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
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 C o m p r e s s i b l e Laminar Boundary Layer with F o r e i g n Gas Injection
b y
-Squadron Leader A. H. Craven, M . S c . , P h . D . , D . C . A e . , (Royal Air F o r c e Technical College, Henlow)
SUMMARY
The equations of the steady c o m p r e s s i b l e two-dimensional laminar boundary l a y e r with foreign gas injection through a porous wall are solved, using an extended form of Lighthill's approximate method, for arbitrary main s t r e a m p r e s s u r e 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 a i r .
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 r e s s 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 r e s s ard heat transfer rate 26
Conclusions 27 Acknowledgements 27
References 28 Figures
LIST O F SYMBOLS a s p e e d of sound A , A , , A J c o n s t a n t s B , B , , Bg c o n s t a n t s c c o n c e n t r a t i o n of f o r e i g n g a s c* c o n c e n t r a t i o n g r a d i e n t I — ) at the wall C specific h e a t at constant p r e s s u r e P M p P 0*^0
D the b i n a r y diffusion coefficient
f d i m e n s i o n l e s s injection p a r a m e t e r = m ( x / p n u ) G(X,i/>) Z - f S(z, t\>) d U ^ ' ( z ) o h specific enthalpy h s t a g n a t i o n e n t h a l p y ^ X ars V^(X) ^ dX o k t h e r m a l conductivity L e L e w i s num.ber p C D . / k '^ P 18
m injection m a s s flow r a t e p e r unit a r e a m(x) 1 + X•y - 1 ^ M«(x)
2
M Mach n u m b e r p p r e s s u r e
q n o r m a l e n e r g y flux due to injection Q^(x) r a t e of heat t r a n s f e r p e r unit a r e a
«
Q, r a t e of heat t r a n s f e r for z e r o injection
s (x) Q ( x ) r x / p M u ~1 . the modified heat transfer rate w w L a a a 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 ( L e - 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 functionwall shear stress
stagnation value
value outside the boundary layer value at the wall
reference condition mainstream
injected gas
1. Introduction
Recent studies* have suggested that injection of a gas into the boundary layer through a porous wall can be used to reduce the skin friction and the rate of heat transfer to the wall. The majority of the work on the laminar boundary layer with injection is theoretical and considers mainly the injection of air into a i r . The analyses are restricted severely by the assumption of particular s t r e a m -wise and injection velocity distributions in obtaining solutions of the equations. Since it is difficult to maintain a laminar boundary layer there is very little experimental evidence but such as exists (Ref. 2) lends support to the theoretical r e s u l t s .
Injection of a foreign gas into a two-dimensional laminar boundary layer has been considered by Smith(3), Eckert and Schneider^^) and Faulders^^). Each shows that injection of a light gas is much more effective than injection of air in reducing skin friction. Smith's solution does not give values of the wall shear s t r e s s explicitly but these can be found from the velocity profiles which are presented. Each solution is subject to some restrictive assumptions. Smith solves the boundary layer equations and the diffusion equation with the boundary conditions appropriate to the impermeable wall. The solution takes account of the foreign gas (the concentration of which is taken to be large at the wall) but paradoxically considers the injection velocity to be z e r o .
The solutions of Eckert and Schneider and of Faulders are restricted to the case of zero heat transfer and assume that the injection velocity varies inversely as x*. A further assumption in Faulder's treatment is that the viscosity of the binary mixture is independent of concentration and varies linearly with temperature. The Schmidt number is taken to be unity.
(16)
The case of non-zero heat transfer is considered by Korobkin in a study to determine which of the properties of the injected gas is of most importance in reducing skin friction and rate of heat transfer. Using the simple rigid sphere model for the molecular collision p r o c e s s e s , the equations of motion are solved numerically for the case when the injection velocity varies inversely as x». In the results presented two of the three properties of the mixture, molecular weight, molecular diameter, and specific heat at constant pressure are given the value for air and the third is varied taking the value corresponding to the calculated concentration. This solution (to an approximate physical problem) shows that variations of C have a negligible effect on skin friction. The greatest reduction in skin friction is to be expected when the injected gas has low molecular weight and large molecular diameter. These properties coupled with high specific heat per unit m a s s should give the greatest reduction in the rate of heat transfer.
A more general formulation and solution of the problem of gas injection into a laminar boundary laver is possible using an approximate method originally developed by Lighthill' ' for the incompressible layer and extended to the
(7)
compressible layer by Lilley . Both these solutions are for the impermeable wall. Stevenson'"' has used Lighthill's approach to solve approximately the equations of the incompressible laminar boundary layer with either suction or air Injection through a porous wall. Arbitrary distributions of main stream velocity,
wall temperature and normal velocity at the wall are included in the solution which is extended in the same paper to the compressible case.
The present paper uses Linens simplified theory for a compressible laminar boundary layer as the starting point to consider foreign gas injection. Approximate solutions a r e obtained for the diffusion equation and the equations of the compressible laminar boundary layer with a r b i t r a r y external p r e s s u r e gradient, wall temperature and injection velocity distributions. Expressions for the wall shear s t r e s s and heat transfer rate to the wall a r e obtained in the form of integral equations involving the concentration of the injected gas at the wall (which is obtained from^ a third integral equation). These.integral equations a r e in a form suitable for numerical iteration.
2. The Boundary Layer Equations appropriate to Injection
It is assmned that both the injected and the mainstream gases are perfect and that chemical reactions are absent. Consequently we may consider the enthalpy h of the binary mixture to be related to the enthalpies of the two constituents by the equation
h = (1 - c) h + ch, (1) e i
where h is the enthalpy of the mainstream gas ƒ C dT
<
h. is the enthalpy of the injected gas / C^ dT *o
and c is the concentration of the injected gas expressed as a m a s s fraction
C and C are functions of T only. Pe Pi
If suffix I denotes local conditions outside the boundary layer, the equations governing the steady two-dimensional compressible boundary layer in the presence of a p r e s s u r e gradient are
(i) continuity ^(,u) ^ ±ip.) - 0 (2) (ii) motion .^ » du 8u , 8u ° " pu -— + pv -— - p u :r-' = r— ^ dx '^ by ^ ' 1 dx 8y
(^IF)
(3)1 ^ = 0
(4)
8y(ill) energy ^ ^ ^ ^ ^^ du, _ / BUV _ 8^
(5) (iv) diffusion p u r— + p v -— + u p . u -T— = pi '^ 8x '^ 8y '^' ' dx '^\ 8£ Ë£ - JL( D —") » — {-H- —\ " 8x ' ' ^ 8y " 8y \ ** '« 8 y / ' 8y \ S c dyJ (6)
where the Schmidt number Sc is defined as p / p D
3
-q in e -q u a t i o n (5) i s the n o r m a l component of the e n e r g y flux c o m p o n e n t . In t e r m s of t h e diffusion v e l o c i t i e s of the two s p e c i e s , q m a y be w r i t t e n in t h e f o r m
o rp
q = p c v^ h^ + p ( l - c)v^ h^ - k — (7) w h e r e v. and v a r e t h e diffusion v e l o c i t i e s of the injected and m a i n s t r e a m
i e •' g a s e s r e s p e c t i v e l y .
In t e r m s of the c o n c e n t r a t i o n and the c o n c e n t r a t i o n g r a d i e n t t h e diffusion v e l o c i t i e s m a y be w r i t t e n , if p r e s s u r e and t h e r m a l diffusion effects a r e n e g l e c t e d , ^ ^ 1 = - ° i a 87 (1 - c)v^ = - D ^ ^ d - c) = D , . | ^ s i n c e D., - D , , l o r e we m a y e x p r e s s — F r o m (1) F u r t h e r m o r e we m a y e x p r e s s —^ in t e r m s of enthalpy and c o n c e n t r a t i o n g r a d i e n t s . Bh ,, , ^ 8 T ^ ^ 8 T , 8 c ^ . 8 c ;—• = (1 - c)C -— + c C r— - h r— + h . r— 8 y p^ 8y p^ 8 y e 8y i 8y = C | 2 : + (h . h ) 1 ^ ; C = c C + (1 - c)C p 8 y i e 8y p p^ p^ o r 8y «r 8y i e «^ 9y M C w h e r e the P r a n d t l n u m b e r a " p k
Substituting t h e s e f o r m s in (7) the n o r m a l e n e r g y flux can be w r i t t e n q = - — r— + — (Le - 1) (h - h.) -— ^ o- 8y a- e i 8y H dh 8 c ,-,. O" 8 y 8 y w h e r e A » — (Le - 1) (h - h.) (T e l and Le i s the L e w i s n u m b e r p C D / k '^ p ia The b o u n d a r y conditions a r e (i) at the wall y = 0 , u = 0
V = V ( x ) w c = c (x) w T = T (x) w
w h e r e the suffix w d e n o t e s the wall v a l u e .
(ii) a t y = » u = u,(x) c = O (10) T = T,(x) ». D _ n _ _ n 8 u _ 8 T _ 8 c _ Q 8 y 8y 8y If we define the s t a g n a t i o n e n t h a l p y h b y s h = h + u*/2 s
it i s p o s s i b l e t o e l i m i n a t e the p r e s s u r e g r a d i e n t in the e n e r g y equation b y m u l t i p l y i n g (3) by u and adding it to (5). T h e r e s u l t i n g e q u a t i o n for t h e s t a g n a t i o n e n t h a l p y i s
^K ^ ^\ 8 r 8 (^x'Sl 8q
o r , on s u b s t i t u t i n g for q f r o m (8) 8h 8h „ / 8h ^% ^ ^% 8 / p ^ M 8 T M n -i» {^'\^ Ö / ' A Ö C \ (11) T h e e x t e r n a l flow i s a s s u m e d t o b e i s e n t r o p i c s o that a* u^ a ' _ L + ^ = h = - ^ (12) 7 - 1 2 si 7 - Iw h e r e y i s the c o n s t a n t r a t i o of the s p e c i f i c h e a t s in the e x t e r n a l flow. 3 . T h e S t e w a r t s o n - lUingworth t r a n s f o r m a t i o n
In the c o m p r e s s i b l e flow the equation of continuity (2) can b e s a t i s f i e d b y a s t r e a m function i^ defined b y
£i3 = 8* ; p v - p V (X) = - p 1 ^ (13) p 8 y '^ "^w w '^o 8x
w h e r e the suffix o d e n o t e s som^e c o n s t a n t r e f e r e n c e condition and p (x) i s the d e n s i t y of the b i n a r y m i x t u r e at the w a l l .
Following S t e w a r t s o n and I l l i n g w o r t h , the x , y c o - o r d i n a t e s of the c o m p r e s s i b l e flow field a r e t r a n s f o r m e d to X , Y c o - o r d i n a t e s r e l a t e d to x , y b y f"" a (x') p^(x') X = / d x ' J a p o o ^^4j a ( x ) 1 a ; p 0 0 0
[ P ( ^ . / ) dy'
J P5
-T h e v e l o c i t y c o m p o n e n t s (U, V) in the X , Y plane a r e now r e l a t e d to t h o s e in t h e X, y p l a n e . T h u s
£ U _ ^ ^ 8 ^ 8 X 8 ^ 8 Y _ fiP_ 8 ^ p " 8 y "^ 8X 8 y 8Y 8 y " a p 8Y "^O «' ^ 1 o o and defining U a s - ^ we have
^ = ^ ) <^^> A l s o - ^ ( p v - p v ) = ^ = ^ 5 X + Ë 1 8 Y p ^'^ ''w w ' 8x 8X 8x 8Y 8x '^o y a p ^ , a u . ^ / a r , «» \ , . _ L I Ë i + _ o _ - L ( _ L P<^'y> d y ' ) a p 8X a, 8x \ a J p ^ / o o ' o o '^o 8X a p p M^ '^w w '^o a . 8x V a i p J \ i^i*^o L ' ^ o o '^o ' -I and t h e r e f o r e If we define V . V = - Ë l w 8X it follows t h a t and ^^gj a p p o o'^w V = V w a , p , p^ w
W r i t i n g suffix o to denote s t a g n a t i o n conditions in the m a i n s t r e a m , equation (15) with u = u^ s u b s t i t u t e d into (12) y i e l d s
/ ( - ^ • ^ )
a ' - B.\ I [1+1^^. ^ ) (17) o
Using the t r a n s f o r m a t i o n e q u a t i o n s (14 - 16) the equation of motion (3) b e c o m e s T I Ë U 4. v ^ u | j „ a u , P' " 8X 8Y " h „ „ ^ , ^ , . S i ' ^'^o o U I H 1 + ^ ,; _Ë.f £if a u " ) (18) ''^ 8 X - ^ p , ''o%Y\pu BY J
which can be simplified by putting
S = 1 - h / h (19) s Si
PQPM
and C(X.Y) = — - (20) '^I'^o o
giving „ „ ^„ 8U
S i m i l a r l y t r a n s f o r m e d the diffusion equation (6) b e c o m e s
u i £ + v ^ = V -L C c . acN (22)
^ 8X "^ 8Y "^o 8Y \ S c 8 Y / ^ ' T h e t r a n s f o r m e d equation for the s t a g n a t i o n enthalpy i s (from 11)
•%fv[ffv^]-„fv[f»-<C4.j]
' ' o 8Y h^^ 8Y U ^ + V ^ 8X 8Y "o " S i (23) In equation 20 we c a n , b y v i r t u e of (4), r e p l a c e p, by p . F o r the c a s e of a i r i n j e c t i o n C can be w r i t t e n T M . w-1^ ^
'TJTo N o /^ ( T~)
if ^ i s t a k e n to be p r o p o r t i o n a l to T . F o r foreign g a s injection fx and p a r e c o n c e n t r a t i o n dependent a s well a s t e m p e r a t u r e dependent, and t h u s no s i m p l i f i c a t i o n of C i s p o s s i b l e .
T h e von M i s e s t r a n s f o r m a t i o n
(24) \SYJ^
^ 80
i s now applied to t r a n s f o r m f r o m the " p s e u d o - i n c o m p r e s s i b l e " s p a c e c o - o r d i n a t e s (X.Y) t o independent v a r i a b l e s (X,ip).
P u t t i n g ^ ^ Z(X,0) = U,(X) - V (X,^)
the e q u a t i o n s of m o t i o n (21), diffusion (22) and stagnation enthalpy (23) b e c o m e r e s p e c t i v e l y
az ^ V
5Z=
s—^
. . .
, .
8X w 80 dX o 80 \ 80, V ^ = s i 5 . . U ^ ( c ^ ) (25)
w 80 dX o 80 \ 80 / 8 £ + V ^ = . ; - L ( U £ 8 £ ) (26) 8X w 80 o 80 \ S c 8 0 /w 80 " "080 [ a 80 J'^''o80 |_o- ^ " ' '' \ 2nig, / J
E q u a t i o n (27) can b e w r i t t e n a l t e r n a t i v e l y in the f o r m
H. V ^ - . -5-( u ^ a s ) ^ ^Jy-^)
±(VCL'-^)w 80 o 80 \ <r 80/ ~ ~ irV 80 V^ 80 y
8S8X
o Ui ^ ^ 2 • a'o 2 a* J ^o -* (28) _8 80In t h e s e e q u a t i o n s the P r a n d t l n u m b e r a , the Schmidt n u m b e r Sc, the L e w i s n u m b e r Le (in A) and the p a r a m e t e r C a r e c o n c e n t r a t i o n d e p e n d e n t . y i s the r a t i o of the specific h e a t s of the m a i n s t r e a m g a s and i s a c o n s t a n t .
4 . An a p p r o x i m a t e solution of the t r a n s f o r m e d equation of motion T h e f i r s t t e r m on the r i g h t hand s i d e of the t r a n s f o r m e d equation of m o t i o n (25) can be w r i t t e n X S(X,0) and t h u s (25) b e c o m e s X d U, (X) dX _8_ 8X
j S(z,0) duj(z).
A ( ^ Z - ƒ S ( z , 0 ) d U ; ( z ) ) = . ^ U ^ ( c | | ) - V ^ | f
(29)
L e t u s now c o n s i d e r the equation 8G 8X (X,0) = v U ^ ( c ^ ) - V o 80 \ 8 0 / 8G w 80 w h e r e G(X,0) = Z / ' S(z,0) d \f{z) (30)
If we r e p l a c e S ( z , 0 ) by s o m e tjuitably chosen a v e r a g e value S*(z) for s m a l l v a l u e s of 0 and b y z e r o for l a r g e v a l u e s of 0 then equation (30) r e d u c e s a p p r o x i m a t e l y to (29) with ^
G(X,0)_^,^ -Z
1
s^(z) d u;(z)
and^<^'*^- ^ - 1 7 ^ ° "^*^
(31)
One f u r t h e r s i m p l i f i c a t i o n can be m a d e to equation (30). We m a y expand 7 ^ { C T— ) s o t h a t (30) b e c o m e s
80 \ 80 /
_8_
8C
C o n s i d e r the t e r m v U — . R e v e r t i n g back to the o r i g i n a l s p a c e c o -o r d i n a t e s ( x , y )
V 3. p .» , P p^< o 80 a, p 8 y \P,P^^^^ )
1 % P o 8 " P P ^ ' a,p, 8y < ^ ^
Now p and M a r e functions of t e m p e r a t u r e and c o n c e n t r a t i o n . T h u s 8 . > 8 , . 8 T . 8 , . 8 c
gj(p/i) = 8T<''^)87 -^ 87<P^^87
It i s shown l a t e r (equation 75) t h a t — i s s m a l l being d i r e c t l y p r o p o r t i o n a l to the injection m a s s flow. F r o m t a b l e s of p r o p e r t i e s of g a s m i x t u r e s ( R e f s . 13, 14) it i s s e e n t h a t 8p i s v e r y s m a l l f o r s m a l l c o n c e n t r a t i o n s of injected g a s and it can
8 8 8 T b e i n f e r r e d t h a t r—(piu) i s not l a r g e . — ( p ^ ) i s s m a l l and — , which i s
r e l a t e d t o t h e h e a t t r a n s f e r r a t e , i s known t o be r e d u c e d by a i r i n j e c t i o n . It i s a s s u m e d (and p r o v e d by the l a t e r a n a l y s i s ) that a g r e a t e r r e d u c t i o n i s obtained b y light g a s i n j e c t i o n . The condition u n d e r which it i s p o s s i b l e to i g n o r e
8C
V U -— can be a s s e s s e d b y c o n s i d e r i n g the c o n c e n t r a t i o n p r o f i l e s found b y o Ö0 .^.
E c k e r t and S c h n e i d e r for h y d r o g e n injected into a i r at z e r o heat t r a n s f e r in i n c o m p r e s s i b l e flow. In t e r m s of the s i m i l a r i t y p a r a m e t e r r/= i y ( U , / i / x) we m a y w r i t e
8 C
V - 1/ U
w o
It & - - i - 2£ T
P l o t t i n g C a g a i n s t 77 for different wall c o n c e n t r a t i o n s of injected h y d r o g e n ( F i g . 1) it c a n b e s e e n t h a t 8 C i s not g r e a t e r t h a n 0.2 . v U 8C c a n b e
8n ° 80
n e g l e c t e d in c o m p a r i s o n with V when 1 2 X w R^ Vu,
» 0.1 We m a y t h e r e f o r e a p p r o x i m a t e to C(X,0) in (32) by i t s value at s o m e v a l u e of 0 . In o t h e r w o r d s we will a s s u m e that C i s a function of X only, i t s v a l u e having to be d e t e r m i n e d l a t e r . The equation of motion (32) b e c o m e s_8_ ^ / ^ .V _ „ ^ , ^ V 8 " G , ^ . , „ /^v 8G
8 X G(X,0) = 1/ U C(X) ^ ( X , A i ) - V (X) 1 ^ (X. 0) (33) o 0,1.» W 81*
with the b o u n d a r y conditions
(i) at the w a l l , 0 = 0 , G(X,0) = U,(X) - ] S (z) d U, (z) o
- 9
(ii) at ^ = c. G(X,0) = O (lil) a s X * O G(X,0) - O
(iv) n e a r the wall .X , * , » h \z) t s /• ^ < ^ ' G(X,0) = U,(0+) - U ( X , 0 ) + / -j^2_^ d U * ( z ) o SI w h e r e " t h e i n t e r m e d i a t e e n t h a l p y " h i s given by S = 1 S I h* s h S I P r o v i d e d c o m p l e t e v e l o c i t y p r o f i l e s a r e not r e q u i r e d we m a y u s e the a p p r o x i m a t i o n t o the y e l o c i t v d i s t r i b u t i o n n e a r the wall u s e d b y F a g e and F a l k n e r ^ l l ) and by Lighthill''^) n a m e l y
T (X)Y I 2 T (X) 1
U =
-^
J
—^!^—
0*
(34)
/^o "^ %
With t h i s s u b s t i t u t i o n the equation of m o t i o n (33) b e c o m e s
| § =J
^-^
. r
(X) C^X)
0^ - ^ -
V^(X)
| 5
(35)
8X N p ' w 8^* w 80 with the b o u n d a r y conditions
(1) a s 0 -» CD , G •• 0
(11) a s X - " , G - 0
^''"^ ^ ^ ^ ^ 0 X h * ( z ) 2 r (X) , / 3 ^ >
G = Uf(0) + f - I d Uf (z) ïi— 0 + O(0'')
j Si o
o
8C
F o r s m a l l v a l u e s of injection v e l o c i t y , ^-7- can be a p p r o x i m a t e d by i t s v a l u e at 80
the w a l l and we m a y r e g a r d it a s a function of X only. T h u s the second t e r m on the r i g h t hand side of equation 35 i s t a k e n a s a function of X only.
P u t t i n g V (X) = 0 in (35) g i v e s the equation for the i m p e r m e a b l e wall
|G . r ^ o ^ ^. i 8G
8X \l p\ w ^ 8^* o . Xor, if t = 1 J -7-^ ^^(X) cNx) dX,
''o^ G ( t . 0 ) = 0^ ^ ( t , 0 ) (a/)
Bt 8 t with the b o u n d a r y conditions at the wallG = 8G 80 F,(X) =
FJX)
X= uf(o) + ƒ
o 2 r ^ ( X ) ^ o h * ( z ) h S id uj(z)
(38)F o l l o w i n g L i g h t h i l l and u s i n g the L a p l a c e t r a n s f o r m m e t h o d , in which F ( P . 0 ) = ƒ e"P F ( t , 0 ) d t , the solution of t h i s equation i s
G = (f p ^ ) ' 0 * r ( i ) I _ | ( q ) F + ( f p * ) ' ^ 0 * r ( | ) I|(q) F, (39) where I2 and I 2 a r e modified B e s s e l F u n c t i o n s
3 " 3
4 i
-and q = j P 0*
T h e solution of the c o m p l e t e equation of m o t i o n (35) for injected flow can be obtained f r o m (39) by the m e t h o d of v a r i a t i o n of p a r a m e t e r s . L e t t h e solution of (35) be G = P, (0) G, + Pg (0) G j (40) _ i w h e r e G = 0^ I 2(q) 3 G = 0 ^ U q ) 3 T h e e q u a t i o n s for I^ and P^ a r e t h e n _ _ _ i d P - G F 0 * 1 d0 " G' G - G G ' ^ 1 2 1 2 and i ~ ~ " 2 d P G F 0 t _ 1 3 ^ '^1' G'. G, - G. G : w h e r e , f r o m (35) and (34) F
; <^) = ''w^^^df) ( 2 . , rjx) c'(x)) ^ • ^ 7 ^
p„V.,(X) | 2 r ^ ( X ) ^o (41) and the p r i m e ' d e n o t e s p a r t i a l differentiation with r e s p e c t to 0.It can r e a d i l y be shown that
1 1 -and t h u s , 2ir r^ 3 o p . = -H - ^ / F I . , ( q ) d0 3 s i n -T- •' o ^
giving t h e o p e r a t i o n a l f o r m of the solution of the equation of m o t i o n in t h e f o r m
G(p,0) = - - ^ 0M^(q) ƒ F, I
3 2(q) d0 3 + - ^ ^ ^ ï^^q) J ^3 I ^<q) d0 (41) 3 s i n ^ 3 J Q ^ 1 1 + A 0 ^ I 2(q) + B0^ l2(q) "3 3 w h e r e A and B m u s t be d e t e r m i n e d f r o m the b o u n d a r y c o n d i t i o n s . In t h e l i m i t a s 0 * 0 , and c o m p a r i n g w i t h (38), equation (41) g i v e sF, = A d p * ) " ^ r(i)
D i f f e r e n t i a t i n g (41) and t a k i n g the l i m i t a s 0 •• 0 F, = B (f p^)V r ( f ).^)V
Hence (41) b e c o m e s 2ir F G(p, 0) = K- 0 ' [ I z(q) f I2(q)d0 - I , ( q ) f I 2(q) d0 |3 s i n ^ L ^ J ^ 3 J -3 J
3 o o (42) + ( | p ¥ r ( i ) F 0 ^ 1 2(q) + (-3 P ^ ) " ' r ( ^ ) F, 0^ M q ) < - 3 •" • 3Since G * 0 a s 0 * <» , the coefficients of l2(q) & I -Aq) m u s t be equal in
3 " 3
m a g n i t u d e and opposite in sign yielding
2 i r F , V J . 1 2 i i
^ j\^2M)-K^^^>)^'^^ + ( | p ^ ) ^ r ( i ) F + ( | p * ) ' * r ( i ) F = o
3 ° (43)
""^ IT'S'" - '-I'^'l 5^^' • - 1 " " T ''>* P"* ƒ" '* "l'^' "'
= - r ( i ) s i n ^ p ' . ^ ( D * (44)
Kj(q) i s a modified B e s s e l function of the t h i r d kind.U s i n g (44), (43) b e c o m e s _ F */ 1 r ( | - ) F + - ^ = - (I) " p ' * ;r7r. F , (45) T a k i n g the i n v e r s e t r a n s f o r m s of (45) we have
^ ""> a 2 f
U,(0+) + I - j _ d U , ( z ) - — / V ^ ( z ) 7-^(z)dz o s i o o- - ^ 2 I C(X,)r'/'(X,)r/' r3(z)C(z)dz1 ^dX
r<^)<Vo)' •'o " Lix " J
° ° ° ^1 (46)
E q u a t i o n (46) i s an i n t e g r a l equation for the wall s h e a r s t r e s s in t e r m s of the e x t e r n a l flow c o n d i t i o n s , and the i n t e r m e d i a t e enthalpy d i s t r i b u t i o n .
5. An a l t e r n a t i v e solution for the e q u a t i o n of m o t i o n V (X) ^ dX w 80 o in e q u a t i o n (35), the equation of m o t i o n b e c o m e s
i"Jïr'«'^"^^=^'** ^ '"'
o with b o u n d a r y conditions (i) a s 0 - « H-. 0 (ii) a s X - 0 H * 0 (ill) a s 0 * 0 r ^ h* (z) r^ V ( z ) r (z) 2 r , , H = U, (0+) + ] - ~ - d U , ( z ) - 2 j — dz ^ 0 + 0(0'^ o S i o o o => H , ( X ) + H,(X)0 . ftCIn defining H(X,0) it i s a s s u m e d that —- i s given i t s wall v a l u e and i s t h u s a 80
function of X only.
U s i n g the o p e r a t i o n a l t e c h n i q u e s of the p r e v i o u s s e c t i o n , (47) b e c o m e s
80 which h a s the solution
H = ( I p * ) ' 0* r ( i ) I 2(q) H, + ( I p ^ ) " ^ 0^ r (f-) l2(q) H .
1 3
-and s i n c e H * 0 a s 0 * • the coefficients of the B e s s e l functions m u s t b e e q u a l in m a g n i t u d e and opposite in s i g n . T h u s H , = - d p * ) ' ' ^ ^ ' ^ H (48)
FTT)
T a k i n g the i n v e r s e t r a n s f o r m s we obtain -^ h * ( z ) . , ^ V (z) rlz) U ^ ( 0 + ) + J - ^ d U ^ ( z ) - 2 J o S 1 o o w w dz-T^f^ r C(X,)r;/-(X,)rfV(z)C(z)dzT dX,
o o o X 1 which i s i d e n t i c a l with equation (46).6. T h e wall s h e a r s t r e s s
We now t r a n s f o r m equation (46) for the wall s h e a r s t r e s s into i t s c o m p r e s s i b l e f o r m b y u s i n g r e l a t i o n s s t e m m i n g f r o m the S t e w a r t s o n - I l l l n g w o r t h t r a n s f o r m a t i o n (14) 3 v - l HV r- -, 2 ( y - l ) V 1 ^ = r m . ( x ) | w h e r e m.(x) = 1 + "V"'^ M*(x) dx L ' J ' 2 1 3y -1 U,(X) = a^M,(x) ; V^(X) = ^ | L j y „ i / < > ' " ^ > (49) . ^ . ^w<^^ ^ V. n I ^ P Q ^ W ^ W T (X) = _, , . m, w h e r e C (x) = C ^ ( x ) "1 w p^p^M^
C o n s i s t e n t with the p r e v i o u s a p p r o x i m a t i o n s we put C = C. E q u a t i o n (46) b e c o m e s X , * / V , x . . 2 y - l h ! (z) . -I „ /• p , . . v . . r ^ ( 2 ) - ^ -r m dz z) 1
t
^ h (z) . -I - f p V T 2 . 3 ' + ~ r(i)(/^„p„)= We define a wall s h e a r s t r e s s p a r a m e t e r t (x) by (50) a" a "a and an injection p a r a m e t e r f (x) by t (x) = T (x) ( x , i f (51) W W \ / p M u / ^ ' a a a ' W f (x) = m / x , V ; m = p V (52) W I / p li u ) ' W W \ a a a /w h e r e the suffix a r e f e r s t o a n a r b i t r a r y r e f e r e n c e condition in the e x t e r n a l s t r e a m and m ( x ) i s the m a s s flow of injected g a s p e r unit a r e a .
F u r t h e r m o r e , m 2 y - l a a a a o / y - 1 , ^ „ . r = —JfT- / na (53) pu a.' /i T / a o^o o o a and 2 - 5 X L o o o J ^ '^o a ' ' S u b s t i t u t i n g in (50) we obtain „ _. Nf(o) f^ h*(z) / M N Z ) . f^ f (z) t (z) . m 7 dz " s i ^ M " ^ o a ^ a
,i r- V' (X,) " ,.„ , ^ ^ r f" C*(.).*(z) , m r'^"--^' r *
M O Si ^ M a a 2.3 = TTa) i Zr^^i \ \ i n / >- x % \ n i _ o X C (x ) a 1 z ' 1 1 a 1 (55) P P M , w h e r e C = — ; m (x) = 1 "t^-— M*(x) a p p u a 2 a •t a a M TIf we put C equal t o i t s w a l l v a l u e for a i r injection, i . e . — = — , equation (55) a w
i s i d e n t i c a l - w i t h S t e v e n s o n s equation B . 6 (Ref. 8). F o r the i m p e r m e a b l e wall f » 0 in which c a s e (55) b e c o m e s the s a m e a s L i l l e y ' s equation 30 (Ref. 7).
E q u a t i o n (55) c a n b e simplified b y a p p r o x i m a t i n g t o the v a l u e of the i n n e r i n t e g r a l in t h e second t e r m on the r i g h t hand s i d e by w r i t i n g
^ 1 X F ( z ) d z " (x - X,) F(x) T h e equation for t (z) b e c o m e s 2 y-1 M ' ( O ) f- h*(z) , M ' ( Z ) , t*- f (z) t (z) / m ( z ) / " ^ dz 1 r h ; ( z ) M-(z) J l^^z) t ^ ( z ) m ( z ) ' 2 M o s i ^ M ' ^ o ^ « C O a ' a , .5y -3 •^ ? / ' ( x ; ( n i , ( z ) / m J < V - l ) o x f ( x - X , ) » C^(z) (56)
An a l t e r n a t i v e f o r m of the wall s h e a r s t r e s s equation can be obtained by w r i t i n g (45) a s
15
-^ (Ip-^)
i >
-r(f-)
F 1r(i)(|,
r ( i ) : ' ' P - * . F, o r equation (48) a s H =r(5-)
'
T a k i n g the i n v e r s e t r a n s f o r m s of e i t h e r e q u a t i o n , we obtainr (X) = <Vo)^
W —J3=*
r'(f)
^1 h* (z)1 * ƒ ( 4 ^<^> "t<^> '^o' ^ [^^""'^ ^L ^^ "^^'^1
X X _2— ƒ T (X ) V (X )( f C(z) T Mz) dz ) ' dX.1
M J W I W I N J W / 1 j (57)R e v e r t i n g to the c o m p r e s s i b l e flow c o - o r d i n a t e s ( x , y ) using the r e l a t i o n s (49) and i n t r o d u c i n g the s h e a r s t r e s s and injection p a r a m e t e r s defined in (51) and (52), equation 57 b e c o m e s x^ C (x) a t (x) = —r
3 ^ r d )
• M ' ( X ) ^ X X ^ 2 y-1 f*' h*(z) , M ' ( Z ) , " I / t (X ) , m r-^ f (x ) m \ x ; r • n \7.i m vz; r i \x i x m ;T ^ "^ i ~h~" ^V~i»~J ' i X c (x) Vi^r j
M O S I ^ M ' o i a i ^ a ' I- a a -^ (58)o r , a g a i n a p p r o x i m a t i n g to the i n n e r i n t e g r a l s , we have an e x p r e s s i o n for le
2i
the w a l l s h e a r s t r e s s which l e n d s i t s e l f to an i t e r a t i v e evdluation.
2 j ^ k i v - i t <x) x^C (x) a — I
*' 3' r d )
m\ 'I
i / « y / 3 ( y - l ) (x - x ) \ a / 7y-3 r M ' ( x ) / ^ h*(z) . M ' ( Z ) 1 r ' t M x ) f (x^) . m ^ < ^ ' ^ ^ -. a a a ' 1 « (59) If conditions in the free s t r e a m a r e known t o g e t h e r with the knowledge of t h e injection m a s s flow and i n t e r m e d i a t e enthalpy, equation (59) can be solved only when C i s known. T h i s r e q u i r e s a solution to the diffusion equation, for with the c o n c e n t r a t i o n of foreign gas d e t e r m i n e d , it i s then p o s s i b l e to c a l c u l a t e the v a l u e s of d e n s i t y and v i s c o s i t y at the wall and from t h e s e to obtain C .7. An a p p r o x i m a t e s o l u t i o n of t h e diffusion equation A g a i n t a k i n g a v a l u e of C and S c h m i d t n u m b e r i n d e p e n d e n t of 0 , the diffusion e q u a t i o n (26) b e c o m e s 9^ + V (X) ?^ = . ^<^) « \ 8 0 / 8X w ' 80 o Sc 80 \ 80-T h e v e l o c i t y n e a r t h e w a l l h a s p r e v i o u s l y (eqn. 34) b e e n t a k e n a s / 2 r (X) 1 U = — ^ ! L _ 0^
l e a d i n g t o the diffusion e q u a t i o n in the f o r m 8 c C(X)
8X p Sc o
h'.'^K ^ (•* IF) • - ^««> If '«<»
for which the boundary conditions a r e (i) as 0 * » , c -. 0 (ii) a s X - . 0 , c - » 0 ^ (ill) as 0 - 0. c = c^(X) + c'(X)(^ T~h)) ^* "^ • • • w 8 c t w h e r e c'(X) = r r ; i . e . at the w a l l . 8Y X 'Y=0 J Sep ^J' If t = / Ö 2 T ( Z ) M d z , (60) b e c o m e s ' Sep \j w o o o 8 8t c ( t . 0 ) -e r -e F^(X)
aT(^^
8 c 8 0 ; = 0 ' V^(X) c'(X) p^. 2 r ^ ( X ) C(X) In t h e n o t a t i o n of t h e L a p l a c e t r a n s f o r m p c-8i(^
1 acV
8 0 / F,(p) F^(t) Sc T h e s o l u t i o n of (61) (62) (63) 8 / i 8cN is given by Lighthill as i" 1 4 i 3 c = a 0 * I i ( q ) + b 0* I i ( q ) ; q = - p * 0^ - 3 Ï J w h e r e a and b a r e c o n s t a n t s t o be found f r o m t h e b o u n d a r y c o n d i t i o n s . U s i n g L i g h t h i l l ' s s o l u t i o n f o r the h o m o g e n e o u s equation we c a n s o l v e (63) by the m e t h o d of v a r i a t i o n of p a r a m e t e r s .17 L e t t h e s o l u t i o n of (63) b e c = P 3 ( 0 ) c , + P^(0) Cg (64) w h e r e c = 0* Ii_(q) ^ 3 i c^ = ,p* 1 Aq) P j and P a r e d e r i v e d f r o m t h e e q u a t i o n s d P 0 0 " F ( p ) ' = - . i 1 _ (65) d ^ d P c 0" F (p) 4 4 d0 _ , _ _ _ , ^ c c - c c 1 2 1 2 w h e r e t h e p r i m e ' d e n o t e s p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t to 0 . • - - 3 Now c ' c - c c ' = - 0 * . 7 — - s i n r r / S 1 2 1 2 ^ 2 i r T h u s , f r o m ( 6 5 ) , 2 . r* P , = - ^ ^ ^ f f * F ^ ( p ) M q ) d 0 3 s i n — J 3 X .«in 3 o 2 ^ > (66) V j 0"^ F^( 3 s i n - J ^ 3 and t h e s o l u t i o n of (63) i s 1 f'^ 27rF(p) , i /•"'' 2^^<P) 4 E ( p . 0 ) = 0* I i ( q ) ^ 0 * I i ( q ) d0 - 0^ U q ) ƒ ^ 0 ' I ^(q) d0 "=> o 3 s i n ^ l Ï » J o 3 s i n f "3 i i + A 0* I i ( q ) + B 0* I i ( q ) (67) ' "3^ ' 1 w h e r e A and B h a v e t o b e d e t e r m i n e d f r o m b o u n d a r y c o n d i t i o n s . T h e b o u n d a r y c o n d i t i o n a s 0 ., 0 c a n b e w r i t t e n in t h e t r a n s f o r m n o t a t i o n a s 1 c ( p , 0 ) = F ( p ) + 2 0 " F ( p ) (68) 9 o w h e r e F (X) = c (X) s w w ^o
F r o m (67) and (68) in the limit a s 0 - 0 A ' = c (p.O) = F (p)
r d ) ? * (1)^
1/ 3 I <69) ^1 P <^^ 4 8-—
^ - ^
B;P' <p' '^^ = F > )
F u r t h e r m o r e a s 0 •» o», c * 0 and hence the coefficients of I , (q) and Ii(q)
"3 s
m u s t be equal in magnitude and opposite in sign, i . e .
2ii¥ <p)
A + B +
' * 3 sin ^
3 o 3 o
Now fTli<q) - I _ i ( q ) ) d q = -iSïtMÏL einir/Z
o
and therefore
A, + B,= I F,(p)p'^ r d ) r(i)
o r using (69)F^(p) = d)^r(i)p"^ F,(p) - d ) ^ p | | ] - P'^ F(p) (70)
Taking the inverse t r a n s f o r m s of (70) gives an equation for the wall concentration of foreign gas ~ i m ) o T ^ n T ^ V - ' v Sc w' ; 1 c (X) = _ a H a ; O c'(X.) C(X ) / [ ^ , , X
w „s m ; o . 1 (X) ^ ^ x
w < 'J,
r(
3) \ 3 < b / •'n ^'^ ^•'x Sc w' ' / 1 3y - 1 p^ P , , ^ , 8c 8c dy ,. , "o ^ 2(y-l) NOW e ( X ) = - ^ = 5 7 dY = ''^''\-V~ "^ Y=0 •' w oand, using the transformations (49) from (X, Y) to (x,y) co-ordinates, equation (71) becomes
1 X
/ ^ ^^o*'o^' TO) I i h c ' ( x , ) "oP, y / 2 ( y - l ) C'(x.)
/ r c W ^ ) d z y i ^ ,.«^ ^ ƒ' p^p,
^ ^ x . , _ y / 2 < y - l ) ^ ^^1 - f ( f ) V 3 ^ y ''o - s f p ^ T ^ <^i>
U . , „ y / 2 ( y - l ) ; ^^1
(72)
19
-It i s now n e c e s s a r y to d e t e r m i n e a n o t h e r equation for the c o n c e n t r a t i o n g r a d i e n t c'(x). T h e diffusion equation (6) can be w r i t t e n
_8_
8x
(puc) + ±ip.c) - i-i^'/)
8y 8y VSc 8 y / which a t the w a l l b e c o m e sp v c w w w
( ^ 9 £ ) = p . v (73) \ s c a y y 1 i
for the injected s p e c i e s , and
p V (1 - c ) - ( ^ ^ ( 1 - c)) = 0 (74) WW w VSc 8 y /
-' y = o for the s t r e a m g a s .
Adding (73) and (74) shows that m = p V = P , v .
W W 1 i
i . e . the m a s s flow n o r m a l to the wall at the wall in the b o u n d a r y l a y e r i s e q u a l to the m a s s flow t h r o u g h the w a l l .
S u b t r a c t i n g gives
Sc
(75) 'y=0 ''w
'''' <'éL
-
^ ^
'=ww
-"
E l i m i n a t i n g c'(x) in (72) and i n t r o d u c i n g the wall s h e a r s t r e s s and injection p a r a m e t e r s t
c o n c e n t r a t i o n i s
p a r a m e t e r s t and f defined by (51) and (52) the equation for the wall y / 2 ( y - l )
''M)
1 rd) ['' W Dlfw^O Sc /"^i X
' w ^ a ' , r ^ A z ) t ^ ( z ) / m ^ ^ ( y - i ) 4 X V x , S'^ 1 ^ ^ " ^ / / ' 3 ^ r d ) ^ o ^ri i ,i(f c'(z) t*(z) ,m y'^^^~'^ y t
X, Sc ^ V m ^ y (76) a n d , a p p r o x i m a t i n g t o the i n t e r i o r i n t e g r a l s , 2 y , , 1 r d ) / • ' ^^:<^> Ü - - w < ^ ) ] s c ^ (^M^) ' ' , c (x) = —T z^rh I —2 r-2 • - 2 • V J dx3^ ^<^> L x > - x ) V ( x ) Jl^^) ^^. ^
1 1 w < a 'f' y^^C^-^w<^jl Sc^ /-(x).3(y-l)
ƒ " 1 1—I • "I— V ~zz—J '^ J^ ^ 3 / „ _ ^ \ 3 + 3 / „ \ r-3l^ X \ m / 3 ' ' r d ) ' o X,' (X - x / t ^ ( x ; C»(x^) . ^ „ )which with equation (59) m a k e s p o s s i b l e the i t e r a t i v e p r o c e s s to d e t e r m i n e the wall s h e a r s t r e s s .
8. An a p p r o x i m a t e solution of the s t a g n a t i o n e n t h a l p y equation
C o n s i d e r i n g the function C, the L e w i s n u m b e r and P r a n d t l n u m b e r to be dependent on X only, being obtained f r o m the i n t e r m e d i a t e e n t h a l p y , the e q u a t i o n f o r the s t a g n a t i o n enthalpy (28) i s
o- 80 V^80/ ^w80 a»TïIzzï~3^ ®*^ ^'''^
^'^-'"^/"°^'Vlf)
8S 8X - V C(l -Ö-) o- U 1 + y-1 u ; 80 (78)with the b o u n d a r y conditions
(i) at 0 = <», S(X,oo) =0 (ii) a s X * 0 . S * 0 (ill) a s 0 * 0, h*(X) S = 1 - s + £ . Ü2. Q (X) h h /u w Si s i w ( X ) (79) w h e r e the r a t e of h e a t t r a n s f e r f r o m the wall to the b o u n d a r y l a y e r i s , in the X , Y c o - o r d i n a t e s ,
w (
Q....X. = - I K. ^^__^
T h e r i g h t hand side of equation 78 can b e w r i t t e n
s i n c e
^^ L80 \''''80/ 2a* \ o" / ^^ V ^ ^ / J
o
1 y - 1 V 9..' V
8c
Consider UA-— near the wall. U has been assumed (eqn. 34) to be
80
(X) *
r-^*)
0 I and A h a s b e e n defined in equation (8) a sA = L e - 1 (h^ - h^)
- 21
, .ifl-i.Uhs-!Ï^z
V
\ S i2 a!
w h e r e , n e a r the wall, the concentration of injected gas i s given by equation (60) 2 p .
c = c
^(X) + c ' ( X ) ^ ^ j 0 J + O(0/*)
and S i s given in equation (79)UA?^= t ^ ^ ^
h - h * s i 8 / T ( X ) . oi
w o wExpanding log c a s a power s e r i e s ,
UA 8c ^ Le 80 "^ c
-(^)[*f(^)k-:-4"')
^ ' ^ o - ' U w ^ w ' a ^ w ^ w ^ w ^ o w N w / w ^ 2a "^ J and thus'Uis) . t - i (^) \±<L . 2(S1% . K* A „51,-4 , „,.,
Also _8_ 80("H) •-(!ƒ(#
(81) (82)Substituting from (81) and (82) into (80), the right hand side of (78) b e c o m e s , , n e a r the wall,
'71'
t (^)0l*
+ 0(1) (83)8X " . o- 80 > C(X) a* , / T v ' / o ^' _i_ 1 -<r / ' w y i h a ^ O ' 2v C(X)
* - 5 — . . ,
si W W 2a o V C(X) a , / T vVz 1. _o 1 1
-<r
/
"^w
V ,
-i
^ 2^7" • T • ~ ^ ( ; r j ^ (84)
s iT h e l a s t t e r m in (84) i s put z e r o b y L i l l e y for the I m p e r m e a b l e wall on the a s s u m p t i o n that the r e c o v e r y enthalpy i s independent of the wall t e m p e r a t u r e d i s t r i b u t i o n .
T a k i n g the a p p r o x i m a t e f o r m for U given by equation (34) and putting X
'I
t = / - ^ <2M„r (z))^ dz ( P ff o w o o and V (X) /^ „ , \ / , ^ . ->cr«/u„ Q „ ( X )V - i r V l l f L e - A c'(X)-]°-^o ^v^l'
% , - " I v ^ C ( X ) "V a ; c ^ ( X ) j h ^ ^ ^ ^ - r ^ ( X )^ A ^ ^ ^w V'^w^^V V ^^ « 2 a' V
^ _ ^ a ^ ^ ) 2h « /< Si a o othe a p p r o x i m a t e s t a g n a t i o n enthalpy equation (84) i s
I f " - * ' - i ^ G ' l f ) •-^w,«'*'*
w h i c h , in t h e notation of the L a p l a c e t r a n s f o r m , b e c o m e s (86)P^-8TG*a-f) = ^>>*"'
(87) w h e r e F (t) = - V (t) T W 1T h i s e q u a t i o n i s s i m i l a r to the t r a n s f o r m e d diffusion equation (63) and i t s solution i s , s i m i l a r l y ,
S(p,0) = 0 * I 1 - ^ / F (p) 0 " ^ IM) d0 - f Ii(q) - ^ / F, (p) 0"^I i(q)dq "'' 3 s i n | •'o ^ » 3 s i n | •* o "=» i i
+ A a 0 * I i(q) + B,0* I i ( q ) (88)
23
-where A and B have to be determined from the boundary conditions. Now, as 0 * 0,
h*(X)
s i
and thus, from (88)
A rc ^1 = ^ (p. 0) = F,(p)
rd)p''d)*
Also, a s 0 * 0 ,i
a c / M X Q ( X ) 1 8S ^ a / '^o \ ^w , - i (89) S l N W / W and t h u s , from (88) 1 / ^ 2B/^
(i)^
U* ^ S(p. 0)1 = F,(p) (90)
rd) L "^ J0
F u r t h e r m o r e a s 0 * » , S * 0 and hence S(p, ") * 0. This implies that the coefficients of I i(q) and Ii(q) in (88) must be equal in magnitude but opposite in sign. "* *
i . e . A, + B^ = I p ' * r ( ^ ) r d ) p ; ( p ) o r using (89) and (90)
F,(p) = d ) ^ r d ) p ' * F,(P) - d ) " ' P* f l l j F ( p ) (91) Taking the Inverse t r a n s f o r m s of (91) we obtain an integral equation for the
r a t e of heat t r a n s f e r in the X, Y co-ordinates
— Ó (X) =
-u w o-r(i) w* s r J o ^ - ' x <r w' / L ^ g / ^ i ) J
\ 3 p / ff r(3) w J <r wi ' w 1
Now
^w<->=(-wi),^,
(• ^ ly\.o Wy^o
= Q ^ x ) P P a p o ' l 0*^0 and hence " w P o ^ P ir ^w<^^ ~- "CUT "^'
w (93)Therefore, substituting for V (X) in (92) from (85) and reverting to the
compressible flow co-ordinates (x,y) using equations (49) and (93), the equation for the heat transfer rate from the wall to the flow is
Q ^ ( x ) (3iU^p^)^ C*(x)h
7 ^ ' <rr(i)mr/2(V-l) J^\J^
TJf^-a 1 w o X crm L 81 _| 2 C^(x) h S i "" "^ C * ( z ) T * ( z ) d Z "* > t ^ _ y / 2 ( y (3|UP ) * o - m ; "o o '_ rd) f (f ^J^K^'^^ \
-1) n¥) I\L . v/2(y-i) ;
m y / 2 ( y - i ) 1 o - 2a-o - Q . (x ) o x o" m^ M c ' ( x )x* 5, / fi c {x ) \ * /^ i \ h - h - u / » / \ , /^ w ' 1 \ (Le - I \ Si s 1 w 1 + ^ « w < ^ ' r L e _ : _ i ^ w " ' < ^ ^ ^ ^ •) 2 r ^ ( x ) l- °- • % < ^ i ) ' ^ ""^ -> C ^ < - i ) ^ 1uV2
It has been shown in equation (75) that c'fx) M w c^(x) = m S c
['-^>1
Hence/ c'(x) ^' /Le - 1 N \ r ^s - "V^ , ^Qw^> r Le - 1 %"'^> 1
{^w n ^ ) ) (~~"^~') — ï T " ; — oTVT [""^^ " s ^ T " " " ^ ^ J
\ w / \ / T (x ) 2r *(x ) w 1 w < w ' w and e q u a t i o n ( 9 4 ) b e c o m e s ^ ^ ' N l - c ) ' ( h - h*) + 2 i ^ ( l - Le - c ) cr w 1 s c w Q w < ^ ) w 225 -Q ^ ( x ) O M -Q P -Q ) ^ C*(x)
T^(x) a r ( i ) m ;
2 C * ( x )I [^ f f"" C*(z) r^(z) dz \"^
i^„y/2<y-ï) i A [ 7 3 7 ^ ) '^ ^ [ ^ X ^ - ^ i ]
o x . cr m 1 1 X X 1 i i 2 / (3M p ) V m y / 2 ( y o o_ rd) f ( f C^(z)rJ(z) dzY^ mf'
-1) rlT) i^ Vi —-72(71) / - x
y / 2 ( y - i ) ° '^1 c r m j ' / 2 ( y - l ) C='(x,) f / n i S c \ L e - 1 ,, «., , * . ^ m S c . , . . ^ w i (-; 1 — j . (1 - c ) (h - h ) + (1 - L e - c ) — y — L \ c / 2/ V w i s e w „ _f, ^ ^ w ' crr (x ) w 2T '{X, w » w * Q„.(x.) ) 1 -o- '/> - ^ • ^ F - ^ w' ( x . ) ] d x ,
We now put Q (x) in a modified form s (x) defined by
w w
s (x) = Q (x)\ xjp M u 1 w w L a a a J
(95)
(96) and i n t r o d u c e the n o n - d i m e n s i o n a l w a l l s h e a r s t r e s a and injection p a r a m e t e r t t and f defined in e q u a t i o n s (51) and (52). In t h i s m a n n e r (95) b e c o m e s ,
W W 3 ( X ) W
t^(x) o - r d )
a y / 2 ( y - i ) , ' ' , , ' ' A, , A , , y / 2 ( y - i )x,(r.) I (/ ^^{<!f"'"4'
\ m / o ^ x , J v m / ' ^ 1 ' 1 a" z 1r -, 2C^(x) , , . , . / m > ' f < y - ^ ) r ^ r ^ c W ^ ( z ) / m \ / 2 ( y - l )
L s ' « ' J ^K r d ) \ m / J ^ \ 4 ^ ^ i \mj J
y / 2 ( y - l )0 •
2 C^(x) a —r-3 V ^ Sc\= o X , 0" z f Sc s . ( x )( - L ^ . ^ .| ( - ^ 1 - ) i ^ 2 ^ . (1-c )'(h -h*) + ^ ! L _ (i-Le-c ) J V ^
. t '\x^ u* dx w ' a J 1 1 -or 2o-(97) A p p r o x i m a t i n g t o t h e i n n e r i n t e g r a l s 1 1 „ y / 2 ( y - l ) .X J / u J s (x)y,^vy-i; ^ ^,12^ (m(x)/mJ/^<^"^>
V ^ ' 3 ^ x ' i , / " ^ a \ /• ^ '^ ^"^/^^^"^a^ ^ r * . , ^ -I
+ _22L_ c ^ ( x ) ( - ^
3 a \ m3 V <
f SC,2 y / 2 ( y - l ) I Xrd) f
o-^C (x ) t"*(x ) , m a ^ (x -x^)^ x^ 6 ( y - l )ITT-.
V - ;
{(
w w W 1 f Sc *v . W ^ ^ " ^ (1-C )'(h - h ' ' ) + W 1 s c s (x ) ( 1 - L e - c ) — ï ^ — ! - + w^ . i l £ t ^ U , ) u '
20" w 1 a 2 t ^ (x,) w ' dx (98)Thus the heat transfer rate can be obtained by an iterative process from the given external flow conditions once the wall shear s t r e s s and wall concentration of injected gas are known.
9. Numerical solutions for the wall shear s t r e s s and heat transfer rate The wall shear s t r e s s and the heat transfer rate must now be found from equations (58) and (98) with (77) using an iterative p r o c e s s . Stevenson' ' was able to obtain, for air injection, relations between f , t and Nusselt number in precise form when it is assumed that the free stream speed, the wall shear s t r e s s and the wall temperature vary as some power of x. This is possible since the viscosity in the boundary layer can be related to the temperature only. In the analysis presented in this paper such a treatment is not possible since the density and viscosity of the boundary layer are dependent also upon the concentration of the injected gas.
To a s s e s s the accuracy of the method the value of t has been calculated for hydrogen injected into an incompressible layer with zero heat transfer and zero p r e s s u r e gradient. The injection velocity is assumed to be proportional to X ' and C is given its value at the wall since, in the absence of complete
CL
concentration profiles, it is not possible to obtain its value elsewhere. The relation of t / t to f is compared in Fig. 2 with the result due to Eckert w . .wo w and Schneider . It is seen that the difference between these results is approximately the same as that between the exact results for air injection found by Donoughe and Livingood^ ' and the approximate results obtained
(8)
by Stevenson . The agreement between the two solutions for hydrogen injection can be improved if the value of C is increased by some 40% above its wall value. Values of t calculated on this basis are given in curve 3
w
of Fig. 2. Even closer agreement would be possible if the percentage increase of C was changed with increase of the. injection parameter. Using the curves
PHl p H obtained from the concentration profiles of Ref. 4 it is seen that the
required 40% increase in C is obtained when rj = 0.8 approximately.
F o r helium injected into the laminar boundary layer on a cooled wall at M = 6, the results of the present paper are compared in Fig. 3 with those of
(•\(i\
Korobkin obtained by considering the variation of the molecular weight of the mixture. Since Korobkin's results are approximate it is not possible to a s s e s s , in this case, the e r r o r of the method at M = 6 or to estimate the alteration necessary to the value of C .
The process of iteration is started by substituting in the concentration equation (77) the value of t for air injection corresponding to the chosen value of f and the external flow conditions. Such substitution gives an integral or the form
/
m-1 ,, .n-1 , X (1 - x) dx
27
-which is the Beta function, the value of -which is immediately obtainable
from tables of the gamma function. The resulting value of the wall concentration is then used to determine the first approximation for wall values of density and viscosity by methods given by Hirschfelder et a P ^ ' . These density and viscosity values a r e substituted in equation (59) to give a second approximation to t and in (98) to obtain a second approximation to the heat transfer r a t e . The higher o r d e r approximations a r e obtained similarly. It was found that five iterations gave an a c c u r a c y of convergence of b e t t e r than five per cent. In Fig. 2 the values of t for helium injection a r e also given. In this calculation the values of viscosity, Prandtl number and t h e r m a l conductivity were obtained from tables recently calculated by E c k e r t , Ibele and Irvine^ "*'.
To illustrate the o r d e r of magnitude of the reduction in skin friction and heat t r a n s f e r rate to be obtained at supersonic speeds with foreign gas injection,
the ratio t / t has been calculated for M - 4 with zero heat t r a n s f e r (Fig. 4)
w wo 1 ^ and for the cooled wall, T =T (Fig. 5). F o r the cooled wall the ratio of heat
W 1
t r a n s f e r r a t e s Q / Q is shown in Fig. 6. In each case the p r e s s u r e gradient
W wo ^ 5 j is z e r o . The corresponding exact r e s u l t s for air injection due to Lew and Fanucci
and Stevenson's approximate r e s u l t s a r e shown for comparison. 10. Conclusions
The equations for foreign gas injection into a compressible steady laminar boundary layer have been solved approximately for a r b i t r a r y p r e s s u r e gradient and a r b i t r a r y distributions of wall t e m p e r a t u r e and injection velocity.
It is shown that substantial reductions in skin friction and heat t r a n s f e r rate can be obtained by injection of a light gas instead of air,
11. Acknowledgements
The author wishes to record his indebtedness to Mr. G. M. Lilley and M r . T. N. Stevenson whose methods (Refs. 7 and 8) have been closely followed in the analysis presented in this paper.
12. References 1. Craven, A.H. 2. Libby. P . A . , Kaufman, L. , Harrington, R . P . 3. Smith, J . W . Eckert, E . R . G . , Schneider, P . J . F a u l d e r s , C R . 6. Lighthill, M . J . 7. Lilley, G.M. 8. Stevenson, T . N . 9. Stewartson, K. 10. Illingworth, C R . 11. Fage," A . , Falkner, V.M.
Boundary layers with suction and injection. College of Aeronautics Report 136, 1960. An experimental investigation of the isothermal laminar boundary layer on a porous flat plate.
J o u r . A e r o . S c i . , vol.19, 1952, pp 127-134. Effect of diffusion fields on the laminar boundary layer.
J o u r . A e r o . S c i . vol. 21,1954, pp 154-162 and pp 640-641.
Effect of diffusion in an isothermal boundary layer.
J o u r . A e r o . S c i . vol.23. 1956, pp 384-387. A note on laminar boundary layer skin friction under the influence of foreign gas injection.
Jour. Aero/Space Sci. vol. 28, 1961, pp 166,167.
Contributions to the theory of heat transfer through a laminar boundary layer.
P r o c . Royal Society, Series A, vol.202, 1950, pp 359-377.
A simplified theory of skin friction and heat transfer for a compressible laminar boundary layer.
College of Aeronautics Note No. 93, 1959. The laminar boundary layer with injection through a permeable wall.
College of Aeronautics Report 145, 1961. Correlated incompressible and compressible boundary l a y e r s .
P r o c . Royal Society (A), vol.200, 1949, pp 84 - 100.
Steady flow in the laminar boundary layer. P r o c . Royal Society (A), vol.199, 1949, pp 533-558.
On the relation between heat transfer and surface friction for lamiinar flow.
29 -12. Donoughe, P . L . , Livingood, J . N . B . 13. Hirschfelder, J . O. , C u r t i s s , C . F . , Bird. R . B . 14. E c k e r t , E . R . G . , Ibele, W . E . , Irvine, T . F . 15. Lew, H.G. , Fanucci, J . B . 16. Korobkin, I.
Exact solutions of laminar boundary layer equations with constant property values for porous wall with variable t e m p e r a t u r e .
NACA T . N . 3 1 5 1 , 1954.
Molecular theory of gases and liquids. John Wiley and Sons Inc. 1954.
P r a n d t l number, t h e r m a l conductity and viscosity of air-helium m i x t u r e s .
NASA T . N . D - 5 3 3 , 1960.
On the laminar compressible boundary layer over a flat plate with suction o r injection.
Jnl. Aero. Sci. , vol.22, 1955, pp 589-597. The effects of the molecular p r o p e r t i e s of an injected gas on compressible laminar boundary layer skin friction and heat t r a n s f e r .
U . S . Naval Ordnance Laboratory Report 7410, 1961.
0 . 7 O. 6 o.s 0 4 O 2
y
^ ^ ^ ^ = = = = ^ / ^ /y ^
^ ^ - ^oX
^
oX
/ i\
y /
o V ""
, . V . y / " ' A «FIG. 1. VARIATION OF ^ IN AN INCOMPRESSIBLE BOUNDARY LAYER
° ° (4)
WITH HYDROGEN INJECTION (from Eckert li Schneider' )
WJECTED GAS SOURCE ® HYDROGEN ECKERT ( SCHNEIDER'' @ HYDROGEN PRESENT RESULTS
HYDROGEN PRESENT RESULTS (MODIFIED) @ HELIUM PRESENT RESULTS
D O N O U G H E I L I V I N G O O D S T E V E N S O N " INJECTED SOURCE GAS 0 AIR KOROBKIN d) AIR STEVENSON* ( D HEUUM KOfOBKIN 0 HELIUM PRESENT RESULTS
FIG. t. EFFECT OF FOREIGN GAS INJECTION ON SKIN FRICTION
(Incompressible flow, zero heat transfer, v proportional
«o x " ' )
FIG. 3. EFFECT OF FOREIGN GAS INJECTION ON SKIN FRICTION ( M . 6 . T / T . 0 . 5 , V proportional lu x~')
INJECTED CAS SOURCE INXCTED GAS SOURCE
>
O 9
(T) AIR LEW t FANUCCI ® AIR STEVENSON ' ( D HELIUM PRESENT RESULT 0 HYDROGEN PRESENT RESULT
O ^
® AIR LEW i FANUCCI ( D AW STEVENSON*
( D HELUIM PRESENT RESUa ® HYDROGEN PRESENT RESULT
FIG. 4. EFFECT OF FOREIGN GAS INJECTION ON SKIN FRICTION (Uniform Injection velocity, M "4, zero heat transfer)
FIG, 5. EFFECT OF FOREIGN GAS INJECTION ON SKIN FRICTION (Uniform Injection velocity M, - 4, T ' T , )
l O ''^ 0 9 0 6 0 7 0 6 0 5 0 . 4
1
* \ \ \ \^._A,
INJECTED GAS ® AIR (3) HELIUM (3) HYDROGEN V 3) 0 1 0 2 0 3FIQ. «. EFFECT OF FOREIGN GAS INJECTION ON HEAT TRANSFER HATE (Uniform Injection velocity, M, • 4, T • T, )
