On motion of fluid in boundary layer near line of intersection of two planes

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Title:
On motion of fluid in boundary layer near line of intersection of two planes
Series Title:
NACA TM
Physical Description:
26 p. : ill. ; 27 cm.
Language:
English
Creator:
Loĭt︠s︡i︠a︡nskiĭ, L. G ( Lev Gerasimovich )
Bolshakov, V. P
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

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

Notes

Abstract:
A motion of a slightly viscous incompressible fluid that flows between two semi-infinite planes which intersect at an angle between 0° and 180° is discussed. The increase in thickness of the boundary layer near the line of intersection of the two planes, the limits of the effect of the corner, and the effect on the drag are determined. The analysis is made for both laminar and turbulent flow by using the momentum theorem and assuming suitable velocity profiles.
Bibliography:
Includes bibliographic reference (p. 23).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by L.G. Loitsianskii and V.P. Bolshakov.
General Note:
"Report date November 1951."
General Note:
"Translation of "O dvizhenii zhidkosti v pogranichnom sloe vblizi linii peresechenia dvukh ploskostei." Rep. No. 279, CAHI, 1936."

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University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779949
oclc - 98954605
sobekcm - AA00006211_00001
System ID:
AA00006211:00001

Full Text
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NACA TM 1308


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL M~IIMORANDUM 1308


ON MOTION OF FLUID IN BOUNDARY LAYER NEAR LINE

OF INTERSECTION OF TWO PLANES*

By L. G. Loitsianskii and V. P. Bolshakov


SUMMARY

In the paper "The Mutual Interference of Boundary Layers," the
authors investigated the problem of the interference of two planes
intersecting at-right angles on the boundary layers formed by the motion
of fluid along the line of intersection of these planes.

In the present paper, the results of the preceding one are general-
ized to the case of planes intersecting at any angle. The motion of a
fluid in an angle less than 1800 is discussed and the enlargement of the
boundary layers near the line of intersection of the planes, the limits
of the interference effects of the boundary layers, and the corrections
on the drag are determined. All computations are conducted by the
Karman-Pohlhausen method for laminar and turbulent boundary layers. The
results are reduced to tabulated form.


INTRODUCTION

The problem of the interaction of the boundary layers formed in the
dihedral angle between two thin plates parallel to the intersection of
the plates was apparently first proposed in reference 1 (for the case
of a right angle). In the discussion of reference 1, the need for a
generalization of results obtained for the case of the right angle to
the case of any angle less than two right angles (1800) was pointed out.
The present paper is concerned with this problem and its solution.
Although the limits of application for the approximate method of the
finite-thickness layer previously used are retained, the problem of the
interaction of the boundary layers near the intersection of a dihedral
angle of any magnitude from 00 to 1800 is solved herein. For a laminar
layer, a first and a second approximation are given and also, for a
check, a sixth approximation (in the terminology of Pohlhausen). It is


*"0 Dvizhenii Zhidkosti v Pogranlchnom Sloe Vblizi Linii Peresechenia
Dvukh Ploskostei." Rep. No. 279, CAHI, 1936, pp. 3-18.








NACA TM 1308


shown that the sixth approximation differs comparatively little from the
second. In conclusion, the case of a turbulent boundary layer is considered
with the assumption of the validity of the '1/7' power law for the velocity
profiles. As in reference 1, all computations can be carried through to
the end, although the procedure is somewhat more cumbersome. The limits
of interference of the layers and the correction on the drag due to the
interference effect are determined.


1. DERIVATION OF FUNDAMENTAL INTEGRAL CONDITION

Consider the flow of a fluid, approaching from infinity with
velocity V, in the dihedral oblique angle 0 between two plates of the
same finite length x along the flow and infinite in the transverse
direction (fig. 1). The Y- and Z-axes are taken along the leading edges
of the plates and the X-axis along the flow parallel to the line of
intersection of the plates. In the oblique system of coordinates thus
obtained, the distribution of the velocities in the boundary layer may
be given in the same manner as for the case of the rectangular system
of coordinates for the flow in a right angle.

If each plate worked independently of the other, there would be
formed on it a layer of thickness 60, which is a function only of x
and the profile of the longitudinal velocities

uO = uo(x, z, sin 0)

or

UO = u0(X, y, sin 8)

depending on whether the boundary layer is considered to lie in the
plane XOY or XOZ.

Because of the retarding effect of one of the plates on the other
near the line of intersection, the layers must not be considered as in
the plane problem. The layers become three dimensional and the thick-
ness will now be a function of two variables:

8 = 8(x,y) on the plate containing the Y-axis

5 = 8(x,z) on the plate containing the Z-axis

The component of velocity parallel to the X-axis, which will be denoted
by u without the subscript 0, will be a function of the three
variables; that is,


u = u(x,y,z)









NACA TM 1308


By the conditions of the problem, under the basic assumption of the
concept of a finite region of influence of the viscosity, this function
must become zero on the surface of the plates, a constant at the outer
limit of the boundary layer, and, particularly, at a finite distance from
the line of intersection of the plates, must become the velocity distri-
bution uo(x,z, sin e) or uO(x,y, sin e), which corresponds to the
isolated plate with boundary layer undisturbed by the adjacent plate.
The boundary separating the region disturbed by the adjacent plate from
the undisturbed region, which corresponds to the isolated plate, shall,
for briefness, be denoted as the "interference boundary" of the layers.
The equation of this boundary in the planes XOY and XOZ will be

ho = ho(x)

A section of the boundary layer cut by the plane x = ( is shown
in figure 2. Inasmuch as the coordinates of the system YOZ are
oblique, the equation of the boundary layer in this section for the
undisturbed region for y>y2((), where
2(S) = h0() zl() cos e = h () 80() cot 0, will have the form


z = l( s sn (1)

Equation (1) holds true both for <- and for 0>-. The non-

coincidence for 0 / of the interference boundary ho0() and the
coordinate y0(r) should be noted, from which starting point the
ordinate z of the outer limit of the boundary layer becomes and remains
constant, independent of y. All that has been stated about the YOX
planes also remains true, of course, for the ZOX plane (because of
the symmetry y1(f) = z1() and Y2(W) = z2(W), which should be remembered
in the following development).

For the present, the question as to the equation of the boundary
layer in the section x = t will be ignored. The fundamental integral
condition of the problem will now be set up. For this purpose, as in
the work previously cited, the momentum theorem is applied for a tube
of flow formed by the coordinate planes, by the surface of the lines of
flow passing through the edge of the boundary layer at section x = x,
and, finally, by the surfaces of the lines of flow passing through the
perpendiculars to the plates located at the points y = h0(x) and
z = h0(x), where the part of the flow tube considered is between the
sections x = x and the section located upstream of the flow at a
sufficient distance from the section x = 0, that is, from the entry of
the fluid on the plates. Then, as is known,







NACA TM 1308


p(V-u) u do = W


f '7)


where W is the drag of the plates applied to the segment of the flow
tube considered and 0 is the section of the boundary layer cut by a
plane perpendicular to the X-axis at distance x from the origin 0.
The double integral on the left may be expressed in the following
manner (fig. 2):


S(0)


(V-u) u da =


sin 0


dy
0o


p(V-u) u dz +


y2 Z
fdy


cos 9


p(V-u) u dz +


y2+z 1 cos -y
cos S
dy
0


p(V-u) u dz


where Z, the ordinate of the outer edge of the boundary layer, is, as
yet, an unknown function of y and x if the angle 9 is considered
as a parameter maintaining a constant value for the given problem. The
first two integrals have a very obvious origin; the presence of the last
integral is due to the obliqueness of the coordinates and the differences
in the direction between the ordinate zl(x) and the thickness of the
layer 60(x), which makes it necessary to take the integration over the
two areas of the triangles shown hatched in figure 2.

The drag W, as is easily seen from figure 3, is determined by the
integral


Y2+zl
2
72








NACA TM 1308


x= -ho()
W = 2
Jo JO


h(x)
Sdy + h(x) h d0 d = 2
h(0)


Lx [ y2(M)+zl() cos e


0 -o


'y2(x)+zl(x) Cos 0


yY2(+zl(t) cos 0


To dy d


where r denotes the friction stress in the region S disturbed by the
adjacent plate and r0 denotes the corresponding stress in the undis-
turbed region SO. The boundary between the regions S and SO is,
of course, the boundary of interference of the boundary layers.

Combining equations (3) and (4) yields, in general form, the
integral condition of the problem:


p(V-u) u dz +


p(V-u) u dz +


SY2+z1 cos e

Y2


y2+z1 cos 0-y
cos e
dy
Jo


p(V-u) u dz


-2 x[ 2()+zl() cos e
= 2 %
0o 0 J


Equation (5) is a generalization of the integral condition derived in
the previously cited paper for the case of a right-angled dihedral
(e = -). The limits of integration are as yet unknown functions of the
arguments. In order to determine the form of these functions, it is
first necessary that the shape of the boundary layer be given.


d dy +


y2(x)+zl(x) cos e
T dy + cos dy d
.y2(g+zl]) cos 9 -








NACA TM 1308


2. SHAPE OF CROSS SECTION OF BOUNDARY LAYER; VELOCITY DIAGRAMS;

SOLUTION OF PROBLEM BY FIRST APPROXIMATION

The section of the boundary layer on the plane x = 5 (fig. 2) is
considered and it is assumed that the equation of the curve defining the
edge of the boundary layer is

f(y,z) = a(,O)

yl() 7 y 72( ) (6)

z l() z z2()

The form of the function f(y,z) cannot be determined unless the
additional assumption is made as to the similarity of the approximate
velocity diagrams in the different planes x = r. The curve is sought
in the form

u f (yz)] (7)


where the function 9(t) is subjected to the conditions

p(0) = 0
y(8)
P(1) = 1 J

and the function f(y,z) is subjected to the conditions

f(O,z) = f(y,0) = 0 (9)

Then the diagram of velocities of equation (7) will evidently satisfy
the end conditions of the problem at the walls and at the edge of the
boundary layer of equation (6). Setting

y = g(t,e) y,

z = Z(C,e) z'

where Z(t,8) is a certain length characteristic for the section x =
and y' and z' are nondimensional magnitudes, yields equation (7) in
the form
u rf(-y', -z')
7 = ^L a(r,)J








NACA TM 1308


From the condition of similarity of the velocity diagrams, the right side
of this equation must not contain t, which can be the case if f(y,z)
is a homogeneous function of the nth degree, so that

U P -e) f(y Z)
v a((,e) (',Z

where


n(,,,) = constant (10)
a(,,e)

the constant depending, of course, on 8. It is easy to see that the
degree of homogeneity should be equal to 2 (n = 2) and the function
f(y,z) must simply be equal to the product of the variables yz because
otherwise, with conditions (9) satisfied, the derivatives of the velocity
with respect to the coordinates would become zero at the walls and would
not give any friction. From equation (6), the following equation of the
edge of the boundary layer at the section x = 5 is obtained:

yz = a(r,e) (11)

where, from equation (10), for example, choosing the coordinate
yl(e) = zl(t) of the undisturbed layer for the characteristic length
2(9,8) yields


a(k,0) = k(e) z2 ()= k(E) 502() (12)
sin2 8

where k(8) is a certain nondimensional constant parametrically
dependent on the angle 8.

It is now easy to obtain the magnitudes Y2(t), h0(t), and the
function Z(y,t,9) for given angle 8. From equations (11) and (12),

2 =a(( ) = k(), ) () 50() (13)
Zl2W = sin----e

where 5O(t) is a known function of t independent of the angle 8
and determined by solving the problem of the boundary layer of the
isolated plate.

From the definition of the interference boundary ho0() given in
equation (1),








NACA TM 1508


ho(,e ) = y2() + z(() cos e =[k() + cos 81.(i) k(O) + cos 0 ()
(14)

Finally, from equation (11), there directly follows that

Z(y,,e) a(,e) _k() 60 )
Z(y,,9) = (15)
Y sin2 Y Y

It is easily verified that, for y = y2(.), Z becomes z. (). The
velocity diagram in the disturbed region of the boundary layer will
therefore be smoothly joined with the velocity diagram in the undisturbed
region, that is, with the diagram of the isolated plate.

When the equation of the boundary layer is found, the required
velocity profile is obtained from equation (7):

u= v(a yz (16)
v -\ e 5

or from equation (15):
u 17)
V= ) (17)

If Y denotes the variable ordinate of the edge of the boundary layer,
that is, the magnitude that, from equation (11), satisfies the equation

Y.Z = a(r,9)

then

S(zr ) (17')

It is important to note that equations (16), (17), and (17') are true
not only for those values of y and z that satisfy inequalities (6):


Zy(,0)< z

but in the entire range of interest:








NACA TM 1508


0 gy < Y2.(91,e)

0o z z2(ge)

When the velocities of the points located in the rhomb

0< ya z ( )
0

are considered, the boundary layer for these points is as though infinite,
but the velocities are determined from computation on the hyperbolic edge
of the boundary layer.

The velocity profile has thus been determined and the edge of the
boundary, layer is known. substitutingg the values of u and Z in the
integral condition (5) yields an ordinary differential equation with one
unkno-wn a(~,e) inasmuch as all the magnitudes involved, including the
friction at the wall


s = 0 in z=0


0 sin 8\ z=


are expressed in terms of this function.

As is easily seen, however, the differential equation reduces to
a simple equation in finite form for determining the coefficient k(8)
appearing in equation (12). In order to obtain this single unknown
coefficient, certain boundary conditions are assigned for the function
P(t) and its derivatives, as in the classical KIrm-n-Pohlhausen method.

A consideration of the first approximation is the first step. The
function p(t) is subjected only to the conditions

p(0) = 0

(1)l = 1

that is, the velocity u(x,y,z) becomes zero at the walls and 1 at the
outer limit of the boundary layer, which leads to the profile








NACA TM 1308


u yz
v a(x,e)


(19)


For substitution in the integral condition, the integrals that enter
this condition are first computed:


dy
0 0J


y2
dy
%1 0


u dz = a(x,0) y dy


udz2 dy
u dz = dy
1


V y12z22 1a
z dz = -4 = a(x,)
Sa(x,9)44


u dz


2(xe) dy = Va(x,e) loge
2y 2


S2+z cos

Y2
72


Y2+Z1 Cos 0-y

dy
Jo


h(x)
V
= 2 a-x,9)
2 vx 2
a Je y2


h0(x)-y

y dy cos 0
y dy I
Jo


h (x)

a(x1B) cos2 0 2


z dz


2(x)-2ho(x) y + y dy


V 1 2 2
V- a(x,0) V12 cos e + y cos2
= a(x,) 1 12 1


Y2
71


a(x,0)
2
Y1


u dz








NACA TM 1308


In a similar manner, the integrals of the square of the velocity are
computed. The development will be limited to computing the integrals
just given and the remaining ones are written out in ready form:


Yl' z2
dy
0 Jo


a(xl) e
Y2 F-
dy
JYI Jo


h0(x) os 8

Y2 dy0
2


u2 dz = V2 a(x,e)
u dz =


u2 dz= 1 V2 a(x,e) loge a(xe)
Y3
Yl


F Y4 c2
u2 dz = V Y12 cos + a(x,) +


1 y16 cos3 e
0 a2(x,0)

'Thus, the left side of the integral condition reduces to the form


J (a)p(V-u) u


do = pV2 sin 0 a(x,e) + a(x,0) loge


a(xe)
yl2


1 2 1 y14 cos2 e 1 y16 cos3 e
Sy cos 8 + 6 a(x,e) 90 a2(x,
6 1 60 a(x,8) 90 a2(x,9)


The computation of the right side of integral condition (5) reduces to











0x h

0 0


NJACA TM 1508


+2
T ay + h 0 d dy 2 x h
,Jho( ) si eJO


pV T- dy
a t Ye~d


sin 0
V. Jo


i V


( ) d +
a( CO)


2 a(x,0) f
71(x)


x x
l Ia


da + 2y l() cos e
Y() 1


According to the first approximation for the isolated plate (according
to Pohlhausen),




S 0 1 --____
Yl() sin 8 sin 8 V


.0


Ssin 8
Jo
0 0


dt
VTi


1 Y 50(x) sin 0


x

W= c os2 Co
sin 0
0O


ya2(9)
a(,8e)


l x
v0


a(f,e) dt +
yl2( )
2 / -
T^Wro


1 a(x e) V sin2 8 + 1 602(x) cos0
J


h (x)
hI W


,i3


so that


a(g,a)
y17Ft









FACA TM 1308 13


All these expressions are immediately simplified as soon as equation (12)
is considered; then

k(9) ,2/
a(5,e) k( ()
sin2 0

After all the simplifications are made, the following equation for
determining k(0) is obtained:

2 3
loge 2 cos 0 2 cos 8 1 cos 0 (20)
loge k() + 5 k2() 15 k3()(20)

It is readily seen that, for 8 = -, equation (20) becomes


log k =
e 3

k 1.95

as given in the previously cited work on the interference of the boundary
layers on mutually perpendicular plates.

The explicit dependence of k on 0 is not given in equation (20).
This relation may be obtained by the following simple device: When

cos 0
Ck os (2 1 )

equation (20) becomes

2 22 13
cos 8 = e3 5 15 (22)

When the values of Y are given in the interval where the absolute value
of the right side of equation (22) will not exceed 1, the corresponding
value of 8 is obtained, Y is determined, and then k(8) is found
from eq.iation (21). It would also be possible, of course, to proceed in
another manner: Equation (22) may be rewritten
2 + t + 2 + + t-

k(t) = e (22-)

From a given value of t, k(t) is obtained and then cos 8, and so forth.
The simplest method is to draw the function








14 NACA TM 1308


2 2 2 1 3
T = =() = .e 1(23)

and obtain its intersection with straight lines parallel to the Y-axis

Tq = cos

and then to obtain k from Y.

The values of 80, k(e), and. (0) are tabulated in -able 1.

The dimensions of the region of interference of the layers are first
determined. According to equation (14),

h0(x,e) = m(e) 0(x) (14')

where

k + cos 0 1+t
m sin cot 0 (24)

The values of m for different 6 can be determined from table 1
The value e = 1800 is somewhat isolated; for this value of P the
value of m becomes indeterminate. The value of m can easily be
obtained, however, by the usual device of analysis:


f+ cos2 2 (- ) in 0
cos 9=(
atn d + e= () -1
9=1

so that, for 0 = 1800, Y = 1 and ms 0. The values of m(e) are
given in the last column of table 1.

The correction in the drag of one side of the plate due to inter-
ference of the layers may be computed by the equation



AW = h (r -) dy d (25)


inasmuch as the difference in the drag may show up only in the region S
(fig. 3). The quantity 2 is the length of the plate in the direction
of the flow.








NACA TM 1308


The integral equation (25) is readily computed and yields

AW f 2Y2()+zl() cos 0




V 2(t)+zl (t) cos e
pV 1y
sin ( ) dy dt
sin 07 1( ) cos 0 a(,


i. 7V cos2 01
S2 sin 2 -T-

Finally,

AW = p(e) t vI
(26)
(e) = k2(e) cos2 e
2k(e) sin 0

If L denotes the width of the plate in the direction transverse
to the flow and the effect of its free end is not considered, the
relative correction due to the interference of the boundary layers may
be computed:

AW p(e) pi Vz p(0) 2lfv
S 0.578 2L v 0.578 L Z

or, when the Reynolds number RE of the plate, which is equal to VZ/v,
is introduced,

AW 2 1
Sq(e) (27)
W L iR

where q(e) denotes the relation

q(e) P(O)
0.578

Generally, the correction obtained is extremely insignificant for plates
that are long in the transverse direction. If, however, the transverse
length L is comparable with the width of the region of the disturbed
la.yr at the end of the plate x = 2, the correction is not insignificant.
Thus, if








16 NACA TM 1308




L = n-ho() = n-m(e) o0(l) = n-m(e) 12

the relative correction will now be equal to

AW q() 1
w -V72 m(0) n

or

AW (e)] percent
W (28)

where
100 q(e) (29)
s V (29)
V12 m(e)

All the magnitudes introduced in the preceding equations may be expressed
in terms of the previously given parameter g:

p(W) cot 0


q(M) = .1g cot e (30)
1.156 Y

s( 0) 100
1. 156 -,F2

In table 1, it is possible to find the values of these magnitudes for
different angles 0 or the corresponding values of the parameter t.
The magnitude s(e) increases with an increase of the angle e. Some-
what paradoxical is the value of 50 percent for s(e) at e0 = 180
and m = 0. It is found that the region of interference is equal to
zero, and that a relative effect occurs on the drag, which is due to the
fact that the width of the region and the absolute correction on the
drag simultaneously approach zero. On the other hand, when 0 decreases
to 0 and m-.m, s decreases to 16 percent. Both these cases are
limiting cases.

In figure 4, the curves of the relation between Y and e are
drawn for the first and other approximations and also for the case of
turbulent layers.









NACA TM 1308


3. SECOND AND SIXTH APPROXIMATIONS

The following approximations differ from the first only in the
boundary conditions that are imposed on the function p(t).

The second approximation corresponds to the assumptions

P(0) = 0

9P(1) = 1

p'(l) = 0

The velocity profile for these conditions will have the form


No new difficulties in principle, as compared with the first approximation,
are obtained. Inasmuch as the basic computations have been explained in
the first approximation, the results are presented with the same
notations.

The magnitudes are all given in table 2.

The final form of the equation for the determination of k(8) is


41 3 cos 8 15 cos2 e
120 + 4k() + 28 k2 +
k (e)


2 cos3 0
21 k3()


5 cos4 e
168 k4()


1 cos5 8
420k4()


(31)


The solution is effected by the same device as in the case of the first
approximation.

All computations were also carried out for the sixth approximation,
by which is meant the results of the assumption of the following
boundary conditions:


log k(0)
e








NACA TM 1308


p(o) 0 (1) = 1

4"(0) = 0 9(1) 0

9"' (0) 9 "(1) = 0

P"'(1) 0

This approximation gives, for the undisturbed layer, the velocity
profile



u0 = 5(g) +6 ) -2(

and the following values of the frictional force and layer thickness,
which are extremely close to the accurate solution of Blasius:


O0 = 0.331 Ip;



80 = 6.048 VJ

It is therefore reasonable to suppose that, for investigating the prob-
lem of the interference of the layers, the sixth approximation is
considerably more satisfactory than the second.

The results are again collected in table 3.

In figure 4, the curve of K against e is given for the second
and sixth approximations.

As before, the equation for the determination of k(e) is also
given

cos e cos2 e cos3 e
loge k = 0.1857 + 0.6370 C + 0.6107 0+ 0.1043
k k2


cos4 9 cos5 9 cos6 9 cos7 e
0.0524 + 0.0053 e + 0.0025 c 0- 0.0001
k k k k7


(32)








NACA TM 1308


4 INTERFERENCE OF TURBULENT BOUNDARY LAYERS

For the solution of this concluding part of the problem, the well-
known '1/7' power law is applied, which, when the boundary layer is
not two dimensional, assumes the form


u = V Yz]17 (33)

As is known, for an infinite plate without interference of the
layers, this law gives excellent agreement with experiment; it may be
expected that the application of the '1/7' power law to this case will
be justified by experiment.

The generalized integral condition (5) is substituted in the
velocity-profile equation (33). Unfortunately, the integrals will now
not be so easily evaluated because the presence of fractional exponents,
particularly in the integrals taken over the hatched areas of the
triangles shown in figure 2, strongly complicates the computations and
leads to the necessity of taking integrals of binomial differentials.
All the computations can be made with a sufficient degree of accuracy,
however, by using converging series and integrating them. The left
side of the integral condition reduces to the following expression:



p(V-u) u do = pV2 a(x,0) sin 0 0.1607 + 0.0972 log a(x) +
e yl2(X)


y2 cos 12 cos 012
0.1360 a ) 0.0231 a ) J +
a(x,e) a(x,0)

y1 2 cos 03 y12 cos ]4
0.0 ( ) J 0.0004 a(x,) + (34)


In computing the right side of the integral condition, the
KnCman formula is used for the expression of the friction at the wall
in the undisturbed region of the layer:


0 = 0.0225 = 0.0225 pv2 v I 5)
U 0\V~o) P 1 in








NACA TM 1308


In the disturbed part of the layer, the analogous formula


= 0.0225 P2 l1/4


= 0.0225 PV2 1/4


= 0.0225 pV2 V. sin 1/4


(36)


is assumed. These values are substituted, as in the previous sections,
in the formula for the resistance of the walls forming part of the
boundary of a tube of flow. Then,


wx- [ hO0)
W = 2
Jo o


T dy + \T~ dy d = 0.1943 pV2 o0() [xx +
I 11( 0)


y (x) cos j +


1 1
4 4
V v
Pi---
1
sin e


0.0090


S0.0090
\-


11 x
yl 4() c4s2 0
a(t,9) dt o0.0014J
a(r,e)


27
4
yl (() cos4
0.0006 d( 0.0003
a3 (,e)


Y1 () cos6 e
d0 (+ .
a5(,e)


a(Ce) d +
5
4


19
Y1 4(e) cos3 e
a2(,O) d +


35
4
Y1 () cos5 e
a4(g,e)


0.0056
^0


x0.0002


0.0002


(37)








NACA TM 1308 21


The Karmn formula was then used:

( 1/5
68(x) = 0.370 5 x

When it is known (see equation (12)) in advance that the solution of the
problem will have the form

2 02(I)
a(x,O) = k(e) y1 (x) = k(e) n2 (38)
1 sin 0

this value is substituted in equations (34) and (37); then, after some
simplifications, the following solution is obtained:

VF F 02x) (
Sp(V-u) u da = p s 0.1607 k+0.0972 k loge k+0.1360 cos e -
SJo sin e

cos2 e cos3 e cos4 e cos5 e
0.0230 + 0.0023 0.0004 + 0.0001 + .
k k k

280 cos2
W = p sin- e .1949 k + 0.1949 cos 0 0.0195 k + 0.0122 C -2
sin 6 k

cos3 e cos4 e cos5 e
0.00305 2 + 0.00133 + 0.00073 k + .
k2 k5 k4

Equating these two expressions yields the following equation for
determining k(e):

cos 8 cos2 e cos3 9
loge k = 0.1518 + 0.6056 + 0.3632 2 0.0556 k3 +
e k k2 k3

cos4 9 cos5 8 cos6 e
0.0175 0e- 0.0082 + 0.0051 + .
k4 k5 k6

As in the preceding sections, the change of variables is made:

cos (39)
kTe() = e) (39)








22 NAOA TM 1308


The transcendental equation then becomes

cos e = ( exp (0.1518 + 0.6056 + 0.3632 t2 0.0556 t3 +

0.0175 4 0.0082 t5 + .) (40)

This equation is easily solved by the tabular method, where

00< O< 1800 0.5540 >t>- 1.0000

The corresponding values of 0, k(e), and t(0) are given in table 4.
The further investigation in no way differs from that for laminar layers.
The values of the coefficients characterizing the boundary of the region
of interference and the corrections on the drag are given in table 4
for various values of 8, with all the computations committed, in the
notation previously used.

In connection with the formulas of turbulent friction, the
coefficients q and p are determined by the formulas:


AW = pp 2 2 ()2/5 (41)


AW v1/5

w L (41')

The dependence of p on t is determined by the following series:

p=- cot e (0.00133 0.00083 t2 + 0.00021 t3 0.00009 +4.+

0.00005 Y5 0.00003 6 + ) (42)



q = (43)
0.036

The dependence of t on e is given for the case of turbulent layers
in figure 4. The boundary of the region of interference (the
coefficient m(e)) in the case of the turbulent layers differs little
from the corresponding boundary of the laminar layers, according to
the sixth- approximation; whereas the relative correction on the drag s
in percent is several times less in the turbulent case.









NACA TM 1308


CONCLUSION

The results obtained have a readily understandable form. In general,
the effect of the interference of the layers on the drag of the plates
is insignificant. The effect assumes an appreciable value only in the
case where the plates in the dimensions transverse to the flow become
comparable with the width of the region of disturbed boundary layer.
Moreover, interference plays a large part in the motion of fluids through
small dihedral angles. Thus, for example, in the motion near the inter-
section of a dihedral angle of about 100, the region of interference
exceeds by 16 times the thickness of the layer at the given section.
At smaller angles, the phenomenon is still more marked.

All the conclusions of the present and preceding papers require
experimental check.

By agreement with the Central Aero-Hydrodynamical Institute, the
aerodynamic laboratory of the Leningrad Industrial Institute is under-
taking an experimental investigation of the phenomenon of the inter-
ference of boundary layers. It is proposed, through use of bhe method
of microtunnels, to observe directly the distortion in the velocity
profiles, and so forth, of the phenomenon.

The present work was carried out at the Aerodynamic Laboratory of
the Leningrad Industrial Institute.


Translated by S. Reiss,
National Advisory Committee
for Aeronautics


REFERENCE

1. Loiziansky, L. G.: Interference of Boundary Layers. CAHI Rep.
No. 249, (Moskow), 1936.









NACA TM 1308


TABLE 1


0 k(e) 1(e) m(e) p(e) q(e) a(e)
(deg) (percent)
0 .2.894 0.3455 6 16.3
10 2.880 .3419 22.257 7.324 12.671 16.4
20 2.840 .3309 11.315 3.697 6.396 16.7
30 2.773 .3123 7.278 2.503 4.330 17.2
40 2.682 .2856 5.364 1.916 5.315 17.8
50 2.567 .2504 4.190 1.570 2.716 18.7
60 2.434 .2054 3.388 1.346 2.329 19.8
70 2.283 .1498 2.793 1.187 2.054 21.2
80 2.120 .0819 2.329 1.069 1.849 22.9
90 1.948 0 1.948 .974 1.685 25.0
100 1.773 -.0980 1.624 .892 1.543 27.4
110 1.601 -.2136 1.340 .813 1.407 30.3
120 1.442 -.3467 1.088 .732 1.266 33.6
130 1.300 -.4945 .858 .641 1.109 37.3
140 1.185 -.6464 .6 2 .536 .927 41.0
150 1.099 -.7880 .466 .417 .721 44.7
160 1.042 -.9018 .501 .285 .493 47.3
170 1.010 -.9750 .144 .142 .246 49.3
180 1.000 -1.0000 0 0 0 50.0


TABLE 2


e k(0) t(e) m(e) P(e) q(e) s(0)
(deg) (percent)
0 2.217 0.4511 13.7
10 2.205 .4466 18.360 10.162 13.921 13.8
20 2.168 .4334 9.090 5.148 7.052 14.2
30 2.108 .4108 5.948 3.584 4.910 14.7
40 2.026 .3781 4.343 2.702 3.701 15.6
50 1.925 .3339 3.352 2.233 3.059 16.7
60 1.808 .2765 2.665 1.928 2.641 18.1
70 1.678 .2038 2.150 1.712 2.345 19.9
80 1.542 .1126 1.742 1.546 2.118 22.2
90 1.407 0 1.407 1.407 1.927 25.0
100 1.284 -0.1353 1.127 1.279 1.752 28.4
110 1.182 -0.2893 .894 1.153 1.579 32.2
120 1.108 -0.4513 .702 1.018 1.395 36.3
130 1.060 -0.6064 .544 .875 1.199 40.2
140 1.032 -0.7422 .414 .721 .988 43.6
150 1.016 -0.8524 .300 .556 .762 46.3
160 1.007 -0.9332 .196 .379 .519 48.3
170 1.002 -0.9832 .097 .192 .263 49.6
180 1.000 -1.0000 0 0 0 50.0








NACA TM 1308


TABLE 3

e k(0) (09) m(e) p(e) q(e) s(e)
(deg) (percent)
0 1.969 0.5079 m g 12.3
10 1.957 .5032 16.940 8.416 12.713 12.4
20 1.921 .4892 8.365 4.272 6.453 12.8
30 1.863 .4648 5.458 2.921 4.412 13.4
40 1.783 .4296 3.965 2.262 3.417 14.3
50 1.685 .3815 3.039 1.879 2.838 15.5
60 1.572 .3181 2.393 1.631 2.464 17.0
70 1.450 .2359 1.907 1.457 2.201 19.1
80 1.327 .1309 1.524 1.324 2.000 21.7
90 1.204 0 1.204 1.204 1.819 25.0
100 1.106 -0.1571 .946 1.094 1.653 28.9
110 1.038 -0.3295 .741 .985 1.480 33.2
120 1.004 -0.4982 .582 .887 1.340 37.5
130 1.000 -0.6428 .466 .766 1.157 41.1
140 1.000 -0.7660 .364 .643 .971 44.2
150 1.000 -0.8660 .268 .500 .755 46.7
160 1.000 -0.9397 .176 .342 .517 48.5
170 1.000 -0.9848 .088 .174 .263 49.6
180 1.000 -1.0000 0 0 0 50.0

TABLE 4

e k(e) t(e) m(6) p(e) q(e) s(8)
(deg) (percent)
0 1.805 0.5540 5 *. 5.35
10 1.796 .5483 16.010 0.0115 0.3189 5.38
20 1.767 .5318 7.915 .0058 .1614 5.51
30 1.720 .5035 5.172 .0039 .1094 5.73
40 1.654 .4631 3.765 .0030 .0839 6.03
50 1.577 .4076 2.898 .0025 .0689 6.45
60 1.485 .3367 2.292 .0021 .0594 7.00
70 1.382 .2475 1.835 .0019 .0525 7.73
80 1.273 .1364 1.469 .0017 .0472 8.68
90 1.164 0 1.164 .0015 .0431 10.00
100 1.065 -0.1631 .905 .0014 .0394 11.76
110 1.000 -0.3420 .700 .0013 .0364 14.05
120 1.000 -0.5000 .577 .0012 .0325 15.22
130 1.000 -0.6428 .466 .0010 .0280 16.24
140 1.000 -0.7660 .364 .0008 .0230 17.08
150 1.000 -0.8660 .268 .0006 .0178 17.95
160 1.000 -0.9397 .176 .0004 .0122 18.75
170 1.000 -0.9848 .088 .0002 .0064 19.65
180 1.000 -1.000 0 0 0 19.97
__ __ _








NACA TMi. 1308


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