method was followed by A. P. Melnikov (refs. 
 41 and 42) . Employing the semi -empirical theory of the turbulent motion 
 between two parallel walls in which the "mixing length" is expressed 
 through the derivatives of the longitudinal velocities along the direc- 
 tion normal to the surface, Melnikov applied this theory to the boundary 
 layer and obtained comparatively simple formulas for the one-parameter 
 family of velocity profiles with the same form parameter which figures 
 in the method of Fedyaevskii. Later Melnikov simplified the method, at 
 the same time, made it more accurate, and confirmed its practical appli- 
 cability by a number of computations. 
 
 In the theory of turbulent boundary layer, there is still a third 
 line of attack considerably more simple from the point of view of its 
 applications, which, in contrast to the above-mentioned semi -empirical 
 methods, might be denoted as empirical. This approach has recently re- 
 ceived the greatest development. 
 
 The underlying basis of all work using the empirical approach is 
 the employment of the momentum equation, which in the case of the turbu- 
 lent boundary layer maintains the same form (eq. (2.3)) as in the case 
 of the laminar layer. The equation contains essentially three unknown 
 magnitudes 5**, 5*, and t . In the semi-empirical theories, having 
 
 chosen a certain one-parameter family of velocity profiles in the sec- 
 tions of the layer, the two unknowns 6* and 5** are expressed in 
 terms of one unknown, the thickness of the boundary layer 5 (see eq. 
 (2.4)); after this there remains only to establish a formula for the 
 resistance connecting t w and 5. For this purpose there is employed 
 the concept of laminar sublayer, introduced, strictly speaking, only for 
 the case of the absence of a longitudinal pressure gradient. 
 
NACA TM 1400 13 
 
 In the investigation using the empirical approach, the family of 
 velocity profiles in the sections of the "boundary layer remains undeter- 
 mined, while the unknown magnitudes 5*, 5**, and t w , or their combi- 
 nations, are connected by approximate relations obtained from tests or 
 from certain assumptions of an intuitive character. Thus, for example, 
 two experimental curves are employed connecting the nondimensional 
 coefficient of resistance 
 
 and the thickness ratio &*/&** = H with the form parameter 
 
 Instead of using experimental curves connecting the resistance 
 coefficient and the magnitude H with a certain form parameter, curves 
 which incidentally are drawn through a very small number of test points 
 and refer to the region of small Reynolds numbers, it is possible, on 
 the basis of certain general assumptions, to construct a method suitable 
 for computations; the accuracy of this method is found to be entirely 
 sufficient in a number of cases. Thus L. G. Loitsianskii (ref. 36) 
 introduced a form parameter T and a reduced resistance coefficient £ 
 according to the formulas 
 
 r = 
 
 U'S** „,„*« , ^ 
 
 G(R**), (,^~ G(R**) 
 
 U "*" " > pU 2 
 
 where G(R**) is a certain function of the number R** = U5**/v deter- 
 mined from tests on plates. In this case, equation (2.3) may be trans- 
 formed to the form 
 
 !4*"")4r (3.3) 
 
 dx U w ' U 
 
 which is entirely analogous to equation (2.1l) for the determination of 
 the form parameter of the theory of laminar boundary layer. The function 
 F(r; R**) entering above and given by 
 
 F(r; R**) = (1 + mK - [3 + m + (l + m)H]r T (3.4) 
 
 is a weak function of R*~* because the number R** enters into it 
 chiefly through the magnitude m, which is equal to 
 
 d In G(R**) R**G'(R**) 
 m= d in R** = G(R^) 
 
14 NACA TM 1400 
 
 Making the simple assumption of similarity of the changes of £ 
 and H as a function of r in the turbulent and laminar (m = l) bound- 
 ary layers easily makes the problem completely determinate, and the 
 functions F(r), C(r), and H(r)> which are the same for different cases 
 of flow, can be tabulated. The function G(R""~ X ") may, however, evidently 
 be well approximated by the empirical formula 
 
 1 
 
 G(R**) = 153. 2R** 6 
 
 whence it follows that m = l/6. The function F(r) is as readily 
 linearized as in the case of the laminar boundary layer. From equation 
 (3.3), which becomes linear, the magnitude T is determined by simple 
 quadrature. Computations show satisfactory agreement with test results. 
 The method may be applied also for determining the abscissa of the point 
 of separation, that is, the value x = x s for which £(x s ) = 0. 
 
 If the turbulent boundary layer is considered for the case of 
 smooth flow without separation about a wing (small relative thicknesses 
 and small lift coefficient), it is sufficient in equation (3.4) to put 
 simply 
 
 m = i, £ = 1, H « 1.4 
 b 
 
 after which equation (3.2)[NACA note: Eq. (3.3).] is easily integrated. 
 For this very simple and also important case from the point of view of 
 practical application a somewhat different, but likewise simple, equation, 
 convenient for solution, was given by L. E. Kalikhman (ref. 15). 
 
 To the empirical methods based, as in the method above, on the mo- 
 mentum equation there may be added the method of computing the boundary 
 layer worked out by L. E. Kalikhman (ref. 14) . 
 
 In the U.S.S.R., as is seen from the previous review, a whole 
 series of original methods of computing the turbulent boundary layer has 
 been developed. The further development of this important field of 
 hydrodynamics requires experimental work on turbulent motion in general 
 and the turbulent boundary layer in particular. 
 
 4. CERTAIN SPECIAL PROBLEMS OF THEORY OF BOUNDARY 
 
 LAYER IN INCOMPRESSIBLE FLUID 
 
 Parallel to the laminar and turbulent internal friction in the 
 boundary layer, the processes of heat transfer occur which are associated 
 with a similar mechanism and which depend on the distribution of the 
 temperatures and velocities in the layer. Investigations along these 
 lines have been conducted principally in U.S.S.R. 
 
NACA TM 1400 15 
 
 G. N. Kruzhilin (ref. 23), making use of the concept introduced by 
 him of a thermal boundary layer of finite thickness, established a 
 simple integral relation for the heat transfer in a laminar layer. Apply- 
 ing a method analogous to that earlier described for the computation of 
 the laminar layer but in a more simplified form, Kruzhilin reduced the 
 problem to quadratures and obtained for N = aZ/\, R = VqZ/v, and 
 P = v/a the following general formula which interconnects them: 
 
 1 1 
 
 N = _L_ F^R 2 (4.1) 
 
 F(x) 
 
 where F(x) , a function of the nondimensional coordinate x = x/z , I 
 being an arbitrary scale dimension of the body, is a quadrature depend- 
 ing on the shape of the body; the magnitudes a, ^, a, and v are 
 respectively, equal to the coefficients of heat transfer, the heat con- 
 ductivity, the thermal diffusivity, and the kinematic viscosity of the 
 fluid. In the case of the flow along a plate, equation (4.l) assumes 
 
 the form 
 
 1 1 
 
 N = O.eVO^R 2 (4.2) 
 
 The coefficient entering it differs little from that of the accurate 
 solution. Equations (4.1) and (4.2) are derived on the assumption that 
 the thermal boundary layer is thinner than the velocity boundary layer, 
 that is, P is greater than 1. The equations retain their form, however, 
 also for P less than 1 but greater than l/2. In his further studies, 
 Kruzhilin applied equation (4.l) to the forward part of a circular cyl- 
 inder (ref. 25) and made a comparison with test data obtained by himself 
 and V. A. Shvab (ref. 26) . The results of the comparison were found to 
 be entirely satisfactory. In one of his subsequent papers (ref. 24), 
 Kruzhilin studied the effect of a longitudinal pressure gradient on the 
 form of the velocity profile in the boundary layer and also the genera- 
 tion of heat arising from the dissipation of energy due to the internal 
 friction in the rapidly moving fluid in the boundary layer. It should 
 be remarked that at the time of the appearance of Kruzhilin 's papers 
 there existed in world literature individual theoretical investigations 
 of the heat transfer of bodies in a forced flow but only for the partic- 
 ular cases of given distribution of the velocities in the outside flow 
 and of the temperatures over the surface of the body and without account 
 taken of the generation of heat due to the dissipation of mechanical 
 energy. 
 
 In the U.S.S.R., the first investigations were carried out in the 
 field of heat transfer in a turbulent boundary layer. V. A. Shvab, in 
 a theoretical paper (ref. 69) dating from 1936, first gave a solution of 
 the problem of the heat transfer under the conditions of the external 
 problem in the presence of a turbulent boundary layer in an incompress- 
 ible fluid. In this paper Shvab makes use of a well-known analogy 
 
16 NACA TM 1400 
 
 between the turbulent transport of momentum and heat and, assuming mono- 
 mials with various powers for the velocity and temperature distribution, 
 he gave formulas for the heat transfer both for a plate and for a cyl- 
 indrical body and body of revolution. For P equal to 1, Shvab obtained 
 an equation connecting the numbers N and R in the form 
 
 1+n 
 
 N = c • R 1+3n 
 
 where n is the exponent in the assumed distribution of the velocities 
 in the sections of the boundary layer. With the usual power law n = l/7 
 
 O Pi 
 
 there is obtained N ~ R in contrast to the previously mentioned law 
 N ~ R for the laminar boundary layer. 
 
 In a second generalizing paper appearing in 1937 (ref. 68), Shvab 
 developed the ideas of the preceding paper, showing how the effect of 
 the point of transition is to be taken into account and comparing the 
 results of the computations with experimental data obtained by him, to- 
 gether with other coworkers, in the aerodynamics laboratory of the 
 Leningrad Polytechnical Institute. 
 
 K. K. Fedyaevskii (ref . 56) generalized his method of computing the 
 turbulent boundary layer to the case of a thermal boundary layer. Making 
 use of a polynomial 'representation of the distribution of the heat trans- 
 port in a section of the layer, he obtained the distribution of the tem- 
 peratures over the cross section and then a new integral formula of the 
 dependence of the local value of N on P and R (the latter enters 
 in nonexplicit form through the coefficient of resistance) . Comparison 
 with the results of the tests of A. S. Chashchikhin showed good agree- 
 ment of theory with experiment. 
 
 Other studies by Soviet investigators in the field of forced heat 
 transfer of bodies in the boundary layer will be discussed in the fol- 
 lowing section devoted to the problems of motion of a gas at large 
 velocities, a case which is inseparably connected with heat transfer. 
 
 There should be mentioned the investigations of Soviet scientists 
 in the field of free convective heat exchange and also on turbulent jet 
 T.heory in which so much progress has been made principally by the work 
 of G. N. Abramovich (ref. l) . In these investigations, practical methods 
 are given for the computation of turbulent jets both with and without 
 heat transfer. 
 
 Together with turbulent jets, there belongs to the number of prob- 
 lems of the so-called "theory of free turbulence" also the problem of 
 the turbulent motion of a fluid in the aerodynamic wake behind a body, 
 that is, in the region of flow formed by the boundary layer coming from 
 the body. We may mention the interesting experimental investigations 
 
NACA TM 1400 17 
 
 of G. I. Petrov and R. I. Shteinberg (ref. 45) who were concerned with 
 the question of the effect of the shape of the body on the frequency of 
 the pulsations, of pressure or velocity in the wake behind the body, and 
 the work of B. Y. Trubchikov (ref. 49) on the measurement of the temper- 
 atures in the wake behind a heated body. These investigations led 
 Trubchikov to establishing a method of measuring the turbulence in wind 
 tunnels. 
 
 In considering the flow about the fuselage of an airplane, the 
 interference of the fuselage with the wing, the flow near the tips of a 
 wing of finite span, and also in studying the phenomena of slip and the 
 flow about a back- swept wing, it is of great importance at the present 
 time to study the three-dimensional flows of a liquid or gas in the 
 boundary layer. The problem of the three-dimensional boundary layer in 
 general presents great theoretical difficulties; the simplest case to 
 solve is that of the flow with axial symmetry. 
 
 In this field, practical application has been made in the U.S.S.R. 
 of the method for computing the frictional resistance of bodies of revo- 
 lution worked out by K. K. Fedyaevskii (ref. 52), based on the applica- 
 tion of power laws of velocity and resistance with variable exponents. 
 The first application of the logrithmic velocity profile to the compu- 
 tation of the boundary layer and the resistance of bodies of revolution 
 for the case of axially symmetric flow about them was made by G. A. 
 Gurzhienko (ref. 6) . 
 
 All new methods of computation of plane laminar flow or of the tur- 
 bulent boundary layer henceforth automatically were carried over to the 
 case of axially symmetric flow about bodies of revolution. The pre- 
 sentation of these methods may be found in the previously cited refer- 
 ences. An approximate method of computing the laminar boundary layer 
 analogous to that described in section 2 is given in a separate paper by 
 L. G. Loitsianskii (ref. 3l) . 
 
 Turning to a consideration of the more difficult problem of the 
 computation of a three-dimensional boundary layer, we may note first 
 that L. E. Kalikhman (ref. 16) gave the derivation of the fundamental 
 integral relations which can serve for the development of approximate 
 methods of solution of the problem analogous to those applied in the two- 
 dimensional case. 
 
 In the period from 1936 to 1938, Loitsianskii published a number of 
 papers in which, by employing various approximate devices, he was able 
 to solve the following three-dimensional problems: 
 
 (l) The laminar and turbulent motion of a fluid in a boundary layer 
 near the line of intersection of two mutually perpendicular planes 
 (there was applied the method of the finite layer (ref. 29) and 
 the method of the asymptotic layer (ref. 33)) 
 
18 NACA TM 1400 
 
 (2) The analogous problem for planes inclined to each other by a 
 certain angle (ref. 38) 
 
 (3) The laminar boundary layer along the line of intersection of 
 two surfaces (ref. 30) 
 
 (4) The laminar boundary layer near the lateral edge of a plate in 
 an axial flow (ref. 37) 
 
 In these papers new phenomena were revealed by mathematical compu- 
 tation, the most interesting of which are: the thickening of the bound- 
 ary layers and the decreasing of the friction in the region of juncture 
 of the planes or surfaces and, coversely, the thinning of the boundary 
 layer and increase in the friction as the lateral edge of the plate was 
 approached. Consequently, there appears the phenomenon of the premature, 
 as compared with the two-dimensional layer, separation of the boundary 
 layer near the line of intersection of the surfaces. The latter phenom- 
 enon, usually aggravated further by the harmful interference of the 
 external potential flows, which are as yet not subject to computation, 
 are actually observed in the region where the wing and fuselage are 
 joined and in other flows where there is an intersection of surfaces in 
 the diffuser region of the layer. 
 
 Very recently V. V. Struminskii (ref. 48) gave a theory of the 
 three-dimensional boundary layer on a cylindrical wing of infinite span 
 moving with constant angle of slip. For this purpose he applied the 
 theory of the boundary layer with finite thickness. 
 
 We now proceed to consider the investigations on the effect of the 
 roughness of the surface on the boundary layer. The effect of surface 
 roughness on the resistance of a body is determined principally by the 
 ratio of the mean height of the roughness protuberance to the thickness 
 of the laminar sublayer. The semi-empirical theory of the turbulent 
 boundary layer near a rough surface was worked out by the combined efforts 
 of several Soviet specialists. Particularly to be mentioned are a num- 
 ber of systematic studies conducted by K. K. Fedyaevskii and his coworker, 
 N. N. Fomina. Fedyaevskii (ref. 53) in his early work, dating from 1936, 
 provided the answer to two fundamental questions of interest to the design- 
 ing constructer: what is the "permissible" roughness which does not 
 appreciably increase the resistance of a wing, and what is the effect of 
 a given over-all roughness on the resistance of a wing. Later on, carry- 
 ing out tests on the resistance of an individual schematized protuberance, 
 Fedyaevskii and Fomina (ref. 61) sharpened the question of the possibility 
 of applying the hypothesis of plane flow to the roughness protuberances. 
 By introducing the notion of the equivalent height of a roughness pro- 
 tuberance, the authors gave a table of conventional heights equivalent to 
 various wing and fuselage surface roughnesses that are encountered in 
 practice. A similar investigation on the roughness of a ship's hull was 
 conducted by I. G. Khanovich (ref. 67). He is also to be credited with 
 a method for computing the boundary layer on a rough surface in the 
 presence of a longitudinal pressure drop. 
 
NACA TM 1400 19 
 
 An analysis of the parameters determining the resistance of a rough 
 surface and also the basis for the derivation of the fundamental formulas 
 of the velocity distribution were given in a note by L. G. Loitsianskii 
 (ref. 34). 
 
 The results of the investigations of our aerodynamicists on the 
 problem of the effect of roughness are widely applied in airplane con- 
 struction practice and in work on the analysis of the effect of roughness 
 on the resistance of ships, on the efficiency of hydraulic turbines (39), 
 and so forth. 
 
 The attention of Soviet investigators was likewise drawn to special 
 problems on the decrease of the friction due to changes in the physical 
 constants of the liquid or gas by having the boundary layer consist of 
 a liquid or gas differing in its properties from those of the approach 
 flow and, also, by heating the surface of the body in the flow. An 
 interesting experimental investigation of the surface of a body in a 
 flow was made by K. K. Fedyevskii and E. L. Blokh (ref. 59) who showed 
 that the coefficient of resistance of a body in an air flow with the 
 surface of the body heated decreases as the square root of the squares 
 of the absolute temperatures of the approach flow and the surface of the 
 body. 
 
 The effect of a boundary layer consisting of a fluid with other 
 constants was investigated in the theoretical note of L. G. Loitsianskii 
 (ref. 32) where it was shown that of fundamental importance for reducing 
 the resistance is a decrease of the ratio of the density of the fluid in 
 the boundary layer to that in the approach flow since this ratio enters 
 as a power close to unity, in contrast to the very small influence of 
 the ratio of the kinematic coefficients of viscosity. 
 
 Fedyaevskii conducted interesting experiments on the effect of the 
 aeration of the boundary layer on the resistance of a body moving in 
 water and showed the practical possibility of decreasing the resistance. 
 Several general considerations on this subject may be found in the 
 theoretical paper (ref. 58) of this author. 
 
 In conclusion, we note the investigations of N. A. Zaks (ref. 11 ) 
 on the control of the boundary layer by suction or blow-off of air on 
 the wing. The theoretical basis of the possibility of obtaining a gain 
 in the lift force from the application of various methods of control of 
 the boundary layer and by adding flaps to the wing was first given by 
 V. V. Golubev. In his investigations on the theory of the slotted wing 
 (ref. 4), Golubev showed that the presence of a forward flap retards 
 the separation of the boundary layer toward the region of larger angles 
 of attack than for the wing without flap and, in connection with this 
 fact, he advanced several general considerations on the structural param- 
 eters of the wing with flap. Later Golubev (ref. 5) occupied himself 
 with the theoretical investigation of other forms of mechanization, in 
 particular, with the suction and blow-off of the boundary layer. 
 
20 NACA TM 1400 
 
 5. BOUNDARY LAYER AND RESISTANCE IN COMPRESSIBLE 
 
 GAS AT LARGE VELOCITIES 
 
 The investigation of the effect of compressibility of a gas on the 
 motion in the boundary layer, the resistance, and the heat transfer is 
 the newest branch of the theory of the boundary layer. 
 
 The first theoretical study in which a method was given for the 
 complete computation of the distribution of the velocities and tempera- 
 tures in a laminar boundary layer in a compressible gas was the work of 
 F. I. Frankl (ref. 63). In this paper, Frankl generalized the usual 
 method of the boundary layer of finite thickness to the case of a com- 
 pressible gas. In his later papers (refs. 64 to 66) dating from 1935 to 
 1937, Frankl solved the problem of the heat transfer and friction in the 
 turbulent boundary layer on a plate. The latter problem, as well as the 
 analogous problem of the laminar boundary layer, presented serious com- 
 putational difficulties but the author carried his investigation far 
 enough to give quantitative conclusions. 
 
 An extremely simple approximate theory of the turbulent friction on 
 a plate in a compressible fluid flow was given by K. K. Fedyevskii and 
 N. N. Fomina (ref. 61) who showed that if the usual quadratic distribu- 
 tion formula for the turbulent friction is assumed for the cross sections 
 of a compressible-flow boundary layer, the effect of compressibility on 
 the resistance of the plate may at first approximation be taken into 
 account through a change in the physical constants in the boundary layer 
 and reduced to the previously mentioned law of the square root of the 
 square of the ratio of the temperatures at the wall to those of the 
 approach flow. 
 
 A fundamental step forward in the solution of the problem of the 
 boundary layer in a compressed gas was the investigation of A. A. 
 Dorodnitsyn (ref. 7) conducted by him even before the war but published 
 only at the beginning of 1942. In this work, Dorodnitsyn showed that at 
 P equal to unity, and in the absence of heat transfer, the system of 
 differential equations of motion of a gas in a laminar boundary layer 
 can be reduced to a form differing slightly from the equations of the 
 boundary layer in an incompressible gas if we pass from the coordinates 
 x and y to the new coordinates £ and r\, connected with the old 
 coordinates by the integral relations 
 
 jf 
 
 ' 4 dx - " - 4 dy {5 - 1] 
 
 where p 00 and pqq are the pressure and density in the gas adiabat- 
 ically brought to rest. 
 
NACA TM 1400 21 
 
 In the particular case of the plate in an axial flow, Dorodnitsyn 
 obtained the following equation for the resistance coefficient: 
 
 c f = 
 
 
 i 
 
 2 
 
 where the magnitude <P~(o) represents a certain function, computed by 
 
 the author, of Mod equal to the ratio of the velocity at infinity to 
 the velocity of sound at infinity, R,,, s= pooVooL/uoo, k = c p /c v , and n is 
 
 the exponent in the assumed law of dependence of the coefficient of 
 viscosity u on the temperature. 
 
 The previous transformation (eq. (5.l)) can be successfully applied 
 also to the turbulent boundary layer if we make use of the usual aver- 
 aged equations or the momentum equation derived from them and carry over 
 the fundamental equations of the semi-empirical theory of turbulence to 
 a compressible gas. By following this method, Dorodnitsyn (ref. 8) 
 obtained the equation for the local coefficient of resistance of a plate 
 in the presence of a turbulent boundary layer over its entire surface 
 
 0^ m VT7IZI M 2 „ |In(R x c f ) + i, ln(l + ^1 M 2 „) + 0.15] 
 
 (5.2) 
 
 where 
 
 R x = 
 
 p V x 
 
 I. A. Kibel (ref. 18) solved the problem of the laminar boundary layer 
 on a plate for P equal to unity and in the absence of heat transport 
 across the wall, but with the presence of radiation. At large values 
 of Moo of the approaching flow the plate temperature established in the 
 
 presence of radiation was found to be much less than in the absence of 
 radiation. 
 
 By employing, in a somewhat generalized form, the method of simpli- 
 fying the fundamental equations given by Dorodnitsyn, L. E. Kalikhman 
 (ref. 17) solved the problem of the laminar and turbulent flow of a com- 
 pressible gas on a plate in the presence of heat transfer. 
 
 The investigations of the boundary layer in a compressible gas on 
 the wi'ng and on a body of revolution were carried out principally in 
 the U.S.S.R. In the work referred to previously (ref. 8), Dorodnitsyn 
 considers not only these cases but, employing the transformation of 
 equation (5.l), also solves much more complicated problems. In the 
 
NACA TM 1400 
 
 general case of a laminar boundary layer, he applies primarily a method 
 analogous to that described in section 2 of this review, while, the 
 turbulent boundary layer, he has recourse to the general devices of the 
 semi-empirical theory. 
 
 To determine the coefficient of profile resistance, a simple formula 
 is established serving as a generalization of the well-known resistance 
 formula of a body in an incompressible fluid. Dorodnitsyn (ref . 9) 
 carried out wide computations of the resistance coefficients of wing 
 profiles at large velocities and brought to light specific effects of 
 the compressibility of the air on the resistance coefficients of wing 
 profiles of various geometric shapes for various conditions of the flow 
 about them. 
 
 The comparative complexity of the method on the one hand, and the 
 impossibility of its application to the solution of the problem of the 
 separation of the boundary layer, on the other, made it necessary to 
 generalize the method of computing the laminar boundary layer, described 
 in section 2 of this review, to the case of the motion of a compressible 
 gas with large velocities. A. A. Dorodnitsyn and L. G. Loitsianskii 
 (ref. 10) showed that equation (2.3), for P equal to unity and in the 
 absence of heat transfer, may be brought to the form 
 
 i - F(f)m -^=: + f £[1* \ (5.3) 
 
 where cxq = v/^SiQ represents the nondimensional velocity at the outer 
 
 boundary of the layer, 1q = J c p T 00 is ^ e total energy, and the form 
 parameter f has the form 
 
 f = 
 
 V'6 
 
 **2 
 
 v 
 
 ♦& 
 
 •oo i 1 - °o) 
 
 In the above equation and equation (5.3), V denotes the^ derivative 
 with respect to x, v 
 mined by the formula 
 
 with respect to x, while the momentum thickness loss 6' is deter- 
 
 It is of interest to remark that the structure of the expression 
 for the function F(f) in terms of £(f) and H(f)(see section 2) in 
 no way differs from the corresponding expression in the case of the 
 
NACA TM 1400 23 
 
 incompressible fluid. If we make the assumption that, at least for not 
 too large values of M,., the functions £(f) and H(f) will be the same, 
 as in the case of the incompressible fluid, we may employ the tables of 
 functions for £(f), H(f), and F(f) computed for the incompressible 
 fluid. For a first approximation we obtain the following generalization 
 of equation (2.17) : 
 
 ~ oV' I „b-l/, 2\m-l r c ,\ 
 
 " vbA — IzNi I v I 1 " a o] 
 
 where a and b are the same constants as in section 2 and the magni- 
 tude m is determined by the equation 
 
 m= 2 + k"TT - 2 
 
 The solution of the problem has thus been reduced, as before, to a 
 simple quadrature. 
 
 L. E. Kalikhman (ref. 13) investigated the laminar and turbulent 
 boundary layers on a wing in two-dimensional flow and on a body of revo- 
 lution with axially symmetric flow for the case of the presence of heat 
 transfer from the surface of the body. Introducing a transformation of 
 coordinates representing a generalization of the transformation of 
 A. A.. Dorodnitsyn (eq. (5.l)), Kalikhman constructed the integral rela- 
 tions of the moments and energies; then assuming a polynomial distri- 
 bution of velocities and temperatures in the cross sections of the 
 boundary layer, he converted these relations into differential equations 
 relative to several complexes containing the thicknesses of loss of 
 momentum and energy. The equations are integrated by the method of 
 successive approximations. In the first approximation, the solution is 
 represented as a simple quadrature. To solve the analogous problem for 
 the turbulent boundary layer, Kalikhman applies a semi-empirical theory 
 of turbulence in which he assumes a linear dependence of the mixing 
 length on the coordinates. The solution of the fundamental differential 
 equations in this case likewise lead to quadratures. At the conclusion 
 of the work an equation is established for the coefficient of profile 
 resistance serving as a generalization of the formula of Dorodnitsyn for 
 the case of a body in a compressible gas flow with the presence of heat 
 transfer. 
 
 The theory of the boundary layer occupies an important place in the 
 Soviet manuals on hydrodynamics (ref. 20) and constitutes a subject of 
 special monographs (ref. 28) . 
 
24 NACA TM 1400 
 
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 the Dependence of the Point of Separation of a Laminar Boundary 
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 Pt. 1 - Theory of a Flap in Plane-Parallel Flow. Trudy CAHI, no. 
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 i mekh., vol. X, no. 4, 1946. 
 
NACA TM 1400 25 
 
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 lence of the Flow on the Maximum Lift of a Wing. Trudy CAHI, no. 
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 the Surface of a Circular Cylinder in a Transverse Air Flow. JTF, 
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 Theory of the Boundary Layer. Trudy CAHI, no. 240, 1935. 
 
26 NACA TM 1400 
 
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 ance of Rough Pipes. Trudy CART, no. 250, 1936. 
 
 35. Loitsianskii, L. G. : Approximate Method of Computing the Laminar 
 
 Boundary Layer on a Wing. DAN, vol. XXXV, no. 8, 1942. 
 
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 no. 6, 1945. 
 
 37. Loitsianskii, L. G. : Three-Dimensional Boundary Layer and Friction 
 
 Near the Lateral Edge of a Plate in a Longitudinal Viscous Flow. 
 Prikl. matem. i mekh., vol. II, no. 2, 1938. 
 
 38. Loitsianskii, L. G. , and Bolshakov, V. P.: The Motion of a Fluid in 
 
 the Boundary Layer Near the Line of Intersection of Two Surfaces. 
 Trudy CART, no. 279, 1936. 
 
 39. Loitsianskii, L. G. , and Simonov, L. A.: The Scale Effect in Hydro- 
 
 turbines. Inzhenernyi sb., vol. I, no. 1, 1941. 
 
 40. Loitsianskii, L. G. , and Shvab, V. A.: Thermal Scale of Turbulence. 
 
 Trudy CAHI, no. 239, 1935. 
 
 41. Melnikov, A. P. : The Theory of the Boundary Layer of a Wing. Trudy 
 
 Leningrad, inst. inzhenerov grazhd. vozd. flota, no. 12, 1937. 
 
NACA TM 1400 27 
 
 42. Melnikov, A. P.: Turbulent Friction on a Wing and Its Computation 
 
 with Account Taken of the Effect of a Pressure Gradient. Trudy- 
 Leningrad, inst. inzhenerov grazhd. vozd. flota, no. 19, 1939. 
 
 43. Minskii, E. M. : The Effect of the Turbulence of the Approach Flow on 
 
 the Transition from the Laminar to the Turbulent State of the 
 Boundary Layer and on the Separation of the Layer. Trudy CAHI, no. 
 415, 1939. 
 
 44. Minskii, E. M. : The Effect of the Turbulence of the Approach Flow on 
 
 the Boundary Layer. Trudy CAHI, no. 290, 1936. 
 
 45. Petrov, G. I., and Shteinberg, R. I.: Investigation of the Nonsmooth 
 
 Flow About Bodies. Trudy CAHI, no. 482, 1940. 
 
 46. Piskunov, N. S. : Integration of the Equations of the Theory of the 
 
 Boundary Layer. Izv. AN SSSR, ser. matem. , vol. 7, no. 1, 1943, 
 pp. 35-48. 
 
 47. Povkh, I. L. : Experimental Investigation of the Boundary Layer on a 
 
 Profile of Biangular Form. JTF, vol. XIV, nos. 10-11, 1944. 
 
 48. Struminskii, V. V.: Slip of a Wing in a Viscous Fluid. DAN. nov. 
 
 ser., vol. LIV, no. 7, 1946. 
 
 49. Trubchinkov, B. Y. : Thermal Method of Measuring the Turbulence in 
 
 Wind Tunnels. Trudy CAHI, no. 372, 1938. 
 
 50. Fedyaevskii, K. K. : Critical Review of the Work on Retarded and 
 
 Accelerated Turbulent Boundary Layers. Tekhn. zametki CAHI, no. 
 158, 1937. 
 
 51. Fedyaevskii, K. K. : Surface Friction in the Turbulent Boundary Layer 
 
 of a Compressible Gas. Trudy CAHI, no. 516, 1940. 
 
 52. Fedyaevskii, K. K. : Boundary Layer and Frontal Resistance of Bodies 
 
 of -Revolution at Large Reynolds Numbers. Trudy CAHI, no. 179, 1934. 
 
 53. Fedyaevskii, K. K. : Approximate Computation of the Intensity of the 
 
 Friction and "Permissible" Heights of Roughnesses for a Wing. 
 Computation of the Friction of Surfaces with Local and General 
 Roughness. Trudy CAHI, no. 250, 1936. 
 
 54. Fedyaevskii, K. K. : Frictional Resistance of Wing Profiles for 
 
 Various Positions of the Place of Transition from Laminar to Turbu- 
 lent Boundary Layer. Tekhn. vozd. flota, no. 7, 1939. 
 
28 NACA TM 1400 
 
 55. Fedyaevskii, K. K. : Frictional Resistance of Wings for Various Posi- 
 
 tions of the Place of Transition from Laminar to Turbulent Boundary 
 Layer. Lecture at Conference on Physical Aerodynamics, Moscow, 
 CART, 1939. (See also Tekhn. vozd. flota, nos. 7-8, 1939.) 
 
 56. Fedyaevskii, K. K. : Thermal Boundary Layer of a Wing. Trudy CART, 
 
 no. 361, 1938. 
 
 57. Fedyaevskii, K. K. : Turbulent Boundary Layer of a Wing. Pt. I - The 
 
 Profile of Frictional Stress and Velocity. Trudy CART, no. 282, 
 1936; Pt. II - The Law of Resistance. Trudy CART, no. 316, 1937. 
 
 58. Fedyaevskii, K. K. : Decrease in the Frictional Resistance by Changing 
 
 the Physical Constants of the Fluid at the Wall. Izv. OTN AN SSSR. , 
 vol. 7, nos. 9-10, 1943. 
 
 59. Fedyaevskii, K. K. , and Blokh, E. L. : Experimental Investigation of 
 
 the Boundary Layer and Frictional Resistance of a Heated Body. 
 Trudy CAHI, no. 516, 1940. 
 
 60. Fedyaevskii, K. K. , and Fomina, N. N. : Effect of the Degree of 
 
 Turbulence of the Flow on the Resistance of Bodies. Tekhn. zametki 
 CAHT, no. 126, 1936. 
 
 61. Fedyaevskii, K. K. , and Fomina, N. N. : Investigation of the Effect 
 
 of Roughness on the Resistance and State of the Boundary Layer. 
 Trudy CAHI, no. 441, 1939. 
 
 62. Fomina, N. N. , and Buchinskaya, E. K. : Experimental Investigation 
 
 of Two-Dimensional Boundary Layer. Trudy CART, no. 374, 1938. 
 
 63. Frankl, F. I.: Theory of the Laminar Boundary Layer in a Compressible 
 
 Gas. Trudy CAHI, no. 176, 1934. 
 
 64. Frankl, F. I.: Heat Transfer in a Turbulent Boundary Layer at Large 
 
 Velocities in a Compressible Gas. Trudy CAHT, no. 240, 1935. 
 
 65. Frankl, F. I.: Friction in a Turbulent Boundary Layer at Large 
 
 Velocities in a Compressible Gas. Trudy CAHI, no. 240, 1935. 
 
 66. Frankl, F. I., and Veishel, V. V.: Friction in a Turbulent Boundary 
 
 Layer About a Plate in Plane -Par all el Flow of a Compressible Gas 
 at Large Velocities. Trudy CAHI, no. 321, 1937. 
 
 67. Khanovich, I. G. : Resistance of Continuously Rough Bodies. Izv. 
 
 VMA VMF, no. 1, 1939. 
 
NACA TM 1400 29 
 
 68. Shvab, V. A.: Theory of Heat Transfer in Turbulent Flow. Trudy 
 
 Leningr. industr. inst., razdel fiz. matem. nauk, no. 1, 1937. 
 
 69. Shvab, V. A.: Heat Transfer Under the Conditions of the External 
 
 Problem in the Presence of a Turbulent Boundary Layer. JTF, vol. 
 VI, no. 7, 1936. 
 
 70. Shvab, V. A., and Tretyankov, P. I.: Choice of Optimal Form of 
 
 Thermal Turbulence Meter. Trudy Leningr. industr. inst., razdel 
 fiz. matem. nauk, no. 1, 1937. 
 
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UNIVERSITY OF FLORIDA 
 
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