Document
M I X I N G I N SUPERSONIC FLOW GASL TECHNICAL REPORT NUMBER 592 by: J. H. Morgenthaler Project 8027 Page i Contract #NAS8-20066
COPY # ‘7
MIXING IN SUPERSONIC FLOW GASL TECHNICAL REPORT # 592 bY J. H. Morgenthaler Prepared for National Aeronautical & Space Administration George C. Marshall Space Flight Center Huntsville, Alabama Prepared by General Applied Science Laboratories, Inc.
Merrick & Stewart Avenues Westbury, L.I. ,N.Y.
Approved by: A. pekri President September 1966 TR 592 Page ii SUMMARY A method was p r e s e n t e d f o r t h e d e t e r m i n a t i o n of t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s i n multicomponent flows w h i c h r e q u i r e s a
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s i n g l e d i f f e r e n t i a t i o n of experimental Y , Vz, H I and p d a t a .
Assumptions which a l l o w s i m p l i f i c a t i o n of t h e g e n e r a l e q u a t i o n s of change, i . e . , c o n t i n u i t y , d i f f u s i o n , momentum, and energy
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e q u a t i o n s , w i t h o u t the r e s t r i c t i o n t h a t a P / a r = 0 w e r e d i s c u s s e d .
A c o n s t a n t s t a g n a t i o n temperature was shown t o be a p a r t i c u l a r s o l u t i o n of t h e energy e q u a t i o n when Le and P r (and hence T T sc ) are u n i t y .
T L i m i t a t i o n s of t h e method w e r e i n v e s t i g a t e d u s i n g a t e s t case, i n which assumed v a l u e s of t h e t r a n s p o r t c o e f f i c i e n t s w e r e - used t o g e n e r a t e downstream Y and V p r o f i l e s , and these computed Z p r o f i l e s t h e n used i n a n attempt t o reproduce t h e o r i g i n a l l y assumed t r a n s p o r t c o e f f i c i e n t s . T h i s t e c h n i q u e allowed d i r e c t comparison of d e r i v e d c o e f f i c i e n t s w i t h t h e i n p u t v a l u e s .
R e s u l t s of these comparisons showed t h e s p a c i n g of t h e d a t a p o i n t s t o be a c r i t i c a l parameter, b u t t h a t i n t e r p o l a t e d v a l u e s c o u l d be used i n c o n j u n c t i o n w i t h o r i g i n a l d a t a p o i n t s ( i f p r o p e r l y smoothed). For t h e t e s t c a s e , i n w h i c h the r a d i u s of t h e mixing r e g i o n c o n s i d e r e d was 2 i n . , p o i n t s p a c i n g s of 0 . 0 2 i n . appeared s u f f i c i e n t l y close t o y i e l d r e a s o n a b l e r e s u l t s ; whereas, s p a c i n g s of 0.06 i n . d i d n o t .
An e s t i m a t e of t h e t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s f o r t h e c a s e of c o a x i a l f r e e - j e t mixing of s u b s o n i c hydrogen w i t h Mach 1 . 6 a i r was o b t a i n e d , using d a t a o f Reference 8, a s an a p p l i - c a t i o n of the method p r e s e n t e d h e r e i n .
TR 592 Page iii TABLE OF CONTENTS S e c t i o n O e s c r i p t i o n Paqe N o .
summary ii I I n t r o d u c t i o n I1 Analysis I11 S i m p l i f i e d Analysis 1 2 P a r t i c u l a r S o l u t i o n o f Energy E q u a t i o n I V V T e s t of N u m e r i c a l Technique V I References Appendix A F i g u r e s Tables
D
\ \ TR 592
S
Page i v
f
s
LIST OF FIGURES F i q u r e N o . T i t l e Paqe N o .
1 Computed Hydrogen C o n c e n t r a t i o n P r o f i l e s , 3 2 T e s t Case
I
2 Computed V e l o c i t y P r o f i l e s , T e s t C a s e 3 3 3 E f f e c t of R a d i a l Grid Spacing on e , T e s t C a s e E f f e c t of Axial P o s i t i o n on 5 , T e s t Case
J
5 Experimental C o n c e n t r a t i o n P r o f i l e s , Case C
a
6 Experimental V e l o c i t y P r o f i l e s , Case C 7 Comparison of Compute C o n c e n t r a t i o n 3 8 P r o f i l e s w i t h Experimental Data 8 Comparison of Computed V e l o c i t y P r o f i l e s 3 9 w i t h Experimental Data TR 592 Page v LIST OF TABLES T i t l e Table N o .
1 T e s t Conditions f o r the Hydrogen-Air C o a x i a l , 40 Mixing Experiments of Reference 8 4 1 Mass and Momentum Balance f o r Hydrogen-Air , 2 C o a x i a l , Mixing Experiments of Reference 8 T u r b u l e n t T r a n s p o r t C o e f f i c i e n t s , C a s e C 3 4 4 3 Turbulent T r a n s p o r t C o e f f i c i e n t s Used I n Numerical I n t e g r a t i o n , C a s e A Comparison Between Experimental and Computed 44 C o n c e n t r a t i o n P r o f i l e s , Case A 6 Comparison Between Experimental and Computed V e l o c i t y P r o f i l e s , Case A 7 Turbulent T r a n s p o r t C o e f f i c i e n t s Used I n Numerical I n t e g r a t i o n , C a s e B 8 Comparison Between Experimental and Computed 47 C o n c e n t r a t i o n P r o f i l e s , Case B 9 Comparison Between Experimental and Computed 48 V e l o c i t y P r o f i l e s , C a s e B 10 49 T u r b u l e n t Transport C o e f f i c i e n t s Used I n Numerical I n t e g r a t i o n , C a s e C 11 Comparison Between Experimental and Computed C o n c e n t r a t i o n P r o f i l e s , C a s e C 5 1 Comparison B e t w e e n Experimental and Computed 1 2 C a s e C V e l o c i t y P r o f i l e s , 13 Comparison of E Obtained from Smooth D a t a d Versus That Obtained from Cosine F i t s , Case A
14 Comparison of 5 Obtained from Smoothed D a t a 53
Versus That Obtained from Cosine F i t s , C a s e A TR 592 Page v i NOMENCLATURE a A r b i t r a r y c o n s t a n t used t o s h i f t o r i g i n f o r L a u r e n t series, f t C S p e c i f i c heat a t c o n s t a n t p r e s s u r e , ft-lbf/lbm-'R pi Molecular d i f f u s i v i t y , o r d i f f u s i o n c o e f f i c i e n t , ft3/eec Di Eddy d i f f u s i v i t y of m a s s , f t a / s e c Ed i Eddy d i f f u s i v i t y of heat, ft2/sec E h a Eddy d i f f u s i v i t y of momentum, f t /sec Em f A r b i t r a r y c o n s t a n t used i n L a u r e n t series Dimensional c o n s t a n t , 32.174 lbm-ft/lbf-sec2 g C H S t a g n a t i o n e n t h a l p y , f t - l b f / l b m h S t a t i c e n t h a l p y , f t-lbf / l b m T o t a l mass f l o w r a t e w i t h i n n t h stream t u b e d i v i d e d by 2a k n
( d e f i n e d by Equation ( 1 2 ) ) , lbm/sec
L e L e w i s number, PC ~ . / k
i P = Le Turbulent Lewis number, E /
T d . Eh
i M Mach number P S t a t i c p r e s s u r e , l b f / f t 2 P r P r a n d t l number , CpLl/k P r T u r b u l e n t P r a n d t 1 number, Em/E T h R a d i a l c o o r d i n a t e , f t r Coordinate of w a l l o r c e n t e r l i n e , f t r* r R a d i a l c o o r d i n a t e of s t r e a m l i n e , f t S sc S c h m i d t number , p / , D i i sc T u r b u l e n t S c h m i d t number , Ti Em/Ed i T Absolute t e m p e r a t u r e , O R V Mass-average or b u l k v e l o c i t y , ft/sec t TR 592 Page v i i
NOMENCLATURE (contd . )
Mass f r a c t i o n yi Axial c o o r d i n a t e , f t z
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Eddy v i s c o s i t y , pEm, lbm/ft-sec 4 3
Eddy thermal c o n d u c t i v i t y , OC %, ft-lbf/sec-ft-'R
x P Molecular shear v i s c o s i t y , lbm/ft-sec I-1 T u r b u l e n t m a s s t r a n s f e r c o e f f i c i e n t , F E d , lbm/ft-sec D e n s i t y , l b m / f t 3 P D i s s i p a t i o n f u n c t i o n , ft-lbf/ft3-sec # A r b i t r a r y f u n c t i o n da S u b s c r i p t s e E x t e r n a l ( a i r ) stream i P a r t i c u l a r molecular (or atomic) species j J e t (Hydrogen) stream r R a d i a l component S S t r e a m l i n e T T u r b u l e n t t T o t a l o r s t a g n a t i o n W W a l l z Axial component Arrows d e n o t e v e c t o r s : bars time-averaged; and primes t u r b u l e n t f l u c t u a t i n g q u a n t i t i e s .
TR 592 Page 1 M I X I N G I N SUPERSONIC FLOW I. INTRODUCTION A basic unders-anding of t u r b u l e n t mixing i s important f o r a wide r a n g e of c u r r e n t a p p l i c a t i o n s , i n c l u d i n g s u p e r s o n i c combustion ramjet e n g i n e s and flows about launch and r e e n t r y v e h i c l e s . For example i n a s u p e r s o n i c combustor employing a d i f f u s i o n flame where mixing i s the c o n t r o l l i n g mechanism, p r e d i c t i o n of t h e mixing i s c r i t i c a l t o an understanding of t h e combustion phenomenon. I n a hydrogen f u e l e d upper stage v e h i c l e , i n which hydrogen i s vented d u r i n g t h e launch phase, knowledge of t h e mixing i s n e c e s s a r y f o r the e v a l u a t i o n of p o t e n t i a l h a z a r d s t o the v e h i c l e .
U n f o r t u n a t e l y , no formal t h e o r e t i c a l development f o r p r e d i c t i n g i s c u r r e n t l y a v a i l a b l e so t h a t a mixing i n complex t u r b u l e n t flows phenomenological approach must be a p p l i e d . Such approaches (e.g., P r a n d t l ' s mixing t h e o r y , R e i c h a r d t ' s i n d u c t i v e t h e o r y , and von Karman's s i m i l a r i t y h y p o t h e s i s ) have been used over t h e y e a r s as a
means f o r t r e a t i n g specific mixing problems . More r e c e n t l y v a r i o u s
2-5 eddy v i s c o s i t y models have been proposed e I n some cases s o l u t i o n of t u - b u l e n t mixing problems have been o b t a i n e d by i n c o r p o r a t i n g t h e s e models i n t o a f i n i t e d i f f e r e n c e t e c h n i q u e f o r s o l v i n g the a p p r o p r i a t e e q u a t i o n s 2,6,7 An a l t e r n a t i v e approach h a s been c o n s i d e r e d by s e v e r a l inves- t i g a t o r s i n which e x p e r i m e n t a l d a t a are used t o determine t u r b u l e n t 8-11 t r a n s p o r t c o e f f i c i e n t s . These c o e f f i c i e n t s g e n e r a l l y are a p p l i c - able o n l y f o r the p a r t i c u l a r experimental c o n d i t i o n s f o r w h i c h t h e y have been determined. They are u s e f u l for e v a l u a t i n g the d e g r e e of mixing o b t a i n e d w i t h a p a r t i c u l a r t e s t geometry, and f o r comparing d i f f e r e n t g e o m e t r i e s and f l o w c o n d i t i o n s ; however, t h e i r major use- f u l n e s s u l t i m a t e l y should be c o r r e l a t i o n of s u p e r s o n i c mixing d a t a s o t h a t p r e d i c t i o n s can be made, a t l e a s t w i t h i n t h e r a n g e of v a r i a b l e s of i n t e r e s t .
TR 592 Page 2 I n References 8 and 9 t h e assumption w a s made t h a t normalized c o s i n e p r o f i l e s a d e q u a t e l y r e p r e s e n t e d b o t h c o n c e n t r a t i o n and v e l o c i t y d a t a a t r e g i o n s downstream of t h e p o t e n t i a l core of a c o a x i a l supersonic j e t . These f i t t e d p r o f i l e s w e r e d i f f e r e n - t i a t e d t w i c e and used i n the d e t e r m i n a t i o n of t h e t r a n s p o r t co- e f f i c i e n t s , Although c o s i n e p r o f i l e s may r e a s o n a b l y w e l l approxi- mate experimental d a t a i n r e g i o n s i n which s i m i l a r i t y between r a d i a l c o n c e n t r a t i o n p r o f i l e s and between v e l o c i t y p r o f i l e s e x i s t s , Hinze12 shows t h a t t r u e s i m i l a r i t y does not e x i s t f o r the g e n e r a l case considered i n References 8 and 9 , i n which the v e l o c i t y of t h e j e t and t h e e x t e r n a l stream are of t h e s a m e g e n e r a l magnitude, i.e., t h e i r v e l o c i t i e s are s i g n i f i c a n t l y d i f f e r e n t b u t n e i t h e r stream i s q u i e s c e n t . S i n c e c o s i n e p r o f i l e s a r e o n l y an approxima- t i o n f o r t h e s e d a t a , s l o p e s o b t a i n e d by d i f f e r e n t i a t i n g t h e m might n o t adequately r e p r e s e n t t r u e l o c a l v a r i a t i o n s of t h e e x p e r i m e n t a l d a t a , and the v a l i d i t y of t r a n s p o r t c o e f f i c i e n t s d e r i v e d by t h i s procedure must be q u e s t i o n e d .
For t h i s r e a s o n , an a l t e r n a t i v e approach w a s s e l e c t e d i n References 10 and 11, i n which t r a n s p o r t c o e f f i c i e n t s w e r e deter- mined by a s i n g l e numerical d i f f e r e n t i a t i o n of e x p e r i m e n t a l con- c e n t r a t i o n , v e l o c i t y , and d e n s i t y p r o f i l e s o b t a i n e d a t t h r e e or more axial s t a t i o n s ; t h i s approach i s n o t l i m i t e d t o r e g i o n s w h e r e s i m i l a r i t y e x i s t s i n t h e flow. Polynomials w e r e f i t t e d through f i v e c l o s e l y spaced d a t a p o i n t s and t h e r e q u i r e d d e r i v a t i v e s o b t a i n e d by d i f f e r e n t i a t i n g t h e polynomial u s i n g a f i v e - p o i n t , second-order running smoothing r o u t i n e 1 3 . The need f o r e v a l u a t i n g second d e r i v a t i v e s w a s overcome by i n t e g r a t i n g the e q u a t i o n s of change once i n the r a d i a l d i r e c t i o n f r o m a boundary t o a stream- l i n e . Experimental r e s u l t s w e r e l i m i t e d t o the case of s o n i c TR 5 9 2 Page 3 r a d i a l and a x i a l i n j e c t i o n of c o l d hydrogen through c i r c u m f e r e n t i a l w a l l s l o t s i n t o c o l d Mach 2 and 3 a i r streams. The energy e q u a t i o n w a s n o t c o n s i d e r e d i n d e t a i l i n t h i s i n v e s t i g a t i o n s i n c e measure- ments showed the s t a g n a t i o n t e m p e r a t u r e remained approximately c o n s t a n t t h r o u g h o u t the mixing r e g i o n . T h e v a l i d i t y o f t h e re- s u l t i n g c o e f f i c i e n t s w a s tested u s i n g a numerical i n t e g r a t i o n t e c h n i q u e (Crank-Nicolson) i n w h i c h t h e t r a n s p o r t c o e f f i c i e n t s and r a d i a l v e l o c i t y w e r e used i n s o l v i n g the d i f f u s i o n and a x i a l momentum e q u a t i o n s both separately and s i m u l t a n e o u s l y . Agreement between com- p u t e d and e x p e r i m e n t a l c o n c e n t r a t i o n and v e l o c i t y p r o f i l e s a t s e v e r a l downstream a x i a l s t a t i o n s w a s c o n s i d e r e d s a t i s f a c t o r y e v i d e n c e t h a t v a l i d eddy c o e f f i c i e n t s had b e e n d e r i v e d from t h e e x p e r i m e n t a l pro- f i l e s . O f c o u r s e , agreement between computed and e x p e r i m e n t a l p r o f i l e s merely d e m o n s t r a t e s the c o n s i s t e n c y o f t h e eddy c o e f f i c i e n t s w i t h t h e o r i g i n a l p r o f i l e s from which t h e y w e r e d e r i v e d .
. .
The a n a l y s i s p r e s e n t e d herein f o r coaxial injsction is m o r e g e n e r a l t h a n t h a t p r e v i o u s l y r e p o r t e d , s i n c e r a d i a l i n t e g r a t i o n o f t h e g e n e r a l axisymmetric d i f f u s i o n and momentum e q u a t i o n s as w e l l as t h e s i m p l i f i e d e q u a t i o n s , and a d e t a i l e d t r e a t m e n t of t h e energy e q u a t i o n are c o n s i d e r e d . I n a d d i t i o n , a n a l y s i s of a t e s t case i s p r e s e n t e d w h i c h c l a r i f i e s c e r t a i n p o i n t s of t h e numerical d a t a h a n d l i n g t e c h n i q u e s .
I n t h i s t e s t case, assumed v a l u e s of t h e t r a n s p o r t c o e f f i c i e n t s w e r e used t o g e n e r a t e downstream c o n c e n t r a t i o n and v e l o c i t y p r o f i l e s , and these computed p r o f i l e s t h e n used i n an attempt t o r e p r o d u c e t h e o r i g i n a l l y assumed t r a n s p o r t c o e f f i c i e n t s . Using t h i s t e c h n i q u e , t h e d e r i v e d c o e f f i c i e n t s c o u l d be compared d i r e c t l y w i t h t h e i n p u t v a l u e s .
U n f o r t u n a t e l y , no completely a d e q u a t e e x p e r i m e n t a l d a t a w e r e a v a i l a b l e f o r u s e f o r t'he d e t e r m i n a t i o n of t r a n s p o r t c o e f f i c i e n t s f o r the case o f i n t e r e s t o f s u p e r s o n i c , c o a x i a l , f r e e - j e t mixing. H o w e v e r , TR 592 Page 4 e x p e r i m e n t a l d a t a p r e s e n t e d i n Reference 8, w h i c h g e n e r a l l y con- t a i n e d f i v e or s i x r a d i a l e x p e r i m e n t a l d a t a p o i n t s a t f i v e o r s i x be used as a first approximation i f a d d i t i o n a l a x i a l s t a t i o n s , could p o i n t s w e r e generated by i n t e r p o l a t i o n . A n a l y s i s of these d a t a i s p r e s e n t e d i n t h e Appendix as an a p p l i c a t i o n of the method p r e s e n t e d h e r e i n . Since any e r r o r s i n the o r i g i n a l p o i n t s would be t r a n s - m i t t e d t o the i n t e r p o l a t e d p o i n t s , d i s c r e p a n c i e s i n t h e o r i g i n a l p o i n t s would be magnified when t h e r e s u l t i n g p r o f i l e s w e r e d i f f e r - e n t i a t e d . For t h i s r e a s o n , no a t t e m p t w a s made t o u t i l i z e t h e s e d a t a f o r o b t a i n i n g even an e m p i r i c a l mixing model; o n l y s i m p l i f i e d t r e n d s , suggested by smoothing the r a w t r a n s p o r t c o e f f i c i e n t s , w e r e obtained. F o r t u n a t e l y , t h e s e t r e n d s w e r e shown t o be r e a s o n - a b l y c o n s i s t e n t w i t h the o r i g i n a l e x p e r i m e n t a l d a t a because com- puted and experimental c o n c e n t r a t i o n and v e l o c i t y p r o f i l e s agreed cpite w e l l a t each downstream a x i a l s t a t i o n a t w h i c h e x p e r i m e n t a l d a t a w e r e a v a i l a b l e . Of c o u r s e closely-spaced a c c u r a t e e x p e r i - mental d a t a w i l l be r e q u i r e d i n f u t u r e w o r k t o o b t a i n d e t a i l e d v a r i - a t i o n s and semiempirical models of the t r a n s p o r t c o e f f i c i e n t s .
TR 592 Page 5 11. ANALYSIS G e n e r a l l y , t h e s t a r t d g p o i n t i n t u r b u l e n t a n a l y s e s i s the hypothesis t h a t the Navier-Stokes e q u a t i o n s and the other equa- t i o n s of change are s a t i s f i e d by i n s t a n t a n t a n e o u s v a l u e s o f the v e l o c i t y , c o n c e n t r a t i o n , and d e n s i t y . However, a group of French s c i e n t i s t s r e c e n t l y has o b j e c t e d t o t h i s h y p o t h e s i s ; t h e y f e e l t h a t s i n c e a t u r b u l e n t v e l o c i t y f i e l d i s i n " p u r e chaos", the i n s t a n - t a n e o u s v e l o c i t y of a p a r t i c l e of f l u i d c o u l d n o t be s u f f i c i e n t l y r e g u l a r t o s a t i s f y a system of p a r t i a l d i f f e r e n t i a l e q u a t i o n s .
O f c o u r s e , the same o b j e c t i o n c a n be a p p l i e d t o u s e o f t h e t u r b u l e n t c o n t i n u i t y , d i f f u s i o n , and energy e q u a t i o n s . U n f o r t u n a t e l y , no s u b s f f t u t e f o r these e q u a t i o n s h a s been proposed, so t h a t it i s n e c e s s a r y t o accept t h e m as the s t a r t i n g p o i n t i n t u r b u l e n t a n a l y s e s , a t l e a s t as t h e best approximation a v a i l a b l e .
P a i states t h a t t h e f i n a l and l o g i c a l s o l u t i o n o f the t u r b u l e n c e problem w i l l r e q u i r e a p p l i c a t i o n of the methods o f s t a t i s t i c a l mechanics. This approach would r e q u i r e e x p r e s s i n g the t u r b u l e n t - t r a n s p o r t r a t e o f a t r a n s f e r a b l e q u a n t i t y c o m p l e t e l y i n t e r m s of s t a t i s t i c a l f u n c t i n n s ~f t3e turbulent v e l o c i t y f i e l d and of boundary or i n i t i a l c o n d i t i o n s . U n t i l such a character- i z a t i o n i s a v a i l a b l e , any s o l u t i o n of t r a n s p o r t problems must be i n c o m p l e t e and a t best approximate (i .e. , semiempirical) 15. Also before a r a t i o n a l s t a t i s t i c a l t h e o r y o f t u r b u l e n c e c a n be developed a l o n g the l i n e s of c l a s s i c a l s t a t i s t i c a l mechanics, it i s n e c e s s a r y t h a t u n i q u e n e s s and e r g o d i c theorems be e s t a b l i s h e d as t h e y have for the case of c l a s s i c a l , s t a t i s t i c a l mechanics .
S i n c e the Navier-Stokes e q u a t i o n s are n o n l i n e a r , the proof o f a g e n e r a l u n i q u e n e s s theorem i s e x t r e m e l y d i f f i c u l t , i . e . , t h a t a g i v e n i n i t i a l s t a t e o f a system a t a p a r t i c u l a r t i m e w i l l u n i q u e l y d e t e r m i n e i t s s t a t e a t any o t h e r t i m e . I n e x p e r i m e n t a l TR 592 Page 6 i n v e s t i g a t i o n s time-average q u a n t i t i e s , w h i c h depend on a p a r t i c u l a r ensemble, are used almost e x c l u s i v e l y because i n p r a c t i c e it i s impossible t o o b t a i n s t a t i s t i c a l averages e x p e r i m e n t a l l y ; however, i n t h e o r e t i c a l i n v e s t i g a t i o n s s t a t i s t i c a l averages ( i - e . , ensemble averages) almost always are used. The e r g o d i c theorem of c l a s s i c a l s t a t i s t i c a l mechanics states the s u f f i c i e n t c o n d i t i o n s f o r the e q u a l i t y of these t w o k i n d s of a v e r a g e s f o r almost a l l samples. Unfortunaely, no e r g o d i c theorem has been proved i n f l u i d mechanics; however, the assumption t h a t t w o a v e r a g e s are e q u i v a l e n t i s f r e q u e n t l y made .
T h e r e f o r e , i n a t t a c k i n g p r a c t i c a l t u r b u l e n t mixing problems, i n s t a n t a n e o u s q u a n t i t i e s are r e s o l v e d i n t o time-averaged and f l u c t u - a t i n g q u a n t i t i e s , s u b s t i t u t e d i n t o t h e a p p r o p r i a t e e q u a t i o n s of change and time-averaged term-by-term. Some s i m p l i - f i c a t i o n of t h e r e s u l t i n g e q u a t i o n s i s o b t a i n e d b y assuming t h a t i n a d d i t i o n t o f l u c t u a t i o n s of v e l o c i t y , d e n s i t y , p r e s s u r e , and t e m p e r a t u r e (or e n t h a l p y ) , there are f l u c t u a t i o n s of m a s s f l u x [i.e., ( p 3 ) ] regarded as a s i n g l e p r o p e r t y . T h i s s i m p l i f i c a t i o n , w h i c h a l l o w s t h e s t e a d y s t a t e c o n t i n u i t y e q u a t i o n t o be s a t i s f i e d b y b o t h time-average and f 1 , u c t u a t i n g components f the m a s s f l u x , w a s f i r s t employed by Van D r i e s t i n h i s a n a l y s i s of t u r b u l e n t c o m -
pressible boundary l a y e r flow (e.g. Reference 16 . A p p l i c a t i o n
of these t e c h n i q u e s t o the s t e a d y , axisymmetric, e q u a t i o n s of change y i e l d s " * D e t a i l s of t h e method are p r e s e n t e d i n Reference 10.
TR 5 9 2 P a g e 7 T u r b u l e n t C o n t i n u i t y E q u a t i o n
-
a -
L a (917 r ) + - (ov,) = 0
r a r r az
T u r b u l e n t D i f f u s i o n E q u a t i o n
r
a?
i I - i 5 ( D + E d )
a
- -
+
r i ' az 3r 1 2 T u r b u l e n t N a v ier -Stoke s Moment urn E q u a t i o n s a. R a d i a l E q u a t i o n I - \ V
av
l - -
r - = - DV a Z g C r I 2 (I+<, )
a
+ -
ar J
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b. Axial E q u a t i o n I -
i av
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z z
+
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(4) T u r b u l e n t E n e r q y E q u a t i o n !-
+ ( p V r ) ' h! r Ti + ( o V r ) ' Y i ' h i ' r '
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( 5 ) TR 592
s
Page 8 where
I
( 7 ) *
I
' I - - 1
I
aV 2v r i, * z
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( D V r ) l V z I f - c3'- \ a r
I
? z -
2 € 4 ' ?VZ ' r 3vr
( O V Z ) ' V Z ' = - - 2 - - - - -
3 az r
ar /
Because o f i t s complexity, Equation ( 5 ) i s w r i t t e n i n t e r m s of Reynolds
I
t r a n s p o r t t e r m s r a t h e r t h a n t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s ; it must be s i m p l i f i e d before it may be a p p l i e d t o p r a c t i c a l problems. For t h e g e n e r a l case of s u b s o n i c flow and b o t h s u b s o n i c and s u p e r s o n i c boundary l a y e r flow, t u r b u l e n t t r a n s p o r t co-
I
e f f i c i e n t s u s u a l l y are d e f i n e d so t h a t t h e Reynolds t r a n s p o r t t e r m s c a n be r e p l a c e d i n t h e t u r b u l e n t e q u a t i o n s of change p r e s e r v i n g t h e I laminar form of these e q u a t i o n s . Of c o u r s e t h i s s u b s t i t u t i o n i s a r b i t r a r y and r e a l l y c a n be j u s t i f i e d o n l y i f these c o e f f i c i e n t s p r o v e 1 t o be a more u s e f u l r e p r e s e n t a t i o n t h a n t h e o r i g i n a l Reynolds t r a n s p o r t t e r m s . Because of t h e complexity of t h e momentum e q u a t i o n s , f o u r a r b i t r a r y c o e f f i c i e n t s of eddy v i s c o s i t y w e r e d e f i n e d i n o r d e r t o pre- s e r v e the laminar form of t h e e q u a t i o n s .
* The a x i a l d i s p e r s i o n c o e f f i c i e n t i s f r e q u e n t l y d e f i n e d i n a s i m i l a r 1 7 manner t o E .
d
i 2 I
i
TR 592 Page 9 U n f o r t u n a t e l y , a t p r e s e n t no e x p e r i m e n t a l p r o c e d u r e has been proposed f o r measur ng t h e t r a n s p o r t c o e f f i c i e n t s d e f i n e d i n Equat,ons ( 6 ) t o (11) d i r e c t l y . O f c o u r s e , i f one o f t h e c o e f f i c i e n t s i n each e q u a t i o n w e r e c o n s i d e r a b l y less important t h a n the other, so t h a t it c o u l d be n e g l e c t e d , each of t h e remaining t e r m s i n t h e e q u a t i o n s m i g h t be e v a l u a t e d u s i n g experimental d a t a , and the m i s s i n g c o e f f i c i e n t d e t e r m i n e d . One i n t e g r a t i o n o f Equations (1) t o (5) would e l i m i n a t e the d i f f i c u l t t a s k of o b t a i n i n g a c c u r a t e second d e r i v a t i v e s of t h e e x p e r i m e n t a l data. T h i s r e s u l t can be accomplished b y i n t e g r a t i n g t h e e q u a t i o n s once i n the r a d i a l d i r e c t i o n between a boundary and a s t r e a m l i n e , i . e . , a l i n e bounding a f i x e d m a s s f l o w d e s i g n a t e d r ( n ) .
S The v a l u e s of r (n) are found f o r a number of t e s t - s e c t i o n S l e n g t h s and v a r i o u s v a l u e s o f t h e c o n s t a n t k by a numerical evalu- n a t i o n of the i n t e g r a l r
*
where r d e s i g n a t e s either t h e c e n t e r l i n e o r a s t r e a m l i n e i n the free
*
stream. The boundary c o n d i t i o n s a t r are
-
ayi 3V
z ar a~ aE -
- - - = o 0 . - - - - - = - - - -
*
- vr a r ? r az ? r ;3z
r
I -
s i n c e no m a s s , momentum, nor energy, d i f f u s e i n t h e free stream, and the c e n t e r l i n e i s an a x i s o f symmetry. Equation ( 1 2 ) shows t h a t there w i l l be no n e t f l u x o f m a s s across r ( n ) b y c o n v e c t i o n , a l t h o u g h b o t h S f u e l and a i r c r o s s it b y d i f f u s i o n ( e q u a l masses i n o p p o s i t e d i r e c t i o n s ) .
TR 592 Page 10 M u l t i p l y i n g each t e r m i n the c o n t i n u i t y e q u a t i o n , Equation (I), b y r d r , i n t e g r a t i n g from either the f r e e stream or the c e n t e r l i n e r and a p p l i c a t i o n of the g e n e r a l i z e d L i e b n i t z formula f o r i n t e r - t o S changing the order of d i f f e r e n t i a t i o n and i n t e g r a t i o n y i e l d s But Equation ( 1 2 ) r e q u i r e s t h a t the second t e r m on the l e f t be z e r o , s o t h a t r S Equations ( 2 ) t o (4) may be i n t e g r a t e d i n a s i m i l a r manner.
Using Equation ( 1 5 ) there r e s u l t s D i f f u s i o n Equation R a d i a l Momentum Equation T R 5 9 2 Page 11 Axial Momentum Equation \ I The momentum f l u x t e r m s i n Equations ( 1 7 ) and (18) are z e r o f o r t h e l i r r i t r* = 0 ; however, t h e y are n o t n e c e s s a r i l y z e r o i n t h e f r e e stream ' e c a u s e c a n be f i n i t e , and therefore, ?r/3z # 0.
r Because of i t s complexity, Equation ( 5 ) , t h e e n e r g y w a s n o t i n t e g r a t e d u n t i l a f t e r t h e s i m p l i f i c a t i o n d i s c u s s e d below.
F o r t u n a t e l y , i n cases where t h e s t a g n a t i o n t e m p e r a t u r e does n o t v a r y s i g n i f i c a n t l y i n t h e mixing r e g i o n , it i s not n e c e s s a r y t o c o n s i d e r t h e energy e q u a t i o n a t a l l .
TR 592 Page 1 2 111, SIMPLIFIED ANALYSIS S i n c e s i x t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s occur i n E q u a t i o n s (15) t o (18), t h e r e are i n s u f f i c i e n t e q u a t i o n s a v a i l a b l e f o r t h e i r direct d e t e r m i n a t i o n , even i f a l l the remaining terms i n t h e s e e q u a t i o n s could be e x p e r i m e n t a l l y e v a l u a t e d . To reduce t h e number of unknowns some assumptions m u s t be made concerning t h e i r r e l a t i o n s h i p s , e . g . , t h a t some a r e e i t h e r e q u a l o r n e g l i g i b l e .
Of c o u r s e , even when such assumptions are made, a c c u r a t e determina- t i o n of t h e remaining t e r m s would be d i f f i c u l t u s i n g e x p e r i m e n t a l d a t a because of the need t o e v a l u a t e b o t h a x i a l and radial de- r i v a t i v e s of v a r i o u s terms. An a l t e r n a t i v e approach t o o m i t t i n g t e r m s , w h i c h l e a d s to c o n s i d e r a b l e s i m p l i f i c a t i o n , i s t o make 10,11 several g e n e r a l assumptions concerning t h e flow
. The
assumptions t h a t appear most r e a s o n a b l e f o r high-speed flow because of t h e importance of a x i a l l y - d i r e c t e d c o n v e c t i v e b u l k f l o w are: 1) Both d i f f u s i o n and energy transfer i n t h e a x i a l d i r e c t i o n by conduction and d i f f u s i o n , are n e g l i g i b l e compared t o t h a t i n t h e r a d i a l d i r e c t i o n : 2) Viscous normal stresses a r e n e g l i g i b l e ; 3 ) Viscous shear stresses depend p r i m a r i l y on t h s r a d i a l g r a d i e n t of a x i a l v e l o c i t y ( 3 v z / a x > > av r h z ) :
4) The t e r m v (ap/az) > > v ( a P / a r j
z r Assumtion 2 ) appears r e a s o n a b l e because a n o r d e r of magnitude a n a l y s i s shows v i s c o u s normal stresses a r e n e g l i g i b l e compared t o t h e p r e s s u r e even i n t h e boundary l a y e r where v i s c o u s f o r c e s a t t a i n t h e i r maxima. A consequence of t h i s assumption i s
-
which a p p e a r s r e a s o n a b l e f o r high-speed > > (pv,) 'Vz'
PITz v z
flow and t h a t mr vr > > (mi; it a l s o a l l o w s s i m p l i f i c a t i o n
r r
of t h e d i s s i p a t i o n f u n c t i o n 3 w h i c h becomes
TR 592 Page 1 3 ( 1 9 ) I f t h e a d d i t i o n a l reasonable assumption i s made t o simplify Equation ( 5 ) t h a t terms containing , p ' and h ' a s products (pVz) '
**
with o t h e r f l u c t u a t i n g terms are n e g l i g i b l e , Equations ( 2 ) t o ( 5 ) become r e s p e c t i v e l y , Diffusion Equation* 3Y b Y .
a y - i --
- a I ~ ( D + E ) r - = I
+ pv, -
pvr ar r or
3 r di Radial Momentum Equation Axial Momentum Equation Energy Equation where - 3V ' Z (pvz) ' V ' '>! -
' 2 br
r * I n Equation (20)E has been w r i t t e n f o r Ed di i **with t h e exception of (pVr) ' .
TR 592 Page 14 and These e q u a t i o n s a l o n g w i t h Equation (1) may be i n t e g r a t e d a s b e f o r e t o g i v e C o n t i n u i t y Equation P V r = PVZ D i f f u s i o n Equation R a d i a l Momentum Equation r r aV
i3V z a r s
z
1 rdr = - ( p + c 2 ) r ar r*
- ( I L +
€ 2 ) ar
r r S
+ s - g P d r - I S
r* C r* A x i a l Momentum Equation r d r = dr TR 592 Page 1 5 Enerqy Equation r S
-
a -
pv 'i r d r =
[ (k + PEh) r g] +
z az r* r S r s z V aV
[p(1- - ) + F E (1- ' ) I
P r m P r T r s
- - --
c 3 u n l e s s V p ' V ' and V p ' V ' a r e s m a l l compared t o S i n c e c 2 r z z r ~
--
pV'V' o r are approximately e q u a l , each of t h e s e c o e f f i c i e n t s must r z -~ be determined independently. However, t h e t r a n s f e r of a x i a l momentum is g e n e r a l l y of g r e a t e r i n t e r e s t t h a n t r a n s f e r of r a d i a l be q u i t e small i n a p p l i c a t i o n s such as momentum, which may f r e e j e t ' m i x i n g ; t h e r e f o r e , € 2 f r e q u e n t l y i s of primary i n t e r e s t .
F o r t u n a t e l y , it may be determined r e a d i l y from Equation ( 2 9 ) and
- - -
e x p e r i m e n t a l V , p , and P p r o f i l e s o b t a i n e d a t v a r i o u s a x i a l z
-
s t a t i o n s , a s long as t h e assumption i s made t h a t p ' v ' i s n e g l i g i b l e
--
compared t o P 3 .
Note t h a t there a r e EO r e s t r i c t i o n s c o n c e r n i n g r a d i a l p r e s s u r e v a r i a t i o n s i n Equations ( 2 2 ) and ( 2 9 ) a s t h e r e are i n t h e boundary l a y e r momentum e q u a t i o n .
I f t h e v i s c o u s t e r m s i n Equations ( 2 1 ) and (28) are n e g l i g i b l e , t h e s e e q u a t i o n s s t i l l would be u s e f u l f o r checking t h e c o n s i s t e n c y of t h e i n e r t i a l and p r e s s u r e t e r m s , and hence, t h e e x p e r i m e n t a l measurements.
If t h e i n e r t i a l t e r m s i n t h e s e e q u a t i o n s a l s o w e r e n e g l i g i b l e ,
Equations ( 2 1 ) and ( 2 2 ) [and ( 2 8 ) and ( 2 9 ) ] reduce t o t h e -
u s u a l boundary l a y e r momentum e q u a t i o n s , since i n t h i s case aP/ar=O.
Equation (3O)can be u s e d t o determine P r and L e i f T T i e x p e r i m e n t a l s t a g n a t i o n temperature p r o f i l e s a r e a v a i l a b l e .
Of
-
c o u r s e , f o r cases i n which T was n o t c o n s t a n t throughout t h e flow, t
- - -
t h e s e p r o f i l e s would be n e c e s s a r y f o r computation o f T , p , and V z .
TR 592 Page 16 The s t a g n a t i o n e n t h a l p y , H could be computed from u s i n g t h e t
i
r e l a t i o n
I
S t a t i c e n t h a l p i e s r e q u i r e d i n Equation (30) c o u l d be computed using t h e relations
I
and
I
h = h Y (33) i i S i n c e Ed and Em a r e determined from Equation ( 2 7 ) and ( 2 9 )
Sc can be o h t a i n e d f r o m t h e r e l a t i o n S c =E /E - L e can be
Ti T i m d ’ i T i e l i m i n a t e d from Equation (30) u s i n g t h e i d e n t i t y
I
L e E P r /scT
( 3 4 ) T i Ti and Equation (30) s o l v e d f o r P r . Once P r T h a s been determined, T Equation (34) c a n be used t o compute L e completing t h e d e t e r m i n a t i o n T i of t h e t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s .
TR 592 Page 1 7 I V . PARTICULAR S O L U T I O N OF ENERGY EQUATION The t u r b u l e n t energy e q u a t i o n , Equation ( 2 3 ) , can be r e w r i t t e n -
t
i n terms of t h e s t a g n a t i o n temperature, T b y using t h e r e l a t i o n s t
I
( 3 5 )
I
- - dH = C dTt
I (36)
i pi
c = c c T
1 (37)
P i P i i which y i e l d s f o r t h e r a d i a l d e r i v a t i v e 2Y
i aE aTt
- -
I + c H - - c -
a r p a r i a r
N e g l e c t i n g molecular t r a n s p o r t compared t o eddy t r a n s p o r t f o r s i m p l i c i t y , s u b s t i t u t i n g Equation (38) i n t o Equation ( 2 3 ) , and
I
using a s i m i l a r r e l a t i o n f o r t h e a x i a l d e r i v a t i v e , g i v e s
- C H
i i
I
t
I f Le and P r are u n i t y , t h e second I_ see Equation ( 2 0 ) j and T i T
I
l a s t t e r m s on t h e right-hand-side of Equation ( 3 9 ) a r e i d e n t i c a l l y zero: t h e r e f o r e , f o r t h i s s p e c i a l c a s e t h e energy e q u a t i o n becomes
I
T R 592 Page 18
1 a
- -
c ZE r
+
J
r a r
P h
c
c
I
i s a p a r t i c u l a r s o l u t i o n t o t h i s e q u a t i o n .
C l e a r l y , a c o n s t a n t T t T h e r e f o r e , if T (r) i s c o n s t a n t a t an i n i t i a l a x i a l s t a t i o n ,
I
t0 L e and P r (and t h e r e f o r e Sc ) are u n i t y , a n d t h e flow i s Ti T T i a d i a b a t i c , t h e s t a g n a t i o n t e m p e r a t u r e w i l l remain c o n s t a n t and
I
e q u a l t o T throughout t h e flow f i e l d * . Note t h a t H i s n o t
t 0 - -
c o n s t a n t a l s o , because E i s a f u n c t i o n of b o t h t h e Y ' s and H ' s
i i I
-
Tt i n g e n e r a l can remain c o n s t a n t throughout t h e flow f i e l d only when P r ScTi, and Le a r e u n i t y ; t h e r e f o r e , t h e procedure
T i -
I
used i n Reference 8 i s a g a i n i n g e n e r a l i n c o n s i s t e n t , - s i n c e T t
was assumed c o n s t a n t i n t h e computation of V and p , and t h e s e
z
I
v a l u e s t h e n used f o r computing v a l u e s o f Sc and L e c o n s i d e r a b l y T i Ti d i f f e r e n t from u n i t y .
I
* T h i s r e s u l t was f i r s t o b t a i n e d from a n a l y s i s of computer o u t p u t : t h e a n a l y s i s p r e s e n t e d h e r e i n was undertaken a t t h e s u g g e s t i o n of D r . R. Edelman o f GASL.
I
TR 592 Page 1 9 V. TEST O F NUMERICAL TECHNIQUE The major d i f f i c u l t y i n o b t a i n i n g t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s from experimental d a t a i s t h e e v a l u a t i o n of t h e a x i a l d e r i v a t i v e s of t h e i n t e g r a l s i n E q u a t i o n s ( 2 7 ) t o ( 3 0 ) . I n o r d e r t o e s t a b l i s h t h e s e d e r i v a t i v e s , e x p e r i m e n t a l p r o f i l e s m u s t be a v a i l a b l e a t a minimum of t h r e e o r f o u r a x i a l s t a t i o n s , so The o r d e r t h a t a polynomial can be f i t t e d and d i f f e r e n t i a t e d .
o f t h e polynomial can be up t o one less t h a n t h e number of a x i a l s t a t i o n s a v a i l a b l e . However, t h e g r e a t e r t h e o r d e r , t h e more f r e q u e n t and extreme can be i t s o s c i l l a t i o n s and t h e more e r r a t i c t h e d e r i v a t i v e s . U s e of a lower o r d e r least s q u a r e s f i t would smooth t h e experimental d a t a , b u t some of t h e r e s u l t i n g d e t a i l s of t h e d i s t r i b u t i o n would be l o s t . Because adequate e x p e r i m e n t a l p r o f i l e s g e n e r a l l y a r e n o t a v a i l a b l e a t more t h a n f o u r or f i v e a x i a l s t a t i o n s , it i s i m p o r t a n t t o determine whether o r n o t s a t i s f a c t o r y a x i a l d e r i v a t i v e s can be o b t a i n e d from such d a t a . T h e r e f o r e , a t e s t c a s e w a s p r e p a r e d by assuming i n i t i a l hydrogen c o n c e n t r a t i o n , and a x i a l v e l o c i t y p r o f i l e s of t h e form -
Y = Oi45 + 0.45 c o s ( 2 r ) (41)
-
( 4 2 )
V = 1000 + l O O r
z where r v a r i e d from 0 t o 2 inches ( F i g u r e s 1 and 2 ) . For s i m p l i c i t y , t h e s t a g n a t i o n temperature w a s assumed c o n s t a n t a t t h e i n i t i a l a x i a l s t a t i o n because i n many c o l d flow mixing s t u d i e s , i n which t h e s t a g n a t i o n temperature of t h e g a s e s t o be mixed are e q u a l p r i o r t o mixing, measured s t a g n a t i o n t e m p e r a t u r e s 8,lO
v a r y o n l y a few p e r c e n t throughout t h e mixing r e g i o n . , A l s o
f o r s i m p l i c i t y , t h e s t a t i c p r e s s u r e was assumed e q u a l t o be atmospheric throughout t h e flow, and no r a d i a l momentum t r a n s f e r Using a GASL program f o r t h e numerical i n t e g r a t i o n was c o n s i d e r e d .
TR 592 Page 2 0 of t h ? d i f f u s i o n , momentum, and energy e q u a t i o n s , c o n c e n t r a t i o n , v e l o c i t y , and d e n s i t y p r o f i l e s w e r e g e n e r a t e d a t numerous down-
I
s t r e a m s t a t i o n s , assuming a c o n s t a n t t u r b u l e n t mass t r a n s f e r -
c o e f f i c i e n t , 5 = pEd = 0 . 0 2 lbm/ft-sec and Sc P r and L e
T ' T T
I
t o b.2 u n i t y * . Three computed c o n c e n t r a t i o n and v e l o c i t y p r o f i l e s a t i n t e r v a l s of approximately 5 i n ( i n a d d i t i o n t o t h e i n i t i a l p r o f i l e s ) were s e l e c t e d f o r t h e t e s t c a s e ; t h e s e p r o f i l e s a r e p r e s e n t e d i n F i g u r e s 1 and 2 .
I
The g e n e r a l geometry and p r o f i l e s s e l e c t e d w e r e s i m i l a r t o t h o s e used i n Reference 18. These p r o f i l e s w e r e u s e d a s i n p u t t o t h e computer program" developed f o r t h e d e t e r m i n a t i o n
of Ed and E (and 4 ) which e v a l u a t e s each t e r m i n Equations ( 2 7 )
m
I
and (29) and s o l v e s f o r t h e t r a n s p o r t c o e f f i c i e n t s . Equation ( 2 8 ) w a s not used i n t h e test case because r a d i a l momentum t r a n s f e r
I
had n o t been considered i n t h e computation of t h e t e s t p r o f i l e s .
The i n t e g r a l s i n Equations (12), ( 2 7 ) , and (29) w e r e
I
e v a l u a t e d numerically by i n t e r p o l a t i n g t h e Y , v and p r o f i l e s
Z a t 250 r a d i a l p o s i t i o n s and u s i n g t h e t r a p e z o i d a l r u l e : t h e i r a x i a l v a r i a t i o n s w e r e d e t e r m i n a t e d by f i t t i n g a second o r d e r ( f o r maximum smoothing) t r u n c a t e d Laurent polynomial i n f / ( z + f a ) The t e r m s ?y/6r a n d and d i f f e r e n t i a t i n g t h e polynomiaf?
a i / a r w e r e determined by numerical d i f f e r e n t i a t i o n o f t h e con-
z c e n t r a t i o n and v e l o c i t y d a t a , u s i n g a f i v e - p o i n t , second-order, - -
running-smoothing r o u t i n e , and 5 was c a l c u l a t e d from Y , V and T
Z u s i n g t h e p e r f e c t g a s law and t h e assumption t h a t and remained t c o n s t a n t .
* The s u b s c r i p t i i s dropped f o r t h e b i n a r y hydrogen-air s y s t e m
c o n s i d e r e d .
**
f and a a r e c o n s t a n t s which depend on t h e magnitude of t h e e x p e r i m e n t a l range of z.
I
I
TR 592 Page 2 1 R e s u l t s of t h e s e computations a r e p r e s e n t e d i n F i g u r e s 3 and 4.
U n f o r t u n a t e l y , t h e computer program used t o g e n e r a t e t h e p r o f i l e s i n F i g u r e s 1 and 2 d i d n o t compute c l o s e l y spaced g r i d p o i n t s
n e a r t h e c e n t e r l i n e because t h e stream f u n c t i o n , a, w a s used
a s t h e r a d i a l c o o r d i n a t e . A l s o computing time was g r e a t l y i n c r e a s e d a s t h e number of r a d i a l g r i d p o i n t s i n c r e a s e d : t h e r e f o r e , t h e number o f g r i d p o i n t s t h a t could be used t o demonstrate t h e e f f e c t o f g r i d s p a c i n g was r a t h e r l i m i t e d . A s shown i n F i g u r e 3 few g r i d p o i n t s r e s u l t e d i n very l a r g e p o i n t s p a c i n g s n e a r t h e c e n t e r l i n e , which y i e l d e d e x c e s s i v e l y l a r g e v a l u e s of a??/ar
and 3v / a r (because o f symmetry t h e s e t e r m s always should e q u a l
z z e r o a t t h e c e n t e r l i n e ) and c o r r e s p o n d i n g l y s m a l l v a l u e s o f 5 i n this r e g i o n .
I n F i g u r e 3 , t h e c a s e d e s i g n a t e d " I n t e r p o l a t e d " was o b t a i n e d by s e l e c t i n g o n l y f i v e p o i n t s from t h e computed p r o f i l e s i n F i g u r e s 1 and 2 , and i n t e r p o l a t i n g an a d d i t i o n a l 3 6 p o i n t s u s i n g a second o r d e r i n t e r p o l a t i o n r o u t i n e , machine p l o t t i n g t h e r e s u l t s t o a l a r g e scale, and smoothing any i n t e r p o l a t i o n e r r o r s by hand.
Although o n l y 41 g r i d p o i n t s were used a t each a x i a l s t a t i o n i n t h e i n t e r p o l a t e d c a s e , t h e i r c l o s e r s p a c i n g n e a r t h e c e n t e r l i n e r e s u l t e d i n much b e t t e r agreement w i t h t h e i n p u t v a l u e of 5 = 0.02 ( F i g u r e 3 ) t h a n d i d t h e 4 1 p o i n t c a s e i n which each g r i d p o i n t was exact ( t a k e n d i r e c t l y from F i g u r e s 1 and 2 ) b u t n o t c l o s e l y spaced a t t h e c e n t e r l i n e . U n f o r t u n a t e l y , because t h e case of t h e 100 g r i d p o i n t s r e q u i r e d e x c e s s i v e computing t i m e - - p r o f i l e s f o r t h e numerical i n t e g r a t i o n w i t h which Y , Vz, and w e r e computed, it was necessary t o l i m i t t h e a x i a l d i s t a n c e over which t h e s e p r o f i l e s were computed t o o n l y 1 . 5 i n . r a t h e r t h a n 15 i n .
a s was o b t a i n e d f o r t h e 2 1 and 41 g r i d p o i n t s .
TR 592
I
Page 22 From Equations ( 2 7 ) t o (30), it i s obvious t h a t v a l u e s of t h e t r a n s p o r t c o e f f i c i e n t s cannot be o b t a i n e d 3 t h e c e n t e r l i n e s i n c e both t h e i n t e g r a l t e r m s as w e l l a s t h e r a d i a l d e r i v a t i v e s a r e z e r o a t t h i s p o i n t . Of c o u r s e , t r a n s p o r t c o e f f i c i e n t s can be o b t a i n e d a s c l o s e t o t h e c e n t e r l i n e a s d e s i r e d a s long as
I
r e l i a b l e d a t a ( o r i n t e r p o l a t i o n s ) a r e a v a i l a b l e . However, t h e c o e f f i c i e n t s could be e v a l u a t e d a t t h e c e n t e r l i n e i f t h e appro- p r i a t e forms of Equations (20) t o (23) a r e used. The symmetry m n d i t i o n s a l l o w s i m p l i f i c a t i o n of t h e s e e q u a t i o n s a t t h e
I
c e n t e r l i n e t o g i v e * , C e n t e r l i n e D i f f u s i o n Equation
I
(43 1 C e n t e r l i n e Radial Momentum Equation
-
v = o
(44) r
I
C e n t e r l i n e Axial Momentum Equation (45 1
I
C e n t e r l i n e Enerqy Equation
I
- aE k a2E +
- = 2 ( ~ + :Eh) -
Pvz az 2
P a r
+
+22; i [I3 (1- F ) +Ed Le T
I
+ 1 [ p ( l - -)+CE 1 (1- - 1 z
P r m P r gC T * I f t h e f u n c t i o n d i f f e r e n t i a t e d i s symmetrical about t h e a x i s , ~ TR 592 Page 23 Values of t h e t r a n s p o r t c o e f f i c i e n t s a t t h e c e n t e r l i n e i n p r i n c i p l e can be o b t a i n e d from t h e s e e q u a t i o n s . Of course, extremely a c c u r a t e c l o s e l y - s p a c e d experimental d a t a ( o r i n t e r p o l t a t i o n s ) would have t o be a v a i l a b l e f o r t h e e v a l u a t i o n o f second d e r i v a t i v e s . The a l t e r n a t i v e of using Equations ( 2 7 ) t o (30) t o determine t r a n s p o r t c o e f f i c i e n t s a s c l o s e t o t h e a x i s as p o s s i b l e and t h e n e x t r a p o l a t i n g smooth continuous c u r v e t o t h e c e n t e r l i n e ( u s i n g t h e symmetry c o n d i t i o n s ) i s v e r y a p p e a l i n g s i n c e i n t h i s procedure t h e d i f f i c u l t problem of t h e e v a l u a t i o n of second d e r i v a t i v e s i s e l i m i n a t e d .
R e s u l t s o b t a i n e d w i t h t h e I n t e r p o l a t e d case i n F i g u r e 3 show t h i s later procedure y i e l d s reasonable r e s u l t s .
I n t h e i n t e r m e d i a t e r e g i o n between 0 . 2 t o 1 . 5 i n . i n F i g u r e 3 , t h e v a l u e of t h e t r a n s p o r t c o e f f i c i e n t s f o r a l l f o u r c u r v e s had a maximum d e v i a t i o n from t h e c o r r e c t v a l u e of o n l y &25%. T h i s agreement i s r a t h e r remarkable c o n s i d e r i n g t h a t f o r t h e c a s e o f 21 g r i d p o i n t s o n l y s l i g h t l y more t h a n a t o t a l of 80 i n p u t p o i n t s w e r e used a t t h e f o u r a x i a l s t a t i o n s , each s e p a r a t e d from t h e o t h e r by 5 i n . , and t h a t i n t h e i n t e r p o l a t e d c a s e a t o t a l o f o n l y 20 o r i g i n a l p o i n t s was used, some of which w e r e more t h a n 0.5 i n .
from n e i g h b o r i n g p o i n t s of t h e p r o f i l e . The o s c i l l a t i o n s t h a t occur i n t h e I n t e r p o l a t e d c a s e ( F i g u r e 3 ) p r i m a r i l y w e r e caused b y t h e d i f f i c u l t y i n d i f f e r e n t i a t i n g i n t e r p o l a t e d d a t a ,and t h e d i s c r e p a n c i e s i n t h e hydrogen mass b a l a n c e s which r e s u l t e d i n
-
i n a c c u r a c i e s i n a Y / a r and i n t h e a x i a l d e r i v a t i v e s of t h e i n - t e g r a l i n Equation ( 2 7 ) . However, t h e s e r e s u l t s c l e a r l y demonstrate t h a t v e r y r e a s o n a b l e approximations of t r a n s p o r t c o e f f i c i e n t s may be o b t a i n e d from r a t h e r l i m i t e d experimental d a t a .
TR 592 Page 24 F i g u r e s 3 and 4 show t h a t a t r a d i a l p o s i t i o n s g r e a t e r t h a n
-
1 . 5 i n . a t which Y approachs z e r o ( F i g u r e 1) v a l i d c o e f f i c i e n t s cannot be obtained. I n t h i s r e g i o n , b o t h t h e a x i a l d e r i v a t i v e
o f t h e i n t e g r a l i n Equation ( 2 7 ) and ayhr approach z e r o a s t h e
f r e e stream i s approached, so t h a t t h e i r r a t i o cannot be a c c u r a t e l y determined. A s r+oo and each o f t h e s e t e r m s becomes z e r o , t h e computer d e s i g n a t e s 0/0 as 0.
F i g u r e 4 i l l u s t r a t e s t h e e f f e c t of a x i a l s t a t i o n on 5 f o r t h e 4 1 p o i n t g r i d . B e s t r e s u l t s a r e o b t a i n e d a t i n t e r m e d i a t e a x i a l s t a t i o n s r a t h e r t h a n a t t h e end p o i n t s : of c o u r s e , t h i s r e s u l t would be expected because of t h e d i f f i c u l t y i n o b t a i n i n g a c c u r a t e s l o p e s from polynomial f i t s a t end p o i n t s . However, v e r y r e a s o n a b l e agreement w a s o b t a i n e d between t h e computed and i n p u t v a l u e s of 5 a t t h e i n t e r m e d i a t e a x i a l s t a t i o n s f o r t h i s c a s e .
The g e n e r a l c o n c l u s i o n s t o be o b t a i n e d from t h i s t e s t c a s e i s t h a t approximate v a l u e s of t h e t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s can be o b t a i n e d from a l i m i t e d number of e x p e r i m e n t a l d a t a p o i n t s , a s long as t h e o r i g i n a l p o i n t s a r e r e a s o n a b l y a c c u r a t e . However, s i n c e p o i n t spacing i s i m p o r t a n t even when t h e d a t a p o i n t s a r e e x a c t , some ambiguity of r e s u l t s i s t o be e x p e c t e d when u s i n g e x p e r i m e n t a l p r o f i l e s . That i s , r e a s o n a b l y c l o s e l y spaced a c c u r a t e experimental d a t a m u s t be used i n o r d e r t o o b t a i n d e t a i l e d v a r i a t i o n s of t h e t r a n s p o r t c o e f f i c i e n t s .
T R 592 Page 25 REFERENCES V I .
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Z a k k a y , V . , K r a u s e , E . , and Woo, S . D . L . , A I A A J., 11, 1 9 3 9
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TR 592 Page 26 15. Hinze, J . O . , Turbulence, M c G r a w - H i l l , N e w York (1959) Chapter 5.
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18. A l p i n i e r i , L . J . , A I A A J. , 2, 1560 (1964).
19. Page, F . , S c h l i n g e r , W . G . , Breaux, D.K. , and Sage, B.H. , Ind. Eng. Chem., 44, 424 (1952).
TR 592 Page 2 7 APPENDIX A EXPERIMENTALLY DETERMINED TURBULENT TRANSPORT COEFFICIENTS One o b j e c t i v e of t h e p r e s e n t i n v e s t i g a t i o n w a s t o u s e t h e numerical t e c h n i q u e p r e s e n t e d h e r e i n t o determine t u r b u l e n t t r a n s - port c o e f f i c i e n t s f o r t h e case of s u p e r s o n i c , c o a x i a l , free-jet mixing. U n f o r t u n a t e l y , no completely adequate (closely-spaced) e x p e r i m e n t a l d a t a w e r e a v a i l a b l e f o r t h i s purpose. However , e x p e r i - mental d a t a p r e s e n t e d i n Reference 8 , which g e n e r a l l y c o n t a i n e d f i v e or s i x r a d i a l e x p e r i m e n t a l d a t a p o i n t s a t each of s i x a x i a l s t a t i o n could be used i f a d d i t i o n p o i n t s w e r e g e n e r a t e d by i n t e r - p o l a t i o n as had been done i n t h e I n t e r p o l a t e d t e s t case p r e v i o u s l y d i s c u s s e d . Of c o u r s e , i n t h e p r e s e n t c a s e the experimental d a t a p o i n t s w e r e n o t n e c e s s a r i l y e x a c t , as t h e y had been i n the t e s t c a s e (where t h e p o i n t s w e r e computed); t h e r e f o r e , any e r r o r s i n t h e o r i g i n a l p o i n t s w e r e t r a n s m i t t e d t o the i n t e r p o l a t e d p o i n t s , so t h a t d i s c r e p a n c i e s i n the o r i g i n a l p o i n t s w e r e magnified when t h e r e s u l t i n g p r o f i l e s w e r e d i f f e r e n t i a t e d . C l e s r l y , d e t a i l e d v a r i a t i o n of t u r b u l e n t t r a n s p o r t c o e f f i c i e n t s o n l y can be o b t a i n e d from c l o s e l y - s p a c e d , accurate d a t a p o i n t s .
me d a t a of Reference 8 obtained f o r the case of c o a x i a l , f r e e - j e t mixing of subsonic hydrogen (M=O.5 t o 0.9) w i t h a surrounding Mach 1.6 a i r j e t (1.1 lb/sec) a t an o v e r a l l e q u i v a l e n c e r a t i o (ER) of 0.10 t o 0.25. S t a g n a t i o n t e m p e r a t u r e and s t a t i c p r e s s u r e w e r e assumed c o n s t a n t throughout the mixing r e g i o n i n t h e computation of v e l o c i t i e s and d e n s i t i e s as had been done i n Reference 8. T y p i c a l c o n c e n t r a t i o n and v e l o c i t y p r o f i l e s o b t a i n e d a t s i x a x i a l s t a t i o n s between 4 and 9 i n . downstream of the i n - j e c t i o n s t a t i o n are p r e s e n t e d i n F i g u r e s 5 and 6 f o r a hydrogen 2 % 592 Page 28' mass flow r a t e of approximately 0.007 lb/sec (M=0.89) i n t o a 1.1 lb/sec, Mach 1.6 a i r s t r e a m (ER = 0 . 2 5 ) . O r i g i n a l d a t a p o i n t s a r e p l o t t e d as symbols i n t h e s e f i g u r e s ; the f i n a l i n t e r p o l a t e d
i
(and somewhat smoothed) p r o f i l e s u s e d t o d e t e r m i n e t h e t u r b u l e n t
*
t r a n s p o r t c o e f f i c i e n t s are p l o t t e d a s s o l i d l i n e s , C o n d i t i o n s
a
f o r t h i s r u n , d e s i g n a t e d Case C , are summarized i n Table 1, along w i t h r u n s A and B. The mass and momentum b a l a n c e s c o m - puted a t each a x i a l s t a t i o n are p r e s e n t e d i n Table 2 . A s u f f i c i e n t number of p o i n t s w e r e i n t e r p o l a t e d f o r each of t h e I e x p e r i m e n t a l p r o f i l e s , so t h a t a t o t a l of more t h a n 40 p o i n t s
I
w e r e a v a i l a b l e a t each a x i a l s t a t i o n . G r i d s p a c i n g a t the c e n t e r l i n e was approximately 0.017 i n . and a t the free s t r e a m 0.020 i n , The t u r b u l e n t mass t r a n s f e r c o e f f i c i e n t , [, a s w e l l a s the
-
I
([=pEd) and t h e eddy d i f f u s i v i t y eddy d i f f u s i v i t y of mass, Ed of momentum, E m w e r e obtained u s i n g Equations ( 2 7 ) and ( 2 8 ) , and
I
t h e procedure p r e v i o u s l y d i s c u s s e d . T y p i c a l r e s u l t s , o b t a i n e d f o r Case C , are p r e s e n t e d i n Table 3 . A s a n t i c i p a t e d c o n s i d e r a b l e v a r i a t i o n occurred i n these t r a n s p o r t c o e f f i c i e n t s because of the i n c o n s i s t e n c i e s i n the mass b a l a n c e s and the d i f f i c u l t i e s i n h e r e n t i n d i f f e r e n t i a t i n g i n t e r p o l a t e d e x p e r i m e n t a l d a t a s t a r t i n g w i t h o n l y a few o r i g i n a l d a t a p o i n t s . I n a d d i t i o n , some e r r o r may have been i n t r o d u c e d i n the v e l o c i t y p r o f i l e s by t h e assumption t h a t t h e l o c a l f r e e s t r e a m s t a t i c p r e s s u r e and the s t a g n a t i o n temper- Because of t h e s e a t u r e w e r e c o n s t a n t throughout the mixing r e g i o n .
problems, better v a l u e s of E w e r e o b t a i n e d by assuming Sc = 1 m T t h a n by d i r e c t d i f f e r e n t i a t i o n of the v e l o c i t y p r o f i l e s , Therefore, e x p e r i m e n t a l v a l u e s of E and Sc are n o t r e p o r t e d . D e s p i t e the m T
I
v a r i a t i o n t h a t occurred i n t h e d e r i v e d t r a n s p o r t c o e f f i c i e n t s , c e r t a i n t r e n d s appeared, which g e n e r a l l y w e r e c o n s i s t e n t f o r each
I
of the c a s e s analyzed.
* For the purpose of t h e i n i t i a l computations, t h e a x i a l symmetry
!
i n d i c a t e d by t h e dashed l i n e s i n F i g u r e s 5 t o 8 was n o t c o n s i d e r e d ; rather t h e b e s t smooth c u r v e s through the e x p e r i e n t a l d a t a w e r e u s e d .
I
~~ ~~~ TR 592 Page 2 9 A s a f i r s t approximation, a model w a s c o n s t r u c t e d s i m p l i f y i n g the major t r e n d s shown i n Table 3 t o i n c l u d e o n l y r a d i a l dependence of [ and E I n o r d e r t o determine whether or n o t these t r e n d s d - w e r e a t l e a s t a v a l i d first approximation, t h e program f o r t h e n u m e r i c a l i n t e g r a t i o n o f the d i f f u s i o n , momentum, and e n e r g y e q u a t i o n s
w a s used t o compute ?, 7 and 0 p r o f i l e s a t v a r i o u s
2 ) downstream l o c a t i o n s u s i n g t h e s i m p l i f i e d t r e n d s a s i n p u t t o t h e program. This t e s t w a s s i m i l a r t o t h o s e p r e v i o u s l y r e p o r t e d , except f o r two i m p o r t a n t d i f f e r e n c e s : 1). I n the p r e s e n t n u m e r i c a l i n t e g r a t i o n t e c h n i q u e , the stream f u n c t i o n , t b , w a s used as t h e r a d i a l c o o r d i n a t e . Because of t h i s t r a n s f o r m a t i o n
-
t h e r a d i a l v e l o c i t y , d i d n o t h a v e t o be s p e c i f i e d i n advance vr as w a s p r e v i o u s l y r e q u i r e d ; only E needed t o be s p e c i f i e d when d the assumption w a s made t h a t Sc = Le = Pr = 1. T h e r e f o r e , T T T the agreement o b t a i n e d between computed and e x p e r i m e n t a l pro- f i l e s w a s a d i r e c t e v a l u a t i o n of t h e v a l i d i t y of the p a r t i c u l a r model b e i n g t e s t e d . 2 ) . A v a r i a b l e r a d i a l g r i d s p a c i n g i n p h y s i c a l c o o r d i n a t e s w a s used i n R e f e r e n c e 10 w h i c h s i g n i f i - c a n t l y reduced the number of r a d i a l g r i d p o i n t s r e q u i r e d , t h e r e b y g r e a t l y s h o r t e n i n g computing t i m e . R e s u l t s are p r e s e n t e d f o r a l l t r i a l s and a l l cases i n Tables 4 t o 1 2 and f o r the best r e s u l t s w i t h C a s e C i n F i g u r e s 7 and 8 ; agreement between e x p e r i m e n t a l and computed c o n c e n t r a t i o n and v e l o c i t y p r o f i l e s i s r e a s o n a b l y good f o r the l a s t t r i a l i n each ease, as shown i n Tables 5,6,8,9, 11, and 1 2 . The t r a n s p o r t c o e f f i c i e n t s used i n these n u m e r i c a l i n t e g r a t i o n s are t a b u l a t e d i n Tables 4,7, and 1 0 ; l i n e a r l y i n t e r - p o l a t e d v a l u e s w e r e used a t r a d i a l p o s i t i o n s i n t e r m e d i a t e t o those t a b u l a t e d . Comparison of t h e v a r i o u s cases shows t h a t a r e l a t i v e l y s m a l l change i n E o r 6 r e s u l t s i n a rather l a r g e d change i n computed c o n c e n t r a t i o n p r o f i l e s , b u t n o t n e a r l y a s TR 592 Page 30..
s i g n i f i c a n t a change i n the computed v e l o c i t y profiles.
Also, r e a s o n a b l e agreement w a s a t t a i n e d u s i n g the simple t r e n d s .
A d d i t i o n a l computer t r i a l s must be made i n order t o d e t e r m i n e
I
whelther E and 6 i s more basic f o r c o r r e l a t i o n of data.
d F u r t h e r evidence concerning t h i s i m p o r t a n t p o i n t could be I
I
o b t a i n e d i n f u t u r e work by a n a l y z i n g the argon and h e l i u m mixing d a t a also a v a i l a b l e i n Reference 8, Because an i n s u f f i c i e n t number of d a t a p o i n t s w e r e a v a i l - able t o a c c u r a t e l y d e f i n e r a d i a l p r o f i l e s , e s p e c i a l l y a t the
I
c e n t e r l i n e where symmetry r e q u i r e d t h a t ;3?/ar and a? /W=O
Z
(see F i g u r e s 5 and 6 ) , c o s i n e p r o f i l e s of the f o r m I
-
Y = Y + A c o s (a) (A-1)
-
V = V + B cos ( p r )
(A-2) z z w e r e f i t t e d through e x p e r i m e n t a l p o i n t s located a t r a d i a l p o s i - t i o n s of 0 , 0.125, and 0.25 i n , These c o s i n e f i t s w e r e more
I
g e n e r a l t h a n t h o s e used i n Reference 8 , i n which t w o rather t h a n three a r b i t r a r y c o n s t a n t s w e r e used. A s shown i n Tables
1 1
1 3 and 14 considerable smoothing of b o t h Ed and 6 occurred u s i n g
these f i t s ; however, o v e r a l l r e s u l t s w e r e n o t d r a s t i c a l l y changed f r o m t h o s e obtained w i t h the smoothed data, e x c e p t t h a t v a l u e s near the c e n t e r l i n e w e r e i n c r e a s e d because the c u r v a t u r e o f the
I
c o s i n e i s maximum a t the o r i g i n .
The good agreement between computed and e x p e r i m e n t a l ? and
v
p r o f i l e s shown i n F i g u r e s 7 and 8 and i n T a b l e s 5,6,8,9,11, z and 1 2 , obtained u s i n g the s i m p l i f i e d t r e n d s , s u b s t a n t i a t e t h e v a l i d i t y of t h e s e t r e n d s and s u g g e s t t h a t f o r these data r a d i a l v a r i a t i o n of the t r a n s p o r t c o e f f i c i e n t s i s more s i g n i c a n t t h a n
I
a x i a l v a r i a t i o n , and E reaches a maximum a t about 20% of the d
I
I
TR 592 Page 31 ‘ d i s t a n c e from the c e n t e r l i n e t o t h e free stream. S i m i l a r t r e n d s w e r e r e p o r t e d i n R e f e r e n c e 19 f o r s u b s o n i c flow between t w o p a r a l l e l plates. F u r t h e r i n v e s i t g a t i o n i s r e q u i r e d t o c o n f i r m these i m p o r t a n t p o i n t s .
' 1
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Radial Distance (inches)
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TR 592 Page 3 3 i I I 1 I i
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I d i a l D i s t a r k e from
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--
i 1 1 I I \ 0 0 - 4 0.8 1.2 1.6 2 .o 2.4 R a d i a l D i s t a n c e ( i n c h e s )
FIGURE 2 - COMPUTED VELOCITY P R O F I L E S , T E S T CASE
' 1
R a d i a l Distance (inches)
6, TEST CASE
FIGURE 3 - EFFECT OF RADIAL GRID SPACING ON
TR 592 I I 0.. 08 0.07 0.06 0.04 0.02 0.01 TEST CASE FIGURE 4 - EFFECT OF AXIAL POSITION ON 4 , TR 5 9 2 Page 36
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Axial Distance from O r i g i n a l I n j e c t i o n S t a t i o n Experimental ( inches) Data P o i n t s
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Axial D i s t a n c e f r o m ! O r i g i n a l E x p e r i m e n t a1 D a t a P o i n t s
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L - , .4 .5 .6 0 .1 . 2 - 3 R a d i a l D i s t a n c e ( i n c h e s )
FIGURE 6 - EXPERIMENTAL VELOCITY P R O F I L E S , CASE C
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TR 592
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Radial D i s t a n c e (inches)
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FIGURE 7 - COMPARISON OF COMPUTED CONCENTRATION PROFILES
WITH EXPERIMENTIG DATA TR 5 F 2 P a e 39--
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I n j e c t i o n S t a t i o n
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I n j e c t i o n S t a t i o n E x p e r i m e n t a l ( i n c h e s ) D a t a P o i n t s
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FIGURE 8 - COMPARISON O F COMPUTED VELOCITY PROFILES
WITH EXPERIMENTAL DATA TR 592 Page 40
a
I n _ _ rn U al rn al m al .f; Q)
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k -n LI a k ?m ?i -rl 3 : Q TR 5 9 2 Page 41
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m o r n \ D * l n d \ D * r l P m o P ( u N c v P P r n \ D \ D * e * m e m eCVNNNcv 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . .
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0 0 0 0 0 0 0 0 0 0 0 0 0 r J ) m d r - l m r - O d m a N d O d m a m d m m d a m d d . o o o o d r l N N N m m m d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . .
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I t TR 5 3 2 Page 4 3
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U a, 0 0 0 0 0 0 cn L n O L n O l I I I I I 0 1 0 a\ O l n b O l n m w r u . . . .
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rl w N * 4J N 0 rcI Id Q 0 -4 I 1 * I \ I I I I I l l .
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-K Ln e r l m e m o r n P r D U t N A 0 0 0 0 0 0 r l d O 0 0 0
I 9 9 9 9 9 9 c o o o o o
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~ ~ m d r t o 3 r l o o o o 0 0 0 0 0 0 ~~ . . . . . .
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m O o P a p m r l 0 0 0 0 0 0 o o m d w o r l r t o o o o 0 0 0 9 9 9 . . .
9 9 9 9 9 9
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m I r l r l O O O O O r l L V d N O ~ 0 0 0 0 0 0 . . . . . .
I
d I \ O N ~ ~ N O l d r l 0 0 0 0 1 9 9 9 9 9 9 d t o m o o 0 > N r l r t 0 0 0 > o o o o o . . . . .
P m m m o o 3 P a d N 0 0 > o o o o o o ~ I n o m o o o 4 r l r l o o o o > o o o o o o > 0 0 0 0 9 0 . . . . . . .
. . . . .
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a * N ~ P ~ N O r l r l o o o o o 0 0 0 0 0 0 0 ~~ . . . . . . .
. . . . . . .
n N e r l m ~ d ~ b r l r l o o o o o m ~ m m b r l o N r l d 0 0 0 0 0 0 0 0 0 0 ~ o o o o o q q . . . . . .
. . . . .
v) N ~ ~ m o m o o m o m m r l o m N r l d O 0 0 N r l r l 0 0 0 0 0 0 9 9 9 9 . .
9 9 9 9 9 9 9 rl r .
m o v l r n o r l o r o m o ~ d ~ b " . - I O 0 0 0 r l r l r l o o o o 9 9 9 9 9 9 9
I 9 9 9 9 9 9 9
rl N ~ n ~ ~ n m o r t o P r l r l O N O N N r l 0 0 0 0 m m ~ r l 0 0 0 9 9 9 9 9 9 9 9 9 9 9 9 9 d p.
\ D r l o r D o + r l o o o o o N O P d d O N N r l O O O O
f
9 9 9 9 9 9 9 9 9 9 9 I 0 0 0 0 r C I a:: a <
I
TR 5 9 2 Page 4 5
I
n
I
N
I
I
I
I
I
LI a, Y W
I
c i i o r l o o i i o r l o o c 0 ~ ~ m d t n u 3 .i o r d M m \t . . . . . . . . . . . .
a -
I
TR 592 Page 46 TABLE 7
I
TURBULENT TRANSPORT COEFFICIENTS USED IN NUMERICAL INTEGRATION, CASE B*
I
TRIAL -L_ _-I i- T--- 2 3 OEd "Ed OEd R lbm/ft-sec lbm/ft-sec lbm/ft-sec inches 0.010 0.01 0.02 - 0.010
-**
. 0 2 0 0.012 . 0 2 6 0.015 .035 0.015 .065 0.020 .085
- 0.030
. l o o
0.030 0.01 .700
*
In all trials the assumption was made that E d = E m ,
i.e., scT = 1
** Linear interpolations was used for evaluation of coefficients 1
at radial positions intermediate to tabulated values.
I TR 592 Page 47 K K 2 g z z g 0 0 0 0 0 0 0 . . . . . . .
N u l r - r l m ~ o e m N O O O O 0 0 0 0 0 0 m w 0 w ~ ~ ' m r n n o a n t v r l o o o o U 0 0 0 0 0 0 . . . . . .
& m o m p . 4 N N r l O O O O a 0 0 9 q 9 . .
w a m r r m m o O N 4 0 0 0 0 0 0 0 0 0 0 . . . . .
ul m o m m ~ o m n r l o o o o
" a
0 0 0 0 0 0 II u a . . . . . .
N h c1 m n o m ~ o m m m 0 0 0 d r l m m m d z z ~ o o o 0 0 0 0 0 0 0 . . . . . .
0 0 0 0 . . . . . .
d m u l ~ m n d o & r N o m~ 3 r l r l o o o o n n N 0 0 0 0 0 0 0 0 0 a 9 9 9 9 9 9 9 . . .
W N 0 - 4 m ~ m L n W N 0 0 0 ~ m n c o o r ( o N r l r l 0 0 0 0 0 0 0 0 0 0 0 . . . . . O 0 0 0 0 0 0 0.
. . . . . .
n m o e m a m n o m N o O u l N 0 0 n n N o o o o 0 0 0 0 0 0 0 . . . . O ~ 0 0 0 0 0 , . . . . . .
N B I
I
I n N - r a 0 4
m l n d N 0 0 0 0 0 0 0 0 0 ~ ~ a r n n o i d . . . . .
N N r l O O O O 0 0 0 0 0 0 . . . . . .
m %4 . - i r n n m n d o L m r l r l m n L n d N 0 0 a, N d d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a a . . . . . . .
. . . . .
w W n m . + m m o ma^ o q o 0 0 0 0 9 9 0 0 0 0 0 3 3 o r l + - l o 3 0 0 + N T d Z % 3 N W d ? 0 o . . . . . .
0 . . . .
C O G O O O O 0 0 0 0 0 0
I
I
TR 592
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Page 40 n
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f
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N a m r "3 t n o r l n m ~ m a r ~ ~ ~ m m m n r l r l r l r l r l r l r l
I
n n m m o m - m n n m b m o N N N N N Y ~ d r l r l r l . - l r l r l
I
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" a
N H
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r l r l o r l o o a o r l ~ n b m a I a ., . . . . . .
I
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TR 592 Page 49 TABLE 1 0 TURBULENT TRANSPORT C O E F F I C I E N T S USED I N NUMERICAL INTEGRATION, CASE C* T R I A L
A - 7
1 2 3 R O E d O E d inches lbm/ft-sec ft /sec lbm/ft-sec 0.0025 0 0.02
- **
0.05 0.0175 0.0250 0.10 0.20 0 . 0 3 0 0 - 0.10
- - -
0.50 0.02 0.70 0.055 0.0300
* In all trials the assumption was made that E =E
d m ' i.e., Sc = 1.
T -
** Lixear interpolation w a s used f o r evaluation of coefficients
at radial positions intermediate to tabulated values.
TR 592 Page 50 r m 4 d a n o " 4 0 0 0 0 0 0 0 0 0 . . . . . .
N O O ~ N O m b r 1 o o o 0 0 0 0 0 . . . . .
. . . . . . .
I '
-
d 1 I , m d n l n ~ o - l P I D f r n O I N N 4 0 0 0 n m N 4 O O O ! 1 0 0 0 0 0 0 3 0 0 0 0 0 . . . . . . .
. . . . . .
i d C I 1 m m b L o N 0 ~ m m m m o t m N 4 0 0 0 ~ m n r ( 0 0 0 > o o o o o o 4 0 0 0 0 0 -~ . . . . . . .
. . . . . .
d
4 C'
u Lo , m 4 m N O ~ O P - ~ O
?I
i m d r 1 0 0 0 l n r l d 4 0 0 0 el 4 0 0 0 0 0 d d 0 0 0 . . . . . .
. . . . .
I
N o 4 m m m d o m o r l d N o 3 d N d 0 0 0 m m b d o o o 3 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . .
. . . . . .
I
d a d l n P r l P - f r l O d 0 0 0 0 0 0 . . . . .
I
m o d o m o d P 4 r n - 4 0 I1 4 4 0 0 0 0 0 . . . . . .
0 0 0 0 0 N
I
4 4 0 4 0 0 4 4 0 4 0 0 I o r l ~ m d m ~ o o d ~ n d r n l O a: . . . . . .
. . . . . . I
I
I
I
I
TR 592 Page 51 " 2 II u a V N H w m u c r ( d O r ( O 0 r l d O r ( O 0 C o r ( ~ m b m w e a .- o r l ~ m b m w . . . . . .
c . . . . . .
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TR 5 9 2
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Page 5 2 L
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d m a i 0
- m
B : :
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c V TR 5 9 2 Page 53 m b a r o ~ m r ~ m o b b m o q r l d r l r l ~ ~ ~ ~ n m m b m a o O O O O O O O O O O O O O O N m . . . . . . . . . . . . . . .
o m a r - m m o o m r n r - a m r l ~ o a m m m m m a a m m m m m m b
&I 9 9 0 0 0 0 0 0 0 0 0 0 0 0 0
. . . . . . . . . . . . .
u m p.1 . . . . . . . . . . . . . . .
W c v1 m 0 5 a, rl U m .d m W r l m m m m m m m m m m o ~ ~ m 0 a, d
P ~ b n o n m m m n n n n b b ~ vi
k ~ q ~ O O O O O O O O O o O O a m . . . . . . . . . . . .
W o m ~ r n o r l r l ~ ~ m a m r l m a h E O N r l d r l N " " N N N I ? O b rl , ~ ~ q q o o o o o o o o o o o C . . . . . . . . . . .
d m rl 0 m rl $ h W
s <
.d ri m N h d h It 8 ~ ? ~ ~ m m . + ~ ? m m m o o o o o h m O N I ? d b b m m m m m w a a w a m
A ? ? 9 9 9 9 9 9 9 9 9 9 9 9 9
u Q W I * a r l b ~ ~ m m m b r - m d m m ~ k o o r l ~ m - r b m m a w a r r - r - w a,
. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I m
P I . . . . . . . . . . . . . . .
s
fi v) m V N c1 0 u
H
4 ri Q n :: W I a N m N o e r - r n , + . + d o r n r - r l V
9 o o ~ ~ n b ~ ~ d m m t n m b b m
W r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 ; I . . . . . . . . . . . . . . .
m m W m r - m ~ m r l o m r n m m m m ~ ~ m rl O ~ ~ ~ M ~ O N N N N N N N N ~ 2 ; . o o q o o o o o o ~ q o ~ ~ q I ? . . . . . . . . u m E c .?I N a, 0 m II
i &.I
y.p2:%~?2xG"wG 0 0 0 0 0 0 0 r l d r l r l r l r l r l N * . . . . . . . . . . . . . . .