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Natural convection in two-dimensional irregular cavities Fournier, Martin 1986

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NATURAL CONVECTION IN TWO-DIMENSIONAL IRREGULAR CAVITIES by MARTIN FOURNIER A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Chemical E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September 1986 © MARTIN FOURNIER, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l - f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the The U n i v e r s i t y of B r i t i s h C o lumbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n permi s s i o n . Department of C h e m i c a l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: September 1986 ABSTRACT N a t u r a l c o n v e c t i o n i n t w o - d i m e n s i o n a l i r r e g u l a r c a v i t i e s was s i m u l a t e d by n u m e r i c a l l y s o l v i n g the s t e a d y - s t a t e c o n s e r v a t i o n e q u a t i o n s w r i t t e n i n terms of stream f u n c t i o n , v o r t i c i t y and te m p e r a t u r e dependent v a r i a b l e s and f o r a g e n e r a l o r t h o g o n a l c o o r d i n a t e system. I t was assumed t h a t the B o u s s i n e s q a p p r o x i m a t i o n s were v a l i d , t h a t the f l u i d was Newtonian and t h a t the p r o p e r t i e s o t h e r than d e n s i t y were c o n s t a n t . The use of o r t h o g o n a l c o o r d i n a t e s and the above s e t of dependent v a r i a b l e s was found t o have s e v e r a l advantages over the use of C a r t e s i a n or n o n - o r t h o g o n a l systems and the s e t of p r i m i t i v e dependent v a r i a b l e s ( v e l o c i t i e s , p r e s s u r e and t e m p e r a t u r e ) . The b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e system was n u m e r i c a l l y g e n e r a t e d by means of the weak c o n s t r a i n t method of R y s k i n and L e a l [ 2 6 ] , S p e c i a l forms of the Wood and second-order v o r t i c i t y boundary c o n d i t i o n s were d e r i v e d f o r a g e n e r a l t w o - d i m e n s i o n a l b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e system. 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 were used t o s o l v e the r e s u l t i n g s e t of d i f f e r e n t i a l e q u a t i o n s . The e f f e c t s of the mapping c h a r a c t e r i s t i c s , the v o r t i c i t y boundary c o n d i t i o n s and the f i n i t e d i f f e r e n c e g r i d s i z e on the a c c u r a c y of the n a t u r a l c o n v e c t i o n s o l u t i o n were i n v e s t i g a t e d f i r s t . For t h e c a v i t y g e o m e t r i e s s t u d i e d , , i t was ob s e r v e d t h a t , except f o r g r i d boundary c o n d i t i o n s which l e d t o u n d e s i r a b l e g r i d s , most c o m b i n a t i o n s of g r i d and v o r t i c i t y boundary c o n d i t i o n s gave r e s u l t s of a c c e p t a b l e i i a c c u r a c y ( r e l a t i v e e r r o r l e s s than one p e r c e n t ) as l o n g as a s u f f i c i e n t l y f i n e g r i d s i z e (28x28 or f i n e r ) was employed. The e f f e c t s of the c a v i t y geometry and the R a y l e i g h number on n a t u r a l c o n v e c t i o n were i n v e s t i g a t e d i n P a r t I I . I t was found t h a t i n c r e a s i n g the R a y l e i g h number always a c t e d t o enhance both the n a t u r a l c o n v e c t i o n c i r c u l a t i o n and the heat t r a n s f e r r a t e , a r e s u l t which was e a s i l y e x p l a i n e d by examining the source term of the momentum e q u a t i o n . The e f f e c t of the c a v i t y geometry was more complex but t h e s e r e s u l t s c o u l d a l s o be i n t e r p r e t e d by e x a m i n i n g the i n f l u e n c e of the c a v i t y shape i n impeding or enhancing f l u i d c i r c u l a t i o n and the opposing e f f e c t s of the d i s t a n c e between i s o t h e r m a l w a l l s on c o n d u c t i v e and c o n v e c t i v e heat t r a n s f e r . The p o s s i b i l i t y of u s i n g a s i m i l a r n u m e r i c a l p r o c e d u r e t o s i m u l a t e a m e l t i n g or a f r e e z i n g p r o c e s s was i n v e s t i g a t e d i n P a r t I I I . N u m e r i c a l p r e d i c t i o n s of the c i r c u l a t i n g f l o w i n the l i q u i d phase of an i c e f o r m i n g p r o c e s s were o b t a i n e d by d i g i t i z i n g the p h o t o g r a p h i c image of a r e a l i c e i n t e r f a c e and u s i n g the t r u e n o n - l i n e a r r e l a t i o n s h i p between d e n s i t y and t emperature f o r water a t low t e m p e r a t u r e . The n u m e r i c a l r e s u l t s were i n r e a s o n a b l e agreement w i t h the f l o w v i s u a l i z a t i o n e x p e r i m e n t s c a r r i e d out by E c k e r t [ 4 2 ] . T a b l e of C o n t e n t s ABSTRACT . i i LIST OF TABLES . v i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS . x i I . INTRODUCTION 1 I I . LITERATURE REVIEW 11 A. N a t u r a l c o n v e c t i o n near a v e r t i c a l i s o t h e r m a l f l a t p l a t e 11 B. N a t u r a l c o n v e c t i o n i n r e c t a n g u l a r e n c l o s u r e s ..12 C. Phase change problems 21 D. N a t u r a l c o n v e c t i o n i n n o n r e c t a n g u l a r e n c l o s u r e s 22 E. B o d y - f i t t e d o r t h o g o n a l t r a n s f o r m a t i o n s 23 I I I . COORDINATE SYSTEM 25 A. B o d y - f i t t e d o r t h o g o n a l mapping u s i n g the weak c o n s t r a i n t method 27 B. Boundary c o n d i t i o n s of the b o d y - f i t t e d o r t h o g o n a l mapping 32 IV. CONSERVATION EQUATIONS 35 A. Assumptions 36 B. C o n s e r v a t i o n e q u a t i o n s i n C a r t e s i a n system ....37 C. C o n s e r v a t i o n e q u a t i o n s i n a g e n e r a l o r t h o g o n a l system 41 D. Boundary c o n d i t i o n s of the c o n s e r v a t i o n e q u a t i o n s 45 V. FINITE DIFFERENCE DISCRETIZATION 49 A. D i s c r e t i z a t i o n of the g r i d g e n e r a t i o n e q u a t i o n s 51 B. D i s c r e t i z a t i o n of the g r i d boundary c o n d i t i o n s 56 C. D i s c r e t i z a t i o n of the n a t u r a l c o n v e c t i o n e q u a t i o n s 57 i v 1. Stream f u n c t i o n e q u a t i o n 60 2. V o r t i c i t y e q u a t i o n 62 3. Temperature e q u a t i o n 68 D. D i s c r e t i z a t i o n of the n a t u r a l c o n v e c t i o n boundary c o n d i t i o n s 68 V I . COMPUTER PROGRAM 75 V I I . TEST PROBLEMS 85 A. F i r s t t e s t 85 B. Second t e s t 88 V I I I . PART I 96 A. N u m e r i c a l e x p e r i m e n t s 96 B. R e s u l t s 100 C. D i s c u s s i o n 118 IX. PART I I 124 A. N u m e r i c a l e x p e r i m e n t s 124 B. R e s u l t s 126 C. D i s c u s s i o n 1 32 D. E m p i r i c a l c o r r e l a t i o n s 143 X. PART I I I 146 A. N u m e r i c a l e x p e r i m e n t s .146 B. D i s c u s s i o n 156 X I . CONCLUSIONS 163 X I I . RECOMMENDATIONS 1 66 NOMENCLATURE 167 REFERENCES 170 APPENDIX A 175 APPENDIX B 216 v LIST OF TABLES T a b l e / Page 1 Independent v a r i a b l e s c o n s i d e r e d f o r the c a v i t y t y p e s C1 and C2 9 2 Flow and heat t r a n s f e r o b s e r v a t i o n s f o r t h e v e r t i c a l square c a v i t y f i l l e d w i t h a i r 14 3 Program a l g o r i t h m 76 4 I n i t i a l c orrespondence between C a r t e s i a n and b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e s a t the domain b o u n d a r i e s 78 5 Comparison of the maximum d e v i a t i o n s ( i n degrees) o b t a i n e d i n the p r e s e n t work and by C h i k h l i w a l a and Y o r t s o s [27] - case 1 87 6 Comparison of the maximum d e v i a t i o n s ( i n d egrees) o b t a i n e d i n the p r e s e n t work and by C h i k h l i w a l a and Y o r t s o s [27] - case 2 87 7 Comparison of the maximum stream f u n c t i o n s o b t a i n e d i n t he p r e s e n t work w i t h the benchmark r e s u l t s of De V a h l D a v i s and Jones [15] 93 8 Comparison of the average N u s s e l t numbers o b t a i n e d i n t he p r e s e n t work w i t h the benchmark r e s u l t s of De V a h l D a v i s and Jones [15] 93 9 D i f f e r e n t g r i d , stream f u n c t i o n , v o r t i c i t y and tem p e r a t u r e boundary c o n d i t i o n s i n v e s t i g a t e d i n P a r t I 97 10 G r i d boundary c o n d i t i o n s i n v e s t i g a t e d i n P a r t I ; D = D i r i c h l e t , N=Neumann 97 11 G r i d s i z e s i n v e s t i g a t e d i n P a r t I 97 12 D i m e n s i o n l e s s a m p l i t u d e s and R a y l e i g h numbers i n v e s t i g a t e d i n P a r t I 99 13 E r r o r s i n m o n i t o r e d v a r i a b l e s f o r ca s e s 1 t o 5 u s i n g a 33x33 g r i d 117 14 G r i d and v o r t i c i t y boundary c o n d i t i o n s used i n P a r t I I 125 15 Average L e f t w a l l N u s s e l t number as a f u n c t i o n of a m p l i t u d e f o r Ra = 0 - c a v i t y C1 131 16 Average L e f t w a l l N u s s e l t number as a f u n c t i o n of a m p l i t u d e f o r Ra = 0 - c a v i t y C2 131 v i 17 P e r c e n t change of the maximum stream f u n c t i o n w i t h i n c r e a s i n g a m p l i t u d e f o r a g i v e n R a y l e i g h number -c a v i t y C1 134 18 P e r c e n t change of the average l e f t w a l l N u s s e l t number w i t h i n c r e a s i n g a m p l i t u d e f o r a g i v e n R a y l e i g h number - c a v i t y C1 135 19 P e r c e n t change of the maximum stream f u n c t i o n w i t h i n c r e a s i n g a m p l i t u d e f o r a g i v e n R a y l e i g h number -c a v i t y C2 136 20 P e r c e n t change of the average l e f t w a l l N u s s e l t number w i t h i n c r e a s i n g a m p l i t u d e f o r a g i v e n R a y l e i g h number - c a v i t y C2 137 21 Curve f i t t i n g c o e f f i c i e n t s of the s i m p l e power law model f o r c a v i t y C1 w i t h d i f f e r e n t a m p l i t u d e s 145 22 Curve f i t t i n g c o e f f i c i e n t s of the s i m p l e power law model f o r c a v i t y C2 w i t h d i f f e r e n t a m p l i t u d e s 145 23 E x p e r i m e n t a l and n u m e r i c a l c o n d i t i o n s used f o r low temperature water n a t u r a l c o n v e c t i o n t r i a l s 150 v i i LIST OF FIGURES F i g u r e Page 1 Two-dimensional r e c t a n g u l a r c a v i t y 3 2 B o d y - f i t t e d o r t h o g o n a l g r i d s g e n e r a t e d by the weak c o n s t r a i n t method of R y s k i n and L e a l [26] f o r s e v e r a l d i f f e r e n t i r r e g u l a r c a v i t i e s 5 3 Three d i f f e r e n t b o d y - f i t t e d o r t h o g o n a l g r i d s g e n e r a t e d by the weak c o n s t r a i n t method of R y s k i n and L e a l [26] f o r a s i n g l e i r r e g u l a r c a v i t y 6 4 G e n e r a l shape of c a v i t y t y p e s C1 (a) and C2 (b) 7 5 T y p i c a l f l o w p a t t e r n s o b s e r v e d i n a v e r t i c a l square c a v i t y , (a) Ra=l0000, u n i c e l l u l a r f l o w , (b) Ra=l00000, " c a t ' s - e y e " p a t t e r n 15 6 D i f f e r e n t c o o r d i n a t e systems which can be used t o map an i r r e g u l a r c a v i t y , (a) C a r t e s i a n , (b) b o d y - f i t t e d n o n o r t h o g o n a l , (c) b o d y - f i t t e d o r t h o g o n a l . 2 6 7 P h y s i c a l domain (a) and t r a n s f o r m e d domain (b) 28 8 U n i f o r m g r i d i n the t r a n s f o r m e d domain used t o d i s c r e t i z e the d i f f e r e n t i a l e q u a t i o n s of g o v e r n i n g the b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e t r a n s f o r m a t i o n . . 5 2 9 G e n e r a l c o n t r o l volume f o r the C a r t e s i a n c o o r d i n a t e d i s c r e t e v a l u e s 54 10 S t a g g e r e d g r i d system used t o d i s c r e t i z e the n a t u r a l c o n v e c t i o n c o n s e r v a t i o n e q u a t i o n s 58 11 G e n e r a l c o n t r o l volume f o r d i s c r e t e stream f u n c t i o n v a l u e s 61 12 G e n e r a l c o n t r o l volume f o r e i t h e r v o r t i c i t y or te m p e r a t u r e d i s c r e t e v a l u e s 63 13 S t a g g e r e d g r i d used t o d e r i v e the d i f f e r e n t e x p r e s s i o n s f o r the v o r t i c i t y boundary c o n d i t i o n 70 14 S e l e c t e d n a t u r a l c o n v e c t i o n r e s u l t s f o r a v e r t i c a l s quare c a v i t y w i t h Ra=l000 and Pr=0.71. (a) Stream f u n c t i o n countour p l o t , (b) Temperature c o u n t o u r p l o t ( i s o t h e r m s range from 0 t o 1 i n i n c r e m e n t s of 0.1), (c) L e f t w a l l N u s s e l t number d i s t r i b u t i o n , (d) R i g h t w a l l N u s s e l t number d i s t r i b u t i o n 90 v i i i 15 S e l e c t e d n a t u r a l c o n v e c t i o n r e s u l t s f o r a v e r t i c a l square c a v i t y w i t h Ra=l0000 and P r = 0 . 7 l . (a) Stream f u n c t i o n countour p l o t , (b) Temperature c o u n t o u r p l o t ( i s o t h e r m s range from 0 t o 1 i n i n c r e m e n t s of 0.1), (c) L e f t w a l l N u s s e l t number d i s t r i b u t i o n , (d) R i g h t w a l l N u s s e l t number d i s t r i b u t i o n 91 16 S e l e c t e d n a t u r a l c o n v e c t i o n r e s u l t s f o r a v e r t i c a l square c a v i t y w i t h Ra=l00000 and P r = 0 . 7 l . (a) Stream f u n c t i o n countour p l o t , (b) Temperature c o u n t o u r p l o t ( i s o t h e r m s range from 0 t o 1 i n i n c r e m e n t s of 0.1), (c) L e f t w a l l N u s s e l t number d i s t r i b u t i o n , (d) R i g h t w a l l N u s s e l t number d i s t r i b u t i o n 92 17 S e l e c t e d r e s u l t s o b t a i n e d f o r case 1 of T a b l e 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n c o u n t o u r s , (e) Temperature c o u n t o u r s , ( f ) and (g) L e f t and r i g h t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y 101 18 S e l e c t e d r e s u l t s o b t a i n e d f o r case 2 of T a b l e 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n c o u n t o u r s , (e) Temperature c o u n t o u r s , ( f ) and (g) L e f t and r i g h t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y 103 19 S e l e c t e d r e s u l t s o b t a i n e d f o r case 3 of T a b l e 12. (a) and (b) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A and B, r e s p e c t i v e l y ( g r i d boundary c o n d i t i o n s C d i d not y i e l d a converged r e s u l t ) , (d) Stream f u n c t i o n c o u n t o u r s , (e) Temperature c o u n t o u r s , ( f ) and (g) L e f t and r i g h t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y 105 20 S e l e c t e d r e s u l t s o b t a i n e d f o r case 4 of T a b l e 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n c o u n t o u r s , (e) Temperature c o u n t o u r s , ( f ) and (g) L e f t and r i g h t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y 107 21 S e l e c t e d r e s u l t s o b t a i n e d f o r case 5 of T a b l e 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n c o u n t o u r s , (e) Temperature c o u n t o u r s , ( f ) and (g) L e f t and r i g h t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y 109 22 P l o t s of the m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r case 1 112 23 P l o t s of the m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r case 2 113 i x 24 P l o t s of the m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r case 3 114 25 P l o t s of the m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r case 4 115 26 P l o t s of the m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c e t e p o i n t s f o r case 5 116 27 P l o t of the maximum stream f u n c t i o n v e r s u s the R a y l e i g h number f o r d i f f e r e n t a m p l i t u d e s - c a v i t y C1.127 28 P l o t of the average l e f t w a l l N u s s e l t number v e r s u s the R a y l e i g h number f o r d i f f e r e n t a m p l i t u d e s -c a v i t y C1 1 28 29 P l o t of the maximum stream f u n c t i o n v e r s u s the R a y l e i g h number and d i f f e r e n t a m p l i t u d e s - c a v i t y C2.129 30 P l o t of the average l e f t w a l l N u s s e l t number v e r s u s the R a y l e i g h number and d i f f e r e n t a m p l i t u d e s -c a v i t y C2 130 31 Low temperature water n a t u r a l c o n v e c t i o n r e s u l t s f o r Tj1=2.3°C. (a) G r i d , (b) Temperature c o u n t o u r s (0°C -2.3°C, 0.23°C i n c r e m e n t s ) , (c) Stream f u n c t i o n c o u n t o u r s , (d) E x p e r i m e n t a l s t r e a k - l i n e s [ 4 2 ] , (e) L e f t w a l l N u s s e l t numbers, ( f ) R i g h t w a l l N u s s e l t numbers 152 32 Low temperature water n a t u r a l c o n v e c t i o n r e s u l t s f o r Th=5.6°C. (a) G r i d , (b) Temperature c o u n t o u r s (0°C -5.6°C, 0.56°C i n c r e m e n t s ) , (c) Stream f u n c t i o n c o u n t o u r s , (d) E x p e r i m e n t a l s t r e a k - l i n e s [ 4 2 ] , (e) L e f t w a l l N u s s e l t numbers, ( f ) R i g h t w a l l N u s s e l t numbers 153 33 Low temperature water n a t u r a l c o n v e c t i o n r e s u l t s f o r : Th=8.6°C. (a) G r i d , (b) Temperature c o u n t o u r s (0°C -8.6°C, 0.86°C i n c r e m e n t s ) , (c) Stream f u n c t i o n c o u n t o u r s , (d) E x p e r i m e n t a l s t r e a k - l i n e s [ 4 2 ] , (e) L e f t w a l l N u s s e l t numbers, ( f ) R i g h t w a l l N u s s e l t numbers 154 34 Low temperature water n a t u r a l c o n v e c t i o n r e s u l t s f o r Th=l5.1°C. (a) G r i d , (b) Temperature c o u n t o u r s (0°C - 15.1°C, 1.51°C i n c r e m e n t s ) , (c) Stream f u n c t i o n c o u n t o u r s , (d) E x p e r i m e n t a l s t r e a k - l i n e s [ 4 2 ] , (e) L e f t w a l l N u s s e l t numbers, ( f ) R i g h t w a l l N u s s e l t numbers 155 x ACKNOWLEDGEMENTS I would l i k e t o e x p r e s s my thanks t o the people who have c o n t r i b u t e d t o t h i s work: my w i f e and f a m i l y , who have s t i m u l a t e d me throughout the c o u r s e of t h i s work; Dr. Bruce Bowen, my s u p e r v i s o r , who has g u i d e d me, p r o v i d e d me knowledge and a s s i s t e d me i n the w r i t i n g of t h i s t h e s i s ; and Frank L a y t n e r , a f r i e n d , who h e l p e d me i n the w r i t i n g of an i n t e r m e d i a t e r e p o r t p r e s e n t e d t o my committee. The f i n a n c i a l a s s i s t a n c e p r o v i d e d by the N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l of Canada i n the form of a Graduate S c h o l a r s h i p i s g r a t e f u l l y acknowledged. x i I . INTRODUCTION N a t u r a l c o n v e c t i o n i s a t o p i c wh;ch has r e c e i v e d growing a t t e n t i o n i n the l a s t few decades. I t i s a s u b j e c t of i n t e r e s t f o r both e n g i n e e r s and p h y s i c i s t s . N a t u r a l c o n v e c t i o n t a k e s p l a c e i n many g e o p h y s i c a l phenomena and has numerous e n g i n e e r i n g a p p l i c a t i o n s . R e c e n t l y , the most i m p o r t a n t e n g i n e e r i n g a p p l i c a t i o n s of n a t u r a l c o n v e c t i o n have been r e l a t e d e i t h e r t o the d e s i g n of s o l a r c o l l e c t o r s f o r p a s s i v e h e a t i n g or t o the s t o r a g e of s o l a r energy by phase change m a t e r i a l s . N a t u r a l c o n v e c t i o n i s promoted by the presence of a d e n s i t y g r a d i e n t i n a body f o r c e f i e l d . The d e n s i t y g r a d i e n t i s u s u a l l y due t o a temp e r a t u r e d i f f e r e n c e , and the body f o r c e i s o f t e n g r a v i t y . N a t u r a l c o n v e c t i o n can be i n v e s t i g a t e d t h e o r e t i c a l l y by s o l v i n g s i m u l t a n e o u s l y the mass, momentum and energy c o n s e r v a t i o n e q u a t i o n s . Because t h e s e e q u a t i o n s a r e s t r o n g l y c o u p l e d and a r e n o n l i n e a r , t h e i r s o l u t i o n g e n e r a l l y r e q u i r e s the a p p l i c a t i o n of n u m e r i c a l methods. In o n l y a few v e r y s i m p l e c a s e s , the n u m e r i c a l a p p r o a c h can be r e p l a c e d by an a n a l y t i c a l approach. E x p e r i m e n t a l and n u m e r i c a l s t u d i e s of n a t u r a l c o n v e c t i o n have been c a r r i e d out f o r many g e o m e t r i e s , but i t i s t he o n e - d i m e n s i o n a l i s o t h e r m a l f l a t p l a t e and the tw o - d i m e n s i o n a l r e c t a n g u l a r c a v i t y which have r e c e i v e d the most a t t e n t i o n . The r e c t a n g u l a r c a v i t y c o n s i s t s of two opp o s i n g i s o t h e r m a l w a l l s a t d i f f e r e n t t e m p e r a t u r e s and two 1 2 a d i a b a t i c w a l l s which complete the e n c l o s u r e ( F i g . 1 ) . Each r e c t a n g u l a r c a v i t y problem i s s p e c i f i e d by f o u r independent v a r i a b l e s : the R a y l e i g h and P r a n d t l numbers ( c a l c u l a t e d w i t h r e s p e c t t o a c h a r a c t e r i s t i c l e n g t h , u s u a l l y t a k e n t o be the l e n g t h of the a d i a b a t i c w a l l ; the c h a r a c t e r i s t i c t e mperature d i f f e r e n c e , T h ~ T c ' a n < ^ fc^e p r o p e r t i e s e v a l u a t e d a t a f l u i d r e f e r e n c e t e m p e r a t u r e , u s u a l l y the average t e m p e r a t u r e of the c a v i t y ) , the c a v i t y a s p e c t r a t i o ( l e n g t h of the i s o t h e r m a l w a l l d i v i d e d by the l e n g t h of the a d i a b a t i c w a l l ) and the a n g l e of t i l t . N a t u r a l c o n v e c t i o n i n t w o - d i m e n s i o n a l n o n r e c t a n g u l a r c a v i t i e s ( o t h e r than a x i s y m m e t r i c c y l i n d r i c a l c a v i t i e s ) has h a r d l y been i n v e s t i g a t e d . The n a t u r a l c o n v e c t i o n i n the l i q u i d phase of a m e l t i n g or f r e e z i n g p r o c e s s t a k i n g p l a c e i n a v e r t i c a l r e c t a n g u l a r c a v i t y c r e a t e s a n o n r e c t a n g u l a r e n c l o s u r e of p a r t i c u l a r i n t e r e s t . Most of the n u m e r i c a l s t u d i e s of n a t u r a l c o n v e c t i o n i n n o n r e c t a n g u l a r e n c l o s u r e s have used e i t h e r the f i n i t e element method or the f i n i t e d i f f e r e n c e method i n c o m b i n a t i o n w i t h a n o n o r t h o g o n a l t r a n s f o r m a t i o n t o s o l v e the c o n s e r v a t i o n e q u a t i o n s . R e c e n t l y , R y s k i n and L e a l [26] have d e v e l o p e d a new method f o r n u m e r i c a l l y g e n e r a t i n g b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e s which are b e t t e r s u i t e d t o the f i n i t e d i f f e r e n c e method. As w e l l as p r o d u c i n g o r t h o g o n a l g r i d s , the method a l l o w s v e r y f l e x i b l e c o n t r o l over the s p a c i n g of g r i d l i n e s as w e l l as nodal c o r r e s p o n d e n c e a t b o u n d a r i e s between c o n t i g u o u s s o l u t i o n domains. I t s a b i l i t y t o map i r r e g u l a r 3 4 c a v i t i e s i s shown i n F i g . 2. Some of the d i f f e r e n t c h a r a c t e r i s t i c s which can be imposed on ,the mapping a r e shown i n F i g . 3. T h i s new mapping p r o c e d u r e has never been used b e f o r e t o study n a t u r a l c o n v e c t i o n i n n o n r e c t a n g u l a r e n c l o s u r e s . The p r i m a r y o b j e c t i v e of the p r e s e n t t h e s i s i s t o de v e l o p a g e n e r a l p r o c e d u r e f o r u s i n g n u m e r i c a l l y g e n e r a t e d b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e t r a n s f o r m a t i o n s t o s o l v e problems of n a t u r a l c o n v e c t i o n i n n o n r e c t a n g u l a r c a v i t i e s . The c a v i t i e s c o n s i d e r e d here a r e tho s e t h a t a r e l i k e l y t o a r i s e i n phase change s i t u a t i o n s and can be seen as e x t e n s i o n s of the v e r t i c a l square e n c l o s u r e problem. In the main p a r t of the work, two g e n e r a l c a v i t y shapes a r e i n v e s t i g a t e d . Both e n c l o s u r e s have n o n f l a t r i g h t w a l l s which are a n a l y t i c a l l y d e f i n e d by the c o s i n e f u n c t i o n s : one c a v i t y employs a h a l f c y c l e , i . e . X = 1 + A - ( A C O S ( T T Y ) ) 0< = Y<=1 ( 1 ) w h i l e the o t h e r uses a f u l l c y c l e , i . e . X = 1 + A - ( A C 0 S ( 2 T T Y ) ) 0< = Y<=1 ( 2 ) For c o n v e n i e n c e , t h e s e two t y p e s of c a v i t i e s a r e r e f e r r e d t o as C1 and C 2 , r e s p e c t i v e l y , and a r e shown i n F i g . 4. N a t u r a l c o n v e c t i o n i n such deformed c a v i t i e s i s a f f e c t e d by f i v e independent v a r i a b l e s ; the R a y l e i g h and 5 F i g u r e 2. B o d y - f i t t e d orthogonal g r i d s generated by the weak c o n s t r a i n t method of Ryskin and L e a l [26] f o r s e v e r a l d i f f e r e n t i r r e g u l a r c a v i t i e s . 6 F i g u r e 3. Three d i f f e r e n t b o d y - f i t t e d orthogonal g r i d s generated by the weak c o n s t r a i n t method of Ryskin and L e a l [26] f o r a s i n g l e i r r e g u l a r c a v i t y . 8 P r a n d t l numbers ( c a l c u l a t e d w i t h r e s p e c t t o the l e n g t h of the c a v i t y bottom w a l l ; the c h a r a c t e r i s t i c t emperature d i f f e r e n c e , T h ~ T c ; a n < 3 t n e p r o p e r t i e s e v a l u a t e d a t the f l u i d r e f e r e n c e t e m p e r a t u r e , ( T ^ - T c ) / 2 ) , the d i m e n s i o n l e s s a m p l i t u d e ( a m p l i t u d e d i v i d e d by l e n g t h of the bottom w a l l ) , the a s p e c t r a t i o ( l e n g t h of the l e f t w a l l d i v i d e d by the l e n g t h of the bottom w a l l ) and t h e a n g l e of t i l t . The independent v a r i a b l e v a l u e s which a r e i n v e s t i g a t e d a r e p r e s e n t e d i n Table 1. Only two para m e t e r s a r e v a r i e d i n the p r e s e n t s t u d y : the R a y l e i g h number and the d i m e n s i o n l e s s a m p l i t u d e . The P r a n d t l number, the c a v i t y a s p e c t r a t i o and the a n g l e of t i l t remain c o n s t a n t . The method of R y s k i n and L e a l [26] i s used t o g e n e r a t e a t w o - d i m e n s i o n a l b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e system. The s t e a d y - s t a t e mass, momentum and energy c o n s e r v a t i o n e q u a t i o n s , w r i t t e n i n terms of the stream f u n c t i o n , v o r t i c i t y and temperature dependent v a r i a b l e s , a r e then s o l v e d i n t r a n s f o r m e d c o o r d i n a t e s u s i n g a f i n i t e d i f f e r e n c e method. The a n a l y s i s assumes t h a t the B o u s s i n e s q a p p r o x i m a t i o n s a r e v a l i d , t h a t t h e f l u i d p r o p e r t i e s o t h e r than d e n s i t y a r e c o n s t a n t , and t h a t the f l u i d i s Newtonian. The study i s s u b d i v i d e d i n t o t h r e e p a r t s . The o b j e c t i v e of the f i r s t p a r t i s t o determine the e f f e c t s of the mapping c h a r a c t e r i s t i c s (which a r e d e s c r i b e d i n a l a t e r c h a p t e r ) , the v o r t i c i t y boundary c o n d i t i o n s ( e i t h e r Wood or second o r d e r ) and the f i n i t e d i f f e r e n c e g r i d s i z e on t h e a c c u r a c y of the n u m e r i c a l r e s u l t s . For t h i s p u r p o s e , o n l y the most Table 1. Independent v a r i a b l e s c o n s i d e r e d f o r the c a v i t v types CI and C2. * Ra y l e i g h P r a n d t l Dimensionless C a v i t y C a v i t y Number Number Ampli tude Aspect Angle of R a t i o T i l t (Degree) 0 1 -0.150 1 0 1000 -0.075 3000 0.000 1 0000 0.075 30000 0. 1 50 100000 10 extreme c a s e s are c o n s i d e r e d . Thus, o n l y the most d i s t o r t e d c a v i t i e s of t y p e s C1 and C2 a r e used. A moderate R a y l e i g h number of 10000 i s employed because i t e n s u r e s t h a t both the c o n d u c t i v e and c o n v e c t i v e heat t r a n s f e r modes a r e i n v o l v e d . From the r e s u l t s of t h e s e s i m u l a t i o n s , the b e s t s e t of mapping c h a r a c t e r i s t i c s and v o r t i c i t y boundary c o n d i t i o n s i s d e t e r m i n e d f o r each extreme c a s e . A l s o , an optimum f i n i t e d i f f e r e n c e g r i d s i z e i s s e l e c t e d which y i e l d s r e a s o n a b l e n u m e r i c a l e r r o r w i t h m i n i m a l c o m p u t a t i o n a l c o s t . In the second p a r t , the o p t i m a l c o n d i t i o n s s e l e c t e d i n the f i r s t s e c t i o n a r e used t o t h o r o u g h l y i n v e s t i g a t e the e f f e c t s of d i m e n s i o n l e s s a m p l i t u d e and R a y l e i g h number on the heat t r a n s f e r by n a t u r a l c o n v e c t i o n i n c a v i t y t y p e s C1 and C2. As w e l l as g e n e r a t i n g f l o w maps and temperature d i s t r i b u t i o n s f o r each c a s e , l o c a l and average N u s s e l t numbers a r e a l s o c a l c u l a t e d and compared w i t h s t a n d a r d c o r r e l a t i o n s f o r r e c t a n g u l a r c a v i t i e s . The o b j e c t i v e of the t h i r d p a r t i s t o demonstrate the a p p l i c a b i l i t y of the n u m e r i c a l p r o c e d u r e t o r e a l phase change problems. S e v e r a l d i f f e r e n t q u a s i - s t e a d y f l o w p a t t e r n s which o c c u r d u r i n g i c e f o r m a t i o n a r e s i m u l a t e d and compared t o a v a i l a b l e e x p e r i m e n t a l s t r e a m - l i n e photographs ta k e n by E c k e r t [ 4 2 ] . I I . LITERATURE REVIEW A. NATURAL CONVECTION NEAR A VERTICAL ISOTHERMAL FLAT PLATE For t h i s s i m p l e c a s e , an a n a l y t i c a l s o l u t i o n of the c o n s e r v a t i o n e q u a t i o n s i s made p o s s i b l e by assuming t h a t the f l u i d i s Newtonian, t h a t a l a m i n a r b o u n d a r y - l a y e r e x i s t s , t h a t the B o u s s i n e s q a p p r o x i m a t i o n s a r e v a l i d and t h a t the f l u i d p r o p e r t i e s o t h e r than d e n s i t y are c o n s t a n t . Under th e s e c o n d i t i o n s a s i m i l a r i t y t r a n s f o r m a t i o n can be a p p l i e d which reduces the s e t 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 t o a p a i r of 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 [ 3 0 , 3 2 , 3 3 ]. A l s o , s i m i l a r i t y p r o f i l e s can be used i n c o n j u n c t i o n w i t h the i n t e g r a l method t o a p p r o x i m a t e l y s o l v e the c o n s e r v a t i o n e q u a t i o n s [ 3 0 , 3 1 ] . The te m p e r a t u r e and v e l o c i t y r e s u l t s from the a n a l y t i c t h e o r y a r e found t o be i n good agreement w i t h e x p e r i m e n t a l measurements [ 3 0 ] . However, the e x p e r i m e n t a l average N u s s e l t numbers a r e s l i g h t l y h i g h e r than those p r e d i c t e d a n a l y t i c a l l y . The d i f f e r e n c e s i n the average N u s s e l t numbers a r e l a r g e r f o r b o t h s m a l l and l a r g e R a y l e i g h numbers. At low R a y l e i g h numbers, the d i s c r e p a n c i e s between p r e d i c t e d and e x p e r i m e n t a l average N u s s e l t numbers a r e l i k e l y due t o the i n c r e a s i n g i n a c c u r a c y of the b o u n d a r y - l a y e r a s s u m p t i o n s . At h i g h R a y l e i g h numbers, the d i f f e r e n c e s a re a t t r i b u t e d t o the development of t u r b u l e n c e . The a n a l y t i c a l average N u s s e l t number i s c o r r e l a t e d t o the R a y l e i g h number by an e q u a t i o n of the form 1 1 12 Nu =a(Ra ) b (3) x,ave x where h x vi ave , . v Nu = (4) x,ave , Ra =g(3 p2 x 3 ( T -T )C /M k (5) and x i s the d i s t a n c e a l o n g the p l a t e i n the f l o w d i r e c t i o n . The v a l u e of the c o e f f i c i e n t a i n Eq. 3 i s weakly dependent on the P r a n d t l number, and the v a l u e of the c o e f f i c i e n t b i s e q u a l t o 0.25 f o r the l a m i n a r b o u n d a r y - l a y e r regime. For P r a n d t l number a p p r o a c h i n g 0 and P r a n d t l numbers l a r g e r than 100, Eq. 3 becomes, r e s p e c t i v e l y , N u x , a v e = 0 ' 8 ( P r R a x ) O ' 2 5 ( 6 ) and N V a v e = 0 - 6 7 ( R a x ) O ' 2 5 ( 7 ) where Pr i s the P r a n d t l number. B. NATURAL CONVECTION IN RECTANGULAR ENCLOSURES The t w o - d i m e n s i o n a l r e c t a n g u l a r c a v i t y i s c a l l e d a v e r t i c a l r e c t a n g u l a r c a v i t y i f the i s o t h e r m a l w a l l s a re v e r t i c a l , and a h o r i z o n t a l r e c t a n g u l a r c a v i t y i f the 13 i s o t h e r m a l w a l l s are h o r i z o n t a l . T h i s c l a s s i f i c a t i o n a r i s e s from the d i f f e r e n c e s i n the c o n v e c t i o n p a t p e r n s o b s e r v e d f o r the two c a s e s [ 1 , 2 ] . In the v e r t i c a l r e c t a n g u l a r c a v i t y , the f l o w s t a r t s i m m e d i a t e l y because the d e n s i t y g r a d i e n t i s p e r p e n d i c u l a r t o the g r a v i t y v e c t o r . On the o t h e r hand, f o r the h o r i z o n t a l r e c t a n g u l a r c a v i t y , two c a s e s a r e p o s s i b l e : 1. i f the d e n s i t y g r a d i e n t i s p a r a l l e l and opposed t o the g r a v i t y v e c t o r , the f l o w s t a r t s o n l y i f the g r a d i e n t i s s u f f i c i e n t l y l a r g e , and 2. i f the d e n s i t y g r a d i e n t i s p a r a l l e l t o and i n the same d i r e c t i o n as the g r a v i t y v e c t o r , no f l o w t a k e s p l a c e i n the c a v i t y . The independent v a r i a b l e s a s s o c i a t e d w i t h the r e c t a n g u l a r c a v i t y are the R a y l e i g h number, the P r a n d t l number, the c a v i t y a s p e c t r a t i o and the a n g l e of t i l t . E x t e n s i v e e x p e r i m e n t a l and n u m e r i c a l i n v e s t i g a t i o n s have been c a r r i e d out on the r e c t a n g u l a r c a v i t y t o e s t a b l i s h f l o w regimes and t o d etermine average N u s s e l t numbers. The r e s u l t s of t h e s e i n v e s t i g a t i o n s a r e t h o r o u g h l y d i s c u s s e d i n two e x c e l l e n t r e v i e w a r t i c l e s [ 1 , 2 ] . The d i f f e r e n t f l o w regimes and dominant heat t r a n s f e r mechanisms o b s e r v e d as a f u n c t i o n of R a y l e i g h number a r e shown i n T a b l e 2 f o r t h e case of the v e r t i c a l square c a v i t y f i l l e d w i t h a i r [ 1 , 3 ] . Two example f l o w p a t t e r n s a r e i l l u s t r a t e d i n F i g . 5. For the v e r t i c a l square e n c l o s u r e , the f o l l o w i n g i d e n t i f i c a t i o n scheme f o r the v a r i o u s heat t r a n s f e r regimes was proposed [ 1 4 ] : Table 2. Flow and heat t r a n s f e r o b s e r v a t i o n s f o r the v e r t i c a l square c a v i t y f i l l e d with a i r . R a y l e i g h Number Range Ob.se rvat ions 0 to 1000 - Weak u n i c e l l u l a r flow p a t t e r n - Conduction heat t r a n s f e r regime 1000 to 10000 ^ U n i c e l l u l a r boundary-layer l i k e flow p a t t e r n - T r a n s i t i o n heat t r a n s f e r regime 10000 to 100000 ^ U n i c e l l u l a r boundary-layer l i k e flow p a t t e r n - Boundary-layer heat t r a n s f e r regime 100000 to 1000000 - Onset of a secondary flow eat's-eye p a t t e r n - Boundary-layer heat t r a n s f e r regime F i g u r e 5. T y p i c a l f l o w p a t t e r n s o b s e r v e d i n a v e r t i c a l square c a v i t y , (a) Ra=1u000, u n i c e l l u l a r f l o w , (b) Ra=1u0000, " c a t ' s - e y e " p a t t e r n . 16 1. the end of the conduction regime i s a t t a i n e d when the h o r i z o n t a l temperature d e r i v a t i v e at the center of the c a v i t y d i f f e r s by 10 percent from the value f o r pure c o n d u c t i o n , and 2. the beginning of the boundary-layer regime i s reached when the h o r i z o n t a l temperature d e r i v a t i v e at the center of the c a v i t y i s equal to zero. The average Nusselt number i s found to be s i g n i f i c a n t l y a f f e c t e d by the R a y l e i g h number [3-8,11,13], low P r a n d t l number [3,7,8], the c a v i t y aspect r a t i o [6,7,13] and the angle of t i l t [ 4-6]. The e f f e c t of l a r g e P r a n d t l number on the average Nusselt number i s n e g l i g i b l e [3,7,8]. The general r e l a t i o n s h i p used to c o r r e l a t e the Nu s s e l t number with the R a y l e i g h number (other independent v a r i a b l e s being constant) i s s i m i l a r to that employed f o r the i s o t h e r m a l f l a t p l a t e (Eq. 3): Nu =a(Ra) b (8) where h L =_ave_c ( g ) and 17 Ra=g0 op o 2L£(T h-T c)C p O/M oko (10) The c h a r a c t e r i s t i c length, L c, i s the length of the adiabatic wall and the f l u i d reference temperature i s the average temperature through the cavity. For the laminar boundary-layer regime, the value of the exponent b i s somewhat greater for the v e r t i c a l rectangular cavity than for the v e r t i c a l isothermal f l a t plate. For the enclosure, b ranges from 0.25 to 0.35 [3,7,9,10]. This fact demonstrates the importance of the core of the cavity and the s i g n i f i c a n t interaction i t has with the boundary-layer at the wall [1,2,4]. Natural convection in a v e r t i c a l square cavity has become a benchmark problem for numerical studies, primarily because t h i s i s the simplest case for which a l l terms of the Navier-Stokes equations must be included and where the energy and motion equations are coupled. In 1982, under the auspices of De Vahl Davis and Jones [15], a comparison exercise was carried out to test the a b i l i t y of a large number of di f f e r e n t numerical techniques to 'successfully solve t h i s problem. Of the 36 contributions, 21 used f i n i t e difference methods while 10 others used f i n i t e element methods. Also, among a l l contributions, 11 considered the primitive variable form (two v e l o c i t i e s , pressure and temperature) of the governing equations while 9 others used the stream function, v o r t i c i t y and temperature form. A large variety of f i n i t e difference d i s c r e t i z a t i o n methods were 18 used. Of the v a r i o u s f i n i t e d i f f e r e n c e s o l u t i o n s s u b m i t t e d , one of the most s u c c e s s f u l was t h a t of Wong and R a i t h b y [ 1 2 ] , who employed the e x p o n e n t i a l d i f f e r e n c i n g , scheme and a second o r d e r a c c u r a t e v o r t i c i t y boundary c o n d i t i o n . I n v e s t i g a t i o n s have a l s o been c a r r i e d out on the v a l i d i t y of the commonly used a s s u m p t i o n s i n n u m e r i c a l n a t u r a l c o n v e c t i o n s t u d i e s . The v a l i d i t y of the B o u s s i n e s q a p p r o x i m a t i o n s and c o n s t a n t f l u i d p r o p e r t i e s ( w i t h the e x c e p t i o n of the d e n s i t y ) assumption was s t u d i e d by e x a m i n i n g the e f f e c t s of v a r i a b l e p r o p e r t i e s on l a m i n a r n a t u r a l c o n v e c t i o n i n a v e r t i c a l square e n c l o s u r e [ 1 0 ] . In t h i s s t u d y , the u n s t e a d y - s t a t e c o n s e r v a t i o n e q u a t i o n s w r i t t e n i n terms of p r i m i t i v e v a r i a b l e s were s o l v e d u s i n g f i n i t e d i f f e r e n c e methods. To d e t e r m i n e the v a l i d i t y l i m i t s of the B o u s s i n e s q a p p r o x i m a t i o n s ( c o n s t a n t p r o p e r t i e s e v a l u a t e d a t the c a v i t y c o l d w a l l t e m p e r a t u r e ) , a c o m p r e s s i b l e Newtonian f l u i d ( a i r ) whose d e n s i t y f o l l o w s the p e r f e c t gas law and whose o t h e r f l u i d p r o p e r t i e s a r e c o n s t a n t was c o n s i d e r e d . I t was found, t h a t t h e B o u s s i n e s q a p p r o x i m a t i o n s were v a l i d i f the o v e r a l l t e m p e r a t u r e d i f f e r e n c e s a t i s f i e d the f o l l o w i n g c r i t e r i o n : T T — — £ < 0 . 1 . (11) T c where the temperatures a r e e x p r e s s e d i n K e l v i n s . C o n s i d e r i n g a c o m p r e s s i b l e Newtonian f l u i d ( a i r ) whose d e n s i t y i s s t i l l 19 given by the p e r f e c t gas law but whose p r o p e r t i e s are temperature dependent, i t was found that 1 . the average N u s s e l t number was u n a f f e c t e d by the assumptions made about the f l u i d p r o p e r t i e s over a range of ( T h ~ T c ) / T c from 0.2 to 2.0 as long as an a p p r o p r i a t e r e f e r e n c e temperature, i . e . T o=T c+0.25(T h-T c) (12) was chosen, and 2. the v e l o c i t y and temperature p r o f i l e s were i n f l u e n c e d to a g r e a t e r or l e s s e r degree by the cho i c e s made concerning the pr o p e r t y assumptions. The Newtonian f l u i d and constant p r o p e r t y assumptions were a l s o i n v e s t i g a t e d by stu d y i n g the n a t u r a l c o n v e c t i o n i n a r e c t a n g u l a r e n c l o s u r e of a f l u i d whose v i s c o s i t y was temperature dependent or whose behaviour was non-Newtonian [ 9 ] . In t h i s study, the s t e a d y - s t a t e c o n s e r v a t i o n equations w r i t t e n i n terms of stream f u n c t i o n , v o r t i c i t y and temperature were s o l v e d u s i n g f i n i t e d i f f e r e n c e methods. I t was assumed that the Boussinesq approximations were v a l i d and that the f l u i d p r o p e r t i e s other than d e n s i t y , v i s c o s i t y and e l a s t i c i t y were con s t a n t . I t was found t h a t , while the v e l o c i t y d i s t r i b u t i o n i s a f f e c t e d s i g n i f i c a n t l y , the o v e r a l l r a t e of heat t r a n s f e r through the enclosure i s n e g l i g i b l y i n f l u e n c e d by assuming 20 Newtonian b e h a v i o u r . N a t u r a l c o n v e c t i o n has been s t u d i e d i n m o d i f i e d r e c t a n g u l a r c a v i t i e s [ 1 6 , 1 7 ] , Chang and a l . [17] n u m e r i c a l l y i n v e s t i g a t e d the e f f e c t s of i n t e r n a l b a f f l e s l o c a t e d o p p o s i t e one another on the t o p and bottom w a l l s of a v e r t i c a l c a v i t y . The u n s t e a d y - s t a t e c o n s e r v a t i o n e q u a t i o n s w r i t t e n i n terms of p r i m i t i v e v a r i a b l e s were s o l v e d u s i n g a f i n i t e d i f f e r e n c e method. I t was assumed t h a t the f l u i d was Newtonian and obeyed the p e r f e c t gas law and t h a t i t s p r o p e r t i e s o t h e r than d e n s i t y were c o n s t a n t . The n u m e r i c a l r e s u l t s a r e d i s c u s s e d but a r e not compared w i t h e x p e r i m e n t a l measurements. Kim and V i s k a n t a [16] s t u d i e d the e f f e c t of w a l l heat c o n d u c t i o n on n a t u r a l c o n v e c t i o n heat t r a n s f e r i n a square e n c l o s u r e . The u n s t e a d y - s t a t e c o n s e r v a t i o n e q u a t i o n s w r i t t e n i n terms of stream f u n c t i o n , v o r t i c i t y and tem p e r a t u r e were s o l v e d w i t h f i n i t e d i f f e r e n c e methods. I t was assumed t h a t the B o u s s i n e s q a p p r o x i m a t i o n s were v a l i d , t h a t the f l u i d was Newtonian, i t s p r o p e r t i e s o t h e r than d e n s i t y were c o n s t a n t and t h a t t h e w a l l p r o p e r t i e s were i s o t r o p i c and t e m p e r a t u r e - i n d e p e n d e n t . T h e i r n u m e r i c a l r e s u l t s were compared w i t h measurements c a r r i e d out by the same a u t h o r s [ 1 6 ] . Very good agreement was found between the n u m e r i c a l and e x p e r i m e n t a l r e s u l t s f o r the temperature d i s t r i b u t i o n i n the s o l i d w h i l e a r e a s o n a b l e agreement was o b t a i n e d f o r the temperature d i s t r i b u t i o n i n the l i q u i d . 21 C. PHASE CHANGE PROBLEMS The problem of m e l t i n g or s o l i d i f i c a t i o n i n an e n c l o s u r e has not been e x t e n s i v e l y i n v e s t i g a t e d e i t h e r n u m e r i c a l l y or e x p e r i m e n t a l l y . The l a c k of t h e o r e t i c a l i n f o r m a t i o n about t h i s case can be a t t r i b u t e d t o the d i f f i c u l t y of s o l v i n g the e q u a t i o n s of motion and energy i n an i r r e g u l a r - s h a p e d c a v i t y w i t h a moving boundary [ 1 9 ] , I t has been amply demonstrated t h a t n a t u r a l c o n v e c t i o n p l a y s a s i g n i f i c a n t r o l e i n such phase change problems [18,19,20]. N u m e r i c a l r e s u l t s a r e a v a i l a b l e f o r m e l t i n g or s o l i d i f i c a t i o n i n a v e r t i c a l r e c t a n g u l a r c a v i t y [18,19] and i n an a n n u l a r c a v i t y [ 2 0 ] . In t h e s e s t u d i e s , a n o n - o r t h o g o n a l c o o r d i n a t e system and a q u a s i - s t e a d y approach were used t o s o l v e e i t h e r the s t e a d y - s t a t e [19] or u n s t e a d y - s t a t e [18,20] c o n s e r v a t i o n e q u a t i o n s . I n a l l c a s e s , i t was assumed t h a t the f l u i d phase p r o p e r t i e s o t h e r than d e n s i t y were c o n s t a n t , t h a t the f l u i d was Newtonian and t h a t the B o u s s i n e s q assumptions were v a l i d w h i l e a l l p r o p e r t i e s of the s o l i d were assumed t o be c o n s t a n t . Both p r i m i t i v e v a r i a b l e s and stream f u n c t i o n , v o r t i c i t y and temperature v a r i a b l e s were used i n thes e n u m e r i c a l i n v e s t i g a t i o n s . Only the n u m e r i c a l r e s u l t s of Ho and V i s k a n t a [18] were e x t e n s i v e l y compared w i t h measurements o b t a i n by the same a u t h o r s [ 1 8 ] . The l i q u i d volume f r a c t i o n , the l o c a l and average N u s s e l t numbers a l o n g the v e r t i c a l hot w a l l , and the shape of the s o l i d - l i q u i d i n t e r f a c e were compared a t d i f f e r e n t times d u r i n g the t r a n s i e n t m e l t i n g p r o c e s s . At 22 e a r l y t i m e s , good q u a n t i t a t i v e agreement was found between the n u m e r i c a l and e x p e r i m e n t a l r e s u l t s , but w i t h i n c r e a s i n g t i m e , the n u m e r i c a l r e s u l t s were found t o d i s p l a y o n l y s i m i l a r q u a l i t a t i v e t r e n d s i n time and space w i t h the measured r e s u l t s . N a t u r a l c o n v e c t i o n phase change problems a l s o occur d u r i n g c a s t i n g p r o c e s s e s . For i n s t a n c e , Kroeger and O s t r a c h [21] have c a r r i e d out a t h e o r e t i c a l study of the c o n t i n u o u s c a s t i n g of a s l a b . A n u m e r i c a l l y g e n e r a t e d c o n f o r m a l o r t h o g o n a l c o o r d i n a t e system was used a l o n g w i t h f i n i t e d i f f e r e n c e methods t o s o l v e the s t e a d y - s t a t e c o n s e r v a t i o n e q u a t i o n s , w r i t t e n i n terms of st r e a m f u n c t i o n , v o r t i c i t y and temperature dependent v a r i a b l e s . I t was assumed t h a t the f l u i d phase p r o p e r t i e s o t h e r t h a t d e n s i t y were c o n s t a n t , t h a t the f l u i d was Newtonian and t h a t the B o u s s i n e s q a p p r o x i m a t i o n s were v a l i d . U n f o r t u n a t e l y , the n u m e r i c a l r e s u l t s were not compared w i t h measured d a t a . D. NATURAL CONVECTION IN NONRECTANGULAR ENCLOSURES N a t u r a l c o n v e c t i o n i n t w o - d i m e n s i o n a l n o n r e c t a n g u l a r e n c l o s u r e s ( o t h e r than c y l i n d r i c a l ) has a l s o been l i t t l e i n v e s t i g a t e d . Some e x p e r i m e n t a l work on n a t u r a l c o n v e c t i o n i n p a r a l l e l o g r a m m i c e n c l o s u r e s has been c a r r i e d out [ 2 3 ] . A major study which i n v o l v e d n u m e r i c a l a n a l y s e s as w e l l as some e x p e r i m e n t a l measurements i n v e s t i g a t e d n a t u r a l c o n v e c t i o n i n t w o - d i m e n s i o n a l v e e - c o r r u g a t e d c h a n n e l s (bottom l e f t image of F i g . 2) [22,24,34-37]. In the n u m e r i c a l i n v e s t i g a t i o n [ 2 2 , 2 4 ] , a S c h w a r t z - C h r i s t o f f e l 23 t r a n s f o r m a t i o n was used to a n a l y t i c a l l y determine a conformal b o u n d a r y - f i t t e d orthogonal c o o r d i n a t e system. The usual assumptions (constant f l u i d p r o p e r t i e s other than d e n s i t y , Boussinesq, Newtonian f l u i d ) were made in order to s o l v e the unsteady-state c o n s e r v a t i o n equations, w r i t t e n i n terms of stream f u n c t i o n , v o r t i c i t y and temperature, using f i n i t e d i f f e r e n c e methods. Average Nusselt numbers along the f l a t s u r f a c e were compared with measurements [34-37] f o r d i f f e r e n t channel aspect r a t i o s , angles of t i l t and R a y l e i g h numbers. S i m i l a r trends were n o t i c e d although q u a n t i t a t i v e d i s c r e p a n c i e s (up to 20 percent) were found. E. BODY-FITTED ORTHOGONAL TRANSFORMATIONS B o d y - f i t t e d orthogonal t r a n s f o r m a t i o n s f o r mapping nonrectangular domains i n t o r e c t a n g u l a r domains can be obtained using e i t h e r an a n a l y t i c a l or a numerical approach. The a n a l y t i c a l approach i n v o l v e s t e d i o u s mathematical manipulations and i s r e s t r i c t e d to the mapping of very simple nonrectangular domains. However, the numerical approach can be used to map n e a r l y any i r r e g u l a r domain and a l l o w s f o r f a r more f l e x i b l e mapping c h a r a c t e r i s t i c s . Numerical c o o r d i n a t e g e n e r a t i o n i s a r e l a t i v e l y new area of numerical a n a l y s i s and i t s p r o g r e s s i o n has been reviewed i n s e v e r a l recent a r t i c l e s [25,26]. Of the numerical mapping procedures a v a i l a b l e f o r e n c l o s u r e s with s p e c i f i e d boundaries, the s o - c a l l e d weak c o n s t r a i n t method of Ryskin and L e a l [26] appears to be most powerful. I t not 24 o n l y g e n e r a t e s o r t h o g o n a l g r i d s which have d i s t i n c t advantages over n o n - o r t h o g o n a l c o o r d i n a t e systems, but i t a l s o a l l o w s f o r the c o n v e n i e n t matching of boundary c o n d i t i o n s a t the i n t e r f a c e of n e i g h b o u r i n g r e g i o n s , an im p o r t a n t c o n s i d e r a t i o n i n phase change problems. D e s p i t e i t s many advantageous f e a t u r e s , the weak c o n s t r a i n t method appears t o have had l i m i t e d use i n s o l v i n g f l u i d f l o w problems. The o n l y r e f e r e n c e t o the p r o c e d u r e t h a t has appeared i n the l i t e r a t u r e so f a r i s a paper by C h i k h l i w a l a and Y o r t s o s [27] who used i t t o map the i r r e g u l a r domains l i k e l y t o a r i s e d u r i n g i m m i s c i b l e f l u i d - f l u i d d i s p l a c e m e n t i n porous media. However, the paper was concerned o n l y w i t h the mapping p r o c e s s ; t h e r e were no accompagning s o l u t i o n s t o the f l u i d f l o w problem. S e v e r a l e a r l i e r s t u d i e s have s o l v e d the N a v i e r - S t o k e s e q u a t i o n s u s i n g l e s s e l a b o r a t e n u m e r i c a l p r o c e d u r e s than the weak c o n s t r a i n t method. For example, Pope [29] used a n u m e r i c a l l y g e n e r a t e d o r t h o g o n a l g r i d ( c o n s t a n t shape f a c t o r ) t o i n v e s t i g a t e t u r b u l e n t f o r c e d c o n v e c t i o n f l o w s i n an i r r e g u l a r c h a n n e l w h i l e Kroeger and O s t r a c h [21] used a n u m e r i c a l l y g e n e r a t e d o r t h o g o n a l g r i d t o stu d y the c o n t i n u o u s c a s t i n g p r o c e s s of a s l a b i n which n a t u r a l c o n v e c t i o n was ta k e n i n t o a c c o u n t . I I I . COORDINATE SYSTEM In mapping a n o n r e c t a n g u l a r two-dj.mensional domain, many c o o r d i n a t e systems c o u l d be c o n s i d e r e d . For example, e i t h e r a C a r t e s i a n , a b o d y - f i t t e d n o n o r t h o g o n a l or a b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e system c o u l d be used ( F i g . 6 ) . Whichever c o o r d i n a t e system i s adopted, the mass, momentum and energy c o n s e r v a t i o n p r i n c i p l e s r e q u i r e d t o s o l v e the n a t u r a l c o n v e c t i o n problem a r e s t i l l a p p l i c a b l e ; however, the m a t h e m a t i c a l f o r m u l a t i o n of the s e c o n s e r v a t i o n p r i n c i p l e s i n a p a r t i c u l a r c o o r d i n a t e system i s c o m p l i c a t e d t o a g r e a t e r or l e s s e r e x t e n t depending on which c o o r d i n a t e system i s chosen. The s t e a d y - s t a t e n a t u r a l c o n v e c t i o n problem which i s d e s c r i b e d by a s e t of e l l i p t i c 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 i s s u b j e c t t o a c o r r e s p o n d i n g s e t of boundary c o n d i t i o n s . These boundary c o n d i t i o n s a r e e i t h e r of the D i r i c h l e t , Neumann or mixed (Robin) t y p e . A D i r i c h l e t boundary c o n d i t i o n s p e c i f i e s the v a l u e of a dependent v a r i a b l e a t the domain boundary. A Neumann boundary c o n d i t i o n s p e c i f i e s the v a l u e s of the d e r i v a t i v e of a dependent v a r i a b l e i n the d i r e c t i o n normal t o the boundary. The mixed boundary c o n d i t i o n a s , i t s name i m p l i e s , r e l a t e s the normal d e r i v a t i v e of a dependent v a r i a b l e w i t h i t s v a l u e a t the boundary. Thus, t h e way the c o o r d i n a t e system matches the c a v i t y boundary i s an i m p o r t a n t f a c t o r t o c o n s i d e r . The e a s i e r and the more a c c u r a t e l y the boundary c o n d i t i o n s can be p r e s c r i b e d , the b e t t e r w i l l be the n u m e r i c a l 2 5 26 F i g u r e 6. D i f f e r e n t c o o r d i n a t e systems which can be used map an i r r e g u l a r c a v i t y , (a) C a r t e s i a n , (b) b o d y - f i t t e d nonorthogonal, (c) b o d y - f i t t e d o r t h o g o n a l . K \ \ J / / V 27 s o l u t i o n [ 2 4 ] . The b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e system i s the o n l y one which matches the domain boundary, a l l o w s easy s p e c i f i c a t i o n of D i r i c h l e t , Neumann and mixed c o n d i t i o n s a t the boundary and reduces the c o n s e r v a t i o n e q u a t i o n s t o a r e l a t i v e l y compact form. The c h o i c e of the C a r t e s i a n c o o r d i n a t e system was e l i m i n a t e d because not a l l of the f i n i t e d i f f e r e n c e nodes l i e on the boundary, nor do the g r i d l i n e s i n t e r c e p t i t a t r i g h t a n g l e s . As a consequence, i t becomes v e r y d i f f i c u l t t o a c c u r a t e l y s p e c i f y any type of boundary c o n d i t i o n . The b o d y - f i t t e d n o n - o r t h o g o n a l c o o r d i n a t e system makes the s e t t i n g of D i r i c h l e t c o n d i t i o n s a s i m p l e m a t t e r , but i t i s a l s o a poor c h o i c e because the Neumann and mixed c o n d i t i o n s cannot be a c c u r a t e l y s p e c i f i e d and f u r t h e r m o r e , the g o v e r n i n g e q u a t i o n s , due t o the f a c t t h a t c r o s s - d e r i v a t i v e terms must be r e t a i n e d , become v e r y cumbersome. A. BODY-FITTED ORTHOGONAL MAPPING USING THE WEAK CONSTRAINT  METHOD In the f o l l o w i n g d i s c u s s i o n , two domains a r e c o n s i d e r e d ( F i g . 7) which are r e f e r r e d t o as the p h y s i c a l and the t r a n s f o r m e d domains, r e s p e c t i v e l y . The c o o r d i n a t e system of the p h y s i c a l domain i s C a r t e s i a n , and i t has an i r r e g u l a r shape whose b o u n d a r i e s can be e x p r e s s e d i n terms of a n a l y t i c f o r m u l a e . R e g a r d l e s s of i t s shape, the boundary of the p h y s i c a l domain can be d i v i d e d i n t o f o u r s e c t i o n s . The Figure 7. Physical domain (a) and transformed domain 29 s e c t i o n s are numbered from 1 to 4 f o l l o w i n g the boundary i n i t s c lockwise d i r e c t i o n . The c o o r d i n a t e system of the transformed domain i s o r t h o g o n a l . The transformed domain always has a r e c t a n g u l a r shape which i s the image, i n the orthogonal c o o r d i n a t e system, of the p h y s i c a l domain. The four boundaries of the transformed domain are r e f e r r e d to as the top, r i g h t , bottom and l e f t w a l l s and are expressed i n a n a l y t i c a l form by the f o l l o w i n g r e l a t i o n s h i p s : l e f t w a l l , £=1; r i g h t w a l l , £=M; bottom w a l l , 77= 1 ; and top w a l l , 77=N; where M and N are the number of rows or columns of nodal p o i n t s i n the £ and rj d i r e c t i o n s , r e s p e c t i v e l y . I f i t i s a r b i t r a r i l y c o n s i d e r e d that the images of the top w a l l and the top r i g h t corner of the transformed domain correspond to s e c t i o n 1 and the j u n c t i o n of s e c t i o n s 1 and 2 of the p h y s i c a l domain, r e s p e c t i v e l y , then the f o l l o w i n g statements can be made about the t r a n s f o r m a t i o n : 1. the images of the top, r i g h t , bottom and l e f t w a l ls of the transformed domain must correspond to s e c t i o n s 1 to 4 of the p h y s i c a l domain boundary, r e s p e c t i v e l y , and 2. the image of the top l e f t , bottom l e f t , bottom 30 r i g h t and top r i g h t c o r n e r p o i n t s of the t r a n s f o r m e d domain boundary must c o r r e s p o n d t o the j u n c t i o n s of the p h y s i c a l domain boundary s e c t i o n s 1 and 2, 2 and 3, 3 and 4, and 4 and 1, r e s p e c t i v e l y . The p o i n t - b y - p o i n t c o o r d i n a t e mapping of the t r a n s f o r m e d domain i n t o the p h y s i c a l domain i s o b t a i n e d n u m e r i c a l l y by s o l v i n g an a p p r o p r i a t e s e t of d i f f e r e n t i a l e q u a t i o n s . In the weak c o n s t r a i n t method of R y s k i n and L e a l [ 2 6 ] , the o r t h o g o n a l mapping i s d e f i n e d by the f o l o w i n g p a i r of e l l i p t i c 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 : H,H 3 a* ax f — a* 977 1 ax' f a?? =o (13) and H,H 3 as 3Y" 3 r j 1 3Y' fa?? = 0 (14) where H f=-a Hs H; "3X" 2 "3Y" + - 9 * _ (15) (16) and 31 2 (17) E q u a t i o n s 13 and 14 a r e L a p l a c e ' s e q u a t i o n s i n g e n e r a l o r t h o g o n a l c o o r d i n a t e s , and they a r i s e from c o n s i d e r a t i o n of the f a c t t h a t a l l c o o r d i n a t e l i n e s i n both the p h y s i c a l and t r a n s f o r m e d domains must meet a t r i g h t a n g l e s . The two e q u a t i o n s must be s o l v e d s i m u l t a n e o u s l y over the whole of the t r a n s f o r m e d domain s u b j e c t t o an a p p r o p r i a t e s e t of boundary c o n d i t i o n s . The q u a n t i t i e s , H,. and H , a r e c a l l e d s c a l e f a c t o r s . ? V They r e l a t e the l e n g t h of i n f i n i t e s i m a l d i s p l a c e m e n t s i n both domains. The shape f a c t o r , f , s p e c i f i e s the a s p e c t r a t i o of an i n f i n i t e s i m a l r e c t a n g u l a r element i n the p h y s i c a l domain which i s the image of an i n f i n i t e s i m a l square element i n the t r a n s f o r m e d domain. The shape f a c t o r i s p r e s c r i b e d over the i n t e r i o r of the t r a n s f o r m e d domain by a f u n c t i o n which i s c a l l e d the d i s t o r t i o n f u n c t i o n . The d i s t o r t i o n f u n c t i o n can be any p o s i t i v e s c a l a r f u n c t i o n i n c l u d i n g , i n i t s s i m p l e s t form, a c o n s t a n t . I f the shape f a c t o r i s everywhere e q u a l t o u n i t y , then t h e mapping which r e s u l t s i s c a l l e d c o n f o r m a l . The mapping of an i r r e g u l a r shape u s i n g a c o n s t a n t d i s t o r t i o n f u n c t i o n i s s a i d t o be " s t i f f " because such a c o n d i t i o n i s a major and unnecessary r e s t r i c t i o n [ 2 6 ] . A d i s t o r t i o n f u n c t i o n which v a r i e s w i t h £ and 77 i s much more d e s i r a b l e because i t g i v e s the user more c o n t r o l over the mapping p r o c e s s . R y s k i n and L e a l [26] recommend the f o l l o w i n g a l g e b r a i c i n t e r p o l a t i o n f o r m u l a 9X 2 "3Y + 877 91? 32 which c o n s i d e r s the shape f a c t o r s along the transformed domain boundary i n order to determine the shape f a c t o r s i n the i n t e r i o r of the transformed domain: f U,i?) = (1 - * ) f O ,r?) + £f ( M , T ? ) + ( 1 - T 7)f U , 1 ) + T j f ($,N) - ( , - £ ) ( l - T ? ) f ( l , l ) - ( l - £ ) 7 ? f ( l , N ) - £ ( 1 - T j ) f (M,1 )-$77f (M fN) (18) I t has been demonstrated that Eq. 18 can be used with great success to map a v a r i e t y of d i f f e r e n t i r r e g u l a r - s h a p e d domains with b o d y - f i t t e d o rthogonal c o o r d i n a t e s [26,27], B. BOUNDARY CONDITIONS OF THE BODY-FITTED ORTHOGONAL MAPPING The i m p o s i t i o n of the boundary c o n d i t i o n s i n b o d y - f i t t e d orthogonal mapping i s of primary importance. Two boundary c o n d i t i o n s , one f o r each C a r t e s i a n c o o r d i n a t e , are r e q u i r e d along each s e c t i o n of the transformed domain boundary. The p o s s i b l e boundary c o n d i t i o n s , which must be e i t h e r of the D i r i c h l e t or Neumann type, are r e p o r t e d below. D i r i c h l e t boundary c o n d i t i o n s : X=F(Y) (19) and 33 Y=F(X) (20) Neumann boundary c o n d i t i o n s : 3X 3Y 3£ 3TJ (21 ) and 3X 3Y 3rj 3 £ (22) In the weak c o n s t r a i n t method, two d i f f e r e n t c h o i c e s of boundary c o n d i t i o n s f o r each boundary s e c t i o n a r e p o s s i b l e . In one c a s e , the co r r e s p o n d e n c e between C a r t e s i a n and o r t h o g o n a l c o o r d i n a t e s i s c o m p l e t e l y s p e c i f i e d a l o n g the t r a n s f o r m e d domain boundary s e c t i o n w h i l e t h e shape f a c t o r s a l o n g the same boundary s e c t i o n r e q u i r e d f o r Eq. 18 a r e o b t a i n e d n u m e r i c a l l y from Eqs. 15 t o 17. T h i s case r e q u i r e s the s p e c i f i c a t i o n of two D i r i c h l e t c o n d i t i o n s . T h i s method of p r e s c r i b i n g boundary c o n d i t i o n s a l l o w s complete c o n t r o l of the s p a c i n g between £ or T? c o o r d i n a t e l i n e s a t a boundary s e c t i o n of the p h y s i c a l domain and a l s o f o r complete matching of £ or 77 c o o r d i n a t e s l i n e s a t the i n t e r f a c e of a d j o i n i n g p h y s i c a l domains. The o t h e r p o s s i b l e c h o i c e i s t o s p e c i f y the shape f a c t o r s a l o n g the t r a n s f o r m e d domain boundary s e c t i o n ( u s u a l l y by l i n e a r l y i n t e r p o l a t i n g the p r e - a s s i g n e d v a l u e s a t the two c o r n e r p o i n t s ) and a l l o w the mapping t o determine the corre s p o n d e n c e between C a r t e s i a n 34 and o r t h o g o n a l c o o r d i n a t e s . T h e r e f o r e , i n t h i s c a s e , i t i s the mapping which d e t e r m i n e s where the £ and 77 c o o r d i n a t e l i n e s i n t e r s e c t on the p h y s i c a l domain boundary f o r the s e c t i o n c o n s i d e r e d . T h i s second c h o i c e r e q u i r e s the s p e c i f i c a t i o n of one D i r i c h l e t and one Neumann c o n d i t i o n a t the boundary. The Neumann c o n d i t i o n (Eqs. 21 or 22) s i m p l y i n d i c a t e s t h a t the c o o r d i n a t e l i n e of £ or 77 must be o r t h o g o n a l t o the p h y s i c a l domain boundary. In p r a c t i c e , the best mapping r e s u l t s a r e u s u a l l y o b t a i n e d by r e q u i r i n g boundary correspondence o n l y a t one or a t most t h r e e boundary s e c t i o n s [ 2 7 ] . A t t e m p t i n g t o impose the corres p o n d e n c e between C a r t e s i a n and o r t h o g o n a l c o o r d i n a t e s on a l l f o u r boundary s e c t i o n s may l e a d t o a mapping i n which the o r t h o g o n a l i t y c o n d i t i o n i s not n e c e s s a r i l y r e s p e c t e d . Such mappings would be i l l - s u i t e d f o r n u m e r i c a l s o l u t i o n s . IV. CONSERVATION EQUATIONS The n a t u r a l c o n v e c t i o n problem i s , d e s c r i b e d by the mass, momentum and energy c o n s e r v a t i o n e q u a t i o n s . These e q u a t i o n s can be w r i t t e n i n terms of d i f f e r e n t s e t s of dependent v a r i a b l e s . One common c h o i c e i s t o use the v e l o c i t y components, the temperature and the p r e s s u r e as the dependent v a r i a b l e s . T h i s s e t i s known as the s e t of p r i m i t i v e v a r i a b l e s . An a l t e r n a t i v e s e t of dependent v a r i a b l e s c o n s i s t s of the stream f u n c t i o n , v o r t i c i t y and tem p e r a t u r e . An advantage of the second s e t over the f i r s t s e t , a t l e a s t i n t w o - d i m e n s i o n a l n a t u r a l c o n v e c t i o n problems, i s t h a t the f o u r c o n s e r v a t i o n e q u a t i o n s a s s o c i a t e d w i t h the p r i m i t i v e v a r i a b l e s a r e reduced t o t h r e e . T h i s advantage i s o f f s e t t o some e x t e n t by the f a c t t h a t t he v o r t i c i t y boundary c o n d i t i o n s a r e d i f f i c u l t t o s p e c i f y and the p r e s s u r e d i s t r i b u t i o n , which i s v e r y i m p o r t a n t i n f o r c e d c o n v e c t i o n , i s not o b t a i n e d e x p l i c i t l y . However, the l a t t e r i n f o r m a t i o n i s not u s u a l l y of d i r e c t i n t e r e s t i n n a t u r a l c o n v e c t i o n s i t u a t i o n s . The s i m p l i c i t y of the c o n s e r v a t i o n e q u a t i o n s w r i t t e n i n g e n e r a l t w o - d i m e n s i o n a l o r t h o g o n a l c o o r d i n a t e s a l s o has t o be taken i n t o account i n the c h o i c e of dependent v a r i a b l e s . Because the stream f u n c t i o n and the v o r t i c i t y a r e s c a l a r q u a n t i t i e s i n two d i m e n s i o n s whereas v e l o c i t y i s always a v e c t o r , t he c o n s e r v a t i o n e q u a t i o n s i n g e n e r a l c o o r d i n a t e s have l e s s complex f o r m u l a t i o n s when the stream f u n c t i o n , v o r t i c i t y and temperature v a r i a b l e s a r e used. 35 36 For the above r e a s o n s , the stream f u n c t i o n , v o r t i c i t y and temperature were adopted as dependent v a r i a b l e s i n the p r e s e n t s t u d y . A. ASSUMPTIONS The assumptions used i n the a n a l y s i s of l a m i n a r n a t u r a l c o n v e c t i o n o c c u r i n g i n an e n c l o s u r e a r e s t a t e d below: 1. Problem i s independent of t i m e . 2. Problem i s t w o - d i m e n s i o n a l . 3. F l u i d i s Newtonian. 4. F o u r i e r ' s law i s v a l i d . 5. B o u s s i n e s q a p p r o x i m a t i o n s a r e v a l i d : a. the d e n s i t y v a r i a t i o n s a r e c o n s i d e r e d o n l y i n s o f a r as they c o n t r i b u t e t o buoyancy, but ar e o t h e r w i s e n e g l e c t e d , and b. the d e n s i t y d i f f e r e n c e which causes the buoyancy i s a p p r o x i m a t e d as a pure temperature e f f e c t . 6. F l u i d p r o p e r t i e s o t h e r than d e n s i t y a r e c o n s t a n t . 7. The body f o r c e i s due o n l y t o the g r a v i t y . 8. V i s c o u s d i s s i p a t i o n i s n e g l i g i b l e . 9. C o m p r e s s i b l e work i s n e g l i g i b l e . 10. Energy i s not g e n e r a t e d . 37 B. CONSERVATION EQUATIONS IN CARTESIAN SYSTEM To o b t a i n the c o r r e c t f o r m u l a t i o n , of the r e l e v a n t c o n s e r v a t i o n e q u a t i o n s of the n a t u r a l c o n v e c t i o n problem i n a g e n e r a l o r t h o g o n a l system, the e q u a t i o n s a r e f i r s t d e r i v e d f o r the C a r t e s i a n system i n the p r e s e n t s e c t i o n and then m o d i f i e d f o r the g e n e r a l system i n the s e c t i o n which f o l l o w s . U s i n g the assumptions s t a t e d i n S e c t i o n A, the r e l e v a n t mass, momentum and energy c o n s e r v a t i o n e q u a t i o n s become, r e s p e c t i v e l y , — (v ) + — ( v )=0 3x x 3y y Po 3 ;3x v — ( v )+v — ( v ) x„. x y 3 y x 3 2 3 2 (v )+ (v ) 3x 2 X 3 y 2 x Po v — ( v )+v — ( v ) . x 3 x y y 3 y y + Mo = - ^ ( e d y n ) + » ° +g/3 0p 0(T-T 0) (23) (24) 3 2 3 2 (v ) + (v ) 3 x 2 y 3 y 2 y (25) PoCpO 3T 3T" "32T 32T" v — + v — = ko + _ x 3 x y3y_ 3x 2 3y 2_ (26) where the g r a v i t y v e c t o r d i r e c t i o n i s opposed t o the Y a x i s , as shown i n F i g s . 1 and 4. The n a t u r a l c o n v e c t i o n d r i v i n g mechanism, which i s r e p r e s e n t e d by the l a s t term on the r i g h t hand s i d e of Eq. 25, r e s u l t s from the i n t e r a c t i o n of 38 the h y d r o s t a t i c pressure g r a d i e n t and the body f o r c e term [30-33]. Thus, the p r e s s u r e i n Eqs,. 24 and 25 i s the dynamic pressure ( l o c a l p r e ssure minus the h y d r o s t a t i c p r e s s u r e ) . Dimensionless v a r i a b l e s and c h a r a c t e r i s t i c groups are i n t r o d u c e d to reduce the number of parameters that need to be s p e c i f i e d i n the numerical experiments. Let T-T 6 = c (27) x X = — (28) L c y Y=— (29) L c x = p 0 C p 0 L c v x / k 0 (30) V y = P 0 C p 0 L c v y / k 0 (31 ) P d y n ~ p d y n p ° C p ° (32) Ra=g/3 0PoL£(T h-T c)C p 0/Mok 0 (33) and Pr = M 0 C r .o /ko (34) where Ra represents the R a y l e i g h number, Pr r e p r e s e n t s the 39 P r a n d t l number and L c , the c h a r a c t e r i s t i c l e n g t h , i s e q u a l t o the l e n g t h of the c a v i t y bottom w a l l . U s i n g t h e s e d i m e n s i o n l e s s v a r i a b l e s and groups, the mass, momentum and energy e q u a t i o n s a r e t r a n s f o r m e d t o — ( V ) + — ( V )=0 3X x 3Y Y (35) a a v — ( v )+v — ( v ) = X 3 X X y3Y x a a v — ( v )+v — ( v ) = x a x y y 9Y y - — ( P , )+Pr ax d y n - — ( P , )+Pr 3Y dyn a 2 a 2 ( V x ) + ( V x ) a x 2 X 3Y 2 x 3 2 3 2 (V )+ (V ) 3X 2 Y 3Y 2 y (36) +RaPr( e - e 0 ) (37) and de be b2e b2e V — + V — = — - + (38) X 3 X y3Y 3X 2 3Y 2 At t h i s p o i n t , l e t us i n t r o d u c e the stream f u n c t i o n and v o r t i c i t y v a r i a b l e s . The stream f u n c t i o n i s d e f i n e d i n such a way as t o s a t i s f y the mass c o n s e r v a t i o n e q u a t i o n . Thus, the str e a m f u n c t i o n d e f i n i t i o n i s V = — (39) X 3Y and 40 3^ V = - — y ax (40) The v o r t i c i t y r epresents the amount of a n t i c l o c k w i s e r o t a t i o n that the f l u i d possesses. I t i s obtained mathematically as the c u r l of the v e l o c i t y v e c t o r , i . e . — (V ) — (V.) ax 3Y (41 ) The v o r t i c i t y i s a vector by d e f i n i t i o n . However, because the vector o r i e n t a t i o n i n a two-dimensional C a r t e s i a n s i t u a t i o n i s everywhere the same ( p e r p e n d i c u l a r to the two-dimensional C a r t e s i a n p l a n e ) , the v o r t i c i t y can be co n s i d e r e d as a s c a l a r v a r i a b l e . Thus, we can wr i t e that J 2 = — ( V )-—(v ) 3X y 3Y X (42) When the stream f u n c t i o n d e f i n i t i o n (Eqs. 39 and 40) i s in t r o d u c e d i n t o the above equation, the f i n a l v o r t i c i t y d e f i n i t i o n becomes + 3X 2 3Y 2 (43) The two momentum equations can be l i n k e d together to c a n c e l the pressure terms by s u b t r a c t i n g the X momentum equation d i f f e r e n t i a t e d with r e s p e c t to Y from the Y momentum equation d i f f e r e n t i a t e d with respect to X. Using the v o r t i c i t y d e f i n i t i o n (Eq. 42), the momentum equations 41 then reduce t o — ( V 0 ) + — ( V Q)=Pr 3X 3Y a 2n d2a' — + — 9X 2 3Y 2 de +RaPr- (44) 3X The v e l o c i t y components t h a t appear i n Eq. 44 (and a l s o i n the temperature Eq. 38) can be r e w r i t t e n i n term of stream f u n c t i o n but a r e r e t a i n e d because t h e i r p resence s i m p l i f i e s the d i s c r e t i z a t i o n p r o c e s s (see Chapter V ) . The s e t of s e m i t r a n s f o r m e d e q u a t i o n s (Eqs. 43, 44 and 38) a r e c a l l e d the stream f u n c t i o n , v o r t i c i t y and temperature e q u a t i o n s , r e s p e c t i v e l y . C. CONSERVATION EQUATIONS IN A GENERAL ORTHOGONAL SYSTEM The c o n s e r v a t i o n e q u a t i o n s w r i t t e n f o r a C a r t e s i a n system cannot be d i r e c t l y a p p l i e d t o a g e n e r a l t w o - d i m e n s i o n a l o r t h o g o n a l system because the domain i s l o c a l l y s t r e t c h e d or shrunk. T h i s f a c t must be t a k e n i n t o a ccount when d e r i v i n g a l t e r n a t e forms of the c o n s e r v a t i o n e q u a t i o n s f o r g e n e r a l o r t h o g o n a l c o o r d i n a t e s . To d e r i v e t h e s e a l t e r n a t e forms, the f o l l o w i n g two s t e p s were used: 1. the C a r t e s i a n c o n s e r v a t i o n e q u a t i o n s were f i r s t r e w r i t t e n i n forms which a r e independent of the c o o r d i n a t e system chosen by u s i n g the C a r t e s i a n g r a d i e n t , d i v e r g e n c e , c u r l and L a p l a c i a n o p e r a t o r s , and then 2. the same v e c t o r o p e r a t o r s d e f i n e d f o r a g e n e r a l 42 o r t h o g o n a l system were s i m p l y s u b s t i t u t e d i n t o the c o o r d i n a t e - f r e e c o n s e r v a t i o n e q u a t i o n s . The d e f i n i t i o n s of the g r a d i e n t , d i v e r g e n c e , c u r l and L a p l a c i a n o p e r a t o r s i n a g e n e r a l o r t h o g o n a l system can be found i n many s t a n d a r d t e x t b o o k s [41,45-47] and, i n o r d e r , a r e 1 30 1 30 V0=- -e +-e z 1 H ,3Z1 H _3Z2 z 1 z2 z2 V«a=-H z 1 H z 2 L ( H z 2 a z 1 ) + ( H z 1 a z 2 } 3Z1 Z ^ 2 1 3Z2 Z ^ Vxa=-H .H _ z 1 z2 z3 (45) (46) ( 4 7 ) and V 20=-H z 1 H z 2 3Z1 H z 2 3 0 H , 3Z1 • z 1 3Z2 H z 1 3 0 H z 2 3 Z 2 (48) The o p e r a t o r d e f i n i t i o n s f o r a C a r t e s i a n system can be o b t a i n e d from the above e q u a t i o n s by s u b s t i t u t i n g u n i t y s c a l e f a c t o r s . L e t us f i r s t r e w r i t e the mass c o n s e r v a t i o n e q u a t i o n i n a g e n e r a l o r t h o g o n a l system i n o r d e r t o d e r i v e the a p p r o p r i a t e d e f i n i t i o n of the stream f u n c t i o n . U s i n g the C a r t e s i a n and o r t h o g o n a l o p e r a t o r d e f i n i t i o n s , the mass c o n s e r v a t i o n Eq. 35 can be r e w r i t t e n i n the g e n e r a l o r t h o g o n a l system as 43 H j.H — (H V t ) + — (H.V ) =o (49) By d e f i n i t i o n , the stream f u n c t i o n must s a t i s f y the mass c o n s e r v a t i o n e q u a t i o n . T h e r e f o r e , i n the g e n e r a l o r t h o g o n a l system, the stream f u n c t i o n i s d e f i n e d by 90 H V F C=— (50) and 9\// H.V =-— (51 ) The v o r t i c i t y , which i s d e f i n e d as the c u r l of the v e l o c i t y v e c t o r , becomes i n the g e n e r a l o r t h o g o n a l system H,H — (H V ) — (H,V,) bi V 71 9T7 * * (52) Because the c o o r d i n a t e system i s t w o - d i m e n s i o n a l , the v o r t i c i t y can once a g a i n be c o n s i d e r e d as a s c a l a r v a r i a b l e f o r reasons s i m i l a r t o thos e d i s c u s s e d f o r the C a r t e s i a n system. T h e r e f o r e , Eq. 52 can be w r i t t e n as 1 0=-H,H — (H V ) — ( H T V T ) 9£ 7 1 7 1 97? * * (53) When the stream f u n c t i o n d e f i n i t i o n f o r an o r t h o g o n a l system i s i n t r o d u c e d i n t o the above e q u a t i o n , the v o r t i c i t y 4 4 d e f i n i t i o n becomes H„H 3£ H dip V 9 9rj H^9i//' H 9T? ( 5 4 ) T h i s e q u a t i o n i s the stream f u n c t i o n e q u a t i o n i n g e n e r a l o r t h o g o n a l c o o r d i n a t e s . To d e r i v e the a p p r o p r i a t e form of the v o r t i c i t y e q u a t i o n (Eq. 4 4 ) i n the g e n e r a l o r t h o g o n a l system, the C a r t e s i a n and g e n e r a l o r t h o g o n a l o p e r a t o r d e f i n i t i o n s a r e r e q u i r e d a l o n g w i t h the r e l a t i o n 9 0 9X H ,H 0 z 1 z 2 9 Y 9 0 9Y 9 0 9 Z 2 9 Z 1 9 Z 1 9 Z 2 ( 5 5 ) which can be found i n the l i t e r a t u r e [ 4 3 ] . In Eq. 5 5 , X i s a C a r t e s i a n c o o r d i n a t e and Z1 and Z 2 are g e n e r a l o r t h o g o n a l c o o r d i n a t e s . A f t e r some m a n i p u l a t i o n , the v o r t i c i t y e q u a t i o n i n general- o r t h o g o n a l c o o r d i n a t e s i s found t o be H,H — ( H v„n)+—(H,V fi) as 71 * 9T, ^ 7 7 Pr H „H RaPr 9? H an v 9T? H^9n' H 9TJ 7? , J H uH 3 Y 3 0 9 Y 3 0 97?aS 3S3»7 ( 5 6 ) S i m i l a r l y , the t e m p e r a t u r e e q u a t i o n (Eq. 3 8 ) , whose t r a n s f o r m a t i o n a l s o r e q u i r e s t h e use of the C a r t e s i a n and g e n e r a l o r t h o g o n a l o p e r a t o r s , becomes 45 H,H — (H V e 0 ) + — ( H t V 0) 3£ * * S r , ^ H„H 3* H 30 _2 LH^9£ 377 "H^90" H 3TJ L 1 " (57) The v e l o c i t y components and can be r e w r i t t e n i n terms of the stream f u n c t i o n but a r e a l l o w e d t o remain i n Eqs. 56 and 57 t o s i m p l i f y the d i s c r e t i z a t i o n p r o c e d u r e d e s c r i b e d i n Chapter V. In summary, the mass, momentum and energy c o n s e r v a t i o n e q u a t i o n s f o r a t w o - d i m e n s i o n a l o r t h o g o n a l system are g i v e n by Eqs. 54, 56 and 57, r e s p e c t i v e l y . D. BOUNDARY CONDITIONS OF THE CONSERVATION EQUATIONS The f o u r types of boundary c o n d i t i o n s needed t o complete the s p e c i f i c a t i o n of the n a t u r a l c o n v e c t i o n problem a r e : 1. the f o u r w a l l s of the c a v i t y a r e impermeable ( i . e . the normal component of the f l u i d v e l o c i t y a t the w a l l i s z e r o ) , 2. t h e t o p and bottom w a l l s of the c a v i t y a r e a d i a b a t i c , 3. t h e l e f t and r i g h t w a l l s of the c a v i t y a re i s o t h e r m a l , and 4. t h e n o - s l i p c o n d i t i o n a p p l i e s a t a l l f o u r bounding s u r f a c e s ( i . e . the t a n g e n t i a l component of f l u i d v e l o c i t y i s z e r o ) . 46 I t i s now ne c e s s a r y t o t r a n s l a t e t he above boundary c o n d i t i o n s , which are g i v e n i n terms of p r i m i t i v e v a r i a b l e s , t o t h e i r e q u i v a l e n t forms w r i t t e n f o r stream f u n c t i o n , v o r t i c i t y and temperature v a r i a b l e s . The f i r s t boundary c o n d i t i o n i s s a t i s f i e d by s p e c i f y i n g a c o n s t a n t stream f u n c t i o n v a l u e a l o n g the domain boundary. By c o n v e n t i o n , t h i s v a l u e i s a r b i t r a r i l y a s s i g n e d as z e r o . Thus, the f i r s t boundary c o n d i t i o n i s g i v e n by ^ d b=0 (58) The second boundary c o n d i t i o n s p e c i f i e s a n u l l heat f l u x t h r o u g h t the t o p and bottom w a l l s . T h i s can be w r i t t e n as de 877 =0 (59) 7J=1 and dd ST? =0 (60) 7? = N where 77=1 and 77=N c o r r e s p o n d t o the bottom and t o p w a l l s , r e s p e c t i v e l y . The i s o t h e r m a l w a l l boundary c o n d i t i o n s a r e g i v e n by a s s i g n i n g the d i m e n s i o n l e s s t e m p e r a t u r e a t the w a l l t o 47 e = 0 (61) *=1 and d = 1 (62) where $=1 and £=M c o r r e s p o n d t o the l e f t and r i g h t w a l l s , r e s p e c t i v e l y . The n o - s l i p boundary c o n d i t i o n i s s a t i s f i e d by s p e c i f y i n g a z e r o stream f u n c t i o n d e r i v a t i v e i n the d i r e c t i o n normal t o each w a l l . Because the f i r s t t h r e e boundary c o n d i t i o n s have a l r e a d y been used t o s p e c i f y the stream f u n c t i o n and temperature v a r i a t i o n s a l o n g the domain boundary, the f o u r t h c o n d i t i o n i s used t o d e r i v e the v o r t i c i t y boundary c o n d i t i o n . Many d i f f e r e n t r e p r e s e n t a t i o n s of the v o r t i c i t y boundary c o n d i t i o n which s a t i s f y the n o - s l i p c r i t e r i o n can be found i n the l i t e r a t u r e [12,38,40,43,44]. Each i s d e r i v e d somewhat d i f f e r e n t l y and each d e s c r i b e s the n o - s l i p c o n d i t i o n w i t h d i f f e r i n g degrees of a c c u r a c y . The Wood and the second o r d e r v o r t i c i t y boundary c o n d i t i o n s a r e c o n s i d e r e d i n t h i s work because they a r e e x p e c t e d t o be the most a c c u r a t e [ 1 2 ] . D e r i v a t i o n s of these two forms of the v o r t i c i t y boundary c o n d i t i o n can be found f o r the t w o - d i m e n s i o n a l C a r t e s i a n system [12,38,40,43,44] and the g e n e r a l t w o - d i m e n s i o n a l c o n f o r m a l o r t h o g o n a l system [ 2 4 ] . However, t h e s e v o r t i c i t y boundary c o n d i t i o n s have never been d e r i v e d f o r the case of 48 the g e n e r a l t w o - d i m e n s i o n a l non-conformal o r t h o g o n a l system used i n the p r e s e n t s t u d y . Such a d e r i v a t i o n i s p r e s e n t e d i n the c h a p t e r on d i s c r e t i z a t i o n which f o l l o w s . V. FINITE DIFFERENCE DISCRETIZATION S e t s of s i m u l t a n e o u s 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 l o n g w i t h t h e i r a t t e n d a n t boundary c o n d i t i o n s a r e v e r y d i f f i c u l t t o s o l v e a n a l y t i c a l l y even under the s i m p l e s t of c i r c u m s t a n c e s . Because of the n o n - l i n e a r i t y of the v o r t i c i t y and temperature e q u a t i o n s and because of the s t r o n g l i n k a g e between a l l t h r e e c o n s e r v a t i o n e q u a t i o n s , an a n a l y t i c a l s o l u t i o n t o the p r e s e n t s e t i s not p o s s i b l e . Thus i t becomes n e c e s s a r y t o r e s o r t t o app r o x i m a t e n u m e r i c a l t e c h n i q u e s . The most common n u m e r i c a l methods used f o r s o l v i n g s e t s of e l l i p t i c 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 f i n i t e d i f f e r e n c e , f i n i t e element and c o l l o c a t i o n methods. In the f i n i t e d i f f e r e n c e method chosen f o r the p r e s e n t s t u d y , the i d e a of o b t a i n i n g a c o n t i n u o u s s o l u t i o n over the whole of the p h y s i c a l domain i s abandoned f o r one of d e t e r m i n i n g approximate v a l u e s of the dependent v a r i a b l e s a t d i s c r e t e p o s i t i o n s w i t h i n and on the b o u n d a r i e s of the domain. These d i s c r e t e p o s i t i o n s a r e not randomly s c a t t e r e d , but a re s p e c i f i e d by, i n the p r e s e n t c a s e , an o r t h o g o n a l g r i d which c o v e r s the domain i n a f a i r l y homogeneous f a s h i o n . At each g r i d p o i n t or node, the d i f f e r e n t i a l e q u a t i o n i s s u b s t i t u t e d by a c o r r e s p o n d i n g a l g e b r a i c e q u a t i o n o b t a i n e d by somewhat c o m p r o m i s i n g l y r e p l a c i n g the e x a c t d i f f e r e n t i a l s by f i n i t e d i f f e r e n c e a n a l o g u e s . T h i s p r o c e d u r e l e a d s t o a s e t of s i m u l t a n e o u s a l g e b r a i c e q u a t i o n s i n which each n o d a l v a l u e i s l i n k e d t o t h o s e of i t s n e i g h b o u r i n g nodes. The s e t i s then s o l v e d u s i n g s t a n d a r d 4 9 50 s p a r s e m a t r i x t e c h n i q u e s . The f i n i t e d i f f e r e n c e form of any 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 can be d e r i v e d u s i n g e i t h e r T a y l o r s e r i e s e x p a n s i o n s or the c o n t r o l volume approach [ 3 9 ] . In the l a t t e r d i s c r e t i z a t i o n t e c h n i q u e , the 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 i s i n t e g r a t e d over the f i n i t e c o n t r o l volume which su r r o u n d s each n o d a l p o i n t . The c o n t r o l volume approach has s e v e r a l d i s t i n c t advantages over the T a y l o r s e r i e s method: 1. i t i s easy t o u n d e r s t a n d and l e n d s i t s e l f t o d i r e c t p h y s i c a l i n t e r p r e t a t i o n , 2. i t ensures t h a t the c o n s e r v a t i o n of each dependent v a r i a b l e i s m a i n t a i n e d over every c o n t r o l volume and hence g l o b a l l y over the whole domain, and 3. i t a l l o w s the use of f a r more r e a l i s t i c dependent v a r i a b l e p r o f i l e s between no d a l p o i n t s . In the p r e s e n t work, the c o n t r o l volume approach i s used t o d i s c r e t i z e a l l of the 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 . However, on a few o c c a s i o n s , T a y l o r s e r i e s e x p a n s i o n s a re used t o approximate f i r s t o r d e r d e r i v a t i v e s i n some boundary c o n d i t i o n s . E q u a t i o n s 63 t o 65 l i s t t h r e e f i r s t o r d e r d e r i v a t i v e a p p r o x i m a t i o n s of second o r d e r a c c u r a c y o b t a i n e d u s i n g s e r i e s e x p a n s i o n s . They a r e r e f e r r e d t o as the f o r w a r d , c e n t r a l and backward f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s , r e s p e c t i v e l y , and a r e w r i t t e n as 51 d0 -30 dZ1 Z1=Z1o -0 d0 d Z l +40 Z1=Z1o 2 1 =Z 1 0 + 2AZ1-0 Z1=Z1o+AZI / ( 2 A Z 1 ) Z1=Z1 0 +AZ1 "0 Z1=Z1 0 +2AZ1 Z1=Z1o (63) / ( 2 A Z 1 ) (64) and d0 dZ1 Z1=Z1Q+2AZ1 30 -40 Z1=Z1o+2AZ1 / ( 2 A Z 1 ) Z 1 = Z 1 0 J Z1= Z1o+AZ1 + 0 (65) A. DISCRETIZATION OF THE GRID GENERATION EQUATIONS The mapping e q u a t i o n s (Eqs. 13 and 14) must be t r a n s f o r m e d i n t o a s e t of a l g e b r a i c e q u a t i o n s i n o r d e r t o gen e r a t e the b o d y - f i t t e d o r t h o g o n a l g r i d . A g r i d which i s u n i f o r m i n the two g e n e r a l o r t h o g o n a l c o o r d i n a t e d i r e c t i o n s ( i . e . A £ = A T ?=1) i s used i n t o s p e c i f y the p o s i t i o n s of the d i s c r e t e v a l u e s i n the t r a n s f o r m e d domain. The g r i d i s r e p r e s e n t e d i n F i g . 8 by the s o l i d l i n e s ; t he dashed l i n e s d e l i m i t the c o n t r o l volumes a s s o c i a t e d w i t h each d i s c r e t e v a l u e . The d i s c r e t e v a l u e s i n t h i s case a r e the C a r t e s i a n c o o r d i n a t e s from which the s c a l e f a c t o r s , t he shape f a c t o r s and v a r i o u s f i r s t o r d e r d e r i v a t i v e s of C a r t e s i a n c o o r d i n a t e s w i t h r e s p e c t t o the o r t h o g o n a l c o o r d i n a t e s can be d e r i v e d . 1 1 i i i i i i T ~ t i i i 1 1 -|- i i i - ] -i i -|- -[-i i i i 1 1 i i i i -|- i i i i i l 1 1 i i i i i i i i r I 1 i i i i i i i i i i I ± i < i i i - i - • i i • i 1 i AS = AT?=1 -f- P o s i t i o n of X, Y, H,, H , f, ax 3S 3X BY 3Y — , — and — d i s c r e t e v a l u e s 3TJ 3S 3T? F i g u r e 8. U n i f o r m g r i d i n the t r a n s f o r m e d domain used t o d i s c r e t i z e the d i f f e r e n t i a l e q u a t i o n s of g o v e r n i n g the b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e t r a n s f o r m a t i o n . 53 The a l g e b r a i c form of the b o d y - f i t t e d o r t h o g o n a l mapping e q u a t i o n i n v o l v i n g X i s o b t a i n e d by i n t e g r a t i n g Eq. 13 over a g e n e r a l c o n t r o l volume ( F i g . 9 ) . Thus, a f t e r i n t e g r a t i n g , Eq. 13 becomes £ o Vo H,H 9 9£ 9X f — 9£ 9r? 1 9X' ton H,H d£dr>=0 (66) The product of s c a l e f a c t o r s appears i n Eq. 66 because the J a c o b i a n must be acco u n t e d f o r when i n t e g r a t i n g i n g e n e r a l o r t h o g o n a l c o o r d i n a t e s . E q u a t i o n 66 can be p a r t i a l l y i n t e g r a t e d t o g i v e 9X f — 9£ c 0 1 9X 9X - f — 9£ d r ? f 9r? 1 9X f 9T? d£ = 0 (67) At t h i s s t a g e , assumptions a r e r e q u i r e d i n o r d e r t o c o n t i n u e . L e t us assume t h a t the normal d e r i v a t i v e and the shape f a c t o r a t a c o n t r o l volume f a c e a r e c o n s t a n t over the e n t i r e f a c e and a r e e v a l u a t e d by assuming a l i n e a r p r o f i l e between the p a i r of n o d a l p o i n t s i n v o l v e d [ 3 9 ] . U s i n g t h e s e a s s u m p t i o n s , i t can be e a s i l y shown t h a t t h e above e q u a t i o n reduces t o the f o l l o w i n g a l g e b r a i c e q u a t i o n a X = G i + 1 X i + 1 + a i - 1 X i - 1 + a j + 1 X j + 1 + a j - 1 X j - 1 (68) where F i g u r e 9. G e n e r a l c o n t r o l v o l u m e f o r t h e C a r t e s i a n c o o r d i n a t e d i s c r e t e v a l u e s . 55 a=a. ( 6 9 ) a • i + 1 (70) a • i-1 = ( f+ f i _ 1 ) / 2 (71 ) j + 1 = 2 / ( f + f j + 1 ) (72) and D-1 = 2 / ( f + f j . , ) (73) In the above e q u a t i o n s , the m i s s i n g i n d i c e s a r e u n d e r s t o o d t o be i and j . T h i s nomenclature i s used thr o u g h o u t the r e s t of t h i s work. Because the mapping e q u a t i o n f o r Y (Eq. 14) i s s i m i l a r t o the one f o r X and because both X and Y share the same c o n t r o l volume, the d e r i v a t i o n of the a l g e b r a i c analogue of the mapping e q u a t i o n f o r Y can be deduced from Eq. 6 8 . Thus, the a l g e b r a i c e q u a t i o n which d e f i n e s the b o d y - f i t t e d o r t h o g o n a l mapping f o r Y i s g i v e n by where the c o e f f i c i e n t s a r e a g a i n g i v e n by Eqs. 6 9 t o 73. E q u a t i o n s 6 8 and 74 d e f i n e a s e t of a l g e b r a i c e q u a t i o n s f o r X and Y which a p p l y a t a l l of the i n t e r i o r nodes of the aY=a- (74) 56 t r a n s f o r m e d domain. Once t h i s s e t of e q u a t i o n s a l o n g w i t h the d i s c r e t i z e d forms of the boundary c o n d i t i o n s have been s o l v e d , the s c a l e f a c t o r s a r e o b t a i n e d by u s i n g second o r d e r a c c u r a t e f o r w a r d , backward or c e n t r a l d i f f e r e n c e s t o r e p r e s e n t the f i r s t d e r i v a t i v e s of the C a r t e s i a n c o o r d i n a t e s w i t h r e s p e c t t o the o r t h o g o n a l c o o r d i n a t e s which appear i n Eqs. 16 and 17. B. DISCRETIZATION OF THE GRID BOUNDARY CONDITIONS The a p p l i c a t i o n of the D i r i c h l e t boundary c o n d i t i o n s of the g r i d g e n e r a t i n g e q u a t i o n s i s s t r a i g h t - f o r w a r d ; the boundary nodes are s i m p l y a s s i g n e d t o the a p p r o p r i a t e v a l u e of X or Y. Two m o d i f i e d Neumann boundary c o n d i t i o n s (Eqs. 75 and 76) have been found t o g i v e b e t t e r r e s u l t s than the two o r i g i n a l Neumann boundary c o n d i t i o n s (Eqs. 21 and 2 2 ) . These new c o n d i t i o n s are d e r i v e d from the o l d ones and a r e g i v e n by 9X 9Y9Y — = - (75) 9TJ 3X97? and 9Y 3X3X — = (76) 3£ 9 Y 9 S These m o d i f i e d Neumann boundary n u m e r i c a l a c c u r a c y because d i f f e r e n c e a p p r o x i m a t i o n of o n l y c o n d i t i o n s y i e l d improved they r e q u i r e the f i n i t e two q u a n t i t i e s : the X and Y 57 d e r i v a t i v e s w i t h r e s p e c t t o e i t h e r £ or 77. The d e r i v a t i v e of X w i t h r e s p e c t t o Y or the d e r i v a t i v e of Y w i t h r e s p e c t t o X can be det e r m i n e d e x a c t l y from the a n a l y t i c a l f u n c t i o n s which d e f i n e the p h y s i c a l domain boundary. However, the o r i g i n a l Neumann boundary c o n d i t i o n s i n v o l v e the approximate e v a l u a t i o n of t h r e e q u a n t i t i e s : the X and Y d e r i v a t i v e s w i t h r e s p e c t t o e i t h e r £ or 77 and the shape f a c t o r (see Eqs. 15, 16 and 17). The a l g e b r a i c form of the m o d i f i e d Neumann c o n d i t i o n s a re o b t a i n e d by u s i n g f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s of second o r d e r . C. DISCRETIZATION OF THE NATURAL CONVECTION EQUATIONS In o r d e r t o s o l v e the n a t u r a l c o n v e c t i o n problem, the stream f u n c t i o n , the v o r t i c i t y and the temperature e q u a t i o n s a l s o have to be d i s c r e t i z e d . The s c a l e f a c t o r s , shape f a c t o r s and v a r i o u s C a r t e s i a n c o o r d i n a t e d e r i v a t i v e s r e q u i r e d i n these e q u a t i o n s must be a v a i l a b l e from the n u m e r i c a l s o l u t i o n of the mapping e q u a t i o n s . A s t a g g e r e d g r i d i s used t o s p e c i f y the p o s i t i o n s of the d i s c r e t e v a l u e s as shown i n F i g . 10. The i n t e r s e c t i o n s of the dashed l i n e g r i d d e termine the p o s i t i o n s of the v o r t i c i t y and tem p e r a t u r e nodes; t h e s o l i d l i n e g r i d d e l i m i t s the v o r t i c i t y and te m p e r a t u r e c o n t r o l volumes. The f i c t i t i o u s d i s c r e t e temperature and v o r t i c i t y nodes o u t s i d e the t r a n s f o r m e d domain a r e used t o s p e c i f y the boundary c o n d i t i o n s . The i n t e r s e c t i o n s of the s o l i d g r i d c o r r e s p o n d t o the stream f u n c t i o n nodes and i n d i c a t e the p o s i t i o n s i • i _•-'- • .'. -'- i i i i -'- -'- • i i -•- i j -i i i -i i -;- i i j - j -i • _ i _ i -!- - ] - -A £ = ATJ=1 - - P o s i t i o n of ft and 6 d i s c r e t e values - j - P o s i t i o n of X, Y, H ?, H , f, 9X 3Y 3Y — f — , — and ii d i s c r e t e 9TJ 3£ 3TJ values 3X 3£ F i g u r e 10. S t a g g e r e d g r i d s y s t e m u s e d t o d i s c r e t i z e t h e n a t u r a l c o n v e c t i o n c o n s e r v a t i o n e q u a t i o n s . 59 where the s c a l e f a c t o r s , shape f a c t o r s and C a r t e s i a n c o o r d i n a t e d e r i v a t i v e s a r e known frpm t h e converged n u m e r i c a l s o l u t i o n of the mapping e q u a t i o n s . The stream f u n c t i o n c o n t r o l volumes a r e d e l i m i t e d by the dashed g r i d . Both the s o l i d and dashed g r i d s a r e u n i f o r m l y spaced i n the two t r a n s f o r m e d c o o r d i n a t e d i r e c t i o n s ( i . e . A £ = A T J = 1 ) . o f c o u r s e , t h e r e i s no l o s s of g e n e r a l i t y by u s i n g a u n i f o r m g r i d i n the t r a n s f o r m e d c o o r d i n a t e system; the n o n - u n i f o r m i t i e s i n the p h y s i c a l domain are a c c o u n t e d f o r i n the t r a n s f o r m a t i o n . S e v e r a l advantages of s t a g g e r e d g r i d s of the type shown i n F i g . 10 over s t a n d a r d g r i d s i n which a l l t h r e e dependent v a r i a b l e s a r e l o c a t e d a t the same no d a l p o s i t i o n s have been c l a i m e d [ 1 2 , 3 9 ] . These advantages i n c l u d e : 1. the f l u i d f l o w a c r o s s any v o r t i c i t y or temperature c o n t r o l volume f a c e i s g i v e n e x a c t l y by the d i f f e r e n c e i n d i s c r e t e stream f u n c t i o n v a l u e s a t the c o r r e s p o n d i n g c o r n e r s , 2. the use of f i c t i t i o u s v o r t i c i t y and tem p e r a t u r e nodes o u t s i d e the s o l u t i o n domain a l l o w s more a c c u r a t e s p e c i f i c a t i o n of d e r i v a t i v e boundary c o n d i t i o n s ( a l l v o r t i c i t y and h a l f of the temperature boundary c o n d i t i o n s i n v o l v e normal d e r i v a t i v e s ; the stream f u n c t i o n i s c o n s t a n t a t the b o u n d a r i e s ) and 3. t h e r e a r e no h a l f or q u a r t e r c o n t r o l volumes f o r v o r t i c i t y or temperature a t the b o u n d a r i e s , 60 a c r o s s which both energy and momentum may be t r a n s f e r r e d . 1. STREAM FUNCTION EQUATION The a l g e b r a i c form of the stream f u n c t i o n e q u a t i o n (Eq. 54) can be deduced from the d i s c r e t i z e d v e r s i o n of the mapping e q u a t i o n f o r X because they both s h a r e the same c o n t r o l volume ( F i g . 11) and have s i m i l a r d i f f e r e n t i a l e q u a t i o n s . In f a c t , the stream f u n c t i o n d i f f e r e n t i a l e q u a t i o n has one a d d i t i o n a l term, the v o r t i c i t y , which a c t s as a sour c e term. I f i t i s assumed t h a t the v o r t i c i t y and the s c a l e f a c t o r s a r e c o n s t a n t over a stream f u n c t i o n c o n t r o l volume, t h i s s ource term can be ap p r o x i m a t e d by Thus, the d i s c r e t i z e d stream f u n c t i o n e q u a t i o n can be w r i t t e n as )/4 (77) o ^ a i + l* i + 1+oi_1*._l+aj + 1^ j + 1+aj_l*j_l+b (78) where the c o e f f i c i e n t s a r e once a g a i n g i v e n by Eqs. 69 t o 73 and the c o e f f i c i e n t b i s d e f i n e d by Eq. 77. F i g u r e 11. General c o n t r o l volume f o r d i s c r e t e stream f u n c t i o n v a l u e s . i-1,j+1 o i-1 , j k- AT? -6 i+1,j+1 i , j + l i + l , j + 1 1 r 3 • i + l , j I i A S i + 1 , j | ^ i - 1 , j - 1 i , j-1 i + 1 , j - 1 S 9X ax BY O P o s i t i o n of X, Y, H t, H , f, — , — , — , * 77 as ar? as 3Y — and t// d i s c r e t e v a l u e s 3r? • P o s i t i o n of and 8 d i s c r e t e v a l u e s 62 2. VORTICITY EQUATION F i r s t , t he v o r t i c i t y e q u a t i o n (Eq., 56) i s i n t e g r a t e d over the g e n e r a l v o r t i c i t y c o n t r o l volume shown i n F i g . 12 t o o b t a i n £ 0 ^o Pr H. H £ r? L RaPr — (H Vj.fi) +—(H tV Q) 9£ 77 * 3T? * 7 1 3r? H h 3fi" H 3T? • »? • H,.H 3Y30 3Y30 3TJ3£ 3 £ 3T? HfcH d£dT7 = 0 ( 7 9 ) A f t e r a p a r t i a l a n a l y t i c a l i n t e g r a t i o n , t he above e q u a t i o n becomes r j o + A r ? Vo £ 0 H an' (H V fl)-Pr-3— 77 5 H ?3£ H 90' (H„V fi)-Pr—S * 77 H 9 77 £ o + A £ T J O +A T ? H 3X2" (H V . f i ) - P r - 2 — 77 * H ?3£. Hj.90' (H>V fi)-Pr—s £ V So' d r ? H 3r? T?0-d£ J £ o J T 7 0 3Y30 9Y30' 9r?9£ 3£3TJ d£dr?=0 (80) At t h i s p o i n t , assumptions a r e r e q u i r e d i n o r d e r t o c o n t i n u e . Thus, the v o r t i c i t y and i t s normal d e r i v a t i v e a t a c o n t r o l volume f a c e a r e assumed t o be c o n s t a n t over the e n t i r e s u r f a c e . I t i s not recommended t h a t a l i n e a r p r o f i l e of the v o r t i c i t y between i t s d i s c r e t e n o d a l v a l u e s be used because i t l e a d s t o a d i f f e r e n c i n g scheme which i s o n l y 63 F i g u r e 12. G e n e r a l c o n t r o l volume f o r e i t h e r v o r t i c i t y or temperature d i s c r e t e v a l u e s . i - 1 , j + 1 i+1,j+1 • i-1 , j i - 1 , j - 1 • P o s i t i o n of fi and 6 d i s c r e t e v a l u e ; ax 3X 3Y 3Y O P o s i t i o n of X, Y, H,, H , f, — , — , — , aS 3T? 3£ 9TJ and \p d i s c r e t e v a l u e s 64 c o n d i t i o n a l l y s t a b l e [ 3 8 - 4 0 ] , Thus, t o d e t e r m i n e the v a l u e s of the v o r t i c i t y and i t s d e r i v a t i v e at. a c o n t r o l volume f a c e , an e x p o n e n t i a l scheme [39] i s employed. The e x p o n e n t i a l scheme uses as a p r o f i l e between two n e i g h b o u r i n g n o d a l p o i n t s the a n a l y t i c a l s o l u t i o n of a o n e - d i m e n s i o n a l pure c o n v e c t i o n - d i f f u s i o n problem. T h i s problem can be w r i t t e n i n a g e n e r a l form as H , H - 3 Z 1 z 1 z 2 ( H z 2 V z 1 * ) = H , H -3Z1 z 1 z 2 H z 2 3 0 H z l 3 Z 1 (81 ) s u b j e c t t o ZI =0 Z1 =L ( 8 2 ) ( 8 3 ) where <j> i s a g e n e r a l dependent v a r i a b l e and r i s a g e n e r a l d i f f u s i o n c o e f f i c i e n t . ^ I f the d i s t a n c e between the d i s c r e t e v a l u e s , L, i s s m a l l , then H 2 l , B 2 and V*z 1 can be taken o u t s i d e the d i f f e r e n t i a l s and a s s i g n e d v a l u e s of H , , 3 z 1 , a v e H ~ o a n d v ~ i ^ „ ~ r r e s p e c t i v e l y . Thus, the above e q u a t i o n z z,ave z i , a v e i s reduced t o z 2 , a v e z 1,ave 90 H ' 3 2 0 y ^ r z 2 , a v e 3Z1 H , 3 Z 1 2 z 1 , a v e ( 8 4 ) or 65 b<t> d2<f> a =b ( 8 5 ) az i a z 1 2 where a=H 0 V . ( 8 6 ) z 2,ave z 1,ave and H 0 b = r_z£ J Lave ( 8 ? ) z1,ave The p r o f i l e of the g e n e r a l dependent v a r i a b l e , 4>, i n the i n t e r v a l between Z1=0 and Z1=L i s o b t a i n e d by s o l v i n g the above d i f f e r e n t i a l e q u a t i o n w i t h i t s boundary c o n d i t i o n s (Eqs. 8 2 and 8 3 ) . T h i s p r o f i l e i s found t o be e x p ( P e Z l / L ) - 1 * = U 2 ~ 0 i ) + 0 i ( 8 8 ) e x p ( P e ) - 1 where aL Pe=— ( 8 9 ) b i s the l o c a l P e c l e t number which r e p r e s e n t s the r a t i o of the r e l a t i v e s t r e n g t h s of c o n v e c t i o n and d i f f u s i o n i n the d i r e c t i o n Z 1 . U s i n g Eq. 88 t o g e t h e r w i t h Eq. 8 9 , b o t h the v o r t i c i t y and i t s d e r i v a t i v e can now be e v a l u a t e d a t each c o n t r o l volume f a c e over which they a r e assumed t o be c o n s t a n t . 66 A l s o , t he s c a l e f a c t o r s a r e assumed t o be c o n s t a n t between the d i s c r e t e n odal v a l u e s of R and over t h e i r common c o n t r o l volume f a c e . The s c a l e f a c t o r s a r e e v a l u a t e d by assuming a l i n e a r p r o f i l e between t h e i r n o d a l v a l u e s . F i n a l l y , t he v o r t i c i t y s ource term ( t h e l a s t term on the r i g h t hand s i d e of Eq. 80) i s assumed t o be c o n s t a n t t h r o u g h o u t the v o r t i c i t y c o n t r o l volume and the v a l u e s of the d e r i v a t i v e of Y w i t h r e s p e c t t o r\ and the d e r i v a t i v e of X w i t h r e s p e c t t o £ are e v a l u a t e d as a m a t h e m a t i c a l a v e r a g e . U s i n g the above a s s u m p t i o n s , i t can be shown t h a t t he d i s c r e t i z e d form of Eq. 80 becomes a O ^ a ^ ^ . ^ + a . ^ O . ^ + a . ^ O . ^ + a . ^ O . ^ + b (90) where a = a i + 1 + a i - 1 + a j + 1 + a j - 1 ( 9 1 ) a. « ^ — (92) exp(Pe )-1 e a..= — 1 1 '3 1 2- (93) 1 exp(Pe )-1 w 1 1 (94) 3 + 1 exp(Pe )-1 a ( » i - 1 > i - 1 ^ j - 1 ) e x P ( P e s )  : _ 1 exp(Pe ) - l (95) 67 Pe = J—!-H H Pe Pe r H, H H s r H 0*f b=RaPr 30 be b , — - b 2 — 3£ br\ b,= bY 3Y 3rj 977 3Y • = i-1 9 7 7 3Y g = i - 1 ,77=3-1-1 /4 (96) (97) (98) (99) ( 1 0 0 ) ( 1 0 1 ) and b,= 3Y 3Y 3£ 3£ 3Y =1-1 3 « 3Y £ = 1-1 ,77=3-1-1 /4 ( 1 0 2 ) where T=Pr. A c e n t r a l f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n of second o r d e r i s used t o approximate t he temperature d e r i v a t i v e s which a r e p r e s e n t i n Eq. 1 0 0 . The advantages of the e x p o n e n t i a l d i f f e r e n c i n g scheme are t h a t i t more r e a l i s t i c a l l y a c c o u n t s f o r the competing r o l e s p l a y e d by c o n v e c t i o n and d i f f u s i o n i n d e t e r m i n i n g dependent v a r i a b l e p r o f i l e s between nodes and t h a t i t has been shown t o be u n c o n d i t i o n a l l y s t a b l e . F u r t h e r m o r e , as sh o u l d be ex p e c t e d , the e x p o n e n t i a l f o r m u l a reduces t o a c e n t r a l d i f f e r e n c e e q u a t i o n when Pe approach z e r o ( i . e . d i f f u s i o n dominates c o n v e c t i o n ) and an upwind or downwind 68 d i f f e r e n c e e q u a t i o n when Pe has e i t h e r a l a r g e p o s i t i v e or l a r g e n e g a t i v e v a l u e ( i . e . c o n v e c t i o n dominates d i f f u s i o n ) . 3. TEMPERATURE EQUATION Because the temperature e q u a t i o n (Eq. 57) has the same form as the v o r t i c i t y e q u a t i o n ( r e c o g n i z a b l e c o n v e c t i o n and d i f f u s i o n terms, but no sou r c e terms i n t h i s c ase) and because they share the same c o n t r o l volume, the d i s c r e t i z e d form of the temp e r a t u r e e q u a t i o n can be deduced from the a l g e b r a i c v o r t i c i t y e q u a t i o n . Thus, the f i n i t e d i f f e r e n c e v e r s i o n of the temp e r a t u r e e q u a t i o n i s g i v e n by °^ ai + i e i + 1 + a i - i e i - 1 + t t j + i e j + 1 + a j - i e j - t { 1 0 3 ) where the c o e f f i c i e n t d e f i n i t i o n s a r e g i v e n by Eqs. 91 t o 97 and r=1. D. DISCRETIZATION OF THE NATURAL CONVECTION BOUNDARY  CONDITIONS The D i r i c h l e t c o n d i t i o n s f o r the stream f u n c t i o n (Eq. 54) a r e s p e c i f i e d s i m p l y by a s s i g n i n g the v a l u e of z e r o to the boundary stream f u n c t i o n nodes. The a l g e b r a i c forms of the temperature boundary c o n d i t i o n s , Eqs. 59 t o 62 a r e o b t a i n e d by u s i n g s econd-order a c c u r a t e T a y l o r s e r i e s a p p r o x i m a t i o n s f o r the former two and m a t h e m a t i c a l a v e r a g e s f o r the l a t t e r two. 69 For the v o r t i c i t y boundary c o n d i t i o n s , two s e p a r a t e r e p r e s e n t a t i o n s , c a l l e d the Wood and second o r d e r e q u a t i o n s , w i l l be t e s t e d . However s i n c e the form of t h e s e e q u a t i o n s a p p r o p r i a t e t o g e n e r a l o r t h o g o n a l c o o r d i n a t e s does not p r e s e n t l y e x i s t , t h i s e x t e n s i o n must f i r s t be c a r r i e d o u t . The p o s i t i o n and n o t a t i o n f o r the v o r t i c i t y , s tream f u n c t i o n and s c a l e f a c t o r v a l u e s i n v o l v e d i n the s e d e r i v a t i o n s a r e shown i n F i g . 13. A l l of the v a r i a b l e s and t h e i r d i f f e r e n t i a l s a r e e v a l u a t e d a t the w a l l u n l e s s o t h e r w i s e s p e c i f i e d . The n o - s l i p boundary c o n d i t i o n and the impermeable w a l l c o n d i t i o n i m p l y , r e s p e c t i v e l y , t h a t 30 =0 (104) 3Zn 30 3 2 0 3 3 0 0= = = =...=0 (105) 3Zt 3 Z t 2 3 Z t 3 where Zn and Zt r e f e r t o the normal and t a n g e n t i a l g e n e r a l o r t h o g o n a l c o o r d i n a t e s , r e s p e c t i v e l y . Thus, i n the v i c i n i t y of the w a l l , the stream f u n c t i o n e q u a t i o n (Eq. 54) i s reduced t o H 3 20 — +(H H .fl)=0 (106) H 3Zn 2 z n z t zn Because a Neumann v o r t i c i t y boundary c o n d i t i o n was found t o g i v e more a c c u r a t e r e s u l t s than a D i r i c h l e t v o r t i c i t y boundary c o n d i t i o n [ 1 2 ] , l e t us d i f f e r e n t i a t e the above ft Zn - Q — w+ 1 AZn o w+ 1 w + 2 w + 2 -a— w + 3 O P o s i t i o n of fi d i s c r e t e v a l u e s • H z n , H z t and * F i g u r e 13. Staggered g r i d used t o d e r i v e the d i f f e r e n t e x p r e s s i o n s f o r the v o r t i c i t y boundary c o n d i t i o n . o 71 e q u a t i o n w i t h r e s p e c t t o the c o o r d i n a t e normal t o the w a l l t o o b t a i n 3 2 — 3Zn H 4. z t H z n J 3 20 H 3 30 3 •+-— +—(H H , ) o 3Zn 2 H 3Zn 3 3Zn z n z t zn 30 + ( H z n H z t > — = 0 ( 1 0 7 ) 3Zn To d e r i v e the Wood and second o r d e r v o r t i c i t y boundary c o n d i t i o n s from Eq. 107, the f o l l o w i n g T a y l o r s e r i e s e x p a n s i o n s a r e r e q u i r e d : 30 13 2 0 13 3 0 0 + 1 = ^ + AZn+ AZn 2+ AZn 3 w 3Zn 23Zn 2 63Zn 3 1 3V + AZn*+0(AZn 5 ) (108) 243Zn" 30 3 2 0 43 3 0 0 9=0+2 AZn + 2 AZn 2+ AZn 3 ^ 3Zn 3Zn 2 33Zn 3 23"0 + AZn'+0(AZn 5) (109) 33Zn" 130 13 20 O =0+ AZn+ AZn 2+0(AZn 3) (110) w 1 23Zn 83Zn 2 and 330 9 3 2 0 0 -=0+ AZn+ AZn 2+0(AZn 3) (111) w ^ 23Zn 83Zn 2 The Wood v o r t i c i t y boundary c o n d i t i o n i s o b t a i n e d by u s i n g one stream f u n c t i o n T a y l o r s e r i e s e x p a n s i o n (Eq. 108), 72 one v o r t i c i t y T a y l o r s e r i e s e x p a n s i o n (Eq. 110), the n o - s l i p boundary c o n d i t i o n (Eq. 104) and Eqs., 106 and 107. The n o - s l i p boundary c o n d i t i o n and Eqs. 106 and 107 a r e f i r s t s u b s t i t u t e d i n t o the stream f u n c t i o n T a y l o r s e r i e s e x p a n s i o n , Eq. 108, t r u n c a t e d a f t e r the f i r s t f o u r terms on the r i g h t hand s i d e . The v o r t i c i t y e x p a n s i o n , Eq. 110, t r u n c a t e d a f t e r the f i r s t two terms on the r i g h t hand s i d e , i s s u b s t i t u t e d i n t o t h i s m o d i f i e d stream f u n c t i o n e x p a n s i o n . The f i n a l r e s u l t of these m a n i p u l a t i o n s i s the e x p r e s s i o n f o r the Wood v o r t i c i t y boundary c o n d i t i o n f o r a g e n e r a l t w o - d i m e n s i o n a l o r t h o g o n a l c o o r d i n a t e system g i v e n below: 3fi 3Zn (112) where a, = — "in AZn • H 3 3 _ zn 12 12H .3Zn z t ( H z n H z t ) A Z n f l H 3 3 6H ,_3Zn z t H z t H. z n J AZn 1 (113) and a2-H 2 zn H AZn 2 _. zn 6H _3Zn z t (H H „)AZn 3 zn z t H 3 3 zn 3H t 3 Z n z t H z t H zn-AZn : (114) The d i s c r e t i z e d form of the Wood boundary c o n d i t i o n i s 73 e a s i l y g e n e r a t e d from Eq. 112 by u s i n g a c e n t r a l d i f f e r e n c e t o approximate the f i r s t d e r i v a t i v e term. The stream f u n c t i o n and s c a l e f a c t o r s needed i n Eqs. 112 t o 114 a r e o b t a i n e d by l i n e a r l y i n t e r p o l a t i n g t h e i r n o d a l v a l u e s . The second o r d e r v o r t i c i t y boundary c o n d i t i o n uses b o t h stream f u n c t i o n T a y l o r s e r i e s e x p a n s i o n s (Eqs. 108 and 1 0 9 ) , both v o r t i c i t y T a y l o r s e r i e s e x p a n s i o n s (Eqs. 110 and 1 1 1 ) , the n o - s l i p boundary c o n d i t i o n (Eq. 104) and Eqs. 106 and 107 . E q u a t i o n s 108 and 109 a r e f i r s t l i n k e d t o g e t h e r t o e l i m i n a t e the f o u r t h o r d e r term from the r e s u l t i n g stream f u n c t i o n s e r i e s e x p a n s i o n . The n o - s l i p boundary c o n d i t i o n and Eqs. 106 and 107 are then s u b s t i t u t e d i n t h i s new stream f u n c t i o n e x p a n s i o n . E q u a t i o n s 110 and 111 a r e a l s o l i n k e d t o g e t h e r t o e l i m i n a t e the second o r d e r term from the v o r t i c i t y s e r i e s e x p a n s i o n s . F i n a l l y , the v o r t i c i t y e x p a n s i o n i s i n t r o d u c e d i n t o the l a s t stream f u n c t i o n e x p a n s ion t o o b t a i n an e x p r e s s i o n f o r the second o r d e r v o r t i c i t y boundary c o n d i t i o n i n a g e n e r a l t w o - d i m e n s i o n a l o r t h o g o n a l c o o r d i n a t e system. T h i s e x p r e s s i o n i s found t o be a, = a 2 n w + 1 + a 3 f l w + 2 + 1 5 0 - 1 6 0 w + 1 + 0 w + 2 ( 1 1 5 ) 9 Z n where 74 a, =• 1 1 H ™ zn 1 2 H AZn 3 _ zn 2H ,.3Zn z t (H H .)AZn" zn z t H 3 3 zn H . 3Zn z t H z t H zn-AZn' a, = — 27H 2 zn 3H. AZn 2 zn 2H t 3 Z n z t (H H t ) A Z n 3 zn z t 3H 3 3 zn H z f c 3Zn H z t H z n J AZn : (116) ( 117) and H 3H 2 z n , . 2, zn a 3 = AZn ^  + 6H . 3Zn z t (H H .)AZn 3 zn z t zn 3H t3Zn z t H z t ^ Hzn. AZn : ( 118) As i n the case of the d i s c r e t i z e d Wood boundary c o n d i t i o n , the a l g e b r a i c form of the second o r d e r boundary c o n d i t i o n i s o b t a i n e d by u s i n g a second o r d e r a c c u r a t e f i n i t e d i f f e r e n c e t o r e p r e s e n t the d e r i v a t i v e , 3fi/3Zn, and l i n e a r l y i n t e r p o l a t i n g the d i s c r e t e stream f u n c t i o n and s c a l e f a c t o r s t o d etermine t h e i r m i d - p o i n t v a l u e s . I f s c a l e f a c t o r s of u n i t y a r e i n t r o d u c e d i n t o Eqs. 112 and 115 above, the Wood and second o r d e r v o r t i c i t y boundary c o n d i t i o n s f o r a C a r t e s i a n system a r e r e c o v e r e d . VI. COMPUTER PROGRAM The computer program i n c l u d i n g the i n i t i a l i s a t i o n p r o c e d u r e s , the t e c h n i q u e s f o r s o l v i n g the a l g e b r a i c f i n i t e d i f f e r e n c e e q u a t i o n s , the a c c u r a c y c r i t e r i a and the performance i n d i c a t o r s employed, and the methods f o r c a l c u l a t i n g l o c a l and average N u s s e l t numbers i s d e s c r i b e d i n t h i s s e c t i o n and l i s t e d i n Appendix A. The c o m p u t a t i o n a l p r o c e d u r e c o n s i s t s of two major p a r t s : the g e n e r a t i o n of the b o d y - f i t t e d o r t h o g o n a l g r i d and the s o l u t i o n of the c o r r e s p o n d i n g n a t u r a l c o n v e c t i o n problem. The g e n e r a l s t e p s of the program a r e l i s t e d i n Table 3. As can be seen from the T a b l e , the p r o c e d u r e c o n s i s t s of 13 s t e p s ; the g r i d g e n e r a t i o n i n v o l v e s s t e p s 3 to 6 w h i l e the n a t u r a l c o n v e c t i o n c a l c u l a t i o n i s c a r r i e d out i n s t e p s 10 t o 12. In the program, t h e t o p , r i g h t , bottom; and l e f t w a l l s of the t r a n s f o r m e d domain c o r r e s p o n d t o the t o p , r i g h t , bottom and l e f t w a l l s of the p h y s i c a l domain, r e s p e c t i v e l y . The a n a l y t i c f u n c t i o n s (as w e l l as the d e r i v a t i v e s needed f o r the m o d i f i e d Neumann boundary c o n d i t i o n s ) which d e f i n e the b o u n d a r i e s of the p h y s i c a l domain, i . e . Y=F(X) or X=F(Y), a r e i n t r o d u c e d i n t o the program t h r o u g h a f u n c t i o n s u b r o u t i n e . The g r i d boundary c o n d i t i o n s , two D i r i c h l e t or one D i r i c h l e t and one Neumann, a r e s p e c i f i e d i n a c o n s i s t e n t manner a t a l l g r i d p o i n t s a l o n g each w a l l of the t r a n s f o r m e d domain. In the program, the d e s i r e d g r i d boundary c o n d i t i o n s a t each w a l l of the t r a n s f o r m e d domain a r e s p e c i f i e d by 75 T a b l e 3. Program a I g o r i t h m . 1) Set up the a n a l y t i c a l f u n c t i o n s and t h e i r d e r i v a t i v e s for each p h y s i c a l domain boundary. Provide the program with the g r i d boundary c o n d i t i o n f l a g values, the shape fa c t o r s at the transformed domain corn e r s , the v o r t i c i t y boundary c o n d i t i o n f l a g value, the g r i d s i z e , the accuracy c r i t e r i a , the allowed maximum number of i t e r a t i o n s and the relaxat ion f a c t o r s . 2) I n i t i a l i z e X, Y, 6, 4> and fi at t h e i r appropriate boundary and i n t e r n a l g r i d p o i n t s . 3) Update X and Y values using Eqs. 68 and 74 (where the c o e f f i c i e n t d e f i n i t i o n s i n both equations are given by Eqs. 69 to 73) and the appropriate boundary c o n d i t i o n s (among Eqs. 19, 20, 75 or 76). One i t e r a t i o n c o n s i s t s of a complete sweep of rows followed by a complete sweep of colurns using-the l i n e - b y - l i n e procedure. A l i n e of X i s followed immediately by the same l i n e of 7 to ensure the s o l u t i o n i s as simultaneous as p o s s i b l e . 4) Update the shape f a c t o r s using Eqs. IS to 17 on walls ( i n c l u d i n g the corner values) where two D i r i c h l e t c o n d i t i o n s are imposed. On walls where one D i r i c h l e t and one Neumann c o n d i t i o n are s p e c i f i e d , the shape f a c t o r s are evaluated by l i n e a r l y i n t e r p o l a t i n g the corner values. The shape f a c t o r s i n s i d e the transformed domain are c a l c u l a t e d by means of Eq. 18. 5) If the maximum absolute d i f f e r e n c e of X or Y i s larger than 2, stop the e x e c u t i o n . Otherwise, go to the next step. 6) If the convergence c r i t e r i a are s a t i s f i e d or i f the maximum number of i t e r a t i o n s for the g r i d generation part i s exceeded, go on to the next step. Otherwise, r e t u r n to step 3. ax ax ay BY 7) C a l c u l a t e — , — , — and — at a l l g r i d d{ dn 3£ di) points in the transformed domain. 8) C a l c u l a t e the g r i d performance i n d i c a t o r s using Eqs. 119 to 12 1. 9) C a l c u l a t e the s c a l e f a c t o r s and the shape f a c t o r s at a l l g r i d p o s i t i o n s in the transformed domain using Eqs. 15 to 17. 10) Update the Q and B d i s c r e t e values using Eqs. 78 (where the c o e f f i c i e n t d e f i n i t i o n s are given by Eqs. 69 to 73 and 77), 90 (where the c o e f f i c i e n t d e f i n i t i o n s are given by Eqs. 91 to 102) and 103 (where the c o e f f i c i e n t d e f i n i t i o n s are given by Eqs. 90 to 99). The boundary c o n d i t i o n s of these dependent v a r i a b l e s are s p e c i f i e d by Eqs. 58; e i t h e r 112 (where the c o e f f i c i e n t s are given by Eqs. 113 and 114) 115 (where the c o e f f i c i e n t are given by Eqs. 116 to 116); and 59 to 62. Again one i t e r a t i o n r e q u i r e s a complete sweep of rows and then columns. Since the g r i d i s staggered, the temperature and the v o r t i c i t y for any given l i n e i s updated before moving on to the adjacent l i n e of stream f u n c t i o n values. 11) If the maximum v o r t i c i t y value i s l a r g e r than 10 1 0 , stop the e x e c u t i o n . Otherwise, go to the next step. 12) If the convergence c r i t e r i o n i s s a t i s f i e d f o r a l l 6, ^ and 0 values, or i f the maximum number of i t e r a t i o n s for the n a t u r a l convection part i s exceeded, go to the next step. Otherwise, r e t u r n to step 10. 13) Find the a b s o l u t e maximum stream fu n c t i o n value and c a l c u l a t e the l o c a l and average Nusselt numbers along the l e f t and r i g h t w a l l s of the enclosure using the Eqs. 124 and 125. 77 a s s i g n i n g a v a l u e of 0 or 1 t o a f l a g v a r i a b l e . On w a l l s where two D i r i c h l e t boundary c o n d i t i o n s a r e used, the i n i t i a l v a l u e s of the C a r t e s i a n c o o r d i n a t e s a t each n o d a l p o i n t remain unchanged d u r i n g the c o m p u t a t i o n . On w a l l s where one D i r i c h l e t and one Neumann boundary c o n d i t i o n s a r e employed, the shape f a c t o r s a r e o b t a i n e d by l i n e a r l y i n t e r p o l a t i n g the c o r n e r v a l u e s ( o t h e r shape f a c t o r p r o f i l e s c o u l d have been u s e d ) . The shape f a c t o r s a t the c o r n e r s of the t r a n s f o r m e d domain a r e a s s i g n e d s p e c i f i c v a l u e s a t the b e g i n n i n g of the program. T h i s c o r n e r shape f a c t o r v a l u e remains c o n s t a n t i f one D i r i c h l e t and one Neumann c o n d i t i o n are s p e c i f i e d on b o t h w a l l s which meet a t t h a t c o r n e r ; o t h e r w i s e , the c o r n e r v a l u e must be r e c a l c u l a t e d n u m e r i c a l l y u s i n g Eg. 15 w i t h second o r d e r f i n i t e d i f f e r e n c e s t o r e p r e s e n t the d e r i v a t i v e s i n Eqs. 16 and 17. Thus, i t i s th r o u g h the c h o i c e of the g r i d boundary c o n d i t i o n s , the i n i t i a l i s a t i o n of the C a r t e s i a n c o o r d i n a t e s and the i n i t i a l i s a t i o n of the shape f a c t o r s a t the c o r n e r p o i n t s t h a t the g r i d c h a r a c t e r i s t i c s a r e s p e c i f i e d . The i n i t i a l c o r r e s p o n d e n c e between C a r t e s i a n and o r t h o g o n a l c o o r d i n a t e s a t each boundary node of the t r a n s f o r m e d domain i s g i v e n i n Ta b l e 4 f o r a MxN g r i d . Note t h a t when two D i r i c h l e t c o n d i t i o n s a r e s p e c i f i e d f o r any w a l l , T a b l e 4 assumes t h a t the nodes a r e t o be u n i f o r m l y spaced i n the d i r e c t i o n of the dependent C a r t e s i a n c o o r d i n a t e of the a n a l y t i c f u n c t i o n ( o t h e r t y p e s of boundary p o i n t c o r r e s p o n d e n c e c o u l d have been p r e s c r i b e d ) . Table 4. I n i t i a l correspondence between C a r t e s i a n and b o d y - f i t t e d orthogonal c o o r d i n a t e s at the domain boundaries. Wall P h y s i c a l Domain New Domain Cav i t y C1 C a v i t y C2 Top T( i-1)(1+2A) , 1 J J " i - i " , 1 (i,N) .L M " 1 M-1 Right F1(Y), N-1 F2(Y), N-1 - -i (M, j ) Bottom " i - i M-1 ,0 "i-1" M- 1 _ ( i , 1 ): L e f t • j - f " o, — N-1_ • j - i y 0, N- lJ_ 0 , j ) 79 Because an i t e r a t i v e method w i l l be used t o s o l v e the f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n of the g r i d g e n e r a t i o n and n a t u r a l c o n v e c t i o n e q u a t i o n s , i t becomes n e c e s s a r y t o s p e c i f y i n i t i a l v a l u e s f o r a l l dependent v a r i a b l e s (X, Y, 8, 4/, and fl) a t the b e g i n n i n g of the program. The i n i t i a l v a l u e s of the C a r t e s i a n c o o r d i n a t e s , X and Y, a t a l l i n t e r i o r nodes of the t r a n s f o r m e d g r i d a r e a s s i g n e d by l i n e a r l y i n t e r p o l a t i n g the boundary v a l u e s . I n i t i a l t e mperature v a l u e s a r e n o r m a l l y o b t a i n e d by a l i n e a r i n t e r p o l a t i o n between the i s o t h e r m a l w a l l s of the e n c l o s u r e s . S i n c e the stream f u n c t i o n and v o r t i c i t y v a l u e s a r e d i f f i c u l t t o s p e c i f y b e f o r e h a n d and, i n f a c t , can be e i t h e r p o s i t i v e or n e g a t i v e , t h e i r i n i t i a l v a l u e s were n o r m a l l y a s s i g n e d t o z e r o . P r o v i s i o n i s a l s o made t o use a converged s o l u t i o n of a s i m i l a r problem (eg. a t a s l i g h t l y d i f f e r e n t R a y l e i g h number) t o o b t a i n s t a r t i n g 8, 0 and 0 v a l u e s f o r a new problem. Of c o u r s e , the i n i t i a l v a l u e s have no e f f e c t on the f i n a l c onverged s o l u t i o n o t h e r than t o modify the number of i t e r a t i o n s r e q u i r e d t o r e a c h the f i n a l answer. D u r i n g each s t e p of the i t e r a t i v e p r o c e s s , the C a r t e s i a n c o o r d i n a t e s ( d u r i n g g r i d g e n e r a t i o n ) and the t e m p e r a t u r e , the stream f u n c t i o n and the v o r t i c i t y ( d u r i n g the n a t u r a l c o n v e c t i o n c a c u l a t i o n s ) are updated u s i n g a l i n e - b y - l i n e s o l u t i o n p r o c e d u r e a l o n g w i t h s u c c e s s i v e r e l a x a t i o n . In the l i n e - b y — l i n e method, the l i n e a r a l g e b r a i c e q u a t i o n s f o r a complete l i n e of new dependent v a r i a b l e 80 v a l u e s a r e w r i t t e n i n i m p l i c i t form. T h i s p r o c e d u r e l e a d s t o a t r i d i a g o n a l m a t r i x which can then be s o l y e d u s i n g the v e r y e f f i c i e n t Thomas a l g o r i t h m [ 3 8 - 4 0 ] , The l i n e - b y - l i n e i m p l i c i t method has much b e t t e r convergence p r o p e r t i e s than e x p l i c i t methods l i k e t he G a u s s - S e i d e l t e c h n i q u e because i n f o r m a t i o n from the b o u n d a r i e s i s t r a n s p o r t e d i n a s i n g l e s t e p throughout the domain. To ensure the r a p i d d i f f u s i o n of a l l boundary i n f o r m a t i o n , the l i n e - b y - l i n e method proceeds by rows i n one sweep through the g r i d and then by columns i n the n e x t . Convergence can be f u r t h e r a c c e l e r a t e d by a p p l y i n g s u c c e s s i v e r e l a x a t i o n t o each s e t of s o l u t i o n v a l u e s . O v e r - r e l a x a t i o n i s used f o r the two C a r t e s i a n c o o r d i n a t e s , the temperature and the stream f u n c t i o n . I t was found t h a t u n d e r - r e l a x a t i o n was n e c e s s a r y i n o r d e r t o ensure the convergence of the v o r t i c i t y . Many c r i t e r i a can be used t o s p e c i f y a d e s i r e d l e v e l of convergence f o r a n u m e r i c a l s o l u t i o n . These c r i t e r i a can be based on e i t h e r a b s o l u t e or r e l a t i v e e r r o r s . I n t h i s work, the a b s o l u t e e r r o r of a dependent v a r i a b l e i s d e f i n e d as the maximum a b s o l u t e d i f f e r e n c e between the d i s c r e t e v a l u e s o b t a i n e d i n the l a t e s t i t e r a t i o n and those o b t a i n e d i n the p r e v i o u s one. The r e l a t i v e e r r o r i s then g i v e n by the r a t i o of the a b s o l u t e e r r o r and the l a r g e s t a b s o l u t e v a l u e of t h a t dependent v a r i a b l e i n the l a t e s t i t e r a t i o n . The n u m e r i c a l p r o c e d u r e f o r g e n e r a t i n g the b o d y - f i t t e d o r t h o g o n a l g r i d i s stopped o n l y when both C a r t e s i a n c o o r d i n a t e s s a t i s f y an a b s o l u t e e r r o r c r i t e r i a . The n u m e r i c a l p r o c e d u r e used t o 81 s o l v e the n a t u r a l c o n v e c t i o n problem i s c o n s i d e r e d t o be converged when the t e m p e r a t u r e , stream f u n c t i o n and v o r t i c i t y d i s c r e t e v a l u e s meet a p r e s c r i b e d r e l a t i v e e r r o r c r i t e r i o n . In most of the n u m e r i c a l e x p e r i m e n t s , b o t h the a b s o l u t e and r e l a t i v e e r r o r s were r e q u i r e d t o be l e s s than 0.00001. The maximum a b s o l u t e d i f f e r e n c e of X and Y a t the end of each g r i d g e n e r a t i o n i t e r a t i o n and the maximum a b s o l u t e v o r t i c i t y v a l u e a t the end of each n a t u r a l c o n v e c t i o n i t e r a t i o n were used t o d e t e c t a d i v e r g e n t s o l u t i o n . I f the maximum d i f f e r e n c e of X or Y were l a r g e r than 2 or the maximum a b s o l u t e v o r t i c i t y v a l u e was g r e a t e r than 1 0 1 0 , the s o l u t i o n was assumed t o be d i v e r g e n t and the pr o c e d u r e was stopped. A l s o , upper l i m i t s on the number of i t e r a t i o n s f o r the g r i d g e n e r a t i o n and the n a t u r a l c o n v e c t i o n c a l c u l a t i o n p a r t s p r e v e n t e d the consumption of e x c e s s i v e CPU t i m e . Once each p a i r of g r i d g e n e r a t i o n and n a t u r a l c o n v e c t i o n problems had s u c c e s s f u l l y c o nverged, the d i s c r e t e r e s u l t s were used t o c a l c u l a t e the f o l l o w i n g i n f o r m a t i o n : 1. the l e n g t h of the c u r v e d r i g h t w a l l , 2. the average and maximum d e v i a t i o n s of the n u m e r i c a l l y g e n e r a t e d g r i d from o r t h o g o n a l i t y , 3. the maximum stream f u n c t i o n v a l u e , and 4. the l o c a l and average N u s s e l t numbers of both the l e f t and r i g h t i s o t h e r m a l w a l l s . The l e n g t h of the r i g h t w a l l , which i s the o n l y c u r v e d boundary of both the C1 and C2 c a v i t i e s , i s c a l c u l a t e d by 82 means of the i n t e g r a l L =/? H rw J 1 7j (119) The i n t e g r a t i o n i n Eq. 119 i s a p p r o x i m a t e d n u m e r i c a l l y by u s i n g the t r a p e z o i d r u l e . The l e n g t h of the r i g h t w a l l i s used as an i n d i c a t o r of the performance of the n u m e r i c a l g r i d g e n e r a t i o n r o u t i n e . The l e n g t h , i n g e n e r a l , i s a f u n c t i o n of the g r i d s i z e . As the number of g r i d p o i n t s i n c r e a s e s , the c a l c u l a t e d l e n g t h c o nverges a s y m p t o t i c a l l y t o the e x a c t v a l u e which can be d e t e r m i n e d by an a n a l y t i c a l i n t e g r a t i o n of the boundary e q u a t i o n . The d e v i a t i o n of the n u m e r i c a l l y g e n e r a t e d g r i d from o r t h o g o n a l i t y i s a n o t h e r i n d i c a t o r of the performance of the program. The d e v i a t i o n from o r t h o g o n a l i t y i s a l s o a f u n c t i o n of the g r i d s i z e . I t s h o u l d a pproach z e r o as the number of g r i d p o i n t s i n c r e a s e s . At each n o d a l p o i n t , the a n g l e , 0, a t which the g r i d l i n e s i n t e r s e c t can be d e t e r m i n e d from COS(0)=-3 X 3 X 3 Y 3 Y + 3 £ 3 T J 9^977 (120) " 3 X " 2 " 3 Y ~ 2" 0 • 5 " 3 X " 2 " 3 Y " 2" + + + .3*. 377 917 To e v a l u a t e 0, the v a r i o u s d e r i v a t i v e s a p p e a r i n g i n Eq. 120 were appro x i m a t e d by second o r d e r a c c u r a t e f o r w a r d , backward and c e n t r a l f i n i t e d i f f e r e n c e s . F i n a l l y , the d e v i a t i o n from o r t h o g o n a l i t y , Dev, was c a l c u l a t e d as 83 Dev=|9O°-0| (121) The maximum stream f u n c t i o n was ta k e n t o be the maximum a b s o l u t e v a l u e of a l l of the d i s c r e t e stream f u n c t i o n v a l u e s on the g r i d . The maximum stream f u n c t i o n v a l u e i s an im p o r t a n t q u a n t i t y because i t i s a measure of the t o t a l c i r c u l a t o r y f l o w i n the e n c l o s u r e . I f the s i g n a s s o c i a t e d w i t h the maximum v a l u e i s p o s i t i v e , the f l o w i s a n t i - c l o c k w i s e ; o t h e r w i s e , the f l o w i s c l o c k w i s e . P r o b a b l y the most i m p o r t a n t d e r i v e d q u a n t i t y i n any c o n v e c t i v e heat t r a n s f e r s i t u a t i o n i s the heat t r a n s f e r c o e f f i c i e n t a t the f l u i d - s o l i d i n t e r f a c e . T h i s q u a n t i t y i s e s s e n t i a l i n the d e s i g n of any u n i t o p e r a t i o n i n which c o n v e c t i o n heat t r a n s f e r o c c u r s . The l o c a l heat f l u x a t a s o l i d - f l u i d i n t e r f a c e i s g i v e n i n g e n e r a l o r t h o g o n a l c o o r d i n a t e s by k 0 9T q =- (122) H L 3Zn zn c For d e s i g n p u r p o s e s , the l o c a l heat f l u x a t the w a l l can a l s o be w r i t t e n as the p r o d u c t of the c h a r a c t e r i s t i c t e m perature d i f f e r e n c e of the system and a c o e f f i c i e n t c a l l e d t he c o n v e c t i o n heat t r a n s f e r c o e f f i c i e n t , i . e . q =h(T.-T ) (123) ^zn n c The d i m e n s i o n l e s s c o n v e c t i o n heat t r a n s f e r c o e f f i c i e n t , 84 c a l l e d the N u s s e l t number, can now be d e r i v e d by combining Eqs. 122 and 123 t o o b t a i n hL 1 9 0 Nu=—-=- (124) k 0 H 9Zn  zn The l o c a l N u s s e l t number i s c a l c u l a t e d n u m e r i c a l l y from the above e q u a t i o n by u s i n g a second o r d e r a c c u r a t e c e n t r a l d i f f e r e n c e t o approximate the temperature d e r i v a t i v e and by l i n e a r l y i n t e r p o l a t i n g the d i s c r e t e s c a l e f a c t o r v a l u e s . The average N u s s e l t number a l o n g a w a l l i s then g i v e n by ; w a l l N u H z t d Z t N u a = - ^ ^ - (125) ' w a l l H z t d 2 t The average N u s s e l t number i s c a l c u l a t e d from Eq. 125 u s i n g the t r a p e z o i d r u l e i n t e g r a t i o n t e c h n i q u e . V I I . TEST PROBLEMS U s i n g r e s u l t s a v a i l a b l e from the t l i t e r a t u r e , two d i f f e r e n t t ypes of t e s t problems were c o n s i d e r e d i n o r d e r t o ensure t h a t the s i m u l a t i o n program was r u n n i n g p r o p e r l y . The f i r s t problem was d e s i g n e d t o t e s t the program's a b i l i t y t o s u c c e s f u l l y g e n e r a t e b o d y - f i t t e d o r t h o g o n a l g r i d s . The second t e s t case examined the program's a b i l i t y t o s o l v e a s t a n d a r d n a t u r a l c o n v e c t i o n problem. A. FIRST TEST C h i k h l i w a l a and Y o r t s o s [27] e x t e n s i v e l y i n v e s t i g a t e d the a b i l i t y of the weak c o n s t r a i n t method t o g e n e r a t e b o d y - f i t t e d o r t h o g o n a l g r i d s i n t w o - d i m e n s i o n a l domains which they suggest a r e e n c o u n t e r e d d u r i n g i m m i s c i b l e f l u i d d i s p l a c e m e n t i n porous media. The g e n e r a l shape of the domain i s i d e n t i c a l t o the c a v i t y C1 w i t h the o n l y d i f f e r e n c e b e i n g t h a t the bottom w a l l l e n g t h i s not B but 0.75B (see F i g . 4 ) . T h e r e f o r e , the f u n c t i o n r e p r e s e n t i n g the r i g h t w a l l i s X=0.75+A C O S ( T T Y ) (126) where the a m p l i t u d e , A, was a l l o w e d t o v a r y from 0.0 ( r e c t a n g u l a r c a v i t y ) t o 0.25. Two of the t h r e e d i f f e r e n t s e t s of boundary c o n d i t i o n s c o n s i d e r e d by C h i k h l i w a l a and Y o r t s o s a r e a l s o i n v e s t i g a t e d i n t h i s work. In case 1, the cor r e s p o n d e n c e between C a r t e s i a n and g e n e r a l o r t h o g o n a l 85 86 c o o r d i n a t e s i s p r e s c r i b e d a t the r i g h t w a l l w h i l e the shape f a c t o r s of the top l e f t and bottom l e f t c o r n e r s a r e a s s i g n e d to u n i t y ( i . e . the shape f a c t o r s a l o n g the l e f t w a l l a r e a l l u n i t y , t h o s e a l o n g the r i g h t w a l l a r e c a l c u l a t e d u s i n g Eqs. 15 t o 17 and tho s e a l o n g the t o p and bottom w a l l s a r e l i n e a r l y i n t e r p o l a t e d from t h e i r r e s p e c t i v e c o r n e r s v a l u e s ) . In case 2, boundary c o r r e s p o n d e n c e between the C a r t e s i a n and g e n e r a l o r t h o g o n a l c o o r d i n a t e s i s p r e s c r i b e d a t the the l e f t , r i g h t and bottom w a l l s (the shape f a c t o r s a r e c a l c u l a t e d f o r the l e f t , r i g h t and bottom w a l l s u s i n g Eqs. 15 t o 17 and a r e det e r m i n e d f o r the t o p w a l l by l i n e a r i n t e r p o l a t i n g i t s c o r n e r v a l u e s ) . As a s t a n d a r d example, the a u t h o r s chose a g r i d h a v i n g 18 rows and 18 columns. U n f o r t u n a t e l y , they g i v e o n l y v e r y c u r s o r y i n f o r m a t i o n about t h e i r d i s c r e t i z a t i o n methods. These two ca s e s were e n t e r e d i n t o the p r e s e n t program. An 18x18 g r i d was employed and the a m p l i t u d e was a l l o w e d t o range from 0.05 t o 0.25 f o r b o t h c a s e s . The maximum d e v i a t i o n s from o r t h o g o n a l i t y computed f o r the i n t e r n a l (non-boundary) g r i d p o i n t s a r e compared t o the r e s u l t s of r e f e r e n c e [27] i n T a b l e s 5 (case 1) and 6 (case 2 ) . I t i s of i n t e r e s t t o note t h a t the g r i d s g e n e r a t e d by the p r e s e n t program a r e more o r t h o g o n a l than those o b t a i n e d by C h i k h l i w a l a and Y o r t s o s . The improved r e s u l t s o b t a i n e d by the p r e s e n t program a r e p r o b a b l y a t t r i b u t a b l e t o the use of second o r d e r a c c u r a t e f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s t h r o u g h o u t , more s t r i n g e n t convergence c r i t e r i a and the 87 Table 5. Comparison of the maximum d e v i a t i o n s ( i n degrees) obtained i n the present work and by C h i k h l i w a l a and Yortsos [27] - case 1. Amplitude Present Work Reference [27] 0.05 0. 1 06 0.300 0.10 0.221 1 .000 0.15 0.352 2.000 0.20 0.809 3.600 0.25 1.510 5.700 Table 6. Comparison of the maximum d e v i a t i o n s ( i n degrees) obtained i n the present work and by C h i k h l i w a l a and Yortsos [27] - case 2. Amplitude Present Work Reference [27] 0.05 0.251 1 .300 0.10 0.743 2.400 0.15 1 .780 3. 700 0.20 3.310 5.000 0.25 6.050 6. 500 88 m o d i f i e d Neumann boundary c o n d i t i o n s (Eqs. 75 and 7 6 ) . T a b l e s 5 and 6 a l s o r e v e a l t h a t the more phe boundary nodes a r e c o n s t r a i n e d , i . e . the l a r g e r the number of bounding s u r f a c e s where complete boundary c o r r e s p o n d e n c e i s r e q u i r e d , the l e s s o r t h o g o n a l i s the g e n e r a t e d g r i d . In f a c t , C h i k h l i w a l a and Y o r t s o s found t h a t , i n some c a s e s , when the boundary nodes were i n i t i a l l y p r e s c r i b e d over the e n t i r e p e r i m e t e r of the domain, i t was e i t h e r i m p o s s i b l e t o o b t a i n a converged s e t of i n t e r i o r g r i d p o i n t s or the o r t h o g o n a l i t y c o n d i t i o n was not s a t i s f i e d . A p p a r e n t l y , the p h y s i c a l g r i d can be o v e r c o n s t r a i n e d a t the b o u n d a r i e s t o the p o i n t where a r e a l i s t i c s o l u t i o n i s no l o n g e r a c h i e v a b l e . However, t h i s f a i l u r e t o o b t a i n complete boundary c o r r e s p o n d e n c e s h o u l d not be c o n s i d e r e d as a f a t a l d e f i c i e n c y of the weak c o n s t r a i n t method. In most c a s e s , i n c l u d i n g t h o s e c o n s i d e r e d i n the p r e s e n t t h e s i s , i t i s o n l y e s s e n t i a l t o have boundary c o r r e s p o n d e n c e at one s u r f a c e where two c o n t i g u o u s r e g i o n s j o i n (see Chapter X ) . The c o n c e n t r a t i o n of g r i d p o i n t s can be v a r i e d by a d j u s t i n g the s c a l e f a c t o r v a l u e s a t the c o r n e r s formed by the o t h e r t h r e e s i d e s . B. SECOND TEST The n a t u r a l c o n v e c t i o n p o r t i o n of the program was t e s t e d by c o n s i d e r i n g the case of a square c a v i t y w i t h v e r t i c a l i s o t h e r m a l w a l l s . As was mentioned e a r l i e r , t h i s problem has been e x t e n s i v e l y i n v e s t i g a t e d i n the l i t e r a t u r e t o the p o i n t where i t i s now c o n s i d e r e d t o be a s t a n d a r d 89 problem f o r comparing d i f f e r e n t n u m e r i c a l heat t r a n s f e r methods. De V a h l D a v i s and Jones [15] have summarized t h e r e s u l t s of a comparison e x e r c i s e f o r the square c a v i t y problem which i n c l u d e d the s o l u t i o n s of 36 i n t e r n a t i o n a l c o n t r i b u t o r s . As w e l l , De V a h l D a v i s and Jones gave a benchmark s o l u t i o n which they c o n s i d e r t o have a r e l a t i v e a c c u r a c y of b e t t e r than 1 p e r c e n t . U s i n g the p r e s e n t program, n u m e r i c a l s i m u l a t i o n s were c a r r i e d out f o r the ca s e s of R a y l e i g h number e q u a l t o 1000, 10000 and 100000. For each c a s e , the P r a n d t l number was s e t e q u a l t o 0.71 ( a i r ) and a u n i f o r m C a r t e s i a n g r i d c o n s i s t i n g of 42 rows and 42 columns was used. Contour p l o t s of the stream f u n c t i o n and temperature d i s t r i b u t i o n s as w e l l as the l o c a l N u s s e l t number v a r i a t i o n s a l o n g the i s o t h e r m a l w a l l s a r e shown on F i g s . 14 t o 16 f o r the t h r e e R a y l e i g h numbers s t u d i e d . The maximum stream f u n c t i o n v a l u e and the average c a v i t y N u s s e l t number f o r each case a r e compared t o the benchmark v a l u e s of De V a h l D a v i s and Jones i n T a b l e s 7 and 8. An e x a m i n a t i o n of T a b l e s 7 and 8 demonstrate t h a t the r e s u l t s of the p r e s e n t program a r e i n e x c e l l e n t agreement w i t h the be s t s o l u t i o n s g i v e n by De V a h l D a v i s and Jo n e s . The l a r g e s t d i s c r e p a n c y was 0.72 p e r c e n t and was a s s o c i a t e d w i t h the maximum stream f u n c t i o n a t Ra=100000. The d i f f e r e n c e s o b s e r v e d i n T a b l e s 7 and 8 a r e s m a l l compared t o the t o t a l range of d e v i a t i o n s l i s t e d i n r e f e r e n c e [ 1 5 ] . Thus, i t can be c o n c l u d e d t h a t , a t l e a s t f o r a v e r t i c a l 90 F i g u r e 14. S e l e c t e d n a t u r a l c o n v e c t i o n r e s u l t s f o r a v e r t i c a l square c a v i t y with Ra=l000 and Pr=0.7l. (a) Stream f u n c t i o n contour p l o t , (b) Temperature contour p l o t (isotherms range from 0 to 1 i n increments of 0.1), (c) L e f t w a l l N u s s e l t number d i s t r i b u t i o n , (d) Right w a l l Nusselt number d i s t r i b u t i o n . AVERAGE NUSSELT NUMBED (RIGHT WALLI-AVERAGE NUSSELT NUMBER (LEFT WALL )* D. 111733SE*01 .1117107E*01 STREAM FUNCTION CONTOUR VALUES MIN" 0.0 MAX. O. M72788E«01 CONTOUR # 1 2 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR O.11737B8E.OO 0.2345577E«CO 0.35I836£E«00 0.4691154£«00 O.S8G3S43E.OO 0.703G731E.OO 0.82O9520E4O0 0.93S2309E.00 O.105S5<OE*Ol a) b) c) flVCRflKr 0.0 0.2 o.i o.6 o.e i.o DISTANCE (TOP TO BOTTOM I d) o.o 0.2 0.1 o.s o.e DISTflNCC (TOP TO BOTTOMI 91 F i g u r e 15. S e l e c t e d n a t u r a l c o n v e c t i o n r e s u l t s f o r a v e r t i c a l square c a v i t y w i t h Ra=l0000 and Pr,= 0 . 7 l . (a) Stream f u n c t i o n c o n t o u r p l o t , (b) Temperature c o n t o u r p l o t ( i s o t h e r m s range from 0 t o 1 i n i n c r e m e n t s of 0.1), (c) L e f t w a l l N u s s e l t number d i s t r i b u t i o n , (d) R i g h t w a l l N u s s e l t number d i s t r i b u t i o n . AVERAGE NUSSELT NUMBER (RIGHT HALL}• O.224I303E-0I AVERAGE NUSSELT NUMBER (LEFT WALL V* 0.274151 IE«01 STREAM FUNCTION CONTOUR VALUES. MIN- O.O MAX- 0.S07OSS6E.O1 CONTOUR * ( CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR , O.S070SB6E*OO O. 1014 (17E*01 0..152 1 I76E*01 0.202S233E.OI 0.2635292E.OI 0.304235IE.01 0.3549409E.O! 0.4O564EBE<0l 0.456352GE«01 a) b) c) • BVCRflGT o.i o.s o.e i.o DISTANCE (TOP TO BOTTOMI d) 0-1 0.6 O.B I 01STRNCC (TOP TO BOTTOM I 92 F i g u r e 16. S e l e c t e d n a t u r a l c o n v e c t i o n r e s u l t s f o r a v e r t i c a l square c a v i t y w i t h Ra=l00000 and Pr=0.71. (a) Stream f u n c t i o n c o n t o u r p l o t , (b) Temperature c o n t o u r p l o t ( i s o t h e r m s range from 0 t o 1 i n in c r e m e n t s of 0.1), (c) L e f t w a l l N u s s e l t number d i s t r i b u t i o n , (d) R i g h t w a l l N u s s e l t number d i s t r i b u t i o n . AVERAGE NUSSELT NUMBER (RIGHT WALLI-AVERAGE NUSSELT NUMBER (LEFT WALL ) a STREAM FUNCTION MIN- 0.0 MAX• 0.9691461 CONTOUR 0 t CONTOUR 0 2 CONTOUR 0 3 CONTOUR 0 4 CONTOUR I S CONTOUR 0 £ CONTOUR 0 7 CONTOUR 0 S CONTOUR 0 9 CONTOUR VALUES. O.96914GOE*O0 0.<938292E«01 O.29O7436E«0l 0.387esa3E-*OI 0.4845728E.OI 0.5BI487CE.O1 0.6784020E«0t 0.7753IG7E.01 O.B7223t4E*OI O.4S17t44E*01 0.45173I2E+01 a) b) 93 T a b l e 7. Comparison of the maximum stream f u n c t i o n s o b t a i n e d i n the p r e s e n t work w i t h the benchmark r e s u l t s of De V a h l D a v i s and Jones [ 1 5 ] . R a y l e i g h Number P r e s e n t Work R e f e r e n c e [15] 1 000 1 . 1 72 1.174 1 0000 5.070 5.079 100000 9.691 9.622 T a b l e 8. Comparison of the average N u s s e l t numbers o b t a i n e d i n the p r e s e n t work w i t h the benchmark r e s u l t s of De V a h l D a v i s and Jones [ 1 5 ] . R a y l e i g h Number P r e s e n t Work R e f e r e n c e [15] 1 000 1.117 1.118 1 0000 2.241 2.238 100000 4.517 4.505 94 square c a v i t y , the p r e s e n t program b e t t e r than s u c c e s s f u l l y s o l v e s s i m u l t a n e o u s mass, momentum and energy b a l a n c e s . The stream f u n c t i o n and t emperature c o n t o u r p l o t s shown i n F i g s . 14 t o 16 i l l u s t r a t e the regimes t y p i c a l of n a t u r a l c o n v e c t i o n heat t r a n s f e r i n e n c l o s u r e s . For low R a y l e i g h numbers l i k e 1000, the b u o y a n c y - d r i v e n f l o w i s v e r y weak and hence the c o n t r i b u t i o n of c o n v e c t i o n t o the o v e r a l l heat t r a n s f e r i s not v e r y s i g n i f i c a n t . T h i s f a c t i s c l e a r l y r e v e a l e d i n the t emperature p l o t where the i s o t h e r m s d e v i a t e o n l y s l i g h t l y from v e r t i c a l i t y ; the s t a t e t h a t would e x i s t i f c o n v e c t i o n were a b s e n t . In the l a t t e r c a s e , the e n c l o s u r e problem reduces t o one of u n i - d i r e c t i o n a l c o n d u c t i o n f o r which case i t i s p o s s i b l e t o prove t h a t the N u s s e l t number s h o u l d be e x a c t l y u n i t y . Thus, a t Ra=l000, the c a l c u l a t e d N u s s e l t number i n d i c a t e s t h a t n a t u r a l c o n v e c t i o n o n l y i n c r e a s e s the o v e r a l l heat t r a n s f e r r a t e by about 12 p e r c e n t . As the R a y l e i g h number i s i n c r e a s e d ( f o r a g i v e n f l u i d , t h i s can be a c h i e v e d by i n c r e a s i n g the c h a r a c t e r i s t i c t e m perature d i f f e r e n c e or the c h a r a c t e r i s t i c l e n g t h of the c a v i t y ) , the f l o w becomes s t r o n g e r and c o n v e c t i o n makes a more i m p o r t a n t c o n t r i b u t i o n t o the o v e r a l l t r a n s f e r of h e a t . At a R a y l e i g h number of 100000, c o n v e c t i o n c o m p l e t e l y dominates c o n d u c t i o n as a heat t r a n s f e r mechanism and the f l o w i s s u f f i c i e n t l y s t r o n g t h a t i t b e g i n s t o t a k e on some of the a t t r i b u t e s of a boundary l a y e r . T h i s i s p a r t i c u l a r l y n o t i c e a b l e i n the t e m p e r a t u r e p l o t where the g r a d i e n t s a r e 95 v e r y s t e e p i n the v i c i n i t y of the i s o t h e r m a l w a l l s and e s s e n t i a l l y z e r o a t the c e n t r e of the e n c l o s u r e . For t h i s c a s e , i t can be o b s e r v e d t h a t secondary f l o w s b e g i n t o d e v e l o p i n the c o r e r e s u l t i n g i n the f o r m a t i o n of the c h a r a c t e r i s t i c " c a t ' s eye" p a t t e r n . VI11 . PART I A. NUMERICAL EXPERIMENTS The n a t u r a l c o n v e c t i o n problem t r e a t e d here i s t o t a l l y d e f i n e d by the s p e c i f i c a t i o n of f i v e independent parameters: the R a y l e i g h number, the P r a n d t l number, the d i m e n s i o n l e s s a m p l i t u d e ( r e f e r r e d t o as the a m p l i t u d e from t h i s p o i n t onwards), the c a v i t y a s p e c t r a t i o and the c a v i t y a n g l e of t i l t . The independent v a r i a b l e v a l u e s i n v e s t i g a t e d a r e p r e s e n t e d i n Table 1. For each c a v i t y type (C1 or C2), the p e r m u t a t i o n of these independent v a r i a b l e v a l u e s d e f i n e s 30 d i f f e r e n t n a t u r a l c o n v e c t i o n problems. Each n a t u r a l c o n v e c t i o n problem i s s o l v e d n u m e r i c a l l y by s a t i s f y i n g the d i f f e r e n c e forms of the b o d y - f i t t e d o r t h o g o n a l mapping, stream f u n c t i o n , v o r t i c i t y and t e m p e r a t u r e e q u a t i o n s a l o n g w i t h t h e i r v a r i o u s boundary c o n d i t i o n s . S i x d i f f e r e n t s e t s of boundary c o n d i t i o n s a r e c o n s i d e r e d i n t h i s work and t h e s e are p r e s e n t e d i n T a b l e s 9 and 10. A l s o , 7 g r i d s i z e s a r e examined and are l i s t e d i n T a b l e 11. The p e r m u t a t i o n of the boundary c o n d i t i o n s and g r i d s i z e s d e f i n e s an a d d i t i o n a l 42 ways t h a t each n a t u r a l c o n v e c t i o n problem c o u l d be s o l v e d . The p r i m a r y o b j e c t i v e of P a r t I i s t o determine the e f f e c t of the g r i d s i z e and the s e t of boundary c o n d i t i o n s used on the a c c u r a c y of the n u m e r i c a l s o l u t i o n s . A l t h o u g h each n a t u r a l c o n v e c t i o n problem r e p r e s e n t s a p a r t i c u l a r c a s e , 60x42 n u m e r i c a l s i m u l a t i o n s were not attempted i n o r d e r t o f u l f i l t h i s f i r s t o b j e c t i v e ; r a t h e r o n l y those 96 97 T a b l e 9. D i f f e r e n t g r i d , stream f u n c t i o n , v o r t i c i t y and te m p e r a t u r e boundary c o n d i t i o n s i n v e s t i g a t e d i n P a r t I . G r i d Stream F u n c t i o n Temperature V o r t i c i t y Case A Case B Case C Eq. 58 Eqs. 59 t o 62 Wood: Eqs. 112 t o 114 Second O r d e r : Eqs. 115 t o 118 T a b l e 10. G r i d boundary c o n d i t i o n s i n v e s t i g a t e d i n P a r t I ; D = D i r i c h l e t , N=Neumann. Case Top W a l l R i g h t W a l l Bottom W a l l L e f t W a l l X Y X Y X Y X Y A N D D N N D D N B N D D ; D . N D D N C D D D D D D D N G r i d S i z e M N 7 7 9 9 1 3 1 3 1 7 1 7 25 25 33 33 49 49 T a b l e 11. G r i d s i z e s i n v e s t i g a t e d i n P a r t I . 98 c a s e s e x p e c t e d t o y i e l d the g r e a t e s t n u m e r i c a l d i f f i c u l t y were i n v e s t i g a t e d i n d e t a i l . Thus, f i v e n a t u r a l c o n v e c t i o n problems were s e l e c t e d f o r t h i s purpose and a r e p r e s e n t e d i n T a b l e 12. These f i v e c a s e s use o n l y the most d i s t o r t e d c a v i t i e s . A R a y l e i g h number of 10000 was chosen i n f o u r of the c a s e s p a r t l y because the heat t r a n s f e r a t t h i s c o n d i t i o n e x h i b i t s both c o n d u c t i o n and c o n v e c t i o n modes, and a l s o because the c o m p u t a t i o n a l c o s t of t a c k l i n g a l a r g e r R a y l e i g h number was p r o h i b i t i v e . However, one case of Ra=100000 was i n c l u d e d because i t p r o v i d e s a more r i g o r o u s t e s t of the a c c u r a c y of the n u m e r i c a l s o l u t i o n p r o c e d u r e . To m i n i m i z e CPU t i m e , the i n i t i a l s o l u t i o n v a l u e s f o r stream f u n c t i o n , v o r t i c i t y and temperature f o r each f i n e r g r i d were o b t a i n e d by l i n e a r l y i n t e r p o l a t i n g the converged answer f o r the p r e v i o u s c o a r s e r g r i d . In each c a s e , once convergence had been a t t a i n e d , p l o t s were made of the g e n e r a t e d g r i d s , the stream f u n c t i o n , v o r t i c i t y and temperature d i s t r i b u t i o n s as w e l l as the l o c a l N u s s e l t number d i s t r i b u t i o n s on the l e f t and r i g h t w a l l s . A l s o , the l e n g t h of the r i g h t w a l l , the maximum d e v i a t i o n of the g r i d from o r t h o g o n a l i t y , the maximum stream f u n c t i o n and the average N u s s e l t numbers of the l e f t and r i g h t w a l l s were r e c o r d e d i n o r d e r t o a s s e s s the a c c u r a c y of the s o l u t i o n s . F i n a l l y , a heat b a l a n c e a t the domain boundary was performed f o r each n u m e r i c a l e x p e r i m e n t . Because the t o p and bottom w a l l s a re a d i a b a t i c , a l l of the heat energy which e n t e r s the c a v i t y from the hot r i g h t w a l l must be removed from the c a v i t y a t the c o l d l e f t T a b l e 12. D i m e n s i o n l e s s a m p l i t u d e s and R a y l e i g h numbers i n v e s t i g a t e d i n P a r t I . Case C a v i t y D i m e n s i o n l e s s A m p l i tude R a y l e i g h Number 1 CI -0.15 1 0000 2 C1 0.15 1 0000 3 C2 -0.15 1 0000 4 C2 0.15 1 0000 5 CI -0.15 100000 100 w a l l i f s t e a d y - s t a t e c o n d i t i o n s a r e t o p r e v a i l . T h i s c o n d i t i o n can be w r i t t e n d i m e n s i o n a l l y as Qlw =Qrw ( 1 2 7 ) The above e q u a t i o n can be r e w r i t t e n i n terms of the average heat t r a n s f e r c o e f f i c i e n t s of each w a l l , i . e . h i L L, ( T , - T )=h L L ( T . - T ) ( 128 ) ave,lw c lw h c ave,rw c rw v h c assuming the c a v i t y has a depth of u n i t y . I f Eq. 128 i s d i v i d e d t h r o u g h by k 0, i t reduces t o Nu , L, =Nu L ( 129 ) ave,lw lw ave,rw rw where i n the p r e s e n t i n v e s t i g a t i o n , L ^ w i s u n i t y . Thus, as an a d d i t i o n a l measure of s o l u t i o n a c c u r a c y , the r a t i o of the d i f f e r e n c e between the l e f t and r i g h t hand s i d e s of Eq. 129 t o t h e i r mean v a l u e was c a l c u l a t e d . B. RESULTS There i s not s u f f i c i e n t space or need t o show a l l of P a r t I r e s u l t s . Some r e p r e s e n t a t i v e examples f o r a 3 3 x33 g r i d a r e g i v e n i n F i g s . 17 t o 21 f o r c a s e s 1 t o 5, r e s p e c t i v e l y . Each f i g u r e i n c l u d e s t h r e e n u m e r i c a l l y g e n e r a t e d g r i d s c o r r e s p o n d i n g t o the t h r e e s e t s of boundary c o n d i t i o n s ( i . e . no boundary c o r r e s p o n d e n c e , boundary F i g u r e 17. S e l e c t e d r e s u l t s o b t a i n e d f o r case 1 of T a b l e 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n c o n t o u r s , (e) Temperature c o n t o u r s , ( f ) and (g) R i g h t and l e f t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y . c) d) e) F i g u r e 18. S e l e c t e d r e s u l t s o b t a i n e d f o r case 2 of T a b l e 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n c o n t o u r s , (e) Temperature c o n t o u r s , ( f ) and (g) R i g h t and l e f t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y . 0-i 0.6 0.6 1.0 DISTANCE (TOP TO BOTTOM I " .0 0.2 0.1 o.c O.B 1.0 1.2 OISTRWCC (TOP TO BOTTOM! F i g u r e 19. S e l e c t e d r e s u l t s o b t a i n e d f o r case 3 of T a b l e 12. (a) and (b) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A and B, r e s p e c t i v e l y ( g r i d boundary c o n d i t i o n s C d i d not y i e l d a converged r e s u l t ) , (d) Stream f u n c t i o n c o n t o u r s , (e) Temperature c o n t o u r s , ( f ) and (g) R i g h t and l e f t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y . F i g u r e 20. S e l e c t e d r e s u l t s o b t a i n e d f o r case 4 of Table 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d with boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n contours, (e) Temperature contours, ( f ) and (g) Right and l e f t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y . 0.< 0.6 0.8 1.0 OI5TfiN(X (TOP 10 UOT70M! F i g u r e 21. S e l e c t e d r e s u l t s o b t a i n e d f o r case 5 of T a b l e 12. ( a ) , (b) and (c) 33x33 g r i d s o b t a i n e d w i t h boundary c o n d i t i o n s A, B and C, r e s p e c t i v e l y , (d) Stream f u n c t i o n c o n t o u r s , (e) Temperature c o n t o u r s , ( f ) and (g) R i g h t and l e f t w a l l N u s s e l t number d i s t r i b u t i o n s , r e s p e c t i v e l y . 110 111 c o r r e s p o n d e n c e on r i g h t w a l l and boundary c o r r e s p o n d e n c e on t o p , r i g h t and bottom w a l l s ) l i s t e d i n Tabl,e 10. A l s o shown f o r purposes of d i s c u s s i o n a r e c o n t o u r p l o t s of stream f u n c t i o n and temperature as w e l l as the l o c a l N u s s e l t number d i s t r i b u t i o n s a l o n g the two i s o t h e r m a l w a l l s of the c a v i t y . Note t h a t these countour p l o t s were g e n e r a t e d u s i n g the r e s u l t s o b t a i n e d w i t h the f i n e s t g r i d (49x49) and were found to be u n a f f e c t e d by the s e t of g r i d and v o r t i c i t y boundary c o n d i t i o n s used. The l e n g t h of the r i g h t w a l l , the maximum d e v i a t i o n from o r t h o g o n a l i t y , the maximum stream f u n c t i o n and the average N u s s e l t numbers used t o m o n i t o r the a c c u r a c y of the s o l u t i o n a re g i v e n i n F i g s . 22 t o 26 f o r c a s e s 1 t o 5, r e s p e c t i v e l y . These parameters a r e p l o t t e d as a f u n c t i o n of the number of g r i d p o i n t s ( T a b l e 11) and the t y p e of boundary c o n d i t i o n s used ( T a b l e s 9 and 10). As can be seen from F i g s . 22 t o 26, a l l of the s e m o n i t o r e d parameters e v e n t u a l l y converge t o a s y m p t o t i c v a l u e s as the number of g r i d p o i n t s i s i n c r e a s e d . C o n s i d e r i n g t h e s e a s y m p t o t i c v a l u e s , r e l a t i v e e r r o r s a s s o c i a t e d w i t h t h e l e n g t h of the r i g h t w a l l , the maximum stream f u n c t i o n and the average N u s s e l t numbers were c a l c u l a t e d f o r the c o a r s e r g r i d s . For example, the e s t i m a t e d r e l a t i v e e r r o r s o b t a i n e d f o r a 33x33 g r i d a r e t a b u l a t e d i n Ta b l e 13 f o r the f i v e extreme c a s e s of i n t e r e s t . A l s o , the maximum d e v i a t i o n from o r t h o g o n a l i t y and the r a t i o of the d i f f e r e n c e between heat t r a n s f e r r a t e s t h r o u g h each i s o t h e r m a l w a l l and t h e i r mean v a l u e s a re 1 12 F i g u r e 22. P l o t s of the m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r case 1 500 1000 1S00 2000 2500 AVERAGE NUSSELT NUMBER - LEFT HALL 1 ]$£ D CASE fi - HOOD W| O CASE B - HOOD i f f & A - SECOND OROER 4} O CASE B - SECOND ORDER JH V CASE C - SECOND OROER 500 1000 1500 2000 2500 AVERAGE NUSSELT NUMBER - RIGHT HALL Q CASE A O CASE B £ CASE A O CASE 6 V CASE C HOOD HOOO SECOND ORDER SECOND OROER SECOND ORDER 500 1000 1500 2000 2500 MAXIMUM DEVIATION FROM ORTHOGONALITt (0EGREES1 — I — 500 D CASE A O CASE B A CASE A o CASE B V CASE C HOOO HOOO SECOND ORDER SECOND OROER SECONO ORDER — i — 2000 1000 1500 2500 LENGTH OF RIGHT HALL O CASE A - HOOO O CASE B - HOOO A CASE A - SECOND OROER o CASE B V CASE C SECONO ORDER SECONO ORDER 500 1000 1500 2000 2500 11 F i g u r e 23. P l o t s of the mo n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r case 2, MAX 1 MUM STREAM PUNCH ON • CASE A - HOOD O CASE B - HOOD a CASE A - SECONO OROER o CASE B - SECOND ORDER 1000 1500 2000 25O0 AVERAGE NUSSELT NUMBER - RIGHT HALL C • CASE A - HOOD O CASE B - HOOD A CASE A - SECOND ORDER o CASE B - SECONO ORDER 1S00 2500 500 1000 1500 2000 2500 MAXIMUM OEVIATION FROM ORTHOGONALITY (OEGREES) • CASE A O CASE B & CASE A o CASE B HOOD HOOO SECOND ORDER SECOND OROER ™ LENGTH OF RIGHT HALL O CASE A - HOOO T O CASE B - HOOO \ A CASE A - SECONO ORDER \ o CASE B - SECOND OROER 500 1000 2000 2500 1000 1500 2000 2500 1 14 F i g u r e 24. P l o t s of t h e m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r c a s e 3 HAXIrttJM STREAM fUNCTlON O CASE B - HOOD O CASE A - SECOND ORDER A CASE B - SECOND ORDER. S00 1000 1500 2000 2500 11 F i g u r e 25. P l o t s of t h e m o n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c r e t e p o i n t s f o r c a s e 4 2500 — 1 — 500 O CASE B - HOOD O CASE A - SECONO OROER A CASE B - SECONO OROER O CASE C - SECOND ORDER 1000 1500 2000 2500 1500 2000 2500 LENGTH OF RIGHT HALL 8 CM _ ~" ° 0 in o in - hi a c a s E B " HOOD J O CASE A - SECONO OROER in CM / A CASE B - SECONO ORDER ~' / O CASE C - SECOND ORDER o o in 2500 500 1000 1500 2000 2500 1 16 F i g u r e 26. P l o t s of t h e mo n i t o r e d v a r i a b l e s as a f u n c t i o n of the number of d i s c e t e p o i n t s f o r c a s e 5, MAXIMUM STREAM FUNCTION • CASE fl - HOOO O CASE A - SECOND ORDER A CASE B - SECONO ORDER © CASE C - SECONO ORDER 1000 1500 2000 2SO0 AVERAGE NUSSELT NUMBER - LEFT HALL • CASE A - HOOD O CASE A - SECONO ORDER 4 CASE 8 - SECOND OROER O CASE C - SECOND ORDER 2500 500 1000 1500 2000 2SO0 MAXIMUM DEVIATION FROM ORTHOGONALITY (DEGREES) O CASE A O CASE A c, CASE B HOOD SECOND OROER SECONO OROER © CASE C - SECOND ORDER 1500 2000 2500 2500 Table 13. E r r o r s i n monitored v a r i a b l e s f o r c a s e s 1 to 5 u s i n g a 33x33 g r i d . Case Gr id Boundary Cond i t i ons Vor t i c 1 t y Boundary Cond I t i on Re 1 a t t ve E r r o r of R Ight Wal 1 Length Max 1 inum Dev1 a t1 on from Or thogona1 1 ty Re 1 a t 1 ve E r r o r of Max 1 mum S tream Func 11 on Re 1 a t1ve E r r o r of Lef t Wal 1 Average Nusse1t Number R e l a t i v e E r r o r of R i ght Wa 1 1 Average Nusse1t Number R e l a t i v e E r r o r of Hea t T r a n s f e r Rate 1 Case A Case B Case A Case B Case C Wood Wood 2nd Order 2nd Order 2nd Order 6 .70E-5 1 .02E-4 6 .70E -5 1 .02E--1 1 .02E-4 0. 386 0. 588 0. 386 0. 58B 0. 225 3.75E-4 4. 18E-4 5.25E-4 4. 18E-4 5.04E-4 7 . 54E-4 1 . I5E-3 8 .97E-4 1 . 16E-3 5.47E-4 9 .09E-4 I . I8E-3 1.05E-3 I.20E-3 8.OIE-4 5.4 IE-5 1.08E-4 5.33E-5 I.OBE-4 1.05E-4 2 Case A Case B Case A Case B Wood Wood 2nd Order 2nd Order 1 . 47E-4 1 . 1 7 E - 4 1 .47E-4 1 . 17E-4 0.47 1 1 .640 0.47 1 1 .640 1 . 15E-3 1.6IE-3 9 .24E-4 1.32E-3 G . 4 7 E - 5 2.45E-4 2 . I7E-4 1 . 12E-5 4 .55E-4 5 .09E-5 7.44E-4 2.05E-4 9.OGE-5 3 . 54E-5 9 . 8 1E-5 3 . 42E-5 3 Case B Case A Case B Wood 2nd Order 2nd Order 1 03E-3 I .8GE-2 1 03E-3 6 . 74 13 . 66 6.74 6 .23E-4 2 .27E-3 8.58E-4 8 . 76E-4 3 . I6E-3 7.24E-4 7 . I6E-4 2.OGE-2 8 . G 1 E - 4 1 . t BE - 4 8 . 28E-6 1 . 26E-4 4 Case U Case A Case B C a s e C Wood 2nd Order 2nd Order 2nd Order 1 . 03E--3 1 .20E-2 1 .03E-3 1 .03E-3 1 1 .00 8.31 1 1 .00 13.37 2 .70E-3 5.48E-4 2 .53E-3 2 .68E-3 0. IGE-4 1 . 5GF.-3 9.08E-4 4 .73E-5 1 .G2E-3 1.05E-2 1 . 7 IE-3 0.00E+0 1.G5E-4 1 . 4 4 E - 5 1 . 7 1E-4 1.59E-4 5 Case A Case A Case B Case C Wood 2nd Order 2nd Order 2nd o r d e r 6 . 3 IE-5 G.3 IE-5 1 .06E-4 1 .OGE -4 0 . 386 0. 386 0.588 0. 225 6 .79E-3 5.G9E-3 8. I9E-3 G.69E-3 9.83E-4 2 .07E-3 2.50E-3 1 .42E-3 1 06E-3 2. I5E-3 2.38E -3 1 .5 IE -3 1.30E-5 9.25E-G I . 47E-5 2 .78E-5 118 l i s t e d f o r a 33x33 g r i d i n the same t a b l e . C. DISCUSSION B o d y - f i t t e d o r t h o g o n a l g r i d s were g e n e r a t e d f o r c a s e s 1 to 5 c o n s i d e r e d i n t h i s e x e r c i s e u s i n g a l l s e t s of g r i d boundary c o n d i t i o n s d e f i n e d i n T a b l e 9 w i t h o n l y one e x c e p t i o n . I t was not p o s s i b l e t o g e n e r a t e a b o d y - f i t t e d o r t h o g o n a l g r i d f o r case 3 w i t h t h e g r i d boundary c o n d i t i o n s C. I t i s not c l e a r why convergence was not reached i n t h i s c a s e , but i t appears t h a t the requirement of boundary correspondence on t h r e e f a c e s o v e r - c o n s t r a i n s the problem f o r t h i s p a r t i c u l a r geometry. A l s o , i n o r d e r t o o b t a i n a converged symmetric g r i d f o r c a s e 4 w i t h the g r i d boundary c o n d i t i o n s A, the n u m e r i c a l p r o c e d u r e had t o be m o d i f i e d s l i g h t l y . The g r i d boundary c o n d i t i o n s A r e q u i r e the use of both Neumann and D i r i c h l e t c o n d i t i o n s a l o n g the e n t i r e t r a n s f o r m e d domain boundary. However, f o r c a s e 4, 2 D i r i c h l e t boundary c o n d i t i o n s were used a t , and o n l y a t , the c e n t r a l node of the r i g h t w a l l i n o r d e r t o o b t a i n the symmetric mapping e x p e c t e d from t h i s geometry. To o b t a i n an a c c u r a t e n u m e r i c a l s i m u l a t i o n u s i n g f i n i t e d i f f e r e n c e methods, the g r i d d e n s i t y s h o u l d be h i g h e s t i n a r e a s of the c a v i t y where e i t h e r the f l o w or the heat t r a n s f e r i s i m p o r t a n t , i . e . where the g r a d i e n t s of t e mperature or stream f u n c t i o n a r e the g r e a t e s t . I t i s c l e a r from the c l o s e p r o x i m i t y of c o n t o u r l i n e s , t h a t the temperature g r a d i e n t i s maximal i n the v i c i n i t y of the 119 i s o t h e r m a l w a l l s . The a r e a s of h i g h e s t heat t r a n s f e r a l o n g the i s o t h e r m a l w a l l s can be l o c a t e d by c o n s i d e r i n g the l e f t and r i g h t w a l l l o c a l N u s s e l t number d i s t r i b u t i o n s g i v e n i n F i g s . 17 t o 21. For c a v i t y c a s e s 1 t o 4, the t o p of the l e f t w a l l and the bottom h a l f of the r i g h t w a l l a r e the a r e a s of h i g h e s t heat t r a n s f e r . The g r a d i e n t s i n t h e s e a r e a s become s t r o n g e r as the R a y l e i g h number i n c r e a s e s because the r e s u l t i n g i n c r e a s e i n c o n v e c t i o n i n t e n s i f i e s the l o c a l heat t r a n s f e r . The f l u i d shear i s g r e a t e s t near the c a v i t y boundary and becomes l e s s i m p o r t a n t i n t h e c o r e . A g a i n , p a r t i c u l a r l y near the b o u n d a r i e s , the v e l o c i t y g r a d i e n t s become s t e e p e r as the R a y l e i g h number i n c r e a s e s and the f l o w assumes a more b o u n d a r y - l a y e r l i k e n a t u r e . An i n s p e c t i o n of the n u m e r i c a l l y g e n e r a t e d g r i d s shown i n F i g s . 17 t o 21 r e v e a l s t h a t most g r i d s have a f a i r l y c o n s t a n t g r i d d e n s i t y over the e n t i r e domain. However, some g r i d s have h i g h e r r e s o l u t i o n s i n r e g i o n s where i t i s not j u s t i f i e d by the g r a d i e n t s (eg. lower l e f t hand c o r n e r of F i g . 17c), r e g i o n s of v e r y low g r i d d e n s i t y (eg. upper and lower r i g h t hand c o r n e r s of F i g s . 19a and i n the m i d d l e of the r i g h t hand w a l l i n F i g . 20a) or have problems s a t i s f y i n g the o r t h o g o n a l i t y c o n d i t i o n ( e . g . F i g . 2 0 c ) . The r e s u l t s of F i g s . 22 t o 26 and T a b l e 13 a l s o d emonstrate t h a t 1. independent of the g r i d and v o r t i c i t y boundary c o n d i t i o n s chosen, n u m e r i c a l s o l u t i o n s h a v i n g a c c u r a c i e s of b e t t e r than 1 p e r c e n t were 120 o b t a i n e d by u s i n g o n l y a 22x22 g r i d f o r a R a y l e i g h number of 10000 w i t h o n l y two e x c e p t i o n s : c a s e s 3 and 4 w i t h g r i d boundary c o n d i t i o n s A, 2. independent of the g r i d and v o r t i c i t y boundary c o n d i t i o n s chosen, n u m e r i c a l s o l u t i o n s h a v i n g a c c u r a c i e s of b e t t e r than 1 p e r c e n t were o b t a i n e d by u s i n g a 28x28 g r i d f o r a R a y l e i g h number of 100000, 3. f o r g r i d s s i z e s l a r g e r than 30x30, the maximum stream f u n c t i o n , average N u s s e l t numbers and r i g h t w a l l l e n g t h m o n o t i c a l l y approach t h e i r a s y m p t o t i c v a l u e s , 4. good a c c u r a c y was o b t a i n e d f o r the g r i d s i z e of 33x33 f o r a l l c a s e s except f o r c a s e s 3 and 4 w i t h the g r i d boundary c o n d i t i o n s A, 5. i t i s d i f f i c u l t t o j u s t i f y which c o m b i n a t i o n of g r i d and v o r t i c i t y boundary c o n d i t i o n s y i e l d s the most a c c u r a t e r e s u l t s f o r c o a r s e r g r i d s e i t h e r because one c o m b i n a t i o n of boundary c o n d i t i o n s does not show c o n s i s t e n t s u p e r i o r i t y over the o t h e r s f o r a l l of the m o n i t o r e d v a r i a b l e s or because a l l c o m b i n a t i o n s l e a d t o s i m i l a r a c c u r a c i e s , and 6. the heat b a l a n c e over the c a v i t y i s always w e l l s a t i s f i e d f o r a 33x33 g r i d w i t h the p r e s e n t a c c u r a c y c r i t e r i a . 121 I t was somewhat s u r p r i s i n g t o d i s c o v e r t h a t the d i f f e r e n t c o m b i n a t i o n s of g r i d ( w i t h the_ e x c e p t i o n of g r i d boundary c o n d i t i o n s A when used w i t h cases 3 and 4) and v o r t i c i t y boundary c o n d i t i o n s gave s i m i l a r s o l u t i o n a c c u r a c i e s . The reasons why the second o r d e r v o r t i c i t y boundary c o n d i t i o n does not show any s u p e r i o r i t y t o the Wood boundary c o n d i t i o n even though the former i s based on T a y l o r s e r i e s of h i g h e r o r d e r can be e x p l a i n e d as f o l l o w s . F i r s t l y , f o r the square c a v i t y n a t u r a l c o n v e c t i o n problem, Wong and R a i t h b y [12 ] showed t h a t the Wood boundary c o n d i t i o n not o n l y gave more a c c u r a t e s o l u t i o n s than any of the o t h e r commonly used v o r t i c i t y boundary a p p r o x i m a t i o n s b u t , i t y i e l d e d r e s u l t s which were o n l y s l i g h t l y l e s s a c c u r a t e than the second o r d e r c o n d i t i o n . S e c o n d l y , i n the p r e s e n t problem, the o r t h o g o n a l c o o r d i n a t e system has t o be ge n e r a t e d n u m e r i c a l l y and hence, a l r e a d y c o n t a i n s i n a c c u r a c i e s . For an o r t h o g o n a l system, t h e second o r d e r c o n d i t i o n has more terms t h a t r e q u i r e n u m e r i c a l m a n i p u l a t i o n of the g r i d c h a r a c t e r i s t i c s than does the Wood boundary c o n d i t i o n . Thus, i t i s s u r m i s e d t h a t the g a i n i n a c c u r a c y a c h i e v e d by u s i n g more terms of the T a y l o r s e r i e s i s o f f s e t , t o some e x t e n t , by a d d i t i o n a l c o m p u t a t i o n a l e r r o r a s s o c i a t e d w i t h the n u m e r i c a l l y g e n e r a t e d g r i d . Thus, even though one of the o b j e c t i v e s of P a r t I was to determine which of the g r i d and v o r t i c i t y boundary c o n d i t i o n s gave the most a c c u r a t e n u m e r i c a l r e s u l t s f o r each c a s e , t h e r e was no c l e a r c o n c l u s i o n . For any g i v e n n u m e r i c a l 122 e x p e r i m e n t , i t was found t h a t the c o m b i n a t i o n of c o n d i t i o n s which m i n i m i z e d the e r r o r f o r one dependent v a r i a b l e o f t e n gave p o o r e r r e s u l t s f o r a n o t h e r v a r i a b l e . However, i t was observ e d t h a t , except f o r boundary c o n d i t i o n s which l e d t o u n d e s i r a b l e g r i d s , most c o m b i n a t i o n s of g r i d and v o r t i c i t y boundary c o n d i t i o n s gave r e s u l t s of a c c e p t a b l e a c c u r a c y as l o n g as a s u f f i c i e n t number of g r i d p o i n t s were used. A l l of the n a t u r a l c o n v e c t i o n r e s u l t s r e p o r t e d i n P a r t s I I and I I I were o b t a i n e d u s i n g the Wood v o r t i c i t y boundary c o n d i t i o n , because i t r e q u i r e s fewer n u m e r i c a l m a n i p u l a t i o n s than does the second o r d e r boundary c o n d i t i o n . For c a v i t y C1, the g r i d boundary c o n d i t i o n s A were chosen because the C c o n d i t i o n s y i e l d e d h i g h g r i d d e n s i t i e s i n r e g i o n s which were i n a p p r o p r i a t e and the B c o n d i t i o n s produced g r i d s which were l e s s o r t h o g o n a l than A. For c a v i t y C2, the g r i d boundary c o n d i t i o n s B were d e c i d e d upon because the A c o n d i t i o n s l e a d t o g r i d s of low d e n s i t y i n a r e a of s u s c e p t i b l e g r a d i e n t s and the C c o n d i t i o n s r e s u l t i n n o n - o r t h o g o n a l g r i d s . The second o b j e c t i v e of P a r t I was t o use the r e s u l t s t o d e t e r m i n e what g r i d s i z e was needed t o g i v e n a t u r a l c o n v e c t i o n r e s u l t s of a c c e p t a b l e a c c u r a c y . I n a l l c a s e s , i t was found t h a t s o l u t i o n s w i t h s u f f i c i e n t a c c u r a c y were o b t a i n e d as l o n g as the g r i d s i z e exceeded 33x33. For example, T a b l e 13 shows t h a t , f o r a 33x33 g r i d , the maximum e r r o r was a 0.0082 i n the stream f u n c t i o n o b t a i n e d a t Ra=100000 f o r a C1 c a v i t y , t ype A g r i d boundary c o n d i t i o n s and second o r d e r v o r t i c i t y boundary c o n d i t i o n s . T h i s maximum 1 23 e r r o r i s a s s o c i a t e d w i t h the l a r g e s t R a y l e i g h number which i s i n agreement w i t h many l i t e r a t u r e o b s e r v a t i o n s , e.g. [ 1 2 ] . T h e r e f o r e , the n u m e r i c a l n a t u r a l c o n v e c t i o n r e s u l t s a r e e x p e c t e d t o have an a c c u r a c y of b e t t e r than 1 p e r c e n t i f a g r i d s i z e of 33x33 or l a r g e r i s used. As a consequence, a 35x35 g r i d was employed t o g e n e r a t e a l l of the r e s u l t s i n P a r t s I I which f o l l o w s . Use of a f i n e r g r i d was r u l e d out because of the e x c e s s i v e c o m p u t a t i o n c o s t s . IX. PART II A. NUMERICAL EXPERIMENTS The p r i m a r y o b j e c t i v e of P a r t I I was t o study the e f f e c t of the a m p l i t u d e and the R a y l e i g h number on the n a t u r a l c o n v e c t i o n f l o w and heat t r a n s f e r i n c a v i t y t y p e s C1 and C2. The c o m b i n a t i o n of the s i x R a y l e i g h numbers (0 t o 100000) and f i v e a m p l i t u d e s (-0.15 t o 0.15) i n v e s t i g a t e d (see T a b l e 1) y i e l d e d a t o t a l of 30 n a t u r a l c o n v e c t i o n problems f o r each c a v i t y . Note t h a t t h e s e 30 e x p e r i m e n t s i n c l u d e s e v e r a l s p e c i a l c a s e s . When the a m p l i t u d e e q u a l s z e r o , the square c a v i t y , f o r which t h e r e i s much c o r r o b o r a t i n g d a t a i n the l i t e r a t u r e , r e s u l t s . When the R a y l e i g h number e q u a l s z e r o , t h e r e can be no n a t u r a l c o n v e c t i o n and a pure c o n d u c t i o n problem r e s u l t s . As was mentioned i n the l a s t s e c t i o n , the r e l a t i v e l y s i m p l e Wood v o r t i c i t y boundary c o n d i t i o n was used i n a l l c a s e s s i n c e i t appears t o cause no l o s s i n n u m e r i c a l a c c u r a c y and r e q u i r e s l e s s computing t i m e . For the c a v i t y C I , Type A g r i d boundary c o n d i t i o n s were employed w h i l e t h e C2 c a v i t y used type B c o n d i t i o n s . The d i f f e r e n t s e t s of g r i d boundary c o n d i t i o n s a r e l i s t e d i n Ta b l e 14. To m a i n t a i n a c c e p t a b l e n u m e r i c a l a c c u r a c y f o r a l l r u n s , the g r i d s i z e was s e t a t 35x35. To c o n s e r v e computing t i m e , the f o l l o w i n g p r o c e d u r e s were u t i l i z e d . F i r s t l y , f o r each c a v i t y shape, the o r t h o g o n a l g r i d was g e n e r a t e d o n l y once w h i l e the R a y l e i g h 124 1 25 Table 14. G r i d and v o r t i c i t y boundary c o n d i t i o n s used i n Part I I . C a v i t y Dimensionless Amplitude G r i d Boundary C o n d i t i o n s V o r t i c i t y Boundary C o n d i t i o n C1 Negat i v e Case A Wood CI P o s i t i v e Case A Wood C2 Negat i v e Case B Wood C2 P o s i t i v e Case B Wood 126 number was i n c r e a s e d from 0 t o 10000. S e c o n d l y , t o p r o v i d e b e t t e r s t a r t i n g v a l u e s each time the R a y l e i g h number was i n c r e a s e d , the stream f u n c t i o n , t e m p e r a t u r e and v o r t i c i t y nodes were i n i t i a l i z e d u s i n g the f i n a l s o l u t i o n s o b t a i n e d f o r the p r e v i o u s R a y l e i g h number. B. RESULTS The computed r e s u l t s f o r a l l 60 n a t u r a l c o n v e c t i o n problems, i n c l u d i n g the g r i d , the stream f u n c t i o n , v o r t i c i t y and temperature c o n t o u r s as w e l l as the l o c a l N u s s e l t numbers f o r the l e f t and r i g h t hand w a l l s , a r e p r e s e n t e d i n compact form i n Appendix B. In the r e s t of the c h a p t e r , o n l y the maximum stream f u n c t i o n and the average N u s s e l t number of the l e f t w a l l a r e c o n s i d e r e d f u r t h e r . The former c h a r a c t e r i z e s the s t r e n g t h of the c o n v e c t i o n f l o w w h i l e the l a t t e r i s a measure of the t o t a l heat t r a n s f e r r a t e t h r o u g h the c a v i t y . Of c o u r s e , as was shown i n Eq. 129, the l e f t and r i g h t w a l l N u s s e l t numbers a r e s i m p l y r e l a t e d t h r o u g h the d i m e n s i o n l e s s l e n g t h of the c u r v e d r i g h t w a l l . The maximum stream f u n c t i o n and l e f t w a l l N u s s e l t numbers a r e p l o t t e d as f u n c t i o n s of the R a y l e i g h number and c u r v e d w a l l a m p l i t u d e i n F i g s . 27 and 28 f o r c a v i t y C1 and F i g s . 29 and 30 f o r c a v i t y C2. The l i m i t i n g l e f t w a l l N u s s e l t numbers f o r R a y l e i g h number e q u a l s z e r o (pure c o n d u c t i o n ) a r e p r e s e n t e d i n T a b l e 15 and 16 f o r c a v i t i e s C1 and C2, r e s p e c t i v e l y . F i g u r e 27. P l o t of the maximum stream f u n c t i o n v e r s u s the R a y l e i g h number f o r d i f f e r e n t a m p l i t u d e s - c a v i t y C1. 1 28 F i g u r e 28. P l o t of the average l e f t w a l l N u s s e l t number v e r s u s the R a y l e i g h number f o r d i f f e r e n t a m p l i t u d e s -c a v i t y C1. F i g u r e 29. P l o t of the maximum stream f u n c t i o n _ v e r s u s the R a y l e i g h number and d i f f e r e n t a m p l i t u d e s - c a v i t y C2. 1 30 F i g u r e 30. P l o t of the average l e f t w a l l N u s s e l t number v e r s u s the R a y l e i g h number and d i f f e r e n t a m p l i t u d e s -c a v i t y C2. 131 T a b l e 15. Average L e f t w a l l N u s s e l t number as a f u n c t i o n of a m p l i t u d e f o r Ra=0 - c a v i t y C1. D i m e n s i o n l e s s A m p l i t u d e Average N u s s e l t Number -0.150 1 .224 -0.075 1 .090 0. 000 1 .000 0.075 0.936 0. 1 50 0.894 T a b l e 16. Average L e f t w a l l N u s s e l t number as a f u n c t i o n of a m p l i t u d e f o r Ra=0 - c a v i t y C2. D i m e n s i o n l e s s A m p l i t u d e Average N u s s e l t Number -0.150 1 .262 -0.075 1 .099 0.000 1 .000 0.075 0.943 0. 1 50 0.914 1 32 C. DISCUSSION The stream f u n c t i o n and tem p e r a t u r e f i e l d s of the 60 n a t u r a l c o n v e c t i o n problems, shown i n Appendix B, a r e examined f i r s t . From the stream f u n c t i o n p l o t s , i t can be seen t h a t the " c a t ' s - e y e " f l o w p a t t e r n (secondary f l o w ) o c c u r s f o r both c a v i t i e s C1 and C2 a t a R a y l e i g h number of 100000 independent of the d i m e n s i o n l e s s a m p l i t u d e . A p p a r e n t l y , f o r a g i v e n R a y l e i g h number, a s i m i l a r f l o w p a t t e r n i s o b t a i n e d i n a l l c a v i t i e s r e g a r d l e s s of t h e i r shape. Perhaps l a r g e r d i f f e r e n c e s would have been ob s e r v e d i f c a v i t i e s w i t h more extreme a m p l i t u d e s had been t r i e d . I f the b e g i n n i n g of the l a m i n a r b o u n d a r y - l a y e r regime i s c o n s i d e r e d t o be reached when the d e r i v a t i v e of the temperature w i t h r e s p e c t t o the X d i r e c t i o n i s e q u a l t o z e r o a t the m i d d l e of the c a v i t y (even though t h i s c r i t e r i o n was d e r i v e d f o r r e c t a n g u l a r c a v i t y ) , an e x a m i n a t i o n of the temperature p l o t s shows t h a t a b o u n d a r y - l a y e r - l i k e regime b e g i n s t o o c c u r a t a R a y l e i g h number of 100001 f o r a l l c a v i t y shapes except the c a v i t i e s C1 and C2 w i t h a m p l i t u d e of -0.15, which seem t o r e q u i r e a R a y l e i g h number between 10000 and 30000. These r e s u l t s a r e s i m i l a r t o t h o s e r e p o r t e d i n the l i t e r a t u r e f o r a square c a v i t y [1,3] ( T a b l e 2 ) . As can be seen from F i g s . 27 and 30, the maximum stream f u n c t i o n and l e f t w a l l N u s s e l t number are both s i g n i f i c a n t l y a f f e c t e d by the R a y l e i g h number and by the a m p l i t u d e of the c a v i t y . (Note t h a t no e r r o r b a r s a r e p l o t t e d i n F i g s . 27 t o 30; i t can be assumed t h a t the e r r o r i n the maximum stream 133 f u n c t i o n s and average l e f t w a l l N u s s e l t numbers a r e , i n a l l c a s e s , l e s s t h a t the h e i g h t of the symbols.) For both t y p e s of c a v i t i e s , C1 and C2, the maximum stream f u n c t i o n and the average N u s s e l t number of the l e f t w a l l i n c r e a s e as the R a y l e i g h number i n c r e a s e s . The maximum stream f u n c t i o n a l s o i n c r e a s e s w i t h i n c r e a s i n g a m p l i t u d e f o r bo t h c a v i t i e s C1 and C2 w i t h one e x c e p t i o n . For c a v i t y C1, the a m p l i t u d e has no n o t i c e a b l e e f f e c t on the maximum stream f u n c t i o n a t Ra=30000. The b e h a v i o r of the average N u s s e l t number w i t h r e s p e c t t o an i n c r e a s e of the a m p l i t u d e i s not so s t r a i g h t f o r w a r d . For c a v i t y C1, the average N u s s e l t number a l o n g the l e f t w a l l d e c r e a s e s m o n o t o n i c a l l y as the a m p l i t u d e i n c r e a s e s w i t h one e x c e p t i o n . The a m p l i t u d e has a n e g l i g i b l e e f f e c t on the average N u s s e l t number a l o n g the l e f t w a l l a t Ra=TO0OO0. However, f o r c a v i t y C2, two d i f f e r e n t t y p e s of b e h a v i o r a re observ e d depending on the R a y l e i g h number. At low R a y l e i g h numbers, the average l e f t w a l l N u s s e l t number d e c r e a s e s as the a m p l i t u d e i n c r e a s e s ; a t h i g h R a y l e i g h numbers, the N u s s e l t number i n c r e a s e s w i t h i n c r e a s i n g a m p l i t u d e . For i n t e r m e d i a t e R a y l e i g h numbers, t h e r e i s a c r o s s - o v e r range where the a m p l i t u d e has no n o t i c e a b l e e f f e c t on the average N u s s e l t number. The t r e n d s o b s e r v e d f o r the maximum stream f u n c t i o n and the average l e f t w a l l N u s s e l t number as a f u n c t i o n of a m p l i t u d e a r e r e c o r d e d i n T a b l e s 17 t o 20 f o r each R a y l e i g h number. To un d e r s t a n d the e f f e c t s of the R a y l e i g h number and the a m p l i t u d e on the f l u i d f l o w and heat t r a n s f e r i n 1 34 Table 17. Percent change of the maximum stream f u n c t i o n with i n c r e a s i n g amplitude f o r a given R a y l e i g h number -c a v i t y C1. R a y l e i g h Number Dimensionless Amplitude Ranges -0.150 to -0.075 -0.075 to 0. 000 0.000 to 0.075 0.075 to 0. 150 - - - -; IOOO I. ... I- I j I 3000 I I ! i I 10000 ' I I 1.12 0.51i \ 30000 0.10 ; -0.93 j -0.09 0.57 100000 - 0.75 1 .93 1 .62 I: i n c r e a s e of 2 percent and more with an i n c r e a s e of the amplitude over the i n d i c a t e d range. D: decrease of 2 percent and more with an i n c r e a s e of the amplitude over the i n d i c a t e d range. -: not a p p l i c a b l e or not d e f i n e d . 1 35 T a b l e 18. P e r c e n t change of the average l e f t w a l l N u s s e l t number w i t h i n c r e a s i n g a m p l i t u d e f o r a g i v e n R a y l e i g h number - c a v i t y C1. R a y l e i g h Number D i m e n s i o n l e s s A m p l i t u d e Ranges -0.150 t o -0.075 -0.075 t o 0.000 0.000 t o 0.075 0.075 t o 0. 1 50 0 D D D D 1 000 D D D ; -1.94 3000- D D D - D; 10000 D D' -1 .53 -1 .07 30000 D : -1.53 ! -0.73 -0.26 100000 - : -1.02 ! -0.16 0.21 I : i n c r e a s e of 2 p e r c e n t and more w i t h an i n c r e a s e of the a m p l i t u d e over the i n d i c a t e d range. D: d e c r e a s e of 2 p e r c e n t and more w i t h an i n c r e a s e of the a m p l i t u d e over the i n d i c a t e d range. -: not a p p l i c a b l e or not d e f i n e d . 136 T a b l e 19. P e r c e n t change of the maximum stream f u n c t i o n w i t h c£vity Sc2 9 a m p l l t u d e f o r a 9 i v e n R a y l e i g h number -R a y l e i g h Number D i m e n s i o n l e s s A m p l i t u d e Ranges -0.150 t o -0.075 -0.075 t o 0.000 0.000 to 0.075 0.075 t o 0.150 0 ! - - ; ! 1000: 1 I I [ I 3000 ! I I ; 1 ! I i 10000 ! 1 , I 1.05 30000 1 I 0.43 1 i ; • I 100000 ! 0.40 ] I I I : i n c r e a s e of 2 p e r c e n t and more w i t h an i n c r e a s e of the a m p l i t u d e over the i n d i c a t e d range. D: d e c r e a s e of 2 p e r c e n t and more w i t h an i n c r e a s e of the a m p l i t u d e over the i n d i c a t e d range. -: not a p p l i c a b l e or not d e f i n e d . 137 T a b l e 20 P e r c e n t change of the average l e f t w a l l N u s s e l t - ? » v ^ v X r 5 I N C R E A S I N 9 a m p l i t u d e f o r a g i v e n R a y l e i g h number R a y l e i g h Number D i m e n s i o n l e s s A m p l i t u d e Ranges - 0 . 1 5 0 to \ - 0 . 0 7 5 - 0 . 0 7 5 t o 0.000 0 . 0 0 0 t o 0 . 0 7 5 0 . 0 7 5 t o 0 . 1 5 0 D D D ; D. 1 0 0 0 D D n. - 1 . 4 5 3 0 0 0 D i - 0 . 6 5 0.23 0.26 1 0 0 0 0 - 0 . 6 2 0.77 1.79 1.66 3 0 0 0 0 I - 0 . 9 2 ' T . 37 i 1 1 0 0 0 0 0 - 0 . 3 6 1.29 ; I I : i n c r e a s e of 2 p e r c e n t and more w i t h an i n c r e a s e of the a m p l i t u d e over the i n d i c a t e d range. D: d e c r e a s e of 2 p e r c e n t and more w i t h an i n c r e a s e of the a m p l i t u d e over the i n d i c a t e d range. -: not a p p l i c a b l e or not d e f i n e d . 138 n o n - r e c t a n g u l a r c a v i t i e s , i t i s u s e f u l t o r e c o n s i d e r the fundamental mechanisms i n v o l v e d i n the n a t u r a l c o n v e c t i o n phenomena. The d r i v i n g f o r c e f o r f l u i d f l o w i n n a t u r a l c o n v e c t i o n i s the d e n s i t y d i f f e r e n c e . I t r e s u l t s from the i n t e r a c t i o n between the g r a v i t a t i o n a l body f o r c e and the h y d r o s t a t i c p r e s s u r e g r a d i e n t and can be ap p r o x i m a t e d i n the p r e s e n t c i r c u m s t a n c e s as a pure t e m p e r a t u r e e f f e c t . A f t e r d i m e n s i o n a l a n a l y s i s , the source term which d r i v e s the f l u i d f l o w i s the body f o r c e term ( l a s t term on the r i g h t hand s i d e ) of Eq. 37. For a d i m e n s i o n l e s s r e f e r e n c e t emperature of 0.5, the body f o r c e term can ta k e on p o s i t i v e , z e r o or n e g a t i v e v a l u e s . These v a l u e s c o r r e s p o n d r e s p e c t i v e l y t o a body f o r c e a c t i n g upwards, a n u l l body f o r c e or a body f o r c e a c t i n g downwards. E q u a t i o n 37 shows t h a t the body f o r c e term 1 i s t he pr o d u c t of the R a y l e i g h number, the P r a n d t l number and the d i f f e r e n c e between the l o c a l and r e f e r e n c e d i m e n s i o n l e s s , t e m p e r a t u r e s . The d i m e n s i o n l e s s groups i n v o l v e d in the body f o r c e term, i . e . Ra and P r , g a t h e r t o g e t h e r the c o n s t a n t p arameters a f f e c t i n g the n a t u r a l c o n v e c t i o n problem: the v a l u e s of the f l u i d p r o p e r t i e s e v a l u a t e d a t the r e f e r e n c e t e m p e r a t u r e p l u s the c a v i t y c h a r a c t e r i s t i c l e n g t h and c h a r a c t e r i s t i c temperature d i f f e r e n c e . For the c a v i t y as a whole, the s t r e n g t h of the n a t u r a l c o n v e c t i o n f l o w i s g i v e n by the maximum stream f u n c t i o n and i s t h e r e f o r e d i r e c t l y i n f l u e n c e d by b o t h Ra and P r . The maximum stream f u n c t i o n i s a l s o a f f e c t e d by the 139 s p e c i f i c geometry of the c a v i t y . However, the r e l a t i o n s h i p between the s t r e n g t h of the f l o w and the c a v i t y geometry i s , i n g e n e r a l , f a r more complex. In a l l c o n v e c t i o n problems, heat i s t r a n s f e r r e d by both c o n d u c t i o n and c o n v e c t i o n . T h e r e f o r e , the r a t e of heat t r a n s f e r , which i s g i v e n i n d i m e n s i o n l e s s terms by the average N u s s e l t number, i s d i r e c t l y i n f l u e n c e d by the f o l l o w i n g two f a c t o r s : 1. the mean d i s t a n c e w hich s e p a r a t e s the two i s o t h e r m a l w a l l s , and 2. the s t r e n g t h of the f l u i d f l o w . The f i r s t f a c t o r a f f e c t s the amount of heat t r a n s f e r t h r ough the c a v i t y by both the c o n d u c t i o n and c o n v e c t i o n mechanisms. I f c o n d u c t i o n dominates, the l a r g e r the d i s t a n c e between the two i s o t h e r m a l w a l l s , the s m a l l e r w i l l be the amount of heat t r a n s f e r r e d t h r o u g h the c a v i t y . A l s o , when c o n v e c t i o n i s the dominant mechanism,, the l a r g e r the d i s t a n c e between the two i s o t h e r m a l w a l l s , the more time the warmer stream f l o w i n g near the upper a d i a b a t i c w a l l has t o t r a n s f e r energy by c o n d u c t i o n through the c o r e t o t h e c o o l e r s t r e a m f l o w i n g a l o n g the lower a d i a b a t i c w a l l . Thus, the g r e a t e r i s the heat exchange between these two c o u n t e r f l o w i n g p a r t s of the l o o p , the l e s s w i l l be the net r a t e of heat t r a n s f e r between the two i s o t h e r m a l w a l l s . C o n s i d e r i n g t h e s e d i f f e r e n t f a c t o r s w hich p l a y a r o l e i n n a t u r a l c o n v e c t i o n f l o w and heat t r a n s f e r , l e t ' s now t r y t o i n t e r p r e t the r e s u l t s of F i g s . 27 t o 30, and T a b l e s 17 t o 140 20. C o n s i d e r c a v i t y C1 f i r s t . There a r e two f a c t o r s which e x p l a i n why the maximum stream f u n c t i o n i n c r e a s e s w i t h i n c r e a s i n g a m p l i t u d e . R e g a r d l e s s of the a m p l i t u d e , a c o l d element from the bottom l e f t c o r n e r of the c a v i t y must move to the bottom r i g h t c o r n e r t o r e p l a c e a f l u i d element a s c e n d i n g because of the buoyancy induced by the hot r i g h t w a l l . As the a m p l i t u d e d e c r e a s e s from 0 t o -0.15, the bottom r i g h t c o r n e r of the c a v i t y a c t s as a dead end t o the f l u i d f l o w . A l s o , the f l u i d element which r i s e s a l o n g the i n w a r d - s l o p i n g r i g h t w a l l works a g a i n i t , e x e r t i n g a p r e s s u r e on the w a l l r a t h e r than a c c e l e r a t i n g the element t o a h i g h e r v e l o c i t y . . C o n v e r s e l y , when the a m p l i t u d e i s r a i s e d from 0 t o 0.15, the dead end d i s s a p p e a r s and now an element of f l u i d warmed by the hot surface: i s f r e e to> r i s e u n h i n d e r e d t o the v e r y t o p of the c a v i t y . As* a consequence of t h e s e two f a c t o r s , the maximum stream* f u n c t i o n g e n e r a l l y i n c r e a s e s as the a m p l i t u d e i s i n c r e a s e d . However,, f o r some? i n e x p l i c a b l e r e a s o n , the a m p l i t u d e has no o b s e r v a b l e e f f e c t on the maximum stream f u n c t i o n a t Ra=30000. Because the body f o r c e term of Eq. 37 i s d i r e c t l y p r o p o r t i o n a l t o the R a y l e i g h number, the f l u i d f l o w s t r e n g t h , as e x p e c t e d , i n c r e a s e s s u b s t a n t i a l l y as the R a y l e i g h number i s r a i s e d . C o n t r a r y t o the f l u i d f l o w b e h a v i o r , the heat t r a n s f e r r a t e depends on f a c t o r s which have o p p o s i n g e f f e c t s . T h e r e f o r e , t o e x p l a i n the b e h a v i o r of the average N u s s e l t 141 number a l o n g the l e f t w a l l w i t h r e s p e c t t o t h e d i m e n s i o n l e s s a m p l i t u d e and the R a y l e i g h number, le£'s i n s p e c t the r e l a t i v e importance of each of t h e s e f a c t o r s s e p a r a t e l y . Without f l u i d f l o w , the average N u s s e l t number of the l e f t w a l l d e c r e a s e s as the mean d i s t a n c e between the i s o t h e r m a l w a l l s i n c r e a s e s s i m p l y because the average l e n g t h of the c o n d u c t i o n p a t h becomes l a r g e r . T h i s t r e n d i s c l e a r l y i n d i c a t e d by T a b l e s 15 and 16, f o r Ra=0. Note t h a t when Ra=0 and A=0 (square e n c l o s u r e f o r both c a v i t i e s C1 and C2), the N u s s e l t number i s e x a c t l y u n i t y . In the p r e s e n c e of f l u i d f l o w , the c h a n g i n g a m p l i t u d e causes two o p p o s i n g e f f e c t s on the N u s s e l t number of the l e f t w a l l . As was j u s t o b s e r v e d , the s t r e n g t h of f l u i d f l o w i n c r e a s e s as the a m p l i t u d e i n c r e a s e s . Thus, on the b a s i s of t h i s f a c t alone,, i t might be e x p e c t e d t h a t the N u s s e l t number of the l e f t w a l l would be m a g n i f i e d as the a m p l i t u d e was i n c r e a s e d . However, as the a m p l i t u d e becomes l a r g e r , the heat t r a n s f e r r e d from the hot stream f l o w i n g a l o n g the a d i a b a t i c t o p w a l l t o the c o l d stream f l o w i n g a l o n g the a d i a b a t i c bottom w a l l by c o n d u c t i o n t h r o u g h the c o r e i n c r e a s e s . F u r t h e r m o r e , the average l e n g t h of t h e c o n d u c t i o n p a t h between the i s o t h e r m a l w a l l s a l s o i n c r e a s e s . These two f a c t o r s would cause the average N u s s e l t number t o d i m i n i s h w i t h i n c r e a s i n g a m p l i t u d e . T h e r e f o r e , t o r a t i o n a l i z e the g e n e r a l t r e n d o b s e r v e d , i . e . t h a t the average N u s s e l t number a l o n g the l e f t w a l l of the C1 c a v i t y d e c r e a s e s as the a m p l i t u d e i n c r e a s e s , the e f f e c t of i n c r e a s i n g the average 142 d i s t a n c e between the two i s o t h e r m a l w a l l s must be d o m i n a t i n g over the e f f e c t of i n c r e a s i n g the s t r e n g t h of c o n v e c t i o n . The average N u s s e l t number of the l e f t w a l l i n c r e a s e s s u b s t a n t i a l l y as the R a y l e i g h number i n c r e a s e s due t o the i n c r e a s i n g s t r e n g t h of the f l o w . C o n s i d e r now the C2 c a v i t y . There a g a i n appear t o be two f a c t o r s which e x p l a i n why the f l u i d f l o w s become s t r o n g e r w i t h i n c r e a s i n g a m p l i t u d e . F i r s t of a l l , f o r n e g a t i v e a m p l i t u d e s , the bottom r i g h t c o r n e r of the c a v i t y a c t s as a dead end t o f l u i d f l o w i n a s i m i l a r f a s h i o n as i t d i d f o r the c o r r e s p o n d i n g C1 c a v i t y . However, t h e r e i s a f a r l e s s a b r u p t change i n the d i r e c t i o n of the f l u i d f l o w i n both the t o p and bottom r i g h t c o r n e r s f o r p o s i t i v e a m p l i t u d e s compared t o n e g a t i v e a m p l i t u d e s . T h e r e f o r e , i n the former c a s e , l e s s momentum i s t r a n s f o r m e d i n t o a p r e s s u r e i n c r e a s e . S e c o n d l y , an element of warm' f l u i d a r i s i n g from the hot r i g h t w a l l can t r a v e l , on a v e r a g e , a much f u r t h e r d i s t a n c e b e f o r e i t s upward motion i s impeded by a h o r i z o n t a l or inward s l o p i n g w a l l . The R a y l e i g h number a f f e c t s the f l u i d f l o w s t r e n g t h i n a s i m i l a r f a s h i o n as was e x p l a i n e d f o r the C1 C a v i t y . Without f l u i d f l o w , the average N u s s e l t number of the l e f t w a l l i n c r e a s e s as the mean d i s t a n c e between i s o t h e r m a l w a l l s i n c r e a s e s , as e x p e c t e d . However, i n p r e s e n c e of f l u i d f l o w , the e f f e c t of the a m p l i t u d e on the average N u s s e l t number a g a i n i n v o l v e s the two f a c t o r s w i t h o p p o s i n g e f f e c t s : f l o w s t r e n g t h and d i s t a n c e between i s o t h e r m a l w a l l s , as were 143 d i s c u s s e d f o r the C1 c a v i t y . As the a m p l i t u d e i n c r e a s e s , the s t r e n g t h of the f l u i d f l o w i n c r e a s e s and c o n s e q u e n t l y the average N u s s e l t number a l o n g the l e f t w a l l i s e x p e c t e d t o r i s e . A l s o , as the a m p l i t u d e i n c r e a s e s , the average l e n g t h s e p a r a t i n g the two i s o t h e r m a l w a l l s i n c r e a s e s , and c o n s e q u e n t l y the average N u s s e l t number a l o n g the l e f t w a l l s h o u l d d e c r e a s e (due t o the combined e f f e c t s of i n c r e a s i n g the c o n d u c t i o n heat t r a n s f e r r e s i s t a n c e between the two i s o t h e r m a l w a l l s and the amount of c o n d u c t i o n heat exchange between the hot and c o l d streams f l o w i n g c o u n t e r c u r r e n t l y near the two a d i a b a t i c w a l l s ) . The f a c t t h a t the c a l c u l a t e d average N u s s e l t numbers a l o n g the l e f t w a l l d e c r e a s e w i t h i n c r e a s i n g a m p l i t u d e a t low R a y l e i g h numbers can be a t t r i b u t e d t o the dominant e f f e c t of an i n c r e a s i n g average d i s t a n c e between the two i s o t h e r m a l w a l l s . The r e v e r s e t r e n d which i s o b s e r v e d a t h i g h R a y l e i g h numbers s u g g e s t s t h a t the e f f e c t of f l u i d f l o w s t r e n g t h becomes the more i m p o r t a n t f a c t o r . Because the i n c r e a s e i n R a y l e i g h number l e a d s t o s u b s t a n t i a l i n c r e a s e s i n the s t r e n g t h of f l u i d f l o w , the average N u s s e l t number a l o n g the l e f t w a l l i n c r e a s e s d i r e c t l y w i t h R a y l e i g h number. D. EMPIRICAL CORRELATIONS When the average l e f t w a l l N u s s e l t number was p l o t t e d as a f u n c t i o n of R a y l e i g h number on l o g - l o g c o o r d i n a t e s , i t was o b s e r v e d t h a t a l i n e a r r e l a t i o n s h i p was o b t a i n e d , a t 1 44 l e a s t f o r c o n d i t i o n s where c o n v e c t i o n was the dominant heat t r a n s f e r mechanism. Thus, f o r t h i s c o n v e c t i o n dominated regime, the average N u s s e l t number r e s u l t s f o r each c a v i t y shape were f i t t e d t o a s i m p l e power law r e l a t i o n s h i p h a v i n g the same form as Eq. 8. The e m p i r i c a l p a r a m e t e r s , a and b, were o b t a i n e d by a l e a s t squares l i n e a r f i t of the l o g a r i t h m i c r e s u l t s and a r e r e p o r t e d i n T a b l e s 21 and 22. No s t a t i s t i c a l a n a l y s i s was performed on the f i t t i n g parameters because i t was r e c o g n i z e d t h a t an u n s u f f i c i e n t number of p o i n t s were used t o o b t a i n t h e s e r e s u l t s . N o n e t h e l e s s , the f i t t i n g parameters d e t e r m i n e d i n the p r e s e n t study f o r d i s t o r t e d c a v i t i e s a r e s i m i l a r i n magnitude t o those l i s t e d i n t he l i t e r a t u r e f o r more r e g u l a r e n c l o s u r e s . The v a l u e s of the c o e f f i c i e n t s a and b c a l c u l a t e d from the r e s u l t s of the p r e s e n t study f o r a square c a v i t y were found to< d i f f e r by o n l y 4 and 2 p e r c e n t , r e s p e c t i v e l y , from the c o e f f i c i e n t s o b t a i n e d by De V a h l D a v i s and Jones [15] i n t h e i r benchmark s o l u t i o n t o t h i s s p e c i a l problem. Note t h a t f o r both t y p e s of c a v i t i e s , the c o e f f i c i e n t a d e c r e a s e s w h i l e the c o e f f i c i e n t b i n c r e a s e s as the a m p l i t u d e i s r a i s e d . A d e c r e a s i n g v a l u e of the c o e f f i c i e n t a r e f l e c t s the f a c t t h a t the N u s s e l t number d i m i n i s h e s as the a m p l i t u d e i n c r e a s e s w h i l e an i n c r e a s i n g v a l u e of c o e f f i c i e n t b i s i n d i c a t i v e of the s t r o n g e r f l o w s t h a t a r e p o s s i b l e w i t h l a r g e r a m p l i t u d e s . 1 45 Table 21. Curve f i t t i n g c o e f f i c i e n t s of the simple power law model f o r c a v i t y C1 with d i f f e r e n t amplitudes. Dimens i o n l e s s Ampli tude Curve F i t t i n g C o e f f i c i e n t s a b -0.150 0. 1 38 0.307 -0.075 0. 1 3 9 0 .304 0.000 0.131 0.308 0.075 0.122 0.314 0. 1 50 0.115 0 . 320 Table 22. Curve f i t t i n g c o e f f i c i e n t s of the simple power law model f o r c a v i t y C2 with d i f f e r e n t amplitudes. Dimensionless Amplitude Curve F i t t i n g C o e f f i c i e n t s a b -0. 150 0. 135 0.305 -0.075 0. 133 0.306 0.000 0.131 0. 308 0.075 0. 1 28 0.312 0. 1 50 0.122 0.320 X. PART III A. NUMERICAL EXPERIMENTS The primary o b j e c t i v e of Part III was to i n v e s t i g a t e the p o s s i b i l i t y of using the present numerical approach to simulate n a t u r a l c o n v e c t i o n flows i n a two-dimensional non-rectangular e n c l o s u r e whose shape r e s u l t s from a phase change pro c e s s . More s p e c i f i c a l l y , an attempt was made to reproduce n u m e r i c a l l y the s t e a d y - s t a t e f l u i d c i r c u l a t i o n p a t t e r n observed e x p e r i m e n t a l l y i n the l i q u i d water phase d u r i n g an i c e formation p r o c e s s . E c k e r t [42] c a r r i e d out a number of i c e forming experiments i n a 5x5 cm 2 v e r t i c a l square c a v i t y . I n i t i a l l y , the h e a v i l y i n s u l a t e d c a v i t y was f i l l e d with water whose temperature was e q u i l i b r a t e d at the hot w a l l value by p a s s i n g the same constant temperature f l u i d through the copper chambers forming the i s o t h e r m a l end w a l l s . At zero time, the f l u i d i n the r i g h t hand chamber was r e p l a c e d by a second f l u i d from a r e f r i g e r a t e d bath whose temperature was w e l l below the f r e e z i n g p o i n t of water. Soon t h e r e a f t e r , i c e began to form on the c o o l e r s u r f a c e . Simultaneously, the changing flow p a t t e r n of the l i q u i d i n the c a v i t y was v i s u a l i z e d by a "streak photography" method. In t h i s technique, n e u t r a l l y - b u o y a n t r e f l e c t i v e p l i o l i t e p a r t i c l e s suspended i n the water were i l l u m i n a t e d by a t h i n sheet of l a s e r l i g h t and photographed at r i g h t angles to the main flow d i r e c t i o n . Thus, the o b j e c t i v e was to t r y and d u p l i c a t e t h i s streak p a t t e r n at a given i n s t a n t i n time f o r 146 147 a few d i f f e r e n t warm w a l l t e m p e r a t u r e s . S i n c e the p r e s e n t program was sef up t o a n a l y z e s t e a d y - s t a t e n a t u r a l c o n v e c t i o n , i t was n e c e s s a r y t o assume t h a t the p r o c e s s was q u a s i - s t e a d y , i . e . , even though the i c e i n t e r f a c e i s growing w i t h t i m e , the f l o w i n the l i q u i d phase i s i n s e n s i t i v e t o t h i s movement. T h i s assumption seems q u i t e r e a s o n a b l e under th e p r e s e n t c i r c u m s t a n c e s as i t was o b s e r v e d t h a t w h i l e the i c e i n t e r f a c e r e q u i r e d many hours t o r e a c h i t s s t e a d y - s t a t e c o n f i g u r a t i o n , d i s t u r b a n c e s i n the l i q u i d phase took o n l y a few minutes t o become c o m p l e t e l y damped. A l s o the temperature boundary c o n d i t i o n s were assumed t o be as f o l l o w s : l e f t w a l l , hot i s o t h e r m a l w a l l ; r i g h t w a l l , c o l d i s o t h e r m a l w a l l ; and t o p and bottom w a l l s , a d i a b a t i c w a l l s . The growing i c e s u r f a c e was, of c o u r s e , a t 0°C. But t h e r e was l i k e l y some v a r i a t i o n i n t e m p e r a t u r e over the h o t t e r " i s o t h e r m a l " w a l l and some heat g a i n t h r o u g h the two " a d i a b a t i c " s u r f a c e s . B e f o r e the e x i s t i n g program c o u l d be; used f o r the P a r t I I I s i m u l a t i o n s , two major m o d i f i c a t i o n s had t o be c a r r i e d o u t . F i r s t , the l i n e a r r e l a t i o n s h i p assumed between the f l u i d d e n s i t y and the temperature does not h o l d f o r water i n the t e m p e r a t u r e range from 0 t o 15 degrees C e l s i u s . In t h i s range, the c o e f f i c i e n t of t h e r m a l e x p a n s i o n i s no l o n g e r c o n s t a n t and f o r the p r e s e n t purposes was r e p r e s e n t e d by a c u b i c p o l y n o m i a l o b t a i n e d by f i t t i n g the d e n s i t y v e r s u s t e m p e r a t u r e data g i v e n i n the Handbook of P h y s i c s and C h e m i s t r y [51] f o r water between 0 and 20°C. A r e a n a l y s i s of 148 the momentum e q u a t i o n s u s i n g s i m i l a r a s s u m p t i o n s as b e f o r e shows t h a t a n o n - l i n e a r r e l a t i o n between d e n s i t y and temperature a f f e c t s o n l y the s o u r c e term and i t can be c o n v e n i e n t l y i n c o r p o r a t e d i n t o the p r e s e n t program t h r o u g h a v a r i a b l e R a y l e i g h number which can be d e f i n e d as g P o ^ C 0 ( T -T ) Ra= ^-^ — — Thus, the R a y l e i g h number becomes dependent on the l o c a l t emperature t h r o u g h the r e l a t i o n s h i p between 9p/9T and T. A l l o t h e r p r o p e r t i e s i n Eq. 130 a r e taken t o be c o n s t a n t and ar e once a g a i n e v a l u a t e d a t a r e f e r e n c e t e m p e r a t u r e . These c o n s t a n t v a l u e s a r e c a l c u l a t e d u s i n g o t h e r e m p i r i c a l r e l a t i o n s h i p s o b t a i n e d by f i t t i n g p r o p e r t y d a t a a v a i l a b l e i n the l i t e r a t u r e . The second m o d i f i c a t i o n c o n c e r n s the; method* of r e p r e s e n t i n g the shape of the l i q u i d - s o l i d i n t e r f a c e . The r e l a t i o n s h i p f o r t h e i n t e r f a c e p o s i t i o n , X=F(Y), can no l o n g e r be g i v e n by a s i m p l e a n a l y t i c a l f o r m u l a . T h i s problem was overcome by f i t t i n g a s e t of c u b i c s p l i n e i n t e r p o l a t i o n f o r m u l a e t o d i s c r e t e p o s i t i o n a l d a t a t a k e n d i r e c t l y from the photographs. In the c u b i c s p l i n e i n t e r p o l a t i o n method [ 4 8 , 5 0 ] , the v a r i a t i o n of the dependent parameter i n the i n t e r v a l between each p a i r of d i s c r e t e p o i n t s i s r e p r e s e n t e d by a c u b i c p o l y n o m i a l whose c o e f f i c i e n t s a r e o b t a i n e d by matching c o n d i t i o n s of c o n t i n u i t y and smoothness a t the d i s c r e t e p o i n t s . The e x t r a p a i r of c o n d i t i o n s needed 3p 3T (130) 149 t o e v a l u a t e the s p l i n e s a r e o b t a i n e d by u s i n g c u b i c e q u a t i o n s p a s s i n g t h r o u g h the f i r s t f o u r and l a s t f o u r p o i n t s t o d e t e r m i n e the i n i t i a l and f i n a l s l o p e s of the f i r s t and l a s t s p l i n e s , r e s p e c t i v e l y . (Note t h a t these end p o i n t d e r i v a t i v e s were a p p r o x i m a t e d n u m e r i c a l l y by u s i n g the d i v i d e d d i f f e r e n c e t e c h n i q u e [ 4 8 - 5 0 ] ) . Of the many e x p e r i m e n t a l r e s u l t s r e p o r t e d by E c k e r t [ 4 2 ] , f o u r c a s e s were chosen f o r n u m e r i c a l s i m u l a t i o n . The s p e c i f i c e x p e r i m e n t a l and n u m e r i c a l parameters used i n each case a r e r e p o r t e d i n T a b l e 23. In a l l f o u r c a s e s , the i c e b l o c k was n e a r i n g i t s f i n a l s t e a d y - s t a t e shape and hence, the i c e - w a t e r i n t e r f a c e was growing o n l y v e r y s l o w l y w i t h t i m e . The p r i m a r y independent v a r i a b l e i n these e x p e r i m e n t s was the warm w a l l temperature which was r a i s e d p r o g r e s s i v e l y from 2.5 t o 15.1°C, a range which encompasses the d e n s i t y extremum. The r e f e r e n c e t e m p e r a t u r e of the l i q u i d phase, i . e . the t emperature a t which the c o n s t a n t p r o p e r t i e s were e v a l u a t e d , was taken t o be the average of the hot w a l l and the i c e - w a t e r i n t e r f a c e t e m p e r a t u r e s . The c h a r a c t e r i s t i c l e n g t h ( i . e . h e i g h t or w i d t h ) of the e x p e r i m e n t a l chamber was 5 cm. U n f o r t u n a t e l y , i t was found t h a t i n o r d e r t o o b t a i n a converged s o l u t i o n w i t h a r e a s o n a b l e amount of computing e f f o r t , a somewhat s m a l l e r c h a r a c t e r i s t i c l e n g t h had t o be used i n two of the f o u r n u m e r i c a l s i m u l a t i o n s . Reducing the c h a r a c t e r i s t i c l e n g t h s i g n i f i c a n t l y reduces the range of the v a r i a b l e R a y l e i g h number (see Eq. 130). Thus, the c h a r a c t e r i s t i c Table 23. Experimental and numerical c o n d i t i o n s used f o r low temperature water n a t u r a l convection t r i a l s . C a s e H o t Wa l1 T e m p e r a t u r e ( D e g r e e s C e l s i u s ) C o l d W a l l T e m p e r a t u r e ( D e g r e e s C e l s 1us ) C h a r a c t e r i s t i c L e n g t h o f t h e V e r t i c a l S q u a r e C a v l ty ( cm) L 1 q u 1 d P h a s e R e f e r e n c e T e m p e r a t u r e ( D e g r e e s C e l s i u s ) P r a n d t 1 N u m b e r R a y l e i g h Numbe r R a n g e s E x p e r 1 m e n t a 1 N u m e r 1 c a 1 S imu1 a t 1 o n E x p e r 1 m e n t a 1 N u m e r 1 c a 1 s1mu1 a 11 on 1 2 . 3 - 9 . 8 5 . 0 5 . 0 1 .15 12 . 69 - 7 . 9 0 E 5 t o - 3 . 4 5 E 5 - 7 . 9 0 E 5 t o - 3 . 4 5 E 5 2 5 . 6 - 9 . 8 5 . 0 5 . 0 2 . 8 1 1 .94 -1 . 92EG t o 8 . 3 9 E 5 - 1 . 92E6 t o 8 . 3 9 E 5 3 8 . G - 1 1 . 8 5 . 0 1 . 5 4 . 33 1 1 . 31 - 2 . 9 5 E 6 t o 3 . 8 2 E 6 - 7 . 9 7 E 4 t o 1 . 03E5 4 15.1 - 1 1 . 8 5 . 0 1 . 5 7 . 55 10. 12 - 5 . 18E6 t o 1 . 74E7 - 1 . 4 0 E 5 t o 4 . 7 0 E 5 151 l e n g t h was reduced to 1.5 cm f o r the t h i r d and f o u r t h cases (see Table 23). Although these c h a r a c t e r i s t i c l e n g t h m o d i f i c a t i o n s w i l l c e r t a i n l y change the s t r e n g t h of the flows i n the d i s t o r t e d c a v i t i e s , i t was hoped that the general c i r c u l a t i o n p a t t e r n s would remain q u a l i t a t i v e l y the same. T h i s would be p a r t i c u l a r l y true i f , at the lower R a y l e i g h number range, a laminar boundary-layer regime had al r e a d y been e s t a b l i s h e d . I t was expected that the flow p a t t e r n s would not change enormously with f u r t h e r i n c r e a s e i n R a y l e i g h number once t h i s regime has been reached. In each s i m u l a t i o n , the type B g r i d boundary c o n d i t i o n s (boundary correspondence on the curved w a l l ) and the Wood v o r t i c i t y boundary c o n d i t i o n were employed. A 15x33 g r i d was used to simulate cases 1 to 3 while a 33x33 g r i d was employed in case 4. These g r i d s i z e s gave s i m i l a r g r i d d e n s i t i e s as i n Part I I . The r e s u l t s are presented in F i g s . 31 to 34 and i n c l u d e , f o r each case, the b o d y - f i t t e d orthogonal g r i d , the temperature and stream f u n c t i o n countour p l o t s as w e l l as the l o c a l N u s s e l t number d i s t r i b u t i o n s along the ice-water i n t e r f a c e and the hot w a l l . A r e p r o d u c t i o n of the experimental s t r e a k p a t t e r n [42] i s a l s o shown f o r comparison with the numerical s t r e a m l i n e p l o t . b) DISTANCE (TOP TO BOTTOM I F i g u r e 31. Low temperature water n a t u r a l c o n v e c t i o n r e s u l t s for Th=2.3°C. (a) G r i d , (b) Stream f u n c t i o n c o n t o u r s , (c) Temperature contours (0°C - 2.3°C, 0.23°C increments), (d) Experimental s t r e a k - l i n e s [42], (e) L e f t w a l l N u s s e l t numbers, ( f ) Right w a l l N u s s e l t numbers. 0.4 O.t O.B OISTHNCT. (TOP TO 80TTOMI r o b) 0.0 0.3 f ) in F i g u r e 32. Low tem p e r a t u r e water n a t u r a l c o n v e c t i o n r e s u l t s f o r Th=5.6°C. ( a) G r i d , (b) Stream f u n c t i o n c o n t o u r s , (c) Temperature c o n t o u r s (0°C - 5.6°C, 0.56°C i n c r e m e n t s ) , (d) E x p e r i m e n t a l s t r e a k - l i n e s [ 4 2 ] , (e) L e f t w a l l N u s s e l t , numbers, ( f ) R i g h t w a l l N u s s e l t numbers. 0.4 O.t 0.1 I OISTRNCL" IT0P TO BOTTOM] 0.0 0.3 0.1 0.8 0.8 1.0 OISTRNCr. IT0P TO BOTTOM I CO d ) \ e ) 0 0 0.2 • .I t.t 0 < l.t o i s r n c c ITOP TO B o n o t i i f ) 5U F i g u r e 3 3 . Low t e m p e r a t u r e w a t e r n a t u r a l c o n v e c t i o n r e s u l t s f o r T h = 8 * . 6 ° C . (a ) G r i d , (b ) S t r e a m f u n c t i o n c o n t o u r s , ( c ) T e m p e r a t u r e c o n t o u r s ( 0 ° C - 8 . 6 ° C , 0 . 8 6 ° C i n c r e m e n t s ) , (d ) E x p e r i m e n t a l s t r e a k - l i n e s [ 4 2 ] , ( e ) L e f t w a l l N u s s e l t n u m b e r s , ( f ) R i g h t w a l l N u s s e l t n u m b e r s . 0.0 0.2 o i s T w e c HOP TO BOTTOMI c) e ) d) 0 0 0.1 f) h 1* f S ? U { ! . ? 5 , l « C W ( I T P r 5 r ? S U r ? M a ^ r natVral * ° n v e c t i o n r e s u l t s Temperature contours ( u V - ' 1 f^c™ U ^ ° ? c o n t o u r s ( <?> S K l i ; e n J 5 } f r ^ k - l i n e s [ 4 2 ] ) (eJ Lett w a l l N u s s e l t numoers, i f ) Right w a l l N u s s e l t numbers. V '•< ••• 0.1 I.I OlSTflNCC I TOP TO BOTTOM I o.o o.i M o.i i ^ ^ OISTf»cc (TOP TO BOTTOM I 156 B. DISCUSSION N a t u r a l c o n v e c t i o n i n the r e g i o n of a d e n s i t y extremum has a f a r more complex f l o w b e h a v i o r than any of the s i t u a t i o n s d i s c u s s e d e a r l i e r . C o n s i d e r the c h a n g i n g s t r e a m l i n e p a t t e r n s r e v e a l e d i n F i g s . 31 t o 34. In case 1 ( F i g . 3 1 ) , the warm w a l l t emperature i s below the extremum temp e r a t u r e and the d e n s i t y i n c r e a s e s m o n o t o n i c a l l y w i t h i n c r e a s i n g t e m p e r a t u r e . Thus, the f l o w i s downward a l o n g the warm l e f t w a l l and upward a l o n g the c o l d r i g h t w a l l c r e a t i n g a u n i c e l l u l a r a n t i c l o c k w i s e c i r c u l a t i o n . Warm water from the l e f t w a l l r e g i o n impinges on the i c e - w a t e r i n t e r f a c e near the bottom and c o o l s as i t r i s e s a l o n g t h i s s u r f a c e by n a t u r a l c o n v e c t i o n . Thus, maximal heat t r a n s f e r o c c u r s near the bottom of the r i g h t w a l l and f a l l s o f f w i t h i n c r e a s i n g e l e v a t i o n . T h i s p r e d i c t i o n i s c o n f i r m e d by the r i g h t w a l l l o c a l N u s s e l t number p l o t and the shape of the i n t e r f a c e ( F i g . 3 1 ) . Because the heat f l u x i n t o the i n t e r f a c e i s h i g h e s t near the bottom, the c o n d u c t i o n r e s i s t a n c e ( i . e . the l e n g t h of the c o n d u c t i o n p ath) t h r o u g h the i c e b l o c k must be s m a l l e s t i n t h i s r e g i o n . In case 2 ( F i g . 3 2 ) , the warm w a l l t e m p e r a t u r e has r i s e n t o 5.6 °C, which i s s l i g h t l y above the maximum d e n s i t y t e mperature of 4°C. In t h i s c a s e , a weaker c l o c k w i s e f l o w can be seen t o be s t a r t i n g i n the lower l e f t hand c o r n e r of the c a v i t y , i n d i c a t i n g the i n f l u e n c e of the r e v e r s a l i n s i g n of the v o l u m e t r i c e x p a n s i o n c o e f f i c i e n t above 4°C. However, the dominant c i r c u l a t i o n i s s t i l l a n t i c l o c k w i s e and because 157 the c a v i t y temperature d i f f e r e n c e i s l a r g e r , i t s f l o w s t r e n g t h i s c o n s i d e r a b l y g r e a t e r than i n c a s e 1. Thus, the N u s s e l t number d i s t r i b u t i o n over the i c e i n t e r f a c e i s even more non-uniform and the i n t e r f a c e assumes an even more n o t i c e a b l e s l o p e . At a warm w a l l t emperature of 8.6°C (case 3, F i g . 3 3 ) , n u m e r i c a l convergence problems were e n c o u n t e r e d . For such a t e mperature d i f f e r e n c e a c r o s s the l i q u i d phase, two c o u n t e r - c i r c u l a t i n g l o o p s of s i m i l a r s i z e d e v e l o p . Because the two l o o p s have almost e q u a l s t r e n g t h , i t i s c o n j e c t u r e d t h a t the f i n a l f l o w p a t t e r n i s o b t a i n e d as a consequence of a v e r y d e l i c a t e momentum b a l a n c e between them. T h e r e f o r e , because the s t e a d y - s t a t e f l o w i s e a s i l y d e s t a b i l i z e d , the c a v i t y c h a r a c t e r i s t i c l e n g t h used i n the n u m e r i c a l i n v e s t i g a t i o n had t o be reduced t o 1.5 cm i n o r d e r t o ensure a converged s o l u t i o n w i t h i n a r e a s o n a b l e CPU c o s t . As a r e s u l t , the n u m e r i c a l stream f u n c t i o n s o l u t i o n does not agree c o m p l e t e l y w i t h the e x p e r i m e n t a l s t r e a k p a t t e r n and a l s o does not e x p l a i n the shape of the i c e - w a t e r i n t e r f a c e . I f the e x p e r i m e n t a l p a t t e r n ( F i g . 33) i s examined c l o s e l y , i t i s c l e a r t h a t the s o l i d i f y i n g i c e mass must be t h i n n e s t a t the t o p because the c l o c k w i s e c i r c u l a t i n g l o o p on the l e f t b r i n g s heat d i r e c t l y from the hot w a l l t o the c o l d one. Over the lower p o r t i o n of the i n t e r f a c e , the i c e i s p a r t i a l l y i n s u l a t e d by an a n t i c l o c k w i s e l o o p d r i v e n by t e m p e r a t u r e s below the d e n s i t y extremum. In t h i s r e g i o n , t h e r e i s no d i r e c t exchange of heat between the l e f t and 158 r i g h t w a l l s . R a t h e r , the c l o c k w i s e l o o p must g i v e up i t s heat by c o n d u c t i o n t o the a n t i c l o c k w i s e l o o p which e v e n t u a l l y t r a n s f e r s t h i s heat t o the i c e . Because the warmest p a r t of t h i s l a t t e r f l o w l o o p s t r i k e s the bottom of the i c e s u r f a c e , the i n t e r f a c e s l o p e s s l i g h t l y t o the l e f t . At a t emperature of 15.1°C (case 4, F i g . 3 4 ) , the r e v e r s a l i n f l o w d i r e c t i o n i s almost complete w i t h the c l o c k w i s e l o o p o r i g i n a t i n g a t the w a l l now almost c o m p l e t e l y d o m i n a t i n g the f l o w i n the c a v i t y . However, a s m a l l c o u n t e r c i r c u l a t i n g f l o w s t i l l e x i s t s a t the v e r y bottom of the i c e s u r f a c e . Once a g a i n , because t h e r e i s no d i r e c t heat exchange between the two w a l l s i n t h i s r e g i o n , the i c e mass i s much t h i c k e r a t the bottom than a t the t o p . N u m e r i c a l convergence problems were a l s o e n c o u n t e r e d i n t h i s c a s e . In c o u n t r a s t t o case 3, the convergence problems of case 4 were due t o the l a r g e t e m p e r a t u r e d i f f e r e n c e t h r o u g h the l i q u i d phase c a v i t y which produced a c o n v e c t i v e f l o w which was a l m o s t t u r b u l e n t i n some r e g i o n s of the c a v i t y . Thus, the c a v i t y c h a r a c t e r i s t i c l e n g t h was once a g a i n reduced t o 1.5 cm i n o r d e r t o a c h i e v e a converged n u m e r i c a l s o l u t i o n . In g e n e r a l , the n u m e r i c a l p r e d i c t i o n s of the f l u i d c i r c u l a t i o n showed r e a s o n a b l y good agreement w i t h the accompanying t i m e - l a p s e s t r e a k p h otographs. In each c a s e , the o v e r a l l f l o w p a t t e r n i s q u a l i t a t i v e l y r e p r o d u c e d . An i n c r e a s e i n the c h a r a c t e r i s t i c l e n g t h used i n the n u m e r i c a l a n a l y s i s would p r o b a b l y l e a d t o the " c a t ' s - e y e " f o r m a t i o n o b s e r v e d i n case 4 and might a l l o w the l e f t c i r c u l a t i o n c e l l 159 t o dominate near the top of the c a v i t y i n case 3. These d i s c r e p a n c i e s seem t o be the o n l y i m p o r t a n t d i f f e r e n c e s between the e x p e r i m e n t a l and n u m e r i c a l r e s u l t s . U n f o r t u n a t e l y , o t h e r i n f o r m a t i o n , such as the e x p e r i m e n t a l temperature d i s t r i b u t i o n s or v e l o c i t y p r o f i l e s were not a v a i l a b l e f o r a d d i t i o n a l v e r i f i c a t i o n of the n u m e r i c a l r e s u l t s . The s u c c e s s of the p r e s e n t s t e a d y - s t a t e program i n s i m u l a t i n g the l i q u i d - s i d e n a t u r a l c o n v e c t i o n f l o w s o c c u r r i n g a t a g i v e n i n s t a n t i n v a r i o u s i c e - f o r m a t i o n e x p e r i m e n t s , s u g g e s t s t h a t the same approach might be used t o model the e n t i r e u n s t e a d y - s t a t e f r e e z i n g p r o c e s s . T h i s i s the i n t e n d e d u l t i m a t e use of the n u m e r i c a l p r o c e d u r e s d e v e l o p e d i n the p r e s e n t t h e s i s , but time d i d not p e r m i t the l a t t e r s i m u l a t i o n s t o be c a r r i e d o u t . However, based on the e x p e r i e n c e g a i n e d i n b u i l d i n g up the p r e s e n t program, a n u m e r i c a l a l g o r i t h m which might a c h i e v e the above-mentioned o b j e c t i v e i s p r e s e n t e d below. F i r s t , the o v e r a l l time i n t e r v a l of i n t e r e s t must be broken up i n t o many s m a l l s u b i n t e r v a l s . These c o u l d be of i n c r e a s i n g d u r a t i o n as the n u m e r i c a l experiment p r o g r e s s e d , r e f l e c t i n g the f a c t t h a t the i c e grows more and more s l o w l y w i t h t i m e . I t would be n e c e s s a r y t o assume t h a t the heat t r a n s f e r p r o c e s s i n both the l i q u i d and the s o l i d i s q u a s i - s t e a d y over any s m a l l s u b i n t e r v a l i n t i m e , i . e . the f l o w and/or tem p e r a t u r e are s t e a d y a t each i n s t a n t . Thus, s t a r t i n g w i t h a f i n i t e l y s m a l l , u n i f o r m l y t h i c k l a y e r of s o l i d phase on the c o l d w a l l 160 of t he c a v i t y , the s o l u t i o n would be extended over each s u b i n t e r v a l by means of the f o l l o w i n g s t e p s : 1. The i r r e g u l a r e n c l o s u r e i n which the l i q u i d phase i s t r a p p e d i s t r a n s f o r m e d i n t o a r e c t a n g u l a r shape u s i n g the o r t h o g o n a l g r i d g e n e r a t i o n r o u t i n e . The c u r v e d i c e - w a t e r i n t e r f a c e would be r e p r e s e n t e d by a s e t of c u b i c s p l i n e s f i t t e d t h r ough the boundary nodes. 2. The i r r e g u l a r shape bounding the s o l i d phase i s s i m i l a r l y t r a n s f o r m e d t o a second r e c t a n g l e . 3. The dependent v a r i a b l e s ( t e m p e r a t u r e , v o r t i c i t y and stream f u n c t i o n ) a r e updated over the f i r s t t r a n s f o r m e d domain. The s o l u t i o n o b t a i n e d a t the end of the p r e v i o u s s u b i n t e r v a l would be used t o dete r m i n e v a l u e s of each dependent v a r i a b l e s f o r the p r e s e n t s u b i n t e r v a l . Note t h a t t h e o n l y new i n f o r m a t i o n s u p p l i e d t o the n a t u r a l c o n v e c t i o n s o l v e r a r e the updated g r i d c h a r a c t e r i s t i c s from s t e p 1. 4. The c o n d u c t i o n problem f o r the s o l i d phase i s r e s o l v e d , i . e . the te m p e r a t u r e i s updated a t each n o d a l p o i n t . Once a g a i n , i n i t i a l t e m p e r a t u r e v a l u e s from the p r e v i o u s s u b i n t e r v a l can be used. Note t h a t because the i n t e r f a c e t e m p e r a t u r e i s f i x e d a t the m e l t i n g t e m p e r a t u r e , the l i q u i d and s o l i d phase s o l u t i o n s a r e uncoupled from one a n o t h e r . 161 5. A heat b a l a n c e i s performed a t each boundary node at the l i q u i d - s o l i d i n t e r f a c e . The d i f f e r e n c e between the heat d i s s i p a t e d by the s o l i d phase at t h a t p o i n t and the heat r e c e i v e d from the l i q u i d phase must e q u a l t o the amount of l o c a l phase change which has t a k e n p l a c e . Knowing the d e n s i t y of the s o l i d phase and the l a t e n t heat of f r e e z i n g , i t then becomes p o s s i b l e t o d e t e r m i n e how f a r the i n t e r f a c e ( i . e . the n o d a l p o s i t i o n ) has l o c a l l y advanced or receded i n the normal d i r e c t i o n t o the s u r f a c e d u r i n g t h a t s u b i n t e r v a l i n t i m e . The new set of the boundary nodes thus c a l c u l a t e d d e f i n e the new i c e - w a t e r i n t e r f a c e . 6. Steps 1 t o 5 a r e r e p e a t e d u n t i l no f u r t h e r change i n the i n t e r f a c e p o s i t i o n i s o b s e r v e d . The n o d a l v a l u e s of a l l dependent v a r i a b l e s then c o r r e s p o n d s t o the f i n a l s t e a d y - s t a t e s o l u t i o n . Note t h a t each time s t e p r e q u i r e s the i t e r a t i v e convergence of two g r i d g e n e r a t i o n problem (one f o r each phase) and two d i f f e r e n t f i n i t e d i f f e r e n c e t r a n s p o r t problems ( t h e d i f f u s i o n and c o n v e c t i o n p r o c e s s t a k i n g p l a c e i n the l i q u i d phase and the c o n d u c t i o n p r o c e s s i n the s o l i d p h a s e ) . Thus, i t i s s u s p e c t e d t h a t the m o d e l l i n g of a time dependent m e l t i n g or f r e e z i n g p r o c e s s i n t h i s manner w i l l r e q u i r e such e x c e s s i v e amounts of computer time t h a t the p o s s i b i l i t y of ever r u n n i n g i t on the p r e s e n t c o m p u t a t i o n a l f a c i l i t i e s i s 162 q u e s t i o n a b l e . However, i f the time s u b i n t e r v a l s a r e s u f f i c i e n t l y s m a l l such the v a r i o u s s o l u t i o n s do not change s i g n i f i c a n t l y over t h a t p e r i o d , the s t a r t i n g dependent v a r i a b l e v a l u e s may be c l o s e enough t o t h e i r next s t e a d y - s t a t e answers t h a t o n l y a s m a l l number of i t e r a t i o n s w i l l be r e q u i r e d t o o b t a i n convergence. In t h i s c a s e , once a s o l u t i o n has been g e n e r a t e d f o r the f i r s t time s t e p , i t may be p o s s i b l e t o o b t a i n a l l of the s u c c e e d i n g ones w i t h r e l a t i v e l y l i t t l e e f f o r t . However, i f t h i s " o p t i m a l " time s u b i n t e r v a l p roves t o be too s h o r t t o be p r a c t i c a l , the s i m u l a t i o n of an u n s t e a d y - s t a t e phase change p r o c e s s may have t o w a i t u n t i l the program can be implemented on a supercomputer, p r e f e r a b l y one w i t h p a r a l l e l p r o c e s s i n g c a p a b i l i t i e s . XI. CONCLUSIONS B o d y - f i t t e d o r t h o g o n a l g r i d s were g e n e r a t e d f o r the most d i s t o r t e d c a v i t i e s of type C1 and C2 u s i n g t h r e e d i f f e r e n t s e t s of g r i d boundary c o n d i t i o n s : c o r r e s p o n d e n c e between C a r t e s i a n and o r t h o g o n a l c o o r d i n a t e s was s p e c i f i e d a l o n g none of the b o u n d a r i e s , the r i g h t w a l l o n l y and the to p , r i g h t and bottom w a l l s . Most of the g r i d boundary c o n d i t i o n s produced g r i d s of r e a s o n a b l y c o n s t a n t d e n s i t y over the e n t i r e domain of th e s e d i s t o r t e d c a v i t i e s . A few boundary c o n d i t i o n s y i e l d e d g r i d s w i t h v e r y low d e n s i t i e s i n r e g i o n s where g r a d i e n t s of stream f u n c t i o n and tem p e r a t u r e were s u b s t a n t i a l . With the e x c e p t i o n of t h i s s m a l l number of u n d e s i r a b l e g r i d s , i t was found t h a t , independent of the g r i d and v o r t i c i t y boundary c o n d i t i o n s used, 1. a g r i d s i z e of 22x22 a s s u r e s n u m e r i c a l r e s u l t s h a v i n g a c c u r a c i e s b e t t e r than 1 p e r c e n t a t a R a y l e i g h number of 10000, 2. a g r i d s i z e of 28x28 a s s u r e s n u m e r i c a l r e s u l t s w i t h a c c u r a c i e s b e t t e r than 1 p e r c e n t f o r Ra=l00000, 3. t h e maximum stream f u n c t i o n , t he average N u s s e l t number and the l e n g t h of the r i g h t w a l l a l l m o n o t i c a l l y approach t h e i r a s y m p t o t i c v a l u e s f o r g r i d s i z e s l a r g e r than 30x30, and 4. a l l of the n u m e r i c a l r e s u l t s o b t a i n e d f o r a g r i d s i z e of 33x33 were a c c u r a t e t o b e t t e r than 1 p e r c e n t . 1 63 1 64 The e f f e c t s of the a m p l i t u d e and the R a y l e i g h number on the n a t u r a l c o n v e c t i o n b e h a v i o u r a r e l i s t e d below: 1. the s t r e n g t h of the f l u i d f l o w i n c r e a s e s w i t h i n c r e a s i n g a m p l i t u d e f o r both the C1 and C2 c a v i t i e s , 2. the s t r e n g t h of the f l u i d f l o w i n c r e a s e s w i t h i n c r e a s i n g R a y l e i g h number, 3. the r a t e of heat t r a n s f e r t h r o u g h the c a v i t y d e c r e a s e s as the a m p l i t u d e i n c r e a s e s f o r the C1 c a v i t y , 4. the r a t e of heat t r a n s f e r t h r o u g h the c a v i t y d e c r e a s e s w i t h i n c r e a s i n g a m p l i t u d e f o r the C2 c a v i t y a t low R a y l e i g h numbers, 5. the r a t e of heat t r a n s f e r t h r o u g h the c a v i t y i n c r e a s e s w i t h i n c r e a s i n g a m p l i t u d e f o r the C2 c a v i t y a t h i g h R a y l e i g h numbers, and 6. the r a t e of heat t r a n s f e r t h r o u g h the c a v i t y i n c r e a s e s w i t h i n c r e a s i n g R a y l e i g h number f o r both the C1 and C2 c a v i t y . The b e h a v i o u r of the s t r e n g t h of f l u i d f l o w w i t h r e s p e c t t o the a m p l i t u d e and the R a y l e i g h number was r a t i o n a l i z e d by c o n s i d e r i n g the e f f e c t of the R a y l e i g h number on the sour c e term i n the v o r t i c i t y e q u a t i o n and the e f f e c t of the c a v i t y shape i n e i t h e r i n h i b i t i n g or enhancing f l u i d f l o w . The b e h a v i o u r of the c a v i t y N u s s e l t number w i t h r e s p e c t t o the a m p l i t u d e and R a y l e i g h number was i n t e r p r e t e d by c o n s i d e r i n g the o p p o s i n g e f f e c t s of the s t r e n g t h of c o n v e c t i o n and the 165 the average d i s t a n c e between the two i s o t h e r m a l w a l l s on the a b i l i t y of the c a v i t y t o t r a n s f e r h e a t . A l s o , the average N u s s e l t number a l o n g the l e f t w a l l was c o r r e l a t e d w i t h the R a y l e i g h number by u s i n g a power law e q u a t i o n of the form Y=aX^ f o r each c a v i t y type (C1 and C2) and f o r each a m p l i t u d e . I t was found t h a t the c o e f f i c i e n t a d e c r e a s e s w h i l e the exponent b i n c r e a s e s as the a m p l i t u d e i s r a i s e d f o r b o th c a v i t y t y p e s . The n u m e r i c a l p r e d i c t i o n s of the f l u i d c i r c u l a t i o n f o r a l i q u i d which does not have a l i n e a r d e n s i t y - t e m p e r a t u r e r e l a t i o n s h i p showed r e a s o n a b l y good agreement w i t h the f l o w v i s u a l i s a t i o n e x p e r i m e n t s c a r r i e d out by E c k e r t [ 4 2 ] . In t h r e e c a s e s out of f o u r , the g e n e r a l f l o w p a t t e r n was r e p r o d u c e d f a i r l y a c c u r a t e l y . N u m e r i c a l convergence problems were e n c o u n t e r e d i n two c a s e s and were caused e i t h e r by the v e r y d e l i c a t e b a l a n c e of momentum which o c c u r s between two c o u n t e r - c i r c u l a t i n g l o o p s of s i m i l a r s i z e or by a v e r y l a r g e imposed te m p e r a t u r e d i f f e r e n c e . X I I . RECOMMENDATIONS S i m i l a r n u m e r i c a l s i m u l a t i o n s t o the one p r e s e n t e d i n t h i s t h e s i s s h o u l d be performed on o t h e r f a m i l i e s of c a v i t i e s t o f u r t h e r i n v e s t i g a t e t he m e r i t s of u s i n g b o d y - f i t t e d o r t h o g o n a l c o o r d i n a t e systems t o s o l v e c o u p l e d t r a n s p o r t phenomena problems and t o l e a r n and und e r s t a n d more about n a t u r a l c o n v e c t i o n phenomena. C a v i t i e s which are even more d i s t o r t e d than the ones employed here s h o u l d be i n v e s t i g a t e d t o h e l p d i f f e r e n t i a t e between the e f f e c t i v e n e s s of the v a r i o u s g r i d and v o r t i c i t y boundary c o n d i t i o n s t r i e d . The use of a d i r e c t l i n e a r e q u a t i o n s o l v e r ( s p a r s e m a t r i x s o l v e r ) s h o u l d be i n v e s t i g a t e d t o h e l p overcome n u m e r i c a l convergence problems which o c c a s i o n a l l y o c c u r when i t e r a t i v e p r o c e d u r e s a r e used t o s o l v e c o u p l e d energy and momentum e q u a t i o n s . To f u r t h e r improve the convergence r a t e of f i n i t e d i f f e r e n c e s o l u t i o n s , e f f o r t s s h o u l d be spent on f i n d i n g a workable p r o c e d u r e which w i l l o p t i m a l l y a d j u s t the r e l a x a t i o n f a c t o r s as the i t e r a t i v e p r o c e s s i s b e i n g e x e c u t e d . 166 NOMENCLATURE a,b G e n e r a l c o e f f i c i e n t s A C a v i t y d i m e n s i o n l e s s a m p l i t u d e B,C C a v i t y w a l l l e n g t h s Cp F l u i d s p e c i f i c heat c a p a c i t y Dev D e v i a t i o n of the n u m e r i c a l l y g e n e r a t e d body f i t t e d o r t h o g o n a l g r i d from o r t h o g o n a l e U n i t v e c t o r f Shape f a c t o r F G e n e r a l a n a l y t i c a l f u n c t i o n g,g G r a v i t a t i o n a l a c c e l e r a t i o n magnitude and v e c t o r Gr Grashof number h C o n v e c t i v e heat t r a n s f e r c o e f f i c i e n t H S c a l e f a c t o r component i , j Node i n d i c e s k F l u i d t h e r m a l c o n d u c t i v i t y L G e n e r a l d i m e n s i o n l e s s l e n g t h L c C a v i t y c h a r a c t e r i s t i c l e n g t h M,N Maximum v a l u e s of £ and rj, r e s p e c t i v e l y Nu. N u s s e l t number Nu x N u s s e l t number c a l c u l a t e d w i t h r e s p e c t t o the d i s t a n c e a l o n g a v e r t i c a l f l a t p l a t e i n the f l o w d i r e c t i o n 0 Order of magnitude of the r e m a i n i n g term i n a t r u n c a t e d s e r i e s p P r e s s u r e P D i m e n s i o n l e s s p r e s s u r e Pe L o c a l P e c l e t number c a l c u l a t e d w i t h r e s p e c t t o the d i s t a n c e between two d i s c r e t e p o i n t s Pr P r a n d t l number 167 168 q H e a t f l u x c o m p o n e n t Q H e a t t r a n s f e r r a t e t h r o u g h , t h e c a v i t y R C a v i t y a s p e c t r a t i o R a R a y l e i g h n u m b e r R a x R a y l e i g h n u m b e r c a l c u l a t e d w i t h r e s p e c t t o t h e d i s t a n c e a l o n g a v e r t i c a l f l a t p l a t e i n t h e f l o w d i r e c t i o n T T e m p e r a t u r e ( K o r o t h e r w i s e s p e c i f i e d ) v V e l o c i t y c o m p o n e n t V D i m e n s i o n l e s s v e l o c i t y c o m p o n e n t x , y C a r t e s i a n c o o r d i n a t e s X , Y D i m e n s i o n l e s s C a r t e s i a n c o o r d i n a t e s Z n , Z t N o r m a l a n d t a n g e n t i a l d i m e n s i o n l e s s g e n e r a l o r t h o g o n a l c o o r d i n a t e s w i t h r e s p e c t t o a w a l l Z 1 , Z 2 D i m e n s i o n l e s s g e n e r a l o r t h o g o n a l c o o r d i n a t e s a G e n e r a l c o e f f i c i e n t o r g e n e r a l v e c t o r c o m p o n e n t a G e n e r a l v e c t o r 0 F l u i d t h e r m a l v o l u m e t r i c e x p a n s i o n c o e f f i c i e n t 7 C a v i t y a n g l e o f t i l t r . G e n e r a l d i f f u s i o n c o e f f i c i e n t A Z n , A Z 1 , A T J , A > ; D i m e n s i o n l e s s s p a c e i n c r e m e n t s D i m e n s i o n l e s s g e n e r a l o r t h o g o n a l c o o r d i n a t e s 6 D i m e n s i o n l e s s t e m p e r a t u r e 0 A n g l e o f d e v i a t i o n f r o m o r t h o g o n a l i t y M V i s c o s i t y p F l u i d d e n s i t y <$> G e n e r a l d e p e n d e n t v a r i a b l e \p D i m e n s i o n l e s s s t r e a m f u n c t i o n 169 SUBSCRIPTS ave bw , lw , rw ,tw c db dyn e , n , s , w h i / j x,y,z w z n , z t z 1 , z2 , z3 0 1,2,3 CO D i m e n s i o n l e s s v o r t i c i t y D i m e n s i o n l e s s v o r t i c i t y v e c t o r Average Bottom, l e f t , r i g h t and t o p c a v i t y w a l l s C a v i t y c o l d w a l l ( E x c e p t i o n : L c ) Domain boundary Dynamic E a s t , n o r t h , s o u t h and west c o n t r o l volume f a c e i n d i c e s , r e s p e c t i v e l y C a v i t y hot w a l l Node i n d i c e s C a r t e s i a n c o o r d i n a t e d i r e c t i o n s ( E x c e p t i o n s : N u x and Ra x) w a l l or node index Normal and t a n g e n t i a l g e n e r a l o r t h o g o n a l c o o r d i n a t e d i r e c t i o n s w i t h r e s p e c t t o a w a l l G e n e r a l o r t h o g o n a l c o o r d i n a t e d i r e c t i o n s R e f e r e n c e t e m p e r a t u r e or p o s i t i o n C o e f f i c i e n t or node i n d i c e s G e n e r a l o r t h o g o n a l c o o r d i n a t e d i r e c t i o n s C o n d i t i o n s f a r from the i s o t h e r m a l f l a t p l a t e REFERENCES C a t t o n , I . , " N a t u r a l C o n v e c t i o n i n E n c l o s u r e s " , P r o c e e d i n g of the S i x t h I n t e r n a t i o n a l Heat T r a n s f e r C o n f e r e n c e , T o r o n t o , Canada, v o l . 6 , pp.13-31, Hemisphere P u b l i s h i n g C o r p o r a t i o n , Canada, 1978. 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Mobley, C D . and S t e w a r t , R.J., "On t h e N u m e r i c a l G e n e r a t i o n of B o u n d a r y - f i t t e d O r t h o g o n a l C u r v i l i n e a r C o o r d i n a t e Systems", J o u r n a l of C o m p u t a t i o n a l P h y s i c s , v o l . 3 4 , pp.124-135, 1980. 29. Pope, S.B., "The C a l c u l a t i o n of T u r b u l e n t R e c i r c u l a t i n g Flows i n G e n e r a l O r t h o g o n a l C o o r d i n a t e s " , J o u r n a l of C o m p u t a t i o n a l P h y s i c s , v o l . 2 6 , pp.197-217, 1978. 30. B u r m e i s t e r , L . C . , " C o n v e c t i v e Heat T r a n s f e r " , John W i l e y & Sons, USA, 1983. 31. Holman, J.P., "Heat T r a n s f e r " , 5th e d i t i o n , M c G r a w - H i l l , USA, 1981. 32. B i r d , R.B., S t e w a r t , W.E. and L i g h t f o o t , E.N., "T r a n s p o r t Phenomena", John W i l e y & Sons, USA, 1960. 173 33. J a l u r i a , Y., " N a t u r a l C o n v e c t i o n " , The S c i e n c e and A p p l i c a t i o n s of Heat and Mass T r a n s f e r , v o l . 5 , Pergamon P r e s s , G r e a t B r i t a i n , 1980. 34. Chinnappa, J.C.V., "Free C o n v e c t i o n i n A i r Between a 60 Degree V e e - c o r r u g a t e d P l a t e and a F l a t P l a t e " , I n t . J . Heat Mass T r a n s f e r , v o l . 1 3 , pp.117-123, 1970. 35. E l S h e r b i n y , S.M., "Free C o n v e c t i o n i n an I n c l i n e d A i r Layer C o n t a i n e d Between V - c o r r u g a t e d and F l a t P l a t e s " , M.A.Sc. T h e s i s , U n i v e r s i t y of W a t e r l o o , Canada, 1977. 36. E l S h e r b i n y , S.M., H o l l a n d s , K.G.T. and R a i t h b y , G.D., "Free C o n v e c t i o n a c r o s s I n c l i n e d A i r L a y e r s w i t h One S u r f a c e V - c o r r u g a t e d " , J o u r n a l of Heat T r a n s f e r , v o l . 1 0 0 , pp.410-415, 1978. 37. R a n d a l l , K.R., "An I n t e r f e r o m e t r i c Study of N a t u r a l C o n v e c t i o n Heat T r a n s f e r i n F l a t P l a t e and V - c o r r u g a t e d E n c l o s u r e s " , Ph.D T h e s i s , U n i v e r s i t y of W i s c o n s i n , USA, 1978. 38. Anderson, D.A., T a n n e h i l l , J.C. and P l e t c h e r , R.H., "C o m p u t a t i o n a l F l u i d Mechanics and Heat T r a n s f e r " , Hemisphere P u b l i s h i n g C o r p o r a t i o n , USA, 1984. 39. P a t a n k a r , S.V., " N u m e r i c a l Heat T r a n s f e r and F l u i d Flow", Hemisphere P u b l i s h i n g C o r p o r a t i o n , USA, 1980. 40. Roache, P . J . , " C o m p u t a t i o n a l F l u i d Dynamics", Hermosa P u b l i s h e r s , USA, 1972. 41. D a v i s , H.F. and S n i d e r , A.D., " I n t r o d u c t i o n t o V e c t o r A n a l y s i s " , 4 t h e d i t i o n , A l l y n and Bacon, USA, 1979. 42. E c k e r t , N., "An E x p e r i m e n t a l Study of N a t u r a l C o n v e c t i o n of Water D u r i n g I c e F o r m a t i o n " , B.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h C olumbia, Canada, 1984. 43. P e y r e t , R. and T a y l o r , T.D., " C o m p u t a t i o n a l Methods f o r F l u i d Flow", S p r i n g e r - V e r l a g New Y o r k , USA, 1983. 44. Gosman, A.D., Pun, W.M., R u n c h a l , A.K., S p a l d i n g , D.B. and W o l f s h t e i n , M., "Heat and Mass T r a n s f e r i n R e c i r c u l a t i n g Flow", Academic P r e s s , G r e a t B r i t a i n , 1 969. 45. A r i s , R., " V e c t o r s , T e n s o r s , and the B a s i c E q u a t i o n s of F l u i d M e c hanics", P r e n t i c e - H a l l , USA, 1962. 46. S p i e g e l , M.R., " V e c t o r A n a l y s i s " , Schaum P u b l i s h i n g Company, USA, 1959. 174 47. C u r r i e , I.G., "Fundamental Mechanics of F l u i d s " , M c G r a w - H i l l , USA, 1974. 48. Hornbeck, R.W., " N u m e r i c a l Methods", P r e n t i c e - H a l l , USA, 1975. 49. Carnahan, B., L u t h e r , H.A. and W i l k e s , J.O. " A p p l i e d N u m e r i c a l Methods", John W i l e y & sons, USA, 1969. 50. F o r s y t h e , G.E., M a l c o l m , M.A. and M o l e r , C.B., "Computer Methods f o r M a t h e m a t i c a l C o m p u t a t i o n " , P r e n t i c e - H a l l , USA, 1977. 51. Weast, R.C., "Handbook of C h e m i s t r y and P h y s i c s " , 56th e d i t i o n , CRC P r e s s , USA, 1975. APPENDIX A The program which was used t o g e n e r a t e the n u m e r i c a l r e s u l t s of P a r t I , P a r t I I and P a r t I I I ( w i t h some minor m o d i f i c a t i o n s ) i s l i s t e d h e r e . The l i s t i n g i n c l u d e s d e s c r i p t i o n s of both the s u b r o u t i n e s and the i m p o r t a n t v a r i a b l e s . The i n i t i a l i s a t i o n p r o c e d u r e , the d i s c r e t i z e d e q u a t i o n s and boundary c o n d i t i o n s and the methods used t o s o l v e them, the a c c u r a c y c r i t e r i a and the d i v e r g e n c e t e s t s were d e s c r i b e d i n p r e v i o u s c h a p t e r s . There i s o n l y one d i f f e r e n c e between the program d e s c r i p t i o n g i v e n i n the main body of the t h e s i s and the program i t s e l f . T h i s d i f f e r e n c e c o r c e r n s the way the l o c a t i o n of the d i s c r e t e g r i d p o i n t v a l u e s a r e i n d i c a t e d on the t r a n s f o r m e d g r i d . In the program, the g r i d p o i n t s a r e numerated u s i n g the i n d i c e s I and J which a r e a s s o c i a t e d w i t h the 77 and £ d i r e c t i o n s , r e s p e c t i v e l y . A l s o , the node 1=1 and J=1 i s a s s o c i a t e d t o the t o p l e f t c o r n e r of the t r a n s f o r m e d domain. The i n d i c e s I and J s t i l l v a r y from 1 t o M (M+1 f o r the s t a g g e r e d g r i d ) and 1 t o N (N+1 f o r . t h e s t a g g e r e d g r i d ) , r e s p e c t i v e l y . O n l y one c a v i t y shape can be h a n d l e d a t a time by the program, but many g r i d c h a r a c t e r i s t i c s , v o r t i c i t y boundary c o n d i t i o n s and g r i d s i z e s can be i n v e s t i g a t e d i n a s i n g l e e x ecut i o n . 1 75 C NAME: MAIN. r C PURPOSE: C C THIS SUBROUTINE READS IN DATA. MANAGES THE D I F F E R E N T I N T E R A C -C TION BETWEEN OTHER IMPORTANT SUBROUT INES . AND SET-UP. MAGNIFY OR C REDUCED ARRAYS. C C C H A R A C T E R I S T I C : C C - L I N E A R TWO -D IMENS IONAL I N T E R P O L A T I O N . C c- — - — - .... C C IMPORTANT V A R I A B L E S : c  C C ' - CO: CURRENT RUN NUMBER. C - COUL : MAXIMUM NUMBER OF RUNS. C - C1UL: MAXIMUM NUMBER OF I T E R A T I O N ALLOWED TO SOLVED C OF THE D I S C R E T I Z E D GR ID D I F F E R E N T I A L EOUAT IONS . C - C2UL: MAXIMUM NUMBER OF I T E R A T I O N ALLOWED TO SOLVED C THE D I S C R E T I Z E D CONSERVAT ION E O U A T I O N S . C - P T C : ARRAY C O N T A I N I N G THE C A R T E S I A N COORDINATE C OF THE P H Y S I C A L DOMAIN CORNER. C - R S F : ARRAY OF THE R A T I O OF SCALE F A C T O R S . C - S F : ARRAY OF STREAM FUNCT ION V A L U E S . C - T : ARRAY OF TEMPERATURE V A L U E S . C - VOR: ARRAY OF V O R T I C I T Y V A L U E S . C - X Y : ARRAY OF C A R T E S I A N V A L U E S . C - S F N : ARRAY OF P R E D I C T E D STREAM F U N C T I O N V A L U E S C FOR A NEW RUN . C - T N : ARRAY OF P R E D I C T E D TEMPERATURE V A L U E S FOR A NEW RUN. C - VORN: ARRAY OF P R E D I C T E D V O R T I C I T Y V A L U E S FOR A NEW RUN. C - X Y N : ARRAY OF P R E D I C T E D C A R T E S I A N COORD INATES V A L U E S C FOR A NEW RUN . C - B T Y P E : ARRAY OF G R I D BONDARY C O N D I T I O N S . C - M: MAXIMUM V A L U E OF THE I N D I C E I FOR THE NON STAGGERED C G R I D . C - N: MAXIMUM V A L U E OF THE I N D I C E J FOR THE NON STAGGERED C G R I D . C - MN: MAXIMUM VALUE OF THE I N O I C E I (NON C STAGGERED G R I D ) FOR THE NEXT RUN. C - N N : MAXIMUM VALUE OF THE I N D I C E <J (NON STAGGERED GR ID ) C FOR THE NEW RUN. C - EPS 1: ABSOLUTE ACCURACY USED TO STOP THE GR ID G E N E R A T I O N . C - R F X Y : R E L A X A T I O N FACTOR FOR THE C A R T E S I A N COORD INATES . C - EPS2: R E L A T I V E ACCURACY USED TO STOP THE NATURAL C CONVECT ION C A L C U L A T I O N . C - R F T : R E L A X A T I O N FACTOR FOR TEMPERATURE . STREAM FUNCT ION C AND V O R T I C I T Y . C - V B C : V O R T I C I T Y BOUNDARY C O N D I T I O N : C - PR : PRANDTL NUMBER. C - RA: R A Y L E I G H NUMBER. C - H: ARRAY OF S C A L E F A C T O R S . C - 0 : ARRAY OF D E R I V A T I V E S OF C A R T E S I A N COORD INATES C WITH RESPECT TO THE ORTHOGONAL COORD INATES . ARRAY OF THE TDMA C O E F F I C I E N T S . ARRAY OF THE TDMA C O E F F I C I E N T S . ARRAY OF THE TDMA C O E F F I C I E N T S . ARRAY OF THE TDMA C O E F F I C I E N T S . ARRAY OF THE TDMA C O E F F I C I E N T S . ARRAY OF THE TDMA C O E F F I C I E N T S . ATDMA: BTDMA: CTDMA: OTDMA: PTDMA: OTDMA: S L N : ARRAY OF S O L U T I O N V A L U E S . MAXSF : MAXIMUM STREAM FUNCT ION V A L U E S . MAXVOR: MAXIMUM V O R T I C I T Y VALUES . D I S T : ARRAY OF D I S T A N C E S ALON THE BOUNDARY SHAPE C A L C U L A T E D FROM THE TOP WALL. NUAV: ARRAY OF AVERAGE NUSSELT NUMBER. NUL : ARRAY OF LOCAL N U S S E L T NUMBER. XY2 : ARRAY OF CONTOUR V A L U E S OF A SCALAR AND I T S C A R T E S I A N C O O R D I N A T E S . 2CNTR : ARRAY OF V A L U E S OF CONTOUR L I N E S . X Y S : ARRAY OF C A T E S I A N P O S I T I O N S OF THE STAGGERED GRID NODES. COND CONV D l FA CONDUCTION STRENGTH THROUGH A CONTROL VOLUME F A C E . CONVECT ION STRENGTH THROUGH A CONTROL VOLUME V A C E . MAXIMUM GR ID D E V I A T I O N FROM ORTHOGONAL ITY ( D E G R E E ) >«:*t>»l I M P L I C I T R E A L * 8 ( A - H . D - 2 ) INTEGER B T Y P E ( 4 ) , C O , C O U L , C 1 U L , C 2 U L , V B C R E A L * 8 P T C ( 2 . 4 ) , R S F ( 5 0 . 5 0 ) R E A L * 8 S F N ( 5 0 . 5 0 ) , T N ( 5 0 . 5 0 ) . X Y N ( 5 0 , 5 0 , 2 ) , V 0 R N ( 5 O , 5 O ) R E A L » 8 S F ( 5 0 . 5 0 ) , T ( 5 0 , 5 0 ) , X Y ( 5 0 , 5 0 . 2 ) . V 0 R ( 5 O . 5 O ) COMMON / B L K 1 / M,N COMMON / B L K 2 / B T Y P E . R S F COMMON / B L K 3 / XY COMMON / B L K G / P R . R A COMMON / B L K 7 / T COMMON / B L K 8 / VOR COMMON / B L K 9 / SF COMMON / B L K 1 1 / EPS 1 . R F X Y , C 1 U L COMMON / B L K 1 2 / E P S 2 . R F T , R F V O R . R F S F , C 2 U L . VBC SET UP THE G R A P H I C SOFTWARE. C A L L DSPDEV ( ' P L O T ' ) P R E L I M I N A I R Y INPUT D A T A . READ ( 5 . 3 G O ) COUL READ ( 5 . 3 7 0 ) P T C ( 1 . 1) READ ( 5 . 3 7 0 ) P T C ( 2 . 1 ) READ ( 5 . 3 7 0 ) P T C ( 1 . 2 ) READ ( 5 . 3 7 0 ) P T C ( 2 . 2 ) READ ( 5 . 3 7 0 ) P T C ( 1 . 3 ) READ ( 5 . 3 7 0 ) P T C ( 2 . 3 ) READ ( 5 , 3 7 0 ) P T C ( 1 . 4 ) READ ( 5 . 3 7 0 ) P T C ( 2 . 4 ) O V E R A L L LOOP. INPUT DATA. DO 3 5 0 C0=1 .COUL READ ( 5 . 3 6 0 ) MM READ ( 5 . 3 6 0 ) NN READ ( 5 . 3 6 0 ) C 1UL READ ( 5 . 3 7 0 ) EPS 1 READ ( 5 . 3 70 ) RFXY READ ( 5 . 3 6 0 ) B T Y P E f 1 ) READ ( 5 . 3 6 0 ) B T V P E{ 2 ) READ ( 5 . 3 6 0 ) ET Y P E ( 3 ) READ ( 5 . . 3 6 0 ) B T Y P E f 4 ) READ ( 5 . . 3 7 0 ) R S F ( 1 . 1 ) READ ( 5 , . 3 70 ) RSF ( 1 . NN ) READ ( 5 , . 3 70 ) R S F ( M N . NN READ ( 5 . , 3 7 0 ) R S F ( M N . 1 ) READ ( 5 , , 3 70 ) PR READ ( 5 . , 3 70 ) RA READ ( 5 . 3 6 0 ) C2UL READ ( 5 . 3 7 0 ) E P S 2 READ ( 5 . 3 7 0 ) RFT READ ( 5 . 3 7 0 ) RFVOR READ ( 5 . 3 7 0 ) RFSF READ ( 5 . 3 6 0 ) VBC PR INTOUT THE INPUT D A T A . WRITE ( 6 , 3 8 0 ) CO. MN, NN . C1 UL , EPS 1 , RFXY , ( B T Y P E ( I ) . I •= 1 . A ), 1 R S F ( 1 , 1 ) , R S F ( 1 . N N ) , R S F ( M N , N N ) , R S F ( M N . 1 ) , P R , R A . C 2 U L . E P S 2 . 2 R F T . R F V O R , R F S F , V B C SET UP THE I N I T I A L C O N D I T I O N . I F ( C O . N E . 1 ) GOTO 9 0 M=MN MP1=M-M MM1=M-1 MM2=M-2 N = NN NP1=N+1 NM1=N-1 NM2=N-2 THE X AND Y VALUES OF BOUNDARY PO INTS ARE C A L C U L A T E D U S I N G EOUAL I N T E R V A L S OF THE DEPENDENT V A R I A B L E . X Y ( 1 . 1 , 1 ) = P T C ( 1 . 1 ) X Y ( 1 , 1 , 2 ) = P T C ( 2 . 1 ) X Y ( 1 , N , 1 ) = P T C ( 1 . 2 ) X Y ( 1 , N , 2 ) = P T C ( 2 , 2 ) X Y ( M . N . 1 ) = P T C ( 1 . 3 ) X Y ( M . N . 2 ) = P T C ( 2 . 3 ) X Y ( M , 1 . 1 )=PTC( 1 . 4 ) X Y ( M . 1 . 2 ) = P T C ( 2 . 4 ) TOP WALL . A M O = ( X Y ( 1 . N , 1 ) - X Y ( 1 . 1 . 1 ) ) / N M 1 DO 10 J=M ,NM2 X Y ( 1 . J + 1 . 1 ) = X Y ( 1 . 1 . 1 ) + J ' A M O X Y ( 1 , J + 1 . 2 ) = F ( X Y ( 1 , J + 1 . 1 ) . 1 ) CONT INUE C * * " R IGHT WALL. C AMO= (XY (M .N .2 ) - XV( 1 .N.2 ) )/MM1 DO 2 0 1=1.MM2 X Y ( I - M . N . 2 ) = X Y ( 1 , N , 2 ) +] "AMO X Y ( ] + 1 . N . 1 ) = F ( X Y ( J + V. N . 2 ) . 2 ) 2 0 CONTINUE C C * » * BOTTOM WALL. C A M O = ( X Y ( M . N . 1 ) - X Y ( M . 1 . 1 ) )/NM1 DO 3 0 J=1 .NM2 X Y ( M . J + 1 , 1 ) = X Y ( M , 1 . 1 ) + J * A M 0 X Y ( M . J + 1 . 2 ) = F ( X Y ( M . J + 1 . 1 ) . 3 ) 3 0 CONT INUE C C * * * L E F T WALL. C A M O = ( X Y ( M . 1 . 2 ) - X Y ( 1 , 1 , 2 ) ) / M M 1 DO 4 0 1=1,MM2 X Y ( 1 + 1 . 1 , 2 ) = X Y ( 1 , 1 , 2 ) + I * A M O X Y ( I + 1 , 1 . 1 ) = F ( X Y ( 1 + 1 . 1 . 2 ) . 4 ) 4 0 CONT INUE C C * » * I N T E R N A L X AND Y V A L U E S . C DO 6 0 I=2,MM1 A M O = ( X Y ( I , N , 1 ) - X Y ( I , 1 . 1 ) ) / N M 1 A M 1 = ( X Y ( I , N . 2 ) - X Y ( I , 1 , 2 ) ) / N M 1 DO 5 0 J=1 .NM2 X Y ( I , J + 1 . 1 ) = X Y ( I . 1 , 1 ) + J * A M O X Y ( I , J+1 , 2 ) -=XY( I , 1 , 2 ) + J - A M 1 5 0 CONT INUE GO CONT INUE C C * * * THE TEMPERATURE I S SET UP CONS IDER ING ONLY THE CONDUCT ION. C * * * S F ( I , J ) = 0 . ( I M P E R M E A B L E W A L L S ) AND V O R ( I . J ) = 0 . C DO 8 0 1 = 1.MP 1 DO 7 0 J =1 ,NP1 S F ( I . J ) = 0 . D 0 T ( I , J ) = 1 . D O - ( 2 . D O * ( J - 1 . D O ) - 1 . D 0 ) / ( 2 . D O * N M 1 ) V 0 R ( I . J ) = O . D O 7 0 CONT INUE 8 0 CONT INUE GOTO 3 4 0 C C * * - M A G N I F I C A T I O N OR R E D U C T I O N OF ARRAYS . C 9 0 I F ( ( M N . E O . M ) . A N D . ( N N . E O . N ) ) GOTO 3 4 0 C C * » * COORD INATES AND STREAM F U N C T I O N . C DO 150 I=1.MN DO 140 J = 1 . N N DO 100 I I = 2 . M I F ( ( 11 - 1 . )/ (M-1 . ) . G T . ( I - 1 . )/ (MN-1 . ) ) GOTO 110 100 CONTINUE 1 10 11 = 1 1 - 1 DO 120 J J = 2 . N 180 I F ( ( J J - 1 . ) / ( N - 1 . ) . G T . ( J - 1 . ) / ( N N - 1 . ) ) GOTO 130 120 CONTINUE 130 d d = d J - 1 AMO = ( ] - 1 . D 0 ) / ( M N - 1 .DO) - (11 - 1 .DO)/ (M-1 .DO) AM1 = ( J - 1 . D O ) / ( N N - 1 . D O ) - ( d d - 1 . D O ) / ( N - 1 .DO) A M 2 = 1 . D O / ( M - 1 . 0 0 ) AM3=1 .DO/ IN -1.DO) AM4 = X Y ( 1 1 + 1 . J d . 1 ) - X Y ( I I . J d , 1 ) AM5=XY( I I . d d + 1 . 1 ) - X Y ( I I , d d , 1 ) A M 6 = X Y ( l ] + 1 . d d + 1 . 1 ) - X Y ( I I + 1 . d d . 1 ) - X Y ( I I , d d + 1 . 1 ) + XY( I I . d d . 1 ) X Y N ( l . d . 1 ) = X Y ( I I . d d . 1 )+AM0 'AM4/AM2+AM1-AM5/AM3 1 + A M O " A M 1 " A M 6 / ( A M 2 * A M 3 ) A M 4 = X Y ( 1 1 + 1 . d d . 2 ) - X Y ( I l . d d . 2 ) A M 5 = X Y ( I I , d d + 1 , 2 ) - X Y ( I I . d J , 2 ) AM6 = XY(11 + 1 . d d + 1 . 2 ) - X Y ( I I + 1 , d d . 2 ) - X Y ( I I . d d + 1 , 2 ) + X Y ( I I , d d . 2 ) X Y N ( I , d . 2 ) = X Y ( I I . d d . 2 ) + A M 0 * A M 4 / A M 2 + A M 1 " A M 5 / A M 3 1+AMO*AM 1 * A M 6 / ( A M 2 * AM3) AM4 = SF(11 + 1 . d d ) - S F ( I I , d d ) AM5 = S F ( I I . d d + 1 ) - S F ( I I , d d ) AM6=SF<11 + 1 , d d + 1 ) - S F (11 + 1 , d d ) - S F ( I I , d d + 1 ) + S F ( I I , 0 J ) S F N ( I , d ) = S F ( I I , d d ) + AMO* AM4/AM2 + AM1 * AM5/AM3 1+AMO*AM1 *AMG/(AM2 * AM3 ) C C * * * IMPERMEABLE W A L L S . C I F ( ( I . E O . 1 ) . D R . ( I . E O . M N ) ) S F N ( I , d ) = 0 . D O I F ( ( d . E O . 1 ) . O R . ( d . E O . N N ) ) S F N ( I , d ) = 0 . D O 140 CONTINUE 150 CONTINUE C C * * * R E A S S I G N THE BOUNDA IRY PO INT COORD INATES . C C * * * CORNER P O I N T S . C X Y N ( 1 . 1 , 1 ) = P T C ( 1 . 1 ) X Y N ( 1 . 1 , 2 ) = P T C ( 2 , 1 ) X Y N ( 1 , N N . 1 ) = P T C ( 1 , 2 ) X Y N ( 1 . N N . 2 ) = P T C ( 2 . 2 ) X Y N ( M N , N N . 1 ) = P T C ( 1 . 3 ) X Y N ( M N . N N . 2 ) = P T C ( 2 , 3 ) X Y N ( M N , 1 , 1 ) = P T C ( 1 , 4 ) X Y N ( M N , 1 , 2 ) = P T C ( 2 , 4 ) C C * * * TOP WALL . C I F ( B T Y P E ( 1 ) . E O . O ) GOTO 170 A M O = ( X Y N ( 1 , N N , 1 ) - X Y N ( 1 , 1 , 1 ) ) / ( N N - 1 . D O ) NNM2 =NN-2 DO 160 d=1.NNM2 X Y N ( 1 , d + 1 . 1 ) = X Y N ( 1 . 1 . 1 ) + d * A M O X Y N ( 1 , d + 1 , 2 ) = F ( X Y N ( 1 . d + 1 , 1 ) , 1 ) 160 CONT INUE C C * * * R IGHT WALL. C 170 I F ( B T Y P E ( 2 ) . E O . O ) GOTO 190 A M O = ( X Y N ( M N , N N . 2 ) - X Y N ( 1 . N N , 2 ) ) / ( M N - 1 . DO ) MNM2=MN-2 DO 180 1=1.MNM2 X Y N ( I + 1 .NN ,2 )= XYN( 1 . N N . 2 ) + I *AMO X Y N ( J + 1 . N N . 1 ) = F ( X Y N ( I + 1 , N N . 2 ) . 2 ) 180 CONTINUE C C * - • BOTTOM WALL . r-190 IF ( E T Y P E O ) . E O . O ) GOTO 2 1 0 AMO = (XYN(MN.NN. 1 ) - X Y N ( M N . 1 , 1 ) )/ (NN-1 . 0 0 ) NNM2 =NN-2 DO 2 0 0 J =1.NNM2 X Y N ( M N . J + 1 . 1 ) = X Y N ( M N , 1 . 1 ) + J * A M O X Y N ( M N . J + 1 . 2 ) = F ( X Y N ( M N . J + 1 . 1 ) . 3 ) 2 0 0 CONTINUE C C - * L E F T WALL. C 2 1 0 I F ( B T Y P E ( 4 ) . E O . O ) GOTO 2 3 0 AM0=(XYN{MN, 1 . 2 ) - X Y N ( 1 . 1 . 2 ) ) / ( M N - 1 . D O ) MNM2=MN-2 DO 2 2 0 1=1.MNM2 X Y N ( I + 1 . 1 , 2 ) = X Y N ( 1, 1 , 2 ) + I * A M 0 X Y N U + 1 . 1 , 1 ) =F (XYN(1+1 . 1 . 2 ) .4 ) 2 2 0 CONTINUE C C * * * TEMPERATURE AND V O R T I C I T Y . C 2 3 0 MP1=M+1 NP1=N+1 MNP 1=MN+1 NNP1=NN+1 DO 2 9 0 1 = 1.MNP 1 DO 2 8 0 J = 1 . N N P 1 DO 2 4 0 11=2,MP 1 I F ( ( I I - 1 . 5 ) / ( M - 1 . ) . G T . ( 1 - 1 . 5 ) / ( M N - 1 . ) ) GOTO 2 5 0 2 4 0 CONTINUE 2 5 0 11=11-1 DO 2 6 0 J J = 2 , N P 1 I F ( ( J J - 1 , 5 ) / ( N - 1 . ) . G T . ( J - 1 . 5 ) / ( N N - 1 . ) ) GOTO 2 7 0 2 6 0 CONTINUE 2 7 0 J J = J J - 1 A M 0 = ( I - 1 . 5 D 0 ) / ( M N - 1 . D 0 ) - ( I I - 1 . 5 D 0 ) / ( M - 1 .DO) A M 1 = ( J - 1 . 5 D 0 ) / ( N N - 1 . 0 0 ) - ( J J - 1 . 5 D 0 ) / ( N - 1 . D O ) AM2 = 1 . D O / ( M - 1 . D O ) A M 3 = 1 . D 0 / ( N - 1 . D O ) AM4 = T ( I I + 1 , J J ) - T ( I I . J J ) A M 5 = T ( I I , J J + 1 ) - T ( I I , J J ) A M G = T ( I I + 1 . J J + 1 ) - T ( I 1 + 1 . J J ) - T ( I I . J J + 1 ) + T ( I I , J J ) T N ( I . J ) = T ( I I . J J ) + A M 0 * A M 4 / A M 2 + A M 1 * A M 5 / A M 3 1+AMO*AM1 * A M 6 / ( A M 2 * A M 3 ) A M 4 = V 0 R ( I I + 1 , J J ) - V O R ( I I , J J ) A M 5 = V 0 R ( I I , J J + 1 ) - V O R ( I I , J J ) AM6«=V0R( 11 + 1 , J J + 1 ) - VOR ( 11 + 1 , J J ) - V O R ( I I . J J + 1 ) + V O R ( I I . J J ) VORN(1 . J ) = V O R ( 1 I , J J ) + AM0*AM4/AM2 + AM1*AM5/AM3 1 + A M 0 * A M 1 - A M 6 / ( A M 2 ' A M 3 ) 2 8 0 CONTINUE 2 9 0 CONTINUE C C » » * G I V E TO X Y . T . V O R AND SF ARRAYS THEIR NEW V A L U E S . C M=MN N=NN 182 DO 3 1 0 I=1,M DO 3 0 0 J= 1 . N XY( ] , J . 1 )=XYN( I . d . 1 ) X V { 1 . J . 2 ) = X Y W ( I . J . 2 ) SF ( I . J ) = 3 F N ( I . J ) 3 0 0 CONT INUE 3 10 CONT INUE MP 1=M+1 NP1= N+1 DO 3 3 0 1 = 1.MP 1 DO 3 2 0 J =1 .NP1 T ( I . J ) = T N ( I . J ) VOR ( I . J ) = V O R N ( I . J ) 3 2 0 CONT INUE 3 3 0 CONT INUE C C - * * GENERATE THE ORTHOGONAL G R I D . C 3 4 0 C A L L GRIDNC C C * * * COMPUTE THE S T E A D Y - S T A T E COND IT ION OF THE NATURAL CONVECT IDN C » * * P R O B L E M . C C A L L NATC C C * * » GRAPH THE R E S U L T S . C C A L L PLOT 3 5 0 CONT INUE C C * * * STOP THE GRAPH IC SOFTWARE. C C A L L DONEPL STOP 3 6 0 F O R M A T ( 1 5 ) 3 7 0 F 0 R M A T ( D 1 3 . 3 ) 3 8 0 F O R M A T ( ' 1 ' , T 5 . ' I N P U T DATA ( ' . 1 3 , ' ) . ' , 2 ( / ) . 1 T 5 , ' W O O D BOUNDARY C O N D I T I O N ' . / , 2 T 5 , ' G R I D S I Z E I S ' , 1 3 . ' B Y ' . 1 3 . / . 3 T 5 , ' M A X . / C OF I T E R A T I O N S ' ' . 1 3 , / . 4 T 5 , ' A C C U R A C Y = ' . D 1 0 . 3 . / , 5 T 5 . ' R F X Y = ' , D 1 0 . 3 . / , 6 T 5 . ' B T Y P E ( T ) = ' . 1 3 . / . T 5 . ' B T Y P E ( R ) = ' . 1 3 . / . 7 T 5 , ' B T Y P E ( B ) = ' . 1 3 . / , T 5 . ' B T Y P E ( L ) = ' . 1 3 . / . 8 T 5 . ' R S F ( T L ) = ' .D 1 0 . 3 . / . T 5 . ' R S F ( T R ) = ' . 0 1 O . 3 . / . 9 T 5 , ' R S F ( B L ) ' ' . D 1 0 . 3 . / . T 5 , ' R S F ( B R ) = ' . D 1 0 . 3 . / . » T 5 . ' P R = ' . D 1 0 . 3 . / . T 5 , ' RA= ' . D 1 0 . 3 , / , 1 T 5 . ' M A X . # OF I T E R A T I O N S ' ' , 1 3 , / , 2 T 5 . ' A C C U R A C Y ' ' . D 1 0 . 3 . / . 3 T 5 . ' R F T = ' . D 1 0 . 3 . / . T 5 . ' R F V O R = ' . D 1 0 . 3 . / . 4 T 5 , ' R F S F = ' . D 1 0 . 3 , / . 5 T 5 , ' V B C = ' . I 3 . 2 ( / ) ) END C C C SUBROUT INE GRIDNC C 183 C NAME : GRID NON CDNFORMAL. C C P U R P O S E : C C T H I S SUBROUTINE HAS B E E N B U I L T TO PRODUCE AN C ORTHOGONAL GRID WHICH DOES NOT HAVE TO BE CONFORMAL. C FOR ANY OF BOUNDARY THE GR ID POINT P O S I T I O N S COULD C EE P R E S C R I B E D (B T Y P E = 1 ) . I N THAT CASE THE I N I T I A L POINT C P O S I T I O N S WILL BE U S E D . C THE RAT IO OF SCALE FACTORS IN EACH CORNER COULD BE C S P E C I F I E D AL SO . THE RAT IO AT A S P E C I F I C CORNER WILL BE CONS IDERED C ONLY IF THE GRID PO INT P O S I T I O N S ARE NOT S P E C I F I E D ON THE C WALL J O I N I N G AT THE CORNER. C C C H A R A C T E R I S T I C : C C - L I N E BY L INE S O L V E R . C - SECOND ORDER ACCURACY NEUMANN BOUNDARY C O N D I T I O N . C - ABSOLUTE ACCURACY C R I T E R I O N . C - MAXIMUM NUMBER OF I T E R A T I O N . C - R E L A X A T I O N FACTOR . C c,...,..»...--....•..«»..».«.«.,...».......„.»»..«.»„....„. C I M P L I C I T R E A L » 8 ( A - H , O - Z ) I NTEGER B T Y P E ( 4 ) , C 1 . C 1 U L R E A L « 8 D ( 5 0 . 5 0 . 4 ) . H ( 5 0 . 5 0 . 2 ) . R S F ( 5 0 . 5 0 ) , X Y ( 5 0 . 5 0 . 2 ) R E A L - 8 A T O M A ( 5 0 ) . B T D M A ( 5 0 ) , C T D M A ( 5 0 ) . O T D M A ( 5 0 ) , S L N ( 5 0 ) COMMON / B L K 1 / M.N COMMON / B L K 2 / B T Y P E . R S F COMMON / B L K 3 / XY COMMON / B L K 4 / H COMMON / B L K 5 / D COMMON / B L K 1 0 / A T D M A . B T D M A . C T D M A , D T D M A . S L N COMMON / B L K 1 1 / EPS 1 . R F X Y , C 1 U L C C * « * SET UP V A R I A B L E S . C C1=0 MM1=M-1 MM2=M-2 NM1=N-1 NM2=N-2 C C * * * O V E R A L L LOOP. C . C * * * C A L C U L A T E THE R A T I O OF S C A L E FACTORS . C 10 C A L L D I S F D I F X = O . D O D I F Y = O . D O C C * * * MOVE ROWS. C DO 120 1=2.MM1 C C * » * SOLVE FOR Y. C C * * * SET UP THE TDMA C O E F F I C I E N T S . C DO 2 0 J = 2.NM-. AT = 2 . D O / ( R S F ( 1 . d )+RSF( I - 1 . d ) ) A R = ( R S F ( I , J > * R S F ( 1 . ) ) / 2 . D O AB = 2 . D 0 / « R S F ! I . J l + R S F ( I + 1 . J ) ) AL = ( R S F ( 1 . J H R S F ( I . J - 1 ) ) / 2 . D 0 ATDMA( J ) = AT + AR+AB+AL B T D M A ( J ) = A R CTDMA( d ) = AL DTDMA( d )=AT * X Y( ] - 1 . J . 2 ) + A B » X Y ( I + 1 . d . 2 ) 2 0 CONTINUE C c . . » sfr-r UP THE BOUNDARY C O N D I T I O N (NEUMAN OR D I R I C H L E T ) . C I F ( B T Y P E ( 4 ) . E O . 1 ) GOTO 3 0 ATDMA( 1 ) = 3.DO B T D M A ( 1 ) = 4 . D O C T D M A ( 1 ) = 0 . D 0 A M 0 = 3 . D 0 * X Y ( I . 1 . 1 ) - 4 . D O * X Y ( I , 2 . 1 ) + X Y ( I , 3 , 1 ) D T D M A ( 1 ) = - A M O * F ( X Y ( I . 1 . 2 ) , 8 ) A C R I 1 = - 1 . D O GOTO 4 0 3 0 A T D M A ( 1 ) = 1.DO B T D M A ( 1 ) = 0 . D 0 C T D M A ( 1 ) = 0 . D 0 D T D M A ( 1 ) = X Y ( I , 1 . 2 ) A C R I 1 = 0 . D O 4 0 I F ( B T Y P E ( 2 ) . E O . 1 ) GOTO 5 0 ATDMA (N )=3 .DO B T D M A ( N ) = 0 . D O CTDMA (N )=4 .DO AMO=3.DO*XY( I , N , 1 ) - 4 . D O * X Y ( I , N M 1 , 1 ) + X Y ( I , N M 2 . 1) DTDMA(N) = - AMO*F ( X Y ( I , N , 2 ) , 6 ) A C R I M = - 1 . D O GOTO 6 0 5 0 ATDMA(N ) = 1.DO B T D M A ( N ) = 0 . D O C T D M A ( N ) = 0 . D O D T D M A ( N ) = X Y ( 1 . N , 2 ) ACRIM=O.DO C C * » * R E V I S E D T R I D I A G O N A L - M A T R I X ALGOR ITHM. C 6 0 C A L L RTDMA ( N . A C R I 1 . A C R I M ) C C * * » STORE THE VECTOR S O L U T I O N . C DO 7 0 d = 1 , N A M O = X Y ( I , d . 2 ) + R F X Y * ( S L N ( J ) - X Y ( I . d , 2 ) ) D I F Y = D M A X 1 ( D I F Y . D A B S ( A M 0 - X Y ( I . d . 2 ) ) ) X Y ( I , d , 2 ) = A M O 7 0 CONTINUE C C ' * * D I R I C H L E T C O N D I T I O N S . C C * * * R IGHT WALL. C I F ( B T Y P E ( 2 ) E O . 1 ) GOTO 8 0 AMO=F (XY (1 . N . 2 ) . 2 ) D I F X = D M A X 1 ( D I F X , D A B S ( A M O - X Y ( I , N . 1 ) ) ) XY ( 1 , N . 1)=AMO c C ' » " L E F T WALL. C BO IF ( B T Y P E C J ) . E O . 1 ) GOTO 9 0 A M O - F ( X Y ( I . 1 . 2 1 . 4 ) D I F X = DMAX 1 ( D l F X . D A B S ( A M O - X Y ( 1 . 1 . 1 ) ) ) X Y ( I . 1 , 1 ) = AMO C C " - * SOLVE FOR X. C SET UP THE TDMA C O E F F I C I E N T S . C 9 0 DO 100 J=2,NM1 AT = 2 . D O / ( R S F ( I . J ) + R S F ( 1 - 1 , J ) ) A R = ( R S F ( I . J ) + R S F ( I . J + 1 ) ) / 2 . D 0 AB = 2 . D O / ( R S F ( I . J ) + R S F ( 1 + 1 , J ) ) A L = ( R S F ( I . J ) + R S F ( I , J - 1 ) ) / 2 . D O A T D M A ( J ) =AT+AR+AB+AL BTDMA ( J ) «= AR C T D M A ( J ) = A L D T D M A ( J ) e A T " X Y ( I - 1 . J . 1 ) + A B * X Y ( I + 1 . J , 1 ) 1 00 CONT INUE C C * « * SET UP THE BOUNDARY C O N D I T I O N S ( D I R I C H L E T ) . C D T D M A ( 2 ) = D T D M A ( 2 ) + C T D M A ( 2 ) * X Y ( I . 1 ,1 ) C T D M A ( 2 ) = 0 . D 0 D T D M A ( N M 1 ) = D T D M A ( N M 1 ) + B T D M A ( N M 1 ) « X Y ( I . N, 1 ) B T D M A ( N M 1 ) = 0 . D 0 C C * * * T R I D I A G O N A L - M A T R I X A L G O R I T H M . C C A L L TDMA ( 2 . N M 1 ) C C * * * STORE THE SOLUT ION V E C T O R . C DO 110 J=2,NM1 A M 0 = X Y ( I , J , 1 ) + R F X Y * ( S L N ( J ) - X Y ( I . J , 1 ) ) D I F X = D M A X 1 ( D I F X , D A B S ( A M O - X Y ( I , J , 1 ) ) ) X Y ( I , J . 1 ) = A M O 110 CONT INUE 1 2 0 CONT INUE C C * « * MOVE COLUMNS. C . DO 2 3 0 J=2.NM1 C c . . » SOLVE FOR X . C C » * * S ET UP THE TDMA C O E F F I C I E N T S . C DO 130 I=2.MM1 AT = 2 . D 0 / ( R S F ( I . J ) + R S F ( I - 1 . J ) ) A R = ( R S F ( I , J ) + R S F ( I . J + 1 ) ) / 2 . D O A B C 2 . D O / ( R S F ( I , J ) + R S F ( I + 1 . J ) ) A L = ( R S F ( I , J ) + R S F ( I . J - 1 ) ) / 2 . D O ATDMA( I ) =AT+AR+AB+AL B T D M A ( I ) = A B C T D M A ( I ) = A T D T D M A ( I ) = A R * X Y ( I . J + 1 . 1 ) + A L * X Y ( I . J - 1 . 1 ) 130 CONT INUE C C * - * SET UP THE BOUNDARY C O N D I T I O N S (NEWMAN OR D I R I C H L E T ) . C I F ( B T Y P E ( 1 ) . E O . 1 ) GOTO 140 ATDMA( 1 ) = 3 . D 0 BTDMA( 1 )=4.DO C T D M A ( 1 ) = 0 . D 0 AMO=3.DO*XY( 1 . J . 2 ) - 4 . D 0 * X Y ( 2 , J . 2 ) + X Y ( 3 . J . 2 ) DTDMA( 1) = -AMO* F ( X Y ( 1 . J . 1 ) . 5 ) A C R I 1 = - 1 . D O GOTO 150 140 ATDMA(1 )=1 .DO B T D M A ( 1 ) = 0 . D 0 CTDMA (1 ) =O .DO D T D M A ( 1 ) = X Y ( 1 . J , 1 ) A C R I 1<=0.D0 1 5 0 I F ( B T Y P E ( 3 ) . E O . 1 ) GOTO 160 ATDMA(M)=3 .DO BTDMA (M )=0 .DO CTDMA(M)=4 .DO A M 0 = 3 . D 0 * X Y ( M , J , 2 ) - 4 . D 0 * X Y ( M M 1 . J , 2 ) + X Y ( M M 2 , d . 2 ) D T D M A ( M ) = - A M O ' F ( X Y ( M . J , 1 ) , 7 ) A C R I M = - 1 . D O GOTO 170 1 6 0 ATDMA(M)=1 .DO BTDMA (M )=0 .DO CTDMA(M)=O.DO D T D M A ( M ) = X Y ( M . J , 1 ) ACR IM=O.DO C C * * - R E V I S E D T R I D I A G O N A L - M A T R I X ALGOR ITHM. C 1 7 0 C A L L RTDMA ( M , A C R I 1 . A C R I M ) C C * * * STORE THE VECTOR S O L U T I O N . C DO 180 I =1 ,M A M O = X Y ( I , d . 1 ) + R F X Y * ( S L N ( I ) - X Y ( I , J . 1 ) ) D I F X = D M A X 1 ( D I F X . D A B S ( A M O - X Y ( I . J . 1 ) ) ) X Y ( I . J , 1 ) = A M O 1 8 0 CONT INUE C C * * * D I R I C H L E T C O N D I T I O N S . C C » * * TOP WALL . C I F ( B T Y P E ( 1 ) . E O . 1 ) GOTO 190 A M O = F ( X Y ( 1 . J . 1 ) . 1 ) D I F Y = D M A X 1 ( D I F Y , D A B S ( A M O - X Y ( 1 , J , 2 ) ) ) X Y ( 1 , J . 2 ) = A M O C C * * * BOTTOM WALL. C 1 9 0 I F ( B T Y P E ( 3 ) . E O . 1 ) GOTO 2 0 0 A M 0 = F ( X Y ( M . J . 1) . 3 ) D I F Y = D M A X 1 ( D I F Y . D A B S ( A M O - X Y ( M . J . 2 ) ) ) X Y ( M , J , 2 ) = A M 0 C C * » * S O L V E FOR Y . C * » " SET UP THE TDMA C O E F F I C I E N T S . C 2 0 0 DO 2 10 I=2.MM1 AT = 2 . D 0 / ( R S F ( 1 , J ) + R S F ( 1 - 1 . J ) ) A R = ( R S F ( I . J ) - > R S F ( 1 . J - M ) ) / 2 . D 0 AB = 2 . D 0 / ( R S F ( I . J ) + R S F ( J - M , J ) ) A L = ( R S F ( I . J ) + R S F ( 1 . J - 1 ) ) / 2 . D 0 ATDMA ( I )=AT->AR--AB + AL BTDMA(1 ) =AB CTDMA ( I ) =AT D T D M A ( I ) = A R » X Y ( I . d + 1 . 2 ) + A L * X Y ( I . d - 1 . 2 ) 2 1 0 CONTINUE C c . » » S E T THE BOUNDARY COND IT IONS ( D I R I C H L E T ) . C D T D M A ( 2 ) = D T D M A ( 2 ) + C T D M A ( 2 ) " X Y ( 1 , d . 2 ) C T D M A ( 2 ) = 0 . D 0 DTDMA(MM 1)=DTDMA(MM1 ) + B T D M A ( M M 1 ) * X Y ( M . J , 2 ) BTDMA(MM1 )=0 .D0 C C * * » T R I D I A G O N A L - M A T R I X A L G O R I T H M . C C A L L TDMA ( 2 . M M 1 ) C C * * * STORE THE VECTOR S O L U T I O N . C DO 2 2 0 I=2.MM1 A M O = X Y ( I , d . 2 ) + R F X Y » ( S L N ( I ) - X Y ( I . 0 . 2 ) ) D I F Y = D M A X 1 ( D I F Y . D A B S ( A M 0 - X Y ( I . d . 2 ) ) ) X Y ( I . d . 2 ) = A M 0 2 2 0 CONT INUE 2 3 0 CONT INUE C C « * » ACCURACY AND P R I N T O U T . C C1=C1+1 WRITE ( G . 2 4 0 ) C 1 . 0 I F X . D I F Y I F ( ( D I F X . G E . 2 . D O ) . O R . ( D I F Y . G E . 2 . D O ) ) STOP I F ( ( ( D I F X . G T . E P S 1 ) . O R . ( D I F Y . G T . E P S 1 ) ) . A N D . ( C 1 . L T . C 1 U L ) ) 1G0T0 10 C C * * * C A L C U L A T I O N OF THE D E R I V A T I V E S AND SCALE F A C T O R S . C C A L L DSF C C * * * ORTHOGONALITY T E S T . C C A L L ORTHO RETURN 2 4 0 F O R M A T ( / , T 5 , ' ( ' . 1 3 . ' ) ' . 3 X . ' D1FX= ' . D 1 1 . 5 . 3 X . ' D I F Y = ' . D 1 1 . 5 ) END C C C SUBROUTINE NATC C c .. C C NAME: NATURAL C O N V E C T I O N . 188 c C PURPOSE • C C T H I S SUBROUTINE COMPUTES THE SOLUTION OF A S T E A D Y - S T A T E C NATURAL CONVECT ION IN A NON-RECTANGULAR C A V I T Y . C C C H A R A C T E R I S T I C : C C - L I N E BY L INE SOLVER C - WOOD V O R T I C I T Y BOUNDARY COND IT ION . C - EXPONENT IAL SCHEME. C - GR ID IS ORTHOGONAL AND NOT N E C E S S A I R L Y CONFORMAL. C - GR ID IS UN IFORMLY SPACE IN THE NEW DOMAIN. C - R E L A X A T I O N F A C T O R . C - STAGGERED G R I D . C - R E L A T I V E ACCURACY C R I T E R I O N . C - D IVERGENCE T E S T . C - MAXIMUM NUMBER OF I T E R A T I O N S . C - WALLS ARE I M P E R M E A B L E . C - TOP AND BOTTOM WALLS ARE A D I A B A T I C . C - R IGHT AND L E F T WALLS ARE I SOTHERMS. C C I M P L I C I T R E A L * 8 ( A - H . D - Z ) I NTEGER C 2 . C 2 U L . V B C R E A L * 8 A T D M A ( 5 0 ) . B T D M A ( 5 0 ) . C T D M A ( 5 0 ) . D T D M A ( 5 0 ) . S L N ( 5 0 ) R E A L - 8 S F ( 5 0 , 5 0 ) , T ( 5 0 . 5 0 ) . V 0 R ( 5 O . 5 O ) R E A L * 8 D ( 5 0 . 5 0 . 4 ) . H ( 5 0 . 5 0 . 2 ) .MAXSF.MAXVOR COMMON / B L K 1 / M ,N COMMON / B L K 4 / H COMMON / B L K 5 / D COMMON / B L K 6 / P R . R A COMMON / B L K 7 / T COMMON / B L K 8 / VOR COMMON / B L K 9 / SF COMMON / B L K 1 0 / A T D M A . B T D M A , C T D M A . D T D M A . S L N COMMON / B L K 1 2 / E P S 2 . R F T . R F V O R , R F S F . C 2 U L . V B C C C « « * SET UP V A R I A B L E S . C C2«=0 MM1=M-1 MM2=M-2 MP1=M+1 NM1=N-1 NM2=N-2 NP1=N+1 C C * * * O V E R A L L LOOP. C 10 MAXV0R=0 .D0 MAXSF=O.DO D I FT=O .DO DIFV0R-=O.D0 D I F S F = O . D O C C * * * MOVE ROWS. C DO 1O0 I=2.M 189 c C * * » ENERGY EQUAT ION. C C * * * COMPUTAT ION OF ENERGY EQUAT ION C O E F F I C I E N T S . C DO 2 0 J = 2 , N COND = '(H( 1 - 1 . 0 - 1 . 1 ) + H ( I - 1 . d . 1 ) ) / ( H ( I - 1 . d - 1 . 2 ) + H ( J - 1 , d . 2 ) ) C O N V = S F ( l - 1 . d - 1 ) - S F ( I - 1 . d ) AT = COND* COEF F ( C O N D . C O N V ) + D M A X 1 ( 0 . D O . - C O N V ) C O N D = ( H ( I - 1 . J . 2 ) + H ( 1 . J . 2 ) ) / ( H ( I - 1 . J . 1 ) + H ( I . J , 1 ) ) C O N V = S F ( I - l . d ) - S F ( I . d ) A R = C D N D » C 0 E F F ( C O N D . C 0 N V ) + D M A X 1 ( O . 0 O . - C 0 N V ) C O N D = ( H ( I . d - 1 . 1 ) + H ( I . d . 1 ) ) / ( H ( I . d - 1 . 2 ) + H ( I . J . 2 ) ) CONV=SF(1 . d - 1 ) - S F ( I . J ) AB = C O N D - C O E F F ( C O N D . C O N V ) +DMAX 1 ( 0 . D O . C O N V ) C 0 N D = ( H ( I - 1 . J - 1 , 2 ) + H ( I , J - 1 . 2 ) ) / ( H ( I - 1 . J - 1 . 1 ) + H ( I . J - 1 . 1 ) ) C O N V = S F ( I - 1 . J - 1 ) - S F ( I . J - 1 ) AL=COND*COEFF ( COND,CONV )+DMAX 1 ( 0 . D O , C O N V ) ATDMA( J )=AT+AR+AB+AL BTDMA( J ) =AR C T D M A ( J ) = A L D T D M A ( J ) = A T * T ( I - 1 . J ) + A B * T ( 1 + 1 , J ) 2 0 CONT INUE C C " * SET UP THE BOUNDARY C O N D I T I O N S (CONSTANT TEMPERATURE W A L L S ) . C ATDMA( 1 ) = 1.DO BTDMA ( 1 )>=-1 .DO C T D M A ( 1 ) = 0 . D O D T D M A ( 1 ) = 0 . 0 0 A T D M A ( N P 1 ) = 1 . D O B T D M A ( N P 1 ) = 0 . D O C T D M A ( N P 1 ) = - 1 . D O D T D M A ( N P 1 ) = 2 . D 0 C C * * * T R I D I A G O N A L - M A T R I X A L G O R I T H M . C C A L L TDMA ( 1 . N P 1 ) C C * * * STORE THE SOLUT ION V E C T O R . C DO 3 0 J =1 ,NP1 A M O = T ( I . J ) + R F T » ( S L N ( J ) - T ( I , J ) ) D l F T = D M A X 1 ( D I P T , D A B S ( A M O - T ( I , J ) ) ) T ( I , J ) = A M O 3 0 CONT INUE C C » « * V O R T I C I T Y E Q U A T I O N . C C * * * COMPUTAT ION OF V O R T I C I T Y EOUAT ION C O E F F I C I E N T S . C DO 4 0 J = 2 . N COND = P R * ( H ( I - 1 , J - 1 . l ) + H ( I - 1 . J , 1 ) ) / ( H ( I - 1 , J - 1 , 2 ) + H ( I - 1 . d . 2 ) ) CONV = S F ( I - 1 . J - 1 ) - S F ( 1 - 1 . d ) AT = CDND *COE F F ( COND . CONV )+DMAX 1 (O .DO. -CONV) COND = P R * ( H ( 1-1 , J , 2 ) + H ( I . J , 2 ) ) / ( H ( 1 - 1 . J . 1 ) + H ( I , J . 1 ) ) C0NV = S F ( I - 1 , J ) - S F ( I . J ) A R = C 0 N D * C 0 E F F ( C O N D , C O N V ) + D M A X 1 ( O . D O . - C O N V ) COND = P R * ( H ( I . J - 1 . 1 ) + H ( I . J . 1 ) ) / ( H ( I . J - 1 . 2 ) + H ( I . J . 2 ) ) C O N V = S F ( I , d - 1 ) - S F ( I . J ) 1 90 A B = C O N D » C O E F F ( COND . CONV) + DMAX 1 ( 0 . DO . CONV ) COND = P R * ( H ( I - 1 , J - 1 , 2 ) +H ( I , d - 1 . 2 ) ) / ( H ( I - 1 , J - 1 , 1 ) + H ( I . d - 1 , 1 ) ) C O N V = S F ( I - 1 . J - 1 ) - S F ( I . J - 1 ) A L = COND * COE F F ( C O N D , C O N V ) + D M A X 1 ( 0 . D O . C O N V ) AMO=(D( ] - 1 . J - 1 . 2 ) + D ( I - 1 . J . 2 ) + D(1 .d -1 . 2 ) + D ( I . d . 2 ) )/4.DO A M 1 = ( D ( I - 1 . d - 1 . 4 ) + D ( l - 1 . d . 4 ) + D ( l , d - 1 . 4 ) + D ( l . d , 4 ) ) / 4 . D O A M 2 = ( T ( I .d+1 ) - T ( I . J - 1 ) ) /2 .DO A M 3 = ( T ( I - 1 . d ) - T ( l - M . d ) ) / 2 . D O B = ( A M 1 * A M 2 - A M 0 * A M 3 ) » P R * RA ATDMA (d)=AT+AR+AB+AL B T D M A ( d ) = AR CTDMA( d )= AL D T D M A ( d ) = A T * V O R ( 1 - 1 . d ) + A B * V O R ( 1 + 1 . d ) + B 4 0 CONT INUE C C * * * SET UP THE V O R T I C I T Y BOUNDARY C O N D I T I O N . C - * » WALLS ARE I M P E R M E A B L E . C I F ( V B C . E O . 1 ) GOTO 5 0 C C » « * WOOD. C A M O = ( H ( I - 1 . 1 . 1 ) +H ( I , 1 . 1 ) ) / 2 . D O AM1 = ( H ( I - 1 , 1 , 2 ) + H ( I . 1 , 2 ) ) / 2 . D 0 A M 2 = ( H ( I - 1 . 2 . 1 ) + H ( I , 2 , 1 ) ) / 2 . D O A M 3 = ( H ( I - 1 . 2 . 2 ) + H ( I . 2 . 2 ) ) / 2 . D O A M 4 = ( H ( I - 1 . 3 . 1 ) + H ( I . 3 , 1 ) ) / 2 . D O A M 5 = ( H ( 1 - 1 . 3 . 2 ) + H ( I . 3 , 2 ) ) / 2 . D 0 A M 6 = ( - 3 . D O * A M O * A M 1 + 4 . D O * A M 2 * A M 3 - A M 4 * A M 5 ) / 2 . D O A M 7 = ( - 3 . D 0 * A M 1 / A M 0 + 4 . D O * A M 3 / A M 2 - A M 5 / A M 4 ) / 2 . D 0 A M 8 = ( A M 0 * * 2 ) / 2 . D O A M 9 = A M 0 * A M 6 / ( 6 . D 0 * A M 1 ) A M 1 0 = - ( A M O * * 3 ) » A M 7 / ( 3 . D O * A M 1 ) A M 1 1 = - ( A M 0 * * 2 ) / 4 . D O A M 1 2 = - A M 0 * A M G / ( 1 2 . D 0 * A M 1 ) A M 1 3 = ( A M 0 * * 2 ) / 6 . D O A M 1 4 = ( A M 0 * » 3 ) * A M 7 / ( 6 . D 0 * A M 1 ) AM15=AM8+AM9+AM10 AM16=AM11+AM12+AM13+AM14 , ATDMA (1 ) =AM16 BTDMA ( 1 )<=AM15 + AM16 C T D M A ( 1 ) = 0 . D 0 D T D M A ( 1 ) = ( S F ( I - 1 . 2 ) + S F ( I . 2 ) ) / 2 . D 0 A M O = ( H ( 1 - 1 , N , 1 ) + H ( I . N , 1 ) ) / 2 . D O A M 1 = ( H ( I - 1 , N . 2 ) + H ( I , N . 2 ) ) / 2 . D O A M 2 = ( H ( 1 - 1 . N M 1 . 1 ) + H ( I . N M 1 , 1) )/2.DO A M 3 = ( H ( I - 1 . N M 1 . 2 ) + H ( I . N M 1 . 2 ) )/2 .DO AM4 = ( H( I - 1 . NM2 , 1 )+H( I . NM2 . 1 ) )/2 .DO A M 5 = ( H ( I - 1 . N M 2 . 2 ) + H ( I . N M 2 . 2 ) ) / 2 . D O A M G = ( - 3 . D O * A M O * A M 1 + 4 . D O * A M 2 * AM3-AM4 * A M 5 ) / 2 . D O AM7 = ( - 3 . D O * A M 1 / A M 0 + 4 . D O * A M 3 / A M 2 - A M 5 / A M 4 ) / 2 . D O A M 8 = ( A M 0 * * 2 ) / 2 . D O AM9=AMO*AMG/ (G .DO*AM1 ) A M 1 0 = - ( A M O * * 3 ) * A M 7 / ( 3 . D 0 * A M 1 ) AM 11 = - ( A M 0 * * 2 ) / 4 . D O A M 1 2 = - A M 0 * A M 6 / ( 1 2 , D O * A M 1 ) A M 1 3 = ( A M 0 * * 2 ) / 6 . D 0 A M 1 4 = ( A M 0 * * 3 ) * A M 7 / ( 6 . D 0 * A M 1 ) AM15=AM8+AM9+AM10 AM16=AM11+AM12+AM13+AM14 191 ATDMA(NP1 )=AM16 B T D M A ( N P 1 )=0 .D0 CTDMA(NP1 )=AM15+AM1S DTDMA(NP1 ) = ( SFI 1-1 .NM1 > + S F ( } .NM1 ) )/2.DO ACR;1=0.DO ACR1M=0.DO GOTO 60 C C**- SECOND ORDER. C 50 . AMO=(H(I - 1, 1, 1 )+H(I . 1 . 1 ) )/2.DO AM 1 = (' H ( 1 - 1 . 1 , 2 ) + H( 1 . 1 , 2 ) ) / 2 . DO AM2=(H(1 -1 . 2 . 1 )+H( I . 2 . 1 ) ) / 2 . DO AM3=(H(I - 1 . 2 . 2 ) + H ( I . 2 . 2 ) )/2.DO AM4=(H(1-1 . 3 , 1 ) + H(1 . 3 . 1 ) ) / 2 .D0 A M 5 = ( H ( 1 - 1 . 3 . 2 ) + H ( 1 . 3 , 2 ) )/2.DO A M 6 = ( - 3 . D O * A M O * AM 1 + 4 . D O ' A M 2 * A M 3 - A M 4 » A M 5 ) / 2 . D O A M 7 = ( - 3 . D 0 * A M l / A M 0 + 4 . D O " A M 3 / A M 2 - A M 5 / A M 4 ) / 2 . D 0 A M 8 = 2 7 . D 0 " ( A M 0 * * 2 ) / 4 . D O A M 9 = 3 . D 0 * A M 0 * A M 6 / ( 2 . D O * A M 1 ) A M 1 0 = - 3 . D 0 * ( A M 0 * * 3 ) * A M 7 / A M 1 A M 1 1 = - 3 . D 0 * ( A M 0 * * 2 ) / 4 . D O A M 1 2 = - A M 0 * A M 6 / ( 6 .D0 *AM1 ) A M 1 3 = ( A M O * * 3 ) * A M 7 / ( 3 . D O * A M 1 ) A M 1 4 = - 9 . D 0 * ( A M 0 * * 2 ) / 4 . D O A M 1 5 = - A M 0 * A M 6 / ( 2 . D 0 * A M 1 ) AM 1 6 = 4 . D 0 * ( A M 0 * " 2 ) / 3 . D 0 A M 1 7 = ( A M 0 * * 3 ) * A M 7 / A M 1 AM 1B = AM8+AM9 + AM10 AM19=AM11+AM12+AM13 AM20=AM14 +AM 15+AM 16 +AM 17 ATDMA(1 ) =AM20 BTDMA(1 ) =AM18+AM20 C T D M A ( 1 ) = 0 . D 0 D T D M A (1 ) = ( 1 6 . D O * ( S F ( I - 1 , 2 ) + S F ( I , 2 ) ) - ( S F ( I - 1 .3 ) + S F ( I , 3 ) ) ) / 2 . D O ACR I 1=AM19 A M O = ( H ( I - 1 , N , 1 ) + H ( I , N , 1 ) ) / 2 . D O A M 1 = ( H ( I - 1 . N , 2 ) + H ( I , N . 2 ) ) / 2 . D O AM2 = ( H (1-1 . N M1 . 1 ) + H ( I . N M1, 1 ) ) / 2 . D O A M 3 = ( H ( I - 1 . N M 1 . 2 ) + H ( 1 , N M 1 . 2 ) ) / 2 . D O AM4«=(H( I - 1 . NM2 . 1 )+H( I . NM2 . 1 ) )/2 .DO A M 5 = ( H ( I - 1 , N M 2 . 2 ) + H ( I . N M 2 . 2 ) ) / 2 . D O A M 6 = ( - 3 . D O * A M O * A M1 + 4 . D O * A M 2 - A M 3 - A M 4 * A M 5 ) / 2 . D O A M 7 = ( - 3 . D O * A M 1/AMO+4.DO*AM3/AM2-AM5/AM4)/2 . DO A M 8 r 2 7 . D O * ( A M O * * 2 ) / 4 . D O A M 9 = 3 . D 0 * A M 0 * A M 6 / ( 2 . D 0 * A M 1 ) A M 1 0 = - 3 . D 0 * ( A M 0 * * 3 ) * A M 7 / A M 1 AM11 = - 3 . D O * ( A M O * * 2 ) / 4 . D O A M 1 2 = - A M 0 * A M 6 / ( 6 . D O * A M 1 ) A M 1 3 = ( A M 0 * * 3 ) * A M 7 / ( 3 . D 0 * A M 1 ) A M 1 4 = - 9 . D 0 * ( A M 0 * * 2 ) / 4 . 0 0 A M 1 5 = - A M 0 * A M 6 / ( 2 . D 0 - A M 1 ) AM16 = 4 . D O * ( A M O * * 2 ) / 3 . D O A M 1 7 = ( A M 0 * * 3 ) " A M 7 / A M 1 AM18=AM8+AM9+AM10 AM19=AM11+AM12+AM13 AM20 = AM 14 +AM15 +AM 16 +AM 17 ATDMA (NP1 ) =AM20 B T D M A ( N P 1 ) = 0 . D 0 CTDMA(NP1 ) =AM18+AM20 D T D M A ( N P 1 ) = ( 1 6 . D O * ( S F ( ] - 1 . N M 1 ) + S F ( I , N M 1 ) ) - ( S F ( I - 1 . N M 2 ) 1 + S F ( I , N M 2 ) ) ) / 2 . D 0 . ACRIM=AM19 C C * * « R E V I S E D TR ID I A G O N A L - M A T R I X ALGOR ITHM. C 6 0 C A L L RTDMA ( N P 1 . A C R I 1 . A C R I M ) C C * * * STORE THE SOLUT ION V E C T O R . C DD 7 0 <J = 1 . NP 1 AM0 = VOR ( I . J )---RFVOR* ( S L N ( d ) - V O R ( I . J ) ) DIFVOR=DMAX1 (D I F V O R . D A B S ( A M O - V O R ( 1 . 0 ) ) ) MAXVOR=DMAX1 (MAXVOR.DABS (AMO) ) V O R ( I . J ) = A M 0 7 0 CONTINUE I F ( I . E O . M ) GOTO 100 C C * * * STREAM FUNCT ION E O U A T I O N . C C * » * COMPUTAT ION OF THE STREAM FUNCT ION EOUATION C O E F F I C I E N T S . C DO 8 0 d=2,NM1 A T = ( H ( I - 1 , d , 1 ) + H ( l , d . 1 ) ) / ( H ( I - 1 , d , 2 ) + H ( I , d , 2 ) ) A R = ( H ( I . d . 2 ) + H ( I . d + 1 , 2 ) ) / ( H ( I . d . 1 ) + H ( I . J + 1 , 1 ) ) A B = ( H ( I , d . 1 ) + H ( 1 + 1 . d . 1 ) ) / ( H ( l . d . 2 ) + H ( I + 1 . d , 2 ) ) AL = ( H ( I , d - 1 , 2 ) + H ( I , d . 2 ) ) / ( H ( I . d - 1 . 1 ) + H ( I , d , 1 ) ) A M 0 = ( V O R ( I . d ) + V O R ( I . d + 1 ) + V 0 R ( I + 1 . d ) + V O R ( 1 + 1 . d + 1 ) ) / 4 .DO B = A M O * H ( I . d . 1 ) * H ( I , d , 2 ) ATDMA(d )=AT+AR+AB+AL B T D M A ( d ) = AR C T D M A ( d ) = A L D T D M A ( d ) = A T * S F ( I-1 , d ) + A B * S F ( 1 + 1 .d)+B 8 0 CONT INUE C C * » * S ET UP THE BOUNDARY C O N D I T I O N S . C D T D M A ( 2 ) = D T D M A ( 2 ) + C T D M A ( 2 ) * S F ( I , 1 ) C T D M A ( 2 ) = 0 . D 0 DTDMA (NM1 )=DTDMA (NM1 )+BTDMA (NM1 ) * S F ( I ,N ) B T D M A ( N M 1 ) = 0 . D 0 C C * * * T R I D I A G O N A L - M A T R I X A L G O R I T H M . C C A L L TDMA ( 2 . N M 1 ) C C * * » STORE THE SOLUT ION V E C T O R . C DO 9 0 d=2.NM1 A M O = S F ( I , d ) + R F S F * ( S L N ( d ) - S F ( l . d ) ) D I F S F = D M A X 1 ( D I F S F . D A B S ( A M 0 - S F ( I , d ) ) ) MAXSF = D M A X 1 ( M A X S F , D A E S ( A M O ) ) S F ( I . d ) = A M 0 9 0 CONT INUE 100 CONT INUE C C * * * MOVE COLUMNS C DO 190 d=2 ,N 193 C « « » ENERGY EOUAT ION. C C * * - COMPUTAT ION OF ENERGY EOUAT ION C O E F F I C I E N T S . C DO 110 I=2.M C O N D = ( H ( I - 1 . d - 1 . 1 ) + H ( I - 1 . d . 1 ) ) / ( H ( I - 1 . d - 1 . 2 ) + H ( ] - 1 . d . 2 ) ) C O N V = S F ( I - 1 . J - 1 ) - S F ( I - 1 . J ) A T = C O N D * C O E F F ( C O N D , C O N V ) + D M A X 1 ( 0 . D O . - C O N V ) C O N D = ( H ( ] - 1 . d . 2 ) - H - ( ( I . d , 2 ) ) / ( H ( I - 1 . d . 1 ) + H ( I . d . 1 ) ) CONV = S F ( I - 1 . d ) - S F ( I . d ) A R = C O N D * C D E F F ( C O N D . C O N V ) + D M A X 1 ( O . D O , - C O N V ) C O N D = ( H ( I . J - 1 . 1 ) + H ( I . d . 1 ) ) / ( H ( I . J - 1 . 2 ) + H ( I . J . 2 ) ) C O N V = S F ( I . d - l ) - S F ( I . d ) A B = C O N D * C O E F F ( C O N D . C O N V ) + D M A X 1 ( 0 . D O . C O N V ) C O N D = ( H ( I - 1 . d - 1 . 2 ) + H ( I , J - 1 , 2 ) ) / ( H ( I - 1 . d - 1 , 1 ) + H ( I . J - 1 . 1 ) ) C O N V = S F ( 1 - 1 , d - l ) - S F ( l . d - 1 ) A L = C O N D * C O E F F ( C O N D . C O N V ) + D M A X 1 ( O . D O , C O N V ) ATDMA( I ) =AT+AR+AB+AL B T D M A ( I ) = A B C T D M A ( I ) = A T D T D M A ( I ) = A R * T ( I ,d*1 ) + A L * T ( 1 . d - 1 ) 1 10 CONT INUE C C * * * SET UP THE BOUNDARY C O N D I T I O N S ( A D I A B A T I C W A L L S ) . C A T D M A ( 1 ) = 1 . D O B T D M A ( 1 ) = 1.DO C T D M A ( 1 ) = 0 . D 0 D T D M A ( 1 ) = 0 . D O ATDMA(MP 1) = 1.DO BTDMA(MP 1 ) = 0 . D 0 C T D M A ( M P 1 ) = 1 . D O DTDMA(MP 1 ) = 0 . D 0 C C*** T R I D I A G O N A L - M A T R I X A L G O R I T H M . C C A L L TDMA ( 1 . M P 1) C C * » * STORE THE S O L U T I O N V E C T O R . C DO 120 I=1 ,MP1 A M O = T ( I . d ) + R F T * ( S L N ( I ) - T ( I , d ) ) D I F T = DMAX 1 (D I F T , D A B S ( A M O - T ( I , d ) ) ) T ( I , d ) = A M O 120 CONT INUE C C « * * V O R T I C I T Y E O U A T I O N . C C * * * COMPUTAT ION OF V O R T I C I T Y EOUAT ION C O E F F I C I E N T S . C DO 130 1=2.M COND = P R * ( H ( 1 - 1 . d - 1 . 1 ) + H ( 1 - 1 . d . 1 ) ) / ( H ( I - 1 . d - 1 . 2 ) + H ( l - 1 . d . 2 ) ) C O N V = S F ( I - 1 . d - 1 ) - S F ( I - 1 . d ) A T = C O N D * C O E F F ( C O N D . C O N V ) + D M A X 1 ( O . D O . - C O N V ) COND = PR * ( H ( I - 1 . d . 2 ) + H ( I . d , 2 ) ) / ( H ( I - 1 . d . 1 )+H( I . d . 1 ) ) C O N V = S F ( 1 - 1 , d ) - S F ( I . d ) AR = COND * COE F F ( C O N D , C O N V ) + D M A X 1 ( 0 . D O . - C O N V ) C O N D = P R * ( H ( I . d - 1 , 1 ) + H ( I . d . 1 ) ) / ( H ( I . d - 1 . 2 ) + H ( I . d . 2 ) ) C O N V = S F ( I , d - 1 ) - S F ( I . d ) AB =COND*COE F F ( C O N D . C O N V ) + D M A X 1 ( 0 . D O , C O N V ) 194 COND=PR* (H ( 1 - 1 . J - 1 . 2 ) + H ( I . J - 1 . 2 ) ) / ( H ( I - 1 , J - 1 . 1 )+H( I . J - 1 . 1 ) ) C O N V ^ S F ( I - 1 . d - 1 ) - S F ( I . d - 1 ) AL = COND* C D E F F ( C O N D . C O N V ) + D M A X 1 ( 0 . D O . C O N V ) A M O = ( D ( ] - 1 . d - 1 . 2 ) + D ( I - 1 . J . 2 ) + D ( I . J - 1 . 2 ) + D ( I . J . 2 ) ) / 4 . D O AM1 = ( D ( I - 1 . J - 1 . 4 ) + D ( I - 1 . J . 4 )+D( I . J - 1 . 4 ) + D ( I , J , 4 ) ) / 4 . D O A M 2 = ( T ( I . d + 1 ) - T ( I . J - 1 ) ) / 2 . D O A M 3 = ( T ( I - 1 . d ) - T ( 1 + 1 , d ) ) /2 .DO B= (AM 1 * AM2-AMO* A M 3 ) * P R * R A A T D M A ( I ) = AT + AR + AB + AL B T D M A ( I ) = A B C T D M A ( I ) = A T DTDMA( I ) = A R * V 0 R ( 1 , 0 + 1 )+AL * V O R ( I . 0 - 1 ) + B 130 CONT INUE C C * « » SET UP THE V O R T I C I T Y BOUNDARY COND IT ION . C * * * WALLS ARE I M P E R M E A B L E . C I F ( V B C . E O . 1 ) GOTO 140 C c » « * WOOD. C A M O = ( H ( 1 , 0 - 1 , 2 ) + H ( 1 . 0 . 2 ) ) / 2 . D 0 A M 1 = ( H ( 1 , 0 - 1 , 1 ) + H ( 1 , 0 , 1 ) ) / 2 . D O A M 2 = ( H ( 2 , J - 1 . 2 ) + H ( 2 . J . 2 ) ) / 2 . D 0 A M 3 = ( H ( 2 , 0 - 1 . 1 ) + H ( 2 . 0 . 1 ) ) / 2 . D 0 A M 4 = ( H ( 3 . J - 1 . 2 ) + H ( 3 , d . 2 ) ) / 2 . D O A M 5 = ( H ( 3 , d - 1 , 1 ) + H ( 3 , d . 1 ) ) / 2 . D 0 A M 6 = ( - 3 . D O * A M O * A M 1 + 4 . D O * A M 2 * A M 3 - A M 4 * A M 5 ) / 2 . D O A M 7 = ( - 3 . D O * A M 1 / A M 0 + 4 . D O * A M 3 / A M 2 - A M 5 / A M 4 ) / 2 . D O A M 8 = ( A M 0 * * 2 ) / 2 . D O A M 9 = A M 0 * A M 6 / ( 6 . D 0 * A M 1 ) A M 1 0 = - ( A M 0 * * 3 ) * A M 7 / ( 3 . D 0 * A M 1 ) AM11 = - ( A M O * * 2 ) / 4 . D O A M 1 2 = - A M 0 * A M 6 / ( 1 2 . D 0 * A M 1 ) A M 1 3 = ( A M 0 * * 2 ) / 6 . D 0 A M 1 4 = ( A M 0 * * 3 ) * A M 7 / ( 6 . D 0 * A M 1 ) AM15=AM8+AM9+AM10 AM16=AM11+AM12+AM13+AM14 ATDMA (1 ) =AM16 BTDMA(1 )=AM15+AM16 C T D M A ( 1 ) = 0 . D 0 D T D M A ( 1 ) = ( S F ( 2 . J - 1 ) + S F ( 2 . 0 ) ) / 2 . D O A M 0 = ( H ( M , d - 1 . 2 ) + H ( M . J , 2 ) ) / 2 . D 0 A M 1 = ( H ( M . J - 1 , 1 ) + H ( M . J . 1 ) ) / 2 . D 0 AM2 = ( H ( M M 1 , J - 1 , 2 ) + H ( M M 1 . J , 2 ) ) / 2 . D 0 A M 3 = ( H ( M M 1 . J - 1 , 1 ) + H ( M M 1 , J , 1 ) ) / 2 . D O A M 4 = ( H ( M M 2 . J - 1 . 2 ) + H ( M M 2 . J . 2 ) ) / 2 . D 0 A M 5 = ( H ( M M 2 . J - 1 . 1 ) + H ( M M 2 , J , 1 ) ) / 2 . D 0 A M 6 = ( - 3 . D O * AMO* AM1+4 .DO *AM2 *AM3 -AM4 * A M 5 ) / 2 . D O A M 7 = ( - 3 . D O * A M 1 / A M O + 4 . D O * A M 3 / A M 2 - A M 5 / A M 4 ) / 2 . D O AM8 = (AMO* * 2 ) / 2 . D O A M 9 = A M 0 * A M 6 / ( 6 . D 0 * A M 1 ) A M 1 0 = - ( A M O * * 3 ) * A M 7 / ( 3 . D O * A M 1 ) AM 11 = - ( A M 0 * * 2 ) / 4 . D O A M 1 2 = - A M 0 * A M 6 / ( 1 2 . D 0 * A M 1 ) A M 1 3 = ( A M 0 * * 2 ) / 6 . D 0 A M 1 4 = ( A M 0 * * 3 ) * A M 7 / ( 6 . D 0 * A M 1 ) AM15=AM8+AM9+AM10 AM16=AM11+AM12+AM13+AM14 ATDMA(MP 1) = AM16 195 BTDMA(MP 1 ) =0 .D0 CTDMA(MP 1 )=AM15+AM16 DTDMA ( MP 1 ) = ( SF ( MM 1 ,J- 1 )•* SF ( MM 1 . J ) .1/2 .DO A C R I i = 0 . D O ACRIM=O.DO GOTO 150 C C * * » SECOND ORDER. C 140 A M O = ( H ( 1 . J - 1 . 2 ) + H ( 1 . J . 2 ) )/2.DO AM1 = (H( 1 . J - 1 . 1 )+H( 1 . J . 1 ) )/2.DO AM2 = ( H ( 2 . J - 1 . 2 ) + H ( 2 . J . 2 ) ) / 2 .D0 A M 3 = ( H ( 2 . J - 1 . 1 ) + H ( 2 . J . 1 ) )/2.DO A M 4 = ( H ( 3 . J - 1 . 2 ) + H ( 3 , J . 2 ) ) / 2 . D 0 A M 5 = ( H ( 3 . J - 1 . 1 ) + H ( 3 . J , 1 ) ) /2 .DO A M 6 = ( - 3 . D O " A M O * A M 1+4 .DO* A M 2 * A M 3 - A M 4 * A M 5 ) / 2 . D O AM7 = ( - 3 . D O * A M 1 / A M O + 4 . D O * A M 3 / A M 2 - A M 5 / A M 4 ) / 2 . D O A M 8 = 2 7 . D 0 * ( A M 0 * * 2 ) / 4 . D O A M 9 = 3 . D 0 * A M 0 * A M G / ( 2 . D 0 * A M 1 ) A M 1 0 = - 3 . D O * ( A M O * * 3 ) * A M 7 / A M 1 AM 11 = - 3 . D 0 * ( A M 0 * * 2 ) / 4 .DO A M 1 2 = - A M 0 * A M G / ( 6 . D O * A M 1 ) A M 1 3 = ( A M O * * 3 ) * A M 7 / ( 3 . D O * A M 1 ) A M 1 4 = - 9 . D 0 * ( A M 0 * * 2 ) / 4 . D O A M 1 5 = - A M 0 * A M 6 / ( 2 . D 0 * A M 1 ) A M 1 6 = 4 . D O * ( A M O * * 2 ) / 3 . D O A M 1 7 = ( A M 0 * * 3 ) * A M 7 / A M 1 AM18=AM8+AM9+AM10 AM 19 = AM11 + AM12 + AM13 AM20=AM14+AM15+AM16+AM17 A T D M A ( 1 ) = A M 2 0 BTDMA(1 ) =AM18+AM20 C T D M A ( 1 ) = 0 . D 0 D T D M A ( 1 ) = ( 1 6 . D 0 * ( S F ( 2 . J - 1 ) + S F ( 2 . J ) ) - ( S F ( 3 , J - 1 ) + S F ( 3 , J ) ) ) / 2 . DO ACR I 1=AM19 A M O = ( H ( M , J - 1 , 2 ) + H ( M . J . 2 ) ) / 2 . D 0 A M 1 = ( H ( M , J - 1 , 1 ) + H ( M , J . 1 ) ) / 2 . D O A M 2 = ( H ( M M 1 . J - 1 , 2 ) + H ( M M 1 . J . 2 ) ) / 2 . D 0 A M 3 = ( H ( M M 1 , J - 1 . 1 ) + H ( M M 1 . J . 1 ) ) / 2 . D O A M 4 = ( H ( M M 2 , J - 1 . 2 ) + H ( M M 2 . J . 2 ) ) / 2 . D 0 A M 5 = ( H ( M M 2 . J - 1 . 1 ) + H ( M M 2 , J . 1 ) ) / 2 .D0 A M 6 = ( - 3 . D 0 * A M 0 * A M 1 + 4 . D O * A M 2 * A M 3 - A M 4 * A M 5 ) / 2 . D O A M 7 = ( - 3 . D 0 * A M 1 / A M 0 + 4 . D O * A M 3 / A M 2 - A M 5 / A M 4 ) / 2 . D O A M 8 = 2 7 . D O * ( A M 0 * * 2 ) / 4 . D O A M 9 = 3 . D 0 * A M 0 * A M 6 / ( 2 . D 0 * A M 1 ) A M 1 0 = - 3 . D O * ( A M O * * 3 ) * A M 7 / A M 1 AM 11 = - 3 . D 0 * ( A M 0 * * 2 ) / 4 . D O A M 1 2 = - A M 0 * A M G / ( 6 . D O * A M 1 ) A M 1 3 = ( A M 0 * * 3 ) * A M 7 / ( 3 . D 0 * A M 1 ) A M 1 4 = - 9 . D 0 * ( A M 0 * * 2 ) / 4 . D O A M 1 5 = - A M 0 * A M 6 / ( 2 . D 0 * A M 1 ) AM 16 = 4 . D O * ( A M O * * 2 ) / 3 . D O A M 1 7 = ( A M O * * 3 ) * A M 7 / A M 1 AM18=AM8+AM9+AM10 AM19 = AM11 +AM 12 +AM 13 AM20 = AM 14 +AM 15 +AM 16 +AM 17 ATDMA(MP 1 ) = AM20 BTDMA(MP 1 ) = 0 . D 0 CTDMA(MP 1)=AM18 + AM20 DTDMA(MP 1 ) = ( 1 6 . D O " ( S F ( M M 1 . J - 1 ) + SF(MM 1 . J ) ) - ( S F ( M M 2 . J - 1 ) 1 -* S F ( MM2 . J ) ) )/2 . DO ACRIM=AM19 C C - - * R E V I S E D T R I D ] A G O N A L - M A T R I X ALGOR ITHM. C 150 C A L L RTDMA (MP 1 .ACR I 1 . A C R I M ) C C * * * STORE THE SOLUT ION VECTOR . C DD 160 1 = 1,MP 1 A M O = V 0 R ( 1 , J l + R F V O R * ( S L N ( I ) - V O R ( 1 . J ) ) D I F V O R = D M A X 1 ( D I F V O R , D A B S ( A M O - V O R ( I , J ) ) ) MAXVOR=DMAX1 (MAXVOR.DABS (AMO) ) V O R ( I . J ) = AMO 160 CONTINUE I F ( J . E Q . N ) GOTO' 1 90 C C - - » STREAM FUNCT ION E O U A T I O N . C C * - * COMPUTATION OF THE STREAM FUNCT ION EOUATION C O E F F I C I E N T S . C DO 170 ' I=2,MM1 A T = ( H ( I - 1 . J . 1 ) + H ( I , J . 1 ) ) / ( H ( I - 1 , J , 2 ) + H ( I , J . 2 ) ) A R = ( H ( I . J . 2 ) + H ( I . J + 1 , 2 ) ) / ( H ( I . J . 1 ) + H ( I . J + 1 , 1 ) ) A B = ( H ( I , J , 1 ) + H ( 1 + 1 . J , 1 ) ) / ( H ( l . J . 2 ) + H ( I + 1 . J . 2 ) ) A L = ( H ( I , J - 1 . 2 ) + H ( I , J . 2 ) ) / ( H ( I . J - 1 , 1 ) + H ( I , J , 1 ) ) AMO= (V0R ( I , J ) + V O R ( I . J + 1 ) + V 0 R ( l + 1 . J ) + V D R ( I + 1 . J + 1 ) ) / 4 . D O B = A M O * H ( I , J , 1 ) * H ( I , J . 2 ) ATDMA( I ) =AT+AR+AB+AL B T D M A ( I ) = A B C T D M A ( I ) = A T D T D M A ( I ) = A R « S F ( I . J + 1 ) + A L * S F ( I , J - 1 ) + B 1 70 CONTINUE C C * * * SET UP THE BOUNDARY C O N D I T I O N S . C D T D M A ( 2 ) = D T D M A ( 2 ) + C T D M A ( 2 ) * S F ( 1 . J ) C T D M A ( 2 ) = 0 . D 0 DTDMA(MM1)=DTDMA(MM 1 ) + B T D M A ( M M 1 ) * S F ( M , J ) BTDMA(MM1 )=0 .D0 C C * » * T R I D I A G O N A L - M A T R I X A L G O R I T H M . C C A L L TDMA ( 2 . M M 1 ) C c » » » STORE THE S O L U T I O N V E C T O R . C DO 180 1=2.MM1 A M O = S F ( I , J ) + R F S F * ( S L N ( I ) - S F ( I . J ) ) D IFSF=DMAX1 ( D I F S F , D A B S ( A M O - S F ( I . J ) ) ) MAXSF = DMAX 1 ( M A X S F . D A B S ( A M O ) ) S F ( I . J ) = A M O 180 CONTINUE 1 9 0 CONT INUE C C * » * ACCURACY , TEST AND P R I N T O U T . C C2=C2+1 AMO=DIFT AM 1=DIFVOR/MAXVOR AM2 = D I F S F / M A X S F WRITE ( G . 2 0 0 ) C 2 . D I F T . D l F V O R . D l F S F .MAXVOR.MAXST.AMO. AMI . 1 AM2 IF ( M A X V O R . G E . 1 . 0 1 0 ) STOP IF ( ( ( A M O . G T . E P S 2 ) .OR . (AM1 .GT .EPS2 > .OR. (AM2.GT . E P 5 2 ) ) 1 .AND. ( C 2 . L I . C 2UL ) ) GOTO 10 C C * " " * C A L C U L A T I O N OF NUSSELT NUMBERS. C C A L L NU RETURN 2 0 0 F O R M A T ( / . T 5 . ' ( ' . I 3 . ' ) ' . T 1 5 . ' ENERGY - . T 2 G . ' V O R T I C I T Y ' . 1 T 3 8 , ' S T R E A M F U N C T I O N ' , / . T 5 . 4 8 ( ' - ' ) . / . T 5 . ' D I F ' , 2 T 1 3 . D 1 1 . 5 . T 2 6 . D 1 1 . 5 . T 4 0 . D 1 1 . 5 . / , T 5 . ' M A X ' , 3 T 2 G . D 1 1 . 5 , T 4 0 . D 1 1 . 5 . / . T 5 . ' R A T I O ' . T 13 .0 1 1 . 5 . 4 T 2 6 , D 1 1 . 5 . T 4 0 . D 1 1 . 5 ) END C C C SUBROUT INE PLOT C c «»«.»»«. ,««. .« . . . . .«»«, . . * , .« ,« .»«. . . .» . .„„*»«, .»„ . *»* * . . . , „ . . . C C NAME: P L O T . C C P U R P O S E : C C T H I S SUBROUTINE PRODUCES THE PLOT OF THE G R I D . C THE TEMPERATURE , THE STREAM F U N C T I O N . AND C V O R T I C I T Y D I S T R I B U T I O N S AND THE C THE LOCAL NUSSELT NUMBER P L O T S . C C N B . T H I S SUBROUT INE I S NOT GENERAL BUT S P E C I F I C TO A G I V E N C P R O B L E M . IT ALSO REOU IRED THE SOFTWARE " D I S S P L A " C c I M P L I C I T R E A L * 8 ( A - H . D - Z ) R E A L * 8 X Y ( 5 0 . 5 0 , 2 ) , T ( 5 0 . 5 0 ) , S F ( 5 0 . 5 0 ) . V 0 R ( 5 O , 5 O ) R E A L * 8 D I S T ( 5 0 , 4 ) , N U A V ( 4 ) . N U L ( 5 0 , 4 ) REAL X ( 1 0 0 ) . Y ( 1 0 0 ) , X Y S ( 5 0 . 5 0 , 2 ) . X Y Z ( 5 0 . 5 0 . 3 ) . Z C N T R ( 5 0 ) R E A L M A X S F . M I N S F . M A X V O R , M I N V O R COMMON / B L K 1 / M.N COMMON / B L K 3 / XY COMMON / B L K G / P R . R A COMMON / B L K 7 / T COMMON / B L K 8 / VOR COMMON / B L K 9 / SF COMMON / B L K 1 3 / D l S T , N U A V . N U L COMMON / B L K A / X Y Z . Z C N T R C C * * » GR ID TO COARSE . C IF ( ( M . L T . 2 0 ) . A N D . ( N . L T . 2 0 ) ) RETURN C C » » * SET UP V A R I A B L E S . C MM1=M-1 MP 1=M+1 198 MP2=M+2 NP1=N+1 C C - - * GRAPH THE G R I D . C C A L L BGNPL ( - 1) C A L L T I T L E ( ' G R I D ' . -4 . ' X A X I S ' . 6 , ' Y A X I S ' . 6 . 7 . . 6 . ) C A L L GRAPH ( 0 . . . 2 . 0 . . . 2 ) C A L L FRAME DO 2 0 1 = 1 . M DO tO J = 1 , N X ( J ) = X Y ( I . J . 1 ) Y ( J ) = X Y ( I , 0 . 2 ) 10 CONT INUE C A L L CURVE ( X . Y . N . O ) 2 0 CONT INUE DO 4 0 U= 1 ,N DO 3 0 1=1.M X ( I ) = X Y ( 1 . U . 1 ) Y ( I ) = X Y ( I , J . 2 j 3 0 CONT INUE C A L L CURVE ( X . Y . M . O ) 4 0 CONT INUE C A L L ENDPL ( - 1 ) C C * * « GRAPH THE LOCAL N U S S E L T NUMBERS (R IGHT W A L L ) . C C A L L BGNPL ( - 2 ) C A L L T I T L E ( ' N U S S E L T NUMBER AT R IGHT WALL ( V S ) D I S T A N C E S ' , - 1 0 0 , 1 ' D I S T A N C E (TOP TO B O T T O M ) ' . 2 4 . ' N U S S E L T N U M B E R ' . 1 4 , 7 . . 6 . ) A M 0 = 1 0 . / 6 . C A L L GRAF (O . . . 2 . 1 . 4 , O. . 1 . , 1 0 . ) C A L L FRAME C A L L GRACE ( O . ) C A L L MESSAG ( ' P R = ' . 3 . 4 . , 5 . 7 ) C A L L REALNO ( P R . 2 . ' A B U T ' . ' A B U T ' ) C A L L MESSAG ( ' R A = ' , 3 . 4 . , 5 . 4 ) C A L L REALNO ( R A , - 2 . ' A B U T ' , ' A B U T ' ) C A L L MARKER ( 1 3 ) C A L L S P L I N E DO 5 0 I=1,MM1 X ( I ) = ( D I S T ( I , 2 ) + 0 I S T ( I + 1 , 2 ) ) / 2 . Y ( I ) = N U L ( I + 1 . 2 ) 5 0 CONT INUE C A L L CURVE ( X . Y . M M 1 . 1 ) X ( 1 ) = 0 . Y ( 1 ) = N U A V ( 2 ) X ( 2 ) = D I S T ( M , 2 ) Y ( 2 ) = N U A V ( 2 ) C A L L RESET ( ' S P L I N E ' ) C A L L CURVE ( X . Y . 2 , 0 ) Y P O S = N U A V ( 2 ) * 6 . / 1 0 . + . 0 2 C A L L MESSAG ( ' A V E R A G E ' , 7 . 0 . 5 , Y P O S ) C A L L ENDPL ( - 2 ) C C * * * GRAPH THE LOCAL N U S S E L T NUMBER ( L E F T W A L L ) . C C A L L BGNPL ( - 3 ) C A L L T I T L E ( ' N U S S E L T NUMBER AT L E F T WALL ( V S ) DI S T A N C E S ' . - 1 O 0 . 1 ' D I S T A N C E (TOP TO B O T T O M ) ' . 2 4 . ' N U S S E L T NUMBER ' . 14 . 7 . . 6 . ) 199 CALL CALL CALL CALL CALL CALL CALL CALL CALL DO 6 0 GRAF (O. FRAME GRACE ( 0 MESSAG REALNO MESSAG REALNO MARKER SPLINE 1 = 1.MM 1 . 2 . 1 .4.0. . 1 . . 10 . ) . ) ( 'PR= ' . 3 . 4 . . i (PR . 2, 'ABUT ' , ( 'RA= ' .3 . 4 . , J ( R A . - 2 . ' A B U T ' ( 13) . 7 ) 'ABUT'I .4 ) . ' ABUT ' . 4 ) ) / 2 . X( I ) = (DIST(I ,4 )+DIST(1 + 1 Y ( I ) = N U L ( 1 + 1 . 4 ) 6 0 CONT INUE C A L L CURVE (X .Y ,MM 1 . 1 ) X ( 1 ) = 0 . Y ( 1 ) = N U A V ( 4 ) X ( 2 ) = D I S T ( M . 4 ) Y ( 2 ) = N U A V ( 4 ) C A L L RESET ( ' S P L I N E ' ) C A L L CURVE ( X . Y . 2 , 0 ) Y P 0 S = N U A V ( 4 ) * 6 . / 1 0 . + . 0 2 C A L L MESSAG ( ' A V E R A G E ' , 7 , 3 . 7 5 , Y P O S ) C A L L ENDPL ( - 3 ) ' "•* THE TEMPERATURE • * * I S NOT DEF INED AT THE G R I D I N T E R S E C T I O N S BUT AT THE CENTER OF E A C H SOUARE D E F I N E BY THE G R I D . C A L C U L A T I O N OF THE COORD INATES OF THE C E N T E R . CORNER P O I N T S . X Y S ( 1 . 1 . 1 ) = X Y ( 1 , 1 X Y S ( 1 . 1 , 2 ) = X Y ( 1 . 1 X Y S ( 1 , N P 1 , 1 ) = X Y ( 1 X Y S ( 1 , N P 1 , 2 ) = X Y ( 1 X Y S ( M P 1 . 1 . 1 ) = X Y ( M 1 , 2 ) = X Y ( M 7 0 8 0 C C " C 1) 2 ) N, 1 ) N . 2 ) 1 . 1 ) X Y S ( M P 1 . . 2 ) = X Y ( . 1 . 2 ) X Y S ( M P 1 . N P 1 , 1 ) = X Y ( M . N . 1 ) X Y S ( M P 1 , N P 1 , 2 ) = X Y ( M . N , 2 ) BOUNDARY P O I N T S . DO 7 0 J = 2 , N X Y S ( 1 . J . 1 ) = ( X Y ( 1 . J X Y S ( 1 . J . 2 ) = ( X Y ( 1 , J X Y S ( M P 1 , J , 1 ) = ( X Y ( M , X Y S ( M P 1 . d . 2 ) = ( X Y ( M , CONT INUE DO 8 0 1=2,M X Y S ( I , 1, 1) = ( X Y ( I , 1, X Y S ( I . 1 , 2 ) = ( X Y ( I . 1 . X Y S ( I , N P 1 . 1 ) = ( X Y ( I , X Y S ( I , N P 1 , 2 ) = ( X Y ( I , CONT INUE I N T E R N A L P O I N T S . 1) + X Y ( 1 . J -2 ) + X Y ( 1 . J -d , 1 ) + X Y ( M , J . 2 ) + X Y ( M . 1) + X Y ( I - 1 2 ) + X Y ( I - 1 N , 1 ) + X Y ( I N . 2 ) + X Y ( I 1. 1 ) ) / 2 . D 0 1 . 2 ) ) / 2 . D 0 J - 1 , 1 ) ) / 2 . D 0 J - 1 . 2 ) ) / 2 . D 0 1 . 1 ) ) / 2 . D 0 1 . 2 ) )/2 .DO • 1 , N , 1 ) ) / 2 . D O • 1. N . 2 ) ) /2 . DO-DO 100 1=2,M DO 9 0 J = 2 . N X Y S ( I , d . 1 ) = ( X Y ( I J . 1 ) + X Y ( I - 1 , J . 1 ) + X Y ( I , J - 1 , 1 ) + X Y ( I - 1 . J - 1 . 1 ) ) 2 0 0 1/4.DO X Y S ( I . J , 2 ) = ( X Y . ( I , d . 2 ) + X Y ( l - 1 . J . 2 ) + X Y ( I . J - 1 . 2 ) + X Y ( I - 1 . J - 1 . 2 ) ) 1/4.DO 9 0 CONT INUE 100 CONT INUE C C * * * GRAPH THE TEMPERATURE F I E L D . C DO 120 1 = 1.MP 1 DO 110 J = 1 , N P 1 X Y Z ( I . J . 1 )=XYS( I . <J . 1 ) X Y Z ( 1 . J . 2 ) = X Y S ( I . 0 . 2 ) I F ( J . E 0 . 1 ) X Y Z ( I . d . 3 ) = 0 . I F ( U . E 0 . N P 1 ) X Y Z ( I . v J .3 ) = 1 . I F ( ( U . N E . 1 ) . AND . ( d . N E . NP 1 ) ) X YZ ( I . vJ. 3 ) = T ( I . U ) 110 CONT INUE 1 2 0 CONT INUE C A L L ESGNPL ( - 4 ) C A L L T I T L E ( ' T E M P E R A T U R E F I E L D S ' . - 1 0 0 . 1 ' X A X I S ' . G . ' Y A X I S ' . 6 , 7 . , 6 . ) C A L L GRAPH ( 0 . . . 2 . O . . . 2 ) C A L L FRAME C A L L GRACE ( 0 . ) C A L L MESSAG ( ' PR= ' . 3 , 4 . , 5 . 7 ) C A L L REALNO ( P R , 2 . ' A B U T ' . ' A B U T ' ) C A L L MESSAG ( ' RA>= ' . 3 , 4 . . 5 . 4 ) C A L L REALNO ( R A . - 2 , ' A B U T ' . ' A B U T ' ) WRITE ( 6 . 2 7 0 ) AM0=0. AM 1 = 1. WRITE ( 6 . 3 0 0 ) AM0.AM1 DO 1 3 0 1=1 .9 Z C N T R ( I ) = I / 1 0 . WRITE ( 6 , 3 1 0 ) I . Z C N T R ( I ) 1 3 0 CONT INUE C A L L CNTR ( MP 1 :, NP 1.9) : X ( 1 ) = 0 . Y ( 1 ) = 1 . DO 1 4 0 1=1.MPT X ( I + 1 ) = X Y S ( I , N P 1 . 1 ) Y ( 1 +1 ) = X Y S ( I , N P 1 , 2 ) 1 4 0 CONT INUE C A L L CURVE ( X . Y . M P 2 . 0 ) C A L L ENDPL ( - 4 ) C C * * * GRAPH THE V O R T I C I T Y F I E L D . C X Y Z ( 1 . 1 .3 ) = ( V 0 R ( 1 . 2 ) + V 0 R ( 2 . 1 ) + V O R ( 2 . 2 ) ) / 3 . X Y Z ( 1 , N P 1 , 3 ) = ( V 0 R ( 1 . N ) + V 0 R ( 2 . N ) + V 0 R ( 2 , N P 1 ) ) / 3 . X Y Z ( M P 1 . 1 .3 ) = ( V O R ( M , 1) + V O R ( M . 2 ) + VOR ( MP 1 . 2 ) ) / 3 . X Y Z ( M P 1 , N P 1 . 3 ) = ( V 0 R ( M . N ) + V 0 R ( M , N P 1 ) + V 0 R ( M P 1 . N ) ) / 3 . DO 150 1=2.M X Y Z ( I . 1 , 3 ) = ( V 0 R ( I . 1 ) + V 0 R ( I . 2 ) ) / 2 . X Y Z ( I . N P 1 . 3 ) = ( V O R ( I , N ) + V O R ( I . N P 1 ) ) / 2 . 1 50 CONT INUE DO 160 J = 2 , N X Y Z ( 1 , vJ.3) = ( V 0 R ( 1 , J ) + V 0 R ( 2 , vJ) )/2 . X Y Z ( M P 1 , J , 3 ) = ( V O R ( M . J ) + V O R ( M P 1 . J ) ) / 2 . 1 60 CONT INUE DO 180 1=2.M DO 170 J = 2 . N XYZ ( 1 , J . 3 )=VOR( I.J.) 1 70 CONTINUE 1B0 CONT INUE C A L L BGNPL ( - 5 ) C A L L T I T L E ( ' V O R T I C I T Y F I E L D $ ' . - 100. 1 ' X A X I S ' . € . ' Y A X I S ' . 6 . 7 . . 6 . ) C A L L GRAPH ( 0 . . . 2 . 0 . . . 2 ) C A L L FRAME C A L L GRACE ( O. ) C A L L MESSAG ( ' P R = ' , 3 . A . .5 . 7 ) C A L L REALNO ( P R , 2 , ' A B U T ' , ' A B U T ' ) C A L L MESSAG ( ' R A = ' . 3 . A . .5.4) C A L L REALNO ( R A . - 2 . ' A B U T ' . ' A B U T ' ) WRITE ( 6 . 2 8 0 ) M A X V O R = X Y Z ( 1 . 1 . 3 ) M I N V O R = X Y Z ( 1 . 1 . 3 ) DO 2 0 0 1 = 1 ,MP1 DO 190 J = 1 , N P 1 M I N V 0 R = A M I N 1 ( M I N V O R , X Y Z ( I . J . 3 ) ) M A X V D R = A M A X 1 ( M A X V O R . X Y Z ( I , J . 3 ) ) 1 90 CONT INUE 2 0 0 CONT INUE WRITE ( 6 . 3 0 0 ) M INVOR.MAXVOR DO 2 1 0 1=1 . 9 Z C N T R ( I ) = ( M A X V O R - M I N V O R ) * I /10.+MINVDR WRITE ( 6 . 3 1 0 ) I . Z C N T R ( I ) 2 1 0 CONT INUE C A L L CNTR (MP 1 , N P 1 . 9 ) X(1)=o. Y ( 1 ) = 1 . DO 2 2 0 I = 1.MP 1 X ( 1 + 1 ) = X Y S ( 1 , N P 1 . 1 ) Y ( I + 1 ) = X Y S ( I , N P 1 , 2 ) 2 2 0 CONT INUE C A L L CURVE ( X , Y , M P 2 , 0 ) C A L L ENDPL ( - 5 ) C c « * « GRAPH THE STREAM F U N C T I O N F I E L D . C MAXSF = S F ( 1 , 1 ) M I N S F = S F ( 1 . 1 ) DO 2 4 0 I = 1 .M DO 2 3 0 J=1 ,N X Y Z ( I . J , 1) = X Y ( I . J . 1) X Y Z ( I , J , 2 ) = X Y ( I . J . 2 ) X Y Z ( I . J . 3 ) = S F ( I , J ) MAXSF = A M A X 1 ( M A X S F , X Y Z ( 1 . J . 3 ) ) M INSF = A M I N 1 ( M I N S F , X Y Z ( I . J . 3 ) ) 2 3 0 CONT INUE 2 4 0 CONT INUE C A L L BGNPL ( - 6 ) C A L L T I T L E ( ' S T R E A M F U N C T I O N F I E L D S ' . - 1 0 0 1 , ' X A X I S ' . 6 . ' Y A X I S ' . 6 . 7 . . 6 . ) C A L L GRAPH ( 0 . . . 2 . O . . . 2 ) C A L L FRAME C A L L GRACE ( 0 . ) C A L L MESSAG ( ' P R = ' , 3 . 4 . . 5 . 7 ) C A L L REALNO ( P R , 2 . ' A B U T ' , ' A B U T ' ) C A L L MESSAG ( ' R A = ' , 3 . 4 . .5 . 4 ) C A L L REALNO ( R A , - 2 , ' A B U T ' , ' A B U T ' ) WRITE ( 6 . 2 9 0 ) WRITE ( 6 . 3 0 0 ) M I N S F . M A X S F DO 2 5 0 1=1 .9 Z C N T R ( 1 ) = ( M A X S F - M I N S F ) * I /10 .+M INSF WRITE ( 6 . 3 1 0 ) l . Z C N T R ( I ) 2 5 0 CONT INUE C A L L CNTR ( M . N . 9 ) X ( 1 ) = 0 . Y( 1 ) = 1 . DO 2 6 0 I=1 .M X ( 3+1 )=XY(1 . N . 1 ) Y ( I + 1 )=XY( 1 . N . 2 ) 2 6 0 CONT INUE C A L L CURVE ( X . Y . M P 1 . 0 ) C A L L ENDPL ( - 6 ) RETURN 2 7 0 F 0 R M A T ( / . T 5 . ' T E M P E R A T U R E CDNTDUR V A L U E S . ' ) 2 8 0 F 0 R M A T ( / , T 5 , ' V O R T I C I T Y COUNTOUR V A L U E S . ' ) 2 9 0 F D R M A T ( / , T 5 , ' S T R E A M F U N C T I O N CONTOUR V A L U E S . ' ) 3 0 0 F 0 R M A T ( T 5 , ' M I N = ' , E 1 5 . 7 . / , T 5 , ' M A X * ' . E 1 5 . 7 ) 3 1 0 F O R M A T ( T 5 . 'CONTOUR * " . 15 . 5X , E 15 . 7 ) END C C C SUBROUT INE TDMA ( I M I N . I M A X ) C c.....,»...,.,.«........,...........»..»............ C C NAME: T R I D I A G O N A L MATR I X ALGOR ITHM. C C P U R P O S E : C C T H I S SUBROUT INE SOLVES A T R I D I A G O N A L M A T R I X . C C INPUT DATA : C C - I M I N : MINIMUM I N D E X . C - IMAX : MAXIMUM I N D E X . C c««.*«.«........................................... C I M P L I C I T R E A L * 8 ( A - H . O - Z ) R E A L * 8 A T D M A ( 5 0 ) , B T D M A ( 5 0 ) . C T D M A ( S O ) , D T D M A ( 5 0 ) R E A L * 8 P T D M A ( 5 0 ) . O T D M A ( 5 0 ) . S L N ( 5 0 ) COMMON / B L K 1 0 / A T D M A . B T D M A . C T D M A . D T D M A . S L N C C * » SET UP V A R I A B L E S . C IM INP1= IM IN+1 IMAXM1 = 1 MAX - 1 N = I M A X - I M I N C C * * * SOLVE THE ARRAY . C P T D M A ( I M I N ) = B T D M A ( I M I N ) / A T D M A ( I M I N ) O T D M A ( I M I N ) = D T D M A ( I M I N ) / A T D M A ( I M I N ) DO 10 I = I M I N P 1 . I M A X DEN = ATDMA( I ) - C T D M A ( I )* P T D M A ( 1 - 1 ) 2 0 3 PTDMA ( I ) =BTDMA ( I ) / 0 E N OTDMA(1 ) ' ( D T D M A ( 1 )+CTDMA( I ) «OTDMA( 1-1 ) )/DEN 10 CONTINUE SLN ( I MAX ) = OTDMA( I MAX) DO 2 0 1 = 1 . N J = I M A X - I S L N ( J ) = P T D M A ( J ) « S L N ( J + 1 K O T D M A ( J I ' 2 0 CONT INUE RETURN END C C C SUBROUTINE RTDMA ( I M A X . A C R 1 1 . ACR IM ) C c.«..-,«.«,,.»...«,..»».»»..»».«.,..»«...«..»».»»»..«»».»....*..*..* C C NAME : R E V I S E D T R I D I A G O N A L MATRIX ALGOR ITHM. C C P U R P O S E : C C T H I S PROGRAM S O L V E S A T R I D I A G O N A L - M A T R I X WHICH HAS TWO ELEMENTS C O U T S I D E OF THE T R I D I A G O N A L . THE F I R S T ONE I S LOCATED ROW 1 AND C COLUMN 3 , AND THE SECOND I S LOCATED ROW IMAX AND COLUMN I M A X - 2 . C C INPUT DATA: C C - I M A X : NUMBER OF ROW AND COLUMN I N THE ARRAY . C - A C R I 1 : A D D I T I O N A L C O E F F I C I E N T RELATED TO THE ROW 1. C - A C R I M : A D D I T I O N A L C O E F F I C I E N T RELATED TO THE ROW IMAX. C c«.»»„..............*..»........»...............*................... C I M P L I C I T R E A L * 8 ( A - H . 0 - Z ) R E A L * 8 A T D M A ( 5 0 ) , B T D M A ( 5 0 ) , C T D M A ( 5 0 ) . D T D M A ( 5 0 ) . S L N ( 5 0 ) COMMON / B L K 1 0 / A T D M A . B T D M A . C T D M A , D T D M A . S L N C C * * * SET UP V A R I A B L E S . C IMAXM1= IMAX-1 G C * * * M O D I F I E D THE TDMA C O E F F I C I E N T S . C A T D M A ( 2 ) ' A T D M A ( 2 ) - C T D M A ( 2 ) * B T D M A ( 1 ) / A T D M A ( 1 ) B T D M A ( 2 ) ' B T D M A ( 2 ) + C T D M A ( 2 ) * ACR I 1/ATDMA ( 1) D T D M A ( 2 ) ' D T D M A ( 2 J + C T D M A ( 2 ) * D T D M A ( 1 ) / A T D M A ( 1 ) A T D M A ( I M A X M 1 ) ' A T D M A ( I M A X M 1 ) - B T D M A ( I M A X M 1 ) " C T D M A ( I M A X ) 1/ATDMA( IMAX ) C T D M A ( I M A X M 1 ) ' C T D M A ( I M A X M 1 ) + B T D M A ( I M A X M 1 ) * A C R I M / A T D M A ( I M A X ) D T D M A ( I M A X M 1 ) ' D T D M A ( I M A X M 1 ) + B T D M A ( I M A X M 1 ) « D T D M A ( I M A X ) 1/ATDMA( IMAX) C T D M A ( 2 ) = 0 . D 0 B T D M A ( I M A X M 1 ) = 0 . D 0 C C * * » TDMA. C C A L L TDMA ( 2 . 1 M A X M 1 ) C C * * * C A L C U L A T E V A L U E S FOR 1 AND IMAX. C 2 0 4 S LN ( 1) = (BTDMA( 1 ) * S L N ( 2 ) + ACR I 1 * S L N ( 3 ) + DTDMA( 1 ) ) / A T D M A ( 1 ) S L N ( I M A X ) = ( A C R I M * S L N ( I M A X - 2 ) + C T D M A ( I M A X ) * S L N ( 1 M A X M 1 ) 1 + D T D M A ( I M A X ) ) / A T D M A ( I MAX ) RETURN END C C C DOUBLE P R E C I S I O N F U N C T I O N COEFF (COND.CONV) C c . . ..»..»..»...»....»....,.*»»*...«..,....,.«.,. »..».«.•,....»»«..», C C NAME: C O E F F I C I E N T . C C P U R P O S E : C C T H I S SUBROUT INE C A L C U L A T E S THE C O E F F I C I E N T A ( P ) C OF THE POWER LAW SCHEME AS PRESENTED BY P A T A N K A R . D C C N B . THE POWER LAW I S USED BELOW, BUT OTHER SCHEMES AS CENTRAL C D I F F E R E N C E SCHEME. UPWIND SCHEME OR H Y B R I D SCHEME COULD BE C USED AS WELL . C C INPUT DATA : C C - COND: CONDUTION S T R E N G T H . C - CONV: CONVECT ION S T R E N G T H . C c**««»,*„««»«.*»„„„.«»»»«»»»»»...«».««..»...«»».«*,.»...«».*.»*»»«, C I M P L I C I T R E A L - B ( A - H . O - Z ) C O E F F = D M A X 1 ( O . D O . ( 1 . D O - O . 1 D O * D A B S ( C O N V / C O N D ) ) - * 5 ) RETURN END C e C SUBROUT INE D I S F C c»*.».».«*................»..,.,....,,...*,.,..».»*.,»*»...,.. C C NAME: D I S T O R T I O N F U N C T I O N . C C P U R P O S E : C C T H I S PROGRAM COMPUTES THE F I R S T ORDER D E R I V A T I V E S OF THE C C A R T E S I A N COORD INATES W ITH R E S P E C T TO THE ORTHOGONAL C COORD INATES AND THE S C A L E FACTORS ALONG BOUNDAR IES C WHERE TWO D I R I C H L E T BOUNDARY COND IT IONS ARE USED . C ALONG BOUNDAR IES WHERE THE PO INT C P O S I T I O N S ARE NOT G I V E N THE R A T I O S OF S C A L E FACTORS ARE C S P E C I F I E D BY L I N E A R L Y I N T E R P O L A T I N G THE CORNER V A L U E S . C THE R A T I O OF S C A L E FACTORS FOR THE I NTERNAL GR ID PO INTS C ARE COMPUTE US ING THE FORMULA PROPOSED BY G . R Y S K I N AND C L . G . L E A L . C C C H A R A C T E R I S T I C : C C - F I N I T E D I F F E R E N C E S FORMULA OF SECOND OROER. C c I M P L I C I T R E A L * 8 ( A - H . 0 - Z ) INTEGER B T Y P E ( 4 ) R E A L ' S D ( 5 0 . 5 0 . 4 ) . H ( 5 0 . 5 0 , 2 ) . R S F ( 5 0 . 5 0 ) . X Y ( 5 0 . 5 0 . 2 ) COMMON / B L K 1 / M.N COMMON / B L K 2 / B T Y P E . R S F COMMON / B L K 3 / XY COMMON / B L K 4 / H COMMON / B L K 5 / D C C - * * SET UP V A R I A B L E S . C MM1=M-1 MM2=M-2 NM1=N-1 NM2=N-2 C C * * * A S S I G N RAT IO OF S C A L E FACTORS AT BOUNDAR I E S . C C * * * TOP L E F T CORNER. C I F ( ( B T Y P E ( 1 ) . E O . O ) . A N D . ( B T Y P E ( 4 ) . E O . O ) ) GOTO 10 D( 1 , 1 , 1 )-=(-3 . D O * X Y ( 1 . 1 . 1 )+4 .DO*XY ( 1 . 2 . 1 ) - XY ( 1 , 3 . 1 ) )/2 . DO D ( 1 . 1 , 2 ) = ( - 3 . D O * X Y ( 1 , 1 , 2 ) + 4 . D 0 * X Y ( 1 , 2 . 2 ) - X Y ( 1 , 3 , 2 ) ) / 2 . D O D ( 1 . 1 , 3 ) = ( 3 . D O * X Y ( 1 , 1 , 1 ) - 4 . D 0 * X Y ( 2 , 1 , 1 ) + X Y ( 3 , 1 , 1 ) ) / 2 . D 0 D ( 1 . 1 , 4 ) = ( 3 . D O * X Y ( 1 . 1 , 2 ) - 4 . D 0 * X Y ( 2 , 1 , 2 ) + X Y ( 3 , 1 , 2 ) ) / 2 . D 0 H ( 1 , 1 , 1 ) - = ( D ( 1 , 1 , 1 ) * * 2 + D ( 1 . 1 , 2 ) * * 2 ) * * 0 . 5D0 H ( 1 , 1 , 2 ) = ( D ( 1 , 1 . 3 ) • * 2 + D ( 1 . 1 . 4 ) - » 2 ) * * 0 . 5 D 0 R S F ( 1 , 1 ) = H ( 1 , 1 . 2 ) / H ( 1 , 1 . 1 ) C C * * * TOP R IGHT CORNER. C 10 I F ( ( B T Y P E ( I ) . E O . O ) . A N D . ( B T Y P E ( 2 ) . E O . O ) ) GOTO 2 0 D ( 1 , N . 1 ) = ( 3 . D 0 * X Y ( 1 . N , 1 ) - 4 . D 0 * X Y ( 1 . N M 1 . 1 ) + X Y ( 1 , N M 2 . 1 ) ) / 2 . D 0 D ( 1 . N . 2 ) = ( 3 . 0 0 * X Y ( 1 , N , 2 ) - 4 . D 0 * X Y ( 1 . N M 1 , 2 ) + X Y ( 1 . N M 2 . 2 ) ) / 2 . D O D ( 1 , N . 3 ) = ( 3 . D 0 * X Y ( 1 , N , 1 ) - 4 . D O * X Y ( 2 , N , 1 ) + X Y ( 3 . N , 1 ) ) / 2 . D 0 D( 1 , N , 4 ) = ( 3 . D 0 * X Y ( 1 , N , 2 ) - 4 . D O ' X Y ( 2 , N , 2 ) + X Y ( 3 , N , 2 ) ) / 2 . D O H ( 1 , N , 1 ) = ( D ( 1 , N . 1 ) * * 2 + D ( 1 , N , 2 ) * * 2 ) * * 0 . 5 D 0 H ( 1 , N , 2 ) - ( D ( 1 , N , 3 ) * * 2 + D ( 1 . N . 4 ) * * 2 ) * * 0 . 5 D 0 R S F ( 1 , N ) = H ( 1 , N . 2 ) / H ( 1 , N . 1 ) C C * * * BOTTOM L E F T CORNER. C 2 0 I F ( ( B T Y P E ( 4 ) . E O . O ) . A N D . ( B T Y P E ( 3 ) . E O . 0 ) ) GOTO 3 0 D ( M , 1 , 1 ) = ( - 3 . 0 0 * X Y ( M . 1 , 1 )+4 . D 0 * X Y ( M , 2 . 1 ) - X Y ( M , 3 . 1 ) ) / 2 . D O D ( M , 1 , 2 ) = ( - 3 . D O * X Y ( M , 1 . 2 ) + 4 . D O * X Y ( M , 2 . 2 ) - X Y ( M , 3 . 2 ) ) / 2 . D O D ( M , 1 , 3 ) = ( - 3 . D O * X Y ( M . 1, 1) + 4 . D O ' X Y ( M M 1 . 1 . 1 ) - X Y ( M M 2 . 1. 1 ) ) / 2 . D O D ( M , 1 , 4 ) = ( - 3 . D 0 * X Y ( M , 1 , 2 ) + 4 . D O * X Y ( M M 1 . 1 , 2 ) - X Y ( M M 2 , 1 , 2 ) ) / 2 .DO H ( M . 1 , 1 ) = ( D ( M , 1 . 1 ) * - 2 + D ( M , 1 , 2 ) * * 2 ) • * 0 . 5 D 0 H ( M , 1 , 2 ) = ( 0 ( M , 1 , 3 ) * * 2 + D ( M , 1 , 4 ) * « 2 ) * * 0 . 5 D 0 R S F ( M , 1 ) = H ( M , 1 . 2 ) / H ( M , 1 , 1 ) C C * * » BOTTOM R IGHT CORNER. C 3 0 I F ( ( B T Y P E ( 2 ) . E O . O ) . A N O . ( B T Y P E ( 3 ) . E O . O ) ) GOTO 4 0 D ( M , N . 1) = ( 3 . 0 0 * X Y ( M . N , 1 ) - 4 . D O » X Y ( M , N M 1 . 1 ) + X Y ( M . N M 2 . 1 ) ) / 2 . D O D ( M . N . 2 ) = ( 3 . 0 0 * X Y ( M . N , 2 ) - 4 .DO*XY(M.NM1 . 2 ) + X Y ( M . N M 2 . 2 ) ) / 2 . D O D ( M . N , 3 ) = ( - 3 . D O * X Y ( M . N . 1 ) + 4 . DO*XY(MM 1 ,N. 1 ) - X Y ( M M 2 . N . 1 ) ) / 2 . D 0 D ( M . N . 4 ) = ( - 3 . D 0 * X Y ( M . N . 2 ) + 4 . D 0 * X Y ( M M 1 . N . 2 ) - X Y ( M M 2 . N , 2 ) ) / 2 . D O H ( M , N , 1 ) = ( D ( M , N , 1 ) * * 2 + D ( M , N . 2 ) * * 2 ) * * 0 . 5 D O H ( M , N . 2 ) = ( D ( M . N . 3 ) - * 2 + D ( M , N , 4 ) - - 2 ) * * O . 5DO R S F ( M . N ) = H ( M . N . 2 ) / H ( M . N . 1 ) C C * * « TOP WALL . C 4 0 I F ( B T Y P E ( 1 ) . E O . O ) GOTO GO DO 5 0 J=2.NM1 D( 1 , J . 1 ) = ( X Y ( 1 . J + 1 . 1 ) - X Y ( 1 . J - 1 . 1 ) )/2 .DO D ( 1 , J . 2 ) = ( X Y ( 1 , J + 1 . 2 ) - X Y ( 1 . J - 1 . 2 ) ) / 2 . D 0 D( 1 . J . 3 ) = ( 3 . D O * X Y ( 1 . J . 1 ) - A .DO* X Y ( 2 . J . 1) + X Y ( 3 , J , 1 ) )/2.DO D( 1 . J . 4 ) = ( 3 . D O * X Y ( 1 . J . 2 ) - 4 . D O * X Y ( 2 , J . 2 ) + XY ( 3 . J . 2 ) ) / 2 .DO H ( 1 . J . 1 ) = ( D ( 1 . J . 1 ) - * 2 + D ( 1 . J . 2 ) * » 2 ) - * 0 . 5 D O H ( 1 . J . 2 ) = ( D ( 1 , J , 3 ) * * 2 + D ( 1 . J . 4 ) * * 2 ) * * 0 . 5 D 0 R S F ( 1 . J ) = H ( 1 . J . 2 ) / H ( 1 . J . 1 ) 5 0 CONT INUE C C * * * BOTTOM WALL . C 6 0 I F ( B T Y P E ( 3 ) . E O . O ) GOTO 8 0 DO 7 0 J=2,NM1 D ( M . J , 1 ) = ( X Y ( M . J + 1 , 1 ) - X Y ( M . J - 1 . 1 ) ) / 2 . D 0 D ( M . J . 2 ) = ( X Y ( M , J + 1 , 2 ) - X Y ( M . J - 1 . 2 ) ) / 2 . D 0 D ( M , J , 3 ) = ( - 3 . D O ' X Y ( M , J , 1 )+4 . D O * X Y ( M M 1 . J , 1 ) - X Y ( M M 2 , J . 1 ) ) / 2 . D 0 D ( M , J , 4 ) = ( - 3 . D O * X Y ( M , J . 2 ) + 4 . D O * X Y ( M M 1 , J . 2 ) - X Y ( M M 2 . J . 2 ) ) / 2 . DO H ( M . J . 1 ) = ( D ( M , J , 1 ) - * 2 + D ( M . J , 2 ) * * 2 ) " * 0 . 5 D 0 H ( M , J . 2 ) = ( D ( M . J . 3 ) * * 2 + D ( M , J , 4 ) * * 2 ) * * 0 . 5 D 0 R S F ( M . J ) = H ( M , J , 2 ) / H ( M , J . 1 ) 7 0 CONT INUE C C * * * L E F T WALL . C 8 0 I F ( B T Y P E ( 4 ) . E O . O ) GOTO 1 0 0 DO 9 0 I=2.MM1 D ( 1 . 1 . 1 ) « ( - 3 . D 0 * X Y ( I . 1 . 1 ) + 4 . D O * X Y ( 1 . 2 . 1 ) - X Y ( I . 3 . 1 ) ) / 2 . D 0 D ( I . 1 , 2 ) = ( - 3 - D O * X Y ( I , 1 , 2 ) + 4 . D 0 * X Y ( 1 , 2 . 2 ) - X Y ( I . 3 . 2 ) ) / 2 . D 0 D ( I . 1 . 3 ) = ( X Y ( I - 1 . 1 , 1 ) - X Y ( I + 1 . 1 . 1 ) ) / 2 . D 0 D ( I , 1 . 4 ) « = ( X Y ( I - 1 . 1 , 2 ) - X Y ( I + 1 , 1 . 2 ) J / 2 . D 0 H ( I . 1 , 1 ) C ( D ( I , 1 . 1 ) * * 2 + D ( I . 1 . 2 ) * * 2 ) * * 0 . 5 D 0 H ( I . 1 . 2 ) = ( D ( I , 1 , 3 ) * * 2 + D ( I , 1 . 4 ) - * 2 ) * « 0 . 5 D 0 R S F ( I . 1 ) = H ( I , 1 . 2 ) / H ( I , 1 . 1 ) 9 0 CONT INUE C C * * * R IGHT WALL . C 100 I F ( B T Y P E ( 2 ) . E O . 0 ) GOTO 1 2 0 DO 110 1=2.MM1 D ( I . N , 1 ) = ( 3 . D 0 * X Y ( I , N , 1 ) - 4 . D O * X Y ( I , N M 1 . 1 ) + X Y ( I . N M 2 . 1 ) ) / 2 . D 0 D ( I , N . 2 ) = ( 3 . D O * X Y ( I . N . 2 ) - 4 . D O * X Y ( I . N M 1 . 2 ) + X Y ( I , N M 2 . 2 ) ) / 2 . D O D ( I . N . 3 ) = ( X Y ( l - 1 . N . 1 ) - X Y ( I + 1 . N . 1 ) ) / 2 . D O D ( I . N , 4 ) = ( X Y ( I - 1 . N . 2 ) - X Y ( I + 1 , N . 2 ) ) / 2 . D 0 H ( I . N . 1 ) = ( D ( 1 , N . 1 ) * * 2 + D ( I . N . 2 ) * * 2 ) * * 0 . 5 D 0 H ( I . N . 2 ) = ( D ( I . N . 3 ) * * 2 + D ( I . N . 4 ) * * 2 ) * * 0 . 5 D 0 R S F ( I . N ) = H ( I , N . 2 ) / H ( I , N , 1 ) 1 10 CONT INUE C C * * * E V A L U A T I O N OF R A T I O OF S C A L E FACTOR AT BOUNDAR IES WHERE C * * * P O I N T S ARE NOT F I X E D , U S I N G A L INEAR I N T E R P O L A T I O N C * * * OF CORNER P O I N T S . 2 0 7 C - * * TOP WALL . C 120 I F ( B T Y P E ( 1 ) . E Q . 1 ) GOTO 140 DO 130 J=2.NM1 R S F ( 1 . J ) = R S F ( 1 , 1 ) + ( J - 1 . D O ) * ( R S F { 1 . N ) - R S F ( 1 , 1 ) ) / N M 1 130 CONTINUE C C * * * R IGHT WALL. 140 I F ( B T Y P E ( 2 ) . E Q . 1 ) GOTO 160 DO 150 I=2,MM1 R S F ( I . N ) = R S F ( 1 . N ) + ( I - 1 .DO) * ( R S F ( M . N ) - R S F ( 1 .N ) )/MM1 150 CONTINUE C C * * " BOTTOM WALL. C 160 I F ( B T Y P E ( 3 ) . E Q . 1 ) GOTO 180 DO 170 J=2,NM1 R S F ( M . J)>=RSF(M. 1 ) + ( J - 1 . D O ) * ( R S F ( M . N ) - R S F ( M , 1 ) )/NM1 170 CONTINUE C r « « » L E F T WALL. C 180 I F ( B T Y P E ( 4 ) . E O . 1 ) GOTO 2 0 0 DO 190 I=2,MM1 R S F ( I , 1 ) = R S F ( 1 . 1 ) + ( 1 - 1 . D 0 ) * ( R S F ( M . 1 ) - R S F ( 1 . 1 ) ) /MM1 190 CONT INUE C C * » * COMPUTAT ION OF THE R A T I O OF SCALE FACTORS FOR THE c » » . I N T E R I O R PO INTS U S I N G THE R E L A T I O N PROPOSED C * * * BY G . R Y S K I N AND L . G . L E A L . C 2 0 0 DO 2 2 0 I=2.MM1 DO 2 1 0 J=2,NM1 A M 0 = ( I - 1 . D 0 ) / M M 1 A M 1 = ( J - 1 . D 0 ) / N M 1 R S F ( I . J ) = ( t . D O - A M O ) - R S F ( 1 , J )+AMO*RSF (M , J ) 1 + ( 1 . D 0 - A M 1 ) « R S F ( I . 1 ) + A M 1 « R S F ( I , N ) 2 - ( ( 1 . D O - A M O ) * ( 1 . D 0 - A M 1 ) * R S F ( 1 . 1 ) 3 + ( 1 . D 0 - A M 0 ) * A M 1 « R S F ( 1 , N ) 4 + A M 0 * ( T . D 0 - A M 1 ) * R S F ( M , 1 ) + A M O * A M 1 » R S F ( M , N ) ) 2 1 0 CONT INUE 2 2 0 CONT INUE RETURN END C C C SUBROUTINE DSF C c C NAME: D E R I V A T I V E S AND S C A L E FACTORS . C C P U R P O S E : C C T H I S SUBROUTINE COMPUTES THE F I R ST ORDER D E R I V A T I V E S OF THE C C A R T E S I A N COORD INATES WITH R E S P E C T TO THE ORTHOGONAL C COORD INATES AND C A L C U L A T E S THE SCALE FACTORS . C C C H A R A C T E R I S T I C : 2 0 8 c C - F I N I T E D I F F E R E N C E S FORMULA OF SECOND ORDER. C c...........*.....,......»«.,»«.«......«.......«...«.. C I M P L I C I T R E A L * 8 ( A - H , D - 2 ) R E A L ' S D ( 5 0 . 5 0 . 4 ) . H ( 5 0 . 5 0 . 2 ) . X Y ( 5 0 . 5 0 . 2 ) COMMON /ELK 1/ M.N COMMON / B L K 3 / XY COMMON / B L K 4 / H COMMON / B L K 5 / D C C * * » SET UP V A R I A B L E S . C MM1=M-1 MM2=M-2 NM1=N-1 NM2=N-2 C C * « * D E R I V A T I V E S AT CORNER. C c « « » TOP L E F T CORNER. C D ( 1 . 1 . 1 ) = ( - 3 . D 0 * X Y ( 1 . 1 . 1 ) + 4 . D O * X Y ( 1 , 2 . 1 ) - X Y ( 1 . 3 . 1 ) ) / 2 . D O D ( 1 . 1 . 2 ) = ( - 3 . D O * X Y ( 1 , 1 . 2 ) + 4 . D O * X Y ( 1 . 2 . 2 ) - X Y ( 1 . 3 . 2 ) ) / 2 . D 0 D ( 1 , 1 . 3 ) = ( 3 . D 0 * X Y ( 1 . 1 . 1 ) - 4 . D 0 * X Y ( 2 . 1 . 1 ) + X Y ( 3 . 1 , 1 ) ) / 2 . D 0 D ( 1 . 1 . 4 ) = ( 3 . D 0 * X Y ( 1 , 1 , 2 ) - 4 . D 0 * X Y ( 2 . 1 , 2 ) + X Y ( 3 , 1 , 2 ) ) / 2 .DO C C * * * TOP R IGHT CORNER. C D ( 1 , N . 1 ) = ( 3 . D 0 * X Y ( 1 , N . 1 ) - 4 . D O * X Y ( 1 , N M 1 , 1 ) + X Y ( 1 , N M 2 , 1 ) ) / 2 . D 0 DC T . N . 2 ) - = ( 3 . D O * X Y ( 1 . N . 2 ) - 4 . D O * X Y ( 1 , N M 1 , 2 ) + X Y ( 1 , N M 2 . 2 ) ) / 2 . D O D ( 1 , N . 3 ) = ( 3 . D O » X Y ( 1 , N , 1 ) - 4 . D O * X Y ( 2 , N . 1 ) + X Y ( 3 . N . 1 ) ) / 2 . D 0 D ( 1 . N , 4 ) = ( 3 . D 0 * X Y ( 1 . N . 2 ) - 4 . D O * X Y ( 2 . N , 2 ) + X Y ( 3 . N , 2 ) ) / 2 . D O C C * * * BOTTOM L E F T CORNER. C D ( M , 1 , 1 ) = ( - 3 . D 0 * X Y ( M . 1 . 1)+4 . D O * X Y ( M . 2 . 1 ) - X Y ( M . 3 . 1 ) ) / 2 . D O D ( M . 1 , 2 ) < = ( - 3 . D O * X Y ( M , 1 , 2 )+4 . DO*XY (M , 2 . 2 ) - XY ( M , 3 . 2 ) )/2 . DO D ( M . 1 , 3 ) = ( - 3 . D 0 * X Y ( M . 1 . 1 ) + 4 . D 0 * X Y ( M M 1 . 1 , 1 ) - X Y ( M M 2 , 1 , 1 ) ) / 2 . D O D ( M , 1 , 4 ) = ( - 3 . D 0 * X Y ( M , 1 . 2 ) + 4 . D O * X Y ( M M 1, 1 . 2 ) - X Y ( M M 2 , 1 . 2 ) ) / 2 . D O C C * * * BOTTOM R IGHT CORNER. C D ( M . N , 1 ) = ( 3 . D 0 * X Y ( M , N . 1 ) - 4 . D O * X Y ( M , N M 1 . 1 ) + X Y ( M , N M 2 . 1 ) ) / 2 . D 0 D ( M . N , 2 ) = ( 3 . D 0 * X Y ( M , N . 2 ) - 4 . D 0 * X Y ( M , N M 1 , 2 ) + X Y ( M , N M 2 . 2 ) ) / 2 . D 0 D ( M , N , 3 ) = ( - 3 . D O ' X Y ( M . N . 1 ) + 4 . D 0 * X Y ( M M 1 . N . 1 ) - X Y ( M M 2 , N . 1 ) ) / 2 . D O D ( M . N , 4 ) = ( - 3 . D 0 * X Y ( M , N . 2 ) + 4 . D 0 * X Y ( M M 1 . N . 2 ) - X Y ( M M 2 . N . 2 ) ) / 2 . D 0 C C * * * D E R I V A T I V E S AT THE B O U N D A R I E S . C DO 10 J=2.NM1 C C * * * TOP WALL . C D ( 1 . 0 , 1) = ( X Y ( 1 ,d+1 . 1 ) - X Y ( 1 . J - 1 . 1 ) ) / 2 . D 0 D ( 1 , J . 2 ) = ( X Y ( 1 . J + 1 , 2 ) - X Y ( 1 , J - 1 , 2 ) ) / 2 . D O 0 ( 1 . J . 3 ) = ( 3 . D O * X Y ( 1 . J , 1 ) - 4 . D O * X Y ( 2 . J , 1 ) + X Y ( 3 . J . 1 ) )/2 .DO D( 1 , J , 4 ) = ( 3 . D O * X Y ( 1 . J . 2 ) - 4 . D O * X Y ( 2 . J , 2 ) + X Y ( 3 , J , 2 ) ) / 2 . D O C C - * * BOTTOM WALL. C D ( M . d , l ) = ( X Y ( M . d + 1 . l ) - X Y ( M . d - 1 . 1 ) ) / 2 . D O D ( M . J . 2 ) = ( X Y ( M . J + 1 , 2 ) - X Y ( M . J - 1 , 2 ) ) / 2 . D O D ( M . J . 3 ) = ( - 3 . D O * X Y ( M . J . 1 ) + 4 . D O * X Y ( M M 1 . J . 1 ) - X Y ( M M 2 , d . 1 ) ) / 2 . D O D ( M , J . 4 ) = ( - 3 . D 0 * X Y ( M , d . 2 ) + 4 . D 0 * X Y ( M M 1 . d . 2 ) - X Y ( M M 2 . d , 2 ) ) / 2 . D O 10 CONT INUE DO 2 0 I=2.MM1 C C * * * L E F T WALL. C D ( I . 1 . 1) = ( - 3 . D O * X Y ( I . 1 , 1 ) + 4 . D O * X Y ( I . 2 . 1 ) - X Y ( I . 3 . 1 ) ) /2 .DO 0 ( 1 , 1 . 2 ) = ( - 3 . D O * X Y ( I . 1 . 2 ) + 4 . D O * X Y ( I . 2 . 2 ) - X Y ( I . 3 , 2 ) ) / 2 . D O D ( I . 1 . 3 ) = ( X Y ( 1 - 1 . 1 , 1 ) - X Y ( 1 + 1, 1. 1 ) ) / 2 . D 0 D ( I . 1 , 4 ) = ( X Y ( I - 1 . 1 , 2 ) - X Y ( 1 + 1 . 1 , 2 ) ) / 2 . D O C C * * * R IGHT WALL. C D ( I , N , 1 ) = ( 3 . D O * X Y ( I , N , 1 ) - 4 . D 0 * X Y ( I . N M 1 , 1 ) + X Y ( I , N M 2 , 1 ) ) / 2 . D 0 D ( I , N , 2 ) = ( 3 . D O * X Y ( I , N , 2 ) - 4 . D O * X Y ( I . N M 1 , 2 ) + X Y ( I . N M 2 . 2 ) ) / 2 . D O D ( I . N , 3 ) - ( X Y ( I - 1 . N . 1 ) - X Y ( 1 + 1 . N . 1 ) J / 2 . D 0 D ( I . N . 4 ) < = ( X Y ( I - 1 , N , 2 ) - X Y ( I + 1 , N , 2 ) ) / 2 . D 0 2 0 CONT INUE C C * « * D E R I V A T I V E S FOR THE INNER GR ID P O I N T S . C DO 4 0 I=2,MM1 DO 3 0 d=2,NM1 D ( I , d . 1 ) = ( X Y ( I , J + 1 , 1 ) - X Y ( I . J - 1 . 1 ) ) / 2 . D 0 0 ( 1 , J " . 2 ) = ( X Y ( I .J+1 . 2 ) - X Y ( I , J - 1 . 2 ) ) /2 .DO D ( I . J , 3 ) = ( X Y ( I - 1 , J , 1 ) - X Y ( 1 + 1 , J , 1 ) ) / 2 . D O D ( I . J . 4 ) = ( X Y ( I - 1 . J . 2 ) - X Y ( I + 1 . J . 2 ) ) / 2 . D O 3 0 CONT INUE 4 0 CONT INUE C C * * * C A L C U L A T I O N OF THE S C A L E FACTOR. C DO GO I =1 .M DO 5 0 J = 1 . N H ( I . J . 1 ) = ( D ( I , J , 1 ) * * 2 + D ( I . J . 2 ) * * 2 ) * * 0 . 5 D 0 H ( I . J . 2 ) = ( D ( I . J . 3 ) * * 2 + D ( I . d . 4 ) * * 2 ) * * 0 . 5 D 0 5 0 CONT INUE 6 0 CONT INUE RETURN END C C C DOUBLE P R E C I S I O N FUNCT ION F ( Z . I ) C c • C NAME: F U N C T I O N . C C PURPOSE : C C T H I S SUBPROGRAM S P E C I F I E D THE D I R I C H L E T BOUNDARY COND IT IONS C OR RETURN THE NORMAL D E R I V A T I V E AT THE W A L L S . C C INPUT DATA: c c c c c c c c c c c c c - Z : INDEPENDENT V A R I A B L E . - I: FUNCT ION NUMBER (1 TO 8 ) . WHERE 1=1: F U N C T I O N TOP Y = F ( X ) X = F ( Y ) Y = F ( X ) X = F ( Y ) 1=2: F U N C T I O N R IGHT  1=3: F U N C T I O N BOTTOM  1=4: F U N C T I O N L E F T  1=5: D E R I V A T I V E TOP . 1=6: D E R I V A T I V E R I G H T . 1=7: D E R I V A T I V E BOTTOM. 1=8: D E R I V A T I V E L E F T . I M P L I C I T R E A L * 8 ( A - H . 0 - Z ) AMP=0 .075DO C C * * » FUNCT ION TOP WALL . C I F ( I . EO.1) F=1.DO C C * * * FUNCT ION R IGHT WALL . C I F ( I . E 0 . 2 ) F=1.DO C I F ( I . E 0 . 2 ) F =1 . D 0 - A M P + A M P * D C 0 S ( 4 . D O * D A T A N (1 . D O ) * Z ) C I F ( I . E Q . 2 ) F=1 .D0+AMP -AMP *DC0S ( 4 .D0 *DATAN(1 .DO ) *Z ) C I F ( I . E Q . 2 ) F=1 .DO-AMP+AMP*DCOS(8 .DO*DATAN(1 .DO) *Z ) C I F ( I . E Q . 2 ) F =1 . D O + A M P - A M P * D C O S ( 8 . D O * D A T A N (1 . D O ) * Z ) I F ( I . E 0 . 5 ) F=O.DO C C * « * D E R I V A T I V E R IGHT WALL . C I F ( I . E 0 . 6 ) F=O.DO C I F ( I . E 0 . 6 ) F = - 4 . D O * D A T A N (1 . D O ) * A M P * D S I N ( 4 . D O * O A T A N (1 . D O ) * Z ) C I F ( I . E 0 . 6 ) F = 4 . D O * D A T A N (1 . D O ) * A M P * D S I N ( 4 . D O * D A T A N (1 . D O ) * Z ) C I F ( I . E O . G ) F = - 8 . D O * D A T A N (1 . D O ) * A M P * D S I N ( 8 . D O * D A T A N (1 . D O ) * Z ) C I F ( I . E 0 . 6 ) F = 8 . D O * D A T A N (1 . D O ) * A M P * D S I N ( 8 . D O * D A T A N (1 . D O ) * Z ) C C « « * D E R I V A T I V E BOTTOM WALL . C I F ( I . E 0 . 7 ) F=O.DO C C * * * D E R I V A T I V E L E F T WALL . C I F ( I . E O . 8 ) F=O.DO RETURN END 21 1 c SUBROUT INE NU C c.«*.**•*.«*»**......• C C NAME: N U S S E L T . C C P U R P O S E : C C T H I S SUBROUT INE COMPUTES THE D I S T A N C E ALONG THE WALLS , THE C L O C A L NUSSELT NUMBERS AND THE AVERAGE N U S S E L T NUMBERS. C C C H A R A C T E R I S T I C : C C - I N T E G R A T I O N BY T R A P E Z E . C c,**«*»**.**«««*«.*.**'******.***.***«**.******•*.**•****„«**.**••• c I M P L I C I T R E A L * 8 ( A - H , 0 - Z ) R E A L * 8 H ( 5 0 , 5 0 , 2 ) R E A L * 8 D I S T ( 5 0 . 4 ) , N U A V ( 4 ) , N U L ( 5 0 . 4 ) R E A L * 8 T ( 5 0 . 5 0 ) COMMON / B L K 1 / M.N COMMON / B L K 4 / H COMMON / B L K 7 / T COMMON / B L K 1 3 / D l S T , N U A V , N U L C C * * * SET UP V A R I A B L E S C NP1=N+ 1 / D I S T ( 1 , 2 ) = 0 . D 0 D I S T ( 1 , 4 ) = 0 . D 0 N U A V ( 2 ) = 0 . D 0 N U A V ( 4 ) = 0 . D 0 C C * « * R IGHT W A L L . C * * * C A L C U L A T E THE D I S T A N C E (TOP TO BOTTOM) AND THE LOCAL NUSSELT C « * * NUMBER. C DO 10 1=2.M D I S T ( I . 2 ) = D I S T ( 1 - 1 . 2 ) + ( H ( I . N . 2 ) + H ( I - 1 . N . 2 ) ) / 2 . D 0 N U L ( I , 2 ) = 2 . D O * ( T ( I , N P 1 ) - T ( I . N ) ) / ( H ( I - 1 , N . 1 ) + H ( I . N . 1 ) ) N U A V ( 2 ) = N U A V ( 2 ) + N U L ( I . 2 ) * ( D l S T ( I . 2 ) - D I S T ( I - 1 . 2 ) ) 10 CONT INUE N U A V ( 2 ) = N U A V ( 2 ) / D I S T ( M , 2 ) C C * * * L E F T WALL . C * * * C A L C U L A T E THE D I S T A N C E S (TOP TO BOTTOM) AND THE LOCAL NUSSELT C * * * NUMBER. C DO 20 I = 2 .M D I S T ( I . 4 ) = D I S T ( I - 1 .4 ) + ( H ( I . 1 . 2 ) + H ( I - 1 . 1 . 2 ) )/2.DO N U L ( I , 4 ) = 2 . D 0 * ( T ( I . 2 ) - T ( I . 1 ) ) / ( H ( I - 1 . 1 , 1 ) + H ( I , 1 , 1 ) ) N U A V ( 4 ) = N U A V ( 4 ) + N U L ( I . 4 ) - ( D I S T ( I . 4 ) - D I S T ( I - 1 . 4 ) ) 2 0 CONT INUE N U A V ( 4 ) = N U A V ( 4 ) / D I S T ( M . 4 ) C C * * * P R I N T O U T . C WRITE ( 6 . 3 0 ) N U A V ( 2 ) . N U A V ( 4 ) , D I S T ( M . 2 ) . D I S T ( M . 4 ) 212 RETURN 3 0 F 0 R M A T ( / , T 5 , ' A V E R A G E NUSSELT NUMBER ( R IGHT WALL)= ' . 0 1 5 . 7 . / , 1 T 5 . ' A V E R A G E NUSSELT NUMBER ( L E F T WALL )= ' . D 1 5 . 7 . / . 2 T 5 , ' L E N G T H OF THE W A L L ( R I G H T ) = ' . D 1 5 . 7 . / , 3 T 5 , ' L E N G T H OF THE W A L L ( L E F T ) = ' . D 1 5 . 7 ) END C C C SUBROUT INE ORTHO C C « * . . » « * . « . « . * * » » . . » , , . * . « . * « . » . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . . . . . . . . C C NAME: ORTHOGONAL. C C P U R P O S E : C C T H I S PROGRAM C A L C U L A T E S THE I N T E R S E C T I N G ANGLE OF TWO C COORDINATE L I N E S AT A L L PO INTS OF THE G R I D . C C N B . THE FORMULA USED HERE IS TAKEN IN THE CH IKHL IWALA AND YOHSOS C A R T I C L E . C c««««.*».,«..«««..«**«.«*«..««...«......**.««.«.«,..,*.....*.**„**..* C I M P L I C I T R E A L « 8 ( A - H . 0 - Z ) R E A L * 8 D ( 5 0 . 5 0 , 4 ) COMMON / B L K 1 / M.N COMMON / B L K 5 / D C C * * * SET UP D A T A . C D I FA=O.DO SUM=O.ODO MI = 1 M<J=1 C C * * « ORTHOGONAL ITY T E S T . C DO 2 0 I = 1 .M DO 10 d = 1 . N A M O = 0 ( I . d , 1 ) * D ( I , d , 3 ) + D ( I . d . 2 ) * D ( I , d , 4 ) A M 1 = ( ( D ( I , d . 1 ) * * 2 + D ( I . J . 2 ) * * 2 ) * " 0 . 5 D 0 ) * ( ( D ( I . J . 3 ) - * 2 1 + 0 ( 1 . d . 4 ) * * 2 ) - * 0 . 5 D 0 ) ANGLE = OARCOS(AMO/AM 1 ) * 1 8 0 . D O / ( 4 . D O * D A T A N ( 1 .DO) ) SUM=SUM+OABS (90 .ODO-ANGLE ) A M O = 0 M A X 1 ( D I F A , D A B S ( 9 0 . D O - A N G L E ) ) I F ( ( A M O - D I F A ) . L T . 1 . O D - 6 ) GOTO 10 MI = I Md = vJ D I F A = AMO 10 CONT INUE 2 0 CONT INUE C C * * * AVERAGE D E V I A T I O N OF ORTHOGONAL ITY . C SUM=SUM/(M*N) C C * « * P R I N T O U T . • C 213 WRITE ( 6 . 3 0 ) D I F A . M I . M d WRITE ( 6 . 4 0 ) SUM RETURN 3 0 F 0 R M A T ( / , T 5 , ' M A X . D E V I A T I O N Or ORTHOGONAL ITY ' . 1 ' ( A L L P O I N T S . D E G R E E ) = ' . 2 D 1 5 . 7 , / . T 5 . ' P O S I T I O N : 1= ' . 1 2 . ' , d = ' . 1 2 ) 4 0 F O R M A T ( / , T 5 . ' A V E R A G E D E V I A T I O N OF ORTHOGONAL ITY ' . 1 ' ( A L L P O I N T S . DEGREE ) = ' . D 1 5 . 7 ) END C C C SUBROUTINE CNTR ( I M A X , J M A X . N C N T R ) C c....................................... C C NAME: CONTOUR. C C P U R P O S E : C C T H I S SUBROUT INE DRAWS THE CONTOUR L I N E S OF A SCALAR C DEPENDENT V A R I A B L E C WHICH I S KNOWN OVER A NON RECTANGULAR G R I O . C C INPUT DATA : C C - IMAX : MAXIMUM V A L U E OF THE INDEX I I N X Y Z ( I . d . K ) . C - J M A X : MAXIMUM V A L U E OF THE INDEX J I N X Y Z ( I . d . K ) . C - NCNTR: NUMBER OF CONTOUR L I N E S . C C C H A R A C T E R I S T I C : C C - L I N E A R I N T E R P O L A T I O N I S USED . C c REAL X Y Z ( 5 0 . 5 0 , 3 ) , X C N T R ( 1 0 ) , Y C N T R ( 1 0 ) , Z C N T R ( 5 0 ) COMMON / B L K A / X Y Z . Z C N T R C C * * « SET UP V A R I A B L E S . C IMAXM1= IMAX-1 dMAXM1=dMAX-1 C C * * * O V E R A L L LOOP . C 0 0 8 0 K=1 .NCNTR 0 0 7 0 1=1.IMAXM1 DO 6 0 d= 1 ,UMAXM1 ZMAX = A M A X 1 ( X Y Z ( I . d . 3 ) . X Y Z ( 1 + 1 . d . 3 ) . X Y Z ( I . d + 1 .3 ) . 1 X Y Z ( I + 1 , d + 1 . 3 ) ) Z M I N = A M I N 1 ( X Y Z ( I . d . 3 ) . X Y Z ( I + 1 . d . 3 ) , X Y Z ( I . d + 1 . 3 ) . 1 X Y Z C I + 1 . d + 1 . 3 ) ) I F ( ( Z C N T R ( K ) . L T . Z M I N ) . O R . ( Z C N T R ( K ) . G T . Z M A X ) ) GOTO 6 0 L=0 Z M A X = A M A X 1 ( X Y Z ( I . d . 3 ) . X Y Z ( I . d + 1 , 3 ) ) ZMIN = AMIN 1 ( X Y Z ( I . d , 3 ) , X Y Z ( I . d + 1 . 3 ) ) IF ( ( Z C N T R ( K ) . L T . Z M I N ) . O R . ( Z C N T R ( K ) . G T . Z M A X ) ) GOTO 10 L = L+1 X C N T R ( L ) = X L I N T ( I . d . I . d + 1 . K ) 214 Y C N T R ( L ) = Y L I N T ( I . J , I , J+1 . K ) 10 ZMAX = AMAX1 ( X Y Z ( I . d + 1 . 3 ) . X Y Z (1+ 1 . J + 1 . 3 ) ) Z M I N = A M I N 1 ( X Y Z ( I . J + 1 ,3 ) , X Y Z (1+ 1 . J + 1 . 3 ) ) I F ( ( Z C N T R ( K ) . L T . Z M I N ) . O R . ( Z C N T R ( K ) . G T . Z M A X ) ) GOTO 2 0 L = L+ 1 X C N T R ( L ) = X L I N T ( I . J + 1 , 1 + 1 , J + 1 ,K ) Y C N T R ( L ) = Y L I N T ( I ,d+1.1 + 1,J+1 .K ) 2 0 Z M A X = A M A X 1 ( X Y Z ( 1 + 1 , d . 3 ) . X Y Z ( I + 1 , d + 1 . 3 ) ) Z M I N = A M I N 1 ( X Y Z ( 1 + 1 . d . 3 ) , X Y Z ( 1 + 1 . d + 1 . 3 ) ) I F ( ( Z C N T R ( K ) . L T . Z M I N ) . O R . ( Z C N T R ( K ) . G T . Z M A X ) ) GOTO 3 0 L = L+1 X C N T R ( L ) = X L I N T ( I + 1 , d . I + 1 . d + 1 , K ) Y CNT R ( L ) = Y L I N T(1 + 1 . d . I + 1 . J + 1 . K ) 3 0 ZMAX = A M A X 1 ( X Y Z ( I , d , 3 ) . X Y Z ( 1 + 1 , J . 3 ) ) Z M I N = A M I N 1 ( X Y Z ( I , J . 3 ) . X Y Z ( 1 + 1 . d . 3 ) ) I F ( ( Z C N T R ( K ) . L T . Z M I N ) . O R . ( Z C N T R ( K ) . G T . Z M A X ) ) GOTO 4 0 L = L+1 X C N T R ( L ) = X L I N T ( I . J . 1 + 1 . J . K ) Y C N T R ( L ) = Y L I N T ( I . J , 1 + 1 . J . K ) 4 0 I F ( L . E 0 . 2 ) GOTO 5 0 X C N T R ( 1 ) = X Y Z ( I , J , 1 ) Y C N T R ( 1 ) = X Y Z ( I . 0 . 2 ) X C N T R ( 2 ) = X Y Z ( I . J + 1 , 1 ) Y C N T R ( 2 ) = X Y Z ( I . J + 1 , 2 ) X C N T R ( 3 ) = X Y Z (1 + 1,J+1 . 1 ) Y C N T R ( 3 ) = X Y Z ( I + 1 . J + 1 . 2 ) X C N T R ( 4 ) = X Y Z ( I + 1 . J . 1 ) Y C N T R ( 4 ) = X Y Z ( I + 1 . J . 2 ) X C N T R ( 5 ) = X Y Z ( I . J . 1 ) Y C N T R ( 5 ) = X Y Z ( I . J . 2 ) X C N T R ( 6 ) = X Y Z ( I + 1 , d + 1 . 1 ) Y C N T R ( 6 ) = X Y Z ( 1 + 1 , 0 + 1 . 2 ) C A L L CURVE ( X C N T R , Y C N T R , 5 . 0 ) X C N T R ( 1 ) = X Y Z ( I + 1 . J . 1 ) Y C N T R ( 1 ) = X Y Z ( I + 1 . J . 2 ) X C N T R ( 2 ) = X Y Z ( I . J + 1 . 1 ) Y C N T R ( 2 ) = X Y Z ( I . J + 1 . 2 ) C A L L CURVE ( X C N T R . Y C N T R , 2 . O ) GOTO 6 0 5 0 C A L L CURVE ( X C N T R , Y C N T R . 2 , 0 ) 6 0 CONT INUE 7 0 CONT INUE 8 0 CONT INUE RETURN END C C C R E A L FUNCT ION X L I N T ( 1 1 . J 1 . I 2 . J 2 . K ) R E A L X Y Z ( 5 0 . 5 0 , 3 ) . Z C N T R ( S O ) COMMON / B L K A / X Y Z . Z C N T R AMO = Z C N T R ( K ) - X Y Z ( I 1 , J 1 .3 ) A M 1 = X Y Z ( 1 2 , J 2 . 3 ) - X Y Z ( I 1 . J 1 . 3 ) AM2 = X Y Z ( 1 2 . J 2 . 1 ) - X Y Z ( I 1 . J 1 . 1 ) X L I N T = A M 0 * A M 2 / A M 1 + X Y Z ( 1 1 , 0 1 , 1 ) RETURN END C C C 215 R E A L FUNCT ION Y L I N T ( I 1 , J 1 . 1 2 . J 2 . K ) REAL X Y Z ( 5 0 . 5 0 . 3 ) . Z C N T R ( 5 0 ) COMMON / B L K A / X Y Z . Z C N T R A M O = Z C N T R ( K ) - X Y Z ( I 1 . J t . 3 ) A M 1 = X Y Z ( 1 2 , J 2 , 3 ) - > Y Z ( I 1 . J 1 . 3 ) A M 2 = X Y Z ( I 2 . J 2 . 2 ) - X Y Z ( I 1 . J 1 . 2 ) Y L I N T = AM0*AM2/AM1 + XYZ ( I 1 . J t . 2 ) RETURN END APPENDIX B The numerical results of Part II are presented in th i s appendix. Plots of the gr i d ; the temperature, stream function and v o r t i c i t y d i s t r i b u t i o n s ; and the l o c a l Nusselt number d i s t r i b u t i o n s along the l e f t and right wall are presented here for each cavity type, dimensionless amplitude and Rayleigh number. The results of cavity C1 are presented f i r s t . For each cavity, the res u l t s are presented in order of increasing amplitude, and for each amplitude, they are given in order of increasing Rayleigh number. Each of the following pages contains the information l i s t e d below: 1. The values of the Prandtl number, Rayleigh number and dimensionless amplitude are written at the top of the l e f t column. D i r e c t l y below, in the same column, the average Nusselt numbers of the isothermal walls, the v o r t i c i t y countour l i n e values and the stream function countour. l i n e values are l i s t e d . The values of the isotherms are not included because they always range from 0 (cavity, l e f t wall) to 1 (cavity right wall) in increments of 0.1. 2. Plots of l o c a l Nusselt number along the right and l e f t walls versus the distances calculated from the top of the cavity are presented at the middle and the bottom of the l e f t column, 216 r e s p e c t i v e l y . P l o t s of the g r i d and t h e t e m p e r a t u r e , stream f u n c t i o n and v o r t i c i t y c o n t o u r s a r e p r e s e n t e d i n o r d e r from the t o p t o the bottom of the r i g h t column. The minimum c o n t o u r l i n e v a l u e f o r the steam f u n c t i o n i s always the one c o r r e s p o n d i n g t o the w a l l . The maximum p o s i t i v e c o n t o u r l i n e v a l u e f o r the v o r t i c i t y i s always the one c l o s e s t t o the c e n t e r . 0.0 DIMENSIONLESS AMPLITDOf. -o ISO • tre WALL(LEFI) . 0 I0O00OOE-.0I § 5 1 : 1 1 1 • , 1 ' • • 0.2 0.1 0.6 0.0 1.0 ' 1 2 01 STANCE (TOP TO BOTTOM! 0 0 i . i 0.6 i V OISTOHCC (TOP TO BOTTOM) PH - 1.0 R A - 1000.0 DIMENSIONLESS AMPLITUDE- -0.150 AVERAGE NUSSELT NUMBER (RIGHT WALL)" AVERAGE NUSSELT NUMBER (LEFT WALL)• >.t331642E+01 .I297287E+01 LENGTH OF THE WALL!RIGHl )• LENGTH OF THE WALL(LEFT)• 3.1053372E+OI . lOOOOOOf»0l VORTICITY COUNT OUR VALUES. MIN- -0 53396516*02 MAX- 0.3O675B0E*02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.4498927E+02 -0.365B206E*02 -O.2817484E*02 -0.197G76tE*02 -O.113G038E+02 -0.2953U0E*0t 0.54540B6E«0( 0. 1366I3IE«02 0.2226854E*02 STREAM FUNCTION CONTOUR VALUES MIN- 0.0 MAX- 0.8O€6495E«O0 CONTOUR * 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.9066492E-0I O. 181329BE*00 O.271994flE*O0 0.3G26597E+00 0.4533247E+00 0.S439897E-.OO 0.6346546E+OO 0.72S3196E*00 0.8159845E«OO U) i n . B.4 0.6 0.8 1.0 DISTANCE (TOP TO BOTTOM) in ° '* .__ HVOfflGr. 0.0 0.2 0.1 0.6 0.6 1.0 1.2 DISTANCE (TOP TO BOTTOM! 2 2 0 PB- 1.0 RA- 3OOO.0 DIMENSIONLESS AMPLITUDE- -0.150 AVERAGE. NUSSELT NUMBER (RIGHT WALL)- 0. I523B46E*01 AVERAQE NUSSELT NUMBER (LEFT WALL)- O.1605336E*01 LENGTH OF THE WALL(RIGHT)• 0.IO53372E+01 LENGTH OF THE WALL(LEFT (• O. lOOOOOOC+01 VORTICITY COUNTOUR VALUES. MIN- -0. 1504 t*ei*03 MAX- 0.7aa2437E*02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -O. 12B15O9E*03 -O.105B870E*03 -0.6362305E<02 -0.6t35913E*02 -0.390952t£*02 -O.16B3130E*02 0.5432617E+OI 0.2769653£*02 0.4996042E+02 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX* 0.2244O84E+01 CONTOUR 0 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 2244084E 44BS 168E 6732253E-B97G337E-1122D42E-13464S0E 1570659E 1795266E-2019674E-Q.4 0.6 O.S 1-0 OISTRNCE (TOP TO BOTTOMt PR- 1.0 RA- 1OOO0.0 0IMENSIONLESS AMPLITUOE• -0.150 AVERAGE NUSSELT NUM8ER (RIGHT WALL )* 0.2236863E+0 I AVERAGE NUSSELT NUMBER (LEFT WALL I - 0.3355936E+01 LENGTH OF THE WALL(RICHT)• 0 1053372E+01 LENGTH OF THE WALL(LEFT)- O.1000000E+01 VORTICITY COUNTOUR VALUES. MIN- -0.4391619E*03 MAX- O.1388923E*03 CONTOUR # 1 -0.3fll35G4E.03 CONTOUR * 3 -O.33355l0E*O3 CONTOUR # 3 -0.2fi57456E»03 CONTOUR t 4 -0.20794031*03 CONTOUR t 5 -0. 1S013S0E*O3 CONTOUR # 6 -O.92329S9E*03 CONTOUR # 7 -O.3452417E*02 CONTOUR * 8 0.2328101E*02 CONTOUR # fi O.81066436*03 STREAM FUNCTION CONTOUR VALUES• MIN- O.O MAX- 0.4654137E*01 CONTOUR # t 0.4654136€*00 CONTOUR # 2 0.93O8273E+OO CONTOUR * 3 0. 1386340E«01 CONTOUR * 4 0. 1BG 165-lE+01 CONTOUR 0 6 0.23270G7E*01 CONTOUR s 6 0.379348\£»O1 CONTOUR # 7 0.3257895E^01 CONTOUR 0 6 0.3723309E+01 CONTOUR tf 9 0.41BB722E«01 0.0 0.2 0.1 O.G 0.8 I. OISTRNCC (TOP TO BOTTOM! •*«.« • a 0.4 0.6 0.0 1.0 OISTRMCE (TOP TO BOTTOM! PH - 1.0 OA- 30000.0 DIMENSIONLESS AMPLITUDE- -O.150 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.3t35B03E«0I AVERAGE NUSSELT NUMBER (LEFT WALL)- 0.3303327E*01 LENGTH OF THE WALL(RIGHT\• O 1053372E*01 LENGTH OF THE WALL(LEFT)• 0 100COOOE*OI VORTICITY COUNTOUR VALUES. MIN- -O.1060230C«04 MAX- 0.2697Q33E«03 CONTOUR # 1 -0.9272361E*03 CONTOUR 0 2 -0.7B42429E*03 CONTOUR 0 3 -0.6GI2493E*03 CONTOUR * 4 -0.52825611*03 CONTOUR 0 5 -0.39526276*03 CONTOUR 0 6 -0.2622688E*O3 CONTOUR 0 7 -O. 1292756E*03 CONTOUR * B 0.37182€2E*01 CONTOUR 0 9 O 1367 117E*03 STREAM FUNCTION CONTOUR VALUES. MIN- O.O MAX- 0.7396093E-01 CONTOUR 0 1 CONTOUR 0 2 CONTOUR 0 3 CONTOUR 0 4 CONTOUR * 5 CONTOUR * e CONTOUR 0 7 CONTOUR 0 6 CONTOUR 0 9 0.7396093E*00 O. 147g218E*Ot 0.2218B27E+O1 0.79S843GE+01 0-369604SE-*01 0.4437655E+OI 0.5t77264E*OJ 0.S9lG874E«01 0-6656483E+01 M « 1.0 BA> 100000.0 OIMINSIONIISI AMPIMUOI* - O t * 0 t V i a i a i N U l l l l I W M I I I IRICM1 W« l l > - O. 4417 l « 3 l « 0 1 IVIBid! NUIttLI MJMfllO U I M W i l l ) ' O . 4 7 K l t O I » 0 I IIHOIM 01 1MI V tL l l R IOH I I - O IOS34IOKOI MUCIN 01 1HI utk.Kt . i r i !• O. I000OOOK0I VOR1ICI1V COUNTOU KIN- -0.9(101071 I U X - 0.(ft4IS4f,l CONTOUR > CONIOUD CONIOUD CONIOUD CONIOUD CONIOUD CONIOUD CONioun CONIOUD V A l U t t . •04 • 03 •0.33473 l l l « 0 4 •0.30I373BK04 •0. KB0140K04 •O. I34C7B0M04 -O.IOI31CIM04 •O.C7S7733t>03 •0.34.3S33K03 • 0 . I 3 7 B 3 7 0 I « 0 3 0.3206BS3I>03 S1BI4M FUNCTION CONIOUD VALUES MIN' 0 .0 U l ' O. 100*743! .03 CONIOUD « I 3 3 CONIOUD CONTOUD CONIOUD CONIOUD CONIOUD CONIOUD CONIOUD 0.IOOC741I <0t 0.30I34D4I.OI O 30303J. I '01 0 < 0 ] E 1 U ! « 0 I 0 60337101-01 O.B0404S3t'OI 0.1047IB4f<0I O.10S3B3CE.OI 0.1 0.0 0.B 1.0 OlSlfMX HOP TO BOTTOM I 0.0 0.3 0.4 0.B 0.B 1.0 1.1 OlSTflMX (TOP TO BOTTOM! 224 ••• t.o 0.0 OIHINSlONLItS »»LIIUOI- -0 078 zzx = 0°,x:™'0r •mm. *•' ol« it 77 OISTBNCC (TOP TO BOTTOM! in CM CM m- i.o •A" 3O0O.0 OlMINSlONLItS AMPLITUDE' -O.OTB AVIBAOI NUS1ELT MJMBEB IBIOHI HALL )• O.IB37IBtf«0I AVERAGE NUSSELT HUMSID (LIFT HALL !• O.IB4B1411*01 LINOTH OF 7HI HALL(BICM7 )• 0. 101)7371-01 LINOTH OF THE HALLlLin)* 0 1000000*«01 VOBTICITT COUNTDUB VALUES• HIM- -0.I48CBB3E>03 MAI. 0 70807331-03 CONTOUD CONIOUD CONTOUD CONIOUD CONIOUD CONTOUD CONTOUR CONTOUD CONTOUD •0.l37StB7C*03 •0. 10SHB3I-03 -0.I3.3B6II-03 -O.CISIDOM'03 -0.38»B04BE«03 •0.17570041-03 0.4441.341-01 0.3t4(130E'03 0.4B4S77CE.03 STREAK FUNCTION CONIOUD VALUES. HIN- 0.0 MAI- 0.3B3SBBCE.OI CONTOUR CONTOUD CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR O.3BS5BBSE 0.B091B71E 0.7S77SS7E 0. I0I03B4E O.I3B3BB3E O.ISISSBIE O.I7CBIB8E 0.30307BBE 0.33733171 00 •00 00 01 01 01 .01 01 •01 0.0 O.J 0.1 0.1 0.B I.B OtSTBNCC HOP TO BOTTOM I Hvnwcr B.B 0.3 0.1 0.S 0.B I.B DISTANCE I TOP TO BOTTOM I 1.0 • A. IOOOO.O DIMIM1IONLISS AMPLITUDE' -0.0'B AVERAGE NUSSELT M M I I (RIGHT W i l l ) . 0.SIT 13431*01 4VIB40E NUSSILT NUMBER ( l i f t HALL!' 0.330113BI•Ol LIMOTH OF TMI »ALL(0IOHT|. O.10117371<0I LENGTH OF THE HALL(LIFT)• 0.10000001*01 VOBTICITY COUNTOUR VALUES. MIN- -0.433t7S3E.03 MAX' 0 . I 3 t l ( ( 0 l * 0 3 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.17(47091«03 •0.330OC((E«03 •0.3(3((3BE»03 -0.3073SB7E<03 -0.IBOBS47i«03 -o.B445o«ai«oa •0.3B04C3BE«03 0. IB3S743E<03 0.747(1731»03 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX • 0.48SS733E»OI CONTOUR ' I CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.4SSB7S3[*O0 O.BBIISOSE'OO 0.14870351.01 0.1BB3700E»OI 0.347S374E.OI 0.3B740B1E«0I 0.34BB737E«OI 0.38(64031*01 O.44(107(1*0I sl. 0.1 0.» 0.0 1.0 0I5TRNCX I TOP TO BOTTOMI 0.< 0.0 0.0 1.0 DISTANCE ITOP TO BOTTOfll NU5SQ.T NjnBCR i.o i.t 1.0 «.o s.o t.o ».o o.o t.o IO.O NUSSO.T NuretR 0 0 1.0 1.0 J.O 4.0 J.O (.0 1.0 1.0 t.o 10.0 I -1 ' 1 1 1 *• I——' n n n o o n o o n x KM n I 3P • • t • I * p a / 2-• UI • ' 1 ? h. 1_ O M % 3 g g g O g O g g O J ; j j g g j j g g g g g " O O O O O O O O O #i ro _ » 4 D U » • « M I > O l i U * 0 U U O J U - — - O U • UU»I**I>IB-o o o o o o.o o o n n n n n o o o K K < oopooppo>«o 55555555:T5 O 0 D H I I 9 S O «•» O O -« u •» • on Ssi • 4 tl tit**** » »oooooooo r I . U D I I U I I I - ft* • • J U D Q o l U l l ) Q u u » a « - 4 « S £ S S 5 " " -41 <4 » t> Z U — •*)•*!«•) *ji **• O I I l * H ID m m m m «• r- r-C C -* 1* mil o • f- • oo» •* " S * " § b • - o o • - o - O v ) — M M M 03 »«• 1.0 «»• 100000.0 0IMENSI0NLESS AMPLITUDE' -0.079 AVERAGE NUSSELT NUMCCP mam *ALL>. o 4(77J»OI-OI AVERAGE NUSSELT NUMBER (LEFT . . I t ) . o ."oi^T ' o i IENOTH Or THI HALL(IISHT)' 0. IOI3737E.0I LINC7H Of THE MALKLCFTI* 0. lOOOOOOE'OI V0RTICIT7 COUNTOUR VALUES •IN- -0.3(7B3B4I-04 MAX- O.C93((4(I<03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.33493B0I.O4 -0.30I3I7BI«04 -0.M7BO70E *04 -0. 049B(9(«04 •0. 10134(01 «04 •0.(7B799IE-03 -0.34.(9041<03 -0 139494l[*03 0.3IB99BBE-03 STREAM FUNCTION CONTOUR MIN* 0.0 "AX- 0.1006550!-03 CONTOUR VALUES. CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 10065501-01 2013IOOI-OI 3018(901-01 40363001-01 903375II-01 (0393001*01 70458511.01 10534031-01 BOS1SS3E*01 OISTBNCC ITOP TO BOTTOM I 1.0 * » • 0 . 0 DIMNSIONLIt t »M-LITUOI- 0 . 0 1VIIA4U MUtt lLT NLMafU (IIOMT W L U - O . I 0 0 0 7 U I . O I • VIIIBI M J l l I l l M M I I U l ' l WALL)- 0 . • • • • M i l •CO LIMOTH Of IMC ¥ « L L < « i e H T | . o. IOOOOOOt'01 LINOTH or I MI W»LL(Liri|. o. 10000001*01 ; l . . p i r o m .< 0.1 0.0 1-0 DISTANCE ITOP TO BOTTOM! rosace •.< «.• 0.0 1.0 01STRNCC ITOP TO BOTTOMI 231 BB* 1.0 •A* IOOO " OtMINSIONLISS 1MPLITU0I* 0.0 AVIRAOI NulStLT NUMBIB (BIGHT M I L ) ' 0.11 179141*01 AVIBAGI NUtltLI NUMBIB ( L I U MALL I* 0.11171111*01 LINCTH Or 7HI HALLIBIGHTI* O. lOOOOOOf *0I LINOTH Or THl MALLIUMl* O.10000001*01 VORTICITY COUNTOUR VALUIJ. MIN* -OB I 114071-03 MAX • 0.31831781*03 CONTOUD CONTOUD CONTOUD CONTOUD CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -O.43808381*0T -0.34804711*03 •0.3S3O0O3E«03 -0.I788S3«I*03 -O.BSBOt.lI-OI -O.I3SSBB0I*0I 0.70117071*01 0.18333381*03 0.33C3B0B!*03 STSfAM rUNCTION CONTOUR VALUCS. MIN- 0.0 MAX* 0.1173 CONTOUR 4* CONTOUR » CONTOUR ' CONTOUR 4 CONTOUR < CONTOUR 4 CONTOUR f CONTOUR > CONTOUR I 0.11731481*00 0.33443881*00 0.38184481*00 0.4«BBBB8I*00 O.B1B0741I*00 0.70338871*00 0.13000471*00 0.81771871*00 0.10848381*01 sl. 0.1 0.1 0.8 1.0 OISTRNCE ITOP TO BOTIOni 0.0 0.1 0.1 0.1 0.8 1.0 DISTANCE ITOP TO BOTTOM! 1.0 BA« 3OOO.0 0IMENSI0NLISS AMPLITUDE- O.O AVCBAOE NUSSELT MUMS ID (RIGHT HALL)' O. I0O3BO7S *0I AVERAGE N U I l l l l NUMBER (LEFT WALK- 0. IB01TTBE»OI LIHCTH OF 1HE VALltBICHT>• 0.10000001*01 LINOTH OF THE VALLIlEFT). O. I0OO0OOE *01 VOBT1CITV COUNTOUR MIN* -0.I47331BE MAX• O.BTTBIBtE CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR VALUES. 03 03 -O. IIU I B B I •0.10431ITE -0.S3B0374E -0.CI38SB4 -0.3BTBTB3 •0.IB3TSBBE 0.3337B3lf 0.34T3SS3E 0.4S34314E 03 03 •03 •03 •03 •03 •01 03 03 STREAM FUNCTION CONTOUR VALUES. MIN" 0.0 MA*. 0.3T0B CONTOUR • CONTOUR » CONTOUR * CONTOUR * CONTOUR I CONTOUR » CONTOUR » CONTOUR > CONTOUR » 37097BBE*00 B4I8SBBE.00 BI3B3B3I*00 IOa3BIBE*OI I3S4BBBE*0I IS39B71i*01 IBBSBSB(*OI 0.3ICTB3TE*0I 0.343B8I(E«0I V. i f ' pvri>flr.r 0.2 0.1 0.B O.B 1.0 DISTANCE ITOP TO BOTTOM I 0.0 0.2 0.1 0.4 0.0 1.0 i.a DISTANCE ITOP TO BOTTOMI •>*• 1.0 •A- toooo.o DIMENSIONLESS AMPLITUDE* 0.0 AVERAGE NUSSELT NUMBIB (RIGHT HAIL)- 0.33BBBBBI«0t AVIBAOI NUStILT NUMBER (LIFT WALL). 0.33BBB7TE*0I LINOTH OF TH1 HALLIIIOMT)* 0. 10000001*01 LINOTH OF THE WALLlLIFTI* 0.10000001*01 VORTICITY COUNTOUR VALUfi. MIN* -0.43S71BB(*O3 M A X - O.I37B07TI*03 CONTOUR / CONTOUR • CONTOUR I CONTOUR I CONTOUR » CONTOUR » CONTOUR » CONTOUR ' CONTOUR I -O.ST 191411*03 •0.3IBS03OE*O3 -0.3B034IBE*03 •0.104BB04I«03 •0.I4B4IS3I*03 -O.B38STTSi*01 -0.3S4BCBBI*01 0.1SBB4(0E*03 O.T243S7BI«03 STREAM FUNCTION CONTOUR VALUES. MIN* 0.0 MAI* 0.BI115I7E-0I CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR O.SIt3SI7f*O0 0.10339O3C*OI 0.1S337B4E*OI 0.3O49O0CE*O1 0.3BBtlB7E*OI 0.30C7B0BE*OI e.>B7B7tll*OI 0.40BOOI3E*OI 0.4S0I3SB[*0I s l . OM 0.S 0.B 1.0 OISTANCC HOP TO BOTTOtll B.6 0.1 0.1 0.0 0.0 1.0 OISTPMX ITOP TO BOTTOMI » R - i . o »»- aoooo.o 0IMIMS1ONLISS AMPLI7U0I- 0.0 AVtRAGC MUSSI11 M M U (SIGHT VALLI* 0.11711311*01 AVIRAGE NUSSELT HUMBta (LIFT MALLI* O.>17Q3MI-01 LENGTH Of THI VALLIRIGHT I* 0. 10000001*01 LINOTH Or THI VALLlLtrTI* 0.10000001-01 VORTICITY COUNTOUR VALUES. MIN- -O.10433191-0* MAX - 0.31971711*03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CDNTOUR CONTOUR CONTOUR •0.91308191*03 •0.71118331*03 -0.1SI(SOCI*01 -0.13144101.03 •0.39134111*03 •0.31104301-03 -0.I3O8.403I-03 •0.13749131*00 0.13951471*03 STREAM fUNCTION CONTOUR VALUES. MIN- 0.0 MAX- O.7338O73E-0I CONTOUR I 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.73390731*00 0.I4C7OI4E-0I 0.330083IE*0I 0.39340391*01 0.31S7S38E»OI 0.440I043E*0I 0.91348801*01 0.81110S1E*0I 0.11018191*01 1.0 MA' 100000.0 OIMENSIOHLISI AMPLIIUOE- 0.0 AVERAGt NUiSILT NUMBER (RIGHT WALL)- O. 4BB3B M l *01 AVIRAGI NUSSELT NUMBIR I LIFT VALL>• O.4SB34BOC*01 LINCTH OF TMl HALL IR IGHT >• 0. 10000001 *01 LINOTH OF THt VALL(LIFT)* 0.10000001*01 VORTICITY COUNTOUR VALUES. MIN' -0.3SB4]43E*04 MAX- O f 3 I I B 3 I E * 0 3 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR •O.33S37O0E' •0.3031IBBE' •0.IBBB6131 -0.I3SB0BBI' -0.I03B93BE' -0.7048BI4f • •0.3T343B2I' •0.4I8B493E' 0.2BBS4B4!• STREAM FUNCTION CONTOUR VALUIS. MIN' 0.0 MAX- 0.10H 16 11-02 CONTOUR I 1 0. 1014 IS 11*01 CONTOUR » 3 0.303B30IE«OI CONTOUR # 3 0.3O434S3E*OI CONTOUR » 4 0.40BCC03E*OI CONTOUR t t O.B070739E*01 CONTOUR > S 0.S0B4B0«E*01 CONTOUR # 7 0.7OBBO9SE*O1 CONTOUR I B O.BI1330BE*OI CONTOUR I • O.BI3739SE*01 0.0 0.2 0.1 0.S 0.0 1.0 1.2 OISTRNCC ITOP TO BOTTOMI \ flVCRfrCC , 0.0 0.3 0.1 0.B 0.0 1.0 i.a DISTHNCC ITOP TO BOTTOMI BB- 1.0 »»• O.O OININSlOHLftl AMPLMUOE- 0.011 AVERAGE W I I U 1 NUMBER (BIOHT WALL I • 0.tlBSBSBI«0O AVIBAOE NUSSELT NUMBER ( l i f t MALL I" O.13SBBB4KO0 LENOTM OF THE MALL I BIGHT )• O. I0I17ME401 LINOTH OF THE MALLILEFT I" O. IO0OOOOE-01 5 5 0.0 0.1 I , 0.1 0.1 I-OISTBNCE ITOP TO BOTTOM) r. 1- •» sL 0.1 0.0 0.0 OlSTrlNCC ITOP TO BOTTOMI »«- 1.0 • A- 1000.0 OlMiNSIONLISS AMPLITUOI' 0.07B AVERAGE NUSSELT NUMBER (RIOHT VILLI* 0.10918641*01 1VIIUI NUSSILt MUM*IB (LIFT MIL)* O. IO«SilTI*OI IMCTH OF THE «ALLIBIGHI>' 0 10137321*01 LENGTH OF THE VALLILEFM' 0. IO0OOO0E*OI VOR.T IC I TV COUNTOUR MIN' -O.B288403E* M4>> 0.3M8074E* CONTOUR ' CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR VALUES. 03 03 -0.4413899E*02 -0.39639O8E*O3 -O.3T3aO(ll*03 •o.iaaaaiai*03 -0.10331(71*03 •0.30(7I73E*OI 0.(3471801*01 0.14781771*03 0.233IG39O03 STREAM FUNCTION MIN' 0.0 MAI- O.133(48: CONTOUR < I CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR , CONTOUR . CONTOUR ; CONTOUR VALUES. { • 0 1 0.12344031.00 O.3473BB(E*O0 0.37084791*00 0.484B873E*0O 0.BIS34((E*0O O.T4IBS38E*O0 0.8(934931*00 0.88818491*00 0. llt3S44E«0l at o.i O.B i.o OlSTflHCC ITOP TO BOTTOMI flVTRflflC 0.0 0.1 o.« as aa i.o i.l OISTRNCC ITOP TO BOTTOMI PH- I.O I f 3000.0 DIM N i l ONI. I IS AMPLITUDE* 0 O H AVIRAGI NUSSELT NUMBER (RIGHT WALL)* 0.I44T7I1E*0I AVER10I NUSSELT NUMSEB (LIFT WALL)• O.I4(7I31E*0I LINOTH OF THE WALL(RIOHT). 0.10137331*01 LINOTH OF THI WALL(LIFT I* 0.10000001*01 VORTICITY COUNTOUR MIN* -O.IB34130E MAX* 0.1BS10731 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR VALUIS. 03 03 -0.I31S«4M»03 -0.IOBM.31-03 -0.•77.7171-03 -O.C»BSB9lt«03 -0 43871131-03 -0.23073771*03 -0.I7440BOE*00 0.3I733BBE*03 0.43C3334E-O3 STBEAM FUNCTION CONTOUR VALUES MIN- 0.0 MAX* 0.3BI10SBE-01 CONTOUR 4 t CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 3BI101SE*00 BC33IIBE*O0 B433I77(*00 I 1344331-01 I403S3BE*0I IBB1134E*0I 0.<B1773BE*0I 0.334BB4CC*0I 0.3S3BBS3I*0I sl. flvrRRGT o.i 0.1 0.0 I. OlSTflNCC ITOP TO BOTTOM I Wo 5' nvofflGc s.4 o . i a l i.o OlSTPiCC ITOP TO BOTTOM I »»• 1.0 RA> IOOOO.O OIMINSIONLISS AMPLITUDE' o m i AVERAGE NUSSELT NUMBER (•ICH1 VILLI' 0 . J IB0T04I '01 AVERAGE NUSSELT NUMBER (LEFT VILLI' 0.23110141*01 LEM01M Of 1MI VALLIII0H1)' O.10137321*01 LINOTH Or THI VALLILIFT)* 0.10000001*01 VORTICITY COUNTOUR VALUES. HIM* -0.44I]B9BI«03 MAX- o o u i i m o i CONTOUR # I CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.3*44i4«l*03 0.337(7311 *03 0.370X331*03 0.31409131*03 0.19734001*01 0.I0043S0E*03 0 41(I7(BE»03 0.13IR33(t*03 0. 700043(1 .03 STREAM FUNCTION CONTOUR VALUES MIN- 0.0 MAX. 0.91700721*01 CONTOUR « I CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR O.9I7O071E O I0340I4E 0. I93I03IE 0.30(80371 0.39aS039E 0.110304 IE o.3(iao4aE 0.4I3C09SE 0 4(5304*1 00 01 01 01 • 01 • 01 01 •01 * o i 1= 0 .0 0 .2 o.i at o.e i.o i.i DISTANCE ITOP TO BOTTOMI AVTRBT,r 01 STANCE ITOP TO BOTTOMI ••• 1.0* OA* 30000.0 DINEMSIONLISS ANfllTUOE* 0.07S AVEBAOE HUltILT NUMBER (BIGHT MALL)* 0.31134311*01 AVEBAOE NUSSELT NLWai* (LEFT Wall)* 0.3IBBB44E.01 LENGTH OF THE VALLdlGHT). 0. 10I37S3E«0I LENOTM OF THI VALKLEFTI* 0.IO0OOO0E«0I VORTICITT COUNT Out) VALUES. MIN* -O.I0C78BBE*04 »»«• 0.1C34I33E»03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -O.B3441B7E*03 -O.BO»3aitE*03 •0.1SB3737E*O3 •O.B3tl<a>l*03 -0.4020S78E*03 -0.3SSB4B9E*03 •0.I3SB4 131*03 -0 37331141.01 0.I30374SC*03 ITOttK FUNCTION CONTOUR VALUES N1H* 0.0 MAI* 0.733777(1*01 CONTOUR 4 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 733777BI. I4.55SSI 3IBB333E' 3B3II0BE' 3M31B7E' 439BC64E' B13B441E< 0.SBC233IE' 0.SSB4SB7E' 1 5' 0.1 0.B 0.B 1.0 OlSTflNCC ITOP TO BOTTOMI Bvrnwnr 0.. 0.4 0.B 1.0 I.J OlSTflNCC ITOP TO BOTTOMI PR- t.O 0INEHS10NLESS AMPLITUOE- 0.150 AVERAGE NUSSELT NUMBER (RIGHT VALLI- O.BBItlBK'OO AVERAGE NUSSELT NUMBER I LEFT W i l l - 0.8B47 17BE «O0 LENGTH OF THE HALL(RIGHT)• 0. IOS347BE401 LENGTH OF THE WALLUEFTI- O IOOOOCOe»OI 0.1 0.8 0-8 * - ° DISTANCE ITOP TO BOTTOM! 0.4 0.6 O.B 1.0 OtSTflNCC (TOP TO BOTTOM) PR- 1.0 R«- IOO0.O 01MENSI0NLESS AMPLITUDE' O 150 AVERAGE NUSSELT NUMBER (RIGHT WALL I• O.87»107 IE'CO AVERAGE NUSSELT NUMBER (LEFT WALL)' 0.1031256E«01 LENGTH OF THE WALL(RIGHT)- O.IOS3476E«0I LENGTH OF THE WALL(LEFT)* 0.1OOOOOOE•O1 VORTICITY COUNTOUR VALUES. MIN* -O.04flS4S1E«O2 MAX' 0.3t14B12E*O2 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -O.4626I33£«02 -0.37S778BE*02 -0.2807.47SE«02 -0.3047148E«02 -O.1I06O21MO2 -0.32S4954E>01 O.S33B3IBE'OI 0.13841SBE»02 0.22544B3E-02 STREAM FUNCTION CONTOUR VALUES. MIN' 0.0 MAX- O.1374205E-01 CONTOUR I 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.I274205E O.25484I0E 0.3B226I5E 0.50B6B2IE 0.637I026E 0.764523 IE 0.6B19436E 0.I01S363E 0. 11467B4E •OO CO 00 •00 • 00 01 I.. 0.4 0.6 0.0 1.0 OlSTflNCC (TOP TO BOTTOM) HVCRflGC 0.0 0.2 0.1 0.6 0.0 1.0 1-2 OlSTflNCC ITOP TO BOTTOM I PR- 1.0 RA- 3000.0 0IMENSI0NLESS AMPLITUDE- O.ISO AVERAGE NUSSELT NUMBER (RIGHT WALL )' 0.138608SE'OI AVERAGE NUSSELT NUMBER (LEFT HALL)' O.14390I9E»0I LENGTH Or THE HALL(RIGHT)' 0 I033476E«0I LENGTH OF THE WALL(LEFT)• O.10OO0OOE»01 VORTICITY COUNTOUR VALUES. MIN- -0.I59S40?E«03 MAX' 0.8393082E*02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.I371837C*03 -O. 114846GE*03 -0.834BS40£'03 -O.TOI5224E*02 -0.4780507E+02 -0.3S45788E<02 -0.31107 IBE>0l O.1923G4SE*02 0.4 15B362E*02 STREAM FUNCTION CONTOUR VALUES. MIN' 0.0 MAX- 0.2B69203E*0t CONTOUR 4 t CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.3B69202E«00 0.S738405E«OO O.B607G07E*00 0.1147680E*OI O.1434601E-01 0.I7215S0E»01 0.2O0B441E401 O.2295361E-01 0.3SB2281E*01 I j 1 1 ~ T 0 2 0.1 0.6 0-8 !'< DISTANCE (TOP TO BOTTOMI 0.0 0.2 0.1 0.6 0.8 1.0 1.2 DISTANCE (TOP TO BOTTOMI PR- 1.0 RA- IOOOO.0 DIMENSIONLESS AMPLITUDE- O ISO AVERAGE NUSSELT NUMBER < RIGHT WALL)• 0.2O85305E-O1 AVERAGE NUSSELT NUMBER (LEFT WALL)- 0.3187 153E-01 LEF4GTH OF THE WALL(RIGHT )- 0. 1053476E-01 LENGTH OF THE WALL(LEFT )• O IOOOOOOE-01 VORTICITY COUNTOUR VALUES -MIN- -0.453I338E«03 MAX- 0.I267906E-03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.3951404E«03 -0.337I479E»03 -0.378I558E»03 -0.231I635E-03 -O 163I71IE-03 -O. I051790E-03 -0.47 186S2E-02 0.1O80566E-02 0.6879785E<02 STREAM FUNCTION MIN- O.O MAX- 0.S196774 CONTOUR 4 1 CONTOUR 4 3 CONTOUR 4 3 CONTOUR 4 4 CONTOUR 4 5 CONTOUR 4 6 CONTOUR 4 7 CONTOUR 4 6 CONTOUR 4 g CONTOUR VALUES. 0.5I9S773E»00 0.I039354£»0I O. 155903 IE-01 O.20787O9E-O1 0.25983B5E-01 0.31180G3E-01 0.3637741E+0I 0.4 1574 18E--01 0.4677094E-01 0.0 0.2 0.1 0.6 0.8 1.0 1.2 DISTANCE (TOP TO BOTTOM) P R . 1.0 R A - 300O0.0 DIMENSIONLESS AMPLITUDE- O 150 AVERAOE NUSSELT NUMBER (RIGHT WALL)-AVERAGE NUSSELT NUMBER (LEFT WALL >-LENGTH OF THE WALL(R1GHT)- O.I0S3476E«0I LENGTH Or THE WALLtLEETI- O.lOOOOOOE•O1 VORTICITY COUNTOUR VALUES. MIN- -O. 10B4 I09E*04 M A X - 0.a(S1B33E<03 0.298B174E«0I 0.3147563E«01 CONTOUR A CONTOUR ' CONTOUR 0 CONTOUR 0 CONTOUR 0 CONTOUR 0 CONTOUR 0 CONTOUR 0 CONTOUR 0 -O.B49I8IBE«03 -O.BI42S46E.03 -0.6793274E.03 -0.544400IE*03 -0.4094734E-03 -0.27494SBE*03 -O.t39SI87E<03 •0.4GB1162E<01 O.I3033S£E*03 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.737O0S3E* CONTOUR 0 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.73700S3E+00 0.I474010E»01 0.221101SE-01 0.394802IE.OI O.368S02GE»01 O 4422030E«OI 0.S199036E.0I 0.S89G042E-01 0.GG33047E-01 0.1 0.6 0.6 1.0 DISTANCE (TOP TO BOTTOMI d AVERAGE 0.0 0.2 0-4 0.6 0.6 1.0 1.2 DISTANCE (TOP TO BOTTOM. PR- t.O e»- looooo.o 01MENS1ONLESS AMPLITUDE- 0.150 AVERAGE I4USSELT NUMBER 1 RIGHT WAIL)- 0.43617I2E*01 AVERAGE NUSSELT NUMBER (LEFT WALL)' O.4594798E-OI LENGTH OF THE WALL( R1GHT )• 0. 1033476E--OI LENGTH OF THE WALL(LEFT)• 0.IOOOOOOE'01 VORTICITY COUNTOUR VALUES. MIN- -0.2749340E-04 MAX- 0.62S2297E-03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR . CONTOUR CONTOUR CONTOUR -0.24118B3E-04 -0.2074427E»O4 -0.173696BE-04 -O.1399512E*04 -0.10620SSE>04 -0.7245964£'03 -0.3B71414E403 -0.4S68433E«03 0.2S77722E*O3 STREAM FUNCTION CONTOUR VALUES. MIN- O.O MAX' 0.1050607E»02 CONTOUR 4 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.1050607E*0! 0.21012I3E-0I 0.3ISISiaE« 0 l 0.4202426E401 O.S2S3033E*OI 0.6303G3BE«01 0.73S4246E*0I 0.64048S4E*0t 0.8455460EtOI 1 1 -i 1 r -0.0 0.2 0.4 0.6 o.e 1-0 OlSTflNCC ITOP TO BOTTOM) RA" 0.0 DIMENSIONLESS AMPLITUDE- -O.ISO AVE.AO ^ « E L T NUMBER (..GMT WALL.- ° ' f « ^ . 0 ° ' AVERAGE NUSSELT NUMBER (LEFT V»LL»- O 13630IBE.01 LENGTH OF THE WALL!RIGHT)• 0.1I95A8SE•Ol LENGTH OF THE WALL(LEFT I* O 9999379E»CO 0.1 0.6 o.e DlSTANCE ITOP TO BOTTOM) 0.1 0.6 0.0 1.0 DISTANCE ITOP TO BOTTOM) PR- I.O HA" 1O0O.0 01MENSIONLCSS AMPLITUDE- -O ISO AVERAGE NUSSELT NUMBER (RIGHT WALL I- 0.1098O93E»01 AVERAGE NUSSELT NUMBER (LEFT WALL I- 0. 1310722C*01 LENGTH OF THE WALL( RIGHT ) - 0. 1102466E-01 LENGTH OF THE WALL(LEFT )• O.9989379E.OO VORTICITY COUNTOUR VALUES. MIN- -0.77B7270E*02 MAX- 0.3696781E*03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.6723864E<03 -0.56SB4S9E«02 -0.4594057E«02 -O.33296SIE«03 -O.246S247E+02 -O.1400842E-02 -0.33S4349E*01 0.7279G94E.OI O.I793374E«02 STREAM FUNCTION MIN- 0.0 MAX- 0.71B431' CONTOUR I 1 CONTOUR 4 2 CONTOUR » 3 CONTOUR 0 4 CONTOUR 0 & CONTOUR I 6 CONTOUR 0 7 CONTOUR 0 6 CONTOUR 0 9 CONTOUR VALUES. 0.7 IB4309E-01 0. 143S863E«0O 0.2155292E-OO 0.2873724E*00 0.3SB31S4E-OO 0.43l0S66E<O0 0.S02BO16E«O0 0.5747448E«O0 O.B46SB79E«00 0.3 0 1 0.6 0.8 1.0 OISTRNCC (TOP TO BOTTOMI 1.3 PR- t . O R » - 3 0 0 0 . 0 01MENS IONI CSS A M P L I T U D E - - 0 . 1 5 0 A V E R A G E N U S S E L T NUMBER ( R I C H ! W A L L ) - 0 . 1 3 0 1 8 2 0 E * 0 1 A V E R A G E N U S S E L T NUMBER ( L E F T W A L L ) - O I S S 2 S 3 6 E - O I L E N G T H OF THE W A L L ( R I G H T ) - O I I 9 3 4 8 6 E - 0 1 L E N G T H OF THE W A L L ( L E F T ) - 0 9 9 9 9 3 7 B E - 0 O V O R T I C I T Y COUNTOUR V A L U E S . M IN- - 0 . 2 2 0 2 4 8 7 E - 0 3 MAX - 0 . 7 3 5 9 8 0 8 E - 0 2 CONTOUR 0 I CONTOUR 4 2 CONTOUR 4 3 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR - 0 . 19OB640E-O3 - 0 . I6 I4793E--03 - 0 . 1 3 2 0 9 4 7 E - 0 3 - 0 . 1 0 3 7 I O 0 E - 0 3 - 0 . 7 3 3 3 5 3 2 E - 0 3 - 0 . 4 3 9 4 O 6 6 E - 0 2 - 0 . I 4 5 5 5 9 8 E 4 0 2 0 . I 4 S 2 8 6 7 E - 0 2 0 . 4 4 2 I 3 2 6 E - 0 3 STREAM F U N C T I O N CONTOUR V A L U E S . M IN- 0 . 0 MAX- O . 1 9 C O I 6 3 E + O I CONTOUR 4 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0 . 1 9 0 0 I 6 3 E 0 . 3 8 0 0 3 2 S E 0 . 5 7 0 O 4 8 8 E 0 . 7 6 O O 6 5 I E O . S S O O S 1 3 E 0 . T I 4 O 0 9 8 E O . 1 3 3 0 1 1 3 E O . 1 5 2 0 I 3 0 E 0 . I 7 I 0 1 4 5 E - C O -OO •OO • OO OO 01 01 01 *o i rWCRBGE 0.4 0.6 0.8 1.0 01 STANCE (TOP TO BOTTOM) 0.4 0.6 0.8 I OISTWCC ITOP TO BOTTOM I PR* t.O RA- 10000.0 DIMENSIONLESS AMPLITUOE- -O.ISO AVERAGE NUSSELT NUMBER (RIGHT WALL )" AVERAGE NUSSELT NUMBER (LEFT WALL)-O.IB6B603E*Ot 0.22-r2446E*01 LENGTH OF THE WALL(RIGHT)• LENGTH OF THE WALL(LEFT)-VORTICITY COUNTOUR VALUES. MIN- -O.GOS2932E-03 MAX- O. 1476740E-03 0.II934B6E-OI .B99B37BE-O0 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -O.S299963E -0.4S46997E -0.379403 IE -0.304 1064E -0.228809BE -O.I33SI33E -O.782I660E -0.29I9B22E 0.7237695E •03 • 03 •03 •03 •03 02 01 02 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.4249744E-O1 CONTOUR A I CONTOUR CONTOUR CONTOUR CONTOUR CDNTOUR CONTOUR CONTOUR CDNTOUR 0.4249744E 0.B4994B9E 0.I274923E O.IG998B6E 0.2I24B7 IE 0.2549B46E O. 29748 I9E C3399796E 0.3824769E 01 •01 • 01 •01 •Ol Ot 01 £tiX<VSC 0.4 0.6 0-0 I. D1STHNCC (TOP TO BOTTOMI P H - 1 .0 RA- 3O00O.0 DIMENSIONLESS AMPLITUOE- -O.ISO AVERAGE NUSSELT NUMBER (RIGHT WALL)* O.26544I3E+0I AVERAGE NUSSELT NUMBER (LEFT WALL)* O.3I65359E*0I LENCTH OF THE WALL(RIGHT)• O.I1924B6E*0t LENGTH OF THE WALL(LEFT)- O.899937SE•00 VORTICITY COUNTOUR VALUES. MIN- -O.1387027E*O4 MAX • 0.2566|50E*03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.t232663£*04 -O.I0SB2BBE*04 -0.8939348E*03 -O.72B570BE*03 -0.5652063E*03 -0.4008423E*03 -0.23G477B£»03 -0.721I377E.03 0.822&024E*O2 STREAM FUNCTION CONTOUR VALUES. MIN- O.O MAX- 0.7 I 1786(E»Ol CONTOUR 4 t CONTOUR 4 2 CONTOUR 4 3 CONTOUR 4 4 CONTOUR 4 5 CONTOUR 4 6 CONTOUR 4 7 CONTOUR 4 8 CONTOUR 4 9 0.71I7B60£*00 0.1423SB2E*01 0.2135387E«OI 0.28*71B3E-01 0.35S8979£*01 0.4270776E*01 0.4SB257IE«01 0 . 6 6 f i 4 3 C 7 E*Ot 0.640€I64£*01 OlSTflNCC ITOP TO BOTTOM! P R - 1.0 R A - tOOOOO.O DIMENSIONLESS AMPLITUOE- -0.150 AVERAGE NUSSELT NUM8ER (RIGHT WALL)" AVERAGE NUSSELT NUMBER (LEFT WALL)" LENGTH Of THE WALL(RICHT)- O.1I924B6E*01 LENGTH OF THE WALLfLEFT )• 0.9999379E *C-0 0.38 1S71TE«01 0.45S0391E*01 VORTICITY COUNTOUR VALUES. MIN- -0.32 36658£«04 MAX- 0.6O98074E+03 CONTOUR CONTDUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.28521938*04 -0.246752GE+04 -0.2082859E*04 -O.1698193E*04 -O.1313526E*04 -O.9288594**03 -0.5441931E*03 -0. 1S95264E*03 0.325i406E*03 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.8B302G7E+O1 CONTOUR * 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 98302G7E+O0 19GG052E*01 2849O80E*01 3932106E*Ol 4915!32E<01 5B96159E*01 €88118GE+01 7664213E*Ot 8847239E+01 CD *o 5 AVERAGE 0.1 0.6 O.B 1.0 DISTHNCC (TOP TO BOTTOMI 0.1 0.6 0.8 I.O DISTANCE (TOP TO BOTTOMI P R - I.O R A - o.o D I M E N S I O N L E S S A M P L I T U D E - - 0 . 0 7 5 A V E R A G E N U S S E L T NUMBER ( R I G H T WALL I - 0 I 0 4 6 5 6 3 E - 0 1 A V E R A G E N U S S E L T N U M B E R ( L E F T W A L L ) ' 0 . I 0 9 B 3 3 B E » 0 1 L E N G T H OF T H E W A L L ( R I G H T ) - 0 1 0 5 2 8 I 3 E « 0 I L E N G T H OF T H E W A L L ( L E F T ) - O . 9 9 9 9 6 2 0 E - 0 0 ! 1 | I 0.0 0.2 0.4 0.6 o.e I DISTANCE (TOP TO BOTTOM) g - 1 —1 —I 1 (— o.i o.t o.e i.o DISTANCE (TOP TO BOTTOM) P B - 1.0 OA- 1000.0 DIMENSIONLESS AMPL 1TUOC• -0.07S AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.1l24238E«OI AVERAGE NUSSELT NUMBER (LEFT WALL)- 0.1IB3872E+OI LENGTH OF THE WALL(RIGHT)• 0.IOS28I3E»0I LENGTH OF THE WALL(LEFT)- O.8999630E-OO VORTICITY COUNTOUR VALUES. MIN- -O.S88862SE-03 MAX- 0.312103IE«02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.5887662E-02 -0.4B86G97E402 -0.3885732E»02 -0.28B4767E«02 -O.1883S01E-02 -0.882B339E-01 O.1I61305E-01 O.111S09BE-02 0.2I2OO64E-02 STREAM FUNCTION CONTOUR VALUES MIN- O.O MAX- O.B7BS647E*0O CONTOUR 4 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR O.978564SE-01 O.1857128E4O0 0.293S693E-00 0.3B142S8E-00 O 4BB2823E«00 0.587I387E-0O 0.G848952E-O0 0.7B265I7E»00 0.880708 t E +O0 RvrRftnr O.i 0.8 0-8 OlSTflNCC (TOP TO BOTTOM) PR- 1.0 BA- 3000.0 DIMENSIONLESS AMPLITUDE- -0.075 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.143770GE*01 AVERAGE NUSSELT NUMBER (LEFT WALL)- O.IBl339SE«0t LENGTH OF THE WALL(RIGHT)- O.I033BI3E40I LENGTH OF THE WALL(LEFT)- O.9999620E*OO VORTICITY COUNTOUR VALUES. MIN- -O.IB31734E»03 MAX- 0.7216054E«O2 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.16574O0E»03 -0.I393067E«03 -0. 1 136733E *03 -O.B643991E-02 -0.600O653E-02 -0.33573I7E402 -0.7I397B6E»01 0. IB2B3SSE-02 0.4S7269EE-02 STREAM FUNCTION CONTOUR VALUES. MIN- O.O MAX- 0.2394I37E-OI CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR , 2394137E400 47B8275E*O0 7183412E1O0 8576550E-00 1197068E*01 143G4B2E*Ot 1G75896E-OI t915309E*Ot 21S4722E-OI 0.1 0.6 0.8 1.0 DISTHNCC (TOP TO BOTTOM) 0.1 0.6 o.o t.o OISTHNCE (TOP TO BOTTOM) pa- i.o B A - IOOO0.0 DIMENSIONLESS AMPLITUDE- -O 075 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.2126877E40I AVERAGE NUSSELT NUMBER (LEFT WALL)- 0.223a388E*01 LENGTH OF THE WALl(RICHT)- O.tOB2fiI3E-0I LENGTH OF THE WALL(LEFT)- O.B8BB630E-OO VORTICITY COUNTOUR VALUES MIN- -0.83040I6E-03 MAA- 0. 13I8381E-03 •0.464IS76E103 -0.3878333E«03 -0.33IG8B3E-03 -0.2654666E-03 -O.18833I4E-03 -O.I328873E-03 -O.B67G343E<03 -0.63864lOE-OO 0.65704E8E402 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.4824135E401 CONTOUR 0 t CONTOUR # 1 CONTOUR 4 2 CONTOUR # 3 CONTOUR 0 4 CONTOUR 0 6 CONTOUR t 6 CONTOUR 0 7 CONTOUR 0 8 CONTOUR 0 8 !»!iSKKSK»:KKi..:.:ii,» CONTOUR 0 CONTOUR 0 CONTOUR 0 CONTOUR 0 CONTOUR I CONTOUR « CONTOUR I CONTOUR 1 4824136E-00 8B48288E«O0 1447340E4-01 1B386S3E*01 34I3066E-01 268448IE-OI 3376883E«OI 3859307E.OI 4341721E-01 0.4 0.6 0.6 1.0 DISTANCE (TOP TO BOTTOM! 0.0 0.2 0.1 0.6 0.6 1.0 01 STANCE ITOP TO BOTTOM) PB- 1.0 «A- 30000.0 DIMENSIONLESS AMPLITUDE- -0.079 AVEBAOE NUSSELT NUMBER (RIGHT WALL)- O.2978*04E-O1 AVERAGE NUSSELT NUMBER ( LEM WALL)- O.313«303E»Ol LENGTH OF THE WALL(RICH7>- O. IOS3Bl3E»OI LENGTH OF THE WALL(LEFT)- O.8BB8B20E«00 VORTICITT COUNTOUR VALUES. MIN- -0.I335B7BE-04 MAX* 0.2937362E<03 CONTOUR 4 t -0. 107B007E404 CONTOUR 4 3 -O.B300334E«03 CONTOUR 4 3 -0.782O64OE*O3 CONTOUR 4 4 -0.634082S£»03 CONTOUR 4 6 -0.486121IE-03 CONTOUR 4 6 -0.3381489E-03 CONTOUR 4 7 -O.180I7BSE«03 CONTOUR 4 6 -0.4220703E-02 CONTOUR 4 8 0. IOf>7644E*03 S7REAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.7303S73E4 CONTOUR 4 I CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.73O3S72E»O0 O. 1460714E-01 0.219!07tE*O1 0.2921428E-01 0.36517B5E-O! 0.4362I42E-O1 0.5tt2499E+Ot 0.B842857E40I 0.65732I5E*OI I- o UJ ^ . ^ O SI i n • 1 ' AVERAGE 0 ( 0.6 0.8 I OlSTRNCE (TOP TO BOTTOM) PR- 1.0 RA- lOOOOO.O DIMENSIONLESS AMPLITUDE- -0.075 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O. 4306437E101 AVERAGE NUSSELT NUMBER (LEFT WALL)- 0.4534001£-01 LENGTH OF THE WALL(RIGHT)- 0. 10528 I3£«01 LENGTH OF THE WALL(LEFT)- O.B899620E-00 VORTICITY COUNT OUR VALUES. MIN- -0.3S64367E-04 MAX- 0.63404IOE-03 CONTOUR A I CONTOUR A 2 CONTOUR A 3 CONTOUR I 4 CONTOUR t 5 CONTOUR A 6 CONTOUR A 7 CONTOUR I e CONTOUR # e -0.2623526E-04 -0.2260G8GE*04 -0.IB88B45E-04 -0.IS37004E<04 -0. I 175IG3E-04 -0.6I33323E*03 -0.45I4617E-03 -0.8964063E-02 O.27220OOE-03 STREAM FUNCTION CONTOUR VALUES. MIN-MAX-CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.0 0.88701I7E« 0.8B701I7E 0.1B74023E 0.396I035E 0.3948047E 0.493S059E 0.5923070E 0.6B0906IE 0.78S6093E 0.8S83I05E 01 »0I in •. RVfRRGC 1.1 0.6 0.6 I DISTANCE (TOP TO BOTTOM) 0.4 O.fi 0.6 I OlSTflNCC ITOP TO BOTTOMI PR- 1.0 RA- 0.0 01MENSIONLESS AMPLITUOE- 0.0 AVERAGE NUSSELT NUMBER (RIGHT HALL)' 0.IO007I3E.0I AVERAGE NUSSELT NUMBER (LEFT HALL)- 0.B8B9542E-00 LENGTH OF THE WALLtRIGHTI- O.lOOOOOOE*01 LENGTH OF THE WALL(LEFT)- 0.lOOOOOOE*01 O.i 0.S 0.0 1.0 OlSTflNCC (TOP TO BOTTOM1 P-o.i o.e o.e i OlSTflNCC (TOP TO BOTTOM) PP- I.O R A- tooo " DIMENSIONLESS AMPLITUDE- O 0 AVERAGE NUSSELT NUMBER (RIGHT WALL)- 0.I I 173I4E*0I AVERAGE NUSSELT NUMBER (LEFT WALL)- 0. I I 17 II6E-01 LENGTH OF THE WALL(RIGHT)- 0.IOOOCOOE<0I LENGTH OF THE WALL(LEFT)- O lOOOOOOE-CM VORTICITV COUNTOUR MIN- -0.5111407E+0 MAX• 0.3193279E-0 CONTOUR 4 1 CONTOUR 4 2 CONTOUR 4 3 CONTOUR 4 4 CONTOUR 4 S CONTOUR 4 6 CONTOUR 4 7 CONTOUR 4 S CONTOUR 4 8 -0.42B0939E -0.345047 IE -0.2620003E -0.1789336E -0.9590668E -O.13B58S0E 0.7018707E 0. 1532339E 0.2363B08E 03 02 02 03 01 •01 01 03 STREAM FUNCTION CONTOUR VALUES. MIN- O.O MAX- 0.1I721SOE.O! CONTOUR 4 1 0. 1 172I49E»00 CONTOUR 4 2 0.2344299E*00 CONTOUR 4 3 0.35 16449E«O0 CONTOUR 4 4 0.46BeS9BE«00 CONTOUR 4 5 O.SB6O74SE.O0 CONTOUR 4 € O.7032B97E«OO CONTOUR 4 7 0.820S047E+00 CONTOUR 4 fi 0.9377I97E+OO CONTOUR 4 9 O.105493SE«01 nvCRBEL. o.i o.6 o.e i.o DISTANCE (TOP TO BOTTOM) 0.0 0.2 0.1 0.6 o.e I.O 1.2 01 STANCE (TOP TO BOTTOM) P P - 1 . 0 R A - 3 0 0 0 . 0 0 I M C N S I 0 N L C S S A M P L I T U O E - O . O A V E R A G E N U S S E L T N U M B E R ( R I G H T W A L L ) A V E R A G E N U S S E L T N U M B E R ( L E F T W A L L ) -L E N G T H O F T H E W A L L ( R 1 G H T ) • O . l O O O O O O E ' O I L E N G T H OF THE W A L L ( L E F T ) - 0 . 1 O 0 O O 0 O E « O 1 .1&03907EMM IS03776E+0I V O R T I C I T Y C O U N T O U R V A L U E S -M I N - - O . 1 4 7 3 2 7 S E + 0 3 M A X - 0 . 6 7 7 5 1 6 B E + 0 2 C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R - 0 . 1 2 5 8 1 9 6 E * - 0 . 1 0 4 3 I I 7 E * - 0 . B 2 8 0 3 7 4 E * - 0 . 6 1 2 9 5 8 4 E * - O . 3 9 7 8 7 9 2 E * - O . 1 8 2 7 9 9 8 E * 0 . 3 2 2 7 9 2 I E * 0 . 2 4 7 3 S 8 2 E * 0 . 4 6 2 4 3 7 4 E * S T R E A M F U N C T I O N C O N T O U R V A L U E S . M I N - 0 . 0 M A X - 0 . 2 7 0 9 7 9 8 E * 0 I C O N T O U R 0 1 C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R 0 . 2 7 0 9 7 9 8 E + 0 0 0 . 5 4 1 9 S 9 S E * 0 0 0 . 8 . 2 9 3 9 3 E + O O 0 . t 0 8 3 9 1 9 E - * 0 1 O . 1 3 5 4 B 9 8 E * 0 1 O . 1 6 2 5 8 7 7 E * 0 1 O . 1 B 9 6 B 5 8 E + 0 1 0 . 2 1 G 7 8 3 7 E + 0 1 O . 2 4 3 8 8 1GE+01 err 0 0.1 0.6 0.8 1-0 OlSTHNtX (TOP TO BOTTOMI HYfRflGC 0.0 Q.2 O.i 0.6 0.6 t.O 1.2 DISTHNCC (TOP TO BOTTOM) PR- 1.0 RA- lOOOO.O OIMENSIONLESS AMPLITUDE- 0.0 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.2255560E»0I AVERAGE NUSSELT NUMBER (LIFT WALL)- 0.3255S77E»OI LENGTH OF THE WALL(RICHT)- 0. 1OOOOOOE•01 LENGTH OF THE WALL ( LEFT )• O.(OOOOOOE• 01 VORTICITY COUNTOUR VALUES. MIN- -O.4267256E»03 MAX- 0.127BB77E-02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.37I2642E -0.315B030E -0.26034I6E -0.204Ba04E -0.1494I92E -0.9395776E -0.3S49656E O.1696460E 0.7242578E 02 02 • 02 02 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.51135I7E-CONTOUR 4 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.5112517E 0.1022503E O.1533754E 0.204S006E 0.25S6257E O. 3C-S750BE 0.357B7G1E 0.4090013E 0.460126SE 01 01 • 01 • 01 •Ol 01 RVrRflGE 0.1 0.6 0.6 1.0 OISTRNCE (TOP TO BOTTOM! 3.1 0.6 0.6 I DISTANCE (TOP TO BOTTOM) PR- 1 .0 RA- 30000.0 DIMENSIONLESS AMPLITUDE- O . O AVERAGE NUSSELT NUMBER (RIGHT WALL)- 0.3 178326E*01 AVERAGE NUSSELT NUMBER (LEFT WALL)- 0.3t7B39SE-01 LENGTH OF THE WALL(RIGHT)- 0.IOOOOOOE-01 LENOTH OF THE WALL(LEFT)- 0.I0OO0O0E-01 VORTICITY COUNTOUR VALUES. MIN- -0.IOA3358E«04 MAX- 0.2B87B78E-03 CONTOUR CONTOUR CONTOUR CDNTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.9I20558E -0.7818S33E -0.SSIBS06E -0.B314460E -0.39134SBE -0.26I0430E - O . I30S403E -0.G374513E 0. T3B5E47E 03 03 •OO 03 STREAM FUNCTION CONTOUR VALUES. MIN-MAX-CONTOUR CONTOUR CONTOUR CONTOUR . CONTOUR . CONTOUR , CONTOUR . CONTOUR i CONTOUR • 0.0 0.7335O73E^OI O.733S073E*OO O.14E7014E^01 O.230053lE^OI 0.383<02QE«OI 0.3SS753SE^OI 0.44OlO43E^O1 O.S1345SOE^01 0.S6BB0SSE+0I 0EE01SESE*01 0 0 0.3 0.4 O.fi O.B 1.0 OISTHNCE (TOP TO BOTTOMI PR- 1 .0 RA- 100OOO.0 DIMENSIONLESS AMPLITUDE- O O AVERAGE NUSSELT NUMBER (RIGHT WALL)- 0.45929 I6E*0 1 AVERAGE NUSSELT NUMBER (LEFT WALL I * 0.4S92490E*01 LENGTH OF THE WALL(RIGHT )* O. 10000001*01 LENGTH OF THE WALL(LEFT >- 0.lOOOOOOE*01 VORTICITY COUNTOUR VALUES. MIN- -0.2694243E*04 MAX- 0.G2I1931E+03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.23627COE -0-3031I5GE -0- 1G996I2E -O .366069E •O.I03GS2SE -O.70498I4E -0.3734382E -0.41694S3E 0.2B9G494E 03 03 02 •03 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.10.4151E+02 CONTOUR t 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR , 0. 1014 151E-»01 0-2028301E+01 0.3042453E«01 04056603E+01 0.50707SSE«01 0-6084906E+01 0-7099O56E+O1 0.B1 13208E«01 0-B127358E«01 2 6 5 SfilV. 0.0 0.2 0.1 0.6 0.6 1.0 1.2 DISTANCE ITOP TO BOTTOMI pa* i.o BA- tooo.o DIMENSIONLESS AMPLITUDE- 0.07S AVERAGE NUSSELT NUMBER (RIGHT WALL )• AVERAGE NUSSELT NUMBER (LEFT WALL I-O. 103MBBE*0l O.t0BS437E*01 LENGTH OF THE WALL!RIOHT)-LENGTH OF THE WALL!LEFT)" :>.tosaa t a t * o i .IOOO04BE*0l VORTICITY COUNTOUR VALUES. MIN* -O.BOS03S5E*03 MAX- 0.3ISS330E«02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR . CONTOUR CONTOUR CONTOUR CONTOUR -0.4338BS4E<03 -0.3407027E*02 -0.3SB53SBE*02 -0.17G36BBE*02 -0.8420I6*E*0( -O.I20390GE*0I o.70i3<aaE*oi o .isaieaat*03 0.2344G98E*03 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.I2BSB32E*0I CONTOUR A t CONTOUR CDNTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR O. 1285632E*00 0.2571GG4E*00 0.38574B7E*00 O.5t4333OE*0O 0.643BIG2E*0O 0.77 1 4 B B 4 E*00 0.B0OOB3GE*O0 0. I02S6G6E*0I 0.1IS734BE*OI 0.4 0.6 0.6 1.0 DISTANCE (TOP TO BOTTOMI 0.1 0.6 0.6 1.0 DISTANCE (TOP TO BOTTOMI PR- I.O RA - JOOO.O OIMIMSIONLISS AMPLITUDE- O.OT5 AVERAGE NUSSELT NUMBER (RIGHT MALL)- 0.1018311-01 AVERAGE NUSSELT NUMBER (LEFT WALL)- 0.150738 IE<01 LENGTH OF THE WALL(RIGHT )• 0. 10528 13E*01 LENGTH OF THE WALL(LEFT)• O.IOOO04BE*0I VORTICITY COUNTOUR VALUES. MIN- -O.146711BE-03 MAI. 0.6423BBaE*02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.I25G180E-03 -O.1045243E403 -0.B34305OE*O2 -O.G233672E*02 -0.41242B5E-02 •0.20I4820E*02 0.844880IE*00 0.2303836E-02 0.43I33IIE-02 STREAM FUNCTION CONTOUR VALUES. MIN-MAX* CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.0 0.3875B43E* 1 2 3 287S842E*00 S7StE84E*O0 B627S26E-00 115033GE*01 1437B21E*01 1725505E»01 20I3088E*01 230OG73E*0l 2S862S7E*01 0.1 0.6 0.8 1.0 OlSTflNCC (TOP TO BOTTOM I d in Q flvf-Rflsr O.t 0.6 0.8 1.0 OlSTflNCC (TOP TO BOTTOM1 P R * 1 . 0 R A - l O O O O . O 0 I M E N S 1 0 N L E S S A M P L I T U D E - 0 . 0 7 5 A V E R A G E N U S S E L T N U M B E R ( R I G H T W A L L I A V E R A G E N U S S E L T N U M B E R ( L E F T W A L L ) -L E N G T H O F T H E W A L L t R I G H T ) - 0 . 1 0 5 2 8 I 3 E - 0 1 L E N G T H OF T H E W A L L ( L E F T I• 0 . I 0 0 0 0 4 5 E « 0 1 0 . 2 1 S O 8 6 6 E * O 1 0 . 2 2 8 8 1 3 B E * 0 1 V O R T I C I T Y C O U N T O U R V A L U E S . M I N - - 0 . 4 3 0 1 1 2 1 E « 0 3 M A X - 0 . I 3 3 5 0 3 5 £ » 0 3 C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R ' - 0 . 3 7 3 7 5 0 S E + 0 3 - 0 . 3 1 7 3 B 8 9 E - 0 3 - 0 . 2 6 1 0 2 7 3 E - 0 3 - 0 . 2 0 4 G G 5 9 E - 0 3 - O . 1 4 B 3 0 4 4 E - 0 3 - 0 . B 1 9 4 2 8 7 E * 0 2 - 0 . 3 5 S 8 1 3 0 E - 0 2 O . 2 0 7 8 O 0 3 E » O 2 0 . 7 7 I 4 I 8 5 E » 0 2 S T R E A M F U N C T I O N C O N T O U R V A L U E S . M I N - 0 . 0 M A X - 0 . 5 2 4 1 2 6 4 E - 0 1 C O N T O U R 4 1 C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R C O N T O U R O . S 2 4 1 2 6 4 E O . I 0 4 8 2 5 2 E O . 1 5 7 2 3 7 9 E 0 . 2 0 9 G 5 0 5 E 0 . 2 G 2 0 6 3 1 E 0 . 3 1 4 4 7 5 8 E 0 . 3 G 6 B B B 4 E 0 . 4 1 S 3 0 I I E 0 . 4 7 1 7 1 3 6 E 0 1 0 1 • 0 1 • 0 1 O J • 0 1 0.< 0.6 0.8 1.0 DISTANCE (TOP TO BOTTOM! PR- 1.0 R»- 30OOO.0 DIMENSIONLESS AMPLITUDE- O.OTS AVERAGE NUSSELT NUMBER (RIGHT WALL)-AVERAGE MJSSELT NUMBER (LEFT WALL)-LENGTH OF THE WALL (RIGHT ) -LENGTH OF THE WALL(LEFT)* 0.3I03331E*01 0.3266882E*Ot 3.IOS28!3E*OI .IOOOO4BE*01 VORTICITY COUNTOUR VALUES. MIN- -0.1062268E*CM MAX- 0.282848 IE-03 CONTOUR CONTOUR CONTDUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR .B277MGE*03 -0.79334SIE*03 -0.6587334E.*03 -O.S3432tfiE*03 -0.38B7102E-03 -0.25S1BaOE«03 -O. 1206873E*03 0.I382446E*02 0.t4S33G2E*03 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- 0.75063B4E*OI CONTOUR A 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.75O63B4£«O0 O.1501376E*01 0.325191SE*01 0.30025S3E*Ot O.3753IB0E.O1 0.4903830E*0I 0.52544S7E*01 0.6005I07E«0I O.G75574SE«0< 0.4 O.G 0.8 1.0 DISTANCE (TOP TO BOTTOM) 0.1 0.6 0.8 1.0 01 STANCE (TOP TO BOTTOMI P R - 1 . 0 HA- 10OO00.0 DIMENSIONLESS AMPLITUDE- 0.075 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.4484457E*01 AVERAGE NUSStLT NUMBER (LETT WALL)- O.472161OE401 LENGTH OF THE WALL(RIGHT)- O.1052BI3E*01 LENGTH OF THE WALL(LEFT)- 0.100004SE40I VORTICITY COUNTOUR VALUES. MIN- -0.2738S46E-04 MAX- 0.6383308E>03 CONTOUR CDNTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -O.240I22OE -0.2063493E -O.172S764E -0.1368037E -0. 1050309E -0.7I25808E -0.3748533E -0.37I25O0E 0.300C026E 04 04 OO • 03 •02 •03 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- O.107B4I5E-02 CONTOUR A 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0. 107S41BE 0.2150B2BE 0.3226243E 0 4 3 0 I 6 5 9 E 0.5377073E O.E452487E 0.7527902E 0.BG03317E 0.BC78732E • 01 01 01 •01 01 PH- 1.0 RA- 0.0 0IMENS10NLESS AMP11TUOE- 0.150 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.7694461E +0O AVERAGE NUSSELT NUMBER (LEFT WALL)- O.91440O6E+00 LENGTH OF THE WALL!RIGHT)• 0.1192486E»01 LENGTH OF THE WALL(LEFT )- O.10O0O84E+01 i i 1 • - i - — — : 1 r— 0.0 0.2 0.1 0.6 o.e I.O 1.2 OlSTRNCe (TOP TO BOTTOM! Wo i n .. i 1 T I 1 T I 0-0 0.2 0.1 0.6 o.e 1.0 1.2 DISTRNCC (TOP TO BOTTOM) P B - 1.0 Rt- 1OOO.0 DIMENSIONLESS AMPLITUDE- 0.150 AVERAGE NUSSELT NUMBER (RIGHT WALL)- 0.897 1367E«0O AVERAGE NUSSELT NUMBER (LEFT WALL I- 0.IOG9689E-0I -LENGTH OF THE WALL(RIGHT )- 0. M92486E-01 LENGTH OF THE WALL(LEFT)- 0. 1000084E«01 VORTICITY COUNTOUR VALUES. MIN- -0.&156377E-02 MAX- 0.310S145E-02 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.4330223E»02 -0.3504073E-02 -0.2677821E-02 -O.IB5I770E-02 -O.1025618E-02 -O.19946S9E-OI 0.63G6861E-Ot O. 14S2S40E-02 0.227B992E-02 STREAM FUNCTION CONTOUR VALUES. MIN- 0.0 MAX- O.1344956E-01 CONTOUR A t CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.134495GE*00 O. 2G89912E-0O 0.4034BG9E-O0 0.537BB25E-00 0.67247B2E-00 O.SOG9738E-0O 0.9414694E-00 0.107596S£»01 0.I2I0461E-0I 0.0 0.2 0.< 0.0 0.0 1.0 DISTflNCC (TOP TO BOTTOMI 5 f- -d 0.4 0.6 0.6 1. DISTHNCC (TOP TO BOTTOMI PB- 1.0 RA- 3O0O.0 0IMENS10NLESS AMPLITUDE- 0.150 AVERAGE NUSSELT NUMBER (BIGHT WALL )- O.1267410E101 AVEBAGE NUSSELT NUMBER (LEFT WALL)- 0.15 11370E«01 LENGTH OF THE WALL(RIGHT)- O.1IS2486E.01 LENOTH OF THE WALL(LEFT)- O.1OOOO84E.01 VORTICITY COUNTOUR MIN- -0.1474016E MAX• O.G307288E CONTOUR I 1 CONTOUR A 2 CONTOUR A 3 CONTOUR A 4 CONTOUR A 3 CONTOUR A C CONTOUR A 7 CONTOUR A B CONTOUR A 9 VALUES. 03 -0.126454SE -O.105S070E -0.8455951E •0.G36I201E -0.42G6454E -O.2171707E -0.7695923E 0.3017790E 0.4 112S38E 103 03 02 -02 00 STREAM FUNCTION CONTOUR VALUES MIN- 0.0 MAX- 0.395S637E*0I CONTOUR I 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.2955637E-»O0 0.39I1374E-O0 O.886691lE-OO O.I182255E+01 O.147781BE-01 O.1773380E*0I 0.2O68945E-OI 0.2364510E-»Ot 0.366O072E-0I 6.0 0.2 0.4 0.6 0.8 1.0 OISTRNCE ITOP TO BOTTOMI 6° CD *° d *»». HVCRflGC + — — — 1 • i- | — 0.0 0.2 0.1 0.6 0.8 1.0 DISTANCE (TOP TO BOTTOM) 274 PR* (.0 RA* ICOOO.O DIMENSION!.ESS AMPLITUDE- 0.150 AVERAGE NUSSELT NUMBER (RIGHT WALL)- O.1987071E*0» AVERAGE NUSSELT NUMBER (LEFT WALL)- O.2334438E*0. LENGTH OF THE WALL(RIGHT)- 0.11924866*01 LENGTH OF THE WALL.LEFT)- O.lOOOOB4£*0. VORTICITY COUNTOUR VALUES• MIN- -O.43627B3E+03 MAX- O.134895IE<03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR -0.37916I9E+03 -0.322O444E*03 -0.2649270E+03 -0.2078096E+03 -0.150692IE«03 -0.9357471E+02 -0.364572BE402 0.20€6016E«02 0.7777734E+02 STREAM FUNCTION CONTOUR VALUES MIN- 0.0 MAX- 0.8296778E+0 CONTOUR # 1 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.52B6777E*00 0. 10S9355E+0I O. I5B9033E+01 0.2118710E+01 0.2648388E«01 0.3178O6SE*O. O.37O7744E+0I 0.4237422E+01 0.476709BE+O1 PB- 1.3 11- 100000.o DIMENSIONLESS AMPLITUDE- O 150 AVERAGE NUSSELT NUMBER (BIGHT WALL)- 0.4082S29E-01 AVERAGE NUSSELT NUMBER (LEFT WALL)- O.ABB1S60E-O1 LENGTH OF THE WALL(RIGHT)- O. 1 1924A6E»OI LENGTH OF THE WALL(LEFT)- O.IOOO0B4E-O1 VORTICITY COUNTOUR VALUES. •IN- -0.3B0B(43E'04 MAX- 0.661304SE-03 CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR , -0.2461630E<04 -0.3M4596E.O4 -O. 1767972E-04 -0.I420S49E-O4 -O. 1073333E-04 -0.7265O07E«03 -O.378477IE-OS -0.324S3I3E'03 O.3I437O3E-03 STREAM FUNCTION CONTOUR VALUES. MIN" MAX-CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR CONTOUR 0.0 O. II3396SE« O. 1 t33863E*OI 0.3267930E-01 0.340t69SE*0t 0.433566IE-04 0.3669S36E»0t 0.6S037B1E-OI 0.79377S6E-0I . 0.90717226-01 0.1020369E-02 R V E R R G E 0.0 0.3 0.4 0.6 0.8 1.0 1.3 01 STANCE (TOP TO BOTTOMI 

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