Approximate method of integration of laminar boundary layer in incompressible fluid


Material Information

Approximate method of integration of laminar boundary layer in incompressible fluid
Series Title:
Physical Description:
21 p. : ill ; 27 cm.
Loĭt︠s︡i︠a︡nskiĭ, L. G ( Lev Gerasimovich )
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Laminar boundary layer   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


A method is given for the approximate solution of the equations of the two-dimensional laminar boundary layer in an incompressible fluid. The method is based on the use of a system of equations of successive moments that is easily solved for simple supplementary assumptions. The solution obtained is given in closed form by simple formulas and is claimed to be no less accurate than the complicated solutions previously obtained, which were based on the use of special classes of flows.
Includes bibliographic references (p. 19-20).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by L.G. Loitsianskii.
General Note:
"Report date July 1951."
General Note:
"Translation of "Priblizhennyi metod integrirovania uravnenii laminarnogo pogranichnogo sloia v neszhimaemom gaze." Prikladnaya Matematika i Mekhanika, USSR, Vol. 13, no. 5, Oct. 1949."

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University of Florida
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By L. G. Loitsianskii

Among all existing methods of the approximate integration of
the differential equations of the laminar boundary layer, the most
widely used is the method based on the application of the momentumI.
equation (reference 1). The accuracy of this method depends on
the more or less successful choice of a one-parameter family of
velocity profiles. Thus, for example, the polynomial of the fourth
degree proposed by Pohihausen (reference 1) does not give velocity
distributions closely agreeing with actual values in the neighbor-
hood of the separation point, so that in the computations a strong
retardation of the separation is obtained as compared with experi-
mental results (reference 2). The more-accurate methods employed
in recent times (references 2 to 4) assume as a single-parameter
family of profiles the exact solutions of some special class of
flows with given simple velocity distributions on the edge of the
boundary layer (single term raised to a power, linear function).

The transition to the more complicated two- and more-parameter
families of profiles would require, in addition to the momentum
equation, the employment of other possible equations (for example,
the equations of energy (reference 5) and others (reference 6)).
A greater accuracy might also then be expected fcr relatively simple
velocity profiles that satisfy only the fundamental boundary con-
ditions on the surface of the body and on the edge of the boundary
layer. This second approach, however, as far as is known, has not
been considered except for very simple solution for the case of
axial flow past a plate (reference 7).

In the present paper, a solution is given of the problem of
the plane laminar boundary layer in an incompressible gas; the
method is based on the use of a system of equations of successive
moments (including that of zero moment, the momentum equation) of
the equation of the boundary layer. Such statement of the problem

"Priblizhennyi Metod Integrirovania Uravnenii Laminarnogo Pogra-
nichnogo Sloia v Neszhimaemom Gaze." Prikladnaya Matematika i Mek-
hanika, USSR, Vol. 13, no. 5, Oct. 1949, p. 513-525.

NACA TM 1293

leads to a complex system of equations, which, however, is easily
solved for simple supplementary assumptions. The solution obtained
is given in closed form by very simple formulas and is no-less
accurate than the previously mentioned complicated solutions that
are based on the use of special classes of accurate solutions of
the boundary-layer equations.

1. Derivation of Fundamental System of Successive Moments of
Boundary-Layer Equation. The well-known equations of the stationary
plane laminar boundary layer in the absence of compressibility
have the form
Ou ou 0 U
u + v -- = jUU' + u

x ~(1.1)

ou ov
-- + E 0

where u(x,y) and v(x,y) are the projections of the velocity at
a section of the boundary layer on the axial and transverse axes of
coordinates x and y, U(x) is a given longitudinal velocity on
the outer boundary U' = dU/dx, and U is the kineratic coeffi-
cient of viscosity. When the equation of continuity is applied,
the first of equations (1.1) may b! given the more convenient form
L(u,v) = [u(U-u)1 + [v(U-u)] + U'(U-u) 0 (U-u) = 0
ox oy 2
The left side of equation (1.2) is multiplied by 9' and
integrated with respect to y from zero to infinity in the case
of an asymptotically infinite layer or from zero to the outer
limit of the layer y = 5(x) for the assumption of a layer of
finite thickness. In either case, the following expression is

L(u,v)yk dy = -U-u) dy + fy [v(U-u)] dy +
fo dx 0 0o oy

U' y (U-u) dy U

I k q2(U-u)
y k 2 dy = 0
y y2


FTACA TM 1295 7

It is assumed in this equation and in what follows that, in
view of the very rapid approach of the velocity difference U-u
to zero as y--4 all integrals with the infinite upper limit have
a finite value.

For k = 0,

d- f u(U-u) dy U' (U-u) dy (1.4)
ax 0 0 .p

where the magnitude

"V = ^(^)}= (1.5)
Ty 1=0

represents the friction stress on the surface of the body.

Equation (1.4), the well-known impulse or momentum equation,
is readily transformed into its usual form

db** UIB** (
---- + (2+H) = (1.6)
dx U p2


j= ( dy
\ U}

,5 > (1.7)

S = 1 -u dy
0 U U

H 5*

For k = 1, a new equation of the 'first moment' is obtained
from equation (1.3)

NACA TM 1293

d- yu(U-u) dy -
dx Jo

v(U-u) dy + U' T

y(U-u) dy = VU

and, in general, for k > 2, the equations of successively increas-
ing moments are obtained

S| yku(U-u) dy k y-v(U-u) dy + U' yk(U-u) dy
U0 0 0

= k(k-l) OT

yk-2(U-u) dy

In all these equations, the transverse velocity
assumed expressed in terms of the axial u(x,y) from
of continuity.

It is now assumed that the family of functions

u = I? (x, y;, 2 X k)

v(x,y) is
the equation


satisfies the boundary conditions of the problem with k param-
eters X, k, which are functions of x, such that the
k successive moments of equation (1.3)


ykL(uo, v0) dy (1.11)

become zero. On the assumption that it is permissible to pass to
the limit k- *, it would then be possible to state that the function

u(x,y) = lim

uo x ; (x), X2(x), ,X k(XA

with parameters Xl(x), K2(x), Xk(x) satisfying the infinite
system of equations



NACA TM 1293

yakL(uo, v) dy 0 (k = 0, 1, 2, )

or, what is equivalent, systems (1.4), (1.83), and (1.9) will be an
exact solution of the fundamental system (1.1) for the assumed
boundary conditions.

For this solution, it is i.;?relV necessary to recall the known
theorem that a continuous functir.n, c-all successive derivatives of
which are eoual to zero, is identically e0ual to zero (reference 8).

The question of the proof ocf the validity of this theorem is
not considered in the case of an infinite interval or of an inter-
val the boundaries of which are fuiinctions of a certain variable
with respect to which the differentiation is effected. A certain
construction, not based it is true on a rigorous proof, of the
solution cof the problem will -be emplc.:.ed with the aid of the suc-
cessive e-uations of the mome-nts of the basic boundar.,

2. Choice of 1-ara:eters of FarJi.l,- of Velocity Profiles at Sec-
tions of Boundary La:cr. Special Form of Equations of Moments.
As is seen from the previously discussed considerations, the funda-
mental difficulty lies in the choice of a family; of vc-locity pro-
files (1.10) and the determination cf the parameters hfk of the
family. One of the simplest :rethods of the solution of this prob-
lem is indicated herein.

In the converging part of the toundar; layer, the velocity
profiles at various sections of the layer are known to be almost
similar; the velocity profile is deformed mainly in the diffuser
part of the boundary layer downstream of the point of minimum pres-
sure. The deformation of the profile consists of the appearance
of a point of flexure that arises near the surface of the body and
then moves awa. from it as tho separation point is approached.

The presence of this deformiation of the profile near the sur-
face should greatly affect the magnitude T prporticnal to the
normal derivative of the velocit& on the surface of the body:,; it
will therefore diminish to zero as the point of separation is
approached. The deformation of the profile will have a smaller
effect cr such Integral magnitudes as Q* Mni 5** And ver,
little esfec+. cn magnItudes that contain under the !rtCilcr.l
s!;n 'runcth-ns t.h'-it. raridll;' de5reas?9 as the srrfaoe of 'r.e 1 :iy
s orprach?'.

NACA TM 1293

For the parameters characterizing the effect of the deforma-
tion of the velocity profile, it is natural to assume those magni-
tudes that depend relatively strongly on the deformation of the
velocity profile. With regard to the magnitudes that vary little
with the deformation of the velocity profile, however, it is natural
to assume that they do not depend on the chosen parameters.

For the fundamental parameters determining a change in the
shape of the velocity profiles, which may be called form parameters,
the nondimensional combination of the magnitudes Tv, B* and 5**
will be employed with the given functions U(x) and U'(x) and
physical constants, namely, the parameters

f U'8o**2

r= Ts** (2.1)
k-(y/b**) }y= 1U

H --

For the computation of the remaining magnitudes in the equa-
tion of moments according to the assumption, the velocity profile
will be assumed in a section of the boundary layer in a form that
does not depend on the parameters f, t, and H:

U ,-).,(,) (2.2)

This assumption permits, as will be subsequently seen, obtain-
ing on the basis of very simple computations a sufficiently accurate
solution of the boundary-layer equations for arbitrary distribution
of the velocity on the edge of the layer. The transformation of
equation (1.6) will now be considered.

If the parameter ( is introduced, then by equation (2.1),

dL** U'8** P
,1, Ufb** +(2+H) =
dx U U 8**
or I2 ^
o U _d (6 *4f2
+ (2+H) f=

NACA TM 1293

It is not difficult to obtain finally

1 U df + H 1 H U "f O (
T x 2+ ~ f = 2 (2
(2 U'dx 2 U,2
For the transformation of the left side of equation (1.8)
the first integral can be written by equation (2.2) (i = y/l*)

yu(U-u) dy = U23**2 1p(l-p) dj = H1U2**2 (2
.0 0

where the magnitude

H1, equal to
er (
H1 = 9p(14) dTi


represents a constant computed by the given function T(j).

In order to compute the following integral, the transverse
velocity v is first expressed by the formula

v = j dy = U% J 9 d)
ox ox
Jo\J0 0

qdrq U
0 0

d JU 5** <9 dx

or, when it is noted that

in d ( ) ")
dx dx 5*

Y dB**
g *2 dx

1 d5**
1 ** dx

the following expression is obtained:

dj U d

/ f I
1 0 1 di /



v = U'F**


= U'3**

NACA TM 1293

There is thus obtained

v(U-u) dy = U25** ) dTI

S UU'**2 ( jp 9 d)(14 ) dti -
uUY'**2Js7J0' Pdn(19 dq


v(U-u) dy = HU2 5** H5 JU' 2 (2.7)
where H2 and H. denote the constants

H2 = ( dT (1-p) dy(
J0 \ 0 \

b = f q~dy](1p) dy (2.8)
H3 d) 1- dq

Finally, the last Integral in equation (1.8) is transformed into
y(U-u) dy = H4U 8**2 (2.9)

where the constant H4 is equal to

H4 = Tl(l-p) dyl (2.10)

NACA TM 1293 9

By substituting the integrals obtained in equation (1.8),

d (HA -KU 2) HU2 d + UU 2 + LUU'8*2 = u
dx L / 2 dx 4

or by replacing 8**2/v = f/U' by equation (2.1) aind carrying out
the transformation,

( H 1 ) = -[1- (2~1+3+)f +37 )f (2.12)

When the nev constants are introduced,

Hi 1 Hz
Bl-J 2
b 2 H, + H+(2.13)

df U1(a-bf U (2.14)

The third equation is obtained from the system (1.9) "by setting
bk = 2.

-d ( y~u(U-u)dyv 2 /yv'U-u~dy + U1 7 2y^U_-dy 2v (U-u)d:y
dxo Jof Jo Jo
the equation of the first moment is reduced to the form

df U' U"
dx u(a-bf) + -- (2.14)

The third equation is obtained from the system (1.9) by Betting
k = 2.

y~uU-udy- 2Jyv(U-u~dy + U'fY2(U-u)dy = 2uJ (U-u)dy

There is obtained, as before,

2 u(U-u)dy = HgU2**3 (2.16)

10 NACA TM 1293
where the constant Hc is equal to

H5 = 2p(1-9)da (2.17)

Further, by analogy with equation (2.7)

| yv(U-vu)dy 6U2r**2 d* H7UU'r**3 (2.18)
Jo dx


H6 f T( P f- T dt d (1-9p)dyl

) (2.19)

H7 = pdj (1-cp)di
0 0C

The last integral on the left side of equation (2.15) is equal

f2(U-u)dy = HgU 5**3 (H8 f 2(l-9)d) (2.20)

The integral in equation (2.15) on the right reduces to the
unknown p-rameter H11

(U-u)y = ) dy = U ** = U ?**H (2.21)
Jo Jb-~d = Ur 8U 3*
By substituting the expressions obtained for the integrals in
the second-moment equation (2.15), there is obtained, after simple

S(3H5-2H6) = ][H (H+H7+ ) f + (3B5-2H6) Z f


NACA TM 1293

The system cf three equations (2.3), (2.14), and (2.22) has
thus been established for determining there three unknown magnitudes
, and H. The solution of this system is now considered.

3. Determination of the Constants H.. Approximate Formulas
for Paranieters f, r, and H. For the determination of the numeri-
cal values of the constants H1, H2, ., H8, the form of the
function (c (1) must be known. The simplest velocity profile in
the theory of the asymptotic boundary layer is the velocity profile
in the sections of the boundary layer of the flow past a plate.
The function (pi) for this case can easily be determined from
the generally known table of values of the velocity ratio u/U as
a function of ; = Ux/2.

Superfluous computations may be avoided by noting that the con-
=*.s to be crin'uted are connected with one another by certain
sim le reltAtions.

First of all, frcm equations (2.3) and (2.14),

= a + + U f (3.1)

5y setting f = C, there is obtained a = 2Co, where t0
denotes the magnitude 6 ccmpuLed for the plate (U = 0, f = 0).
From' the definition of C and frcm the known relations for the

S= 2 ,- 0.664 .32 = .664 = 0.4408
U i U 11 VU

Further, by comparing with one another the magnitudes H1, H2,
H_, and 84,

H3 = Bl H2 (3.3)

H4 H3 = I( IdTI (1-.;p)dy = (1-p)dl 2 ( 02


NACA TM 1293

where H0 is the value of H for f = 0, that
for a plate is equal, as is known, to

H0 = 12 = 121= 2.59
6** 0.664

is, the ratio

It is
(3.3), and,

then easy to obtain the value of b

by equations (2.13),

b = 2H1+H3+4 = a (2H1+H1-H2+H1-H2+ "H62)

= a (4H1-2H2+ 1H2) = 4 + a H02 = 5.48 5.5 (3.5

When df/dx is eliminated from equations (2.22) and (2.14),

i ~3H5-2H6
H = (5+H7+ -8) f + (2H1+H3+H4) f
4(H1- 2H2)

= (H5+H7+ H) f + (3H5-2H6) 1 f (3.6

By setting f = 0,

S(3H5-2H6) = = 5.89 (3.7
= -a =0.4-4Z58

The only magnitude that must be computed again
of values 4(i) is the magnitude H5 + H7 + H8/2,.
gration gives

H5 + H7 + H8 = 24.73

after which there is immediately obtained

from the table
Numerical inte-


H = 2.59 7.55 f





NACA TM 1293

Substituting this expression for R in equation (3.1) gives

C = 0.22 + 1.85 f 7.55 f2 (3.10)

Finally, integrating the simple linear equation (2.14) gives

aU' b-1 0.44U' U4.5
f ub Uo )u 5 Jo (3.11)
U 0 ou.5 Fo

Equations (3.9), (3.10) and (3.11) give the required solution.

The simple, approximate solution just obtained is now compared
with the actual values. The almost complete agreement of the val-
ues of f obtained with the first approximation (which is practic-
ally the only one that is applied) of the preceding works (refer-
ences 2 and 3) will be noted. The closed-form relation between (
and f likewise differs little from the corresponding tabulated
functions in the references cited.

For comparison, the curves C(f) and H(f) obtained accord-
ing to the formulas of reference 2 and by the formulas (3.10) and
(3.9) are shown in figure 1. The results obtained will also be
compared with the formulas of Wright and Bailey (reference 9). An
approximate method of computation of the laminar boundary layer is
proposed therein in which the equation of momentum (1.6) is employed
with T. and 6** substituted by the formulas for the flow past
a plate. By expressing the results of Vright and Bailey in the
parameters of the present report, the analogs of equations (3.9),
(3.10), and (3.11) are obtained.

H = 2.59

C = 0.22 + 4.09 f (3.12)

f = 0.44 UJ

It is easily seen that this formula for f corresponds to
equation (3.11) for b = 1. The straight lines for ( and H
shown dotted in figure 1 indicate the considerable deviation of
the formulas of Wright and Bailey from more accurate formulas
presented herein.

NACA TM 1293

For confirmation, the particular case of the laminar boundary
layer corresponding to the so-called single-elope velocity distri-
bution at the outer boundary of the layer U = 1- x will be con-
sidered. This case has been theoretically solved and an exact
solution in a tabulated form (reference 10) is available. The
results of the recomputation of these accurate solutions in the
form assumed by the parameters are given in figure 2. Also shown
for comparison are the corresponding curves obtained by the pro-
posed approximate method and by the method of Wright and Bailey.

4. Possible Methods of Rendering the Foregoing Solution More
Accurate. The method described in the preceding sections was based
on the assumption of a slight dependence of Hi on the form param-
eters f, C, and H. This assumption may be eliminated and the
method rendered more accurate, although it thereby becomes con-
siderably more complicated.

In order to discuss the possible generalizations of the method,
the complete system of equations, for example, for the three-
parameter case is written out; that is, a three-parameter family
of velocity profiles is assumed in place of equation (2.2).

f, H) (4.1)

By substituting this velocity profile in the system of the
three equations of successive moments (1.6), (1.8) and (2.15),
there is obtained a system of three ordinary nonlinear differential
equations that determine the magnitudes of the parameters f, (,
and H:

1 U df 1 UU"
ZI Uf+ 2 f + Ef (4.2)
F U'-dx 2 U'

r [1 ( ++4 + (x1- )H2 ) f (4.3)
H2 \+ o+ )f +dx +2+ dx (5 + / dx

U [1 (2H,+H3+H4)f] + '1(]l H2) f (4.3)

MACA TM 1293

["1 ~ ~ M5 f H.
It (35-2E6) + K4 ) 2

(K6 "+2H dx U

f (4
dx +K)


f +

+ R8f]

- (3H5-2H6) f

in which, in addition to the previous notations, the following
definitions are chosen:


K1 =

K2 *o

K3 L

K4 T Jf

K5 = 0 0

K6 = f9 (1-fq)d)
6 0 0



di)j (1- )dTq

7f d ) (1 p)dTi

a d: 1 ( 1 rp ) d y l

L d) (1-cp)dl
6f 1)

NACA TM 1293

It is noted that, in the system of equations (4.2), (4.3),
and (4.4), Hi and Ki are not constant magnitudes as pre-
viously, but known functions of the form parameters f, C, and H;
the form of these functions depends on the chosen family of
profiles (4.1).

The equations (2.3), (2.14), and (2.22) earlier employed evi-
dently represent a particular case of the system (4.2), (4.3), and
(4.4) on the assumption that the family of velocity profiles at
the different sections of the boundary layer has the form of equa-
tion (2.2); in other words, these profiles are similar to one
another. All values of Ki are of course then equal to zero and
Hi is constant.

The proposed method may be rendered considerably
by assuming, for example, the single-parameter family

= h(e; f)


=K3 =-- =K
3 OH 1

more accurate
of velocity


S= K6 = = 0
ac 2EO

and the system
as follows:

of equations (4.2), (4.3) and (4.4) is transformed

1 5df 1UU
SU + 2 f +
2 U,'dx ( 2 U12)


S 2 U1+ f f L1

- (2H1+H3+H4)f] +

(3H-25H6 z
= -H56 + f V 5 dx

UH (R5+H7+i 1: ])fl + IU"35-2H6
T \2 4j 411





i l H 2 f

NACA TM 1293

Equation (4.8) can be given the form

U' 1 (2H1+H3+H4)f U
ul (K++H/f U-
U 1 H}2 + (Kl+6H1/f~ U'

H1 1


which represents a generalization of equation (2.12) where equa-
tion (4.10) approximates equation (2.12) because of the small change
in Hi with change in the parameter f and the smallness of the
magnitude (Kl+3H1/of)f in comparison with H1 H2/2. This gen-
eralization permits obtaining the integral of equation (4.10) by
introducing a correction to the solution of equation (2.12).
By dividing both sides of equation (4.9) by the corresponding
sides of equation (4.8) and thus eliminating df/dx, there is
H = (H5+H7+: H8)f+

(3H5-2H6) +

H 1 H2 +

1K + f
Kl* F

1 (2Hl+H3+H4)f] .+

(H12 H 2)

1 (3H5-2H6) + + 1 Z-) 1f ( 1 f
-- 1 ( 3 H5 2 H6 ) 4
1 H2 + (K1 f(.

By similar considerations on the smallness of the magnitudes
(K4+1/2 oH5/6f)f in comparison with (3H5-2H6)/4 and of
(Ki+oH1/6f)f in comparison with H, H2/2 and on the slight
variability of Hi, it may be concluded that the value of H
determined by equation (4.11) is an improvement in the accuracy of
the approximate value of H according to equation (3.6).
It may be remarked that in this more accurate approximation
there is no longer that universal relation between the parameters H
and f, independent of the form of the function U(x), character-
izing the given particular problem. The presence in equation (4.11)


NACA TM 1293

of a second term with the factor JU't/U'2 shows that in the more
accurate approximation the magnitude H in a given section of the
layer depends not only on the value of the form parameter f in
this section, as was the case in the rougher approximation of equa-
tion (3.6) or (3.9), but also on the value of the magnitude UU"/U'
in the section considered, that is, on the values of the func-
tion U(x) and its first two derivatives. It is readily observed
that the second term on the right side of equation (4.11) will
give a small correction to the solution (3.6) for relatively small
values of the magnitude UU"/U'2.

The same considerations hold for the expression for t, which
may be obtained by substituting df/dx from equation (4.10) and
H from equation (4.11) into equation (4.7):

S1 ~ (21+8*4 1l 1 /, H^JC.1 ^ f8
1- (2E+H3+H4)f F+ 1 (3H5-2H6)f +('K4 1 f 2] +
1 1 H2 L2f 4 2 Zif
51"2 2 + Si Kl)g~

1 2
2f + (H5+7+ H8)f2-
/f[ +H, or1 f,3/ i H i \-
U K (+ f2 + (3H5-26) K4 + 2- --2)
U12 1 (3H
HJ l 2 2 + 1+
2 \ (4.12)
As is seen, in this new approximation, in contrast to the pre-
ceding one, there is no universal relation between t and f. The
presence of a term with the factor UU'"/U,2 makes the magnitude t
depend not only on the value of the parameter f but also on the
form of the function U(x) and its first two derivatives in the
given section of the boundary layer.

It is of interest to remark that in this approximation the
position of the point of separation of the boundary layer, that is,
the value of x = x. for which C is equal to zero, will no
longer be determined by some universal value of the form param-
eter f., but in each individual case the value of x = X8 must
be determined for which the right side of equation (4.12) become

NACA TM 1293

By assuming a particular form of a family of velocity profiles
(4.6), employing, for example, the sets of velocity profiles
applied in the previous investigations (references 2 to 4), the
values of the functions Hi and Ki are determined; the form
parameters f, (, and H, that is, the thickness of the momentum
loss **, the friction stress Tw and the displacement thick-
ness 5* may then be found without difficulty. The solution
of equation (4.10) and the determination of H and t by equa-
tions (4.11) and (4.12) offers no particular difficulty. Further
improvement in the accuracy requiring the solution of a system of
the type of equations (4.2), (4.3) and (4.4) is hardly of practical

In the previous discussion, the scheme of the asymptotically
infinite boundary layer was used, but similar equations may be
obtained also for the case where the boundary layer is assumed to
be of finite thickness.

The method here proposed may evidently also be applied to the
case of the thermal boundary layer. The characteristic feature of
the method for the cases of both the dynamic and the thermal bound-
ary layer lies in the fact that the friction stress and the quantity
of beat given off by a unit area of the body are expressed in inte-
gral form and not in terms of the derivatives of functions that
represent the approximate velocity and temperature distributions
in the sections of the boundary layer.

Translated by S. Reiss
National Advisory Committee
for Aeronautics.


1. Loitsianskii, L. G.: Aerodynamics of the Boundary Layer.
GTTI, 1941, pp. 170, 187.

2. Loitsianskii, L. G.: Approximate Method for Calculating the
Laminar Boundary Layer on the Airfoil. Comptes Rendus
(Doklady) de l'Acad. des Sci. de L'URSS, vol. XXXV, no. 8,
1942, pp. 227-232.

3. Kochin, N. E., and Loitsianskii, L. G.: An Approximate Method
of Computation of the Boundary Layer. Doklady AN SSSR,
T. XXXVI, No. 9, 1942.

NACA TM 1293

4. Melnikov, A. P.: On Certain Problems in the Theory of a Wing in
a Nonideal Medium. Doctoral dissertation, L. Voenno-
vozdushnaia inzhenernaia akademia, 1942.

5. Leibenson, L. S.: Energetic Form of the Integral Condition in
the Theory of the Boundary Layer. Rep. No. 240, CAHI, 1935.

6. Kochin, N. E., Kibel, I. A., and Roze, N. V.: Theoretical
Hydrodynamics, pt. II. GTTI, 3d ed., 1948, p. 450.

7. Sutton, W. G. L.: An Approximate Solution of the Boundary Layer
Equations for a Flat Plate. Phil. Mag. and Jour. Sci.,
vol. 23, ser. 7, 1937, pp. 1146-1152.

8. Carslaw, H., and Jaeger, J.t Operational Methods in Applied
Mathematics. 1948.

9. Wright, E. A., and Bailey, G. W.: Laminar Frictional Resistance
with Pressure Gradient. Jour. Aero. Sci., vol. 6, no. 12,
Oct. 1939, pp. 485-488.

10. Howarth, L.: On the Solution of the Laminar Boundary Layer
Equations. Proc. Roy. Soc. (London), vol. 164, no. A919,
Feb. 1938, pp. 547-579.

NAjA TM 1293


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