Detailed Modelling of Combustion: A ... - American Chemical Society

27 Detailed Modelling of Combustion: A Noninterfering Diagnostic Tool ELAINE S. ORAN, JAY P. BORIS, and M. J. FRITTS

Downloaded by UNIV OF LIVERPOOL on May 15, 2018 | https://pubs.acs.org Publication Date: September 23, 1980 | doi: 10.1021/bk-1980-0134.ch027

Laboratory for Computational Physics, Naval Research Laboratory, Washington, D.C. 20375

D e t a i l e d m o d e l l i n g , o r n u m e r i c a l s i m u l a t i o n , p r o v i d e s a met h o d we c a n u s e t o s t u d y c o m p l e x r e a c t i v e f l o w p r o c e s s e s ( 1 ) . P r e d i c t i o n s about the b e h a v i o r o f a p h y s i c a l system a r e o b t a i n e d by s o l v i n g n u m e r i c a l l y t h e m u l t i - f l u i d c o n s e r v a t i o n e q u a t i o n s f o r mass, momentum, and e n e r g y . Since the success o f d e t a i l e d modell i n g i s c o u p l e d t o one's a b i l i t y t o h a n d l e an abundance o f t h e o r e t i c a l and n u m e r i c a l d e t a i l , t h i s f i e l d has matured i n p a r a l l e l w i t h t h e i n c r e a s e i n s i z e and speed o f c o m p u t e r s a n d s o p h i s t i c a t i o n of numerical techniques. I t i s i m p o r t a n t t o d i s t i n g u i s h b e t w e e n e m p i r i c a l , phenomenol o g i c a l and d e t a i l e d m o d e l s . E m p i r i c a l m o d e l s a r e c o n s t r u c t e d f r o m d a t a o b t a i n e d by e x p e r i m e n t s , summarized i n a n a l y t i c a l o r n u m e r i c a l form, and subsequently t e s t e d a g a i n s t proven t h e o r e t i c a l laws o r o t h e r data. Phenomenological models a r e e x t r a p o l a t i o n s f r o m t h e o r y b a s e d o n o u r p h y s i c a l i n t u i t i o n w h i c h must b e tested against experimental data. The i n t u i t i v e a n d e x p e r i m e n t a l b a s i s o f e m p i r i c a l and p h e n o m e n o l o g i c a l m o d e l s h a v e l e d t o t h e i r w i d e s p r e a d i n c o r p o r a t i o n i n t o s i m u l a t i o n s o f combustor systems i n s p i t e o f v e r y s e r i o u s shortcomings i n the n a t u r e of the models themselves. D e t a i l e d m o d e l l i n g s e e k s t o overcome t h e s e d e f i c i t s by means o f i m p r o v e d n u m e r i c a l t e c h n i q u e s and t h e i n c r e a s e d power o f modern c o m p u t e r s . T h i s p a p e r w i l l i n no way c o n s t i t u t e a r e v i e w o f c u r r e n t comb u s t o r models, but w i l l i n s t e a d attempt t o e l u c i d a t e the extens i o n s and i m p r o v e m e n t s made p o s s i b l e t h r o u g h d e t a i l e d m o d e l l i n g . The p u r p o s e o f t h i s p a p e r i s t o f a m i l i a r i z e t h e r e a d e r w i t h t h e g o a l s , t e r m i n o l o g y and i n h e r e n t p r o b l e m s i n m o d e l l i n g f u n d a m e n t a l combustion processes. The e m p h a s i s i s n o t on p r e s e n t i n g a f u l l s e t o f c o m p l i c a t e d m u l t i - f l u i d e q u a t i o n s o r on e x p l a i n i n g t h e numerical algorithms r e q u i r e d t o s o l v e the governing equations. I n s t e a d we hope t o i m p a r t a s e n s e o f t h e power and r o l e o f d e t a i l e d m o d e l l i n g , a n u n d e r s t a n d i n g o f why p h y s i c a l i n s i g h t must be b u i l t i n t o n u m e r i c a l a l g o r i t h m s , and a n i n d i c a t i o n o f how t o This chapter not subject to U . S . copyright. Published 1980 A m e r Jean Chemical Society

Crosley; Laser Probes for Combustion Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


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t e s t these models a t every s t a g e o f c o n s t r u c t i o n a g a i n s t b o t h theory and experiment. The s h o r t c o m i n g s o f t h e e m p i r i c a l m o d e l s l i e i n t h e i r l i m i t e d r a n g e o f v a l i d i t y , w h i l e p h e n o m e n o l o g i c a l m o d e l s become more tenuous as t h e y approach t h e c o m p l e x i t i e s o f r e a l p h y s i c a l s y s tems. D e t a i l e d m o d e l s u s u a l l y c o n t a i n p a r t s w h i c h may be e m p i r i c a l o r phenomenological i n o r i g i n . However, d e t a i l e d m o d e l l i n g a t t e m p t s t o overcome t h e s e s h o r t c o m i n g s by i n c o r p o r a t i n g t h e o r e t i c a l d e t a i l r i c h enough t o a p p r o x i m a t e r e a l i t y ; d e t a i l f a r r i c h e r t h a n c o u l d b e summarized i n any s u c c i n c t a n a l y t i c a l m o d e l , y e t more t h e o r e t i c a l l y sound t h a n s t a n d a r d p h e n o m e n o l o g i c a l m o d e l s o r empirical f i t s . From e x p e r i m e n t a l o b s e r v a t i o n a n d a p p r o x i m a t e t h e o r e t i c a l m o d e l s we c a n p o s t u l a t e q u a n t i t a t i v e p h y s i c a l l a w s w h i c h we e x p e c t an e f f e c t t o obey. These " l a w s " c a n be t e s t e d a g a i n s t r e a l i t y by i n c o r p o r a t i n g them i n a d e t a i l e d m o d e l w h i c h makes q u a n t i t a t i v e p r e d i c t i o n s f o r s e r i e s o f e x p e r i m e n t a l measurements. Each c a l c u l a t i o n p e r f o r m e d w i t h a d e t a i l e d m o d e l i s l i k e a u n i q u e e x p e r i m e n t p e r f o r m e d w i t h one s e t f r o m a n i n f i n i t y o f p o s s i b l e s e t s o f g e o m e t r i c , b o u n d a r y , and i n i t i a l c o n d i t i o n s . Just as v a l i d r e s u l t s can be e x t r a c t e d from an experiment o n l y through an u n d e r s t a n d i n g o f t h e e f f e c t s and l i m i t a t i o n s o f t h e i n s t r u m e n t s used i n c o l l e c t i n g t h e d a t a , r e s u l t s o b t a i n e d u s i n g d e t a i l e d m o d e l l i n g must b e examined i n t h e l i g h t o f t h e l i m i t a t i o n s i n h e r e n t i n i t s t o o l s , b o t h a n a l y t i c a l and n u m e r i c a l . The f i r s t s e c t i o n o f t h i s paper w i l l t h e r e f o r e d e a l w i t h an e x p o s i t i o n o f the problems i n h e r e n t i n d e t a i l e d m o d e l l i n g o f combustion p r o c e s s e s s o t h a t a s we p r o c e e d we h a v e a h e a l t h y r e s p e c t b o t h f o r t h e m a g n i t u d e o f t h e p r o b l e m s and t h e l i m i t a t i o n s o f o u r methods. The n e x t s e c t i o n w i l l c o n c e n t r a t e o n t h e c h o i c e o f n u m e r i c a l a l g o r i t h m s used i n t h e models. This process corresponds t o t h e c o n s t r u c t i o n a n d d e s i g n o f a n e x p e r i m e n t a l a p p a r a t u s w h i c h must r e f l e c t a good k n o w l e d g e o f t h e p h y s i c s t h e e x p e r i m e n t i s t o s t u d y . M o d e l l i n g c o m b u s t i o n s y s t e m s h a s i t s own p a r t i c u l a r p r o blems because o f t h e s t r o n g i n t e r a c t i o n between t h e energy r e l e a s e d f r o m c h e m i c a l r e a c t i o n s and t h e d y n a m i c s o f t h e f l u i d motion. R e l e a s e o f c h e m i c a l energy g e n e r a t e s g r a d i e n t s i n temperat u r e , p r e s s u r e , and d e n s i t y . These g r a d i e n t s , i n t u r n , i n f l u e n c e t h e t r a n s p o r t o f mass, momentum, a n d e n e r g y i n t h e s y s t e m . On a l a r g e s c a l e , t h e g r a d i e n t s may g e n e r a t e v o r t i c i t y o r a f f e c t t h e d i f f u s i o n o f mass and e n e r g y . On a more m i c r o s c o p i c s c a l e , t h e y may g e n e r a t e t u r b u l e n c e w h i c h d r a s t i c a l l y a f f e c t s m a c r o s c o p i c m i x i n g and b u r n i n g v e l o c i t i e s . I n modelling shocks, detonations, o r f l a m e p r o p a g a t i o n , t i m e a n d s p a c e s c a l e s o f i n t e r e s t c a n span as many a s t e n o r d e r s o f m a g n i t u d e . Thus t o o b t a i n a d e q u a t e r e s o l u t i o n , t h e n u m e r i c a l methods must b e c o m p u t a t i o n a l l y f a s t a s w e l l a s a c c u r a t e . Methods must b e d e v e l o p e d w h i c h r e l y on asympt o t i c s o l u t i o n t e c h n i q u e s t o f o l l o w s h o r t time and space s c a l e phenomena o n a m a c r o s c a l e . I ti s i n t h i s aspect that d e t a i l e d m o d e l l i n g most c l o s e l y a p p r o x i m a t e s e x p e r i m e n t . I f our numerical



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apparatus cannot r e s o l v e the b a s i c c o n t r o l l i n g p h y s i c a l p r o c e s s e s , no m e a n i n g f u l c a l c u l a t i o n s c a n be made o f t h e i r e f f e c t s . A l t h o u g h d e t a i l e d m o d e l l i n g does n o t d i r e c t l y p r o v i d e t h e types of u s e f u l a n a l y t i c r e l a t i o n s h i p s which guide our i n t u i t i o n and a l l o w us t o make q u i c k e s t i m a t e s , i t g i v e s us t h e f l e x i b i l i t y t o e v a l u a t e t h e i m p o r t a n c e o f a m o d e l l e d p h y s i c a l e f f e c t by s i m p l y t u r n i n g i t o f f o r on o r c h a n g i n g i t s s t r e n g t h . The m o d e l c a n a l s o be u s e d t o t e s t t h e s e n s i t i v i t y o f t h e computed r e s u l t s t o i n d e p e n d e n t t h e o r e t i c a l a p p r o x i m a t i o n s . Those a n a l y t i c r e s u l t s w h i c h a r e a v a i l a b l e a r e v a l u a b l e i n benchmarking the model i n various l i m i t s . A s e r i e s o f t e s t s w h i c h compare a n a l y t i c r e s u l t s t o n u m e r i c a l s i m u l a t i o n s may c a l i b r a t e t h e s i m u l a t i o n b e f o r e i t i s compared t o e x p e r i m e n t s o r u s e d f o r e x t r a p o l a t i o n . C o n v e r s e l y , a w e l l - t e s t e d m o d e l s e r v e s as a v e r y u s e f u l means o f c a l i b r a t i n g unknown p a r a m e t e r s and f o r m f a c t o r s i n a p p r o x i m a t e t h e o r i e s . The l a s t two s e c t i o n s o f t h i s p a p e r w i l l d i s c u s s t h i s i n t e r p l a y b e t w e e n d e t a i l e d m o d e l l i n g and b o t h t h e o r y and e x p e r i m e n t . The t h i r d s e c t i o n d e s c r i b e s how a m o d e l must be t e s t e d i n v a r i o u s l i m i t s f o r p h y s i c a l consistency to insure i t s accuracy. The s p e c i f i c example c h o s e n h e r e i s a c o m p a r i s o n b e t w e e n an a n a l y t i c s o l u t i o n and a d e t a i l e d n u m e r i c a l s i m u l a t i o n o f a p r e m i x e d l a m i nar f l a m e . The l a s t s e c t i o n shows how a c o m p a r i s o n b e t w e e n model r e s u l t s and e x p e r i m e n t s c a n be u s e d t o c a l i b r a t e t h e m o d e l and t o g u i d e f u r t h e r e x p e r i m e n t s . The e x a m p l e c h o s e n i s a c a l c u l a t i o n o f f l o w o v e r an immersed o b j e c t w h i c h i s compared t o b o t h e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s . Problems

i n M o d e l l i n g R e a c t i v e Flows

E r r o r s and c o n f u s i o n i n m o d e l l i n g a r i s e b e c a u s e t h e c o m p l e x set of c o u p l e d , n o n l i n e a r , 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 a r e not u s u a l l y an e x a c t r e p r e s e n t a t i o n o f t h e p h y s i c a l s y s t e m . As examp l e s , f i r s t c o n s i d e r t h e i n p u t p a r a m e t e r s , s u c h as c h e m i c a l r a t e constants or d i f f u s i o n c o e f f i c i e n t s . These i n p u t q u a n t i t i e s , u s e d as submodels i n t h e d e t a i l e d m o d e l , must be d e r i v e d f r o m more f u n d a m e n t a l t h e o r i e s , m o d e l s o r e x p e r i m e n t s . They a r e u s u a l l y n o t known t o any a p p r e c i a b l e a c c u r a c y and o f t e n t h e i r v a l u e s a r e s i m p l y g u e s s e s . Or c o n s i d e r t h e g e o m e t r y u s e d i n a c a l c u l a tion. I t i s o f t e n one o r two d i m e n s i o n s l e s s t h a n needed t o c o m p l e t e l y d e s c r i b e the r e a l system. M u l t i d i m e n s i o n a l e f f e c t s w h i c h may be i m p o r t a n t a r e e i t h e r c r u d e l y a p p r o x i m a t e d o r i g n o r e d . T h i s l a c k o f e x a c t c o r r e s p o n d e n c e b e t w e e n t h e m o d e l a d o p t e d and the a c t u a l p h y s i c a l s y s t e m c o n s t i t u t e s t h e b a s i c p r o b l e m o f d e t a i l e d m o d e l l i n g . T h i s p r o b l e m , w h i c h must be overcome i n o r d e r t o a c c u r a t e l y m o d e l t r a n s i e n t c o m b u s t i o n s y s t e m s , c a n be a n a l y z e d i n t e r m s o f t h e m u l t i p l e t i m e s c a l e s , m u l t i p l e s p a c e s c a l e s , geom e t r i c c o m p l e x i t y , and p h y s i c a l c o m p l e x i t y of the systems t o be m o d e l l e d . M u l t i p l e Time S c a l e s .

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t h e r e s u l t o f t r y i n g t o r e p r e s e n t phenomena c h a r a c t e r i z e d b y v e r y d i f f e r e n t time s c a l e s . I n o r d i n a r y flame and d e t o n a t i o n problems t h e s e s c a l e s r a n g e o v e r many o r d e r s o f m a g n i t u d e . When phenomena a r e modelled t h a t have c h a r a c t e r i s t i c times o f v a r i a t i o n s h o r t e r t h a n t h e t i m e s t e p one c a n a f f o r d , t h e e q u a t i o n s d e s c r i b i n g t h e s e phenomena a r e u s u a l l y c a l l e d " s t i f f . E q u a t i o n s d e s c r i b i n g sound waves a r e s t i f f w i t h r e s p e c t t o t h e t i m e s t e p one w i s h e s t o employ when m o d e l l i n g a s u b s o n i c f l a m e s p e e d . Many c h e m i c a l r e a c t i o n r a t e equations are s t i f f w i t h respect t o convection, d i f f u s i o n , or e v e n sound wave t i m e s t e p c r i t e r i a . Two r a t h e r d i s t i n c t m o d e l l i n g approaches, g l o b a l i m p l i c i t and t i m e s t e p - s p l i t a s y m p t o t i c , have been developed t o t r e a t these t e m p o r a l l y s t i f f e q u a t i o n s . These two a p p r o a c h e s a r e b r i e f l y d e s c r i b e d l a t e r i n t h i s p a p e r .


1 1

M u l t i p l e S p a c e S c a l e s . The s e c o n d c l a s s o f p r o b l e m s i n v o l v e s t h e huge d i s p a r i t y i n s p a c e s c a l e s o c c u r r i n g i n c o m b u s t i o n problems. To m o d e l t h e s t e e p g r a d i e n t s a t a f l a m e f r o n t , a c e l l s p a c i n g o f 1 0 " cm o r s m a l l e r m i g h t be r e q u i r e d . To m o d e l c o n v e c t i o n , g r i d s p a c i n g s o f 1 t o 10 cm m i g h t be a d e q u a t e . Complex phenomena s u c h a s t u r b u l e n c e w h i c h o c c u r on i n t e r m e d i a t e s p a t i a l s c a l e s present a p a r t i c u l a r m o d e l l i n g problem. I t w o u l d be a pipedream t o expect a numerical c a l c u l a t i o n t o f a i t h f u l l y r e p r o d u c e p h y s i c a l phenomena w i t h s c a l e l e n g t h s s h o r t e r t h a n a c e l l size. Therefore, t o calculate r e a l i s t i c p r o f i l e s of p h y s i c a l v a r i a b l e s , a c e r t a i n c e l l spacing i s required t o obtain a given accuracy. C h o o s i n g a method w h i c h m a x i m i z e s a c c u r a c y w i t h a minimum number o f g r i d p o i n t s i s a m a j o r c o n c e r n i n d e t a i l e d modelling. 3

Geometric Complexity. The t h i r d s e t o f o b s t a c l e s a r i s e s because o f the geometric c o m p l e x i t y a s s o c i a t e d w i t h r e a l systems. Most o f t h e d e t a i l e d m o d e l s d e v e l o p e d t o d a t e h a v e been oned i m e n s i o n a l , b u t t h i s g i v e s a v e r y l i m i t e d p i c t u r e o f how t h e e n e r g y r e l e a s e a f f e c t s t h e h y d r o d y n a m i c s . E v e n t h o u g h many p r o c e s s e s i n a c o m b u s t i o n s y s t e m can be m o d e l l e d i n o n e - d i m e n s i o n , t h e r e are o t h e r s , such as boundary l a y e r growth, o r the f o r m a t i o n of v o r t i c e s a n d s e p a r a t i n g f l o w s , w h i c h c l e a r l y r e q u i r e a t l e a s t two-dimensional hydrodynamics. R e a l combustion systems are a t l e a s t t w o - d i m e n s i o n a l , w i t h u n u s u a l boundary c o n d i t i o n s and i n t e r n a l s o u r c e s and s i n k s . However, e v e n w i t h s i x t h g e n e r a t i o n p a r a l l e l p r o c e s s i n g c o m p u t e r s a v a i l a b l e , what c a n b e a c h i e v e d w i t h t w o - d i m e n s i o n a l d e t a i l e d m o d e l s i s s t i l l l i m i t e d by computer time and s t o r a g e requirements. In the c u r r e n t s t a t e - o f - t h e - a r t , o n e - d i m e n s i o n a l models c a n b e s t be used t o l o o k i n d e t a i l a t the c o u p l i n g o f a v e r y l a r g e number o f s p e c i e s i n t e r a c t i o n s i n a g e o m e t r y t h a t i s a n a p p r o x i mation t o r e a l i t y . P r o c e s s e s such as r a d i a t i o n t r a n s p o r t , t u r b u l e n c e , o r t h e e f f e c t s o f h e t e r o g e n e i t y o f m a t e r i a l s c a n be i n c l u ded e i t h e r a s e m p i r i c a l l y o r t h e o r e t i c a l l y d e r i v e d s u b m o d e l s . Two- and t h r e e - d i m e n s i o n a l m o d e l s a r e b e s t u s e d t o s t u d y e i t h e r


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gross flow p r o p e r t i e s or d e t a i l e d r a d i a t i o n transport. l a t t e r m o d e l s , t h e c h e m i c a l r e a c t i o n scheme i s u s u a l l y i d e a l i z e d or parameterized.

In these quite

P h y s i c a l Complexity. The f i n a l s e t o f o b s t a c l e s t o d e t a i l e d m o d e l l i n g concerns p h y s i c a l c o m p l e x i t y . Combustion systems usua l l y h a v e many i n t e r a c t i n g s p e c i e s . T h i s l e a d s t o s e t s o f many c o u p l e d e q u a t i o n s w h i c h must be s o l v e d s i m u l t a n e o u s l y . C o m p l i cated ordinary d i f f e r e n t i a l equations d e s c r i b i n g the chemical reactions or large matrices describing the molecular d i f f u s i o n p r o c e s s a r e c o s t l y and i n c r e a s e c a l c u l a t i o n t i m e o r d e r s o f m a g n i t u d e o v e r i d e a l i z e d o r e m p i r i c a l m o d e l s . T a b l e I l i s t s some o f t h e m a j o r c h e m i c a l and p h y s i c a l p r o c e s s e s w h i c h h a v e t o be c o n s i d e r e d f o r an a c c u r a t e d e s c r i p t i o n o f a c o m p l i c a t e d c o m b u s t i o n system. The d a s h e d l i n e i n t h e t a b l e i n d i c a t e s t h a t m u l t i p h a s e p r o c e s s e s s u c h a s s u r f a c e c a t a l y s i s and s o o t f o r m a t i o n c a n be i m p o r t a n t e v e n when we a r e p r i m a r i l y i n t e r e s t e d i n g a s p h a s e combustion. F o r most i n t e r e s t i n g s y s t e m s , one f i n d s t h a t t h e b a s i c c h e m i c a l r e a c t i o n scheme, t h e i n d i v i d u a l c h e m i c a l r a t e s , t h e o p t i c a l o p a c i t i e s , or the e f f e c t s of surface r e a c t i o n s are not w e l l known. B e f o r e a m o d e l o f a w h o l e c o m b u s t i o n s y s t e m c a n be a s s e m b l e d , e a c h i n d i v i d u a l p r o c e s s must be s e p a r a t e l y u n d e r s t o o d and m o d e l l e d . These submodels a r e e i t h e r i n c o r p o r a t e d i n t o t h e l a r g e r d e t a i l e d m o d e l d i r e c t l y o r , i f t h e t i m e and s p a c e s c a l e s a r e t o o d i s p a r a t e , t h e y must be f i t i n p h e n o m e n o l o g i c a l l y . F o r e x a m p l e , d i f f u s i o n and t h e r m a l c o n d u c t i v i t y b e t w e e n a w a l l and t h e r e a c t i n g gas c a n be s t u d i e d s e p a r a t e l y and t h e n i n c o r p o r a t e d d i r e c t l y i n t o a d e t a i l e d combustion model. T u r b u l e n c e , however, c a n be m o d e l l e d on i t s own s p a c e s c a l e s o n l y i n i d e a l i z e d c a s e s . T h e s e i d e a l i z e d , f u n d a m e n t a l , "ab i n i t i o " t u r b u l e n c e c a l c u l a t i o n s must be u s e d t o d e v e l o p p h e n o m e n o l o g i c a l m o d e l s f o r u s e i n t h e m a c r o s c o p i c d e t a i l e d m o d e l . R e s o l u t i o n and c o m p u t a t i o n a l c o s t prevent i n c o r p o r a t i n g t h e d e t a i l e d t u r b u l e n c e model d i r e c t l y . Table I Fundamental P r o c e s s e s i n Combustion gas p h a s e Chemical k i n e t i c s Hydrodynamic s - l a m i n a r Thermal c o n d u c t i v i t y , v i s c o s i t y Molecular d i f f u s i o n Thermochemistry Hydrodynamics-turbulent Radiation Nucleation Surface Effects Phase T r a n s i t i o n s (Evaporation, condensation...)

multi-phase



336


O f t e n t h e r e a r e c a s e s w h e r e t h e s u b m o d e l s a r e p o o r l y known or m i s u n d e r s t o o d , s u c h a s f o r c h e m i c a l r a t e e q u a t i o n s , t h e r m o chemical data, or transport c o e f f i c i e n t s . A t y p i c a l example i s shown i n F i g u r e 1 w h i c h was p r o v i d e d b y D a v i d G a r v i n a t t h e U. S. N a t i o n a l Bureau o f Standards. The f i g u r e shows t h e r a t e c o n s t a n t a t 300°K f o r t h e r e a c t i o n HO + O3 HO2 + O2 a s a f u n c t i o n o f t h e y e a r o f t h e measurement. We n o t e w i t h amusement and c h a g r i n t h a t i f we w e r e m o d e l l i n g a k i n e t i c s scheme w h i c h i n c o r p o r a t e d t h i s r e a c t i o n b e f o r e 1970, t h e r a t e w o u l d b e u n c e r t a i n by f i v e o r d e r s of m a g n i t u d e ! As shown most c l e a r l y b y t h e p a i r o f r a t e c o n s t a n t v a l u e s w h i c h h a v e a n e q u a l u p p e r bound a n d l o w e r b o u n d , a s e n s i t i v i t y a n a l y s i s u s i n g s u c h p o o r l y d e f i n e d r a t e c o n s t a n t s w o u l d be useless. Y e t t h i s case i s not a t y p i c a l o f the u n c e r t a i n t y i n r a t e c o n s t a n t s f o r many m a j o r r e a c t i o n s i n c o m b u s t i o n p r o c e s s e s . Gedanken Flame E x p e r i m e n t . I n o r d e r t o i l l u s t r a t e how t h e problems caused by the r e q u i r e m e n t s o f temporal and s p a t i a l r e s o l u t i o n and g e o m e t r i c and p h y s i c a l c o m p l e x i t y a r e t r a n s l a t e d i n t o c o m p u t a t i o n a l c o s t , we h a v e c h o s e n t o a n a l y z e a gedanken f l a m e experiment. C o n s i d e r a c l o s e d t u b e one m e t e r l o n g w h i c h c o n t a i n s a c o m b u s t i b l e g a s m i x t u r e . We w i s h t o c a l c u l a t e how t h e p h y s i c a l p r o p e r t i e s such as t e m p e r a t u r e , s p e c i e s d e n s i t i e s , and p o s i t i o n of t h e f l a m e f r o n t change a f t e r t h e m i x t u r e i s i g n i t e d a t one end. The b u r n i n g g a s c a n b e d e s c r i b e d , we assume, b y a c h e m i c a l k i n e t i c s r e a c t i o n r a t e scheme w h i c h i n v o l v e s some t e n s o f s p e c i e s and h u n d r e d s o f c h e m i c a l r e a c t i o n s , some o f w h i c h a r e " s t i f f . " We w i l l assume o n e - d i m e n s i o n a l p r o p a g a t i o n a l o n g t h e t u b e . Bound a r y l a y e r f o r m a t i o n a n d t u r b u l e n c e w i l l be i g n o r e d . We f u r t h e r assume t h a t t h e f l a m e f r o n t moves a t a n a v e r a g e v e l o c i t y o f 100 cm/ s e c . T a b l e I I summarizes the p e r t i n e n t time and space s c a l e s i n t h i s problem. Assuming the speed o f sound i s 10 cm/sec, a t i m e s t e p o f about 1 0 ~ sec would be r e q u i r e d t o r e s o l v e the motion o f sound waves b o u n c i n g a c r o s s t h e chamber. C h e m i c a l t i m e s c a l e s , a s m e n t i o n e d a b o v e , a r e a b o u t 10"~ s e c . T h i s number may b e r e d u c e d d r a s t i c a l l y i f t h e r e a c t i o n r a t e s o r d e n s i t y changes a r e v e r y fast. I t t a k e s a sound wave a b o u t 10"" s e c o n d s t o c r o s s t h e 1 m e t e r s y s t e m a n d i t t a k e s t h e f l a m e f r o n t a b o u t one s e c o n d t o c r o s s . We f u r t h e r assume t h a t t h e f l a m e zone i s a b o u t 1 0 ~ cm w i d e a n d t h a t i t t a k e s g r i d s p a c i n g s o f 1 0 " cm t o r e s o l v e t h e s t e e p g r a d i e n t s i n d e n s i t y and temperature i n t h i s flame zone. In t h o s e p o r t i o n s o f t h e tube on e i t h e r s i d e o f the flame f r o n t , we assume t h a t 1 cm s p a c i n g s a r e a d e q u a t e . To e s t i m a t e t h e c o m p u t a t i o n a l e x p e n s e o f t h i s c a l c u l a t i o n , we u s e 1 0 " s e c o n d s o f computer t i m e a s a r e a s o n a b l e e s t i m a t e o f t h e t i m e i t t a k e s t o i n t e g r a t e e a c h g r i d p o i n t f o r one t i m e s t e p (a s i n g l e p o i n t s t e p ) . This estimate includes a solution of the c h e m i c a l a n d h y d r o d y n a m i c e q u a t i o n s and i s b a s e d on a d e t a i l e d model o f a hydrogen-oxygen flame problem o p t i m i z e d f o r a p a r a l l e l p r o c e s s i n g c o m p u t e r . F i g u r e 2 shows t h e i n f o r m a t i o n i n T a b l e I I 5

9

6

3

2

3

3


27.

Detailed Modelling

ORAN ET AL.

of

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RATE C O N S T A N T S AT 300 Κ FOR -12 | _ H O + 0 - H 0 + 0 3

2

2

337

-Θ

Τ


13h

• 14

+

©++ + ~© ί

Θ μ

NBS RECOMMENDED VALUE •15

ι

•16

1970

1960

1950

τ

YEAR

Figure 1. Measured values of the rate constant for HO + O -> H0 + O as a function of the year of measurement. The arrows with overbars and underbars indi cate measured upper and lower bounds, respectively. The NBS-recommended value is the value with the smallest error bars (12). s

2

x

1000 100

105 PT.

(/)

I \ CONVECTIVE SCALES -100 SEC j

10

Œ

HI

È

1

} DIFFUSION SCALE

1014 p i . STEP -3000 YEARS

κ 10-1

I

-SYSTEM SIZE

CTCD

} FLAME ZONE -FLAME RESOLUTION

10-2

10-3 10-10 I 10-4

At,

10-7

At

10-4 c

10-1

100 SEC

t

SOUND FLAME

Figure 2. Space and time scales in the gedanken flame calculation. A naive direct solution of the problem could take 3000 years of computer time. The calculation should be possible in 100 sec.



338

c a s t i n t o a graph of space v e r s u s time. Since the s c a l e s are l o g a r i t h m i c , a c a l c u l a t i o n o f t h e number o f p o i n t s t e p s and t h e n o f t h e needed c o m p u t e r t i m e r e q u i r e s e x p o n e n t i a t i o n . Thus i t a p p e a r s t h a t 3000 y e a r s o f computer t i m e i s r e q u i r e d t o c a l c u l a t e the 1 0 p o i n t s t e p s needed t o r e s o l v e t h e f i n e s t s p a c e and t i m e s c a l e s everywhere! 1 4

Important

Table I I S c a l e s i n Gedanken F l a m e E x p e r i m e n t Spacescales


Timescales At

sec

Δχ

Flame R e s o l u t i o n F l a m e Zone Diffusion Scale Convective Scales System S i z e = 100 cm/sec

Sound Speed 10 Chemistry 10 Sound T r a n s i t Time 10 F l a m e T r a n s i t Time 1

cm 10ΙΟ" 1010 100

3

2

1

Of c o u r s e , t h i s i s u n a c c e p t a b l e . I d e a l l y such a s i m p l e c a l c u l a t i o n s h o u l d t a k e a b o u t 100 s e c o n d s (See F i g u r e 3 ) . What a r e needed a r e n u m e r i c a l a l g o r i t h m s w h i c h h a v e t h e r e s o l u t i o n i n t i m e and s p a c e o n l y where i t i s r e q u i r e d . F u r t h e r m o r e , t h e s e a l g o r i t h m s s h o u l d be o p t i m i z e d t o t a k e a d v a n t a g e o f what i s known a b o u t t h e p h y s i c s and c h e m i s t r y o f t h e p r o b l e m . T h i s w i l l be d i s c u s s e d f u r t h e r b e l o w w h e r e i t i s shown how t h e a p p l i c a t i o n o f v a r i o u s n u m e r i c a l a l g o r i t h m s c a n be u s e d t o r e d u c e t h i s f l a m e system to a t r a c t a b l e computational problem. Turbulence. T u r b u l e n c e i s one o f t h e o u t s t a n d i n g p r o b l e m s o f r e a c t i v e f l o w m o d e l l i n g and i s a n o t h e r e x c e l l e n t e x a m p l e o f t h e d i f f i c u l t y we h a v e i n r e s o l v i n g h i g h l y d i s p a r a t e t i m e and s p a c e s c a l e s . Our u n d e r s t a n d i n g and e v e n t u a l a b i l i t y t o p r e d i c t the complicated i n t e r a c t i o n s o c c u r r i n g i n t u r b u l e n t r e a c t i v e f l o w p r o b l e m s i s i m p e r a t i v e f o r many c o m b u s t i o n m o d e l l i n g a p p l i c a t i o n s . The p r e s e n c e o f t u r b u l e n c e a l t e r s m i x i n g and r e a c t i o n t i m e s and h e a t and mass t r a n s f e r r a t e s w h i c h i n t u r n m o d i f y t h e l o c a l and g l o b a l d y n a m i c p r o p e r t i e s o f t h e s y s t e m . What we need t o r e s o l v e t h e s e p r o b l e m s a r e a c c u r a t e y e t compact phenomenological t u r b u l e n c e m o d e l s w h i c h c a n be u s e d t o d e s c r i b e r e a l i s t i c combus t o r s y s t e m s , open f l a m e s , and o t h e r t u r b u l e n t r e a c t i v e f l o w s c o n f i d e n t l y and e f f i c i e n t l y . These c o m p u t a t i o n a l m o d e l s must d e c o u p l e t h e s u b g r i d t u r b u l e n c e and m i c r o s c o p i c i n s t a b i l i t y mecha n i s m s f r o m c a l c u l a t i o n s o f t h e m a c r o s c o p i c f l o w . B e l o w we l i s t t h e i m p o r t a n t p r o p e r t i e s o f an i d e a l t u r b u l e n c e m o d e l (2) . 1. C h e m i s t r y - H y d r o d y n a m i c s C o u p l i n g and F e e d b a c k . Explicit e n e r g y f e e d b a c k mechanisms f r o m m i x i n g and r e a c t i o n s t o t h e t u r b u l e n t v e l o c i t y f i e l d and t h e m a c r o s c o p i c f l o w must be f o r m u l a t e d .



27.

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Detailed Modelling

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339

The " l a m i n a r " m a c r o s c o p i c f l o w e q u a t i o n s c o n t a i n p h e n o m e n o l o g i c a l terms which r e p r e s e n t averages over the m a c r o s c o p i c dynamics t o i n c l u d e the e f f e c t s of t u r b u l e n c e . Examples of these terms a r e eddy v i s c o s i t y and d i f f u s i v i t y c o e f f i c i e n t s and a v e r a g e c h e m i c a l h e a t r e l e a s e t e r m s w h i c h a p p e a r as s o u r c e s i n t h e m a c r o s c o p i c flow equations. Besides p r o v i d i n g these phenomenological terms, t h e t u r b u l e n c e m o d e l must u s e t h e i n f o r m a t i o n p r o v i d e d by t h e l a r g e s c a l e f l o w dynamics s e l f - c o n s i s t e n t l y to determine the energy which d r i v e s the t u r b u l e n c e . The m o d e l must be a b l e t o f o l l o w r e a c t i v e i n t e r f a c e s on t h e m a c r o s c o p i c s c a l e . 2. M o d e l l i n g Onset and O t h e r T r a n s i e n t T u r b u l e n c e Phenomena. The m o d e l s h o u l d be a b l e t o p r e d i c t t h e o n s e t of t u r b u l e n c e i n i n i t i a l l y l a m i n a r f l o w s i n c e b u r s t s and o t h e r h i g h l y t r a n s i e n t phenomena seem t o be t h e r u l e i n r e a c t i v e f l o w t u r b u l e n c e . Grad i e n t s i n d e n s i t y , t e m p e r a t u r e , and v e l o c i t y f i e l d s i n t h e r e a c t i n g f l u i d d r i v e the m a c r o s c o p i c f l u i d dynamic i n s t a b i l i t i e s which i n i t i a t e turbulence. Thus t h e s e g r a d i e n t s f r o m t h e m a c r o s c o p i c c a l c u l a t i o n a r e bound t o be key i n g r e d i e n t s i n d e t e r m i n i n g t h e energy a v a i l a b l e to d r i v e the t u r b u l e n c e . 3. C o m p l i c a t e d R e a c t i o n s and F l o w . The i d e a l t u r b u l e n c e m o d e l must d e a l w i t h m u l t i s c a l e e f f e c t s w i t h i n t h e s u b g r i d m o d e l . I f t h e r e i s a d e l a y as v e l o c i t y c a s c a d e s t o t h e s h o r t w a v e l e n g t h end o f t h e s p e c t r u m due t o c h e m i c a l k i n e t i c s o r b u o y a n c y , f o r e x a m p l e , t h e m o d e l must be c a p a b l e o f r e p r e s e n t i n g t h i s . Otherw i s e b u r s t s and i n t e r m i t t e n c y phenomena c a n n o t be c a l c u l a t e d . 4. L a g r a n g i a n Framework. An i d e a l s u b g r i d m o d e l s h o u l d be c o n s t r u c t e d on a L a g r a n g i a n h y d r o d y n a m i c s f r a m e w o r k m o v i n g w i t h the macroscopic f l o w . This requirement reduces p u r e l y n u m e r i c a l d i f f u s i o n t o z e r o so t h a t r e a l i s t i c t u r b u l e n c e and m o l e c u l a r m i x i n g phenomena w i l l n o t be masked by n o n - p h y s i c a l n u m e r i c a l smoothing. T h i s r e q u i r e m e n t a l s o removes t h e p o s s i b i l i t y o f m a s k i n g p u r e l y l o c a l f l u c t u a t i o n s by t r u n c a t i o n e r r o r s f r o m t h e n u m e r i c a l r e p r e s e n t a t i o n of macroscopic c o n v e c t i v e d e r i v a t i v e s . The t i m e - d e p e n d e n t ( h y p e r b o l i c ) L a g r a n g i a n f r a m e w o r k s h o u l d a l s o g e n e r a l i z e t o t h r e e d i m e n s i o n s as w e l l as r e s o l v e r e a c t i v e i n t e r faces dynamically. 5. S c a l i n g . Breaking a c a l c u l a t i o n i n t o macroscopic scales and s u b g r i d s c a l e s i s an a r t i f i c e t o a l l o w us t o m o d e l t u r b u l e n c e . The i m p o r t a n t p h y s i c s o c c u r s c o n t i n u o u s l y o v e r t h e w h o l e s p e c t r u m f r o m k , t h e wavenumber c o r r e s p o n d i n g t o t h e s y s t e m s i z e , t o k-diss » number c o r r e s p o n d i n g t o a mean f r e e p a t h o f a molecule. Thus t h e m a c r o s c o p i c and s u b g r i d s c a l e s p e c t r a o f any p h y s i c a l q u a n t i t y must c o u p l e s m o o t h l y a t k - Q , t h e c e l l b o u n d a r y wave number. I f t h i s number were t o be c h a n g e d , as m i g h t happen i f n u m e r i c a l r e s o l u t i o n were h a l v e d o r d o u b l e d , t h e p r e d i c t i o n s of t h e t u r b u l e n c e m o d e l must n o t change. 6. Efficiency. Of c o u r s e , t h e m o d e l must be e f f i c i e n t . The number of d e g r e e s of f r e e d o m r e q u i r e d t o s p e c i f y t h e s t a t u s o f t u r b u l e n c e i n e a c h s e p a r a t e l y r e s o l v e d s u b g r i d r e g i o n has t o be k e p t t o a minimum f o r t h e m o d e l t o be g e n e r a l l y u s e a b l e . The Q

t n e

w

a

v

e

c e



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r e a l f l u i d h a s e s s e n t i a l l y a n i n f i n i t e number o f d e g r e e s o f f r e e dom t o r e p r e s e n t t h e s t a t e o f t h e g a s i n e a c h s m a l l e l e m e n t . We w o u l d l i k e t o be a b l e t o do t h e j o b w i t h a m i n i m a l number o f d e grees o f freedom.


Choosing an A l g o r i t h m Based on t h e P h y s i c s o f t h e Problem I n r e a c t i v e f l o w c a l c u l a t i o n s we a r e c o n c e r n e d w i t h two f l o w r e g i m e s w h i c h depend on t h e r a t e o f e n e r g y r e l e a s e . When e n e r g y i s r e l e a s e d q u i c k l y , s h o c k s a n d d e t o n a t i o n s a r e f o r m e d . When e n e r g y i s r e l e a s e d s l o w l y , f l a m e s a r e f o r m e d . The f o r m e r r e q u i r e s t h a t t h e n u m e r i c a l a l g o r i t h m used f o l l o w t h e changes o f t h e s y s t e m on t i m e s c a l e s d e t e r m i n e d b y t h e s p e e d o f sound i n t h e m a t e r i a l ( C o u r a n t c o n d i t i o n ) . I f we f o l l o w t h i s same a c o u s t i c wave t r a n s i t t i m e s c a l e i n t h e f l a m e c a s e where t h e p h y s i c a l t i m e s c a l e s o f i n t e r e s t a r e much l a r g e r , t h e c o s t i s e x h o r b i t a n t . The g e d a n k e n f l a m e c a l c u l a t i o n d e s c r i b e d above c o s t s o much p a r t l y b e c a u s e we p o s t u l a t e d t h e u s e o f a n e x p l i c i t a l g o r i t h m b a s e d on t i m e s t e p s determined by t h e Courant c o n d i t i o n . F o r flame c a l c u l a t i o n s , t h e n , t h e answer i s t o u s e t e c h n i q u e s i n w h i c h t h e e n e r gy c o n s e r v a t i o n e q u a t i o n i s c o n v e r t e d t o a p r e s s u r e e q u a t i o n which i s solved i m p l i c i t l y . A major area o f m o d e l l i n g concern i s that o f c o u p l i n g i n t o one c a l c u l a t i o n a l l o f t h e p e r t i n e n t p h y s i c a l and c h e m i c a l p r o c e s s e s c h a r a c t e r i s t i c o f a c o m b u s t i o n s y s t e m . Two d i s t i n c t approaches have e v o l v e d . I n the f i r s t of these, often c a l l e d " g l o b a l i m p l i c i t " d i f f e r e n c i n g , t h e complete s e t of n o n l i n e a r coupled equations d e s c r i b i n g t h e p h y s i c a l system o f i n t e r e s t i s c a s t i n t o a s i m p l e f i n i t e - d i f f e r e n c e f o r m . The s p a t i a l and tem p o r a l d e r i v a t i v e s a r e d i s c r e t i z e d and t h e n o n l i n e a r t e r m s a r e l i n e a r i z e d l o c a l l y about t h e s o l u t i o n s o b t a i n e d n u m e r i c a l l y a t the previous timestep. T h i s p r o c e s s i s v a l i d o n l y when t h e v a l ues o f t h e p h y s i c a l v a r i a b l e s change s l o w l y o v e r a t i m e s t e p . A r i g o r o u s l y c o r r e c t treatment o f t h e n o n l i n e a r terms r e q u i r e s i t e r a t i o n a n d l a r g e m a t r i c e s must b e i n v e r t e d a t e a c h t i m e s t e p t o guarantee s t a b i l i t y . I n one s p a t i a l d i m e n s i o n , s a y x , t h e p r o blem u s u a l l y appears as a b l o c k t r i d i a g o n a l m a t r i x w i t h M i n d e p e n d e n t p h y s i c a l v a r i a b l e s t o be s p e c i f i e d a t N g r i d p o i n t s . Then a n Μ Ν b y Μ Ν m a t r i x must b e i n v e r t e d a t e a c h i t e r a t i o n o f each t i m e s t e p . The b l o c k s on o r a d j a c e n t t o t h e m a t r i x d i a g o n a l are M Χ M i n s i z e so t h e o v e r a l l m a t r i x i s q u i t e sparse. Never t h e l e s s , a n enormous amount o f c o m p u t a t i o n a l w o r k goes i n t o a d v a n c i n g t h e s o l u t i o n even a s i n g l e t i m e L t e p . Multidimensional pro b l e m s , i n t h i s a p p r o a c h , l e a d t o m a t r i c e s w h i c h a r e M N N by Μ Ν Ν i n two d i m e n s i o n s and Μ Ν Ν Ν by M N N N i n t h r e e d i m e n sions. I n c o m p l e x k i n e t i c s p r o b l e m s w i t h no s p a t i a l v a r i a t i o n , t h e M i n d e p e n d e n t v a r i a b l e s a r e t h e s p e c i e s number d e n s i t i e s and t e m p e r a t u r e i n t h e homogeneous v o l u m e o f i n t e r e s t . The Gear method ( 3 ) i s a n e x a m p l e o f t h i s g l o b a l i m p l i c i t a p p r o a c h f o r pure k i n e t i c s problems. x

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The s e c o n d a p p r o a c h i s a f r a c t i o n a l - s t e p method we c a l l asymptotic t i m e s t e p - s p l i t t i n g . I t i s d e v e l o p e d by c o n s i d e r a t i o n of the s p e c i f i c p h y s i c s of the problem b e i n g s o l v e d . Stiffness i n t h e g o v e r n i n g e q u a t i o n s c a n be h a n d l e d " a s y m p t o t i c a l l y as w e l l as i m p l i c i t l y . The i n d i v i d u a l t e r m s , i n c l u d i n g t h o s e w h i c h l e a d t o t h e s t i f f b e h a v i o r , a r e s o l v e d as i n d e p e n d e n t l y and a c c u r a t e l y as p o s s i b l e . E x a m p l e s o f s u c h methods i n c l u d e t h e S e l e c t e d A s y m p t o t i c I n t e g r a t i o n Method (4,5) f o r k i n e t i c s p r o b l e m s and t h e a s y m p t o t i c s l o w f l o w a l g o r i t h m f o r h y d r o d y n a m i c p r o b l e m s where t h e sound s p e e d i s so f a s t t h a t t h e p r e s s u r e i s e s s e n t i a l l y constant (6,7). The t r a d e o f f s b e t w e e n t h e s e two a p p r o a c h e s a r e c l e a r . The i m p l i c i t a p p r o a c h p u t s maximum s t r a i n on t h e c o m p u t e r and m i n i mal s t r a i n on t h e m o d e l l e r . F o r t h i s method, convergence of the computed s o l u t i o n s i s e a s y t o t e s t w i t h i m p r o v e d t e m p o r a l and spatial resolution. N o n - c o n v e r g e n c e o f any p a r t i c u l a r c a l c u l a t i o n may be h a r d t o s p o t s i n c e s e v e r e n u m e r i c a l damping has b e e n i n t r o d u c e d t o m a i n t a i n n u m e r i c a l s t a b i l i t y and p o s i t i v i t y . This damping changes t h e d e s i r e d p r o f i l e s q u a n t i t a t i v e l y , a l t h o u g h q u i c k l y d e t e c t e d q u a l i t a t i v e e r r o r s a r e o f t e n smoothed o u t . S o l u t i o n s may be wrong y e t s t a b l e . I n c o n t r a s t , t h e a s y m p t o t i c a p p r o a c h p u t s m i n i m a l s t r a i n on t h e c o m p u t e r b u t demands more o f t h e m o d e l l e r . The c o n v e r g e n c e o f t h e computed s o l u t i o n s i s u s u a l l y e a s y t o t e s t w i t h r e s p e c t t o s p a t i a l and t e m p o r a l r e s o l u t i o n , b u t s i t u a t i o n s e x i s t where r e d u c i n g t h e t i m e s t e p c a n make an a s y m p t o t i c t r e a t m e n t o f a " s t i f f " phenomenon l e s s a c c u r a t e r a t h e r t h a n more a c c u r a t e . T h i s f o l l o w s b e c a u s e t h e d i s p a r i t y o f t i m e s c a l e s b e t w e e n f a s t and s l o w phenomena i s o f t e n e x p l o i t e d i n t h e a s y m p t o t i c a p p r o a c h r a t h e r t h a n tolerated. F u r t h e r m o r e , t h e n o n - c o n v e r g e n c e o f any p a r t i c u l a r s o l u t i o n i s o f t e n e a s i e r t o s p o t i n t i m e s t e p s p l i t t i n g w i t h asymp t o t i c s b e c a u s e t h e manner o f d e g r a d a t i o n i s u s u a l l y c a t a s t r o p h i c . I n k i n e t i c s c a l c u l a t i o n s , l a c k o f c o n s e r v a t i o n o f mass o r atoms signals inaccuracy rather c l e a r l y . The a s y m p t o t i c a p p r o a c h u s u a l l y l e a d s t o more m o d u l a r s i m u l a t i o n models than the g l o b a l i m p l i c i t approach. Hydrodynamics, t r a n s p o r t , e q u a t i o n o f s t a t e c a l c u l a t i o n s , and c h e m i c a l k i n e t i c s a r e t i e d n e a t l y i n t o i n d i v i d u a l p a c k a g e s . What i s e v e n more i m p o r t a n t , s p e c i a l i z e d t e c h n i q u e s f o r e n h a n c i n g a c c u r a c y c a n be i n c o r p o r a t e d a t e a c h s t a g e and f o r e a c h p h y s i c a l phenomenon b e i n g m o d e l l e d s e p a r a t e l y . T h e r e i s no need t o u s e s i m p l e r methods which a r e s u i t a b l e f o r i n c l u s i o n i n t o a s i n g l e g i a n t f i n i t e difference formula. S i n c e e a c h phenomenon i s t r e a t e d as an i n d e p e n d e n t p a c k a g e , t h e f u l l s p e c t r u m o f n u m e r i c a l t r i c k s i s applicable. T h e s e p a c k a g e s a r e r e l a t i v e l y e a s y t o t e s t i n d i v i d u a l l y and can be v e r y s o p h i s t i c a t e d . They c a n a l s o be u s e d d i r e c t l y i n a number o f t o t a l l y d i f f e r e n t p h y s i c a l p r o b l e m s w i t h l i t t l e o r no change and a r e h e n c e more f l e x i b l e t h a n e q u i v a l e n t p o r t i o n s o f a g l o b a l i m p l i c i t a l g o r i t h m . The p r i c e f o r t h i s f l e x i b i l i t y i s


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t h e need t o t r e a t c a r e f u l l y a l l t h e c o u p l i n g s b e t w e e n t h e i n d i v i d u a l p h y s i c a l terms and e f f e c t s . Using the asymptotic approach one c a n n o t s i t b a c k a n d t u r n a m a s s i v e m a t h e m a t i c a l c r a n k t o g e t an a n s w e r . A t t h i s p o i n t t h e p r o s a n d cons o f t h e two a p p r o a c h e s seem to r o u g h l y c o u n t e r b a l a n c e . T h i s a p p a r e n t e q u i t y e x t e n d s t o most accuracy c r i t e r i a as w e l l . I f a t i m e s c a l e i s not r e s o l v e d , n e i t h e r s o l u t i o n method can g i v e d e t a i l e d p r o f i l e s o f phenomena o c c u r r i n g on t h a t s c a l e . S i m i l a r i l y , t o compute s p a t i a l g r a d i e n t s a c c u r a t e l y t h e y must b e r e s o l v e d w i t h enough s p a t i a l g r i d p o i n t s i n e i t h e r type of c a l c u l a t i o n . The f a c t t h a t t h e a s y m p t o t i c a p p r o a c h demands more w o r k o f the modeller i s c o u n t e r b a l a n c e d by t h e w o r k t h a t must be done t o reduce the c o m p u t a t i o n a l expense o f u s i n g the g l o b a l i m p l i c i t method. T h i s calculâtional e x p e n s e , above a l l e l s e , i s t h e f a c t o r w h i c h has c a u s e d u s t o employ a s y m p t o t i c r a t h e r t h a n g l o b a l i m p l i c i t formulations. For example, s o l v i n g a c h e m i c a l k i n e t i c s scheme f o r M s p e c i e s r e q u i r e s i n v e r t i n g a g e n e r a l m a t r i x o f s i z e M Χ M. T h i s i n v o l v e s a p p r o x i m a t e l y M o p e r a t i o n s . In contrast, the s e l e c t e d a s y m p t o t i c approach t o s o l v i n g the k i n e t i c s equa t i o n s g e n e r a l l y s c a l e s a s M. I t i s one g o a l o f d e t a i l e d m o d e l l i n g t o be a b l e t o i n c l u d e t h e f u l l d e t a i l s o f e x t r e m e l y c o m p l e x k i n e t i c s systems coupled t o time-dependent f l u i d dynamics. Since more c o m p l e x p r o b l e m s can b e s o l v e d f o r t h e same c o s t u s i n g asympt o t i c s , we a r e w i l l i n g t o i n v e s t t h e e f f o r t i n t h e p h y s i c s mod u l e s a n d t h e i r c o u p l i n g i n o r d e r t o b e a b l e t o expand o u r compu tational abilities. U s i n g t h e i n f o r m a t i o n d i s c u s s e d s o f a r , we can now r e t u r n t o the gedanken f l a m e experiment w i t h the i d e a o f c o n s i d e r i n g modi f i e d n u m e r i c a l methods i n o r d e r t o r e d u c e t h e c o m p u t a t i o n a l c o s t . The g o a l i s t o c a l c u l a t e t h e p r o p a g a t i o n o f a f l a m e f r o n t a c r o s s a one-meter t u b e u s i n g a o n e - d i m e n s i o n a l g e o m e t r y w i t h a f i x e d d e t a i l e d c h e m i c a l r e a c t i o n r a t e scheme. F i r s t , we r e c o g n i z e i m m e d i a t e l y t h a t we a r e i n t e r e s t e d i n c a l c u l a t i n g a f l a m e f r o n t m o v i n g a t l e s s t h a n t h e l o c a l sound s p e e d . Thus e i t h e r a s l o w f l o w a p p r o x i m a t i o n o r any method w h i c h t r e a t s p r e s s u r e i m p l i c i t l y w o u l d e l i m i n a t e t h e sound s p e e d c r i t e r i o n on t h e t i m e s t e p . By u s i n g t h e a s y m p t o t i c s l o w f l o w t e c h n i q u e d e s c r i b e d below and s t i l l assuming a u n i f o r m g r i d s p a c i n g , t h e number o f p o i n t s t e p s i s r e d u c e d f r o m 1 0 to 10 . Thus F i g u r e 3 shows t h a t t h e t i m e r e q u i r e d f o r t h e c a l c u l a t i o n i s r e d u c e d f r o m 3000 t o 3 y e a r s ! But t h i s i s s t i l l a t r o c i o u s , and we must now f a c e t h e p r o blem o f e l i m i n a t i n g unnecessary g r i d p o i n t s . Adaptive g r i d d i n g is currently a f r o n t i e r i n reactive flow modelling. As yet there a r e no g e n e r a l , e x c e l l e n t t e c h n i q u e s . The b l o c k o n t h e g r a p h i n F i g u r e 4 shows t h e r e g i o n spanned i n t h e gedanken f l a m e p r o b l e m by a n a d a p t i v e l y g r i d d e d c a l c u l a t i o n . H e r e 100 c e l l s o f 1 cm l e n g t h a r e used and the r e g i o n s u r r o u n d i n g the flame f r o n t i s f i n e l y g r i d d e d w i t h 100 a d d i t i o n a l c e l l s o f 1 0 ~ cm l e n g t h . The 3

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t i m e s t e p i s s t i l l g o v e r n e d b y t h e s m a l l e s t c e l l s , b u t b y now o n l y 200 c e l l s a r e n e e d e d r a t h e r t h a n 1 0 . The s a v i n g , a b o u t a f a c t o r o f 500, r e d u c e s t h e c o m p u t a t i o n a l t i m e t o 2 X 1 0 p o i n t s t e p s , o r a b o u t two d a y s . F i n a l l y , F i g u r e 5 summarizes the c o m p u t a t i o n a l expense o f p e r f o r m i n g the flame p r o p a g a t i o n problem u s i n g the p o s s i b l e , b u t as y e t u n e x p l o i t e d , t e c h n i q u e o f a d a p t i v e i n t e r m i t t e n t g r i d d i n g . The i d e a h e r e i s t h a t a f i n e l y g r i d d e d r e g i o n i s i n j e c t e d i n t o the c a l c u l a t i o n a t i n t e r m i t t e n t timesteps. T h i s i s done o f t e n enough t o u p d a t e t h e p r o p e r t i e s o f t h e f i n e l y - s p a c e d r e g i o n w h i c h are then used as i n t e r i o r boundary c o n d i t i o n s f o r the c o a r s e l y spaced r e g i o n . This i s t r u l y an asymptotic o u t e r - i n n e r - o u t e r m a t c h i n g p r o c e d u r e . Now assume t h a t 100 c e l l s a r e needed t o r e s o l v e t h e f l a m e z o n e . F u r t h e r , 100 s h o r t t i m e s t e p s a r e enough t o r e s o l v e c h a n g e s i n t h e f l a m e z o n e b r o u g h t a b o u t by t h e r e l a t i v e l y s l o w l y changing outer boundary c o n d i t i o n s . D u r i n g t h e imbedded c a l c u l a t i o n , t h e f l a m e f r o n t moves o n l y 10 o f t h e f i n e z o n e s , w h i c h i s s u f f i c i e n t t o d e t e r m i n e f l a m e speed and boundary c o n d i t i o n s t o b e u s e d i n t h e c o a r s e l y s p a c e d c a l c u l a t i o n . The imbedded c a l c u l a t i o n n e e d b e p e r f o r m e d a t most once i n e a c h l a r g e c e l l . Thus a t o t a l o f 100 + ( 1 0 0 ) ( 1 0 ) = 1100 s e c o n d s o f c o m p u t a t i o n a l time i s r e q u i r e d f o r the l a r g e s c a l e s i m u l a t i o n , a cost a t l e a s t a p p r o a c h i n g o u r o r i g i n a l n a i v e e s t i m a t e o f 100 s e c o n d s . T h i s e x a m p l e has i l l u s t r a t e d t h e i m p o r t a n c e o f u s i n g t h e a p p r o p r i a t e a l g o r i t h m m o t i v a t e d by c o n s i d e r a t i o n s o f the a c t u a l p r o b l e m t h a t must be s o l v e d . I t h a s f u r t h e r i l l u s t r a t e d how much may b e a c c o m p l i s h e d b y d e v e l o p i n g t h e methods o f a d a p t i v e g r i d ding. One p o i n t t h a t h a s n o t b e e n m e n t i o n e d , h o w e v e r , i s t h a t much o f t h e c o s t o f a d e t a i l e d r e a c t i v e f l o w c a l c u l a t i o n i s t a k e n up b y t h e i n t e g r a t i o n o f t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g the c h e m i c a l k i n e t i c s . U s i n g the l a t e s t a s y m p t o t i c t e c h n i q u e s i m p r o v e s t h e p i c t u r e p a i n t e d a b o v e b y a f a c t o r o f two t o four. But f u r t h e r improvements i n t h e s e i n t e g r a t i o n times w i t h o u t s a c r i f i c i n g a c c u r a c y i s c e r t a i n l y a n a r e a where d e v e l o p ment i s n e e d e d . 5


8

Testing

the Model A g a i n s t

Theoretical

Results

A n a l y t i c a l s o l u t i o n s , w h i l e o f t e n approximate, are extremely u s e f u l i n p r o v i d i n g f u n c t i o n a l r e l a t i o n s h i p s and g e n e r a l i z i n g trends. B e l o w we show t h a t b y c o m p a r i n g n u m e r i c a l a n d a n a l y t i c r e s u l t s , we c a n g a i n new i n s i g h t s i n t o t h e c o n t r o l l i n g p h y s i c a l processes. I g n i t i o n o f a f u e l - o x i d i z e r m i x t u r e o c c u r s when a n e x t e r n a l s o u r c e o f e n e r g y i n i t i a t e s i n t e r a c t i o n s among t h e c o n t r o l l i n g conv e c t i v e , t r a n s p o r t and c h e m i c a l p r o c e s s e s . Whether t h e p r o c e s s r e s u l t s i n d e f l a g r a t i o n , d e t o n a t i o n , o r i s s i m p l y quenched depends on t h e i n t e n s i t y , d u r a t i o n , and v o l u m e a f f e c t e d b y a n e x t e r n a l heat source. I g n i t i o n a l s o w i l l depend on t h e i n i t i a l a m b i e n t p r o p e r t i e s o f the m i x t u r e which determine the chemical i n d u c t i o n



27.

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Detailed

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345

t i m e and t h e h e a t r e l e a s e p e r gram. Thus i g n i t i o n i s a c o m p l i c a t e d phenomena and i t s p r e d i c t i o n f o r a s p e c i f i c m i x t u r e o f homo g e n e o u s , p r e m i x e d g a s e s depends s t r o n g l y on i n p u t p a r a m e t e r s w h i c h a r e o f t e n v e r y p o o r l y known. A c o n v e n i e n t , i n e x p e n s i v e way to estimate whether a mixture w i l l i g n i t e g i v e n a heat source i n t e n s i t y , d u r a t i o n , and v o l u m e w o u l d be a v a l u a b l e l a b o r a t o r y t o o l and a u s e f u l l e a r n i n g d e v i c e . A c l o s e d form s i m i l a r i t y s o l u t i o n f o r the n o n l i n e a r timedependent s l o w - f l o w e q u a t i o n s has been used as t h e b a s i s f o r a s i m p l e , time-dependent, a n a l y t i c model o f l o c a l i z e d i g n i t i o n w h i c h r e q u i r e s m i n i m a l c h e m i c a l a n d p h y s i c a l i n p u t ( 8 ) . As a f u n d a m e n t a l p a r t o f t h e m o d e l , t h e r e a r e two c o n s t a n t s w h i c h must calibrated: t h e r a d i i , or f r a c t i o n o f t h e time-dependent s i m i l a r i t y s o l u t i o n r a d i u s , a t w h i c h t h e t h e r m a l c o n d u c t i v i t y and i n d u c t i o n parameters a r e evaluated. This c a l i b r a t i o n i s achieved by c o m p a r i s o n w i t h t h e r e s u l t s o f a d e t a i l e d t i m e - d e p e n d e n t numer i c a l flame s i m u l a t i o n model w h i c h i s a f u l l s o l u t i o n o f t h e m u l t i f l u i d conservation equations. The d e t a i l e d m o d e l i t s e l f h a s been checked e x t e n s i v e l y w i t h r e s p e c t t o i t s v a r i o u s c h e m i c a l , d i f f u s i v e t r a n s p o r t , and h y d r o d y n a m i c components. The b a s i c s i m i l a r i t y s o l u t i o n f o r t h i s i g n i t i o n p r o b l e m i s d e r i v e d f r o m t h e s l o w f l o w (6,^7) a p p r o x i m a t i o n , c h a r a c t e r i z e d by (1) f l o w v e l o c i t i e s w h i c h a r e s m a l l compared t o t h e s p e e d o f s o u n d , and ( 2 ) a n e s s e n t i a l l y c o n s t a n t p r e s s u r e f i e l d . The e n e r gy and v e l o c i t y e q u a t i o n s may t h e n be w r i t t e n a s HP

-k^CMr v

0 = - yPV · ν + V · yNkgKVT + S ( t ) e

}

2

,

(1)

f r o m w h i c h we c a n d e r i v e a n a l g e b r a i c e q u a t i o n f o r V · v . Here Ρ i s t h e t o t a l p r e s s u r e , ν i s t h e f l u i d v e l o c i t y , Τ i s t h e tem perature, γ i s the r a t i o of heat c a p a c i t i e s C /C (assumed here t o be a c o n s t a n t ) and κ i s a f u n c t i o n o f t h e m i x t u r e t h e r m a l c o n d u c tivity, λ , m p

v

£yNk è - \ _ 'm (T),

w

ο

Ν i s t h e t o t a l p a r t i c l e d e n s i t y and k g i s B o l t z m a n n ' s c o n s t a n t . The l a s t t e r m on t h e r i g h t hand s i d e o f Eq. ( 1 ) i s t h e s o u r c e t e r m . P r o p e r c h o i c e o f S ( t ) e n s u r e s t h a t a g i v e n amount o f e n e r gy, E , i s d e p o s i t e d i n a c e r t a i n v o l u m e , R^, i n a t i m e T . The c h o i c e o f t h i s G a u s s i a n p r o f i l e a l l o w s u s t o o b t a i n a " c l o s e d " form s i m i l a r i t v s o l u t i o n . I f t h e f l u i d v e l o c i t y ν i s then expan ded s u c h t h a t Q

0

v(r,t) «

V l

(t)r

and s p h e r i c a l symmetry i s assumed, E q . (1) may be s o l v e d cally to obtain


(3) anyti-


346

T(r,t) = Τ

A(t)ee

p(r,t) = ρ

e

k

2

(

t

)

r

2

(4)

A ( t ) e

and -A(t)eM

t

;

k 2 ( t ) r 2

e

,

(5)


where T^ and a r e t h e b a c k g r o u n d t e m p e r a t u r e and d e n s i t y f a r f r o m t h e h e a t s o u r c e . S o l u t i o n s f o r A ( t ) and k ( t ) may t h e n be o b t a i n e d by s o l v i n g two c o u p l e d o r d i n a r y d i f f e r e n t i a l e q u a t i o n s ^ at

= -

k

v

2

fjA . l i t l dt

3

Kk

(6)

1 _

6 K k

,

A i

γρ

I n v o k i n g e n e r g y c o n s e r v a t i o n y i e l d s an e x p r e s s i o n f o r t h e v e l o c i t y c o e f f i c i e n t ν , and i s e f f e c t i v e l y t h e f i r s t c a l i b r a t i o n i n the model: _ S _ F' (O)-F' (A) βγΡ^ F (A)

= V

l

AF' ( A ) - F ( A ) F (A)

2 +

Z

K

K

. w

where t h e f u n c t i o n F(A) i s d e f i n e d as F(Α)

Ξ Γ

Λ 4πχ [1-β" 2

"

χ 2

Α β

]dx.

(9)

The m o d e l r e q u i r e s one f u r t h e r d e f i n i t i o n i n o r d e r t o p r e dict ignition. A c u r v e o f c h e m i c a l i n d u c t i o n t i m e as a f u n c t i o n o f t e m p e r a t u r e must b e i n c l u d e d i n o r d e r t o d e f i n e t h e i n d u c t i o n parameter,

™

- 4

χ

m i t ' ) )

10

c I g n i t i o n " o c c u r s " when I ( t ) = 1 i n t h i s m o d e l , w h i c h i s a n e x a c t r e s u l t i n t h e l i m i t o f l a r g e h e a t s o u r c e and c o n s t a n t t e m p e r a t u r e near the center of the heated r e g i o n . A simple a n a l y t i c expres s i o n f o r T ( T ) d e p e n d i n g on t h r e e c o n s t a n t s h a s b e e n d e r i v e d and c a n be c a l i b r a t e d u s i n g a s few a s t h r e e d i s t i n c t v a l u e s o f T a t d i f f e r e n t temperatures. The c h e m i c a l r e a c t i o n scheme u s e d i n t h e d e t a i l e d m o d e l was used t o generate a curve f o r T ( T ) . The v a l u e s o f t h e r m a l c o n d u c t i v i t y used i n t h e d e t a i l e d model were used t o g e n e r a t e a f u n c t i o n κ. Then a s e r i e s o f c o m p a r i s o n s w e r e made, i n w h i c h t h e d e t a i l e d m o d e l was c o n f i g u r e d i n s p h e r i c a l symmetry w i t h a Gaus s i a n energy d e p o s i t i o n . We show r e s u l t s f o r s e v e r a l t e s t c a s e s . I n o n e , R = 0 . 1 cm and τ = 1 χ 1 0 " * s e c . The s i m p l e m o d e l p r e d i c t s t h a t 3 . 3 χ 1 0 ^ e r g s i s t h e minimum i g n i t i o n e n e r g y and t h e s e r e s u l t s a g r e e w e l l w i t h t h e s i m u l a t i o n ( F i g u r e s 6 and 7 ) . B o t h m o d e l s p r e d i c t c

c

c

0

1

0


Detailed

ORAN ET AL.

Modelling

of

347

Combustion

χ 1000

L

105 PT. STEPS

10010-

g

100 SEC

1

Ρ 10-1

10 SEC


10-3 _j

10-4 10-10

L

10-7

ι

Figure 5.

j FLAME ZONE

x 100

10-4 ι

CONVECTIVE SCALES } DIFFUSION SCALE

104

g 10-2

SYSTEM SIZE

-FLAME RESOLUTION 10-1

100 SEC

SOUND FLAME

The as yet undeveloped injected adaptive gridding method will reduce the computational time to 1100 sec.

TIME (sec x 10 )

TIME (sec x 10~ )

4

4

Figure 6. Calculated results from the similarity solution plotted as a function of time. Here R is the characteristic radius for energy deposition, A is the nonlinear amplitude of the temperature and density functions, Τ (R = 0) is the central tem perature, and I is the induction parameter. The indicates the predicted induction time; r = 1.0 χ W sec; E «= 4.0 Χ 10 ergs; R — 0.1 cm. c

0

4

0

4

0

American Chemical Society Library 1155 16th St N. W. Crosley; Laser Probes for Combustion Chemistry ACS Symposium Series; American Chemical Washington, DC, 1980. WttttiRflton, D. C.Society: 20036

348


i g n i t i o n a t e s s e n t i a l l y t h e same t i m e f o r a r a n g e o f i n p u t e n e r gies. I n t h e s e c o n d e x a m p l e , R = 0.025 cm and τ = 1 χ 1 0 ~ sec. The s i m p l i f i e d m o d e l p r e d i c t s a minimum i g n i t i o n e n e r g y o f ~ 8 χ 1 0 ergs. The f u l l s i m u l a t i o n d o e s n o t show i g n i t i o n , b u t p r e d i c t s t h a t some b u r n i n g does o c c u r a n d t h e f l a m e i s e v e n t u a l l y quenched ( F i g u r e 8 ) . Thus i n t h e r e g i m e f o r w h i c h b o t h m o d e l s a g r e e , we h a v e i n f a c t t e s t e d them b o t h . I n t h e r e g i m e where t h e y do n o t a g r e e , we must t h e n f i g u r e o u t what p h y s i c s i s m i s s i n g f r o m t h e s i m i l a r i t y m o d e l . When t h i s i s d o n e , we c a n , i n e f f e c t , u s e t h e d e t a i l e d m o d e l t o b u i l d a c c u r a t e phenomenology i n t o the s i m i l a r i t y s o l u t i o n . The s i m i l a r i t y s o l u t i o n h a s t e s t e d t h e f u l l d e t a i l e d m o d e l , a n d t h e f u l l d e t a i l e d m o d e l h a s shown and h e l p e d e x t e n d t h e l i m i t s o f t h e s i m i l a r i t y s o l u t i o n . 4

Q

0


2

T e s t i n g the Model w i t h Experiment A l t h o u g h comparisons between a n a l y t i c t h e o r y and model r e s u l t s can b e u s e d t o e x t e n d o u r u n d e r s t a n d i n g o f t h e c o n t r o l l i n g p r o c e s s e s i n a s y s t e m w i t h l i m i t e d p h y s i c a l c o m p l e x i t y , many s y s tems may p r e c l u d e any a n a l y t i c f o r m u l a t i o n . Then e x p e r i m e n t a l d a t a p r o v i d e t h e o n l y means o f c h e c k i n g t h e a c c u r a c y o f t h e model. B e l o w we show a n o n - r e a c t i n g c a s e i n w h i c h t h e r e s u l t s f r o m a n experiment were used t o t e s t a n u m e r i c a l model . The m o d e l r e s u l t s t h e n s u g g e s t e d new d i r e c t i o n s f o r t h e e x p e r i m e n t s . The d e t e r m i n a t i o n o f t h e e f f e c t s o f s u r f a c e waves o n s u b merged s t r u c t u r e s h a s many p r a c t i c a l a p p l i c a t i o n s , p a r t i c u l a r l y i n a n o c e a n e n v i r o n m e n t . Due t o t h e c o m p l e x i t y o f t h e p r o b l e m , a n a l y t i c r e s u l t s a r e l i m i t e d t o i d e a l i z e d f l o w s and g e o m e t r i e s . A major p a r t o f the complexity a r i s e s from the e x i s t e n c e o f t h e free surface i t s e l f . N o t o n l y does t h e f r e e s u r f a c e d o m i n a t e t h e f l o w , b u t i t may become m u l t i p l y c o n n e c t e d when s p r a y s , wavebreaking o r c a v i t a t i o n occur. There i s u s u a l l y no steady s t a t e , and a m o d e l f o r t r a n s i e n t f l o w o f t h e f l u i d o v e r t h e o b s t a c l e must be u s e d . H e r e we d e s c r i b e t h e r e s u l t s o f a m o d e l c a l c u l a t i o n o f t h e w a v e - i n d u c e d p r e s s u r e f o r c e s o n a submerged h a l f - c y l i n d e r , a n d compare t h e r e s u l t s w i t h e x p e r i m e n t a l d a t a . The i m p l i c a t i o n s o f the comparisons f o r b o t h the v a l i d i t y o f the model and the e x p e r i m e n t a l procedure w i l l be examined. F i n a l l y , the a p p l i c a t i o n o f the model t o o t h e r f l u i d f l o w s and t o combustion problems w i l l be discussed. F i g u r e 10 i l l u s t r a t e s t h e i n i t i a l c o n d i t i o n s f o r t h e n u m e r i c a l m o d e l . A h a l f - c y l i n d e r o f r a d i u s " a " i s submerged i n a f l u i d whose u n d i s t u r b e d f r e e s u r f a c e s t a n d s a t a h e i g h t h = 2 a o v e r t h e b o t t o m s u r f a c e . A p r o g r e s s i v e wave w i t h w a v e l e n g t h λ = 5 a i s i n c i d e n t on the c y l i n d e r from the l e f t . The s i d e s o f t h e compu t a t i o n a l region are p e r i o d i c ; t h a t i s , the p h y s i c a l system being s i m u l a t e d i s a c t u a l l y t h a t o f p r o g r e s s i v e waves o v e r a s e r i e s o f half-cylinders. P e r i o d i c boundary c o n d i t i o n s were chosen t o a v o i d n u m e r i c a l damping a n d r e f l e c t i o n a t a n o u t f l o w b o u n d a r y .


27.

ORAN ET AL. 2,000

*

Detailed

I I

I


of

s

^*\4* ° Ί

/

349

Combustion

I I II

I

1,500 h

oc D Ι α

Modelling

4

I ill

I e r

9

s

J 3.3*10 ergs 4

1,000

UJ Q.

/ 500

10"

I I

^ ^ \ ^ 3 x 10 ergs 4

I

I I II

I 10"

I III

I

10-

io-

TIME (sec) Figure 7. Comparisons of the similarity ( ) and detailed model ( ) solu tions for the central temperature as a function of time. The marks the induction time predicted by the similarity solution; τ = 1.0 X 10 sec; R = 0.1 cm. 4

0

0

3,000

.06

.08

POSITION (cm) Figure 8. Temperature as a function of position at four times in the detailed flame model. The flame dies out even though the similarity solution indicates it should not, R = 0.025 cm; E = 4 χ 10 ergs; T — 1 X 10* sec; (%), 7.127 X 10 sec;(X), 1.113 X W ; (O), 2.256 χ 10 ;(Α), 7.123 X 10\ 0

2

0

4

s

0

4




350

Figure 9. The triangular grid early in a calculation of wave flow over a halfcylinder. The first wave of a wave train has passed over the cylinder from the left.

0 0I

I

-80

'

I

-60

«

I

-40

ι

L_J

-20

I

0

ι

I

20

,

I

40

ι

I

60

ι

l_l

80

A Z I M U T H Θ, (deg) Figure 10. A comparison of numerical results with experiment and with linear theory for the magnitude of pressure variations over a half-cylinder owing to a progressive wave with ka = 2.5 and kh = 5: ( ), linear theory (3); (X), SPLISH results (NRL); (θ, ·, Δ , Α λ experiments (NRL).



27.

ORAN ET

AL.

Detailed Modelling

of

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The c a l c u l a t i o n s e e k s t o f i n d t h e p r e s s u r e s a t e v e r y p o i n t i n t h e f l u i d as a f u n c t i o n o f t i m e , and i n p a r t i c u l a r t h e p r e s s u r e s and p r e s s u r e g r a d i e n t f o r c e s a t p o i n t s on t h e c y l i n d e r s u r f a c e . The n u m e r i c a l m o d e l i s b a s e d on 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 s f o r s o l v i n g the equations f o r i n v i s c i d , i n c o m p r e s s i b l e f l u i d f l o w u s i n g a t r i a n g u l a r g r i d which extends throughout the i n t e r i o r of t h e f l u i d ( 9 ) . The f r e e s u r f a c e and r i g i d b o u n d a r y s h a p e s a r e a p p r o x i m a t e d by s t r a i g h t l i n e s w h i c h e x t e n d b e t w e e n p o i n t s on t h o s e s u r f a c e s and w h i c h d e f i n e t h e edges o f t h e c o m p u t a t i o n a l grid. The g o v e r n i n g e q u a t i o n s a r e c a s t i n a L a g r a n g i a n f o r m a l i s m so t h a t p o i n t s o r i g i n a l l y l y i n g on a s u r f a c e w i l l r e m a i n t h e r e a t a l l times d u r i n g the c a l c u l a t i o n . P o i n t s i n t e r i o r to the f l u i d w i l l f o l l o w L a g r a n g i a n p a t h l i n e s as i f t h e y w e r e e x p e r i m e n t a l marker p a r t i c l e s i n a r e a l f l u i d . The e q u a t i o n s a r e d i f f e r e n c e d such t h a t v o r t i c i t y i s conserved i d e n t i c a l l y at a l l times. Ver tex p r e s s u r e s a r e chosen t o keep the l o c a l f l u i d volumes d i v e r gence-free. T h e s e new p r e s s u r e s a r e i n t u r n u s e d t o a d v a n c e t h e v e l o c i t i e s and u p d a t e t h e g r i d p o s i t i o n s . The p h y s i c a l b e h a v i o r o f t h e g o v e r n i n g e q u a t i o n s can be p r e s e r v e d i n t h e a p p r o x i m a t e d i f f e r e n c e e q u a t i o n s b e i n g s o l v e d numer i c a l l y by u s i n g a t r i a n g u l a r g r i d . A Lagrangian g r i d w i l l d i s t o r t i n any n o n - t r i v i a l f l o w f i e l d , and as g r i d d i s t o r t i o n becomes severe the c a l c u l a t i o n q u i c k l y l o s e s accuracy. However, a t r i a n g u l a r g r i d c a n be m a n i p u l a t e d l o c a l l y i n s e v e r a l ways t o e x t e n d r e a l i s t i c c a l c u l a t i o n s of t r a n s i e n t f l o w s (9,10). Each g r i d l i n e r e p r e s e n t s a q u a d r i l a t e r a l d i a g o n a l , and t h e o p p o s i t e d i a g o n a l can be c h o s e n whenever v e r t i c e s move i n t h e f l o w t o p o s i t i o n s w h i c h favor that connection. Such a r e c o n n e c t i o n i n v o l v e s j u s t t h e f o u r v e r t i c e s d e s c r i b i n g the q u a d r i l a t e r a l . No f l u i d moves r e l a t i v e t o t h e q u a d r i l a t e r a l , e l i m i n a t i n g one f o r m o f n u m e r i c a l d i s s i p a t i o n . V e r t i c e s may a l s o be added o r d e l e t e d t o p r e s e r v e t h e d e s i r e d r e s o l u t i o n by l o c a l a l g o r i t h m s w h i c h i n v o l v e o n l y t h o s e v e r t i c e s i n t h e v i c i n i t y o f t h e g r i d anomaly. M a j o r a d v a n t a g e s o f t h i s t e c h n i q u e a r e t h a t t h e a l g o r i t h m s c a n be c o n s e r v a t i v e , t h e y p e r m i t a minimum o f n u m e r i c a l d i s s i p a t i o n and y e t t h e y r e q u i r e v e r y l i t t l e computer t i m e s i n c e most o f t h e g r i d r e m a i n s u n a l t e r e d . D a t a f o r t h e e x p e r i m e n t a l c o m p a r i s o n was o b t a i n e d t h r o u g h w a v e - t a n k e x p e r i m e n t s p e r f o r m e d w i t h a bottom-mounted h a l f - c y l i n d e r so t h a t p r e s s u r e measurements c o u l d be compared d i r e c t l y t o t h e n u m e r i c a l r e s u l t s ( 1 1 ) . The o b s t a c l e was p l a c e d o n e - t h i r d o f t h e t a n k l e n g t h f r o m a m e c h a n i c a l wavemaker and a t t h e o t h e r end of t h e c h a n n e l a s l o p i n g p o r o u s b e a c h a b s o r b e d 95% o r more o f t h e i n c i d e n t wave e n e r g y . R e s u l t s o f t h e e x p e r i m e n t and o f t h e n u m e r i c a l s i m u l a t i o n a r e shown i n F i g u r e 10, t o g e t h e r w i t h t h e r e s u l t s o f l i n e a r t h e o r y . The m a g n i t u d e o f t h e p r e s s u r e f l u c t u a t i o n s as m e a s u r e d by t h e e x p e r i m e n t a t d i f f e r e n t p o i n t s on t h e c y l i n d e r (Θ = 0° a t t h e t o p of t h e c y l i n d e r ) a r e compared w i t h t h e p r e d i c t i o n s o f t h e m o d e l . As shown i n F i g u r e 10, t h e c o m p a r i s o n i s q u i t e good. F i g u r e 11 compares t h e c a l c u l a t e d and m e a s u r e d i n s t a n t a n e o u s p r e s s u r e



352


d i s t r i b u t i o n around the c y l i n d e r f o r the s i t u a t i o n i n which t h e c r e s t o f t h e p r o g r e s s i v e wave i s n e a r t h e l e f t s i d e o f t h e h a l f cylinder. A g a i n t h e c o m p a r i s o n s l o o k g o o d , b u t now some d i f f e r e n c e s become e v i d e n t . I t was f o u n d t h a t t o w i t h i n e x p e r i m e n t a l e r r o r , a l l o f t h e o b s e r v e d d i s c r e p a n c i e s c o u l d be e x p l a i n e d b y two f a c t o r s . The f i r s t f a c t o r i s t h a t the model d i d not e x a c t l y d e s c r i b e t h e p h y s i c a l s i t u a t i o n i n the experiment: t h e wave t a n k had a s i n g l e c y l i n d e r , whereas the c a l c u l a t i o n i s f o r a s e r i e s o f c y l i n d e r s . The s e c o n d f a c t o r was t h e s u r p r i s i n g r e s u l t t h a t t h e r o u g h l y 5% r e f l e c t e d wave f r o m t h e wave t a n k s i g n i f i c a n t l y a f f e c t e d t h e e x p e r i m e n t a l r e s u l t s due t o m o d i f i c a t i o n s i n t h e d y n a m i c p r e s s u r e fluctuations. I n t h i s instance a d e t a i l e d examination of t h e model and e x p e r i m e n t a l r e s u l t s has i n d i c a t e d t h a t an e x p e r i m e n t a l e f f e c t thought t o be s m a l l c o u l d i n f a c t cause n o t i c e a b l e d e v i a t i o n s i n the d a t a measured. The a p p l i c a t i o n o f t h i s n u m e r i c a l t e c h n i q u e t o r e a c t i v e f l o w i s r e l a t i v e l y s t r a i g h t f o r w a r d . A l t h o u g h the example p r e s e n t e d above i s f o r homogeneous f l o w s , t h e e x t e n s i o n t o i n c l u d e i n t e r f a c e s i n v o l v e s no b a s i c c h a n g e s t o t h e u n d e r l y i n g g r i d d i n g scheme, b u t o n l y t h e p r o v i s i o n t h a t no i n t e r f a c e s i d e s a r e a l l o w e d t o be reconnected. Instead l o c a l g r i d d i n g anomalies a t the i n t e r f a c e s a r e r e s o l v e d by a d d i n g o r s u b t r a c t i n g v e r t i c e s a t t h e i n t e r f a c e and r e c o n n e c t i n g g r i d l i n e s l e a d i n g t o t h o s e v e r t i c e s . That t h i s s o l u t i o n i s v i a b l e i s most e a s i l y shown b y F i g u r e 12 w h i c h shows stages i n the c o l l a p s e o f a Rayleigh-Taylor unstable f l u i d l a y e r c a l c u l a t e d w i t h t h e same m o d e l . H e r e t h e c a l c u l a t i o n c a n c o n t i n u e even though the o r i g i n a l l y simply-connected l i g h t e r f l u i d performs a t r a n s i t i o n t o a m u l t i p l y connected f l u i d which i n cludes "bubbles w h i c h have been e n t r a i n e d by the h e a v i e r f l u i d . Of c o u r s e , f o r r e a c t i v e f l o w c a l c u l a t i o n s a new m o d e l w o u l d h a v e to be c o n s t r u c t e d based on these t e c h n i q u e s w h i c h used i n s t e a d the e q u a t i o n s g o v e r n i n g c o m p r e s s i b l e f l u i d s and w h i c h c o n t a i n e d t h e added c h e m i c a l r e a c t i o n s a n d d i f f u s i v e t r a n s p o r t e f f e c t s . 11

Conclusion D e t a i l e d m o d e l l i n g o f l a m i n a r r e a c t i v e f l o w s , even i n f a i r l y c o m p l i c a t e d g e o m e t r i e s , i s c e r t a i n l y w e l l w i t h i n o u r c u r r e n t capabilities. I n t h i s p a p e r we h a v e shown s e v e r a l ways i n w h i c h t h e s e t e c h n i q u e s may be u s e d . A s t h e p h y s i c a l c o m p l e x i t y we w i s h t o m o d e l i n c r e a s e s , o u r f o o t i n g becomes l e s s s u r e and more phenomenology must be a d d e d . F o r e x a m p l e , we m i g h t h a v e t o a d d e v a p o r a t i o n laws a t l i q u i d - g a s i n t e r f a c e s o r l e s s well-known chemic a l r e a c t i o n r a t e s i n complex hydrocarbon f u e l s . P e r h a p s t h e b i g g e s t p r o b l e m f a c i n g c o m b u s t i o n m o d e l l i n g now i s turbulence: t h e r e a r e no e x c e l l e n t o r e v e n good methods o f i n c l u d i n g such e f f e c t s i n our c a l c u l a t i o n s . A t b e s t we h a v e a number o f p h e n o m e n o l o g i c a l m o d e l s w i t h l i m i t e d r a n g e s o f v a l i d i t y and w h i c h i m p l y a s t e a d y s t a t e .



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Modelling

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-1

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0

+1

ΔΡ/ΔΡΜΑΧ

ΔΡ/ΔΡΜΑΧ

Figure 11. A comparison of numerical results with theory and experiment for instantaneous pressure distributions on a half-cylinder (ka = 2.5 and kh = 5) with the wave crest just to the left of the cylinder: ( ), linear theory (3); (X), SPLISH results; (%), experiments (NRL)

1.5

PATIIKINKS

1.5

P A T H LINKS

' · ,"\* » % , · ' * 1

%

%

V ····'

Λ

P A T H I. INKS

Figure 12. Lagrangian path lines at various stages of a Rayleigh-Taylor collapse for the case of two inviscid, incompressible fluids having a density ratio of 2:1. A free surface is present above the dense fluid and the interface between the fluids is indicated for each stage. The simulation shows how later evolution of thefluidflow is dominated by the strength and dynamics of the vortex pair created during the early stages of collapse.


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We b e l i e v e t h a t d e v i s i n g a way t o h a n d l e t h i s d i f f i c u l t p r o b l e m of s t r o n g l y c o u p l e d m u l t i p l e time and space s c a l e s i s t h e c h a l l e n g e we c u r r e n t l y f a c e . Acknowledgment s T h i s work has been funded by t h e N a v a l R e s e a r c h L a b o r a t o r y through the O f f i c e of Naval Research.


Abstract A major goal of detailed numerical modelling of combustion is to develop computational tools which can be used interactively with laboratory diagnostic techniques to interpret and understand experiments. Only through this quantitative interaction of prediction and measurement can we calibrate and extend our understanding of the fundamental physical and chemical processes involved. Modelling of flames and detonations is particularly challenging because of the strong feedback between the energy released due to chemical reactions and the evolution of the hydrodynamic flow. Since it is generally not possible to decouple these processes completely, special numerical techniques are required to solve the problems. In this paper, the basic steps and decisions which must be made when constructing and using a numerical model will be discussed. These involve 1) choosing the numerical algorithms based on the physics and chemistry of the problem, 2) testing the model based on these algorithms in different limits against theoretical and analytical results, and 3) using the model interactively with experiments. With the third step specifically in mind, it will be shown how the species concentration, velocity, and temperature profiles obtained from laser probe diagnostics can provide information for use in and comparison with detailed calculations. Literature Cited 1.

Oran, E. S and Boris, J. P., Detailed Modelling of Combustion Processes, to appear in Prog. in Energy and Comb. Sci, 1980.

2.

Boris, J. P. and Oran, E. S., Modelling Turbulence: Physics or Curve-Fitting, to appear in the Proc. of the International Symposium on Gas Dynamics of Explosions and Reactive Systems, Gottingen, 1979, AIAA; also NRL Memorandum Report, Naval Research Laboratory, Washington, D. C. 20375, 1979.

3.

Gear, C. W.,"Numerical Initial Value Problems in Ordinary Differential Equations," Prentice-Hall, Englewood Cliffs,N.J., 1971.



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4.

Young, T. R., and Boris, J. P., A Numerical Technique for Solving Stiff Ordinary Differential Equations Associated with the Chemical Kinetics of Reactive Flow Problems, J. Phys. Chem., 1977, 81, 2424.

5.

Young, CHEMEQ - A Subroutine for Solving Stiff Ordinary Differential Equations, NRL Memorandum Report 4091, Naval Research Laboratory, Washington, D. C. 20375, 1979.

6.

Jones, W. W. and Boris, J. P., FLAME--A Slow-Flow Combustion Model, NRL Memorandum Report 3970, Naval Research Laboratory, Washington, D. C. 20375, 1979.

7.

Boris, J. P. and Oran, E. S., Detailed Modelling of Reactive Flows, Proc. of the GAMNI International Conference on Numeri cal Methods for Engineering, Paris, France, 1978.

8.

Oran, E. S. and Boris, J. P., Theoretical and Computational Approach to Modelling Flame Ignition, Proc. of the Inter national Symposium on Gas Dynamics of Explosions and Reactive Systems, Gottingen, 1979; also NRL Memorandum Report, Naval Research Laboratory, Washington, D. C. 20375, 1979.

9.

Fritts, M. J. and Boris, J. P., The Lagrangian Solution of Transient Problems in Hydrodynamics Using a Triangular Mesh, J. of Comp. Phys., 1979, 31, 173.

10. Crowley, Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics, Springer-Verlag, New York/Berlin, 1971. 11. Miner, E. W., Griffin, O. M., Ramberg, S. Ε., and Fritts,M.J., Numerical Calculation of Wave Effects on Structures, Pro ceedings of the Conference on Civil Engineering in the Oceans, IV, San Francisco, CA, 10-12 September 1979. 12. Garvin, David, personal communication. RECEIVED

February 1, 1980.


Detailed Modelling of Combustion: A ... - American Chemical Society

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