The applicability of a sine series velocity profile in a two-dimensional incompressible laminar boundary layer

z

.;

THE APPLICABILITY OF A SINE SERIES VELOCITY PROFILE IN A TWO-DIMENSIONAL INCOMPRESSIBLE

LAMINAR BOUNDARY LAYER

By

G. KURYLOWICH

IN A TWO-DIMENSIONAL INCOMPRESSIBLE LAMINAR BOUNDARY LAYER

BY

G. KURYLOWICH

ACKNOWLEDGEMENT

The author is grateful to Dr. G. N. Patterson, Director of the Institute of Aerophysics, for the opportunity to pursue this interest-ing topic.

Special thanks go to Dr. J. A. Steketee for his valuable guidance and for his patient and thorough review of the final manuscript.

An account is given of the adaptibility of a sine velocity pro-file by developing a solution similar to the fourth degree expansion of Pohlhausen. The trigonometrie results are presented in tabular form for the use in the determination of flow quantities in two dimensional boundary layers and a eomparison is made with Pohlhausen and Thwaites. Finally, the results are used in the analysis of the flow over an infinite cylinder.

(i)

TABLE OF CONTENTS

NOTATION I. INTRODUCTION

1I. THEORETICAL DERIV A TION

2. 1 Develop!Dent of the Velocity Profile 2. 2 The M = 6 Profile

2.3 The M

=

3, M

=

5 and M

=

10 Profiles 2.3.1 The M

=

3 Profile

2.3.2 The M

= 5 Profile

2.3.3 The M

= 10 Profile

2. 4 Method of Solution and Observations of the Universal Function Behaviour

2. 5 Flat Plate Analysis ( "

=

0) 2. 6 The Flow Over a Cylinder lIl. SUMMARY REFERENCES FIGURES 1 to 13 Page ii 1 1 1 4 6 7 7 8 11 11 13 14

x U_oo

u

Re f(n) K NOTATION

Schubauer distances defined by the following diagram

U-o

running variabie along the surface of the body with the origin being at the stagnation point

the flow velocity at infinity

the potential flow velocity over the surface of the body viscosity

Reynold's number

non-dimensional displacement thickness non-dimensional momentum thickness

non-dimensional velocity profile through the boundary layer

shear stress at the wall of the body friction coefficient

a non-dimensional direction

non-dimensional parameter (a shape factor) a second shape factor

( 1 ) 1. INTRODUCTION

In his solution of an incompressible two-dimensional boundary layer, Pohlhausen assumed a polynomial approximation for the velocity profile in order to solve for quantities as separation point, momentum thickness, displacement thickness and shear stress. If the pressure distribution over a continuous body is considered, this approximation yields satisfactory results only up to that point on the body where the velocity becomes a maximum; in other words, the Pohlhausen

approxi-mation provides accurate results in regions of favourable pressure gradients and accelerated flows. In areas of unfavourable pressure gradients or

retarded flows, the method diverges from the exact solution producing large errors between experimental and predicted results. This error is proportional to the distance between the separation point and the point of minimum pressure and in instanees where this distance is large,

Pohlhausen prediets separation much too late or sometimes not at all. As an exam ple of this, consider an elliptical cylinder whose major axes is parallel to the direction of the free stream. G. B. Schubauer (Ref. 1) found that for an ellipse with ratio a:b

=

2.96: 1, separation took place at x:b

= 1.

99. A calculation based on Pohlhausen's results showed very good agreement with measurements for velocity profiles up to the point of

minimum pressure but predicted no separation at all. Thus, the point of separation can only be calculated with some degree of uncertainty.

On exam ining the various flat plate coefficients obtained by the use of different profiles (see Fig. 1), it becomes evident that the Lamb assumption approximates the exact flat plate profile more accurately than any of the others; the polynomial results for

7ó

are 3% in error while the sine assumption differs by only 1. 2% from that of the Exact Blasius Solution. This suggests that a trigonometrie profile may be more suitable with regard to a boundary layer flowapproximation and if so, a better prediction of the separation point may possibly be obtained.

Il. THEORETICAL DERIV ATION

2. 1 Development of the Velocity Profile

If the equations of motion for the flow in a two-dimensional incompressible boundary layer are integrated over the width of the boundary layer, then the Von Karman Momentum Equation is obtained

(see Ref. 2). This equation may be written in the form:

U"1.~

+

(2

s

+

<b")

U~U

-=

":fi,

d~ X

;0

( 1 )

To obtain the significant boundary layer parameters from Eq. (1), a velocity profile must be selected, which to some extent

equals the exact velocity profiles found in the boundary layer. We shall, therefore, require that the velocity profile takes the form

r(~

)

=

~/

U

where ~ .. y/~

and the following conditions are satisfied:

( i ) ( ii ) ( iii ) ( iv ) ( v ) ( vi ) ( 2 ) at separation

Condition (i) requires that the velocity profile vanishes at the wall, in other words; there is no slip flow on the surface of the body. The following three conditions join the boundary layer profile smoothly to that of the free stream. Finally, condition ( v ) follows from the ordinary boundary layer assumption in that the pres-sure difference across the boundary layer is negligible.

The form selected is the following:

.f(~

)

=

~ S~(f.f1)

-i-

b

s/",2(fit) .;

Co.

S,.~

(f

'1)

(3)

in the range 0 '

~ ~

I.

fbr

~

>

I , we are in the free stream so that

te,)

'= ,

It is evident from Eq. (3) that condition (Eq. (2) ( i

»

is satisfied automatically so that constants a, band care determined by satisfying the other conditions of Eq. (2).

( 3 )

Therefore, in order to satisfy Eq. (2) (ii) and (iii), etc., a, b, and c must be selected such that they satisfy the following equations:

yd~

:-udu:=r'èJ.Jb

.

à

'ë}"2.

~ '2. ~2. .

If a new parameter

~

is introduced such that

~-=-b=-~

dU

"2

'f->n2.ël)(.

th en the constants become on solving Eqs. (6), (7) and (8)

'>.. (

2.

H- "") M-I a

=

tl

+

t1-l b

=

- 2.X

c

=

.

1.

~ t'\- \

H-I

(6 ) ( 7 ) ( 8 ) ( 9 )

Substituting these values for a, band c into Eq. (3) gives the velocity profile as the following:

M

Y../tj ::

f(r"\)

=

(~

\)

~

(~ ~)

-(Fb/)..6\

n

(trI,

~)

+}..

r

(2H-4)~iY\(1I.~)_

2

$1~2(!I:~\+

2.

s"nM(Ir..t\)]

(

10 )

L (

M _I) '2. 'Z. IJ

(1'1-1)

"'Z.

The velocity profiles so obtained form a one-parameter family of curves with the dimensionless parameter

~

determining the shape of the velocity profile. The parameter

À

can be physically interpreted as the ratio of pressure to viscous forces and is a function of the shape of the body over which computations are being made.

It is easily found from condition (vi) and Eq. (10) that this profile provides a separation point when

( 11 )

So far, the parameter M has been left free. However, if M is taken to be very large, Eq. (10) becornes

This profile seem s to be the ideal profile for in the case of no pressure gradient, it gives the Lamb solution. But, if

À..,

0

it is noticed that condition f" (1)

=

0 is violated. It follows then that this profile will be useful only for

,,=

0 which is the Lamb case. To

investigate the influence of the choice of M further, consider, in

the next sections , the values M = 3, 5, 6 and 10.

'Another generalization of Lamb 's profile would be a Fourier -sine series of the type

f(

ru

-=

-m#

~

S,"" (

m

f

~)

(m.

I.,

3,S····N)

This type of velocity profile, however, cannot satisfy the condition of the pressure gradient at the wall (vi) and has no use in the case of curved surfaces .

2.2 The M = 6 Profile

With M ;;; 6, E q. (10) takes the form

f(I1)"O&

t

sÎn

(f '1)

~

t

SIn

'(~17)

+À[ism(ifl) -

2.

S/I")2.(1[17)

fo-}Sln'(!,,)]

~

13)

while it is found from Eq. (11) that separation occurs for

~~

=: ...,.

0.75

( 14 )

To use the momentum equation, we need to know

U,

~

, ', . and

"

S'"

in order that

e

may be found as a function X.

U

and~' are found from the potential flow around the body so that

1.",

~... and

e

have to be computed from Eq. (13).

and By definition I

t*-

r[l-t(~)Jd~

öi'ö:=

.

'2

9855to -

•

/435'72

À

( 15 ) ( 17 ) ( 18 )

The first integral of Eq. (18) has been computed above, the second integral becomes 011 using Eq. (13)

..

.

'

Substituting Eq. (19) and (16) into Eq. (18) produées

B/~

==

,1/2114 - ,o/'S008À -,

o348b3

~2

;( 20)

Further, we have

-{ 21)

and one finds

I.SB4CjS~

+

2.S13274À

( 22 )

We now multiply the momentum equation by.JZ.. so th at

it appears in the non-dimensional form \0>

If one lets where

+

(z

+

l;'e)

12..2.

dIJ

==

~Tx ( 23 ) ( 24) ( 25 ) ( 26 ) ( 27) ( 28 )

K is now a second shape factor connected with the

momen-turn thickness just as· the first shape factor ~ was connected with the

boundary layer thickness

b .

.

To use Eq. (28) the expressions (24) to (27) respectively

Now

K _

e'2.

dO

13'ax

A substitution of Eq.· ( 9 ) gives

K

~ (~)2.7T2

À

Substituting Eq. (20) into Eq. (30) gives

<

29 )

~ 30 )

2

K-=

TT'l.À[

,

112/14

-,OIS008À-,034e~3À2.J

(31)

In a similar fashion one obtains

f:

(I-() ..

fb;/$

~

,

2'955' -

11435~2

À

J

e

b

.112114 -,0150(8)\-,0348103)\'1

'32)

;'2. (K)

==

~

(E) -- (

/,884

CJSCe,

+

2,5/3274

À)

~u

T

\2( 33 )

x(,

112114- ,0150(8).-.0348'3/\)

Equation (28) may now be written ih the form

~-~

( 34 )

where

K=r~

and

_r(K)==-

2~(k)-4K-2K[-f(K)J

Equation (34) is a non-linear differential equation of the first order for

e"

as a function of x.

~

Although F (k) is very complicated, tj:üs does 'not const:j..tute a major difficulty insofar as the ~olution is concerned as it is an

universal function whi.ch has only to be worked out once; the other functions which appear are also universal and all fundions may be plotted against the variable K. Separation occurs at ~ . . . 0.75 and a variation of

À

in the range between stagnation and separation should provide an adequate working range for boundary layer

compu-tations (See Fig. 5). '

2.3 The M

=

3, M

=

5 and M

=

10 Profiles'

Since the reasoning, basic equations , and methods of evaluation are the same as those in Sec~ 2:. 2, the fulLpresentation of the M

=

3, M

=

5 and M

=

10 profiles is omitted in order to avoid repetition. Only the important functions required to evaluate (Eq. 34) are presented.

( 7 )

2.3.1 The M = 3 Profile (see Fig. 3)

lfi.,).

(3/2.t~)Sjr"l(~rt) -2Àsinl(~rt) +(~-~)Sih3(l"')

<

35)

~

. .

/~

. . I

2572.

77 - .

Q ,

I

033

À

( 36

,)

e/~

...

,/020"9 - . DoC)o48

~

-, 00teJ724

À2.

(

37 )

t i

=

2.

35~'~S

+

/.5'707'97'\

(

38 )

~u

_K'Ia

_7T2.À[

_,/020~B-

~]'

,

oo~04BA-,oo'7241\

(

39)

f;

(K) ..

I

2S72~~-, Q~/Q3~À

1/02.0 -,009049)\-, OO~7J.~)\'2..

(2.35~/~5+ /

.

'57

D

_7'7\)

)<.. (,

lo2DCJB-

I

oo~o48À-.

0 0 '

7.24)f)

~~

=

-/,50

forseparation ( 40 ) ( 41 ) ( 42 ) The tables provided eaU for a working range of values between stagnation and separation . '

2.3.2 The M = 5 Profile (see Fig. 4)

f(IJ)=

(5/4

+">V4)S""(~IJ)-2À5i.,2(i~) .,.(~-t)5

ih

5

(ft!) ( 43 )

_

<b'~

'C:. ,

28'/~

-

,/24t:,te,

À

. .

(

44 )

S/~

=

,lo9~4

-

,01408)...-,02680

\2.

(

45 )

l.t

=

-

j.

"34'}

+

2

,

55"'\

-{

46)

K

-=

TT'1~[

,./0"54

-,OI40B~-,02'SOÀ2.Jl.

(47)

t,(K)=

12e'J3-,124'~À

(48)

./0'54-

,o/40S>\-,02.6801)\%O

f~(K)-=

(/.

"':54'

+2,

!5'/c}~)

(49 )

)(. (

,

/0-'54-,

0140S\-

,02"SO>?-)

~'5

. . - I

e

3

3(3)

for separation

I

50 )

Thetables provided have a working range of

À

between stagnation and separation.

2.3.3 The M = 10 Profile (see Fig. 6)

tUt)

=(IOI~

+ /

~A4)5

i

Y'\

(Ii

'1) ...

2

À

5'

n(lf.'l)

+(

~

...

~)

S',

~(lil'l)

(

51 )

ç

'*/~

= •

3/

~ ~c:;

- •

/B~4fa\

(

52 )

7ö~-== /,74537+2.7~2S'~À

(54) ~U 4

K

=

7T

2

\[.//Bt,O - .

D/742À -_

QS~2B\'2.J

c

55 ) . /18~O-. O/742)\-.D5"~2B~ ( 56 )

~

(K)

=

(I.

74~

31

-t-

2.

7

~

2

Ö

3

~

)

.

(

57 ) X

(.118"0 - . 0/742

À-.

OS~~8À'2.)

\~--.c:'25

for separation ( 58 )

The tables provided show a w orking range of

X

between s tagna tion and s epar a tion .

2.4 Method of Solution and Observations of the Universal Function Behaviour

and

Before using the equations

d=t: ...

F(

KJ

orx

K=è~LJ

dx.

some remarks must be made .

( 59 )

( 60 )

. The v'alues. of U (x)" and

du(~~

to be used are obtained by considering the flow of a perfect fluÜ:P!round the object being con-sidered. The variabIe x is the running variabIe over the surface of the body. lts range is bounded by the s'tagnation point upstream and the downstream point of separation. However, at the stagnation point, U = 0 and oI~

...

00 unless F (H> = O.

~-If F (K) = 0 at stagnation, ~ becomes indeterminate so that its value can be obtained by going

o~er

to the limit and using Eq. (60). Since ~+O at stagnation unless the body possesses a sharp

. cusped velocity distribution with zero angle at the stagnation point, we have

( 9 )

where das hes denote derivatives with respect to X while the subscript o indicates stagnation values.

An exariünation of the Pohlhausen F (K) curve shows

F (K) = 0 at stagnation. Thus no difficulty is encountered in following the above method. However, the sine F (K) solutions behave

irreg-. ularly near and at this point. Most important of all, F (K)

=ti::

0 at

stagnation and this irregularity increases with increasing M. Therefore E q. (59) cannot be integrated num erically as ~ __ c::c at stagnation so that computations cannot be started at this bo&<'ndary positïon.

However, the original momentum equation wasformulated from the assumption that ca1culations are made at some distance from the stagnation point. At and near stagnation, this assumption breaks down so that the momentum equation becomes invalid. It is indeed fortunate that this 'cliscontinuity does not reveal itself in the Pohlhausen results , nevertheless, the sine profiles seem to be very sensitive to its effect.

If the "stagnation irregularity" is neglected, it is found that the F (K) as functions of K curves are practically straight lines so that they can be approximated by

( 6 2 )

With this value for F (K), the integration can be performed straight a way and we find

2.

r)(.

b-I

U~

..

US

=

.fL..

U

0/><.

(

63 )

y:-

U

b

-I 0 !

. On approximating F (K) by straight lines and forcing F (K) =

o

at the stagnation point, equations (59) and (60) become the following for t~e various profiles assumed.

M = 3 Profile )I..

r

-=

.~

f

USdx

U&c:I

Ko=

.0802

_\ ( 64 ) \

Ks

-== -,

J4~7

at x-=o

M = 5 Profile

peK') -

.440 -

Ct,

K

~ ~

..

4~

(US"cI)(

U c ( 65 )

.

Ko

==

·,0733

Ks--'

-.8So

.

,.

_Zo'::

jO

0733

Iud

at- )(. ....

0

M

=

6 Profile

F(K)

=

.

•

44o-taK

~

=

.440

.

!u

5

dx

U1i

~

~

Kè)=.O,33

(66) M = 10 ;Profile Ks~

-,

,9,

La-='

.0733/U~

at X--o

F(K)-== .414 -

Co

K

)(

~

=-

.~

~

u

5

d><.

U

0

1<0=.

OIO

K

_s

= -

0.1'3

~o

-=

. 0 , 0

lu

0'

~t ) ( - 0 ( 67 )

For purposes of comparison Thwaites and P ohlhausen results are also given:

Thwaites (see Ref. 4)

Co '/..

e

2 ₌

_.45\-"'

_U-

~ u'Sd~

o

K

_s

= -

0.082

( 11 )

Pohlhausen (simplified by A. Walz, ;Ref. 2)

)<.

e'"=

.470YU-~~U5dx

o

I<~

=- -

o.

/510

7

- (69)

It is noticed, especially' for a larger M, that there is good agreement with Thwaites resultp,. Since Thwaites values for separation give gdod agreement with known exact solutions , we may conclude

that the M

=

5, M

=

6 and M

=

10 cases will produce results 'which are equally as satisfactory.

The above statement can be verified further if the shear curve is examined (Fig. 10). In the region of accelerated flow, the shear stress growth is roughly equivalent in each of the cases-~ In the region of retarded flows, the shear stress does not build up to the same',extent as that predicted by PCJblhaÎlSen. Therefore, the sine assumption will predièt separation much earlier than the Pohlhausen results . This is

à

definite improvement as the Pohlhausen approxi-mation fails there.

Tables provided for computational purposes (see Figs. 3, 4, 5 and 6) are the corrected values insofar as the "stagnation irreg-ularity" is concerned. Figure 7 shows the general trend of this

irregularity as

Ä

approaches the stagnation value. The "approximate" and "irregular" curves are shown on each graph presented.

,.

2.5 Flat Plate Analysis (À = 0 ) (see Fig. 2)

Figure 2 contains a comparison between the exact Blasius values of displacement thickness, shear stress, momentum thickness, and friction with that of the approximate results developed. -It is seen that agreement is very satis~actory especially ih the M = 5 case.

~. 6 The Flow Over a Cylinder

We now propose to compare the approximate res-ults of the M = 5 and M = 10 cases with those obtained by Pohlhausen, Thwaites and the exact Blasius solution for the two-dimensional

cylinder. The Pohlhausen curves shown in Figs. 11 to 13 respectiv~ly

are based on the straight line approximation developed by A. Waltz (Ref.2); the exact curves are obtained by the use of the potenti-al velocity, distribution expanded into a· series containing five terms

(up to X 9).

The potential velocity distribution over a cylinder is as follows:

U(><.}-=

2L.Jcocslh(~)

(71)

-,

=

~Ue{~-·~l~)3+t(~)S

(72)

-

~L

(

~

')

f-

~( ië)~j

(used for the exac{ solution) Substituting Eq. (71) into Eq. (63) produces , :. ' :. ij

Uoo

e'2.,.

2 b

ç

.

.6... -

cos

cp[

~

sit"\q,

+..ft_

+

sin

'4ç~o

J1 (

73 )

'R'(>

Sin''P

2

15

's

l5' S .

S

wher.e. 'f-~ and R = radius of the cylinder.

As X approaches zero,

qï

approaches zero so,_ that equation ( 73 ) becomes indefinite . However, Eq. (61) produces the stagnation

, limit and this becomes on using $q. (71)

f-o-=

Kc R

a..t

x-

0

,

<

74)

2.Ucc

and. as a re.sult of this

4 0

0092.

-=

2

Ko

at

x-a

(

75 )

R'?

Equations (73) and (75) then yield the momentum distri-bution over the cylinder as a function of the angle in degrees.

Since

( 76 ) the universal function K can be ca1culated and for each réspective K, a value for .f 1 (K) and f 2 (K) can be se le.cted from the given tables .

The displacement thickness and shear buildup over the cylinder can then be obtained as the following re lationships are true

~JuooRI

-R

'r'

2.

7;,

JUcaR

=

8

f.

O<JSihtf)

( 77 ) ( 78 )

r

Uooz.

'r

2SJUooR

R

-,::r-

,

The parameters governing the flow over a two-dimensional cylinder can now be plotted and are shown in Figs. 11, 12 and 13.

A study of the graphs reveals that in the region of favour-able pressure gradients, the exact and approximate results are

. almost in complete agreement, hOWever, in the region of reta:r;ded flow, the approximate values of

e

and ~ ~are higher than the

respective exact values and the deviation increases as separation is approached.

, .' \

\ ... J.

:i

' J )

( 13 )

The stress curve reveals that the M = 5 profile predicts a separation point ,equivalent to that of Thwaites while the M = 10 profile predicts a separation point which occurs much earlier. Thus, even though the sine profiles produce the best approximation to the exact momentum distribution, the separation point for a cylinder is predicted too soon.

lIL SUMMARY

In this note, a new ass umption for the ve locity profile has befi:n used for solving incompressible 'larninar boundary layers as de-scribed mathematically by the momentum equation.

The velocHy profile assumed is of the form

M

+{- -

.p(~

')

:=

..tL

S;~(lI:t't\

-

-L

'Sin

(rr.~)

u 1'1-/ "2:')

H-I

2.

The M

=

6 profile solution is presented in' full detail and tables are prepared for each M selected.

In the case of the flat plate (

À

= 0), the sine appr-oximation coefficients show a noticeable improvement over previous m ethods (especially in the M = 5 solution) .

For À:~O, the results agree well with Thwaites solution and produce a separation prediction which is roughly equivalent to or earlier than the Thwaites I separation point.

If one considers flows where the-distance between the point of maximum velocity and the separation point is large, the divergence between exact _and calculated results may not be as great as that ob-tained in the cylindrical results so that a more reliable prediclion of separation could be obtained.

N evertheless, the method is a definite improvement over the Pohlhausen solution as it does predict separation much earlier while still retaining good accuracy in regions of accelerated flow s .

1. Schubauer, G. B.

2. Schlichting, H.

3. Goldstein, S.

4. Thwaites, B.

REFERENCES

Airflow in a Separating Laminar

Boundary Layer, NACA Rep. 527, 1935. Boundary Layer Theory.

Published by McGraw-Hill Book Co. Inc. Modern De,velopments in Fluid Dynamics, Published by' the Oxford University Press. Approximate Calculations of the Laminar Boundary Layer, Aeronautical Quarterly, Vol. I, p. 245-280, 1949.

Profile ~~J1Ç'

~YJ

Cf

f(p

.

~"'*Ie

.p(f1

)-fl

1. 732 .289 1.155 3.00

-F

(~

) _

'3/'2..J't -

Y2.1'l3

1. 740 .323 1.292 2.70

f(r0-

~'1-';).~'5+'74

1. 752 .343 1.3-92 2.55

f(l'J)

=-

s/n(.fi}) [

La.mbJ

1. 741 .327 1. 310 2.66

The exact Bfasius solution 1.729 .332 1. 328 2.61

FIG. 1 FLAT PLATE COEFFICIENTS

Pref,

Ie.

stt

J1;

_p.x

_...u

Xi

_U

J~x

_U

Cf

'Re

<i~/e

Pohl. 1. 752 .343 1.392 2.550 Lamb 1. 741 .327 1. 310 2.660 M = 3 1.748 .255 1.000 2.520 M = 5 1. 731 .328 1. 312 2.639 M = 6 1. 734 .324 1. 296 2.670 M = 10 1.736 .322 1.288 2.698

.

The Blasius Exact Solution 1. 729 .332 1.328 2.610

À

K

~(K)

fa(K)

reK)

-1.50 -;14965 3.46949 .0000000 1. 63702 -1.45 -.146218 3.42079 .. 0079387. 1.60111 -1.40 -.142592 3.37371 .0159570 1.56441 -1.35 -.138781 3.32821 .0240468 1. 52700 -1.30 -.134792 3.28419 .0322002 1. 48893 -1.25 -.130633 3.24161 .0404093 1.45028 -1.20 -.126315 3.20041 .0486662 1.41111 -1.15 -.121844 3.16053 .0569629 1. 37149 -1.10 -.117232 3.12191 .0652915 1. 33149 -1. 05 -.112486 3.08452 .0736441 1.29117 -1.00 -.10761'1 3.04830 .0820128 1. 25059

-

.950 -.102635 3.01321 .0903896 1. 20984

-

.900 -.0975487 2.97922 .0987667 1.16896

-

.850 -.0923685 2.94627 .107136 1.12803

-

.800 -.0871047 2.91435 .115489 1. 08710

-

.750 -.0817675 2.88341 .123820 1. 04624

-

.700 -.0763673 2.85342 . .132118 1.00552

I

-

.650 -.0709145 2.82436 .140378 .964990

-

.600 -.0654195 2.79619 .148590 .924709

I

.550 -.0598928 2.76888 .156747 .884739 I

-

I

-

.500 -.0543450 2.74243 .164840 .845137

-

.,450 -.0487866 2.71679 .112863 .805960

-

.400 -.0432281 2.69195 .180807 .767263

-

.350 -.0376799 2.66790 .188664 .729101

-

.300 -.0321526 2.64460 .196426 .691524

-

.250 -.0266565 2.62205 .204085 .654585

-

.200 -.0212018 2.60022 .211633 .618333 I

-

.150 -.0157988 2.57910 .219063 .582816

I

-

.100 -.0104575 2.55869 .226367 .548079

-

.050 -.0051880 2.53895 .233536 .514168 .000' .000000 2.51990 .240562 .:481125 + .050 .0050968 2.50150 .247439 .448991 + .100 .0100931 2.4.8377 .:254157 .417804 + .150 .0149794 2.46667 .260709 .387601

I

+ .200 .0197468 2.4:5022 .267087 .358418 + .250 .0243865 2.43441 .273283 .330286 + .300 .0288898 2.41922 .279289 .303237 + .350 .0332485 2.40466 .285097 .• 2:77.298 + .400 .0374545 2.39072 .290700 .. 252495 + .450 .0415000 2.37741 .296089 .228852 + .500 .0453778 2.36472 .301256 .206390 + .550 .0490806 2.35266 .306194 .185126 + .600 .0526017 2.34122 .310895 .165077 + .650 .0559349 2.33042 .315350 .146256 + .700 .0590739 2.32026 .319552 .128674 + .750 .0620134 2.31074 .323492 .112337 + .800 .0647480 2.30189 .327164 .0972510

+

.850 .0672730 2.29370 .33055.9 .0834171

+

.900 .0695842 2.28619 .333668 .0708343

+

.950 .0716776 2.27938 .. 336485 .0595982

~

K

,,( K)

f

2

(K)

F(\<)

+ 1. 00 .0735500 2.27328 .339001

-+ 1. 05 .0751985 2.26793 .341208

-+ 1. 10 .0766206 2.26333 .343099

-+1. 15 .0778146 2.25952 .344665

-+ 1. 20 . 0787793 2.25652 .345899

-I + 1. 25 .0795139 2.25438 .346792

-i + 1. _{30 .0800184} 2.25312 .347337

-i + 1. 35 .0802931 2. 25280 .34752

-!I FIG. 3 M

=

3

-

.

'"

K

f,(

K)

f

a.(

K)

F(K)

,

.0736 2.2733 .3618 .0717 2.2793 .2581 .0696 2.2862 .3539 .0673 2.2937 .3494 .0647 2.3019 .3443 .0620 2.3107 .3389 .0591 2.3203 .3332 .0559 2.3304 .3269 .0526 2.3400 .3204 ~ .625 .050266 2.33980 .31018 .184069 .600 .049517 2.34388 .30882 .187450 .575 .048644 2.34870 .30721 .191358 .550 .047647 2.35424 .30536 .195801 .525 .046528 2.36049 .30328 .200788 .500 .045289 2.36743 .30096 .206327 .475 .043933 2.37504 .29841 .212421 .450 .042460 2.38331 .29565 .219075

I

..400 425 .040874 .039178 2.39222 2.40178 .29267 .28949 .226291 .234069 1 .375 .037375 2.41197 .28610 .242409 l

l

.350 .035467 2.42279 .28252 .251308 t _{.325 .033459 2.43422 .27874 .260762}

I

_{.300 .031353 2.44628 .27478 .270766} i , .275 .029153 2.45894 .27065 .281314 .250 .026864 2.47223 .26634 .292398 I .225 .024489 I 2.48612 .26186 .304009 .175 _.019~ 2.51577 .25244 .328770 .150 .016 4 2.53153 .24750 .341896 .125 .014220 2.54791 .24242 .355502 .100 .011482 2.56493 .23720 .369571 I .075 .008687 2.58259 .23185 .384089 .050 .005838 2.60089 .22638 .399039 : , .025 .002940 2.61986 .22078 .414403

.

_{000 00000 2.63949 .21508 .430161} I : - ~ 025 -.002978 2.65980 .20926 .446294 ; - .050 - .004990 2.68080 .20335 .462782 i - .075 - .009029 2.70251 .19734 .479602 I -.012090 2.72493 .19124 .496733

1

-

.100 i - • 125 - .015167 2.74809 .18505 .514151 I i - .150 - .018256 2.77201 .17879 .531833 i

1

-

.175 - .021350 2.79669 .17246 .549752 t _I i - 200 - .024445 2.82216 .1.6606 .567886 \

.

I

1

-

.225 - .027534 2.84845 .15960 .586206 I - .250 - .030612 2.87556 .15309 .604687

I

1

-

.275 - .033673 2.90353 .14653 .623301 - .300 - .036712 2.93239 .13992 .642020 - .325 -.039724 2.96215 .13328 .660817 - .350 - .042703 2.99285 .12662 .679662

!

-

.375

I

- .045643 3.02452 .11992 .698525 ..

" ,

~

K

f,(K)

f

₂

(K)

F(K)

-.400 -.048539 3.05719 .113210 .717378 -.425 - .051386 3.09089 .106480 .736189 - .450 - .054179 3.12566 .099758 .754926 -.475 -.056912 3.16154 .093025 .773567 -.500 -.059580 3.19858 .086299 .792070 -.525 -.062179 3.2368a .079583 .810409 -.550 -.064703 3.27626 .072884 .828551 - .575 -.067147 3.31701 .066209 .846464 - .600 - .069507 3.35911 .059562 .864117 -.625 - .071777 3.40259 .052950 .881477 -.675 - .076035 3.49400 .039855 .915189 -.700 -.078013 3.54204 .033383 .931476 -.725 -.079885 3.59174 .026970 .947342 -.750 -.081648 3.64318 .020621 .962753 -.775 -.083297 3.69642 .014342 .977677 -.800 -.084829 3.75157 .00814'1 .992083 -.825 -.086240 3.80871 .002021 1.00593 -.850 -.087529 3.86794 -.004011 1. 01920 F IG 4 M

=

5

.0736 2.2733 .3618 .0717 2.2793 .3581 .0696 2.2862 .3539 .0673 2.2937 .3494 . 0647

.

2.3019

_.

'.3443 .0620 2.3107 : .3389 .0591 2.3203 .3332 .0559 2.3304 ' .3269 .0526 2.3400. .3204 .550 .047266 2.,35314 .304879 .198244 -.525 .046395 2.35845 .303220 .202015 .500 .045379 2.36468 .301260 .206390 .475 .044218 2.37182 .299006 .211382 .450 .042916 2.37983 .296467 .217000 .425 .041476 2.388"'0 .293851 .223250 .400 .039900 2.39841 I .290566 .230137 .375 .038192 2.40894 .287220 .237664 .350 .036356 2.42029 .283621 .245831 .325 .034396 2.43243 .279779 .254635 .300 · 03231'1 2.44537 .275700 .264073 .275 .030123 2.45911 .271393 .274139 .250 .027819 2.47363 .266867 . 2,84825 · 225 • 025411 2.48894 .262129 .296120 · 200 .022903 2.50504 .257188 .308012 · 175 .020302 2.52192 .252052 .320488 · 150 .017614 2.53961 .246729 .333533 .125 · 01484'~ 2.55810 .241228 .347129 .100 .011999 2.57740 .235556 .361257 .075 .009086 2.59753 .229722 .375897 .050 .006110 2.61849 .223734 .391026 .025 .003079 2.64029 .217601 .406622 000 000000 2.66296 .211329 .422659 -.025 -.003120 2.68651 .204929 .439112 -.050 -.006276 2.71096 .198408 .455952 -.075 -.009459 2.73633 .191773 .473151 -.100 -.012661 2.76265 .185034 .490678 -.125 -.015877 2.78993 .178199 .508503 -.150 -.019098 2.81821 .171275 .526593 -.175 -.022317 2.84751 .164272 .544915 -.200 -.025527 2.87786 .157196 .563435 -.225 -.028721 2.90931 .150057 .582117 -.250 -.031890 2.94188 .142863 .600925 -.275 -.035028 2.97561 .135621 .619823 -.300 -.038128 3.01055 .128341 .638772 -.325 -.041182 :3'004673 .121030 .657735 -.350 -.044183 3.08422 .113696 .676673 -.375 -.047125 3.12304 .106348 .695546 -.400 -.049999 3.16328 .098994 . .714313 -.425 -.052800 3.20497 .091642 .732935

~

_K

_i

_0<)

f

₁

(K)

F(K)

-.450 -.055520 3.24818 .0843011 .751370 -.475 -.058154 3.29298 .0769781 .769577 -.500 -.060694 3.33944 .0696819 .787515 -.525 -.063135 3.38764 .0624207 .805142 -.550 -.065470 3.43765 .0552027 .822415 -.575 -.067693 3.48957 .0480362 .839292 -.600 -.069800 3.54350 .0409292 .855731 -.625 - .071783 3,59952 .0338901 .871693 -.650 -.073639 3.65775 .0269271 .887127 -.675 -.075363 3.71832 .0200484 .902000 - .700 -.076949 3.78134 .0132621 .916265 -.725 -.078393 3.84695 .0065766 .929882 -.750 -.079692 3.91530 .0000000 .942810 FIG 5 M = 6

1 .u70 2.2862 .3553 .0673 2.2937 .3497 .0647 2.3019 .3442 .0620 2.3107 .3385 .0591 2.32.03 .3324 .0559 2.3304 .32572 .0526 2.3400 .3188 .0495 2.3439 .3125 .0476 2.3542 .3083 -.0453 2.3674 .3035 .0425 2.3833 .2976 .0409 2.3922 .2942 .0392 2.4018 .2907 .0374 2.4120 .2869 .0355 2.4228 .2829 .300 .03473 2.4379 .27977 .25126 .275 .03257 2.4528 .27534 .26058 .250 .03025 2.4689 .27056 .270·74 .225 .02777 2.4862 .26545 .28172 .200 .02514 2.5047 . 26002 .29351 .175 .02238 2.5245 .25429 .30609 .150 .01948 2.5455 .24828 .31944 .125 .01647 2.5677 .24199 .33353 .100 ' .01335 2.5912 .23545 .34834 . 0750 .01013 2.6159 .22866 .36382 . 0500 . 006823 2.6420 .22165 .37995 .0250 . g03443 2.6693 .21442 .39669 .000 .00000 2.6981 .20700 .41400 -. D250 -. 003494 2.7281 .19939 .43182 -. 0500 -. 007027 2.7597 .19161 .450~1 -. 0750 - .01059 2.7927 .18368 .46883 --- .100 - .01416 2.8272 .17560 .48792 - .125 - .01773 2.8632 .16741 .50732 - .150 -.02130 2.9009 .15910 .52698 - .175 -.02484 2.9403 .15070 .54685 -.200 - .02834 2.9815 .14222 .56665 - .225 -.03180 3.0245 .13367 .58693 -.250 - .03520 3.0694 .12507 .60702 -.275 - .03852 3.1164 .11644 .62706 - .300 - .04176 3.1654 .10778 .64699 -.325 - .04490 3.2167 .09912 .66673 -.350 - .04794 3.2703 .09046 .686-23 -.375 - .05085 3.3264 .08183 .70540 - .400 - .05364 3.3851 .07324 .72418 -.425 - .05628 3.4465 .06469 ,74250 ~.450 - .05878 3.5109 .05622 .. .76029 - .475 - .06111 3.5783 .04783 .77748

I

À

K

-t,

(K)

-Ç'l.(

K)

F(K)

-.500 -.06328 3.6491 . O~ 9529 .79400 -.525 -.06527 ;5.7233 . 031342 .80978 -.550 -.06707 3.8013 .023281 .82475 -.575 -.06868 3.8834 .015360 .83884 -.600 -.07009 3. 9697 .075953 .85198 -.625 -.07129 4. 0606 .00000 .86410 -.650 -.07228 4. 1565 -.00741 .87514 -.675 -.07;:$05 4. 2577 -.014621

.

.88502 -.700 -.07360 4. ;)647 -.021618 .89369 -.800 -.07355 4.8609 -.047167 . 91493 -. 900 -.06987 5.5000 -.068111 . 91186 -1. 00 -.06276 6.3513 -.083505 .88116 FIG. 6 M :: 10

When ).. and r'k')are dashed in the above tables,

\~)

ft, ,,)

and

.ç.z..(K)

are the approximate "stagnation irregularity free" results. The only exception to this occurs in the case of M

=

3 (Fig. 3). Here, only

r{\c.~

is affected by the irregularity; the universal functions

K)

f.O<)

and ~'K) are not affected by the stagnation problem.

""

K

+L(

KJ

F(K)

M

=

3 -1. 00 .0736 2.2733 .3390 .0494 1. 10 .0766 2. 2633 .3431 .0329 1. 20 .9787 2.2565 .3460 .02'11 1. 30 .0800 2. 2531 .3473 .0140 1. 40 .0803 2.2535 .3474 .0113 1. 50 .0798 2.2580 .3459 .0126 1. 60 .0783 2.2671 .3429 .0176 1. 80 .0728 2.3025 .3319 .0371 2.00 .0644 2.3677 .3140 .0656 M

=

5 0.50 .0453 2.3674 .3010 .2063 0.60 .0495 2.3439 .3088 . 1875 0.70 .0518 2.3323 .3127 .1770 0.80 .0520 2.3347 .3122 .1739 O. 90 .0502 2.3541 3070 .1770 1. 00 .0465 2.3621 .2966 .1842 M

=

6 .40 .0399 2.3984 .2906 .2301 .50 .0454 2.3647 .3013 .2064 .60 .0486 2.3455 .3073 .1925 .70 .0494 2.3430 .3080 .1873 .80 .0478 2.3611 .3031 .1893 .90 .0440 2.4063 .2198 .1960 1. 00 .0382 2.4897 .2738 .2042 M

=

10 .30 .0347 2.4380 .2798 .2513 .40 .0416 2. 39-12 .2938 . 2223 .50 .0453 2.3665 .3010 .2064 .60 .0457 2.3680 .3007 .2017 .70 .0429 2.4035 .2917 .2053 .80 .0372 2.4885 .2732 .2123 .90 .0292 2.6541 .2442 .2165 1. 00 .0199 2. 9739 .2038 .2096

/:':"

> . '. :" ",." : ... : ...

. : .. :~, ::' .. ."::' : .. :-'::: _{I ~ :-:-:.' -:=-:..~. ~::::::::C -:-:::?:7c~':: r-~-::.I:} .;;.: ..

:; ::>

:~~:~'-~;r: :,;,~; :"i . _{.:~'~ :1118 : ~: ...:.:. -=-~~:~} ::':'':'-;'; ':';:-"-:-:+ ,-+..o.lili~~~ _{''2;j-'" -F;;,-:;,;,..:.: Ji::'ji:} .':!~",;r.p{

t~=~':; T.;:: ':+;: f-".t::;f~ ~'"h ,

11

I

•

•

lil

::

b~~H·~~rT::~-:l:L.iiI'~; :;!.~I:1-h~~

Ió-· l~

FIG 9 THE UNIVERSAL FUNCTION f (K)

.,.:;~ :_" I::':::'::: I::: _~::: '. = I~?~:·I-:~::

.

::""-:0-'cc t~·.; ~-':::~ f-Ä:-F~ = 1:/ VERSUS K CURVE

e

:.1 ' r..:':···:

:::

I1ftrj:~ ;:;= .;:: ~ :.; ': ei! I Fi tt;-: ~! I , {;, "i1fu! 1 iiH-;$!: h=1;tt;~

FIG 10 THE UNIVERSAL FUNCTION f_2(K) = 67

0 VERSUS K CURVE ,u. U

R

~ 1.2 1.0 Pohlhousen

0.8

0.6

I

~

0.4;

<.:~

0.2

o

60

70

80

90

14\0 'I' - - - ' J .. ~

FIG 11 THE MOMENTUM DISTRIBUTION OVER THE SURFACE OF AN INFINITE CY LINDER

1.4 1.2 1.0

0.8

0.6

0.4

0.2

o

ÎJu:;;

RIT

10

20

30

40

Pohlhouse M=IO

50

60

70

80

90

o

~

~

FIG 12 THE DISPLACEMENT THICKNESS DISTRIBUTION OVER THE SURFACE OF AN INFINITE CYLINDER

1.3 1.1

0.5

0.1

o

2;' ju..R·

pU:O

T

10

20

30

40

50

60

70

80

90

,.

FIG 13 THE STRESS DISTRIBUTION OVER THE SURFACE OF AN INFINITE CY LINDER