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# The boundary layers in fluids with little friction

## Material Information

Title:
The boundary layers in fluids with little friction = Grenzschichten in flüssigkeiten mit kleiner reibung
Portion of title:
Grenzschichten in flüssigkeiten mit kleiner reibung
Physical Description:
57 p. : ; 28 cm.
Language:
English
Creator:
Blasius, H
United States -- National Advisory Committee for Aeronautics
Publisher:
Place of Publication:
Washington, D.C
Publication Date:

## Subjects

Subjects / Keywords:
Aeronautics   ( lcsh )
Turbulent boundary layer   ( lcsh )
Genre:
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

## Notes

Statement of Responsibility:
H. Blasius.
General Note:
"Technical memorandum 1256."
General Note:
"Report date February 1950."

## Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003690109
oclc - 76876089
sobekcm - AA00006218_00001
System ID:
AA00006218:00001

Full Text
NRC T-nvM 1VIr

U
7Cc Le 740 f

TECHNICAL MEMORANDUM 1256

THE BOUNDARY LAYERS INl FLUIDS

WITH LITTLE FRICTION*

By H. Blasius

ITRODUCTION

1. The vortices forming in flowing water behind solid bodies are
not represented correctly by the solution of the potential theory nor
by Helmholtz's jets. Potential theory is unable to satisfy the condi-
tion that the water adheres at the wetted bodies, and its solutions of
the fundamental hydrodynamic equations are at variance with the obser-
vation that the flow separates from the body at a certain point and
sends forth a highly turbulent boundary layer into the free flow.
Helmholtz's theory attempts to imitate the latter effect in such a wa--
that it joins two potential flows, jet and still water, nonanalytical
alon:- a stream curve. The admissibility of this method is based on
the fact that, at zero pressure, which is to prevail at the cited
stream curve, the connection of the fluid, and with it the effect of
adjacent parts on each other, is canceled. In reality, however, the
pressure at these boundaries is definitely not zero, but can even be
varied arbitrarily. Besides, Helmholtz's theory with its potential
flows does not satisfy the condition of adherence nor explain the
origin of the vortices, for in all of these problems, the friction
must be taken into account on principle, according to the vortex
theorem.

When a cylinder is dipped into flowing water, for example, the
flow corresponds, qualitatively, to the komwn potential, but as the
water adheres to the cylinder, a boundary layer forms on the cylinder
wall in which the velocity rises from zero at the wall to the value
given by the potential flow. In this boundary layer, the friction
plays an essential part because of the marked velocity difference;
on it also depends the extent of the velocity-decreasing wall effect,
which must be conveyed by shearing forces into the fluid, that is, the

*"Crenzschichten in Fludssigp-eiten mit kleiner Reibung."
Zeitschrift fur Mathematik urnd Physik, Band 56, Heft 1, 1903, pp. 1 37

2 IACA TM 1256

thickening of the boundary layer. That the outer flow separates at a
certain place, and that the water, set in violent rotation at the
boundary, leads into the open, must be explainable from the processes
in the boundary layer.

The exact treatment of this question was undertaken originally
by Prandtl (Verhandlungen des intern. Math. Congress, 1904). This
explanation of the separation is repeated below. Since the integration
of the hydrodynamic equations with friction is a too difficult
problem, he assumed the internal friction as being small, but retained
the condition of adherence at the boundary surface. In the present
report, several problems, based upon the simplified hydrod.rnamic
equations resulting from Prandtl's article, are worked out. They
refer to the formation of boundary layers on solid bodies and the
origin of separation of jets from these boundary layers suggested b,
Prandtl. The writer wishes to thank Prof. L. Prandtl for the sugges-

2. The constant of the internal friction is assumed small as in
Prandtl's report. The boundary layers then become correspondingly
thin; the fluid maintains its normal (potential) velocity up to
near the boundary surface. Nevertheless, the decrease in velocity
to value zero, and, as the calculation will show, the separation
in this boundary layer must, naturally, continue, and so the potential
flow is not completely regained, even at arbitrarily little friction;
rather the separation and the transformation of the flow effected
through it behind the body must prevail even at arbitrarily small
friction.

Figure 1

The procedure is limited to two-dimensional flow and coordinates
parallel and at right angles to the boundary (arc length and normal
distance). In spite of its curvature, the type of the basic equations
in the narrow space of the boundary does not differ perceptibly from
that for rectangular coordinates. With e as order of magnitude of
the boundary-layer thickness

iACA TM 12.6

1 c&u 1
e -:.
LI -

as the velocity u over this distance is to increase from zero-to

normal values; u, -,a O, and
ot JX

-u have normal value; from the
Ok-

equation of continuity follows then -. 1, and by integration, v ~ e.
oy

The terms in the fundamental
order of magnitude

p

equations obtain then the following

Ou ru I
+ -- + v-
,x oy

xop

1 i-i E.-

(uV
\Pt

+u-- v-
Ox ",'I,

E 1.iE E.l

jou u
+ I +
7x2 + y2,

1
1 1
E2

+k +
.2^

2v)
y_2

Oy

E
E

+cv o
ox 6y
1 1

The friction gains influence when it is put at k ~ E2; this
gives the relationship between boundary-layer thickness and smallness
of friction constant. In the first equation, the term 62u/2x2
cancels out; in the second equation, only op/5y e or, when
allowing for the coordinate curvature, ~ 1 remains. In both cases,

LAllowance for the curvature of the coordinates produces, as is
apparent when reforming the differential quotients, only in the second
equation a not-to-be-neglected term pu2/r if r is the radius of
curvature. This term is of the order of magnitude, unity.

NACA TM 1256

the effect of the pressure on y is to be disregarded since, in the
narrow space of the boundary layer, the integration of 6p/6y can,
at the most, produce pressure differences of the order of magnitude
o2 or e, or, in other words, pressure and pressure difference
6p/Ix are independent of y, hence, are "impressed" by the outer
flow on the boundary layer. The velocity of the outer flow next to
the boundary layer is denoted by 9 and is to be regarded solely
as function of x because the really existing dependence on y, when
compared with the substantial variations in the boundary layer itself,
can be ignored in the sense of the foregoing omissions; v is accord-
ingly e = i, hence becomes zero with k. The remaining fundamental
equations for the boundary layers are then:

pcu + u- + v- p + u--- + k--
t x yt ox o y2

ox 6y

Boundary conditions are

for y = O: u = 0 v =0

for y = m: u = a

These equations establish, to a certain extent, a basis for a
special mechanics of boundary layers, since the outer flow enters only
in "impressed" manner.

3. The qualitative explanation for the separation of flow
according to Prandti is as follows: the pressure difference, and
with it the acceleration, is, apart from the friction term, constant
throughout the boundary layer, but the velocity near the wall is
lower. As a result, the velocity here drops sooner below the value

IACA TM 125r6

zero for pressure rise than outside, thus giving rise to return flow
and jet formation, as indicated by the velocity profiles in the figure
below.

Figure 2

The region of separation itself is therefore characterized by

= 0
o j/

for y = 0

This explanation does not work like the'Helmholtz jet theory with
an ad hoc asumilMption, but only with the concepts forming the basis
of the present hydrodynamic equations. The stream line, which bounds
the separated part of the flow, departs at a certain angle from the
area of separation since the stream function develops
around the separation point [x] in the following manner:

t = c y3 + c2(x [x] )y2

As a less important effect, it is to be foreseen that, as a
consequence of the stagnation of water effected by adhesion, the flow
is pushed away, from the body. Through this and the reformed flow aft
of the body:, the flow upstream from the body is, of course, affected
also, so that the assumption of potential flow is insufficient for
quantitative accuracy of results and must be replaced by experimental
recording of the pressure distribution.

6 HACA. TM 125':

I. BOUNDARY LAYER FOR THE STEADY MOTION ON A FLAT

PLATE IMMERSED PARALLEL TO THE STREAM LINES

The flow proceeds parallel to the x-axis. The plate starts in
the origin of the coordinates and lies on the positive r-axis.

In this very elementary case,
hence, no separation is expected.
carried out to illustrate the mode

there is no pressure difference;
However, the calculation is
of calculation to be used later.

u cluf l 2u
pu- + v- = k--
\o oy I ir-

ayx dy
Clu C-V 0

The equation of continuity is
function :

integrated by introducing the stream

u = -

V -
ox

Boundary conditions are:

for y = 0:

for y = :

u = 0, v = 0

u = U, constant

1. According to the principle of mechanical similitude, the
equations can be Eimplified when a similitude transformation converting
differential equations and boundary conditions are known: multi--
plying x, y, u, v, 4i by the factors Xo, Jo, Uo, Vo, and *o results

Puo
Xo

k
y2
0c

UoYo
"o=

*0 = uoyo;

Lo = T4

as conditions that the problem and its solution are transformed, and
that, through the transformation, p, k, U = 1 are created. The four

IIACA TM 1256 7

equations still leave a degree of freedom in the choice of the
factors xo, yo, uo, Vo, and Io. The last three equations define
the factors assumed by u, v, and \ through the transformation; the
first states that the desired solution of the problem transforms in
itself, provided only that

L
k xo

or in other words, with consideration of the factors which u, v,
and 1 assume, the condition can depend only on

pu y2
k x

By this arg;unent, the number of independent variables is reduced.
Next

= 1/2 .

are introduced; 5 is then sole function of ; and

u = 1/2 ut'

v =/2 1- ( -

Insertion in the differential equation gives

r'" =- c',

Boundary conditions:

for E = 0: 0' = 0 0 = 0 frca u = 0; v = 0;

for = m: = 2 from u = u

8 NACA TM 1256

2. The integration of these and subsequent equations is effected
by expansion in series: expansion in powers for a = 0, asymptotic
approximations for a = o. The boundary conditions at both points
being given, one and two integration constants, respectively, occur
In the expansions. They are defined by the fact that both expansions
must agree, at an arbitrary point in the function value to the
first and second differential quotient. The agreement of all differen-
tial quotients is then assured by the differential equation.

3. Solution of the above equation by expansion in powers

C = -5'"

for t = 0 with the boundary conditions at this point

=0 =

is effected by

S (--i)n ncLn+l 3n+2
n=O (3n + 2)!

which is so chosen that the coefficients cn to be defined are
whole positive numbers, which simplifies calculation. The
n+l
factor a brings out the nature of entry of the integration
constant; co, which otherwise would occur as such, can then be
put as co = 1. The recursion formula for cn reads

n-1
cn = 3I n-v1) CvCn-1-v
v=0-- \ 3V

The first of the thus computed coefficients are:

co = 1 c c2 = 11 = 375 c = 27,897

c5 = 3,317,137 2c6 = 865,874,115 C7 = 298,013,289,795

On account of the convergence, the denominator (3n + 2)! was used in
the previous equations; and t" are easily formed.

`The coefficients c6 and c7 in the original thesis are
incorrect. This error has no effect until the fourth decimal.

IIACA TM 1256.

4. There is an additive integration constant for
asymptotic approximation of F because

for 5 = m:

' = 2

hence,

( = 2 + const. = 2T|

so that T appears as new coordinate shifted toward

To compute a first correction i1, put

t = 2r + t1

which gives

2Tiwith = -l"ar

with the squares of the corrections disregarded, hence by integration:

1 = 7 d e-f dri

T1 =2e--rl
di' = 7 e- 2 dl
Y 2

fq '2 2
= 7 e"-r drl + 7e-

2
SI :" I

The general procedure for computing the other terms is such
that further minor corrections tn are added and its squares dis-
regarded. The result is a set of linear differential equations for :
the left, homogeneous side always the same; at the right, the error
appears as "impressed force" which the sum of the preceding approxima-
tions, inserted in the differential equations, leaves.

5. The object is reached much quicker by the following argument:
The differential equation for (t

2.q = -5l't

arises from the original equation

tt = -tin

in the

NACA TM 1256

when the roughest approximation t
Obviously, t has the least effect
equation is then integrated as if

= 2I is inserted at the left for t.
at this point, and the differential
t were known at this point.

S= fIdTq fdTenfd

The three integration constants are contained in the arbitrary low
limits. Putting t = 2rj at the right gives (1 at the left, but
putting C = 2T + 51 at the right gives

C J f TI de -- e-i f t dri

or with consideration to the boundary conditions

= 2q7 + 7 drj dnel2 1 )d

= 2T + y 7 / dTi I dTe-2/TI dT

Hence, the second asymptotic approximation

2 = -y2 dTl d -e 12 e Ld7

/dn /e-t 2dI

By partial integration

S(22 1)e 2 lTI 2d
2" = &(22 + l)e /- 2d -
2co

72 2
4

Se- 2 d
cco

e-272
4

O e d

+ 2 -2"2
+ --e

-7,2 2 i -.r2
Cg2 =e o

72
--.i
4

,.

T I 2 2
I e- dr
a/C

+ 2- /Tie-22dn
2 tca

NACA TM 1256

According to the formula

e-T1 = n--le-2 + -- e2 d
t' 2 2 (/,,
to be gained by partial integration, each integral of this form can be
2 Tjl 2
reduced to the functions e- and / er' dr multiplied by powers

of q. After several such integrals are obtained, the innermost is
transformed, if necessary, in the indicated manner. The integral e-T2
or fIe-2 dn multiplied by powers, appears then below the penultimate
integral sign. The former gives no new difficulty; the latter can be
reduced by partial integration to the two functions e-T2 and e-T .r2dT

jI;b f e-r'i2dri = TI f e-Adr + .e-'1*

T T1e--2d i L le--2d 1 /2 + 2 1e---e2
Tdr e-r = d -r 92d e d + -r -

T d 3 1 1- T12 122 1 2
T r2dr e-j dT 2 3 =_2 / + 2 eTi + 2 e-T + and so forth.
J3 0

If, as above, the integral can be quadratic in e-2 four types
must be distinguished:

e-- f eI2 (e / r e2 2 fTe-2
e e-TI2 e-d d, e2d j 2e

multiplied by powers of iT. The first and fourth types give nothing
new. Partial integration provides for the second the formula
7 1 r f-I '.2 /ITl -r2 / 2e
e-21drl e-1 dr, = e, d -- j Te 2d e-T 2dT
(Jo I.000 (U CO

NACA TM 1256

.Ie-2dT e 2d = 2 e--2d 2
'1.O 1.

and

T 2 re2d =
n6_92 dT) L e-11 = -2T

.e-72 j
Jdo

+ e-2id

r e-d = .e-1 2
Co2

e-- 2dT +

and so forth.
Likewise for the third type

1 e-- 2dT 2 d =2 2
Z/o coc I"

+ e-T2 e T2d

- e-2T2dr
0Ggo

-- -2 2 e oT2 l 2

I- 12
a'-?-44.

+ e-2 e-q2

+ e-212 and so
8

forth.

Since no new types for integrals are introduced by these formulas,
the indicated tables of formulas govern all integral in which e-T2
occurs no more than twice. Any number of successive integration
over such functions are possible; the powers of i involved are
unrestricted. The formulas for 2 in section 5 were obtained by
this method. These integration will be met again later. With the
type of integration results thus known, the calculations can be made
by utilizing a formula with indeterminant coefficients.

7. With this differential equation, it is possible also to define
the error that afflicts the present solution as a result of the effected

S 2e--2

- e-2q2
go

NACA TM 1256

omissions. It is easily verified that 2T9, 2r + Si, 2n] + +1 + 2
remain below the true value of t. An upper limit can also be found
by employing the previously given (see Section 5) form, somewhat
modified

S= 2Tr + 7' dr 1 d-Tie 1e 1
u/OO

of computing
chosen upper
expansion of

a finer from a rougher approximation: a rather arbitrarily
limit, such as the first term of the semiconvergent
SJ, for instance, is entered for t 2n, thus

- 2-9 < e-'2
4 2

and an asymptotically
from this assumption.
below the assumed one.
formula)

finer upper limit
It is insured so
The calculation

for t", t', t is computed
long as the latter remains
gives (according to the general

Se-q12 .
J100 e I .

e-T 2

+ 00
v + 1 e-n2 d
2 1 v+2

1 e-12
2 v+l

Ti j

I- 2d ( i)d -Ti

NACA TM 1256

.0 a

Figure 3

The upper limit for e' is then found according to the above figure as

e') ,' 1 + ao

e 0-
So

where Io is the highest existing value of 'J, hence corresponds to
the value of the coordinates for which t is to be computed. It
results in

S< 2r + ? f dTI d e-l2 1( +
f/T- I/T,

a8 e-'2 /

< l2
<2] + +
1 "

IACA TM 1256

similarly for (' and. ":

e-'-212
'> 2 + ay eo
32 A 4

2
+ a2 e-al
11 8 3

A more accurate execution of the integrals affords
result.

8. The connection of the two developments and
of the integration constants (a, y, and r i) is
separate the integration constant a,

z = -,

a more accurate

the determination
as follows: To

X =/at

is introduced in the power development (3), which results in

S=
,Zf

d _- '
dX2 a-

-(- "1)n cn x3n+2
n=o (3n + 2)!

= (- 1)n Cn 13n+1
n=O (3n + )

1)n -(3n)--

The displacement of I
the integration constant 3

relative to

E is expressed by introducing

Qr = x 0

NACA TM 1256

The formulas are completed by inserting

S= 2tq- + 1 + 2

s' = 2 + t + 2

= I + t2"

from the asymptotic approximation (4) and
and its differential quotients fran which
quoted:

(5). A graph is made for
the following values are

X = 0 0.8 1.0 1.2 1.4 1.9 2.0 2.05 2.1

Z = 0 0.317 0.492 0.701 0.938 1.63 1.79 1.8561 1.94

d = 0 .784 .961 1.121 1.257 1.50 1.53 1.5479 1.56
dX

d2Z = 1 .639 .34 .28 .2582 .23

c7X23
The terms of the power series are computed up to ; further terms
23'
are extrapolated, in part, from the difference series of the logarithms
of the coefficients. The location of the asymptotes is already quite
apparent in figure 4.

IJACA TM 1256

1.0 2.0

Figure 4

Since owing to 2 = 2r

asymptotic,

Z = 1, (X ), rough approximation
a2/3

values can already be read for a and P: a = 1.30, P = 0.96,
with X = 2.05 as connecting coordinate for = 1.00. The corresponding

values of 0- and (" give for 7: 7 = 0.92. The calculation is

more rigorous when a, 7, and the connecting coordinate related
to T = 1 are varied by minor corrections and these then computed
from linear equations. To judge the accuracy, it is stated that our
calculation for X = 2.05 gave

2 = 1.8.61,,

S= 1.5479,
d1C

d28
d2 = 0.2582,
dX-

d3 = -0.479
dX3

where the fourth decimal is no longer certain. Asymptotic approximation

for i- = 1 gives (using Markoff's e t2dt)
t

18 NACA TM 1256.

S=2+ 0.04454 + J0.00012 2
0.OO010bJ

0.00076 2
S= 2 0.13940 0.00423

0" = 0.36788 7 + 1.0 } 2

the top numerals in the { stemming from %2, the bottom numerals from
the upper limit defined in (7). (The latter is, as stated before,
rather rough.) The "temporary assumption" about the upper limit of
gives: < < 2 + 0.092 7. The upper limits are therefore guaranteed
(reference 7). Hence, the result

a = 1.3266, X = 2.0494, 7 = 0.9227, (B = 0.9508)

It can be safely assumed that a ranges between 1.326 and 1.327.

9. From it, it can be computed, for example, what drag a plate of
width b and length Z is subjected to when dipped parallel to the
flow lines into a flow moving at velocity U. The drag per unit of
surface is

X = k =- kkV',fl 1
Xy y 2 2Vk j

Integration over the plate gives

b Xy dx = -b kpu
o y 2 V

hence, when the water flows at both sides of the plate

drag = 1.327- b \kpu3

NACA TM 1256 19

II. CALCULATION OF REGION OF SEPARATION BEHIND

A BODY DIPPED INTO A UNIIFORM FLOW

1. The following problem is treated: In an otherwise parallel
flow, a cylindrical body is immersed symmetrically to the direction
of flow. The boundary-layer coordinates are computed from the point
of division of the flow. The quantity u is expanded as function
of x in a power series. For the integration of the fundamental
equations

,Ou i u -Uu k 62u
u- + 7--- = +
ox oy ox p 6y2

u + v 0
ox oy

u =jI_ q-,x2
l=o

the formula

,4 = > -(., 2 +-
Z=o

is used, with due regards to the symmetrical conditions for the
stream function \$i; u and v are obtained then by differentiation.

NACA TM 1256

Figure 5

Consistent with the general boundary conditions, the functions 'X(y)
must then satisfy the boundary conditions

=0 = 0 for y = 0

2'. = q for y = m

hence

X = qy + r

r, is the constant of integration. From insertion in the first
fundamental equation, the differential equations for X are obtained
as:

-(2, ('2%")= +- (2% + l)q q 1 +k
k=O \=0

which for 2 = 0 is quadratic, for I > 0 linear in the Y, function
to be defined. This equation can, like the proceeding problem, be
solved by expanding y = 0 in powers, for y = m approximating
asymptotically and joining both. Subsequently, it is shown that the
asymptotic approximation can be omitted, since the power series already
identifies the asymptote and therefore the integration constant with
sufficient accuracy. The calculation is restricted to y and X1,

IACA TM 1256

that is, the first and third powers of x. Because, since the corre-
sponding coefficients qo and q1 in u already indicate a first
increasing, then decreasing velocity the case, in which presumably
separation occurs, is characterized by qo > O, ql < 0 the type
of pressure distribution required in the introduction (3) is already
supplied by the first two powers; hence, it is to be expected
that Yo and Xl, even though not quantitatively exact, already
represent the effect of the separation. In one of the problems
treated in similar manner later on the next approximation was also
computed; and it substantiated the admissibility of the limitation
to the first two powers of x.

2. The equations for x and N, are

'2 Xo- = qo 2 + k~Xo,

o1 ,oi" + 3(ki = 4qqI + k

The manner of entry of qo' q1' k, P can be established by mechanical
similarity. Here also, the first two terms indicate universal
significance in same respects. Hence, writing

i = q x3 f = x X X3
o 1 o 1

and introducing the following quantities

1 r i, _,, X1
-= 1 n-= k -" Xo- o

for x, y, ., yI gives

u- qF 3)
u V1

22 NACA TM 1256

= (to +- ti3)

V2pql

u o l+ '3) etc.

to and t1 satisfy, as functions of Tr, the differential equations

o12 o- o = + to"

4o i o 3C 1 = 16 + l1"'

Boundary conditions

for 0 = 0: =o = 0 = = 0
o -1
for 1 = : = 2 C' = 2

3. For 0 the power series
o

Sa-9+1b
p=2 9 '
is entered.

Insertion in the differential equation gives:

b2 arbitrarily = 1, since a already is integration constant.

a4p = -4; since, in the formula of the integration constant a,
no allowance was made for the homogeneity of the equation for (o,
a appears again in this equation.

b4 = 0; the curvature of the velocity profile does not change,
at first, since the friction in its effect is two terms ahead of the

NACA TM 1256 23

inertia; starting from m =.5, it is

A--
bm =) (m- 3] b'bm--_
m =2 \[ ~ 9-n

The coefficients of these recurrence formulas can, like all numbers
combined this way from binomial coefficients, be computed from a
diagram similar to Pascal's triangle, whose start is the following:

/ -5 o
5 -1/-/ / //
6 /--1,2/0 2 / / 1
7 .-1 3/-0/2/31 / /
8 /-1,, /o0 / 54 X1 /

and in which each term is the sum of those above it. Only the
framed-in portion, consistent with the foregoing limits of sums,
is counted.

The first 13 coefficients are

2 =1 b = b=0

b- =1 b = 2b b = 2b 2
0 3 7 3
b9 =-4b3 b10=--ib3
b8 = -1 b = -4b bio = -1ob 2

bll= 27 16b33 b12 = 181b b13 = 840b 2

4. Besides a, two more integration constants due to the
asymptotic approximation are involved, which, as in the preceding
problem, should join the computed power series. For the present

NACA TM 1256

purposes (calculation of point of separation), it is, however,
sufficient to know a, and, as stated before, it will be seen that a
can already be computed with sufficient accuracy by means of the
power series.
dZ
Put Zo a-, H = ar and plot as function of H from
a ddZ

the power series. Z0 itself is still dependent on b and
dEH 3 a
dZo 2
shall, for the correct value of a, approach the asymptote -
dE O2
For other values of a, it approaches no asymptote at all, as a result
of which as fig. 6 shows, the method for defining a is very
sensitive. The value a = 1.515 is obtained; the last cipher is no
longer certain.

dZ
o
dH
1.0
0.87

Figure 6

5. The calculation of C 1
boundary conditions is effected

by the above linear equation and the
in similar manner: power formula

c
CC.

NACA TM 1256

02 = 1, since 5 already is integration constant; 6c3 = -16;
c4 = 0; and for m > 5:

cm. An 3) + 4 3 -3 )]3 "G+14
p=2 -^ 2) gf 1 t-/

Here also the coefficients
from a diagram whose first line
+4, -3, while the others follow

in these formulas can be computed
(m = 3) consists of the numbers -1,

6 /- 1/ Z6/ / Y Z-3
8 / --I /-7 / / / /

The first coefficients are

c2 = 1; 5c3 = -16;

c4 = 0; 0 = 4a3;

C6 = 6a3c3 8;

cy = -32c3; C8 = 17cLa6

010 = -576a3c3 256;

09 = 30a6,c3 2243;

1 = 2048c3 + 294x9;

7839c3 6 6 3
S= 783 c3 50922 13 = -17392a C3 + 59648a

c14 = 221952a33 315a. 136192;

15 = -11025a12c3 1024000c 54864a9;

3 3
Cl6 = 174168a9c3 221296m6

6. The asymptotic approach is again disregarded, the integration
constant 5 being defined by the condition that il' mist have the
asymptote 11 = 2. Figure 7 shows the terms of the power series

NACA TM 1256

for t1', those free from c3 and those multiplied by c3, as
curves A and B, that is

-1' = 5- A 16- B

after which l' is plotted for different values of 5.

Figure 7

This curve indicates that the convergence of the series is rather poor
in spite of the great number of computed coefficients c, even
at r = 1.6. In any event, the terms indicate, when identical powers
of a are combined, a satisfactory variation so that the series are
still practicable. The correct value of 5 ranges between 8,20
and 8.30. The curve rises, at first, very quickly and approaches
its asymptote fron above. This marked influence on u near T = 0
compared to = oo permits u in the case of separation to change
signs at the boundary before it does on the outside.

7. Proceeding to the calculation of the point of separation, it
will be remembered from (1) that, quantitatively, the results are not
exact, since only the first and third powers of x were taken into
consideration. The point of separation [t] is defined by

NACA TM 1256 27

0 0= u ( o~ l J3) for = 0
:OY 8kql

or by (3) and (5)

cL3 & = 0

By (4) and (6), respectively, a = 1.515, 5 = 8.25. Thus, in the case
of the lower prefix, the only one of interest, the coordinate of the
point of separation is

hence

[x] = 0.65

with

S= qx qlx3

The maximum of the velocity (minimum pressure) lies therefore at

S= 0.577 q

while zero velocity in the outside flow would not be reached

till x = 1 q* Accordingly, the point of separation is 12 percent

of the total boundary-layer length behind the pressure maximum. The
obtained figures are independent of friction constant, density, and a
proportional increase of all velocities.

According to Prandtl's diagram (section 3 of Introduction) the stream
line 0 = 0 diverges from the boundary at a certain angle, which is
computed as follows: In the vicinity of the point of separation, the
development of the expression for given in (2) reads

IACA TM 1256

t= vq,3i ^((st"B ~C11M [3)T3 +3 ( to" 3t 2~)(^ -] )n2)

= 0 gives for the divergent stream line
2
3& -l a
S = 3I = 11.5
S- B 16N3 -400

or in the not-reduced coordinates

y = 1.5

These formulas are characterized by considerable uncertainty because
only two terms of the development of were computed and the higher
differential quotients, which represent more subtle processes, are
always less accurately computed than the former.

III. FORMATION OF THE BOUNDARY LAYER AND OF THE ZONE

OF SEPARATION AT SUDDEN START OF MOTION FROM REST

1. The two preceding problems treated stationary flows. The
problem of the growth of the boundary layer is now treated. Assume
that a cylinder of arbitrary cross section is suddenly set in motion
in a fluid at rest and from t = 0 is permanently maintained at
constant velocity. At first, the state of potential flow is reached
under the single action of the pressure distribution. The thickness
of the boundary layer is zero to begin with, so far as the sudden
velocity distribution can be obtained at all. The boundary layer
develops in the first place under the effect of friction, then
through the convective terms. The result is that, after a certain
time, the separation starts at the rear of the body and, from there,
progresses gradually. Since the fundamental equations refer only to
thin boundary layers, they, naturally, represent only the start of the
separation process, just as the previous problems dealt with the boundary
layer only as far as the zone of separation.

NACA TM 1256 29

2. The equations involved here are

ou Au Mu 32u
+ u-- + v--- = u. +
ot ox oy ox 6y2

U = V -
oy ox

K substitutes for -
P

The potential flow which is set up first gives the boundary
value u as function of x. Since the process for t = 0 is singular,
the type of development is, for the time being, still unknown; it
must be established by successive approximation. The principal influence
on the changes has (at small t) the friction, hence, for the first
approximation uo

uo a 2uo
at oy2

The integral of this equation

UO = f e7

2
Y -

satisfies the conditions of supplying a vanishing boundary layer
for t = 0 and of joining the outside flow uo = u for y = c. The
subsequent approximation is obtained by inserting uo in the
convective terms, while time and friction terms obtain u = uo + ul.
The resultant equation for ul reads

dU1 -0
---= + u- (function of rI)
Ot Jy2 ox

NACA TM 1256

According to mechanical similarity, this equation is satisfied by the
formula

u f
l = tu f()

which is also not contradictory to the boundary condition ul = 0
for y = 0 and y = o.

After further considerations, which in particular refer to the
insertion of x, the quantity u is represented in an expansion in
powers of t, the coefficients of which are functions if q, that is,
still contain t. These functions are also still dependent on x,
but this time x enters the differential equations only as parameter.

3. The formula for is accordingly

t = 2 tt Z X (XI)
TI=0

u.= .tV
v=o

u tvV
V = OTI

and hence the differential equations for X

+. 2 -- = 0 I 12 4

for g = 1, the right-hand side contains -4u .
ox

As before, the calculation is limited to the first two terms,
that is

xo = o(n) x -=

NACA TM 1256

hence

The equations for

u = u + tu:-i

to'" + 2Tt = 0

t" + 2T1 4 = (2 t "- 1)

Bodary conditions
Boundary conditions

for q = 0:

o=0
0o =0
(1l=-

(o' = 0
0o 0

= 0

for n = o: o' = 1 y1 = 0

4. The solutions of the above differential equations, which are
to be used in the subsequent problem, are obtained by quadrature when
the homogeneous equations are integrated. The latter integral were
obtained by the following consideration: The homogeneous parts of
the equations stem from the time and friction term which together
form the heat conduction equation

au = -u
at y2

Of this equation integrals of the form

Un = tnfn()

2 =-

P.r-

NACA TM 1256

exist, according to similarity considerations, whereby fn
the differential equation

fn" + 2Tfn' 4nfn = 0

which the above form possesses.

Thus, for example (see above)

o -2
UO = f o = -
Tit;

T e- 2d
fl^j
LIe

uo = 1 when t > 0

Uo = 0 when t < 0

For n = 0, hence, for y = 0, un is proportional
be representable by superposition of solutions uo
form

un = n uo t--t T
S2 (t -to)

to tn, hence must
in the following

- t dto

=n Uq ( y ) -to0n-ldto
Uo 2 \(t to)

since for y = 0 it is

= n tn-dto = t

For the evaluation of this integral, put t to = T

un = -.n uo ) -(t-T T
"t l2 Y

satisfies

For Tj = 0

NACA TM 1256

insert herein

T-- = T
2^ 2

t1^
Wk 1 2

T =1= y;

1 2'
T=^4k

d 2T
2k 3

and finally obtain

u- = r 2n

F3 1

-1 n-I PeS2d
^ / Uco

Calculation of this integral by the binomial theorem and the
previously cited method of partial integration finally gives

n n
+ -12 2V-le7 12

v=1 = (2p )...(2 2 + 1)

The other integral is algebraic and equal to the above factor

of r e- 2dl

n 2(n

n =o(21 1)...3.1

5. Quantity to is determined as follows: With the boundary
conditions taken into consideration

TACA TM 1256

S 1 + ee- d
a +ar e S d

whence by integration

1
0 f it4 1 (iI

eT11 +2
d T 1+ce 1 1

while utilizing

o =
50 T

The second differential equation (of the second order for
assumes then the form

8 _2
\mTI

L" + 2j"1 LO' 16
1 1 f 7

I' )

16

- .e4T2 'e-I2d 2e-22

The integral of the homogeneous equation for '

fl (22 + 1) + 2 + (22 + 1)

The integral of the nonhomogeneous equation would then be
obtainable by quadratures. But it is also true that, by twice
differentiating, the differential equation finally becomes

t "H' + 2Ti "" = function of 1T

is by (4)

e- 2dj --
em dT

IIACA TM 12-,6

which is easier to integrate as an equation of essentially first
order. Hence

C "" = e-T2

feq2[function of r] d

Since the impressed force of the differential equation contains e- 2

in each term after twice differentiating, eI2 cancels out, and s""
and then r can be integrated, because the functions behind the
2
integrals contain, at the most, e twice, and in addition, powers
of r, and must be integrated several times, which can be accom-
plished by the methods discussed previously (1,6). The result of

6 I = e2 dle2d 2(2 1)
1 x Soo V

2 2
+ 2-2r2
it

+ -12 _2 4

+ ca(2T2 + 1) +

" = -- (2n2 1)e-

- (2r12 + 3)e-12
fn

Seq2dn _-_2
+- 32

P [ne-q2 + (212 + 1) I e-62dn]

TI 2 d +8 ) 2 { nI 2 _2 -2T2
0 df +- dS -7e

+8 e-- 2
32

+ 4aq + 2e-21 + 4 e d

The reason that the equation here could be integrated in closed
form, despite its affinity with the previously stationary problem, is

3b NACA TM 1256

due to the fact that ou is simpler than uS-, although both have,
)t dx
according to the order of the differential quotient, "heat conduction
character."

The determination of a and 0 fran the limiting
conditions = 0 for r = 0 and q = = gives

a=0 P = + .

6. For computing the zone of separation, there is

0 = -U 1 ut + 1t]u C"
2y 2 \Jt \ o ax 1

for q = 0

-" =22
o f t=2

an2 + 8
1
--l 7#

The condition for the time of separation [t] is

1 + 1 +-L) [t] u=

hence, must be negative. The separation occurs first where
ax ox
has the greatest magnitude. The result applies to cylinders of any
cross section; f is the corresponding potential flow.

IV. DEVELOPMENT OF ZONE OF SEPARATION FROM REST

AT UNIFORMLY ACCELERATED MOTION

1. Against the physical principles of the foregoing problem, the
objection may be raised that the sudden shock might be accompanied

Then

NACA TM 1256 37

by an interruption of the fluid. Hence, let the solution of the
problem assume that, starting from the time t = 0, the immersed
body is subjected to constant acceleration.

In that case

= tw(x)

1 op u 67u ,2 aw
-+ u-+ = w + t w-
p 8x at 6x x

2. From considerations similar to those made before, the solution
of the differential equation

6u bu 2u 1 Sp ?u
+ + v- 1 + K
it 3x dy p x .y2

is based on the formula

S= 2 st It2V+lX2v+1(xT)
V=o

v=o ^T

2 t

Insertion in the basic equation gives

d3X 2X oX
21 + 2- 2X 4(2X + 1)x+
a3+3 N2 0-YT

NACA TM 1256

xl Tax 8~2x
=- 2p+l 2X 2p-13
-=o bZ xy

-Px- c2+1
ox6T2

for X = 0, the right-hand side contains -4w, for X = 1, -4w-.
ox

The calculation of the state is again limited to the first two
terms, while it should be noted that through those two terms, the two

terms of the pressure w + t2wv are also taken into consideration.
ox
The impressed force of the nexr equations contains only earlier
development coefficients. For the final equation, however, which
supplies the zone of separation, the coefficient of the next term is
computed also. For X1 and the relationship of x can be
introduced in the following manner:

1 = w (T),

3 Ox '

The differential equations for ( are then:

3 1 62 1 =1
-- + 2TrI -- -- -4
273 oT2 lj

a3
-;3

23
+ 2T--.
ST,2

- 12-3 -4 + 4
oil

Boundary conditions:

for I = 0:

for t = oc.

tl = 0,

t3 = 0,
3

= 1,

61

0

0^
? 0

=OT
--= 0
OTf

K

:1 ,2
Oil _1

U =0
from
v =

from u = tw

\OT,

NACA TM 125r6

3. According to the general solutions of the present type of

differential equations discussed in III (4), can be written
ZOTl
a1
forthwith, since the nonhomogeneous term -4 is disposed of by = 1;
C is obtained by integration by the repeatedly cited method (16)

1 .+" + 2q e-Iq2 d-9

6T2 2 2
S1 + e + (1 + 22)e

n= + 2-- 1 + (1 + T2)e7,2

+ (31 + 213) S e-q2dT]

These functions are quantitatively plotted in figure 8 and given in a
table (see Section 6 following).

NACA TM 1256

Figure 8

The impressed force on
is then

the right-hand side of the second equation

Se-92dd

+ 16
3A

[4i e-i 2d + 2e-92

(- + 2)e-22 ++ (-49 + 43)e

4. The
succeeds by

+ (3 + 4T4) e- 2dl 2

integration of the second equation, in closed form, again
the same methods as in III (5). For the part of the

2
impressed force quadratic in e 2 a formula with indeterminate

e-- 2di
J o

NACA TM 1256

(a + bi12 + cT4)e-22

+ (drT + en3 + fi5)e-T12 J e--T2d
00

+ (g + h2 + iT4 + kTj6) e-2dl 2

This formula fails when the impressed force contains terms which exceed

6 -2T2 5e- 2 de-2 Ti4 {s eT d 2 (compare III (5)). The
oo co
coefficients are determined from linear equations. The other
portions of -: are easier to compute; 3 and -- follow by
as t3 o2
integration and differentiation. So, when the integration constants
are correctly computed, the final result is

2t 3- 4 _-.2 32 eqe2drl
Oa2 3 j- 15x

+ (-2 + 203)e-2.2
31f

+ (1 + 2Ti2 + 8T4)e-2 fne-2 i

+ (6ri + 8-3 + 8Ti5) fi e- 2 }T

+ 1- 6 45'1 ) (16 + 36n2 + 8 )e-12

+ (60o + 80o 3 + 165) fe--2d1L
S.o T

NACA TM 1256

4. e-q2drl
3/1 a,

1- e- 2 + 2q
151 L

+ --(8
9i

/ e T d
Li a,

+ 12 + 24 )e-22

+ (24n + 8r3+ 85)e-2

Se--ledy

+ (-9 + 18n2 + 12n4 + 86)

1 56
15 6 f?

(33n + 28,N3 +

45

+ (15 + 90n2

+ 60o4 + 8r6)

l e- 2

hence, by integration
3 =2- e2 + 2T re-L2d
3 Cif L u< -I

8 e-2
157

+ (1 + 2Y2)

9I 21
a, J

+ -- (49 + l113 + 10o5)e-'22 + 768
315 L-
L1n(,il~ a~e22 6

f e-2n2dn
fm

"T
6TI

T 2

4T5)e-2

NACA TM 1256

+ (-537 + 198n2 + 64T4 + 40T6)ea2 .e-S2d

+ (-315I + 21o13 + 84n5 + 40o17)

- 2 dTI
dTI}2
00""

+ -.L- (o
105 V6F

h16 ) 24 + 87T2
45f/3 )

+ 404 + 416)e- 2

+ (105sl + 210rT3 + 84T5 + 8T17)

( 128
+ f-10a.73
U1575 ,3

128
105 1/2

9)

These three functions, plotted in figure 9, rigorously satisfy
the differential equations and the boundary conditions for the
coefficient X.
5. The condition for the zone of separation has the form

= 1 L
- jv w

_2 ox6 J2
x6,2 Trj =

whence, by the foregoing formulas

2 t 3 31 256
O3n2 '15if 225V3

0 by

a2-ir h

NACA TM 1256

The equation for the separation time [t] reads

64
225n

[t 2 = 0
ox

The next term in the separation equation T- = 0
dy

would

2X

coefficient in this separation
variation of X5, is computed.

0.138

to be able to allow for it, too, the

equation, rather than the total

0.25

Figure 9

The development term X5

+ 2I2

20--
6TI

satii

sfied t

6x

= 4 3 __a2X
Z) ax6N

the equation

32,3 ",xr
62 aTI
+ .-.l
TT O1

Yx23 ax l
8x 1

The entry of x
right-hand side

in X1 and X3 is known, and calculation Df the
confirms that X assumes the form

o;3
o^3~

1 + 3-
( 60

NiACA TM 1256

X5 Ta X

+ -w2Z
6X2 5m

Since tw cancels out, the condition of separation reads

+ t23
ox 2 -

+ t42 J
w 2
TI=0

+ _A_
+ t l 2 t
--q0

6. This leaves the calculation of the coefficients P-2

and --- I
and

For 5 the differential equation reads

+ 21__2
N2

=l 8~,

- t 12
l ''

- 32

and the boundary conditions

= 0, --=0
5 =0

for i\ = 0 -- for
Ni

The impressed force f(i) is given by the previously written

functions. The desired coefficients -- are computed by

Green's method as follows:

-
- 20--
OTI

32

_I Tj=0o

NACA TM 1256

'8

-2O-)

A2r
dTI = i --
-T 12

+ 2^. 0

M+ L- -0 2L 20)d
sJo 6-q F12 6t q

2 r 22, = 0

and the boundary conditions

-(0) = -1

,5( ) = 0

the result is

.12 1=0

= f dr
o

f(n) is given previously; the influence coefficient a (Green's
function) is obtained by integration of his differential equation

T(Oq) = ( 28959 + 5280"3 + 2352i5 + 35217+ 106T9)
945 t L

+ (945 + 9450nr2 + 126,00i + 50O4n60 + 720T18 + 321310)el 2 "se-2d]

The curve of a is shown in figure 10, along with the
product 9 f. The area of this last curve gives the desired
coefficient.

+2 .
an;?

C)1 ""

IJACA TIM 12 6

0.-5

0

-0.5

-1.0

For computing

6n13

25 0.50

Figure 10

'2 =
6_ 5m )=

the equations

-2 = g d T
L_72 ]_ -=0

are available; ? g is plotted in figure 10 according to the values
indicated below.

= g(l)

,2
(I r
+ 2Tr-~~4
6T12

HACA TM 1256

The computed values are the following:

all
821

3
323

(

N12
*C f

f(T)

6(Tl)

3 f

g(T)

3 g

0

0

2.26

0

0

.964

0

-1

0

0

0

0.25

.061

.450

1.396

.022

.137

.231

.315

-.327

-.103

.124

-.041

0.50

.211

.720 1

.799

.060

.150

-.092

.750

-.12

-.084

.240

-.027

1.00

.638

.943

.201

.115

.020

-.156

.457

-.018

-.008

-.016

.0003

1.50

0.035

-.05

S- o.376

1

0

.138

0

0
o

0

0

0

0

0

The area of the two curves is approximately

1h2 Ti0O

= -0.058

LI
[2 -m0

7o The equation of separation therefore reads

4L + [t]2.~ -_ 256\_ [t]Y ) 0.058 [t]4w 0.023 = 0
TIC L x 15{ii 225TAf3 \ 2

= -0. 02 3

NACA TM 1256

1 + 0.427 [t]2w 0.026 [t] 2 0.01 t] 2w = 0

Since the newly added correction term is even negative, the existence
of the zero position appears to be certain.

The position and time of separation is according to the earlier
approximation (without the term computed last)

2 t = -2.34

For the case of a cylinder symmetrical to the direction

flow, = 0 at the rear point where the separation starts, the
0x2
newly computed correction gives

[t] 2 = -2.08

equivalent to an error of about 10 percent. From this the quality of
the approximation made in the other problems, where only the first
powers were taken, can probably be also appraised.

V. APPLICATION OF THE RESULTS OF THE SEPARATION PROBLEM

TO THE CIRCULAR CYLINDER

1. On the circular cylinder

S= 2V sin R

x is called the reduced coordinate X; V is the velocity at which
R
the parallel flow flows toward the right, and the cylinder moves

50 NACA TM 1256

toward the left, respectively. In the steady case, the separation

starts according to Part II, Section 7 at xeep, = 0.65 1; the

maximum velocity lies at

mx = 0577
I1

where

l3
u= q x qx
ol

Figure 11

Taking the ordinary development in powers of sine

S2V

x = 1.59 R, X = 911
sep. sep. 4

2V
6R3

x = 1.41 R; X = 810
max max

IACA TM 12 5u

But approximating the sine in the interval 0 x by the method of
least squares, gives

2v= o.856,
lo R "

ep. = 1.97.R,
sep.

x = 1.75 R,
max

a = L 0.093
R3

X = 113
sep.

X = 1010
max

In any case, the point of separation lies, by the present
calculation, at from 11 percent to 12 percent of the total boundary-
layer length behind the maximum of the velocity. This statement makes,
of course, no claim to accuracy, since only the first two powers of x
are taken into consideration. Besides, test records of the pressure
difference indicate that the state near the separation is difficult
to attain by development from starting point of the boundary layer,
because it is too strongly affected by the pressure distribution of
the turbulent bodies behind the cylinder. The sole purpose of the
present calculations is to indicate that separation is actually
obtained by the hydrodynamic equations. Further development of the
calculating methods, especially for the more important problems of
solids of revolution, promises, therefore, success.

2. If the cylinder with constant velocity is suddenly set in
motion

u = 2V sin X

The time of separation

= cos X
8x R

[t] is, according to III (6), given by

+ A

Bz

t] = -0.35 R
V cos X

NACA TM 1256

The separation starts for X = i, cos X = -1 at time

to 0.395

Up to that, the cylinder has travelled a distance

S = Vto = 0.35 R

All this is independent of velocity, density, and friction coefficient
(little friction assumed).

3. At constant acceleration

u = tw(x) = 2Vt sin x
R

where V is then the acceleration of the cylinder in the flow. The
separation time is (IV,7) for the start of separation

or = -2.08

respectively, or

t]2 = --117 R
V cos X

or = R
V cos X

respectively. The distance covered by the cylinder is

S = lt2
2

at start of separation (X = t)

S = 0.99 R or = 0.52 R

respectively.

- = -2.34

NACA TM 1256

4. The resistance which the cylinder experiences at constant
acceleration is computed next. The stress components are

Y -- + 2k1 y

6-a 6v

Owing to the smallness of the friction, 2y and
respect to leaving as force in direction of
C7

6v cancel with
the outside flow
the outside flow

K = 2 .B p co X + k sin X RdX

B is the width of the layer (height of immersed part of cylinder).

The pressure portion is computed as follows;

Pressure = 2BR fp cos XdX
\$---~uo

= -2BR2 x sin XdX
-"so opx

Then

u = tw

(= p +2

w = 2V sin X

NACA TM 1256

The second term cancels out in the integration; the first gives

Kpressure = 23pBR2

hence, an increase in inertia by twice the amount of displaced fluid.
The friction portion is

Kfriction 2kBR
2 V-i

. 2

c 2

o2+ \
+ t3- -) sin XAX
6x 2 /

where K = k/p. Again, the second term disappears because --

S2^3
and -2 are merely constants, leaving

Friction = 4"pkt BRV

5. To give a picture of the flow conditions corresponding to
these formulas, the flow curves for a specific state of motion of the
uniformly accelerated cylinder are represented in a diagram. The
parameters R, V, r are arbitrary; hence, necessitate the introduction
of reduced quantities for x, y, t, *, and u, so that R, V, K
disappear. It is accomplished by

x = RX, t = -,

x=B ,

y = YjR

u = U

NACA TM 1256

by which the formulas (compare IV (2) and V(3))

* = 2t3/12w( 1 +

TI = tv-w-

S2V Kt

w 2V Bin

+ t2
x

_3\

t2 2Vt2 cos
ex R R

become the following reduced equations

3 = 4T3/2 sin X ( + 2T2

U = 2T sin X -' + 2T2 cos X

Y
2

The curve I is then plotted against
for a number of coordinate values X,
values T = constant read from these

Y = 2/T 1 for a fixed time T
and the position of the
curves. In figure 12, the

cylinder is shown from X = w/2 to X = x. The separation time is
given by 2T2 cos X = -2.34, that is, the start of the separation

cos X 3)

UACA TM 1256

by T = 1.08. In figure 12 2T2 = 5, hence T = 1.58, was chosen.
For this chosen time, the separation point has already progressed up
to beyond 600 at the cylinder; nevertheless the boundary layer still
is fairly thin, the relative sizes correspond to the values R = 10 cm,
2
S= 0.01c2 (water), V = 0.1----, that is, to a very small acceleration.
sec sec2
Accordingly, t = 15.8 sec.

Figure 12

NACA TM 1256

The picture obtained by the previous reduction fa orn as

far V 10-SL after 1.58 sec. is represented in figure 12. The
2
sec
thickening of the boundary layer would be diminished in the ratio

of 1 : .

Translated by J. Vanier
for Aeronautics

NACA-Langley 4-22-55 75

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