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
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