O F D I F F E R E N T GASES IN A PRESSURE GRADIENT
T h e s i s by Manuel R. Rebollo
In P a r t i a l F u l f i l l m e n t of the R e q u i r e m e n t s F o r the D e g r e e of
Doctor of P h i l o s o p h y
C a l i f o r n i a I n s t i t u t e of Technology P a s a d e n a , California
1973
ACKNOWLEDGMENTS
The a u t h o r w i s h e s to e x p r é s s h i s a p p r e c i a t i o n to h i s a d v i s o r , P r o f e s s o r Anatol R o s h k o , for h i s guidance and d i r e c t i o n of the p r e s e n t r e s e a r c h . In a d d i t i o n , the t i m e l y a d v i c e of P r o f e s s o r Donald Coles i s gratefully acknowledged.
Special thanks go to D r . G. L . B r o w n for h i s a s s i s t a n c e during the e a r l y p a r t of the e x p e r i m e n t s . The author a p p r e c i a t e s the c o o p e r a t i o n of M r . J i m é n e z Sendin on the d e v e l o p m e n t of the a s y m p -totic b e h a v i o r of the n u m e r i c a l solution.
It i s a p l e a s u r e to thank M r s . G e r a l d i n e K r e n t l e r for typing the m a n u s c r i p t .
The author i s indebted to the California I n s t i t u t e of T e c h -nology, the E u r o p e a n Space R e s e a r c h O r g a n i z a t i o n , the National A e r o n a u t i c s and Space A d m i n i s t r a t i o n , and the Spanish G o v e r n m e n t for their financial a s s i s t a n c e . The r e s e a r c h was s u p p o r t e d by the D e p a r t m e n t of the Navy, Office of Naval R e s e a r c h .
ABSTRACT
An a n a l y t i c a l and e x p e r i m e n t a l study h a s b e e n m a d e of the t u r b u l e n t mixing l a y e r in a p r e s s u r e g r a d i e n t . T h e o r y p r e d i c t s the p o s s i b l e e x i s t e n c e of e q u i l i b r i u m flows, and this was c o n f i r m e d e x p e r i m e n t a l l y for t u r b u l e n t s h e a r l a y e r s b e t w e e n s t r e a m s of h e l i u m and n i t r o g e n .
T h e only c a s e for which s i m i l a r i t y is p o s s i b l e is for
p2U2 2 = pi Ui2 , s i n c e then P2 (x) = P i (x). T h e s e e q u i l i b r i u m flows a r e
(y x d U
of the f o r m Ui ~ x and 6 ~ x , w h e r e a - 77—-r-3*" is a n o n - d i m e n s i o n a l
Ui dx p r e s s u r e g r a d i e n t p a r a m e t e r .
The e x p e r i m e n t a l i n v e s t i g a t i o n was conducted in the facility d e s i g n e d by B r o w n to p r o d u c e t u r b u l e n t flows at p r e s s u r e s up to 10 a t m o s p h e r e s . The a d j u s t a b l e w a l l s of the t e s t s e c t i o n of the a p p a r a t u s w e r e modified in o r d e r to s e t the p r e s s u r e g r a d i e n t .
Shadowgraphs of the m i x i n g zone for a - 0 and a - - 0. 18, at different Reynolds n u m b e r s , r e v e a l e d a l a r g e s c a l e s t r u c t u r e n o t i c e -ably different for e a c h a.
The s i m i l a r i t y p r o p e r t i e s of the s h e a r l a y e r w e r e e s t a b l i s h e d f r o m m e a n p r o f i l e s of total head and d e n s i t y . In addition, the r m s d e n s i t y fluctuations w e r e found to be s e l f - p r e s e r v i n g . F r o m the m e a n p r o f i l e s , the s p r e a d i n g r a t e , t u r b u l e n t m a s s diffusion, Reynolds s t r e s s and Schmidt n u m b e r d i s t r i b u t i o n s w e r e c a l c u l a t e d f r o m the equations of m o t i o n .
m á x i m u m s h e a r i n g s t r e s s i s 7 0 $ l a r g e r and the m á x i m u m valué of the t u r b u l e n t m a s s diffusion i s 2 0 $ l a r g e r than t h e i r a = 0 c o u n t e r p a r t s . The m á x i m u m r m s d e n s i t y fluctuations a r e a p p r o x i m a t e l y 0. 2 in both flows.
T A B L E O F CONTENTS T I T L E
Acknowledgme nts A b s t r a c t
T a b l e of Contents N o m e n c l a t u r e L i s t of T a b l e s L i s t of F i g u r e s I n t r o d u c t i o n
1. 1 P r e v i o u s I n v e s t i g a t i o n s 1. 2 Goals of the P r e s e n t Study 1. 3 E x p e r i m e n t a l T e c h n i q u e s
A n a l y t i c a l Study of E q u i l i b r i u m F l o w s
2. 1 D e r i v a t i o n of the Equations of Motion for H e t e r o g e n e o u s Flow
2. 2 H e t r o g e n e o u s T u r b u l e n t Mixing L a y e r : E q u i l i b r i u m F l o w in a P r e s s u r e G r a d i e n t 2. 3 S h e a r S t r e s s and T u r b u l e n t M a s s Diffusion
Di s t r i b u tions
2. 3a L o c a t i o n of the Dividing S t r e a m l i n e 2. 3b M a s s F l o w E n t r a i n m e n t
N u m e r i c a l Solution of the E q u a t i o n s
3. 1 Eddy V i s c o s i t y and Eddy Diffusivity Model 3. 2 Z e r o P r e s s u r e G r a d i e n t C a s e : a - 0
3. 3 P r e s s u r e G r a d i e n t C a s e : a 4 0
PAGE
í i
m
IX X I
X l l
13
T A B L E O F CONTENTS (cont'd. )
P A R T T I T L E IV. E x p e r i m e n t a l F a c i l i t y , I n s t r u m e n t a t i o n and
Equipme nt
4. 1 F l o w A p p a r a t u s 4. 2 I n s t r u m e n t a t i o n
4. 2a Static and P i t o t Tubes 4. 2b A s p i r a t i n g P r o b é
4. 3 D e s c r i p t i o n of the E x p e r i m e n t a l E q u i p m e n t 4. 3a E l e c t r o n i c M a n o m e t e r and C o n s t a n t
T e m p e r a t u r e A n e m o m e t e r
4. 3b A / D C o n v e r t e r and C o n t r o l E q u i p m e n t 4. 3c M a g n e t i c T a p e R e c o r d e r s
V. E x p e r i m e n t a l P r o c e d u r e s 5. 1 F a c i l i t y T e s t s
5. 2 P r o c e d u r e to Set Up the E q u i l i b r i u m F l o w 5. 3 S e l e c t i o n of F l o w and T r a v e r s i n g P r o c e d u r e s 5. 4 D y n a m i c P r e s s u r e T r a v e r s e
5. 5 D e n s i t y T r a v e r s e VI. P h o t o g r a p h i c I n v e s t i g a t i o n
6. 1 F l o w S t r u c t u r e
6. 2 F l o w S t r u c t u r e at Different Reynolds N u m b e r Z e r o and A d v e r s e P r e s s u r e G r a d i e n t s
T A B L E O F CONTENTS (cont'd. )
T I T L E PAGE
Data P r o c e s s i n g 47 7. 1 P r e s s u r e M e a s u r e m e n t s 47
7. 2 D e n s i t y M e a s u r e m e n t s 49
R e s u l t s 51 8. 1 A d v e r s e P r e s s u r e G r a d i e n t E x p e r i m e n t 51
8. l a D y n a m i c P r e s s u r e P r o f i l e s 51
8. Ib D e n s i t y P r o f i l e s 53 8. l e M e a s u r e m e n t s at Different R e y n o l d s N u m b e r s 55
8. Id C a l c u l a t i o n of Reynolds S t r e s s and 56 T u r b u l e n t M a s s Diffusion
8. 2 Z e r o P r e s s u r e G r a d i e n t 59 8. 2a D y n a m i c P r e s s u r e and D e n s i t y P r o f i l e s 59
8. 2b C a l c u l a t i o n of T u r b u l e n t T e r m s 60
D i s c u s s i o n and Conclusión 62 9. 1 C o m p a r i s o n with. N u m e r i c a l Solutions 62
9. 2 D i s c u s s i o n of the R e s u l t s 63 9. 2a Effect of D e n s i t y on Spreading R a t e and 64
Eddy V i s c o s i t y
9. 2b D e n s i t y M e a s u r e m e n t s and L a r g e 66 S t r u c t u r e Model
9. 2c Effect of A d v e r s e P r e s s u r e G r a d i e n t 68 on the L a r g e S t r u c t u r e
T A B L E O F CONTENTS (cont'd. )
P A R T T I T L E A p p e n d i c e s
A. A s y m p t o t i c B e h a v i o r of N u m e r i c a l Solution B . A S m a l l , F a s t R e s p o n s e P r o b é to M e a s u r e
C o m p o s i t i o n of a B i n a r y Gas M i x t u r e
C. An E s t i m a t e of the E r r o r in the M e a s u r e m e n t s of D y n a m i c P r e s s u r e
N O M E N C L A T U R E
C , c o n s t a n t i n e q u a t i o n ( 3 . 1. 2)
C c o n s t a n t i n e q u a t i o n ( 3 . 1.2)
&, e d d y d i f f u s i v i t y
f n o r m a l i z e d n o n - d i m e n s i o n a l p r o f i l e of m e a n v e l o c i t y U
h d i m e n s i o n l e s s m á x i m u m s l o p e t h i c k n e s s ( e q u a t i o n ( 9 . 2 . 1))
p p r e s s u r e
P m e a n p r e s s u r e
^/v72*
p v
5 n o n - d i m e n s i o n a l t u r b u l e n t m a s s diffusion = " TT
v P iUi
v t
Sc„_ t u r b u l e n t S c h m i d t n u m b e r = -r—
u v e l o c i t y i n d i r e c t i o n x
y] u '3 r m s v e l o c i t y f l u c t u a t i o n i n d i r e c t i o n x
U m e a n v e l o c i t y i n d i r e c t i o n x
Uio h i g h s p e e d v e l o c i t y a t t h e e x i t of t h e n o z z l e
Ugo l o w s p e e d v e l o c i t y a t t h e e x i t of t h e n o z z l e
v v e l o c i t y i n d i r e c t i o n y
r m s v e l o c i t y f l u c t u a t i o n i n d i r e c t i o n y
V m e a n v e l o c i t y i n d i r e c t i o n y
x s t r e a m w i s e c o o r d i n a t e
x d i s p l a c e m e n t of v i r t u a l o r i g i n f r o m c o o r d i n a t e o r i g i n
y t r a n s v e r s e c o o r d i n a t e
a n o n - d i m e n s i o n a l p r e s s u r e g r a d i e n t p a r a m e t e r = y^—-r-1
3 c o n s t a n t i n e q u a t i o n (2. 2 . 8) = -r— •y c o n s t a n t = —g—
77-8 u ^
NOMENCLATURE (cont'd. )
6X t h i c k n e s s i n t h e d y n a m i c p r e s s u r e p r o f i l e ; d e f i n e d b y d i s t a n c e
b e t w e e n m á x i m u m a n d m i n i m u m
52 t h i c k n e s s i n t h e d e n s i t y p r o f i l e ; d e f i n e d b y d i s t a n c e b e t w e e n
p o i n t s c o r r e s p o n d i n g to —*— - 1 . 6 0 a n d -*— = 6 . 4 0
F P i P i
AU r e f e r e n c e v e l o c i t y d i f f e r e n c e = Ux - U3
TI s i m i l a r i t y v a r i a b l e = / 3~~
T\ l o c a t i o n of t h e d i v i d i n g s t r e a m l i n e
X c o n s t a n t i n e q u a t i o n (2. 2 . 9) = jz —-r-^
[i viscosity
£ similarity variable = -^•
3 7 x
p density
VplS rms density fluctuation
non-dimensional Reynolds shear stress = -°—TTS
PIUI
non-dimensional stream function =
6piU! Y stream function
Subscripts and Superscripts
i free stream conditions on high speed side
2 free stream conditions on low speed side
' fluctuating quantity about the time average
time average
LIST O F T A B L E S
I. S u m m a r y of flow conditions for e x p e r i m e n t s . U10 H e l i u m v e l o c i t y at the nozzle exit.
Ugo N i t r o g e n v e l o c i t y at the nozzle exit.
II. A s p i r a t i n g p r o b é t r a v e r s e ; different p o s s i b i l i t i e s with.
- the Data S l i c e r .
LIST O F FIGURES 1. Mixing l a y e r m o d e l .
2. N u m e r i c a l solution for a - 0 (piUi = p2U2 c a s e ) Se = 0. 1,
0 . 2 , 0 . 3 , 0 . 4 , 0. 5, 1. 0.
3. C o m p a r i s o n of n u m e r i c a l solutions for f a v o r a b l e and a d v e r s e p r e s s u r e g r a d i e n t s for Se = 0 . 3 0 .
4. Sketch of the t e s t s e c t i o n .
5. Adjustable s l a t s and p e r f o r a t e d p í a t e . 6. A r r a y of s t a t i c p r e s s u r e t u b e s .
7. Sketch of the new p r o b é .
8. A p p a r a t u s r i s e t i m e : o s c i l l o s c o p e photo of the r e s p o n s e of a hot w i r e and pitot tube to s t a r t i n g p r o c e s s . H o r i z o n t a l s c a l e : 100 m s e c / d i v ; V e r t i c a l s c a l e : l v o l t / d i v . Tank p r e s s u r e = 7 a t m .
9. T i m e r e s p o n s e of the pitot tube and S c a n i v a l v e . H o r i z o n t a l s c a l e : 20 m s e c / d i v ; V e r t i c a l s c a l e : . 5 v o l t / d i v . Tank p r e s s u r e = 6 a t m .
10. T w o - d i m e n s i o n a l i t y t e s t : o s c i l l o s c o p e photos of the r e s p o n s e of two hot w i r e s 2 " a p a r t aligned s p a n w i s e at s e v e r a l down-s t r e a m l o c a t i o n down-s . T a n k p r e down-s down-s u r e = 3 a t m .
11. Block d i a g r a m of the set up to w r i t e p r e s s u r e data on tape. 12. T i m i n g of o p e r a t i o n to w r i t e p r e s s u r e data on t a p e .
13. P h o t o g r a p h i c c o m p a r i s o n of mixing l a y e r s for a < 0 and
a - 0. LiOwer(U2o= 393 c m / s e c ) s t r e a m i s 3NL; u p p e r
(U10 = 1040 c m / s e c ) s t r e a m is He. Tank p r e s s u r e = 4 a t m .
14. Shadowgraphs of mixing l a y e r for a - 0 at different Reynolds n u m b e r s . (piU12= p2U22 c a s e . ) L o w e r (low s p e e d ) s t r e a m i s
N~; u p p e r (high speed) s t r e a m i s H e .
15. Shadowgraphs of mixing l a y e r for a < 0 at different Reynolds n u m b e r s . L o w e r (low speed) s t r e a m i s N?; u p p e r (high
speed) s t r e a m i s He.
17. A s p i r a t i n g p r o b é c a l i b r a t i o n c u r v e at 4 a t m .
18. D y n a m i c p r e s s u r e t r a v e r s e s a t s e v e r a l s t a t i o n s . Tank p r e s s u r e = 4 a t m ; a = - 0. 18; (a) x = 0. 7 5 " ; (b) x = 1. 00"; (c) x = 1. 50"; (d) x = 2. 00".
19. D y n a m i c p r e s s u r e t r a v e r s e s at s e v e r a l d o w n s t r e a m s t a t i o n s . T a n k p r e s s u r e = 4 a t m ; a = - 0. 18; (e) x = 2.5 0"; (f) x = 3. 00"; ( g ) x = 3 . 2 5 " .
20. V a r i a t i o n of m i x i n g l a y e r t h i c k n e s s with x c o o r d i n a t e . Tank p r e s s u r e = 4 a t m ; a - - 0. 18.
2 1 . F r e e s t r e a m v e l o c i t y d e c a y a s a function of d o w n s t r e a m p o s i t i o n . Tank p r e s s u r e = 4 a t m ; a - - 0. 18.
22. S i m i l a r i t y profile for d y n a m i c p r e s s u r e . Tank p r e s s u r e = 4 a t m ; a = - 0 . 18; x = 0 . 7 5 " .
o
23. D e n s i t y t r a v e r s e ; a s p i r a t i n g p r o b é voltage a s a function of t i m e , and p r o b a b i l i t y d i s t r i b u t i o n at four data p o i n t s . Tank p r e s s u r e =• 4 a t m ; a - - 0. 18; x = 3. 00".
24. S i m i l a r i t y profile for d e n s i t y . T a n k p r e s s u r e = 4 a t m ;
a = - 0 . 18; x = 0. 7 5 " . o
25. S e l f - p r e s e r v a t i o n for r m s d e n s i t y fluctuations. Tank p r e s s u r e = 4 a t m ; a - - 0. 18; x = 0. 7 5 " .
v o
26. F r e e s t r e a m v e l o c i t y d e c a y a s a function of d o w n s t r e a m p o s i t i o n . T a n k p r e s s u r e = 7 and 2 a t m ; a - - 0 . 18.
27. S i m i l a r i t y profile for d y n a m i c p r e s s u r e . Tank p r e s s u r e = 7 a t m ; a = - 0 . 18; x = 0. 6 5 " .
o
28. S i m i l a r i t y profile for d e n s i t y . Tank p r e s s u r e = 7 a t m ;
a = - 0 . 18; x = 0. 6 5 " . o
29. S e l f - p r e s e r v a t i o n for r m s d e n s i t y f l u c t u a t i o n s . Tank p r e s s u r e = 7 a t m ; a - - 0. 18; x = 0. 6 5 " .
30. S i m i l a r i t y profile for d y n a m i c p r e s s u r e . Tank p r e s s u r e = 2 a t m ; a - - 0 . 18; x = 0 . 82".
o
31. S i m i l a r i t y profile for d e n s i t y . Tank p r e s s u r e = 2 a t m ;
LIST O F FIGURES (cont'd. )
32. S e l f - p r e s e r v a t i o n for r m s d e n s i t y fluctuations. Tank p r e s s u r e = 2 a t m ; a - - 0. 18; x = 0. 8 2 " .
o 33. Velocity p r o f i l e ; a = - 0 . 18.
34. Shear s t r e s s d i s t r i b u t i o n ; a = - 0 . 18.
35. T u r b u l e n t m a s s diffusión d i s t r i b u t i o n ; a - - 0 . 18.
36. Eddy diffusivity, eddy v i s c o s i t y , and Schmidt n u m b e r d i s t r i -butions a c r o s s the mixing l a y e r ; a - - 0. 18.
37. V a r i a t i o n of m i x i n g l a y e r t h i c k n e s s with x c o o r d i n a t e ; Tank p r e s s u r e = 4 a t m ; a - 0.
38. S i m i l a r i t y profile for d y n a m i c p r e s s u r e . Tank p r e s s u r e = 4 a t m ; a = 0, x = - 0 . 2 0 " .
o
39. S i m i l a r i t y profile for d e n s i t y . Tank p r e s s u r e = 4 a t m ;
a = 0; x = - 0 . 2 0 " . o
40. S e l f - p r e s e r v a t i o n for r m s d e n s i t y fluctuations. Tank p r e s s u r e = 4 a t m ; a = 0; x = - 0. 20".
r o
4 1 . Velocity p r o f i l e ; a = 0.
42. Shear s t r e s s d i s t r i b u t i o n ; a - 0.
4 3 . T u r b u l e n t m a s s diffusion d i s t r i b u t i o n ; a = 0.
44. Eddy diffusivity, eddy v i s c o s i t y , and S c h m i d t n u m b e r d i s t r i -butions a c r o s s the mixing l a y e r ; a - 0.
4 5 . C o m p a r i s o n of n u m e r i c a l solutions with e x p e r i m e n t a l r e s u l t s ; d y n a m i c p r e s s u r e p r o f i l e s ; a - - 0 . 18.
46. C o m p a r i s o n of n u m e r i c a l solutions with e x p e r i m e n t a l r e s u l t s ; d y n a m i c p r e s s u r e p r o f i l e s ; a - 0.
47. D e n s i t y t r a v e r s e ; a s p i r a t i n g p r o b é voltage a s a function of t i m e , and p r o b a b i l i t y d i s t r i b u t i o n at data points 1, 2, 3 and 4. Tank p r e s s u r e = 4 a t m ; a = 0.
48. D e n s i t y t r a v e r s e ; a s p i r a t i n g p r o b é voltage a s a function of t i m e , and p r o b a b i l i t y d i s t r i b u t i o n at data points 5, 6, 7 and 8. Tank p r e s s u r e = 4 a t m ; a - 0.
I. INTRODUCTION
The p r e s e n t w o r k had i t s p r i m e m o t i v a t i o n f r o m a continuing effort a t the California I n s t i t u t e of Technology d i r e c t e d t o w a r d s the u n d e r s t a n d i n g of h e t e r o g e n e o u s t u r b u l e n t m i x i n g .
"Within the f r a m e w o r k of that a i m a new facility was c o n s t r u c t e d s e v e r a l y e a r s a g o , and the f i r s t i n v e s t i g a t i o n s w e r e c a r r i e d out by
B r o w n and R o s h k o (Ref. 1) on t u r b u l e n t mixing l a y e r s b e t w e e n two s t r e a m s of different g a s e s .
During the c o u r s e of those i n v e s t i g a t i o n s it was r e a l i z e d that i t should be p o s s i b l e to e s t a b l i s h e q u i l i b r i u m t u r b u l e n t mixing l a y e r s in p r e s s u r e g r a d i e n t s , for p a r t i c u l a r c o m b i n a t i o n s of the f r e e s t r e a m p a r a m e t e r s . T h i s led to the p r e s e n t r e s e a r c h .
1. 1 P r e v i o u s I n v e s t i g a t i o n s
E q u i l i b r i u m flows a r e r a t h e r s c a r ce due to the fact that they e x i s t only for p r o p e r l y adjusted e x t e r n a l p r e s s u r e g r a d i e n t s , and for s p e c i a l c o m b i n a t i o n s of the p a r a m e t e r s involved in the p r o b l e m . Two e x a m p l e s a r e well known in the c a s e of b o u n d a r y l a y e r s :
a) In the l a m i n a r b o u n d a r y l a y e r c a s e the o r d i n a r y differential equation was f i r s t deduced by F a l k n e r and Skan, and is wldely r e p o r t e d in the l i t e r a t u r e ; i t s s o l u t i o n s w e r e l a t e r i n v e s t i g a t e d in d e t a i l by
D. R. H a r t r e e (Ref. 2).
F o r t u r b u l e n t j e t s and w a k e s in p r e s s u r e g r a d i e n t s the t h e o r e t -i c a l cond-it-ions for the e x -i s t e n c e of s -i m -i l a r -i t y solut-ions of the boundary l a y e r e q u a t i o n s w e r e s e t out by Townsend (Ref. 4), Wygnanski and F i e d l e r (Ref. 5), and G a r t s h o r e and N e w m a n (Ref. 6), but the only r e l e v a n t e x p e r i m e n t s w e r e c a r r i e d out by G a r t s h o r e (Ref. 7) on a two-d i m e n s i o n a l w a k e , antwo-d F e k e t e (Ref. 8) on a t w o - two-d i m e n s i o n a l j e t in
s t r e a m i n g flow.
V e r y l i t t l e r e s e a r c h h a s been done on the c a s e of a h o m o g e -neous mixing l a y e r in a p r e s s u r e g r a d i e n t . One of the few w o r k s in this á r e a h a s b e e n that of Sabin (Ref. 9) who found "a s e l f - s i m i l a r solution to an a p p r o x i m a t e equation which is not dependent upon a p a r t i c u l a r choice of e i t h e r the eddy v i s c o s i t y or p r e s s u r e g r a d i e n t " . It should be noted h e r e that no e q u i l i b r i u m flow o r s i m i l a r i t y solution, in the p r e c i s e s e n s e of the w o r d , e x i s t s for the p l a ñ e , h o m o g e n e o u s , t u r b u l e n t mixing l a y e r in any type of p r e s s u r e g r a d i e n t . T h e r e a s o n for this i s that the only c a s e for which s i m i l a r i t y i s p o s s i b l e i s for P2^22 = P i ^ i2, which can occur only for p2/ px*.
During the c o u r s e of B r o w n and R o s h k o ' s i n v e s t i g a t i o n s on the c a s e p2U22 = piUj2 for z e r o p r e s s u r e g r a d i e n t it w a s r e a l i z e d , by
a r g u i n g in p h y s i c a l t e r m s , that the only way an e q u i l i b r i u m mixing l a y e r in a p r e s s u r e g r a d i e n t could be e s t a b l i s h e d w a s by having p2U22 = PiUi2, s i n c e then P2(x) = Pi(x) and the r a t e of change of the
free s t r e a m v e l o c i t i e s with the d o w n s t r e a m c o o r d í n a t e is s u c h that
YT,—v = c o n s t a n t for all x. Doing the t h e o r e t i c a l a n a l v s i s on the U ^ x )
e q u a t i o n s of motion* it t u r n s out that t h e s e e q u i l i b r i u m flows a r e of
c¿
the f o r m X5 X ~ x and 6 ~x, w h e r e 6 is a c h a r a c t e r i s t i c t h i c k n e s s ; i. e. , the s h e a r l a y e r still g r o w s l i n e a r l y in x. The n o n d i m e n s i o n a l p r e s
-, . -, . x d U i x d U p i. . i-i i
s u r e g r a d i e n t p a r a m e t e r i s° a - rr~ J - Tf ~¿ - c o n s t a n t . C l e a r l y ,
c Ui dx U2 dx J
for a - 0, Ux = c o n s t and U2 = const ( z e r o p r e s s u r e g r a d i e n t ) ; for a > 0
trié'"fiow i s a c c e l e r a t e d (favorable p r e s s u r e g r a d i e n t ) ; and for a < 0 the flow is d e c e l e r a t e d ( a d v e r s e p r e s s u r e g r a d i e n t ) .
1. 2 Goals of the P r e s e n t Study
Much attention h a s been given to the p r o b l e m of f r e e turbulent m i x i n g , due to the fact that a l a r ge n u m b e r of flow configurations of e n g i n e e r i n g significance a r e r e l a t e d to this p r o c e s s , a s i s the c a s e for fully s e p a r a t e d flows, w h e r e an i m p o r t a n t e l e m e n t i s the s h e a r l a y e r which develops behind the s e p a r a t i o n point. T u r b u l e n t m i x i n g with l a r g e d e n s i t y n o n u n i f o r m i t i e s p l a y s a v e r y i m p o r t a n t r o l e in c o m b u s tión, c h e m i c a l mixing of different s p e c i e s , and m o r e r e c e n t l y in c h e m -i c a l l á s e r s . In m o s t of t h e s e -i m p o r t a n t flows the p r e s s u r e v a r -i e s along the s t r e a m w i s e d i r e c t i o n ; t h e r e f o r e the a n a l y s i s of t h e s e c a s e s r e q u i r e s knowledge of the p r o p e r t i e s of the h e t e r o g e n e o u s t u r b u l e n t mixing l a y e r in a p r e s s u r e g r a d i e n t .
Our m a i n i n t e r e s t was in trying to find e q u i l i b r i u m s o l u t i o n s , s i n c e the study of t h e s e configurations i s s i m p l e r than that of
non-p r e s e r v i n g flows; the f o r m e r a r e of fundamental i m non-p o r t a n c e to non-p r o v i d e i n s i g h t for the m o r e c o m p l i c a t e d flow c a s e s .
F r o m the equations of m o t i o n we found the conditions for the e x i s t e n c e of s i m i l a r i t y . Once the b o u n d a r y - l a y e r equations w e r e r e d u c e d to o r d i n a r y d i f f e r e n t i a l equations we obtained n u m e r i c a l s o l u -tions by using the h y p o t h e s e s of c o n s t a n t eddy v i s c o s i t y and eddy diffu-sivity. The p r o c e d u r e contains two e m p i r i c a l c o n s t a n t s left free to be adjusted f r o m e x p e r i m e n t .
In the e x p e r i m e n t a l w o r k , e x t e r n a l p r e s s u r e g r a d i e n t of the f o r m p r é s c r i b e d by the t h e o r y was i m p o s e d upon a t w o - d i m e n s i o n a l t u r b u l e n t mixing l a y e r between s t r e a m s of n i t r o g e n and h e l i u m with equal d y n a m i c p r e s s u r e s , with the following goals in m i n d . F i r s t , p r o f i l e s of m e a n total head and d e n s i t y a t s e v e r a l d o w n s t r e a m s t a t i o n s w e r e d e s i r e d so that the p o s s i b l e s i m i l a r i t y flow found a n a l y t i c a l l y could be v e r i f i e d ; m e a s u r e m e n t s of r m s d e n s i t y fluctuations w e r e sought to the end that s e l f - p r e s e r v a t i o n of the t u r b u l e n t q u a n t i t i e s a s s e t out by T o w n s e n d ' s c r i t e r i a could be e s t a b l i s h e d . Second, a d e t e r -m i n a t i o n of the b a s i c flow p a r a -m e t e r s , e. g. , s p r e a d i n g r a t e , s h e a r
s t r e s s , t u r b u l e n t m a s s diffusion and S c h m i d t n u m b e r d i s t r i b u t i o n s w e r e r e q u i r e d . F i n a l l y , m e a s u r e m e n t s for the c a s e of z e r o p r e s s u r e g r a d i e n t w e r e d e s i r e d for r e a s o n s of completen.ess.
1. 3 E x p e r i m e n t a l T e c h n i q u e s
The e x p e r i m e n t a l i n v e s t i g a t i o n was c a r r i e d out in the facility d e s i g n e d by B r o w n to p r o d u c e t u r b u l e n t flows at p r e s s u r e up to 1 0 a t m o s p h e r e s with v e r y s h o r t running t i m e s .
B e c a u s e of the s h o r t d u r a t i o n of the flow, only a few s e c o n d s , h i g h - s p e e d m e a s u r e m e n t t e c h n i q u e s w e r e u s e d .
The side w a l l s of the t e s t s e c t i o n of the a p p a r a t u s (Ref. 1) w e r e changed for a d j u s t a b l e s l a t s and a p e r f o r a t e d p í a t e was added at the channel exit.
The h i g h e s t s p e e d s w e r e 1000 c m / s e c for the light gas and 378 c m / s e c for the h e a v y one. E x p e r i m e n t s w e r e m a d e at t h r e e dif-f e r e n t tank p r e s s u r e s (7, 4 and 2 a t m o s p h e r e s ) to study the b e h a v i o r
of the mixing l a y e r at different R e y n o l d s n u m b e r s . Mean d y n a m i c p r e s s u r e p r o f i l e s w e r e obtained a t s e v e r a l d o w n s t r e a m l o c a t i o n s u s i n g a f a s t e l e c t r o n i c m a n ó m e t e r ( B a r o c e l ) and a pitot tube. Analog s i g n á i s f r o m the B a r o c e l w e r e c o n v e r t e d through an A / D c o n v e r t e r , to digital f o r m and w r i t t e n on m a g n e t i c t a p e . A Kennedy I n c r e m e n t a l Tape R e c o r d e r was u s e d for this p u r p o s e .
C o m p o s i t i o n m e a s u r e m e n t s of the b i n a r y m i x t u r e w e r e m a d e a t s e v e r a l d o w n s t r e a m l o c a t i o n s u s i n g an a s p i r a t i n g p r o b é developed by B r o w n and Rebollo (Ref. 10)*. A v e r y fast data a c q u i s i t i o n s y s t e m (Data S l i c e r ) , d e s i g n e d by C o l e s , was u s e d ; f i r s t , to t r a v e r s e the a s p i r a t i n g p r o b é a c r o s s the s h e a r l a y e r ; s e c o n d , to c o m m a n d the A / D c o n v e r s i ó n of the analog voltage coming f r o m the feedback b r i d g e of
the hot w i r e ; and t h i r d , to c o n t r o l the w r i t i n g of the digital s i g n a l on m a g n e t i c tape for l a t e r c o m p u t e r p r o c e s s i n g . The r e c o r d i n g of the data was by m e a n s of a Kennedy Synchronous Tape R e c o r d e r .
2. 1 D e r i v a t i o n of the Equations of Motion for H e t e r o g e n e o u s F l o w
The equations of m o t i o n for an i n c o m p r e s s i b l e , t w o - d i m e n s i o n a l flow in a n o n - u n i f o r m m é d i u m can be w r i t t e n as follows:
x- m o m e n t u m
9t 3x 8y 3x ox r9 x 9y ro y
y - m o m e n t u m
(2. 1.1)
t + 8x 9v 3 v+3 x V + 8v V [ ' L ' '
Continuity
^ + ^ + ^ = 0 ( 2 . 1 . 3 )
G r a v i t a t i o n a l f o r c e s have b e e n neglected in the m o m e n t u m equation. Following Reynolds (1895) we divide the flow q u a n t i t i e s into t h e i r m e a n and fluctuating p a r t s :
u = U + u' v = V + v1
p = P + p!
p = p + p'
• ^ ( p U2) + | - ( p U V + Up'v') = - | | - A - ( p u ' v ' ) (2.1.4) ap
^ ( p v '2) = - w (2.1.5)
8
• ^ ( p U ) + ^ (PV + p'v') = 0 ( 2 . 1 . 6 )
after the following approximations have been made:
a) gradients in x a r e small compared -with gradients in y
(boundary layer approximation) i. e. , I Q /Q 1 < < : 1
b) valúes o í ^ u '2 and'y/v'2 a r e comparable
cj —\—l \—¿ >> x^ where 6(x) is a m e a s u r e of the width of the
shear zone, and AU(x) is a reference velocity difference at
each c r o s s section.
As a consequence of the above assumptions, the Reynolds
s t r e s s e s a r e supposed to be very large relative to the viscous s t r e s s
for sufficiently high Reynolds numbers.
The only remaining equation is the diffusion equation which
reduces to
9x oy '
if the molecular diffusion is neglected, which is consistent, for a
turbulent flow, with the approximations already made (Ref. 1). F r o m
equation (2. 1. 5) we get
| - ,
pt f » ,
+^ ,
p„ V ) = - ^ i S Í . . ^ ( ¿ 5 ^ r , (3.1.9
)
^ ( PU ) + | - ( p V ) = 0 ( 2 . 1 . 1 0 )
9 x oy dy p v '
w h e r e pV = pV + p ' v ' (2. 1. 12)
I t s h o u l d b e n o t e d t h a t w i t h t h i s s u b s t i t u t i o n t h e e q u a t i o n of
c o n t i n u i t y r e c o v e r s i t s n o r m a l f o r m , b u t a m a s s - w e i g h t e d t u r b u l e n t
m a s s d i f f u s i o n t e r m a p p e a r s i n t h e r i g h t h a n d s i d e of t h e d i f f u s i o n
e q u a t i o n .
2 . 2 H e t e r o g e n e o u s T u r b u l e n t M i x i n g L a y e r : E q u i l i b r i u m F l o w i n a
P r e s s u r e G r a d i e n t
L e t u s c o n s i d e r t h a t a t x = 0 t h e r e i s a m e e t i n g of two p a r a l l e l
s t r e a m s of d i f f e r e n t g a s e s , d e n s i t i e s px a n d p2 , w h o s e v e l o c i t i e s a r e
U-L a n d U2 r e s p e c t i v e l y , i t b e i n g a s s u m e d t h a t l ^ > U2 . D o w n s t r e a m
of t h e p o i n t of e n c o u n t e r t h e s t r e a m s w i l l f o r m a m i x i n g z o n e s u b j e c t
to a p r e s s u r e g r a d i e n t i n t h e s t r e a m w i s e d i r e c t i o n ( F i g . 1). d P (x)
T o find t h e c o n d i t i o n s f o r s i m i l a r i t y w h e n ^!p—• 4 0, w e
a s s u m e f o l l o w i n g T o w n s e n d (Ref. 4 ) , t h a t
u
u
2_P_ Pi
p u ' v ' P i Ux y
= Mr\)
= PÍTl)
=
T{T\)-1-T7-I
( 2 . 2 . 1)
(2. 2.2 )
(2.2. 3 )
¡*Tr = S ( n ) ( 2 . 2 . 4 ) P iui
w h e r e U 1 - Ux(x)
TI = y / 6(x )
6(x) i s a c h a r a c t e r i s t i c d i m e n s i ó n i n t h e t r a n s v e r s e d i r e c t i o n , i . e . ,
a m e a s u r e of t h e w i d t h of t h e m i x i n g r e g i ó n .
T h e b o u n d a r y c o n d i t i o n s a r e a s f o l l o w s
ÍU(T1) -• 1
'p(Tl) -• 1
U ( n ) -> YT2
(pin) ^ - e a .
P l
H e n e e 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 w e d e d u c e t h a t -=r^¡ —7 h a s Ui(x)
to b e a c o n s t a n t .
W i t h i n t h e b o u n d a r y l a y e r a p p r o x i m a t i o n , w e w i l l h a v e o u t s i d e
t h e l a y e r , w h e r e -K— i s v e r y s m a l l , _ ¿ P l (x) . p u d l i
d x ~ P l U l d x
After i n t e g r a t i o n
U T3( X ) , U£_. „ \ (Pi " P2 ü ^ ") = C
w h e r e C i s a c o n s t a n t ; but s i n c e \3± - U^x) and ye2 = c o n s t , this i m p l i e s
ui
that C = 0; t h e r e f o r e in o r d e r to have P3(x) = P ^ x )
PiU1» = P aU8 a , or ^ » = ^ ( 2 . 2 . 5 )
In h o m o g e n e o u s flow, px = p2 , the only way to s a t i s f y this
condition i s with Ux = U2 , i. e. , "no s h e a r b e t w e e n the two s t r e a m s . If
L e t us define a s t r e a m function Y(x, y ) , such that the continuity equation is i d e n t i c a l l y s a t i s f i e d .
TT 9 Y
v 9 Y
PV = - K
w h e r e Y(x, y) i s m a d e d i r a e n s i o n l e s s , and a s s u m e d to be only a function of r\, by the s u b s t i t u t i o n
* / i Y ( x , y )
x)
C o n s e q u e n t l y , the r e l a t i o n between the v e l o c i t y c o m p o n e n t s and §(r|) i s given by
# = pu ( 2 . 2 . 6 ) a n d
pv = (S) n p u - y - g £ - «(Tí ( 2 . 2 . 7 )
~ V where v = j=- •
ui
Introducing the non-dimensional quantities from (2. 2. 1),
(2. 2. 2), (2. 2. 3), and (2. 2. 4) into the equations of motion, and after
replacing v for its expression in (2. 2. 7) we obtain
6(x) dU x(x) , a n rd5(x) 6(x) d ü1( x )1I d u _ dr
Ux(x) dx v p u " l) " L dx Ui(x) dx •'Idn ~ dn
Ldx Ux(x) dx Ji d«n d r ]l i > / p j
Similarity solutions exist only if
- ^ ^ = const = B (2.2.8)
6W d ^ í x )
= c o n s t = x ( 2 > 2 - 9 )Ui(x) dx
From these two equations the potential velocity U^x) and the
scale factor 6(x) for the ordinate can be evaluated.
6(x) - x (2.2. 10)
Ui(x) - Xa (2. 2. 11)
where a = \/f3
x dUn Ux dx
In conclusión, the only case for which similarity is possible
is when pgUg2 = piUi3 and the equilibrium flows are of the form given
by the equations (2.2. 10) and (2.2. 11).
d(pu) , 1 d(pv) . _ ._ _ , _ .
*-&r 8 d ^- °P U = (2.2.12)
.. ¿(pu3) , 1 d(puv) 2 1 dT , , , , , ,
1 1 ^ " + B~V + f f ( 2 p U " 1 ) = B d ^ ( 2 . 2 . 1 3 )
^ + 0 d^ + «U =B d ^( S / p ) ( 2 . 2 . 1 4 )
2. 3 S h e a r S t r e s s and T u r b u l e n t M a s s Diffusion D i s t r i b u t i o n s I n t e g r a t i n g the equation (2. 2. 6) we find that
f (TI) = J pu dx o
w h e r e at ri = 0 , $(r| ) = 0
and v(n ) = 0 'o
n is the dividing s t r e a m l i n e of the flow. A c r o s s this line the t r a n s -'o °
p o r t of m a s s is equal to z e r o .
To get the d i s t r i b u t i o n s of s h e a r s t r e s s and t u r b u l e n t m a s s diffusion a c r o s s the l a y e r we i n t é g r a t e the s i m i l a r i t y e q u a t i o n s .
Continuity y i e l d s
- (§u + v) p + (1 + a) J pu dx = 0 (2. 3. 1)
o w h e r e 5 = y /x •
F r o m the m o m e n t u m equation (2. 2. 13) and after u s i n g continuity we ha ve
? ? ? - (1+or) u P pudx + (1+a) J pu2dx + a \ (pu2-l) dx
\ =0 § =0 \ =0
o °o ^o
S i m i l a r l y , i n t e g r a t i n g (2. 2. 14) a n d a f t e r u s i n g e q u a t i o n (2. 3 . 1),
w e o b t a i n f o r t h e t u r b u l e n t m a s s d i f f u s i o n d i s t r i b u t i o n
- (r+or) J p u d x + (1+a?) f u d x = ( S / p ) - ( S / p ) ( 2 . 3 . 3 ) 5 =0 § - 0 5 = 0
°o °o °o
H e n e e , if w e m e a s u r e t h e v e l o c i t y a n d d e n s i t y p r o f i l e s , a n d l ó c a t e
t h e d i v i d i n g s t r e a m l i n e on t h e m , t h e s h e a r s t r e s s a n d t u r b u l e n t m a s s
d i f f u s i o n c a n b e c a l c u l a t e d f r o m e q u a t i o n s (2. 3 . 2) a n d (2. 3 . 3 ) .
We s h o u l d n o t e t h a t s i n c e
(|£) = a (pu2 - 1) 1 (2.3.4 )
s§ = 0 k = 0
3 o ° o
T h e m á x i m u m s h e a r i n g s t r e s s o c e u r s o n t h e d i v i d i n g s t r e a m
-l i n e o n -l y f o r two c a s e s : e i t h e r
a - 0, o r (pu2 - 1)_ _ Q = 0
o~
h o w e v e r
(d (^p )) = 0 f o r a n d ( 2 . 3 . 5 )
Úl ?o=0 atO
p v
A m a s s - w e i g h t e d t r a n s v e r s e v e l o c i t y f l u c t u a t i o n — i s a
m á x i m u m a t £ = § i n d e p e n d e n t l y of t h e p r e s s u r e g r a d i e n t .
2- 3 a L o c a t i o n of t h e D i v i d i n g S t r e a m l i n e
C o n s i d e r i n g t h a t T ( ? ) s h o u l d t e n d t o w a r d s z e r o a t b o t h e d g e s
of t h e m i x i n g l a y e r ; w e d e d u c e f r o m e q u a t i o n (2. 3 . 2) t h a t : a s
T ( ? )F _ = (1 + or) T pu (u - f f ) d x + a [ (pu2 - 1 )dx (2. 3 . 6)
° ^ o "U E =0c; = 0 ~ U lx F =0 §
^o ^ o
a n d w h e n ? -• +0 0
00
- T ( 5 ) - = n = ( l + a ) J pu(u - l ) d x + <* f ( p u2- l ) d x ( 2 . 3 . 7 ;
° 9o 5 =0 5 =0
o o
C o n s e q u e n t l y
00 c o
(1 + a) [ pu(u - l ) d x + a J (pu2 - l ) d x
? = 0 § = 0
o o
— oo — oo
= (1 + a) f pu(u - ^ ) d x + a P (pu2 - l ) d x ( 2 . 3 . 8)
? =0 U* J§ =0
o 3o
w i l l d e f i n e ^ =0 i n o u r e x p e r i m e n t a l p r o f i l e s .
F o l l o w i n g t h e s a m e p r o c e d u r e a n e q u a t i o n f o r § =0 c a n a l s o b e
f o u n d f r o m e q u a t i o n (2. 3 . 3)
- ( S / p ) - = n= (1 +<*) I* u ( l - p)dx ( 2 . 3 . 9 )
So U ? = 0 •
^o
- 0 0
- ( S / p ) - _n = (1 + a ) J u ( l - ( - t - ) ) d x (2. 3 . 10)
H e n e e
00 ^ o = 0
u ( p - l ) d x = f u ( l -, —?—r)dx ( 2 . 3 . 1 1
o
i s a l s o a n e q u a t i o n f o r d e t e r m i n i n g t h e p o s i t i o n of § =0 f r o m o u r
Naturally both equations ought to give the same location for £ •
2. 3b Mass Flow Entrainment
We can define the m a s s entrainment as pv(5) when | ^ | -* oo ,
but subtracting from it the valué which exists in conjunction with the
p r e s s u r e gradient. ' '
F r o m equation (2. 3. 1) we ha ve
pv = § pu - (1 + a) \ pudx
5 =0 ^o
as ^ -• ± oo and taking into account the boundary conditions we a r r i v e
at
. 0 0 0 0
p l V ( +° ° ) = - r (pu - l ) d x - a f pudx (2.3.12)
pi U l J? =0 ? =0
^o ^o Entrainment
- 0 0 _„ . - o o
p8V(-oo) = _ f ( £ s ¿ k) d x_ „ r pudx (2.3.13)
Pi ui ¿ P -nF =0 P iui - ~
P l l £ =0
^o ^o Entrainment
The last t e r m in each equation is connected with the existence
of the p r e s s u r e gradient, and is left out of our definition of mass
III. NUMERICAL SOLUTION O F THE EQUATIONS 3. 1 Eddy V i s c o s i t y and Eddy Diffusivity Model
In o r d e r to p r e d i c t any t u r b u l e n t flow a c e r t a i n n u m b e r of a s s u m p t i o n s have to be m a d e about the Reynolds t r a n s p o r t t e r m s . A c c u r a c y and scope of the p r e d i c t i o n s a r e n o r m a l l y dependent upon the equations c h o s e n for the c l o s u r e of the s y s t e m .
During the L a n g l e y Working Conference on T u r b u l e n t F r e e S h e a r F l o w s in 1972 a wide and r e p r e s e n t a t i v e s p e c t r u m of t u r b u l e n c e m o d e l s w e r e p r e s e n t e d * . Two g e n e r a l t e c h n i q u e s that h a v e b e e n u s e d e x t e n s i v e l y to e v a l ú a t e the needed t u r b u l e n c e input a r e the
eddy-v i s c o s i t y and the t u r b u l e n t k i n e t i c e n e r g y a p p r o a c h e s .
Of the c u r r e n t a l t e r n a t i v e s a v a i l a b l e for m o d e l i n g the t u r b u l e n t t r a n s p o r t t e r m s we should i n d i c a t e t h a t , s i n c e our p r i m a r y i n t e r e s t i s devoted to e s t a b l i s h i n g the p o s s i b i l i t y of e q u i l i b r i u m flows which have b e e n found in s e c t i o n II, we have u s e d a v e r y s i m p l e eddy v i s c o s i t y and diffusivity m o d e l in o r d e r to give us a q u a l i t a t i v e i d e a of what to e x p e c t f r o m the e x p e r i m e n t s . N a t u r a l l y , the e l e c t i o n of this or any o t h e r t u r b u l e n t m o d e l would yield the s a m e f o r m of the g r o w t h laws for the e q u i l i b r i u m flows.
We t h e r e f o r e a s s u m e
(a)
JW^=-
»f £
y ( 3 . 1 . 1 )
I t T - t
p u ' V = - p v
au
t 8y(b) fi = C , 6 AU t a v = C 6 AU
t p,
(3. 1.2;
Substituting t h e s e a s s u m p t i o n s into the s i m i l a r i t y f o r m of the equations we g e t :
Continuity
AX + A' + u (1 + a) = 0 (3. 1. 3)
M o m e n t u m
C
AY + a(pM3 - l ) = - J i - ^ y Y' ( 3 . 1 . 4 )
Diffusion
A ' + u ( l +a) = - ( - J L ^ S - ) - i - X' ( 3 . 1 . 5 )
w h e r e
/~*J
A = - n u + | ( 3 . 1 . 6 )
X = - 3£- ( 3 . 1 . 7 )
Y = p | H ( 3 . 1 . 8 )
v t
S ct = r— (turbulent Schmidt n u m b e r )
t At
with b o u n d a r y conditions (P - • ! fu -4 1
( p -*P3/Pl
n - > -oo s .
1 u -^Ua/Ux
3 . 2 Z e r o P r e s s u r e G r a d i e n t C a s e : a - 0
W h e n no p r e s s u r e g r a d i e n t i s i m p o s e d u p o n t h e m i x i n g l a y e r ,
Ui = c o n s t a n t a n d a = 0, h e n e e , r e p l a c i n g t h i s v a l u é of a i n t o t h e
e q u a t i o n s a n d a f t e r e l i m i n a t i n g A ' + u (1 + a) o u t of t h e c o n t i n u i t y a n d
d i f f u s i o n e q u a t i o n s w e find
w h e r e
A X = r L X ' S ct
A Y = V Y '
C AU
y = —^3 Ui —
C o n s e q u e n t l y
S ct — X ' X
I n t e g r a t i n g t w i c e , a n d a f t e r u s i n g t h e b o u n d a r y c o n d i t i o n s to c a l c ú l a t e
t h e t w o c o n s t a n t s of i n t e g r a t i o n , w e w i l l h a v e a d i r e c t r e l a t i o n s h i p
b e t w e e n p{r\) a n d u(rj); if w e n o w s e t
U(T}) = o T T ( 1 + T r 2 ~
2 Ui Ux + U2 f(Tl))
w e f i n a l l y a r r i v e a t
-Se. - 1 / S <
p(*n) =
(•fia) - 1 oo Sct
l+£z S e — - J C f * ( x ) ] d x J [ f ( x ) l t d x T]
( 3 . 2 . 1)
F r o m t h e m o m e n t u m e q u a t i o n , a n d t a k i n g i n t o a c c o u n t t h a t
1 r»
A(rj) = - - J P ^ d x
w e d e d u c e
«J = 0 P [ Ux + U2 ^ x ^a x ¿ 8 lUx+ U2 ' d nl p ;
After multiplying and dividing the left hand side by p(r|),
defining |3 such that,
_ J L /ui - US N _ i
B VUX + Ua ; 2
and integrating we get
T) , X
T) = 0 r O x 3
d x
t
' h ) = ™ e " ° ,3.2.2
)
We should note that the linearized result * ~ jr ?i << 1 for
p = const reduces to the solution obtained by Gdrtler (Ref. 23); if
our definition of 3 is made compatible with his, namely
fk.
(ui
-
v*
)
=I
3 ^ + U2 4
t h e n
f'(ri) = f'(0) e_ r i
* - x 2
f(r]) = f(0) + f'(0) f e ~ * d x Jo
and from the boundary conditions
w e g e t
f(0) = O
f'(0)=-¿-• /T T
T h e r e f o r e when (Ui - U2) -> 0 and p = const the solution i s given by the
e r r o r function
f(Tl) = J- Jo e " d x
To solve our p r o b l e m for a - 0 we t r y an i t e r a t i v e p r o c e d u r e ; f i r s t , we i n t r o d u c e f(r|) = e r f (r\) into (3. 2. 1); s e c o n d , with the c a l c u -l a t e d profi-le p{r\) we go to (3. 2. 2) and compute V{r^ which i s i n t e g r a t e d and back again to the s a m e loop till c o n v e r g e n c e is a c h i e v e d .
A s an i l l u s t r a t i o n of the s o l u t i o n s , the c a s e p2U22 =• piU]3 for
different v a l ú e s of the t u r b u l e n t Schmidt n u m b e r i s shown in figure 2.
3. 3 P r e s s u r e G r a d i e n t C a s e : a 4 0
In our f i r s t a t t e m p t to solve this p r o b l e m we m a d e the a s s u m p -tion that the v e l o c i t y profile would n e v e r e x c e e d its f r e e s t r e a m valué in a s i m i l a r i t e r a t i o n p r o c e d u r e to the one a l r e a d y followed in s e c t i o n 3. 2. T h i s s u p p o s i t i o n p r o v e d to be f a l s e , a s shown in Appendix A. F o r the c a s e - 1 < a < 0, and Se < 1, the v e l o c i t y will a p p r o a c h a s y m p t o t i c a l l y i t s low speed side f r o m below, and that of high speed f r o m abo v e .
With this knowledge of the a s y m p t o t i c behavior a different p r o c e d u r e was adopted, w h e r e i n no r e s t r i c t i o n s w e r e i m p o s e d upon the i t e r a t i v e p r o c e d u r e .
we have
X' S ct
X y A
TI
But A(n) = - ( 1 + a) J pu dx
p
v°
Integrating twice and using the boundary conditions to find the
constants, this expréssion yields
00
-B(x)
J e -
B Wd x
i n ( ^ ) \ R, , (3.3.1)
p l
J e "
B ( x )d x
Ph) = e
where, as before
JL ,U* - U ^ = 1
3 VUX + U2 ' 2
a n d
x
B(x
> =
Sc
tI
0
{
(
wr J
0
p<
1 +
ttfto*}<c
The momentum equation can be put in the form
y, _ A. y = £ (pu2 - 1)
and multiplying left and right hand sides by
1 ^
- - \ A(x) dx
we get
TI T I
- • i j1 A(x) dx - I j A(x) dx
Yri =0 .. Y = 0
b i a , Q ' o
*[*•
v
K
(pu¿ - 1) eWe can now i n t é g r a t e to a r r i v e at
f'(Ti) = D(Ti){p(0)f'(0) + - ^ ü - F ( T I ) } ( 3 . 3 . 2 )
w h e r e B(n)
"S ct
D(T
»
-
^
r
a n d
B(x) (pu2 - 1) e
F(n) = J (pu2 - 1) e dx
We s t a r t our i t e r a t i v e p r o c e d u r e i n t r o d u c i n g p{r\) and í{r\) f r o m the
<x - 0 solution into (3. 3. 1). Getting a new f'(r|) f r o m equation (3. 3. 2) and i n t e g r a t i n g we will have f i r s t i t e r a t i o n p r o f i l e s for p(r)) and í{r\) which, s u b s t i t u t e d back into the e x p r e s s i o n s for p(r\) and f'(r|) will give us a second a p p r o x i m a t i o n , and so on till c o n v e r g e n c e i s a c c o m p l i s h e d .
Taking into a c c o u n t the w o r k of o t h e r i n v e s t i g a t o r s , e. g. , B r o w n and R o s h k o (Ref. 1), Way and Libby (Ref. 24), we deduced that a v e r y p l a u s i b l e t u r b u l e n t S c h m i d t n u m b e r for our e x p e r i m e n t s was going to be about 0. 3 0. Having this in m i n d we k e p t c o n s t a n t Se and c o m p a r e d our n u m e r i c a l solutions for different v a l ú e s of a. F r o m t h e s e c o m p a r i s o n s i t can be c l e a r l y s e e n that the effect of an a d v e r s e p r e s s u r e g r a d i e n t (a < 0) is far m o r e p r o n o u n c e d than that of a
f a v o r a b l e one {a > 0).
B a s e d upon this finding we d e c i d e d to c o n c é n t r a t e our e x p e r i -m e n t s on the a d v e r s e p r e s s u r e g r a d i e n t .
F i g u r e 3 shows a c o m p a r i s o n for the u and pu2 p r o f i l e s
IV. EXPERIMENTAL, F A C I L I T Y , I N S T R U M E N T A R O N , AND E Q U I P M E N T
4. 1 Flow A p p a r a t u s
The e x p e r i m e n t s w e r e p e r f o r m e d in the h i g h - p r e s s u r e flow facility (Ref. 1) d e s i g n e d to p r o d u c e a t u r b u l e n t s h e a r flow between two s t r e a m s of different g a s e s .
B a s i c a l l y it c o n s i s t s of two supply u n e s e a c h one coming f r o m eight 2000 p s i b o t t l e s ; the gas s t r e a m s a r e b r o u g h t t o g e t h e r at the exit of two 4 " x 1" n o z z l e s in the t e s t s e c t i o n ; a s c h e m a t i c r e p r e s e n t a t i o n is shown in figure 4. T h e t e s t s e c t i o n i s e n c l o s e d by a cylinder which s l i d e s over and s e á i s a g a i n s t O - r i n g s p l a c e d on c i r c u l a r p l a t e s at both ends of the s e c t i o n , and the whole tank can then be p r e s s u r i z e d up to
10 a t m o s p h e r e s . The u p s t r e a m and d o w n s t r e a m r e g u l a t o r s and v a l v e s which c o n t r o l the flow r a t e s and p r e s s u r e s in the tank ha ve v e r y fast t i m e r e s p o n s e c h a r a c t e r i s t i c s . O p e r a t i n g the s y s t e m at 7 a t m ,
s t e a d y flow in the t e s t s e c t i o n is e s t a b l i s h e d in about 150 m i l l i s e c o n d s with v e l o c i t i e s up to 50 f t / s e c . N o r m a l running t i m e s v a r y f r o m 1 to 3 s e c o n d s . The t u r b u l e n c e l e v e l is l e s s than 0. 5 ^ . Reynolds n u m b e r
5
at 10 a t m o s p h e r e s can be as high as 10 / c m . Adjustable side w a l l s which span the t e s t s e c t i o n a r e u s e d to adjust or r e m o v e p r e s s u r e g r a d i e n t s in the flow.
The t h i c k n e s s of the s p l i t t e r p í a t e , which s e p a r a t e s the two s t r e a m s till they m e e t at the nozzles exit, is a p p r o x i m a t e l y . 002" at i t s d o w n s t r e a m end.
Some m o d i f i c a t i o n s w e r e m a d e in the t e s t s e c t i o n in o r d e r to i m p o s e an a d v e r s e p r e s s u r e g r a d i e n t on the m i x i n g l a y e r . The 4 " x 1" n o z z l e s w e r e r e p l a c e d by 4 " x-|", so that the t r a n s i t i o n r e g i ó n needed in adjusting the flow to the d e s i r e d free s t r e a m conditions would be s h o r t e n e d . The side w a l l s of the working s e c t i o n , u s e d for the z e r o p r e s s u r e g r a d i e n t c a s e , h a d b e e n a solid wall for the high v e l o c i t y side and a l O ^ s l o t t e d wall for the o t h e r s i d e . To set the a d v e r s e p r e s s u r e g r a d i e n t (cf. F e k e t e , Ref. 8), we changed the se w a l l s for ones with adjustable s l a t s and i n s t a l l e d a p e r f o r a t e d p í a t e at the d o w n s t r e a m exit of the channel (Fig. 5).
4. 2 I n s t r u m e n t a t i o n
B e c a u s e of the v e r y s h o r t d u r a t i o n of the flow and the i n t r i n s i c difficulties of unknown c o m p o s i t i o n , high t u r b u l e n c e l e v é i s and f r e -q u e n c i e s , v e r y s o p h i s t i c a t e d and fast d a t a a c -q u i s i t i o n s y s t e m s w e r e u s e d for the c o l l e c t i o n of the d a t a ; this e q u i p m e n t , d e s c r i b e d in
s e c t i o n 4. 3, h a n d l e d the i n f o r m a t i o n coming f r o m t h r e e different types of p r o b e s : an a r r a y of s t a t i c p r e s s u r e t u b e s , a pitot t u b e , and an a s p i r a t i n g p r o b é .
4. 2a Static and P i t o t T u b e s
The a r r a y of s t a t i c p o r t s c o n s i s t s of six s t a t i c p r e s s u r e
the s e n s o r h o l e s of the l o n g e s t tube as r e f e r e n c e , e a c h of the s u c c e s -sive ones i s 1" l o w e r so that, by placing the a r r a y in the free s t r e a m of our t w o - d i m e n s i o n a l t u r b u l e n t s h e a r l a y e r o r i e n t e d in the s p a n w i s e d i r e c t i o n , the s t a t i c p r e s s u r e , or Ui(x) and U2(x), can be m e a s u r e d a t
e v e r y i n c h down to 5 " f r o m the s p l i t t e r p í a t e .
T h i s p r o b é w a s u s e d in the p r e l i m i n a r y m e a s u r e m e n t s m a d e to s e t up the d e s i r e d e x t e r n a l flow field; d u r i n g t h e s e m e a s u r e m e n t s one of the inputs of the p r e s s u r e t r a n s d u c e r w a s d i r e c t l y connected to the r e f e r e n c e tube; the o t h e r five s t a t i c p r e s s u r e tubes w e r e joined by m e a n s of p l á s t i c tubing to the e n t r i e s of a fast p r e s s u r e s c a n n e r
(Scanivalve), a device that s e q u e n t i a l l y c o m m u n i c a t e s e a c h one of the s t a t i c p o r t s to the c o l l e c t o r , which in t u r n goes to the o t h e r input of the p r e s s u r e t r a n s d u c e r .
The pitot t u b e , in connection with a D a t a m e t r i c s e l e c t r o n i c m a n o m e t e r and B a r o c e l differential p r e s s u r e s e n s o r , was u s e d to obtain d y n a m i c p r e s s u r e p r o f i l e s a c r o s s the mixing l a y e r , and to take the final m e a s u r e m e n t s in o r d e r to e s t a b l i s h an a d v e r s e p r e s s u r e g r a d i e n t .
4. 2b A s p i r a t i n g P r o b é
Due to the n e c e s s i t y of knowing the l o c a l c o m p o s i t i o n in our p l a ñ e t u r b u l e n t m i x i n g l a y e r , a novel p r o b é was developed for this study by B r o w n and Rebollo (Ref. 10). A c o m p l e t e a c c o u n t of the way i t w o r k s i s given in Appendix B .
figure 1 of Appendix B w e r e f i r s t , the r e p l a c e m e n t of the w i r e when-e v when-e r i t b r o k when-e ; s when-e c o n d , too l a r g when-e l when-e n g t h / d i a m when-e t when-e r r a t i o s whwhen-en using s m a l l e r w i r e s ; in addition, s o m e m i n o r p r o b l e m s w e r e r e l a t e d to the w a r m i n g u p of the g l a s s . B a s e d upon t h e s e c o n s i d e r a t i o n s our final d e s i g n was that of a n o r m a l hot w i r e i n s i d e a hollow h o l d e r . The hot w i r e is e n c l o s e d by a long tipped g l a s s hood which s l i d e s o v e r and is s e a l e d with epoxy a g a i n s t the outside s u r face of the h o l d e r (Fig. 7). To avoid flow i n s t a b i l i t i e s i n s i d e the p r o b é a v e r y g r a d u a l á r e a e x p a n -sión w a s c h o s e n for the g l a s s c o v e r , and in o r d e r to have f a s t e r t i m e r e s p o n s e * a s m a l l e r w i r e d i a m e t e r (. 0002") was u s e d .
4. 3 D e s c r i p t i o n of the E x p e r i m e n t a l E q u i p m e n t
T h e e q u i p m e n t u s e d d u r i n g the e x p e r i m e n t s could be divided into different c a t e g o r i e s a c c o r d i n g to i t s function. Some c o m p o n e n t s of the a p p a r a t u s r e c e i v e a p h y s i c a l signal f r o m the m e a s u r i n g p r o b e s and c o n v e r t i t into an analog v o l t a g e ; e. g. , e l e c t r o n i c m a n o m e t e r ( B a r o c e l ) , c o n s t a n t t e m p e r a t u r e a n e m o m e t e r ( T h e r m o - S y s t e m s ) ; that v o l t a g e is d i g i t i z e d by m e a n s of an A / D c o n v e r t e r , and it is d e l i v e r e d to an output unit; e. g. , i n c r e m e n t a l or s y n c h r o n o u s tape r e c o r d e r . A v e r y i m p o r t a n t function i s that of r e g u l a t i n g the flow of d a t a ; this t a s k i s p e r f o r m e d by the c o n t r o l e q u i p m e n t ; e. g. , Scanivalve and c o n t r o l l e r , e l e c t r o n i c pulsing c i r c u i t and c o u p l e r for the i n c r e m e n t a l tape r e c o r d e r , coupler (Data S l i c e r ) for the s y n c h r o n o u s tape r e c o r d e r .
4. 3a E l e c t r o n i c M a n o m e t e r and C o n s t a n t T e m p e r a t u r e A n e m o m e t e r -The b a s i c s y s t e m i s m a d e u p of the B a r o c e l p r e s s u r e differ-e n t i a l s differ-e n s o r and thdiffer-e Typdiffer-e 1014A D a t a m differ-e t r i c s differ-e l differ-e c t r o n i c m a n o m differ-e t differ-e r . The Type 1014A p r o v i d e s the e l e c t r i c a l e x c i t a t i o n for the p r e s s u r e s e n s o r and in t u r n a c c e p t s the p r e s s u r e g e n e r a t e d s i g n á i s for voltage c o n v e r s i ó n .
The B a r o c e l w a s u s e d to m e a s u r e the s t a t i c p r e s s u r e f r o m the a r r a y of tubes at s e v e r a l d o w n s t r e a m l o c a t i o n s in the f r e e s t r e a m when e s t a b l i s h i n g the p r e s s u r e g r a d i e n t , and to m e a s u r e the d y n a m i c p r e s s u r e a c r o s s the l a y e r when t r a v e r s i n g the pitot tube.
The a s p i r a t i n g p r o b é was u s e d in connection with a high
f r e q u e n c y and low n o i s e c o n s t a n t t e m p e r a t u r e a n e m o m e t e r Model 105 0 ( T h e r m o - S y s t e m s ) which p r o v i d e d the u s u a l feedback b r i d g e .
4. 3b A / D C o n v e r t e r and C o n t r o l E q u i p m e n t
The A / D c o n v e r t e r (Raytheon Model ADC M u l t i v e r t e r ) i s capable of o p e r a t i n g a t 3 3 , 000 c o n v e r s i o n s p e r s e c o n d while m u l t i -plexing up to 16 channels of analog data. The analog input via BNC
g
c o n n e c t o r s is b i p o l a r , + 1 0 v, into 10 o h m s . The r e s o l u t i o n i s 11 bits and sign; i. e. , 2048 counts on e i t h e r side of z e r o . T h e output i s
b i n a r y i n t e g e r , p o s i t i v e - t i m e l o g i c .
The Scanivalve is a scanning type p r e s s u r e s a m p l i n g valve for switching m ú l t i p l e p r e s s u r e p o i n t s . The p r e s s u r e t r a n s d u c e r is
s e q u e n t i a l l y connected to the v a r i o u s P p o r t s via a r a d i a l hole in the r o t o r which t e r m i n a t e s a t the c o l l e c t o r h o l e . As the r o t o r r o t a t e s , this c o l l e c t o r hole p a s s e s u n d e r the P p o r t s in the s t a t o r . Our
Scanivalve h a s a wafer s w i t c h with 24 e n t r i e s plus c o l l e c t o r , and it is d r i v e n by a L e d e x solenoid.
The solenoid c o n t r o l l e r r e g u l a t e s the stepping speed of our L e d e x solenoid d r i v e n S c a n i v a l v e . It will d r i v e the s o l e n o i d m o t o r a t any r a t e up to 20 s t e p s p e r s e c o n d . The c o m m a n d to a d v a n c e the Scanivalve one s t e p can be given r e m o t e l y , by an e x t e r n a l s w i t c h c l o s u r e of 5 m i l l i s e c o n d s m i n i m u m , or through a m a n u a l or l o c a l c o m m a n d p u s h button.
In o r d e r to c o n t r o l the flow of data coming f r o m the B a r o c e l , which is going to be w r i t t e n on tape by m e a n s of an i n c r e m e n t a l tape r e c o r d e r , a pulsing c i r c u i t , and an 8 b i t - 1 byte c o u p l e r b e t w e e n the R a y t h e o n M u l t i v e r t e r and the Kennedy i n c r e m e n t a l r e c o r d e r w e r e u s e d .
The e l e c t r o n i c p u l s i n g c i r c u i t p r o v i d e d the n e c e s s a r y p u l s e s to s t e p the stepping m o t o r , so that it would t r a v e r s e the pitot p r o b é a t any r a t e u p to 500 p u l s e s / s e c o n d (-¿ i n c h / s e c o n d ) , or 1000 p u l s e s / s e c -ond when slewing up the f r e q u e n c y of the p u l s e s ; t h e s e p u l s e s w e r e a l s o u s e d as a d o c k for the o p e r a t i o n of w r i t i n g the d a t a . The e l e c t r o n i c coupler s y n c h r o n i z e s the t r a v e r s i n g m e c h a n i s m with the R a y t h e o n M u l t i v e r t e r and the d i g i t a l i n c r e m e n t a l tape r e c o r d e r .
T h e c o u p l e r c o n t r o l s p r o v i d e a l i m i t e d choice of data w o r d l e n g t h , r e c o r d length, and file length. A s h o r t r e c o r d containing a t w o - d i g i t f i l e - i d e n t i f i c a t i o n n u m b e r m a y be w r i t t e n if d e s i r e d .
4. 3c M a g n e t i c Tape R e c o r d e r s
All data s i g n á i s w e r e w r i t t e n on a 9 t r a c k 800 B P I m a g n e t i c tape u s i n g two types of r e c o r d e r s . The Kennedy Model 1600/360 d i g i t a l i n c r e m e n t a l r e c o r d e d all p r e s s u r e m e a s u r e m e n t s . Data can be c o m m a n d e d to be w r i t t e n , by the c o u p l e r , at any r a t e u p to 500 b y t e s p e r s e c o n d , or 1000 b y t e s / s e c o n d if we s l e w up the frequency of the w r i t e p u l s e s coming f r o m the pulsing clock.
V. E X P E R I M E N T A L PROCEDURES
5. 1 F a c i l i t y T e s t s
A l a r g e n u m b e r of t e s t s w e r e p e r f o r m e d to know and, when-e v when-e r w a s p o s s i b l when-e , to i m p r o v when-e thwhen-e flow quality of thwhen-e t when-e s t s when-e c t i o n as well as s o m e of the c h a r a c t e r i s t i c s of the i n s t r u m e n t s we w e r e going to u s e .
During the e x p e r i m e n t s c a r r i e d out in r e f e r e n c e 1 the turbu-l e n c e turbu-l e v e turbu-l in the f r e e s t r e a m w a s thoroughturbu-ly m e a s u r e d and r e d u c e d f r o m about l 4 to 0. 3 $ u s i n g h o n e y c o m b s and s c r e e n s . A b s o r b i n g m a t e r i a l to d a m p the a c o u s t i c l e v e l w a s u s e d in the supply s e c t i o n of the s y s t e m . Side wall p o s i t i o n s to r e m o v e p r e s s u r e g r a d i e n t s w e r e i n v e s t i g a t e d for different v e l o c i t y r a t i o s .
To p r o v i d e a r e f e r e n c e e x p e r i m e n t for the e x p e r i m e n t s with two g a s e s , B r o w n and R o s h k o m a d e m e a s u r e m e n t s in the s h e a r l a y e r between two s t r e a m s of n i t r o g e n t r a v e r s i n g a pitot tube and a hot w i r e side by side {-¿ u a-part). Since a c c u r a t e e x p e r i m e n t a l m e a s u r e m e n t s for h o m o g e n e o u s flows had b e e n m a d e p r e v i o u s l y by L i e p m a n n and Laufer
(Ref. 25), S p e n c e r and J o n e s (Ref. 26), M i l e s and Shih (Ref. 27), and o t h e r s , sufficient c o m p a r a t i v e i n f o r m a t i o n e x i s t e d to v e r i f y the validity of the e x p e r i m e n t a l data p r o c e s s i n g p r o c e d u r e s . No significant differe n c differe s w differe r differe found b differe t w differe differe n thdiffere r differe s u l t s of B r o w n ' s and R o s h k o ' s h o m o -geneous e x p e r i m e n t s and those of o t h e r i n v e s t i g a t o r s .
F u r t h e r t e s t s of the facility have been u n d e r t a k e n during the p r e s e n t i n v e s t i g a t i o n and we s h a l l d e s c r i b e s o m e of t h e m .
hot w i r e (upper t r a c e ) and a pitot tube (lower t r a c e ) , p l a c e d side by s i d e , to the p r o c e s s of s t a r t i n g the flow in the t e s t s e c t i o n . The h o r i z o n t a l s c a l e is 100 m i l l i s e c o n d s / c m , v e r t i c a l s c a l e is 1 v o l t / c m . The t i m e needed to e s t a b l i s h s t e a d y flow is about 15 0 m i l l i s e c o n d s for a tank p r e s s u r e of 7 atm. and v e l o c i t i e s of 30 f t / s e c .
When m e a s u r i n g the s t a t i c p r e s s u r e in the f r e e s t r e a m s (Sec. 5. 2) a v e r y i m p o r t a n t q u e s t i o n a r i s e s : how long should we k e e p the Scanivalve connecting a c e r t a i n p r e s s u r e s t a t i c tube with the
p r e s s u r e t r a n s d u c e r ? A m i n i m u m valué is s e t by the t i m e r e s p o n s e of the line once the s t e p c o m m a n d h a s b e e n given to the S c a n i v a l v e , a m á x i m u m t i m e i s l i m i t e d by the d u r a t i o n of the flow.
To m e a s u r e the t i m e r e s p o n s e of the l i n e , the total head p r o b é w a s connected to, s a y , e n t r y No. 2 of the fluid wafer switch; the c o l l e c t o r joined to one of the inputs of the p r e s s u r e s e n s o r , and the o t h e r input was d i r e c t l y c o m m u n i c a t e d to the s t a t i c p r o b é ; the r o t o r of the p r e s s u r e s c a n n e r was facing e n t r y No. 1 at this t i m e .
t i m e r e s p o n s e of the line going through the Scanivalve i s a p p r o x i m a t e l y 100 m i l l i s e c o n d s when u s i n g a p l á s t i c tubing of 2 / l 6 " in d i a m e t e r .
Although the t w o - d i m e n s i o n a l i t y of the flow h a d a l r e a d y been shown in r e f e r e n c e 1, s e v e r a l t e s t s w e r e m a d e to c o n f i r m it.
Two p l a t i n u m hot w i r e s . 0002" in d i a m e t e r , and a l m o s t the s a m e cold r e s i s t a n c e , w e r e s e p a r a t e d 2 " a p a r t and o p e r a t e d at the s a m e o v e r h e a t r a t i o . The two w i r e s w e r e p l a c e d a p p r o x i m a t e l y in the m i d d l e of the m i x i n g l a y e r , and aligned in the s p a n w i s e d i r e c t i o n . The outputs f r o m a two channel c o n s t a n t t e m p e r a t u r e a n e m o m e t e r a r e
shown, for s e v e r a l d o w n s t r e a m locations, in the o s c i l l o s c o p e p i c t u r e s of figure 10. The p h o t o g r a p h s r e v e a l a high d e g r e e of c o r r e l a t i o n b e t w e e n the two s i g n á i s , as would be e x p e c t e d f r o m a t w o - d i m e n s i o n a l flow. The e x p e r i m e n t s w e r e p e r f o r m e d at a tank p r e s s u r e of 3 a t m . , but it should be noted that the s a m e t e s t s done by B r o w n and R o s h k o at 7 atm. c o r r e l a t e d equally well the outputs of the two p l a t i n u m h o t w i r e s .
5. 2 P r o ce d u r e to Set Up the E q u i l i b r i u m F l o w
P r e l i m i n a r y m e a s u r e m e n t s to s e t up the d e s i r e d p o t e n t i a l flow w e r e m a d e with the a r r a y of s t a t i c p r e s s u r e p r o b e s ; a s c h e m a t i c r e p r e s e n t a t i o n of the way this w a s done is shown in figure 11; the
r e q u i r e d timing of the o p e r a t i o n is p r e s e n t e d in figure 12. F i v e r e c o r d s w e r e w r i t t e n for e v e r y r u n , i. e. , one r e c o r d for e a c h s t a t i c p r e s s u r e tube; the i n t e r v a l of t i m e between r e c o r d gaps w a s 470 m i l l i
