The Laminar Boundary Layer with Injection Through a Permeable Wall

10 rD"1961

THE COLLEGE OF AERONAUTICS

CRANFIELD

LAMINAR BOUNDARY LAYER WITH INJECTION

THROUGH A PERMEABLE WALL

R E P O R T NQ.145 January, 1961

T H E C O L L E G E OF A E R O N A U T I C S

C R A N F I E L D

The Laminar Boundary Layer with Injection Through a Permeable Wall

b y

-T. N. Stevenson, B . S c , D . C . A e .

SUMMAR\

The steady incompressible laminar boundary layer equations for a perfect gas with a r b i t r a r y distributions of wall temperature, main

stream velocity and normal velocity at the wall are solved approximately by a method similar to that of Lighthill (1950). Equations for the skin friction and the rate ol heat transfer are obtained.

In order to a s s e s s the accuracy of these equations, solutions are presented and compared with exact solutions for wedge type flow with a wall temperature distribution such that the difference between the wall and the stream temperatures is proportional to a power of the distance from the leading edge. A Prandtl number of 0 7 is used. A solution for the skin friction with a constant normal velocity at the wall and constant main stream velocity is also presented.

The method is probably accurate enough for engineering purposes in regions of negau^ve p r e s s u r e gradient.

A solution for compressible flow using a similar method is outlined.

* The incompressible solution was submitted in partial fulfilment of the requirements for the Diploma of the College of Aeronautics.

P a g e S u m m a r y

L i s t of Symbols

Introduction 1 T h e B o u n d a r y L a y e r E q u a t i o n s with

Flow t h r o u g h the P o r o u s Wall 3 The Solution of the Momentxim Equation

( 2 . 1 0 ) 4 T h e Solution of the E n e r g y Equation ( 2 . 1 1 ) 7

A G e n e r a l Equation for the Heat T r a n s f e r

Rate Q (x) 12 w

The N u s s e l t N u m b e r 13 Solution of the I n t e g r a l Equations when

the Wall S h e a r S t r e s s r (x) = C o n s t . x x w

and the F r e e S t r e a m Velocity u,(x) = ex 14

7 . 1 . The m o m e n t u m equation 14 7 . 2 . A p a r t i a l i n v e r s i o n of the

m o m e n t u m equation (3.14) 16 7 . 3 . The f i r s t i t e r a t i o n of the e n e r g y

equation 18 7 . 4 . The second i t e r a t i o n of the e n e r g y

equation 18 7 . 5 . I t e r a t i o n s u s i n g the g e n e r a l equation 5 . 3 19 7 . 6 . T h e l o c a l N u s s e l t n u m b e r , nu 20 7 . 7 . Wall t e m p e r a t u r e v a r i a t i o n T = B X 22 w 1 A p p r o x i m a t e Solution to the I n t e g r a l E q u a t i o n s when the F r e e S t r e a m i s

Uniform and the Velocity, v i s a constant 23

D i s c u s s i o n 27 C o n c l u s i o n s 29 Acknowledgements 29

R e f e r e n c e s 30 Appendix A - A P a r t i a l T r a n s f o r m a t i o n

of the Momentum Equation 33 Appendix B - The C o m p r e s s i b l e L a m i n a r

B o u n d a r y L a y e r with Suction

o r Injection 34 T a b l e s 1, 2 and 3

LIST O F SYMBOLS a , a , A C o n s t a n t s ( s e e equations 8 . 8 , 8 . 1 8 , and 4 . 1 8 r e s p e c t i v e l y ) b^ , B , B, C o n s t a n t s ( s e e equations 8 . 8 , 4 . 1 8 , and 7.37 r e s p e c t i v e l y ) m c Constant defined by u, = c x C Specific heat at constant p r e s s u r e D Constant ( s e e equations 7.40 and 7.41) E Constant defined by equation 7.31

f D i m e n s i o n l e s s m e a s u r e of the flow through the wall ( s e e equation 7.6)

f Dimiensionless skin friciion (see equation 7.7)

mx) - ƒ -^r^(x')v^(x')dx' +u^(x)

1-1 / \ 2 r (x) Fg (x) = w UP T-. / \ crv (x) Q (x) F (x) = - w ^ w T (x) 2k W 1^

G Constant defined by equation 7.3

G(xf ) = Z(x, f) + r S(x', $ )dx'

o

Iji-) Modified B e s s e l function of f i r s t kind k T h e r m a l conductivity

K, K^ C o n s t a n t s defined by equations 7.4 and 8.17 r e s p e c t i v e l y Ky (-) Modified B e s s e l function of second kind

L Length of plate

n - l , n - 2 nu Nu P P q Q Q^(x) R, R^ Re S ( x , i ) T T _{•••1 » -^2} u W

J

y

z

Defined by equation 5. 2 L o c a l N u s s e l t nximber O v e r a l l N u s s e l t n u m b e r P r e s s u r e ; H e a v i s i d e o p e r a t o r F u n c t i o n of $ in equation 4 . 1 2

4 i

3 P ?*, w h e r e p i s H e a v i s i d e o p e r a t o r F u n c t i o n of * in equation 4 . 1 2

Rate of heat t r a n s f e r to the wall

C o n s t a n t s ( s e e equations 8.7 and 8.18 r e s p e c t i v e l y ) Reynolds num.ber

az

8$

_{9^ A}

Independent v a r i a b l e ( s e e equations 3 . 8 and 4 . 6 a ) ; t^ d u m m y v a r i a b l e of i n t e g r a t i o n T e m p e r a t u r e (A t e m p e r a t u r e s c a l e with the f r e e stream, z e r o ) Defined by equation 4 . 1 3 Velocity in x - d i r e c t i o n ; u^ i s f r e e s t r e a m v e l o c i t y Vp^ocity in y - d i r e c t i o n W r o n s k i a n

C o - o r d i n a t e from stagnation point along the p l a t e ; x^ , and x' , d u m m y v a r i a b l e s of i n t e g r a t i o n

C o - o r d i n a t e n o r m a l to the plate

2 2

u^ - u

L i s t of Symbols (Continued) a Defined by equation 7. 32 i? Exponent from r (x) = K x y Exponent from v (x) = Gx^ r (-) Gamnaa function e Exponent from T (x) = B x ; e = ; ^ w 1 T d T X w 2 W V X w U, V dx p D e n s i t y U D y n a m i c v i s c o s i t y V K i n e m a t i c v i s c o s i t y C n P P r a n d t l n u m b e r , w ^ S t r e a m function * Modified s t r e a m function, f ~ i T (x) Wall s h e a r s t r e s s w Svxbscripts

w R e f e r s to conditions at the wall th

n R e f e r s to the n i t e r a t i o n

1. Introduction

In order to predict the skin friction or the heat transfer at the

surface of a body moving through the atmosphere, it is essential to know the behaviour of the boundary layer over the body. It may be n e c e s s a r y to heat a surface to prevent icing or alternatively to cool the surface to prevent it reaching high t e m p e r a t u r e s . The high temperatures at high Mach numbers a r e due to the heat transferred from the boundary layer which is heated by viscous s t r e s s e s . An effective method of cooling heated bodies is to inject a gas through a permeable wall into the

boundary layer, thus modifying the velocity and temperature profiles and, thereby reducing the heat transfer to the surface. Suction through a

permeable wall niay be used to control the boundary layer by increasing the skin friction and delaying separation.

Many solutions to the boundary layer equations have been reported for flow over a flat plate which has a no^'mal velocity at the surface. Fage and Falkner (Ref. 1) solved the problem for conditions of constant fluid properties, variable wall temperature and a linear velocity increase normal to the wall. Emm^ons and Leigh (Ref. 2) used the Blasius

transformation and obtained numerical solutions to the laminar boundary layer equations for the conditions of constant wall temperature and a normal velocity at the wall which is proportional to the inverse half power of the distance from the leading edge. These assumptions lead to

•similar* velocity and temperature profiles. Brown and others (Refs. 3, 4 and 5) used a modified Blasius transformation and obtained solutions for a wedge-type flow (flow for which the main stream velocity is proportional to a power of the distance fromi the stagnation point) with variable fluid properties and constant wall t e m p e r a t u r e . Donoughe and Livingood (Ref. 6) used a similar method for wedge-type flow with constant fluid properties but a variable wall t e m p e r a t u r e . In the numerical solutions for the wedge-type flow, the normal velocity at the wall is limited to one of the form:

m - 1 a constant x(tne distance from the stagnation point) 2

where m is the Euler number.

Iglisch (Ref. 7) obtains exact numerical solutions when there is

constant suction along the wall. Approximate results for constant suction are presented by Curie (Ref. 8) who extends a method used by Stratford (Ref. 22) in which the p r e s s u r e forces in the boundary layer are equated to the viscous forces.

2

-A little experimental work has been performed for the low speed cases by Duwes and Wheeler (Ref. 9), Libby and others (Ref. 10) and by Mickley and others (Ref. 11).

Solutions of the compressible laminar boundary layer including the effects of transpiration cooling have been presented by Low (Ref. 12) who uses a method similar to that of Chapman and Rubesin (Ref. 13). Yuan (Ref. 14) has assumed that a polynomial of the fourth degree may be used to represent the velocity profile and solves the problem by a Karm^n-Pohlhausen method. Lew and Fanucci (Refs. 15 and 16) have used a modified Karman momentum method and also an exact method to solve the equations.

In the solutions which have been obtained so far, the normal velocity at the wall has been severely limited by the transformations used. In this paper an extension of Lighthill's method (Ref. 17) has been used to obtain an approximate solution to the incompressible laminar boundary layer with a r b i t r a r y distributions of wall temperature, of main siream velocity and of normal velocity at the wall. Two integral equations are obtained for the skin friction and the heat transfer rate to the wall. In order to estimate the accuracy of these integral equations, solutions are obtained for the particular case of wedge-type flow. The solutions a r e then compared with the exact results of Donoughe and Livingood (Ref . 6 ) . A further solution for the skin friction is obtained for the case of uniform suction at the wall and is then compared with the exact results of Iglisch (Ref. 7). The accuracy of the integral equations is of the

same order as that of Lighthill's equations which were for an impermeable wall,

The uses of an incompressible solution are extremely limited but the method has been extended to the compressible case by using a paper by Lilley (Ref. 18). The results are presented in Appendix B. The stability of the laminar boundary layer under conditions of flow injection is not considered.

2. The Boundary L a y e r E q u a t i o n s with Flow through the P o r o u s Wall R e f e r r i n g to F i g . 1, the s u b s c r i p t , d e n o t e s conditions o u t s i d e the b o u n d a r y l a y e r and the s u b s c r i p t ^ d e n o t e s conditions at the w a l l . F o r a p e r f e c t g a s the equations of continuity, m o m e n t u m and e n e r g y for a s t e a d y l a m i n a r i n c o m p r e s s i b l e b o u n d a r y l a y e r flow in two d i m e n s i o n s a r e 9x 9 y

u|u + v . ^ = ^ A - T. 1^ (2.2)

9x 9 y If d Y P ' dx and 2 - 2 ^ BT 3 T /J fdnY ^ Ai 9 T .„ „V u ^— + V ^— = -rpr- a— + "fr • —'2 ( 2 . 3 ) 9 x a y PC \ 9 y y Po- 9 ^ P ^ The Von M i s e s ' t r a n s f o r m a t i o n which changes the independent

v a r i a b l e s (x, y) to {x,\ir), w h e r e f i s the s t r e a m function, i s given by

|JÏ^ = - p V and 1 ^ = ^u (2.4)

9x 9y

T h e s e r e l a t i o n s s a t i s f y the continuity equation, 2 . 1 . On applying the t r a n s f o r m a t i o n to the m o m e n t u m and e n e r g y equations then

and 9Z

ax

ar

ax

= =

a^z

lip 9 u 2 C • d^' P ( 2 . 5 ) Af P u — ] . ( 2 . 6 )

w h e r e Z = u^(x) - u^(x i^^). ( 2 . 7 )

A modified s t r e a m function ^ = \jr - ^ ( 2 . 8 ) w i s now i n t r o d u c e d , A X 3 V i s defined by (r—) = - Pv = - r — . ( 2 . 9 ) w *' \ 9 x / - w 9 X y=0

H e n c e , u s i n g t h e independent v a r i a b l e s (x, $), equations 2 . 5 and 2 . 6 m a y be w r i t t e n

'^^ ^9*^ + (^ - ,pn CÖ-') (2.10)

9V^ \ 9 x y ^ \dxj^ V9$ ,^ and / 9 T / 9 T \ fd§\ ^ fdT\ 1 d f „ dT\ .„ , , .

VWJ W , / U A - ^' 9-l.^''"a¥J <2.ii)

In equation 2 . 1 1 the frictional heating t e r m h a s been a s s u m e d negligible and h a s been o m i t t e d . Lighthill (Ref. 17) obtained a p p r o x i m a t e

solutions to t h e s e equations when the v e l o c i t y at the wall, v , was z e r o . L i g h t h i l l ' s method w a s extended b y B e r n a r d L e F u r (Ref. 19) to include +he f r i c t i o n a l heating t e r m .

3 . T h e Solution of the Momentum Equation (2.10)

In o r d e r to solve equation 2 . 1 0 , a method v e r y s i m i l a r to that of L i l l e y (Ref. 18) will now be u s e d ,

Let

'1') (§) - «==•*> '3-'>

• ' X ^ '^f I X

Now S(x, §) = g— / S(x', § )dx', so that equation 2.10 m a y be w r i t t e n : •'o I - (Z + f S ( x ' , $ ) d x ' ) = / i P U ^ , (Z + /* S(x', 0)dx'). ( 3 . 2 ) ox ,' : 9$"^ / ^ o *'o 3: x At * = 0 then f S ( x ' , § ) d x ' = f S{x', 0)dx' and o o X at $ = oo then [ S(x', §)dx' = 0 since S(x, «) = 0 . •'o A l s o —; / S ( x ' , 5 ) d x ' = 0 at § = « . 9* ] O H e n c e , n e a r $ = 0, equation 3 . 2 m a y b e w r i t t e n a p p r o x i m a t e l y a s

- | ' . G(x, $) = upvi ^ . G(x, $) ( 3 . 3 ) ax a$

w h e r e G ( x , $ ) = Z ( x , i ) + ƒ S(x', 0)dx'

o X

F o r l a r g e r v a l u e s of $ it h a s a s i m i l a r form with G(x, $ ) = Z + / S(x', $)dx' such that G(x, oo) = 0.

Following P a g e and F a l k n e r (Ref. 1), an e x p r e s s i o n for the velocity u, T (x). y

w

n a m e l y u = — , which i s m o s t c l o s e l y a c c u r a t e n e a r the w a l l , will now be s u b s t i t u t e d into the e q u a t i o n s .

F r o m equation 2 . 4 : /•y r (x) f = p\ u d y =^'-"27 • y^ + ^^<x) o and t h u s / 2 r (x) Nf u = \ ^ — ^ — §y a s $ •» 0 . ( 3 . 4 ) On s u b s t i t u t i n g 3.4 into equation 3 . 3 , then

F r o m e q u a t i o n s 3 . 4 and 2 . 7 2 r (x)

\ ' ' \

^ a s $ - 0 . ( 3 . 6 ) ;

UP

Using t h i s equation t o g e t h e r with 2 . 9 and 3 . 1 then X X 2 r (x')

S(x', 0)dx' = / . v ^ ( x ' ) d x .

The b o u n d a r y conditions for G(x, $ ) , for which 3 . 3 i s to be s o l v e d , a r e t h e r e f o r e

6 -# » 00 G ( x , $ ) = O , X - O G ( x , $ ) - O , X 2 r (x') 2 r (x) - - O G ( x , $ ) = < ( x ) + / ^ . v^(x')dx' - - ^ = i ; ( x ) + F^(x). § . ( 3 . 7 ) The functions F^ (x) and F2(x) a r e defined by equation 3 . 7 .

If t = f i2np T^(z)) dz I ( 3 . 8 ) o then equation 3 . 5 m a y be w r i t t e n

|S,t,„ -. ,i. g£(t.i, . (3.9)

T h e b o u n d a r y conditions at $ = 0 a r e G = F ( t ) and | ^ = F (t) . ( 3 . 1 0 ) • 0 $ 2

Using the L a p l a c e t r a n s f o r m notation F ( p , i ) = / e ^ F ( t , $ ) d t ( 3 . 1 0 a ) o

and the condition G = 0 when t = 0, equation 3,9 m a y be w r i t t e n

PG= # 0 . (3.11)

Equation (66) of L i g h t h i l l ' s p a p e r (Ref. 17) i s analogous to equation 3 . 1 1 and both s a t i s f y s i m i l a r b o u n d a r y c o n d i t i o n s .

The solution of equation 3 . 1 1 i s t h u s

G = ( l ) ' p " ^ # . r ( i ) . I 2 ( | P^ * * ) F + ( ! p ^ ) " ' # r ( | ) .

Izép^ h p ( 3 . 1 2 )

3 '3

w h e r e I 2 and I2 a r e modified B e s s e l functions. Now G •* 0 a s § - 00 " 3 3

t h e r e f o r e , from the p r o p e r t i e s of B e s s e l functions, the coefficients of I 2 and 12 m u s t be equal and o p p o s i t e . Hence

o r , taking the i n v e r s e t r a n s f o r m

,t F (t )

I^ (t) = - ( f ) - -jr. f ^ - ^ 1 dt^ r 2 \ " 3 _ 1 r 2 1 " ' ^ ' ' ( t - t , ) ^

r(i)

Using equation 3 , 8 and s u b s t i t u t i n g for F^ (t) and F2(t) from equation 3 . 7 the following equation i s obtained after simplifying:

V\z),{up) o x, ^

! i v^i^- y^i>^^i • <3-i4)

o

Solutions to t h i s i n t e g r a l equation a r e obtained in Section 7 . 1 . 4 . T h e Solution of the E n e r g y Equation 2 . 1 1

A method s i m i l a r to that used by Lighthill (Ref. 17) will be used to find the heat t r a n s f e r to the wall in t e r m s of an a r b i t r a r y wall t e m p e r a t u r e and skin friction d i s t r i b u t i o n and a n o n - z e r o v e l o c i t y , at the w a l l .

In finding the a p p r o x i m a t e solution to the m o m e n t u m equation in the p r e v i o u s s e c t i o n , conditions which w e r e a c c u r a t e at the wall w e r e s u b s t i t u t e d for o v e r a l l c o n d i t i o n s . T h i s p r o c e d u r e will again be adopted. T h e a p p r o x i m a t i o n for u, equation 3 . 4 , will again be u s e d .

( 4 . 1 ) ( 4 . 2 ) Now The r a t e of heat t r a n s f e r Q , ( x )

K'Vx

= l^

tl

/3T> (sy\ F r o m the a p p r o x i m a t i o n for u (d7\ . V^Vx

/ " V

( , 2 . ^ ( x ) ; to the wall is • • 1 $ - 2 • given by ( 4 . 3 )

H e n c e , from, the above t h r e e equations t o g e t h e r with 2 . 9

W ^ W ; / k V^2rjx)

$^ p v (x). _w

On substituting t h i s into the e n e r g y equation 2.11 then

'aT^ C (x) w p 2 r w 1 2 i *"^ P V + . „ w V ax Let F 3 ( x ) = - " w w r 2 k w

ê'(4V^^>^ ïï$<*^ 5F)

^ a then equation 4 . 4 s i m p l i f i e s to v 9 x / f "2 F3 ( 2 / i p r ) ' w 3§ ( 5' 9 T ÏÏF' • Now if

t = r

i:(2

.P

r^)'

dx^, w 1

(this i s different from the t used in Section 3) equation 4 . 6 m a y be w r i t t e n ^ ^ F, -['^ $ J - (i^ 9T)

a$ ^ a$' •

The b o u n d a r y conditions a r e : (a) T = 0 at t = 0 and at 5= = <« , (b) T = T^.(t) at §. = 0 and, (c) ( k ~ - ) = Q (t) at ^ = 0 k ^ ) = Q (t) ( 4 . 4 ) ( 4 . 5 ) ( 4 . 6 ) ( 4 . 6 a ) ( 4 . 7 ) ( 4 . 8 )

Using the t r a n s f o r m notation of equation 3 . 1 0 a equation 4 . 7 b e c o m e s p T _9_

_a$

_{•" -ST} .^ 9 t

1

•2 F_ ( 4 . 9 )

The homogeneous form of t h i s equation i s equivalent to equation (21) in L i g h t h i l l ' s p a p e r (Ref. 17) and the solution i s

_ 1 / A i. 1\ 1. /A X

T = a §^I 1 ( i p ^ § M + b éh '-^^ '^

" 3 V "^ J I 3

(4.10)

d P d i and dQ d $ TS"^ F3(p) TJTS-T^T, T ^"-^ F , (p)

The c o m p l e t e solution of the equation 4, 9 will now be found by the method of v a r i a t i o n of p a r a m e t e r s . Let q = i p 2 i t . ( 4 . 1 1 ) A solution of 4 . 9 i s a s s u m e d to be T = P ( * ) T^ + Q ( i ) T ( 4 . 1 2 ) _ i. w h e r e T = $* I i(q) ^ ~3 and T^= $ ^ I i ( q ) . ( 4 . 1 3 ) 3

The equations to d e t e r m i n e P and Q, a r e thus

(4.14)

T T - T T

1 2 2 1

w h e r e the p r i m e i n d i c a t e s a differential with r e s p e c t t o $ F r o m 4 . 1 3 T ; % - %T,= $ M M q ) I'j_(q) - I i(q). ^ ( q ) ^ 3 - 3 - 3 3 Now the W r o n s k i a n , W ( I i ( S ) , I i ( ^ ) ) = - ^ | - S i n J . 3 " 3 - <i'" 3 ^ Hence T ; T , - T,'T^ = - * ' [ ^ • | Sin | J . (4.15) The equations 4 . 1 4 m a y now be w r i t t e n :

and f / F 2 7r \ j^

The solution to equation 4 , 9 i s obtained by substituting 4 . 1 6 and 4 . 1 7 into equation 4 . 1 2 , t h u s :

10 -T ( p $ ) = $* I i(q) 2 vrF 3 3 Sin'VS, I i ( q ) *" d t $ * I i ( q ) 2w F 3 3 Sinw/3 o 1 ( 4 . 1 8 ) I i(q) $••* d§ + Af^ I i_(q) + B ** M q )

w h e r e A and B a r e to be found from the b o u n d a r y conditions 4 . 8. They a r e not functions of * .

As f •« cc then T •• 0, t h e r e f o r e the coefficients of Ij^ and I j^

3 ~ 3 m u s t be equal and o p p o s i t e . H e n c e : I 27r F p

T

_{sibw J ( V ' - i-i'"*) """ "^ •" ^ ' " •}

o r

I F. p " W

Ki(q)

dq + A + B = 0 .

3 ^ o ^ Kj^(q) i s a modified B e s s e l function o 3 F r o m Ref. 20,

the second kind.

(4.19) Ky ( / 9 c ) c ' ' " ^ dc = 2°"'^ y9"^r(icx + i i . ) r ( i a - i y ) providing (a +v) > 0 and fi > 0. T h u s equation 4 . 1 9 m a y be w r i t t e n - 1 r ( f ) r ( i ) p " ^ F + A + B = 0 . F r o m equation 4 . 1 8 , a s f • 0 then -T ( p . ) . A ( | ) ^ ^ . T h u s by u s i i g the b o u n d a r y condition 4 . 8(b) A = (1)^ r ( | ) p ^ / ^ T ( p ) .

Now from equation 4 . 3 and the b o u n d a r y condition 4. 8(c),

( 4 . 2 0 )

(4.21)

( 4 . 2 2 )

F will be defined by the equation

On differentiating equation 4 . 1 8 with r e s p e c t to $ and c o n s i d e r i n g the l i m i t a s $' -» 0 then i 9T ^ „ , 3 , ' p 1/6

9f ^ 4 ' TTir •

T h e r e f o r e from 4 . 2 3 2 B = p " ^ ^ ^ r ( i ) . ( I ) ' F ( p ) . (4.24) After s u b s t i t u t i n g equation 4 . 24 and 4 . 22 into 4 . 20 ;

i 1 3 _ i _ q 3 -p / 2 \ i f 2 \ T i / 2 \ _ 3 -rn / • 5 \ i \ 3 ' _ 3 F^ = (!) r ( | ) . p 3 F^ _ (|.) i _ ^ p3 T^ . (4.25) 3n\ i s given by By t h e Convolution T h e o r e m the i n v e r s e t r a n s f o r m of p T J ^ w . / 3 T (t ) 1 /• V T T - ^ + S { t ) T ( t ) > l t , ^^^^ o ( t - t , ) ^

w h e r e 6 i s the Delta function (an i m p u l s e function). F o r s h o r t n e s s equation 4 . 26 will be w r i t t e n in Stieltjes f o r m . Using equation 4 . 6 a then

P^ T (p)c 4 , (IMS^ ^ r ( C rjzf- dz'"'

T (x') w ° ""' ( 4 . 2 7 ) S i m i l a r l y •3- ^/„V r- J _ / ' 2 i i p \ f ^ Ï i f -r /„\i "" O X V ( x ' ) Q ( x ' ) d x ' -^ \ . (4.28) r ( x M ' w

12

-Using 4 . 27 and 4 . 28, the i n v e r s e t r a n s f o r m of equation 4 . 25 i s

w o w X 1 \ - l V ( x ) Q ( x ' ) d x ' ( V 2 .1 1 W W z) dz ) T-,Ti , w ; r ^ ( x ' ) 2 ( 4 . 2 9 ) X

w h e r e P Q (X) 1 . I S the heat t r a n s f e r to the wall when the •^ v =0

u J, -v^.

v e l o c i t y , v (x), at the wall i s z e r o . It i s the e x p r e s s i o n which w a s w obtained by Lighthill (1950) and i s given by

T Q (X)"] _ I w V =0 •- -^ w .o \ i r (x)^ 1 / o x T ( z ) M z ; d T (x') , ' w ( 4 . 3 0 ) After changing the o r d e r of i n t e g r a t i o n the t o t a l heat t r a n s f e r

r a t e o v e r an a r e a of s u r f a c e between x = 0 and L i s given by :

.L . L I 1 L

, Q (x)dx = /

FQ (X)1

-dx

J W J L W J v =0 o o W T (z)2 dz w

oVT r(f)

' 2 i • 3 I

r(T)

V (x') Q , ( x ' ) d x ' W VJ r ix')^ w (4.31)

Equation 4 . 29 with equation 4 . 3 0 i s a V o l t e r r a tjrpe i n t e g r a l equation and can be solved by a s t a n d a r d i t e r a t i v e p r o c e d u r e (Ref, 24). Hov/ever it w a s found m o r e convenient in the p r e s e n t c a s e to use a slightly modified p r o c e d u r e and t h i s i s explained below.

5. A G e n e r a l Equation for the Heat T r a n s f e r Rate Q (x)

Let (Q ) be the e x p r e s s i o n for Q after the n i t e r a t i o n , w n w The equation for (Q ) i s obtained d i r e c t l y from equation 4 . 29.

p W f r(i) . 2 rx/ /x ^ v f

3 , ; rrt) V / W r^^dz

(Q ) „ = (Q ) 1 w n w n - 1 V - T N , -dx'' ( 5 . 1 )

w h e r e N , ^ = (Q ) , - (Q ) ^ . ( 5 . 2 ) n - l , n - 2 w n - 1 w n - 2

On i n t e g r a t i n g equation 5 . 1 from 0 to L and changing the o r d e r of i n t e g r a t i o n then

t

(Q ) dx =

A Q

) . dx - ( ^ f ï|i-> ƒ V . *dz J

•I w n •' w n - 1 \ ^ / r ( f ) J \ i A w / o X V w _{, . N , „ d x ' , ( 5 . 3 )} ^ 2 n - 1 , n - 2 w

T h i s equation will be used in Sections 7.4 and 7 . 5 . 6. The N u s s e l t N u m b e r An o v e r a l l N u s s e l t n u m b e r , Nu, m a y be defined by dx

f Ï

NU = - / - ^ ( 6 . 1 ) o m w h e r e T i s a m e a n wall t e m p e r a t u r e . m A L o c a l N u s s e l t n u m b e r , nu, i s defined b y Q X nu = - - ^ 5 ^ ( 6 . 2 ) k T _w

( w h e r e x i s the d i s t a n c e from the leading edge). Equation 5 . 1 m a y t h e r e f o r e b e w r i t t e n : 1.

P V Y

r(f) ^ V rV r^ ^ï.y ^w

'n --"'n-1 - VTTTy r(F) I T T - / W w ^V r i •

W J J I, w o X k T ^ ' (nu) , - ( n u ) „ )} dx' ( 6 . 3 ) X \ n - 1 n - 2

14

-7. Solution of the I n t e g r a l Equations when the Wall S h e a r S t r e s s w (x) = Const. X XT and the F r e e S t r e a m Velocity u (x) = ex 7 . 1 . The m o m e n t u m equation

The initial a s s u m p t i o n s a r e that u^(x) _{c X} m

2 /? r (x) = K X

w

( 7 . 1 ) and r (x) = K x . ( m , i3, c and K a r e c o n s t a n t s ) .

On substitutirig equations 7.1 into equation 3.14 and i n t e g r a t i n g : ^zm 2.3.^ K^^^ r(t).(/^p) _ 2 _ /9+ 2 2 4-/g+2 J3/9+21 ,2V

3 3— ^ W | 7 _

^7^..ö+2 , 2

i i r- + 3 X 2 + —

V^+ 2

V Kf xf dx, w 1 1

It i s obvious from t h i s equation that the e x p r e s s i o n for v (x) i s r e s t r i c t e d to one of t h e f o r m V (x) = Gx'^ w w h e r e y i s d e t e r m i n e d from 7 . 2 . TT m - 1 , ^ 3m - 1 Hence y = — - — and /? = providing /5 + y 3^ - 1 . The a s s u m p t i o n s 7.1 t h e r e f o r e i m p l y that u^(x) = e x . . .,2 3m - 1 r^,x)- K X — 2 — V ( x ) = G x 2 1 - 1 1 w 2 m i^ 0 . ( 7 . 2 ) ( 7 . 3 ) ( 7 , 4 )

T h e s e conditions, excluding the condition m / 0, a r e the s a m e a s t h o s e u s e d by Brown and Donoughe (Ref. 4), E m m o n s and Leigh (Ref. 2) and Donoughe and Livingood (Ref. 6). (They a r e the conditions for s i m i l a r i t y between the velocity and t e m p e r a t u r e p r o f i l e s when cr= 1).

E q u a t i o n s 7.2 can now be w r i t t e n 4 2 . 3 ^ K ^ ^ ^ f \3 (u^y 3m + 3

r(l)

9 m + l ' \ m + 3 /

r(i)r'

l l m + 3 _3m+3 + K^ G U m ( 7 . 5 )

The function, f (a d i m e n s i o n l e s s m e a s u r e of the coolant flow through the p o r o u s wall) ana the function, f (the d i m e n s i o n l e s s skin f r i c t i o n ) ,

w a r e now i n t r o d u c e d . and w w m + 1 V (x) _w r (x) w f' u{x)u 1 u , ( x ) y 1 2 Hence equation 7 . 5 i s (m + 1) , 1 + 2m w w ,7/3 3 ^ ( m + l ) '

_{r(i) r'}

,i7~7Tïm+3

3m+3 When When m = 0. 5 then equation 7. 8 i s 1 + 1. 5 f . f w w " 4 / 3 1.284 (f ) ' ^ w m = 1. 0 then equation 7. 8 i s 1 + f f w w 0.843 ( f " ) ^ / ^ w

(f)^f'

w ( 7 . 6 ) ( 7 . 7 ) ( 7 . 8 ) ( 7 . 9 ) (7.10) f in equation 7 . 9 and 7 . 1 0 h a s b e e n evaluated for c e r t a i n v a l u e s

w ^

of f . The r e s u l t s a r e p r e s e n t e d in T a b l e 1, and a r e c o m p a r e d with the exact r e s u l t s p r e s e n t e d by Donoughe and Livingood (Ref. 6). The e r r o r s a r e v e r y l a r g e r a n g i n g from 8% to 24%. The r e a s o n for the l a r g e e r r o r i s that it contains the combined e r r o r s of both t e r m s on the right hand side of equation 3 . 1 4 .

16

However an i m p r o v e d solution for f can be obtained if we a s s u m e w

that equation 7 . 8 i s only used to d e t e r m i n e the difference b e t w e e n (f ) - and f w h e r e (f ) . i s that calculated by Lighthill, o r

w v = O w w v = 0 ^ & >

w w A(f" ) = ( f " ) - if" )

w w ' e q u . 7 . 8 w'Lighthill

(7.11) Whea t h i s difference i s obtained the absolute value of f can be found by

w "^ using the exact v a l u e s for ( f ; ) ^ __^ ^^ calculated by H a r t r e e (Ref. 21).

o r f" = ( f " ) „ , + Af" w w H a r t r e e w ( 7 . 1 2 ) 7. 2 . A p a r t i a l i n v e r s i o n of the m o m e n t u m equation 3.14 Equation 3,14 m a y be w r i t t e n ( s e e Appendix A) r (x) = w (P^)

r(f).

3

u J T d z •' / w x V r d x ' + w w o X i r 2 dz w u,(x') (7.13)

If e x p r e s s i o n s for T , u, and v a r e substituted into 7.13 then

w w

the equation obtained will be the s a m e a s that obtained from 7 . 5 . The r e a s o n for w r i t i n g equation 3 . 1 3 in the above f o r m i s to i n v e s t i g a t e t h e effect of introducing the a p p r o x i m a t i o n

x T ( z ) ' dz " (x - x ' ) Ï- ^ (x') w w x' (7.14) which Lighthill (1950) s u g g e s t e d .

T h i s a p p r o x i m a t i o n will only be used in evaluating the t e r m containing

V ( x ) .

w

The e x p r e s s i o n s f o r r (x), u (x) and v (x) of equations 7.4 a r e w 1 w ^

X X 2 . . 2 3 m - l / O X

(i)r(|^"^^3

^ ^ ^ V ' V 3 ( m + l ^ ^—TT^— ( 7 . 1 5 ; P , 9m+l 3m+3

and on using the a p p r o x i m a t i o n 7.14 for the t e r m which contains v (x)

then 2 . ,„ 3 m - l I f^ / f^ i \~3 K^ G 2 — / ( / r ^ d z ) ' v T dx' ^ X U \ W / W W M _{' o ' ^ x '}

r ( i ) r ( ^

J, / 3 m + l (7.16)

If equations?. 15 and 7.16 a r e s u b s t i t u t e d into 7 . 1 3 then:

When m = 0 then t h i s equation r e d u c e s to / \ 4 / 3 ( When m ( When m ( f = 0.211 + 2 . 1 5 f . f ' ^ wy W W = 0 . 5 then

^ A^^

f ) = 0.779 + 1.20 f f" . W / W w = 1.0 then / # / 3 f" ) = 1.19 + 1.24 f . f" , vrj w w (7.17) (7.18) (7.19) ( 7 . 2 0 ) The value of f i s modified a s in Section 7 . 1 . , equation 7 . 1 2 .

w

18

-7 . 3 , The f i r s t i t e r a t i o n of the e n e r g y equation

The total heat t r a n s f e r r a t e o v e r an a r e a between x = 0 and x = L i s given by equation 4 . 3 1 in t e r m s of Q . An i t e r a t i v e p r o c e d u r e will

v/

be used and the f i r s t a p p r o x i m a t i o n for QL will be L i g h t h i l l ' s e x p r e s s i o n

(equation 4 . 30). w

The condition of constant wall t e m p e r a t u r e , T (or constant

difference between the free s t r e a m t e m p e r a t u r e ana the wall t e m p e r a t u r e ) will be c o n s i d e r e d .

E x p r e s s i o n s <'oi' T ( X ) , U,(X) and v (x) f r o m equations 7 . 4

w W

and Q (x) from 4 , 3 0 a r e substituted into equation 4 . 3 1 and, after w i n t e g r a t i n g : L m+1 \ tr^ ^^ ^ ^ "^w /3a-p V / 4 K 4' 2 ^ o rp , / c r p , ^ , (O ) dx = - -r-rrn-T- I —T- TTi—TTx L + 2 T k - = ^ G. L w i 2Vt,\) \u.^ J \ 3 ( m + l ) y w \ u j ( 7 . 2 1 ) m+1 2

On introducing the o v e r a l l N u s s e l t n u m b e r , Nu, defined by equation 6 . 1 :

>4/3 (Nu)

F i r s t i t e r a t i o n 2. r ( ^ )

.^vf p^'-^l)]"' -^-^w • <^-22)

On substituting n u m e r i c a l v a l u e s it i s found that t h i s equation o v e r

-e s t i m a t -e s th-e c o r r -e c t i o n r -e q u i r -e d . M o r -e i t -e r a t i o n s will b-e n -e c -e s s a r y . 7 . 4 . The second i t e r a t i o n of the e n e r g y equation

N , „(x) i s defined by equation 5 . 2 | such that n - 1 , n-<5 N (X) . ( Q ^ ) , - ( Q ^ ) ^ , „ w T h e r e f o r e from equation 7.21 X 2 T. ^k o- p G

ƒ ^^'i,0^^^^^^ = im+l),

o ïB:~^

N, Ax) = ^ G x ^ . T . k. m + 1 and • 1 , 0 ' _w ( 7 . 2 3 ) ( 7 . 2 4 )

P r o c e e d i n g a s before : L L / (Q ) , dx = [ (Q ). dx -J w ^ j w ^ p5/3 ^5/3 u

TJT

3(m+l) 4/3 and (Nu) Second i t e r a t i o n „ , _2 , i v „3 m + 1 T ^ . k G r ( i ) 3^

-J-1 L

2 K^ 0 . 8 0 7 . 0^ r f ^ T ( | ( m + l ) ) - 5 / 3 , , , , f „2 •cr f + w 1.02 cr ' (m+1)^ f w

m

7 . 5 . I t e r a t i o n s using the g e n e r a l equation 5.5

F r o m the p r e v i o u s Sections 7. 3 and 7 . 4 , it i s s e e n that N , „ dx' h a s the f o r m n - 1 , n - 2 m+1 , N . „(x ) dx J n - 1 , n - 2 o w h e r e M i s independent of x . M X ( 7 . 2 5 ) (7.26) (7.27) Equation 5. 3 m a y now be w r i t t e n (Q ) dx w n (Q ) , dx w n - 1 p 2 ^ \ 3 r ( | ) T . . G M 3u w 3 ( m + l )

r(i) K^

On i n t e g r a t i n g and r e a r r a n g i n g then Nu Re n Nu Re 3(m+l) 3(m+l) 4 4 - X , / m L m+1 2 'n-1 _{\ k Re^}

r(i)

"T—T 2^ 3^ m - 3 4 dx. 2 2 _3 ƒ _ „ , 1 \ 3 j a' (m+l)'f . f f ' T w L wj (7,28)

20 F r o m equation 7. 27 and 5. 2 .L (Nu) M L m+1 2 n - 2 - <^^)n-l ^ ^^ • (7.29) P r o v i d i n g the value of (Nu Re ^) when v (x) = 0 i s known, a l l o t h e r

w

t e r m s in the s e r i e s m a y be obtained from. 7 . 2 8 . F r o m equation 7.28 and 7.29

1 i

(Nu R e " ' ) = (Nu Re"^) _{V (x)=0} w 1 + E f w w E f \2 w E f w ^f )3 w w h e r e E o r Nu Re

r ( i ) a^'(m+l)'

T—T ,3 o 3 2^ 3' (Nu Re"2)

v^(x)=0L Ef

(7.30) ( 7 . 3 1 ) w 1? f providing — 1 < * w , J— < i .

if )'

w (t" )* w

L i g h t h i l l ' s e x p r e s s i o n for (Nu Re ) . . _, from equation 4 . 3 0 i s w 2 1 (Nu Re 2) v (x)=0 w 3 ' 2 ' ( m + 1)

nv)

w,

After substituting t h i s equation into 7.30 it i s found that the f i r s t t h r e e t e r m s a r e the s a m e a s t h o s e of equation 7 . 2 6 .

7 . 6 . The l o c a l N u s s e l t n u m b e r , nu

By c o m p a r i s o n with equation 7. 27, a r e l a t i o n between the l o c a l N u s s e l t n u m b e r s after consecutive i t e r a t i o n s i s found to be

" % - l - ' ^ ^ - 2

a x m - 1

T h e r e f o r e equation 6 . 3 s i m p l i f i e s to m+1 ( n u R e " ^ ) = ( n u R e " ^ ) . + ( ""^ i ) r ( i ) J"(m+1)^ - ^ " • ^ ^ R e ^ '^ ( f " ) ' ' w ( 7 . 3 3 ) _ i

L i g h t h i l l ' s e x p r e s s i o n for (nu Re ^) /^\_o» from equation 4 . 3 0 w

i s

_2 2 i W

F r o m equations 7.34 and 7 . 3 3 , the s e r i e s for the l o c a l N u s s e l t n u m b e r i s 1 1 E f / E f \ 2 nu R e " ^ = (nu Re"^) , , n^ '^ + ^ + ( ^ o r nu Re ^ = (nu Re~^) ' w^ w ( r ) 3 p r o v i d i n g E f w ( f ' ' ) ^ w

V^^=° V(f'')^-Ef ^ <^-3'^

w w < 1 . E i s defined by equation 7 . 3 1 .

In the equations for the o v e r a l l and the local N u s s e l t n u m b e r s , the skin friction p a r a m e t e r , f , naust be evaluated at the a p p r o p r i a t e f .

_ i i

B e c a u s e (nu Re ^) . . i s p r o p o r t i o n a l to (f" )^ (from equation 7.34) v \X/=U w

w

the c o r r e c t i o n whirh i s used to c o n v e r t the f i r s t t e r m of equation 7 . 3 5 to one having the c o r r e c t skin friction p a r a m e t e r i s

( f " ) f ~ , „ - 2 \ / w At p a r t i c u l a r w ^ (nu Re '') t \ n v (x) = 0 I . // w at f =0 w so that equation 7. 35 m a y be w r i t t e n : 2 ^ -\ / nu Re"2 \ / w \ ,„ „^, nu Re ^ = ( T ) X ( T ) . (7.36) ^ ( f " ) ' \ {x)--0 Mf")3 _ E f W W W W

- 22

Values for nu Re a r e p r e s e n t e d in T a b l e 3 . During the calculation the exact v a l u e s obtained by Donoughe and Livingood (Ref. 6) w e r e used

_ l

for (nu Re ^) / v r» and for f" .

V (x)=0 w w

7 . 7 , Wall t e m p e r a t u r e v a r i a t i o n T = B^x

Let the v a r i a t i o n in T be B.x ; B and e a r e c o n s t a n t s .

w 1 ' 1

( 7 . 3 7 ) When t h i s equation t o g e t h e r with the equations 7.4 a r e substituted into equation 4 . 3 0 and the r e s u l t i n g equation i s i n t e g r a t e d and n o n -d i m e n s i o n a l i s e -d t a e n

( n „ R p - ^ ) - 2 ^ / 3 c r 3 f " 3 / ^ + 1 ) - t i j l l i l ^ l f e + r ) .

( n u R e )^ ^^j^Q - 2 cr f^ (m+1) -^——j-^

w 1

3' ^<i>^3feTi)*

( 7 . 3 8 ) By c o m p a r i s o n with equation 7 . 3 2 , the r e l a t i o n between the l o c a l N u s s e l t n u m b e r s after consecutive i t e r a t i o n s i s m - 1 +2e nu , - nu w n - 1 n - 2 a X B, ( 7 . 3 9 )

If t h i s i s substituted into the g e n e r a l equation 6. 3 then m+1 "2" w h e r e Hence (nu Re ^) n (nu Re ^) , + n - 1 ax Re D = cr3(ni+l)3 r ( | ) r 4 e (m+1) + f •i / 4e' 3 ^ 2 ^ T ' — ^ (^3(m+l) ^ (nu Re ^) = nu Re i £« 3 w W V (x) = 0 - v x W j " 3 w providing Df w 1 // v 3 ( f " ) w < 1 . D. f w ( f " ) ^ V.' ( 7 . 4 0 ) (7.41) ( 7 . 4 2 )

When e = O^this equation is identical with 7.35. The numerical values for nu Re^^ for various values of f and m are presented together with those of the previous section in Table 3 and also in Figs. 4, 5, and 6.

8. Approximate Solution to the Integral Equations when the F r e e Stream is Uniform and the Velocity, v is a constant

The partially inverted momentum equation which is derived in Appendix A is 2 r^ / i^ \ 2 (8.1) r (x) = - i i 2 1 i _ ^ ) . ±. / U r (z)^ dz i v (x.) r (x.) dx + 2 i \"3 ^ " " r ( | ) 3 ^ ^ ' 'o^x, ^ ' " ' " ' ' w - ^ ' W o X, u](x,)

When m = 0 in the equations 7.4 and when v = 0 , then u = a constant

2 1 w ' 1

and r (x) = K x"^ (8.2) w

where from equation 7. 5

K = ^1

4 o 1 / 8 / v4

u* 3 (^p^ 27/8^r(f))3/^

( 8 . 3 )

Equation 8.2 will be used as a first approximation to r (x).

w

On substituting 8. 2 into the right hand side of 8.1 and integrating then the second approximation for r (x) becomes

2 f~ W

< V > . n a A p p r . x . = , - g l [ - ^ K * ' ^ | . * - . r < i . H f . ^ i c " K - % ) » x - ^ (8.4) Now the dimensionless form of the flow through the wall is defined by equation 7.6 as j_ 2 V Re^ f = (8.5) w u^ where Re = u,x 1

24

-Thus equation 8.4 may be written

(T ( X ) ) - , . = (1 - nx^)K x~^ (8. 6)

w 2nd Approx.

i i ' - 2*r(-) r^(-)

where ftx^ = - ^ ;J ^ f . R is not a function of x. (8.7)

3^

If it were now possible to substitute equation 8.6 into the right hand side of equation 8.1 and integrate, then a third approximation for

T (x) would be found. If this procedure were continued then an

w ^ equation for r (x) of the form

w

r „ ( x ) = K x " M l -a,(Rx^) - b , ( R x ^ ) ' ), j . (8.8)

^ | R x ^ | < l ,

could be obtained.

In this paper only the first two t e r m s will be considered. This i

implies that Rx^ is small, i . e . the velocity through the wall is very small, The equation for r (x) thus obtained will then be compared with those of Iglisch (Ref. 7) and Curie (Ref. 8). ^

In order to find the value of a^ in equation 8 . 8 , this expression for

T (x) will be substituted into the t e r m s on the right hand side of equation 8 . 1 .

Three different approximate methods will be used. F i r s t Method

i

r (z)^dz will be put equal to

w

^1 4 i 1 3 3

| r ^ ( x , ) ^ x,^ (x^ - x f ) (8.9) which is correct only when v = 0.

*' w

2 - i i

On substituting x (x) = K x ^ (1 - a^ Rx^) (8.10) into the integrals of equation 8 . 1 :

' • ' ' ^ ^ - ^"^ ,4,^ ^ 4 / 3 r . / u . . 2 . , 1 . . „ J r"(i) / ( i ' w ' ^ 0 " "w V ^ ^ - - / / ^ ' ' ' [ r ( i ) r ( | ) - ( | ) a , R x =

and ^ \ \ Tjz)^dz) ' - | ( | ) ^ K " ^ X " M 1 + i a R x ^ . . . ) ( 8 . 1 2 ) o

Hence on s u b s t i t u t i n g 8.11 and 8 . 1 2 into 8.1

2 i i

r (x) = K x M l + ( i a^- l ) R x ^ . . ) . (8.13) T h e r e f o r e from 8. 8 and 8.13 a^ = | and so

2 i i

r (x) = K X M l - I R x ^ . . ) . (8.14) Second Method

If the a p p r o x i m a t i o n which Lighthill (1950) s u g g e s t e d i s u s e d , i . e .

X 1 I

r ^ ( z ) 2 dz - (x - x ^ ) 7 - ^ ( x ^ ) M ( 8 . 1 5 ) then ^ from equation 8.1 the value of 7 (x) when v (x) = 0 i s

2 1 w w modified to T (X) = K X " ^ (8,16) W 3 ^ i Where K^ = < ^^'^" . ( 8 . 1 7 ) 2 3 / 8 ^ r ^ 2 ) ) 3 / 8 Equation 8.10 i s modified to 2 - i i r (x) = K X M l - a, ( R x ^ ) , . . ) , ( 8 . 1 8 ) Vv * 1 w h e r e R x ^ = - r ( j ) ( r ( f ) ) ^ ^ f « - 1 . 3 1 f . (8.19) 2^ 3^ w w

On substituting 8.18 into 8.1 then the equation obtained i s T = K X M l + i(a^ - 1)R x ^ . . . ) .

Hence r = K^ x " M l - I R, x ^ . . ) . ( 8 . 2 0 )

26

T h i r d Method

In the f i r s t and second m e t h o d s the r e s p e c t i v e a p p r o x i m a t i o n s 8.9 and 8.15 w e r e used for both t e r m s on the right hand side of 8 . 1 . The second t e r m can however be evaluated without using an

a p p r o x i m a t i o n .

The e x p r e s s i o n which i s obtained in place of 8.12 i s thus :

2 ^3

O K " ' x " M l + ^ a ^ R x ^ . . )

and the final equation for r (x) i s given by

T ( x ) w

K x"^ (1 Rx^. . . ) . (8.21)

In o r d e r to c o m p a r e equations 8.14 8. 20 and 8. 21 with p r e v i o u s work, the function S i s introduced such that

v^ . X

w

U^ V

Now from equation 8.7 Rx "^ 3 ^ S i m i l a r l y from 8.19 T h e r e f o r e equation 8.14 i s R^x r (x) w K x ^ 1 + 2 . 2 5 5 2 . 6 l i ; ^

l,cl<i

( 8 . 2 2 ) ( 8 . 2 3 ) equation 8. 20 i s r (x) w I^.x-1 1 + 1.96 e^, |.'i[<.38 and equation 8.21 i s ( 8 . 2 4 ) T ( X ) W I^ 1 = 1 + 2 . 5 s ^ 2

kl<i

( 8 . 2 5 )

T h e s e equations a r e p r e s e n t e d in F i g . 7 and c o m p a r e d with the r e s u l t s of C u r i e (Ref. 8) and Iglisch (Ref. 7).

9. Discussion

Two approximate integral equations have been obtained for the skin friction and for the heat transfer rate to the wall. The approximation used in the momentum equation consisted of replacing the velocity u with its asymptotic form as y •• 0, and in the energy equation the

B T

velocity u and -— were replaced with their asymptotic forms as y -• 0.

The argument is that the most important part of the velocity and temperature profiles are those parts near the wall, since the precise way in which they approach the main stream values will not greatly influence the wall

conditions. The e r r o r s introduced by this method must be obtained by comparing the final equations with exact numerical solutions for special c a s e s ,

r (x) y _w

Lighthill (Ref. 17) only used the approximation u

u

and this is correct in the limit for a very large velocity boundary layer thickness and a very small temperature boundary layer thickness. In this paper the heat transfer equation with fluid injection at the wall is not asymptotic for large cr because the further approximation,

G (x).v

T = "w had to be introduced.

In the momentum equation only the appioximation for u was used. If this had been substituted immediately into equation 2.10 then the resulting equation would not have satisfied the boundary condition at infinity. Because of this, a method very similar to that of Lilley

(Ref. 18) was used. A function S(x, §) is introduced which by definition does satisfy the boundary conditions at § = <» and at $ = 0. In this way the difficulty is overcome. The approximation for u has to be used in a t e r m which is equivalent to that in Lighthill's paper and also in a t e r m which includes the velocity of blowing or suction at the wall. It is therefore reasonable to expect that there will be a greater e r r o r in the final equation for the skin friction with a velocity, v (x), at the wall. The resulting equation (3.14) was compared with the exact solutions of Donoughe and Livingood (Ref. 6) for the particular case of wedge type flow. A singularity exists when the Euler number, m, is z e r o . This is probably due to the physically imipossible condition of a constant free stream velocity being unaffected by an infinite velocity, v , which exists at the leading edge. The numerical values for the skin friction parameter f ", a r e presented in Table 1 and they show large e r r o r s . However in a practical application of the integral equation, the conditions when v is

28

-zero will probably be known. If this is so then the modified equation 7.11 may be used. The results are given in Table 2 and the e r r o r s a r e now less than 11%. In Section 7. 2, f" , is evaluated by an approximate method in which there is no singularity at m = 0. The values for f" using this method, together with those of the previous method are

presented in F i g s . 2 and 3. For conditions of negative p r e s s u r e gradient the results are reasonably good.

The equation for the local Nusselt number is obtained from the heat transfer integral equation for the particular case of wedge-type flow. The results in Table 3 and the curves of Figs. 4, 5 and 6 are calculated using the naodifier. equation 7.42. The results are poor when m is zero but this probably reflects the singularity which existed in the skin friction equation. The local Nusselt num.ber is reasonably accurate over the region for which the equation 7.42 is valid.

In the curves of Fig. 7 the equations predicted in this paper a r e only valid for ^ « i , therefore the curves are extended past this limit empirically.

An analysis similar to that of Section 8 for a flat plate with an impermeable portion from the leading edge to a point x,, but with a constant velocity v thereafter has been considered. All the t e r m s

1 •' w

2

of order (Rx^) and higher have been neglected and the final equation is: r (x) w , - — 3 i = 1 ; X < x^ K x 2 1 = 1 - l ( R x ^ ) ; X > X 1

This type of solution may be sufficiently accurate for engineering purposes but a much more difficult analysis could probably include t e r m s of higher o r d e r .

The frictional heating term was neglected in the solution of the energy equation but the method of Bernard Le Fur (Ref. 19) in which the frictional heating term is included, could be used for the case of fluid injection.

If improved velocity and temperature distributions near the wall were used, then the accuracy of the method would be improved. A suitable improved velocity distribution is given by Spalding (Ref. 23).

Solutions to the compressible boundary layer are outlined in Appendix B. The simplified skin friction equation for a uniform free stream, compares reasonably well with those of Low (Ref. 12) and Lew and Fanucci (Ref. 16).

The calculations a r e for a Prandtl number of 0. 7, 10. Conclusions

The compressible steady laminar boundary layer equations have been solved approximately for a r b i t r a r y , p r e s s u r e gradient, wall temperature distribution and noi^mal velocity through the permeable wall, v .

The method is probably sufficiently accurate for practical purposes for negative p r e s s u r e gradients providing the heat transfer rate and the

skin friction are known when the velocity, v is zero. w

11. Acknowledgements

The writer is indebted to Mr. G.M. Lilley and Dr. J . F . C l a r k e for their very helpful suggestions and criticisms throughout the investigation of this problem.

12. R e f e r e n c e s 30 -1. Fage, A. , F a l k n e r , V . M , 2. E m m o n s , H . W . , L e i g h , D . C. 3, 4 . 5, 7. 8. B r o w n , W . B . Brown, W . B . , Donoughe, P . L . Brown, W . B . , Livingood, J . N. B . 6. Donoughe, P . L . , Livingood, J . N. B . I g l i s c h , R. C u r i e , N.

On the r e l a t i o n between heat t r a n s f e r and s u r f a c e friction for l a m i n a r flow. R & M 1408, 1931.

Tabulation of the B l a s i u s Function with blowing and suption.

A . R . C. C. P . No. 157, 1953.

E x a c t solution of the l a m i n a r b o u n d a r y l a y e r equations for a p o r o u s plate with v a r i a b l e fluid p r o p e r t i e s and a p r e s s u r e g r a d i e n t in the m a i n stream.. P r o c e e d i n g s of the F i r s t U. S. National C o n g r e s s of Applied M e c h a n i c s , p p 8 4 3 - P ' ^ l , A . S . M . E . , 1952, T a b l e s of exact l a m i n a r b o u n d a r y l a y e r solutions when the wall i s p o r o u s and fluid p r o p e r t i e s a r e v a r i a b l e , NACA T N . 2 4 7 9 , 1 9 5 1 . Solutions of l a m i n a r b o u n d a r y l a y e r equations which r e s u l t in s p e c i f i c -weight-flow p r o f i l e s l o c a l l y exceeding f r e e s t r e a m v a l u e s . NACA T N . 2 8 0 0 , 1952. Exact solutions of l a m i n a r b o u n d a r y l a y e r equations with constant p r o p e r t y v a l u e s for p o r o u s wall with v a r i a b l e t e m p e r a t u r e .

NACA T N . 3 1 5 1 , 1954.

E x a c t calculation of l a m i n a r b o u n d a r y l a y e r in longitudinal flow o v e r a flat plate with homogeneous s u c t i o n . NACA TM, 1205, 1944.

The e s t i m a t i o n of l a m i n a r skin f r i c t i o n , including the effects of d i s t r i b u t e d

s u c t i o n .

The A e r o n a u t i c a l Q u a r t e r l y , v o l . 1 1 , F e b r u a r y , 1960, pp 1 - 2 1 .

9 . 10. 1 1 . Duwez, P . , Wheeler, H . L . Libby, P , Kaufmann, L. , Harrington, R. P . Mickley, H . S . , Ross, R. C. , Squyers, A. L. , Stewart, W.E.

Experimental study of cooling by injection of a fluid through a porous m a t e r i a l .

Jnl. Aero. Sciences, vol.15, 1948, pp 509-521.

An experimental investigation of the isothermal laminar boundary layer on a porous flat plate.

Jnl. Aero. Sciences, vol.19, 1952, pp 127-134,

Heat, Mass and Momentum transfer for flow over a flat plate with blowing or suction.

NACA TN.3208, 1954.

12. Low, G.M. The laminar compressible boundary

layer with fluid injection. NACA TN.3404, 1955.

1 3 . Chapman, D. R. , Rubesin, M.W.

Temperature and velocity profiles in the compressible lamiinar boundary layer with a r b i t r a r y distribution of surface temperature.

Jnl. Aero. Sciences, vol. 16,1949, pp 547-565.

14. Yuan.S.W. Heat transfer in laminar compressible

boundary layer on a porous flat plate with fluid injection.

Jnl. Aero. Sciences, vol.16, 1949, pp 741-748. 15. 16. Lew, H.G. Lew,H.G. , Fanucci, J. B.

The cooling of a flat plate in a laminar compressible flow by uniform fluid injection.

P . l . B . A . L . Report No. 131, Institute of Brooklyn, 1948.

On the laminar compressible boundary layer over a flat plate with suction or injection,

Jnl. Aero.Sciences, vol.22, 1955, pp 589-597.

References (Continued) 32

-17. Lighthill, M . J . Contributions to the theory of heat

transfer through a laminar boundary layer.

P r o c . Royal Society, Series A, vol.202, 1950, p.359.

18. Lilley, G.M. A simplified theory of skin friction and

heat transfer for a compressible laminar boundary layer.

College of Aeronautics Note No. 93, 1959.

19. Le F u r , B . Heat transfer and recovery factor in a

laminar boundary layer with a r b i t r a r y p r e s s u r e gradient and wall temperature distribution.

Jnl. Aero/Space Sciences, vol.26, 1959, pf. 8B2-683.

20. _{Erdelyi, A.} Higher transcendental functions.

vol.2, McGraw-flill, 1953.

21, H a r t r e e , D . R. On an equation occuring in Falkner

and Skan's approximate treatment of the equations of the boundary layer. P r o c . Cambridge Philosophical Society, vol.33, 1937.

22. _{Stratford, B . S . Flow in the laminar boundary layer near}

separation.

R & M.3002, 1954.

23. Spalding, D . B . Heat transfer frona surfaces of

non-uniform temperature.

Jnl. Fluid Mech. , vol.4,1958, pp 22-32.

24. _{Mikhlin,S.G. Integral equations.}

A P a r t i a l T r a n s f o r m a t i o n of the Monaentum Equation E q u a t i o n 3 . 1 3 i s 1 2 -p / 2 \ f 2 \ ~ 3 „ ~ 3 •*• ^ 3

F = -(f) ' p ' iqfy F^ (A.l)

R e a r r a n g i n g : F. = - ( ! ) ^ J^\ . P"^ F (A. 2) The i n v e r s e t r a n s f o r m of t h i s equation, using the Convolution T h e o r e m , i s t h e r e f o r e , 2 T (t) i , f' (t - t , ) - ^ ^ o 1 2i 2' i M P^ Hence <^^' TIW f if^' '' r ^ < ^ ) > ^ d z ) - ' [ d ( u ^ ( x ^ ) ) - | y x , ) v ^ ( x ^ ) d x ^ 2 r (x) 1 »x / X ^ ~, ^ _ / 2 \ 3 A I I I / •> /r,\\2^„\-3) ^ / „ 2 / UP O x^ On r e a r r a n g i n g t h i s equation then ^ ,x ,x 1 - I . .X .X 1 r w o X^ O X . v (x.) r (x, )dx. W ^ W ' 1 J (A. 3) The t e r m involving u^(x) i s a 'Stieltjes i n t e g r a l ' and it h a s a value when the free s t r e a m v e l o c i t y , u^ , i s a c o n s t a n t . The equation i s f i r s t used in Section 7. 2 a s equation 7 . 1 3 .

34

-APPENDIX B

The Compressible Laminar Boundary Layer with Suction or Injection

The method used in this report has now been extended to the compressible boundary layer by using the paper by Lilley (Ref. 18). The integral equations for the skin friction and the heat transfer, which are equivalent to equations 3.14 and 4. 29, a r e :

a ^ h (z) X r (X ) ëllsl) a I M : ( O ) + r ^ dM:(z)i = ^ ƒ J ^ ^ MX,) %^(x,)v^(x,)dx,+ J ^ -J. ^ o o * * w o o / o o , , o O 1 1 1

3^.2 ,^ vüi!l' , r^ ((" v'"'V""^" ':..

o o •' w ^ o X, ( B . l ) a n d 2 r. t \ ^ ( t ) o-^ # <x). r (x)^ ^ , s ff f ^ ( z ) ' ' ' (z)^ .. i ' Q (x) = r • w w ^w m(x) \ w w dz ) w r ( i ) ^ ^ T 1 j \ J /

V ^ ^ ^w<f i) ^w<^i>^ ^^1 33 <^o ^o>' V < ^ ^ ' ^w<^) - < ^ ) ^

'^w V^>'^<^i^ •r"<i") c r ' c ( x ) ^ _{. . . u}

w o o

i 1 i "3 c {zV T (zr dz w w , . y ^ ... ^^. h (X ) m(z) / \ w 1 w 1 o x^ o

(K

<^^"

(B.3) m(x) = ( 1 + ^ . M^ixfy^^''^^ (B.4) U T ^w o

M i s the Mach number, c (x) is( = - ) and h i s the stagnation enthalpy o w

Convenient d i m e n s i o n l e s s f o r m s of t h e s e equations a r e obtained by i n t r o d u c i n g f and f' which a r e defined:

° w w i 1 f = - 2 P v (—^ ] and f" = r f ^ , w w w v u p ^ y w w v p ^ i ^ \ a a a / \ a a a ( B , 5 ) T h e suffix ' a ' d e n o t e s an a r b i t r a r y constant r e f e r e n c e condition.

E q u a t i o n s B . 1 and B . 2 b e c o m e , (when y = 1 . 4 ) "" i ( z ) dMf(z) f"" f j ( x , ) f (x,) i "''^ w 1 / w ' w ^ / a T\/r^ •' J^i * T\/r^ •' X , ( i (x ) M o ^ M o 1 \ w ^ _{a a} 2 9 / 2 3 / 2 (J^L 11/4 5 _ + M ^ 3i^ . f;(x^) i^ 2 ^ 5 + M ; ( X )

(TT^T")

'^^' "^^o " x ~ vy^O

viT^iT-a - 1 viT^iT-a f f" ( z ) * 1 " ^ 5 .HM^ ' / * - *

4, IJ-VW ^IT^J 'V'''

( B . 6 ) 1 - C d Q (x) „ _ A ^ w f X 3 ' ( h ( + o ) - h (+0) x l ^ a ^ a ^ r ( i ) c r ^ V w < ^ V W W . 1 , .X f (z)2 / i \ 2 m ( z ) d z ^ d ( h (x ) - h (x )) 1 / w / a ^ a j w i W r , i

T~ VTID;

r(f) a f V - ' A J J j> ,A „ ,.., / ( / r V

_{. 33 i-r: x^ cr3 fMx)^ m (x).} X* J z* " V " ' " (h (+0) - h (+0)) o X^ , W W ^ w-1 ,0 '2> .1 / i „ . ( x ) \ A 2 .. 1 r / f / i . J z ) 2 IXi) \ 1 / w a J ^ J \ 1 a o x^ a i (Z)^ \ " Y p-X-— Q (x ) 1 M x ) W / V J ^ W 1 W 1 T- m (z) dz z^ , a a a w w f ( x J ^ x , ^ \ ^ P u /J * (h (+o)-h (+o))J 7 7 ) 1 ~ | 1 -w o w ^ ^ -1 / i (x ) a

36 -5 + M^ \ r / 4 2 w h e r e m (x) = ( g / • i i s the enthalpy (h = i + — ) ^ ^5 + M,(x) '' ^ co It i s a s s u m e d that i^ ~ T w h e r e cj i s a c o n s t a n t .

E q u a t i o n s B . 6 and B . 7 naay be solved by m e t h o d s sinailar to t h o s e used in t h i s r e p o r t for the i n c o m p r e s s i b l e flow t o g e t h e r with the

m o d i f i c a t i o n s u s e d in L i l l e y ' s p a p e r .

F l a t plate with z e r o p r e s s u r e g r a d i e n t and constant wall enthalpy C o n s i d e r i n g the velocity v (x) to v a r y a s — and using the

T/X"

a p p r o x i m a t e naethod of Section 7. 2 then the equation B . 6 r e d u c e s to

/ i Y - 1 I/O / i \ l ( ' ^ - l )

0.211 + 2 . 1 5 f f" ( ^ = f " 4 / 3 ( ^ 1 . ( B . 8 )

W W \ l / W \ 1 /

W W The energy equation reduces to

k ^ R e * = (k Re*) / _ i _ _ _ _ w \ , w / w \ ^ 3 ^ / ( B . 9 ) 1 r f "3 \ ^ 2 W r ( i ) cr' w h e r e E = —r—i— and 2^ 3^ - Q (x) w

k, i s the Stanton heat t r a n s f e r coefficient given by k, = T, r- » • h ^ -^ h ^ u (h -h )

a a w w o The c o m p r e s s i b l e equations r e d u c e d to the i n c o m p r e s s i b l e form when the wall enthalpy i s constant and the free s t r e a m i s uniform

(Q ) , (f ) a n d ( f " ) will be defined by the following e q u a t i o n s :

-w c -w c -w c y < , CÜ-1 co-1 co-1 • ~ ? " • " T

Q = (Q) C^Y . f =<f) ( - ) . f" =(f")

CA

w w c \ i ^ / w w c \ i , / w w c \ i ^ / / i ^ \ (B.IO) When ( - ^ ) and M^ a r e constant then equations B . 6 and B . 7 r e d u c e to:

m)

27 O X^* "" (f (z))2 ^ \ ^ - 1 - ^ w' "c dz ) dx^ ( B . l l ) and " \ 2 "3 Q A 1 1 X f" ^ w c i X 3^ / / X' 4 / r w c dz (f ) X* w o (h (+o)- h _ (+o)) J p /J u „ / i \ 3 w c w_ N 1 1 1 r ( 3 ) cr \ J _ z T 2

^ÜiL, x^crMf''/ r r f (f")^ . _ ? r (Q )

2r (i)3^ ^ M V i -J5L^^ ^^ ^ > '^ ^

p /J u X, ^4 (^(h^(+o)-h^ ^ ^ ^ j ^ (f ) " ^ ^ 3 dx, ( B . 1 2 ) ^ P M U It" \2 4 ^' 1 1 I -* ( f ) X * w c ^

T h e s e a r e equivalent to the i n c o n a p r e s s i b l e equations when m = 0. T h e r e f o r e the solutions m a y be w r i t t e n d i r e c t l y from the naain text of t h i s r e p o r t .

(a) When the v e l o c i t y v (x) i s p r o p o r t i o n a l to —r then

w /"x

( f " ) ^ ' ^ = 0.211 + 2 . 1 5 (f ) ( f " ) ( B . 1 3 ) w c w c w c

and

. 2 _ / I , T i „ 2 l

k^Re^ = K'^X (x)=0 ËaT— 1 «•'•I*'

w \ , w c

1 - 1— ( f " ) '

w c ( T h e s e a r e the s a m e a s equations B . 8 and B . 9 ) .

Equation B . 13 c o m p a r e s favourably with the solution of the c o m p r e s s i b l e b o u n d a r y l a y e r by Low (Ref. 12). In F i g . 3 the c u r v e of Donoughe and

Livingood for ra = 0 i s i d e n t i c a l with L o w ' s c u r v e and the o t h e r c u r v e for m = 0 i s i d e n t i c a l with that of equation B . 13, except that f and f"

^ W W

a r e now r e p l a c e d by (f ) and (f" ) -^ --^ w c w c

38

(b) When the v e l o c i t y v i s constant then, from Section 8,

equation 8. 24 l-(o r (x) / i \ ^ , ^ , >, = l + 0 . 9 8 2 ( f / = 1 + 0.982 f . i - ^ J ( B . 1 5 ) (7^ (x)) / X rt w c w \ i , / w V (x)=0 1 w o r equation 8 . 2 5 = 1 + 1.25 (f ) ( B . 1 6 ) w c

Lew and F a n u c c i (Ref. 16) p r e s e n t a c u r v e for the skin friction with uniform suction for the c o n a p r e s s i b l e b o u n d a r y l a y e r .

j-~-A i s plotted against J M . ^ _ L J w h e r e g = ( - - ^ j ( ^

W W -^ W 1 1 T and C = 7j— . -^ T h e s e t e r m s m a y be c o m p a r e d with t h o s e in t h i s p a p e r if C i s a s s u m e d equal to V-pp~ Ï- 2 ( f " ) rZ / T rnu w w c , 2 ? / 1 \ V2 T h e n - = . , > and \ T T L r ^ = -r- . {i ) . P V u (f ) ^' C VT y 2 w c w w 1 w c ^ w

E q u a t i o n s B . 15 and B . 16 a r e c o m p a r e d with Lew and F a n u c c i ' s r e s u l t s in F i g . 8.

m 1.0 0.5 f w 0 0.5 1.0 0 0.5 1.0 F f w r o m E q u . 7. 8 1.136 .784 .586 .830 .554 .408 f w F r o m Ref. 6* 1.233 .969 .756 .900 .697 .534 % E r r o r 8 19 22 8 20 24 T A B L E 2 na 1.0 0.5 w w w % E r r o r 0 -0.5 -1.0 0 -0.5 -1.0 F r o m Equ. 1.233 0.881 .683 .900 .624 .478 7. 12 F r o m Ref.6 1.233 .969 .757 .900 .697 .534 -9 10 -10 11

w -0.5 m 0.5 1.0 -1.0 O 0.5 1.0 TABLE _ i e nu Re~^ F r o m Equ 0 0.5 1.0 0 0.5 1.0 0 0.5 1,0 0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 .133 .226 .284 .240 .361 ,445 .279 .393 .478 . 0 4 0 " .081 .108 _ L . 1 5 2 ^ .251 .318 " . 1 7 7 ' .267 ,336 3 nu Re"^ . 7 . 4 2 F r o m Ref.6 .166 .261 ,321 .259 .383 .471 .293 .413 .503 .052 .105 .138 .139 .253 .331 .146 .255 .336 % E r -20 -13 -12 -7 -6 -6 -5 -5 -5 -23 -23 -22 10 -1 -4 21 5 0

The B r a c k e t s , [' 2» denote that

D f w f" ' w 1 . _ i

c

EQUATION 7.17. - ".Cx) ^ « C x y ) 0 - 8 VwCx) O N J E C T I O N + V E )

c

FIG. I.

e 15 WALL TEMPERATURE-GRADIENT PARAMETER (VU IS LOCAL NUSSELT NUMBER

f IS DIMENSIONLESS FORM OF FLOW THROUGH •' POROUS WALL O'S 0 - 6 0 - 4 0 - 2 0 - 6 0 - 4 0 - 2

FIG.2. THE VARIATION OF THE SKIN FRICTION f^RAMETER,fw^ WITH THE EULER NUMBER,m

0 - 6 0 - 5 0 - 4 tui 0 - 3 0 - 2 O l 1 PRESENT PAPER ; EQUATION 7 . 4 2 . EQUATION 7. 42.WHEN I / / ' / r

11/

/ / • • / / ^ ^ / )UGHEI Di, r— . 1

1

/ ^ ^ ^ ^ J w -' /J.-0-5

:|i-'-°

FIG. 3. THE EFFECT O F ^ AND THE EULER NUMBER, m, ON THE SKIN FRICTION PARAMETER J j '

-o-s O 0-5 c

EULER NUMBER - O

RG. 4. HEAT TRANSFER TO A POROUS WALL WITH VARIABLE WALL TEMPERATURE

o 7

EXACT SOLUTION CDONOUGHE & LIVINGOOD) PRESENT P A P E R :

EQUATION 7. EQUATION 7.42,

0 7

EULER NUMBER.0-5

FIG. 5. HEAT TRANSFER TO A POROUS WALL WITH VARIABLE WALL TEMPERATURE

e IS WALL TEMPERATURE-GRADIENT PARAMETER

nu IS LOCAL NUSSELT NUMBER

EULER NUMBER'I'O

FIG. 6. HEAT TRANSFER TO A POROUS WALL WITH VARIABLE WALL TEMPERATURE

e IS WALL TEMPERATURE-GRADIENT PARAMETER lUI IS LOCAL NUSSELT NUMBER

3 0 ZS 2 0 r , ( x ) l-^i^-O I S

• EXACT SOLUTION IGLISCH,R. 0944) REF. 7 APPROKItulATE SOLUTION. C U R L , N . Q 9 6 0 3 R E F . 8 . PRESENT RIVPER I EQUATION B.2a ] I , EQUATION 8.24. I N C L U D I N G | 3 | > ' 3 . - EQUATION 8.25. 0-5

RG.7. SKIN FRICTION FOR FLAT PLATE WITH UNIFORM SUCTION

f = Vwfx «5 U, V Wc ( ^ ^ : ^ = 0 - 0 - 4 - 0 - 2 O 0-2 0-4 0-6 OS 1-0

-/WAi)

FIG.8. SKIN FRICTION WITH UNIFORM SUCTION AND INJECTION (COMPRESIBLE SOUJTION)