Boundary layer

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Title:
Boundary layer
Series Title:
NACA TM
Physical Description:
29 p. : ; 26 cm.
Language:
English
Creator:
Loĭt︠s︡i︠a︡nskiĭ, L. G ( Lev Gerasimovich )
United States -- National Advisory Committee for Aeronautics
Publisher:
National Advisory Committee for Aeronautics
Place of Publication:
Washington
Publication Date:

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Subjects / Keywords:
Aerodynamics   ( lcsh )
Boundary layer -- Research   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The author reviews the scope of papers on boundary layer contained in seventy references for the period from 1917-1948.
Bibliography:
Includes bibliographic references (p. 24-29).
Funding:
Sponsored by National Advisory Committee for Aeronautics.
Statement of Responsibility:
by L.G. Loitsianskii.
General Note:
"Report date May 1956."
General Note:
"Translation of "Pogranichnyi sloi." From Mechanics in the U.S.S.R. over Thirty Years, 1917-1947, pp. 300-320."

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University of Florida
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oclc - 81184795
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AA00009203:00001


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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1400


BOUNDARY LATER*

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"l 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.
1Joukowsky, N. E.: 0 forme sudov. Trudy Otdeleniya fizicheskikh
nauk Obshchestva lyubitelei estestvoznania, 1890. (See also N. E.
Joukowsky, Collected Works. Vol. II. Gostekhizdat, 1949, pp. 627-639.)







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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:
6u 6u dU 02u
U r + V = + V -
ox Uy dx 4 2

3 6w = o (2.1)
-- + -= 0
ox oy

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 = p/P
is the kinematic coefficient of viscosity. The boundary conditions of
the problem have the form

u = 0, v = 0 for y = O
(2.2)
u -U(x) for y J

where at times there is the further requirement of satisfying a given
distribution of velocities u = uO(y) at the initial section of the
layer x = 0.

The conditions of existence and uniqueness of solutions of equations
(2.1) have been considered by N. S. Piskunov (ref. 46).

The question of an effective method for solving equations (2.1) 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







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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 = 6(x), where the function
5(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

d5-+ VU (2* + *) = (2.3)
dx U pU2

where the prime denotes differentiation with respect to x. (This equa-
tion may also be derived strictly from equations (2.1) by employing the
accurate boundary conditions (eq. (2.2)). The two conventional boundary-
layer thicknesses 5*(x) and 5*(x) are defined by the integrals

S(1 )dy

(2.4)
5 =} Q 1 ) dy
so U ( U

denoted, respectively, as the displacement thickness and loss of momentum
thickness, while the magnitude T defined by the equation
w
/u\


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 n = y/Sc with
coefficients which are functions of x. Then, by satisfying the
conditions


u O, = v for y = 0
U, O for y(2.5)
(u 2u
us U, Cx-= 0, ay 2 0 for y W a






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the polynomial approximating the velocity distribution may be given the
form

S= X(,\) = 21 213 + 14 + X(l 1)3 (2.6)

where

X 2 (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:

dX = U' 8(X) + U1 k(x) (2.8)
dx U U

where we must put (ref. 35)

bl (2H** + H*)x
g(X) = --- *-- -1
X dB +1 H
dh 2




dX 2

with the notation


H= (1 cp)dTl, H** = 1 p(l cp)dl

U (2.9)
bl = (i1t d



For the given form (eq. (2.6)) of the function (p(rj,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 = 10, where X0 is determined from the






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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),
5(x) also. Then there is no difficulty in computing the magnitudes
6*, 6**, and r, whereby the problem is solved.

The abscissa of the point of separation is determined from the
condition
6u
S= SX o


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. 41) 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. (1.6)) 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 d3u/dy3 = 0 for y = 0 and dropping the old
condition d2u/dy2 = 0 for y = 6. This device gave good agreement of
the computation with experiment for the case of the flow about an







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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**2, F(X) = 2H**[b1 X(2H** + t*)] (2.10)

The functions g(X) and k(X) entering equation (2.8) are expressed in
terms of them as follows:

SF(X) f(X)
g() = df -, k(X) = df
dX dX

Equation (2.8) can then be reduced to the form

df U' F(f) + U f (2.11)
Cx U U'

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 *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 8 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.







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The problem is thus reduced to that of determining once and for all
the functional relations

F Fff),

H- H= H(f)

u (2.13)




after which, by solving equation (2.11), it is possible to find succes-
sively f(x), then by equation (2.12) to find 6**(x), by the third part
of equations (2.13) to find t,(x), and finally, if required, by the
second part of equation (2.13) to find 5*(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):

u = 1 + al(l ,)n + a(1 )n+l + a.3( )n+2 (2.14)

where the coefficients al, a2, and a3 are determined from the boundary
conditions on the surface of the cylindrical body:

)2u LUU' 3u
u = 0, 2 2 -V U u = 0 for y = 0 (2.15)
dy 6y"

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 = ex 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 t 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.







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The same idea was more consistently carried out in the cooperative
work of N. E. Kochin and L. G. Loitsianskii (ref. 21). 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".

A. M. Basin (ref. 3) proposed employing the family of velocity pro-
files for the same purpose in place of equation (2.14)

+ 1 sin sin i
U I ( 2 \ 2 q

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) may be integrated in quadratures
and the solution has the form

f U= b- l(F)d (2.17)
ub /


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), 5* (x), and so forth. The condition of







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separation will be f = fs = constant, where this constant likewise has
different values depending on the approximation used; there may be
assumed, for example, fg = 0.085 or other values of f, close to it.
To compute and 8*, it is necessary to have recourse to tables for
C(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.


9







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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 refss. 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 exchange 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. C. 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:

( l + U)(B 2= fs (3.1)


where y is a certain constant, characteristic of the given flow, and
is determined by the equation

b(3.2)


where f5 is the separation value of the form parameter that is given
by fs = -0.085. The expression in parentheses on the right side of
equation (3.2) is computed, once and for all, for a given aerodynamic







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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 1/7 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,






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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 refss.
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 E**, b*, and tw. 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 B* 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, 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.







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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*, 8**, and Tw, 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
1

pU2

and the thickness ratio s*/**= H with the form parameter

1
U'S U6** 4


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 r and a reduced resistance coefficient C
according to the formulas

U'b G(R), !w-(R5)


where G(R**) is a certain function of the number R** = UB**/v deter-
mined from tests on plates. In this case, equation (2.3) may be trans-
formed to the form


dl =U F(F; R ) + F (3.3)
dx U U'

which is entirely analogous to equation (2.11) 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; RB') = (1 + m)! [3 + m + (1 + m)H]r' (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**)
d In R G(R**)







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Making the simple assumption of similarity of the changes of 5
and H as a function of r in the turbulent and laminar (m = 1) bound-
ary layers easily makes the problem completely determinate, and the
functions F(F), (r), and H(r), which are the same for different cases
of flow, can be tabulated. The function O(R *) may, however, evidently
be well approximated by the empirical formula

G(R**) = 155.2R* 6

whence it follows that m = 1/6. 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 r 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
:cf separation, that is, the value x = xc for which C(x3) = 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 = -, 5 = 1, H = 1.4


after which equation (3.2)[NACA note: Eq. (35.3).] 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 IHCOrMPRESSIBLE 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


G. II. Kruzhilin (ref. 23), making use of the concept introduced by
him of a thermal bow.uidary; 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 1I = al/X, R = VOl/v, and
P = v/a the following general fc-ormula which interconnects them:

1 1
1 = R2 (4.1)
F(x)

where F( ), 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, X, 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 (4.1) assumes
the form
11
I = 0.670OP3R2 (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 1/2. In his further studies,
Kruzhilin applied equation (4.1) 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







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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
i =c R1+3

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 = 1/7
there is obtained 1H R0. in contrast to the previously mentioned law
1 R.05 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
theory in which so much progress has been made principally by the work
of G. N. Abramovich (ref. 1). 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







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of C. 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. 31).

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:

(1) 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))







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(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 lamininar 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. 3. 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


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. C. 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 (53),
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
NV. 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 Colubev (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.







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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. Frank (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 refss. 64 to 66) dating from 1935 to
1937, Frank! 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 r and 71, connected with the old
coordinates by the integral relations


C= dx, x== dy (5.1)
SP00 P00


where p00 and poo 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:


2 "(0) ( I
c -f 1 + I

where the magnitude c9&(o) represents a certain function, computed by
the author, of M.a equal to the ratio of the velocity at infinity to
the velocity of sound at infinity, R. = pV.L/pe, k = cp/cv, and n is
the exponent in the assumed law of dependence of the coefficient of
viscosity u on the temperature.

The previous transformation (eq. (5.1)) 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

02= 42- 1 / 7 M [n(R,c,) + r ln(l + M.) + 0.151


(5.2)
where

R, -


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 M. 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 wing 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 (5.1), 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

df = F(f)ln V + Vf n (5.3)
dx k
2 k-1


where ao = V/-2TO represents the nondimensional velocity at the outer
boundary of the layer, i1 = JcpToo is the total energy, and the form
parameter f has the form




( 2)1+ -1

In the above equation and equation (5.3), V' denotes the derivative
with respect to x, while the momentum thickness loss E& is deter-
mined by the formula


5 1 vd,


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


incompressible fluid. If we make the assumption that, at least for not
too large values of M., the functions ((f) and R(f) will be the same,
as in the case of the incompressible fluid, we may employ the tables of
functions for t(f), H(f), and F(f) computed for the incompressible
fluid. For a first approximation we obtain the following generalization
of equation (2.17):


f VV Vb-l 22 m-1 :x (5.4)



where a and b are the same constants as in section 2 and the magni-
tude m is determined by the equation

K b
m = 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. (5.1)), 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).







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REFERENCES

1. Abramovich, G. N.: Turbulent Free Streams of Liquids and Gases.
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2. Aleksandrov, A. N.: Theoretical and Experimental Investigation of
the Dependence of the Point of Separation of a Laminar Boundary
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VIII, no. 22, 1938.

3. Basin, A. M.: An Approximate Method of Computing the Laminar Boundary
Layer. DAN, vol. XL, no. 1, 1943.

4. Golubev, V. V.: Investigations on the Theory of a Slotted Wing.
Pt. 1 Theory of a Flap in Plane-Parallel Flow. Trudy CAHI, no.
147, 1933.

5. Golubev, V. V.: Theoretical Foundations of the Methods of Increasing
the Lift of a Wing. Trudy VVA im. UI. E. Zhukovskogo, no. 46, 1939.

6. Gurzhienko, G. A.: Application of the Universal Logarithmic Law of
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Numbers. Trudy CAHT, no. 257, 1936.

7. Dorodnitsyn, A. A.: Laminar Boundary Layer in a Compressible Gas.
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8. Dorodnitsyn, A. A.: Boundary Layer in a Compressible Gas. Prikl.
matem. i mekh., vol. VI, no. 6, 1942.

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10. Dorodnitsyn, A. A., and LoitsiansKii, L. G.: Transition of the
Laminar into the Turbulent Boundary Layer and Laminar Profiles.
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11. Zaks, N. A.: Aerodynamics of a Wing with Suction and Blowoff of the
Boundary Layer. Trudy VVA im. N. E. Zhukovskogo, no. 54, 1940.

12. Kalikhman, L. E.: Effect of Shape of Profile on the Frictional Re-
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Region of Separation. Trudy CARHI, no. 333, 1937.

13. Kalikhman, L. E.: Gasdynamic Theory of Heat Transfer. Prikl. matem.
i mekh., vol. X, no. 4, 1946.






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14. Kalikhmnan, L. E.: New Method of Computing the Turbulent Boundary
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15. Kalikhman, L. E.: A simple Method for Computing the Turbulent Bound-
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16. Kalikhman, L. E.: Three-Dimensional Boundary Layer. Trudy CAHI,
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17. Kalikhman, L. E.: Resistance and Heat Transfer of a Flat Plate in
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18. Kibel, I. A.: Boundary Layer in a Compressible Fluid with Account
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19. Kosmodemyanskii, A. A.: Theory of Frontal Resistance. Pt. 1 -
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20. Kochin, II. E., Kibel, I. A., and Roze, N. V.: Theoretical Hydro-
mechanics, pt. II. Gostekhizdat, 31448, p. 427-520.

21. Kochin, N. E., and Loitsianskii, L. '.: An Approximate Method of
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22. Krasilshchikov, P. P.: Effect of the Reynolds Number and the Turbu-
lence of the Flow on the Maximum Lift of a Wing. Trudy CAMI, no.
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23. Kruzhilin, (G. Nl.: Investigation of the Thermal Boundary Layer. JTF,
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24. Kruzhilin, G. N.: Determination of the Temperature of the Surface of
a Non-heat Conducting Body in a Very Rapidly Moving Stream of an
Incompressible Fluid. JTF, vol. VI, no. 9, 1936.

25. Kruzhilin, C. 1.: Theory of the Heat Transfer of a Circular Cylinder.
JTF, vol. VI, no. 5, 1936.

26. Kruzhilin, G. 1., and Shvab, V. A.: Investigation of the a-Field on
the Surface of a Circular Cylinder in a Transverse Air Flow. JTF,
vol. V, no. 4, 1935.

27. Leibenzon, L. S.: Energy Form of the Integral Condition in the
Theory of the Boundary Layer. Trudy CART, no. 240, 1935.







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28. Loitsianskii, L. G.: Aerodynamics of the Boundary Layer.
Gostekhizdat, 1941.

29. Loitsianskii, L. G.: Interaction of Boundary Layers. Trudy CAHI,
no. 249, 1936.

30. Loitsianskii, L. G.: Motion of a Fluid in Boundary Layer Along the
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31. Loitsianskii, L. C.: Laminar Boundary Layer on a Body of Revolution
DAN, vol. XXXVI, no. 6, 1942.

32. Loitsianskii, L. C.: Change in the Resistance of Bodies by Having
the Boundary Layer Consist of a Fluid with Other Physical Constants.
Prikl. matem. i mekh., vol. VI, no. 1, 1942.

33. Loitsianskii, L. G.: On a Problem of the Three-Dimensional Boundary
Layer. Trudy LII, razdel fiz.-matem. nauk, no. 1, 1937.

34. Loitsianskii, L. G.: Universal Formulas in the Theory of the Resist-
ance of Rough Pipes. Trudy CARI, no. 250, 1936.

35. Loitsianskii, L. G.: Approximate Method of Computing the Laminar
Boundary Layer on a Wing. DAN, vol. XXXV, no. 8, 1942.

36. Loitsianskii, L. G.: Approximate Method of Computing the Turbulent
Boundary Layer on a Wing Profile. Prikl. matem. i mekh., vol. IX,
no. 6, 1945.

37. Loitsianskii, L. C.: 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. C., and Bolshakov, V. P.: The Motion of a Fluid in
the Boundary Layer Near the Line of Intersection of Two Surfaces.
Trudy CAHI, 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


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, C. I., and Shteinberg, R. I.: Investigation of the Nonsmooth
Flow About Bodies. Trudy CARHI, 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 CABI, 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.






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,
CAMI, 1935. (See also Tekhn. '.'ozd. flota, nos. 7-8, 1939.)

56. Fedyaevskii, K. K.: Thermal Boundary Layer of a Wing. Trudy CARI,
no. 361, 1938.

57. Fedyaevskii, K. K.: Turbulent Boundary Layer of a Wing. Pt. I The
Profile of Frictional Stress and Velocity. Trudy CAHI, no. 282,
1936; Pt. II The Law of Resistance. Trudy CAHI, 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. II.: Effect of the Degree of
Turbulence of the Flow on the Resistance of Bodies. Tekhn. zametki
CAHI, no. 126, 1936.

61. Fedyaevskii, K. K., and Fomina, HI. 1.: Investigation of the Effect
of Roughness on tie 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 CAHI, no. 374, 1938.

63. Frankly, F. I.: Theory of the Laminar Boundary Layer in a Compressible
Gas. Trudy CAHI, no. 176, 1934.

64. Frankly, F. I.: Heat Transfer in a Turbulent Boundary Layer at Large
Velocities in a Compressible Gas. Trudy CAHI, no. 240, 1935.

65. Frank, F. I.: Friction in a Turbulent Boundary Layer at Large
Velocities in a Compressible Gas. Trudy CAHI, no. 240, 1935.

66. Frank, F. I., and Veishel, V. V.: Friction in a Turbulent Boundary
Layer About a Plate in Plane-Parallel 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


68. Shvab, V. A.: Theory of Heat Transfer in Turbulent Flow. Trudy
Leningr. industry. 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. industry. inst., razdel
fiz. matem. nauk, no. 1, 1937.


Translated by S. Reiss
National Advisory Committee
for Aeronautics


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