lithiêi d« Kuyierwaij CoA. Report No. 155

THE 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 COMPRESSIBLE LAMINAR BOUNDARY LAYER

WITH FOREIGN GAS INJECTION

by

A. H. Craven

^

REPORT NO. 155

January, 1962.

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

C R A N F I E L D

The Compressible Laminar Boundary Layer

with Foreign Gas Injection

- b y -

Squadron Leader A. H. Craven, M.Sc . , P h . D . , D . C . A e . ,

(Royal Air Force Technical College, Henlow)

SUMMARY

The equations of the steady compressible two-dimensional laminar boundary layer with foreign gas injection through a porous wall are solved, using an extended form of Lighthill's approximate method, for arbitrary main stream pressure gradient, wall temperature and injection velocity. The wall shear s t r e s s and heat transfer rate are obtained in the form of equations suitable for iteration.

It i s shown that substantial reductions in skin friction and heat transfer rate can be obtained by the injection of a light gas instead of air .

CONTENTS

Page

Summary

List of Symbols

Introduction 1

The boundary layer equations appropriate

to injection 2

The Stewai-tson-Illingworth transformation 4

An approximate solution of the transformed equation of motion 7 An alternative solution for the equation

of motion 12

The wall shear s t ress 13

An approximate solution of the diffusion equation 18 An approximate solution of the stagnation enthalpy equation 20

Numerical solutions for the wall shear

s t ress ard heat transfer rate 26

Conclusions 27

Acknowledgements 27

References 28

Figures

LIST O F SYMBOLS

a speed of sound

A, A, , A J constants

B , B , , Bg constants c concentra t ion of foreign gas

c* concentrat ion gradient I — ) at the wall

C specific heat at constant p r e s s u r e P

M p P 0*^0

D the b ina ry diffusion coefficient

f d imens ion less injection p a r a m e t e r = m (x/p n u )

G(X,i/>) Z - f S(z, t\>) dU^ ' (z )

o

h specific enthalpy

h stagnation enthalpy

^

X ars

V^(X) ^ dX

o

k t h e r m a l conductivity Le Lewis num.ber p C D . / k

'̂ P 18

m injection m a s s flow r a t e pe r unit a r e a m(x) 1 + X ^ M«(x) •y - 1

2

M Mach number

p p r e s s u r e

q no rma l energy flux due to injection

Q^(x) r a t e of heat t r a n s f e r pe r unit a r e a « Q, r a t e of heat t r ans fe r for z e ro injection

s (x) Q ( x ) r x / p M u ~1 . the modified heat transfer rate w w L a a a J

S 1 - h / h ,

Sc Schmidt number p/pD,j

t (x) non-dimensional wall shear stress, w I'."»''. J

t non-dimensional wall shear stress for zero injection wo

T temperature

u, V velocity components in the compressible flow

U, V velocity components in the transformed flow

V , V normal velocity at the wall in the compressible and transformed flows.

respectively

X, y co-ordinates in the compressible flow

X, Y co-ordinates in the transformed flow

z u;-u'

•y ratio of specific heats C /C

A i (Le - l)(h^ - h.) ff e l

(i viscosity

V kinematic viscosity

p

«r

•

' • w

Subscripts

o

i

w

a

e

i

density

Prandtl number ju C /k

stream function

wall shear stress

stagnation value

value outside the boundary layer

value at the wall

reference condition

mainstream

injected gas

A bar over a quantity denotes its Laplace transform