engineering - fundamentals of computational fluid dynamics

8/7/2019 Engineering - Fundamentals of Computational Fluid Dynamics

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F u n d a m e n t a l s o f C o m p u t a t i o n a l F l u i d

D y n a m i c s

H a r v a r d L o m a x a n d T h o m a s H . P u l l i a m

N A S A A m e s R e s e a r c h C e n t e r

D a v i d W . Z i n g g

U n i v e r s i t y o f T o r o n t o I n s t i t u t e f o r A e r o s p a c e S t u d i e s

A u g u s t 2 6 , 1 9 9 9


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4/276

C o n t e n t s

1 I N T R O D U C T I O N 1

1 . 1 M o t i v a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 . 2 B a c k g r o u n d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 . 2 . 1 P r o b l e m S p e c i c a t i o n a n d G e o m e t r y P r e p a r a t i o n . . . . . . . 2

1 . 2 . 2 S e l e c t i o n o f G o v e r n i n g E q u a t i o n s a n d B o u n d a r y C o n d i t i o n s . 3

1 . 2 . 3 S e l e c t i o n o f G r i d d i n g S t r a t e g y a n d N u m e r i c a l M e t h o d . . . . 3

1 . 2 . 4 A s s e s s m e n t a n d I n t e r p r e t a t i o n o f R e s u l t s . . . . . . . . . . . . 4

1 . 3 O v e r v i e w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 . 4 N o t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 C O N S E R V A T I O N L A W S A N D T H E M O D E L E Q U A T I O N S 7

2 . 1 C o n s e r v a t i o n L a w s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 . 2 T h e N a v i e r - S t o k e s a n d E u l e r E q u a t i o n s . . . . . . . . . . . . . . . . . 8

2 . 3 T h e L i n e a r C o n v e c t i o n E q u a t i o n . . . . . . . . . . . . . . . . . . . . 1 2

2 . 3 . 1 D i e r e n t i a l F o r m . . . . . . . . . . . . . . . . . . . . . . . . . 1 2

2 . 3 . 2 S o l u t i o n i n W a v e S p a c e . . . . . . . . . . . . . . . . . . . . . . 1 3

2 . 4 T h e D i u s i o n E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

2 . 4 . 1 D i e r e n t i a l F o r m . . . . . . . . . . . . . . . . . . . . . . . . . 1 4

2 . 4 . 2 S o l u t i o n i n W a v e S p a c e . . . . . . . . . . . . . . . . . . . . . . 1 5

2 . 5 L i n e a r H y p e r b o l i c S y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . 1 6

2 . 6 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7

3 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 2 1

3 . 1 M e s h e s a n d F i n i t e - D i e r e n c e N o t a t i o n . . . . . . . . . . . . . . . . . 2 1

3 . 2 S p a c e D e r i v a t i v e A p p r o x i m a t i o n s . . . . . . . . . . . . . . . . . . . . 2 4

3 . 3 F i n i t e - D i e r e n c e O p e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . 2 5

3 . 3 . 1 P o i n t D i e r e n c e O p e r a t o r s . . . . . . . . . . . . . . . . . . . . 2 5

3 . 3 . 2 M a t r i x D i e r e n c e O p e r a t o r s . . . . . . . . . . . . . . . . . . . 2 5

3 . 3 . 3 P e r i o d i c M a t r i c e s . . . . . . . . . . . . . . . . . . . . . . . . . 2 9

3 . 3 . 4 C i r c u l a n t M a t r i c e s . . . . . . . . . . . . . . . . . . . . . . . . 3 0

i i i


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3 . 4 C o n s t r u c t i n g D i e r e n c i n g S c h e m e s o f A n y O r d e r . . . . . . . . . . . . 3 1

3 . 4 . 1 T a y l o r T a b l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1

3 . 4 . 2 G e n e r a l i z a t i o n o f D i e r e n c e F o r m u l a s . . . . . . . . . . . . . . 3 4

3 . 4 . 3 L a g r a n g e a n d H e r m i t e I n t e r p o l a t i o n P o l y n o m i a l s . . . . . . . 3 5

3 . 4 . 4 P r a c t i c a l A p p l i c a t i o n o f P a d e F o r m u l a s . . . . . . . . . . . . . 3 7

3 . 4 . 5 O t h e r H i g h e r - O r d e r S c h e m e s . . . . . . . . . . . . . . . . . . . 3 8

3 . 5 F o u r i e r E r r o r A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . . . 3 9

3 . 5 . 1 A p p l i c a t i o n t o a S p a t i a l O p e r a t o r . . . . . . . . . . . . . . . . 3 9

3 . 6 D i e r e n c e O p e r a t o r s a t B o u n d a r i e s . . . . . . . . . . . . . . . . . . . 4 3

3 . 6 . 1 T h e L i n e a r C o n v e c t i o n E q u a t i o n . . . . . . . . . . . . . . . . 4 4

3 . 6 . 2 T h e D i u s i o n E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . 4 6

3 . 7 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7

4 T H E S E M I - D I S C R E T E A P P R O A C H 5 1

4 . 1 R e d u c t i o n o f P D E ' s t o O D E ' s . . . . . . . . . . . . . . . . . . . . . . 5 2

4 . 1 . 1 T h e M o d e l O D E ' s . . . . . . . . . . . . . . . . . . . . . . . . 5 2

4 . 1 . 2 T h e G e n e r i c M a t r i x F o r m . . . . . . . . . . . . . . . . . . . . 5 3

4 . 2 E x a c t S o l u t i o n s o f L i n e a r O D E ' s . . . . . . . . . . . . . . . . . . . . 5 4

4 . 2 . 1 E i g e n s y s t e m s o f S e m i - D i s c r e t e L i n e a r F o r m s . . . . . . . . . . 5 4

4 . 2 . 2 S i n g l e O D E ' s o f F i r s t - a n d S e c o n d - O r d e r . . . . . . . . . . . . 5 5

4 . 2 . 3 C o u p l e d F i r s t - O r d e r O D E ' s . . . . . . . . . . . . . . . . . . . 5 7

4 . 2 . 4 G e n e r a l S o l u t i o n o f C o u p l e d O D E ' s w i t h C o m p l e t e E i g e n s y s t e m s 5 9

4 . 3 R e a l S p a c e a n d E i g e n s p a c e . . . . . . . . . . . . . . . . . . . . . . . . 6 1

4 . 3 . 1 D e n i t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1

4 . 3 . 2 E i g e n v a l u e S p e c t r u m s f o r M o d e l O D E ' s . . . . . . . . . . . . . 6 2

4 . 3 . 3 E i g e n v e c t o r s o f t h e M o d e l E q u a t i o n s . . . . . . . . . . . . . . 6 3

4 . 3 . 4 S o l u t i o n s o f t h e M o d e l O D E ' s . . . . . . . . . . . . . . . . . . 6 5

4 . 4 T h e R e p r e s e n t a t i v e E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . 6 7

4 . 5 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 8

5 F I N I T E - V O L U M E M E T H O D S 7 1

5 . 1 B a s i c C o n c e p t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2

5 . 2 M o d e l E q u a t i o n s i n I n t e g r a l F o r m . . . . . . . . . . . . . . . . . . . . 7 3

5 . 2 . 1 T h e L i n e a r C o n v e c t i o n E q u a t i o n . . . . . . . . . . . . . . . . 7 3

5 . 2 . 2 T h e D i u s i o n E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . 7 4

5 . 3 O n e - D i m e n s i o n a l E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . 7 4

5 . 3 . 1 A S e c o n d - O r d e r A p p r o x i m a t i o n t o t h e C o n v e c t i o n E q u a t i o n . 7 5

5 . 3 . 2 A F o u r t h - O r d e r A p p r o x i m a t i o n t o t h e C o n v e c t i o n E q u a t i o n . 7 7

5 . 3 . 3 A S e c o n d - O r d e r A p p r o x i m a t i o n t o t h e D i u s i o n E q u a t i o n . . 7 8

5 . 4 A T w o - D i m e n s i o n a l E x a m p l e . . . . . . . . . . . . . . . . . . . . . . 8 0


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5 . 5 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3

6 T I M E - M A R C H I N G M E T H O D S F O R O D E ' S 8 5

6 . 1 N o t a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6

6 . 2 C o n v e r t i n g T i m e - M a r c h i n g M e t h o d s t o O E ' s . . . . . . . . . . . . . 8 7

6 . 3 S o l u t i o n o f L i n e a r O E ' s W i t h C o n s t a n t C o e c i e n t s . . . . . . . . . 8 8

6 . 3 . 1 F i r s t - a n d S e c o n d - O r d e r D i e r e n c e E q u a t i o n s . . . . . . . . . 8 9

6 . 3 . 2 S p e c i a l C a s e s o f C o u p l e d F i r s t - O r d e r E q u a t i o n s . . . . . . . . 9 0

6 . 4 S o l u t i o n o f t h e R e p r e s e n t a t i v e O E ' s . . . . . . . . . . . . . . . . . 9 1

6 . 4 . 1 T h e O p e r a t i o n a l F o r m a n d i t s S o l u t i o n . . . . . . . . . . . . . 9 1

6 . 4 . 2 E x a m p l e s o f S o l u t i o n s t o T i m e - M a r c h i n g O E ' s . . . . . . . . 9 2

6 . 5 T h e ; R e l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3

6 . 5 . 1 E s t a b l i s h i n g t h e R e l a t i o n . . . . . . . . . . . . . . . . . . . . . 9 3

6 . 5 . 2 T h e P r i n c i p a l - R o o t . . . . . . . . . . . . . . . . . . . . . . . 9 5

6 . 5 . 3 S p u r i o u s - R o o t s . . . . . . . . . . . . . . . . . . . . . . . . . 9 5

6 . 5 . 4 O n e - R o o t T i m e - M a r c h i n g M e t h o d s . . . . . . . . . . . . . . . 9 6

6 . 6 A c c u r a c y M e a s u r e s o f T i m e - M a r c h i n g M e t h o d s . . . . . . . . . . . . 9 7

6 . 6 . 1 L o c a l a n d G l o b a l E r r o r M e a s u r e s . . . . . . . . . . . . . . . . 9 7

6 . 6 . 2 L o c a l A c c u r a c y o f t h e T r a n s i e n t S o l u t i o n ( e r

j j e r

!

) . . . . 9 8

6 . 6 . 3 L o c a l A c c u r a c y o f t h e P a r t i c u l a r S o l u t i o n ( e r

) . . . . . . . . 9 9

6 . 6 . 4 T i m e A c c u r a c y F o r N o n l i n e a r A p p l i c a t i o n s . . . . . . . . . . . 1 0 0

6 . 6 . 5 G l o b a l A c c u r a c y . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 1

6 . 7 L i n e a r M u l t i s t e p M e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . 1 0 2

6 . 7 . 1 T h e G e n e r a l F o r m u l a t i o n . . . . . . . . . . . . . . . . . . . . . 1 0 2

6 . 7 . 2 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 3

6 . 7 . 3 T w o - S t e p L i n e a r M u l t i s t e p M e t h o d s . . . . . . . . . . . . . . 1 0 5

6 . 8 P r e d i c t o r - C o r r e c t o r M e t h o d s . . . . . . . . . . . . . . . . . . . . . . . 1 0 6

6 . 9 R u n g e - K u t t a M e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 7

6 . 1 0 I m p l e m e n t a t i o n o f I m p l i c i t M e t h o d s . . . . . . . . . . . . . . . . . . . 1 1 0

6 . 1 0 . 1 A p p l i c a t i o n t o S y s t e m s o f E q u a t i o n s . . . . . . . . . . . . . . 1 1 0

6 . 1 0 . 2 A p p l i c a t i o n t o N o n l i n e a r E q u a t i o n s . . . . . . . . . . . . . . . 1 1 1

6 . 1 0 . 3 L o c a l L i n e a r i z a t i o n f o r S c a l a r E q u a t i o n s . . . . . . . . . . . . 1 1 2

6 . 1 0 . 4 L o c a l L i n e a r i z a t i o n f o r C o u p l e d S e t s o f N o n l i n e a r E q u a t i o n s . 1 1 5

6 . 1 1 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 7

7 S T A B I L I T Y O F L I N E A R S Y S T E M S 1 2 1

7 . 1 D e p e n d e n c e o n t h e E i g e n s y s t e m . . . . . . . . . . . . . . . . . . . . . 1 2 2

7 . 2 I n h e r e n t S t a b i l i t y o f O D E ' s . . . . . . . . . . . . . . . . . . . . . . . 1 2 3

7 . 2 . 1 T h e C r i t e r i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3

7 . 2 . 2 C o m p l e t e E i g e n s y s t e m s . . . . . . . . . . . . . . . . . . . . . . 1 2 3


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7 . 2 . 3 D e f e c t i v e E i g e n s y s t e m s . . . . . . . . . . . . . . . . . . . . . . 1 2 3

7 . 3 N u m e r i c a l S t a b i l i t y o f O E ' s . . . . . . . . . . . . . . . . . . . . . . 1 2 4

7 . 3 . 1 T h e C r i t e r i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 4

7 . 3 . 2 C o m p l e t e E i g e n s y s t e m s . . . . . . . . . . . . . . . . . . . . . . 1 2 5

7 . 3 . 3 D e f e c t i v e E i g e n s y s t e m s . . . . . . . . . . . . . . . . . . . . . . 1 2 5

7 . 4 T i m e - S p a c e S t a b i l i t y a n d C o n v e r g e n c e o f O E ' s . . . . . . . . . . . . 1 2 5

7 . 5 N u m e r i c a l S t a b i l i t y C o n c e p t s i n t h e C o m p l e x - P l a n e . . . . . . . . . 1 2 8

7 . 5 . 1 - R o o t T r a c e s R e l a t i v e t o t h e U n i t C i r c l e . . . . . . . . . . . 1 2 8

7 . 5 . 2 S t a b i l i t y f o r S m a l l t . . . . . . . . . . . . . . . . . . . . . . . 1 3 2

7 . 6 N u m e r i c a l S t a b i l i t y C o n c e p t s i n t h e C o m p l e x h P l a n e . . . . . . . . 1 3 5

7 . 6 . 1 S t a b i l i t y f o r L a r g e h . . . . . . . . . . . . . . . . . . . . . . . . 1 3 5

7 . 6 . 2 U n c o n d i t i o n a l S t a b i l i t y , A - S t a b l e M e t h o d s . . . . . . . . . . . 1 3 6

7 . 6 . 3 S t a b i l i t y C o n t o u r s i n t h e C o m p l e x h P l a n e . . . . . . . . . . . 1 3 7

7 . 7 F o u r i e r S t a b i l i t y A n a l y s i s . . . . . . . . . . . . . . . . . . . . . . . . 1 4 1

7 . 7 . 1 T h e B a s i c P r o c e d u r e . . . . . . . . . . . . . . . . . . . . . . . 1 4 1

7 . 7 . 2 S o m e E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 2

7 . 7 . 3 R e l a t i o n t o C i r c u l a n t M a t r i c e s . . . . . . . . . . . . . . . . . . 1 4 3

7 . 8 C o n s i s t e n c y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 3

7 . 9 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 4 6

8 C H O I C E O F T I M E - M A R C H I N G M E T H O D S 1 4 9

8 . 1 S t i n e s s D e n i t i o n f o r O D E ' s . . . . . . . . . . . . . . . . . . . . . . 1 4 9

8 . 1 . 1 R e l a t i o n t o - E i g e n v a l u e s . . . . . . . . . . . . . . . . . . . . 1 4 9

8 . 1 . 2 D r i v i n g a n d P a r a s i t i c E i g e n v a l u e s . . . . . . . . . . . . . . . . 1 5 1

8 . 1 . 3 S t i n e s s C l a s s i c a t i o n s . . . . . . . . . . . . . . . . . . . . . . 1 5 1

8 . 2 R e l a t i o n o f S t i n e s s t o S p a c e M e s h S i z e . . . . . . . . . . . . . . . . 1 5 2

8 . 3 P r a c t i c a l C o n s i d e r a t i o n s f o r C o m p a r i n g M e t h o d s . . . . . . . . . . . 1 5 3

8 . 4 C o m p a r i n g t h e E c i e n c y o f E x p l i c i t M e t h o d s . . . . . . . . . . . . . 1 5 4

8 . 4 . 1 I m p o s e d C o n s t r a i n t s . . . . . . . . . . . . . . . . . . . . . . . 1 5 4

8 . 4 . 2 A n E x a m p l e I n v o l v i n g D i u s i o n . . . . . . . . . . . . . . . . . 1 5 4

8 . 4 . 3 A n E x a m p l e I n v o l v i n g P e r i o d i c C o n v e c t i o n . . . . . . . . . . . 1 5 5

8 . 5 C o p i n g W i t h S t i n e s s . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 8

8 . 5 . 1 E x p l i c i t M e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 8

8 . 5 . 2 I m p l i c i t M e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 9

8 . 5 . 3 A P e r s p e c t i v e . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 0

8 . 6 S t e a d y P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 0

8 . 7 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 1


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9 R E L A X A T I O N M E T H O D S 1 6 3

9 . 1 F o r m u l a t i o n o f t h e M o d e l P r o b l e m . . . . . . . . . . . . . . . . . . . 1 6 4

9 . 1 . 1 P r e c o n d i t i o n i n g t h e B a s i c M a t r i x . . . . . . . . . . . . . . . . 1 6 4

9 . 1 . 2 T h e M o d e l E q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . 1 6 6

9 . 2 C l a s s i c a l R e l a x a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 8

9 . 2 . 1 T h e D e l t a F o r m o f a n I t e r a t i v e S c h e m e . . . . . . . . . . . . . 1 6 8

9 . 2 . 2 T h e C o n v e r g e d S o l u t i o n , t h e R e s i d u a l , a n d t h e E r r o r . . . . . 1 6 8

9 . 2 . 3 T h e C l a s s i c a l M e t h o d s . . . . . . . . . . . . . . . . . . . . . . 1 6 9

9 . 3 T h e O D E A p p r o a c h t o C l a s s i c a l R e l a x a t i o n . . . . . . . . . . . . . . 1 7 0

9 . 3 . 1 T h e O r d i n a r y D i e r e n t i a l E q u a t i o n F o r m u l a t i o n . . . . . . . . 1 7 0

9 . 3 . 2 O D E F o r m o f t h e C l a s s i c a l M e t h o d s . . . . . . . . . . . . . . 1 7 2

9 . 4 E i g e n s y s t e m s o f t h e C l a s s i c a l M e t h o d s . . . . . . . . . . . . . . . . . 1 7 3

9 . 4 . 1 T h e P o i n t - J a c o b i S y s t e m . . . . . . . . . . . . . . . . . . . . . 1 7 4

9 . 4 . 2 T h e G a u s s - S e i d e l S y s t e m . . . . . . . . . . . . . . . . . . . . . 1 7 6

9 . 4 . 3 T h e S O R S y s t e m . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 0

9 . 5 N o n s t a t i o n a r y P r o c e s s e s . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 2

9 . 6 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 7

1 0 M U L T I G R I D 1 9 1

1 0 . 1 M o t i v a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 1

1 0 . 1 . 1 E i g e n v e c t o r a n d E i g e n v a l u e I d e n t i c a t i o n w i t h S p a c e F r e q u e n c i e s 1 9 1

1 0 . 1 . 2 P r o p e r t i e s o f t h e I t e r a t i v e M e t h o d . . . . . . . . . . . . . . . 1 9 2

1 0 . 2 T h e B a s i c P r o c e s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 9 2

1 0 . 3 A T w o - G r i d P r o c e s s . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 0

1 0 . 4 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 2

1 1 N U M E R I C A L D I S S I P A T I O N 2 0 3

1 1 . 1 O n e - S i d e d F i r s t - D e r i v a t i v e S p a c e D i e r e n c i n g . . . . . . . . . . . . . 2 0 4

1 1 . 2 T h e M o d i e d P a r t i a l D i e r e n t i a l E q u a t i o n . . . . . . . . . . . . . . . 2 0 5

1 1 . 3 T h e L a x - W e n d r o M e t h o d . . . . . . . . . . . . . . . . . . . . . . . . 2 0 7

1 1 . 4 U p w i n d S c h e m e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0 9

1 1 . 4 . 1 F l u x - V e c t o r S p l i t t i n g . . . . . . . . . . . . . . . . . . . . . . . 2 1 0

1 1 . 4 . 2 F l u x - D i e r e n c e S p l i t t i n g . . . . . . . . . . . . . . . . . . . . . 2 1 2

1 1 . 5 A r t i c i a l D i s s i p a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 3

1 1 . 6 P r o b l e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 4

1 2 S P L I T A N D F A C T O R E D F O R M S 2 1 7

1 2 . 1 T h e C o n c e p t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 7

1 2 . 2 F a c t o r i n g P h y s i c a l R e p r e s e n t a t i o n s | T i m e S p l i t t i n g . . . . . . . . . 2 1 8

1 2 . 3 F a c t o r i n g S p a c e M a t r i x O p e r a t o r s i n 2 { D . . . . . . . . . . . . . . . . 2 2 0


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C T H E H O M O G E N E O U S P R O P E R T Y O F T H E E U L E R E Q U A T I O N S 2 6 5


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2 C H A P T E R 1 . I N T R O D U C T I O N

a r e t h e o n e s a c t u a l l y b e i n g s o l v e d b y t h e n u m e r i c a l p r o c e s s , a r e o f t e n r e f e r r e d t o a s

t h e m o d i e d p a r t i a l d i e r e n t i a l e q u a t i o n s . S i n c e t h e y a r e n o t p r e c i s e l y t h e s a m e a s

t h e o r i g i n a l e q u a t i o n s , t h e y c a n , a n d p r o b a b l y w i l l , s i m u l a t e t h e p h y s i c a l p h e n o m e n a

l i s t e d a b o v e i n w a y s t h a t a r e n o t e x a c t l y t h e s a m e a s a n e x a c t s o l u t i o n t o t h e b a s i c

p a r t i a l d i e r e n t i a l e q u a t i o n . M a t h e m a t i c a l l y , t h e s e d i e r e n c e s a r e u s u a l l y r e f e r r e d t o

a s t r u n c a t i o n e r r o r s . H o w e v e r , t h e t h e o r y a s s o c i a t e d w i t h t h e n u m e r i c a l a n a l y s i s o f

u i d m e c h a n i c s w a s d e v e l o p e d p r e d o m i n a n t l y b y s c i e n t i s t s d e e p l y i n t e r e s t e d i n t h e

p h y s i c s o f u i d o w a n d , a s a c o n s e q u e n c e , t h e s e e r r o r s a r e o f t e n i d e n t i e d w i t h a

p a r t i c u l a r p h y s i c a l p h e n o m e n o n o n w h i c h t h e y h a v e a s t r o n g e e c t . T h u s m e t h o d s a r e

s a i d t o h a v e a l o t o f \ a r t i c i a l v i s c o s i t y " o r s a i d t o b e h i g h l y d i s p e r s i v e . T h i s m e a n s

t h a t t h e e r r o r s c a u s e d b y t h e n u m e r i c a l a p p r o x i m a t i o n r e s u l t i n a m o d i e d p a r t i a l

d i e r e n t i a l e q u a t i o n h a v i n g a d d i t i o n a l t e r m s t h a t c a n b e i d e n t i e d w i t h t h e p h y s i c s

o f d i s s i p a t i o n i n t h e r s t c a s e a n d d i s p e r s i o n i n t h e s e c o n d . T h e r e i s n o t h i n g w r o n g ,

o f c o u r s e , w i t h i d e n t i f y i n g a n e r r o r w i t h a p h y s i c a l p r o c e s s , n o r w i t h d e l i b e r a t e l y

d i r e c t i n g a n e r r o r t o a s p e c i c p h y s i c a l p r o c e s s , a s l o n g a s t h e e r r o r r e m a i n s i n s o m e

e n g i n e e r i n g s e n s e \ s m a l l " . I t i s s a f e t o s a y , f o r e x a m p l e , t h a t m o s t n u m e r i c a l m e t h o d s

i n p r a c t i c a l u s e f o r s o l v i n g t h e n o n d i s s i p a t i v e E u l e r e q u a t i o n s c r e a t e a m o d i e d p a r t i a l

d i e r e n t i a l e q u a t i o n t h a t p r o d u c e s s o m e f o r m o f d i s s i p a t i o n . H o w e v e r , i f u s e d a n d

i n t e r p r e t e d p r o p e r l y , t h e s e m e t h o d s g i v e v e r y u s e f u l i n f o r m a t i o n .

R e g a r d l e s s o f w h a t t h e n u m e r i c a l e r r o r s a r e c a l l e d , i f t h e i r e e c t s a r e n o t t h o r -

o u g h l y u n d e r s t o o d a n d c o n t r o l l e d , t h e y c a n l e a d t o s e r i o u s d i c u l t i e s , p r o d u c i n g

a n s w e r s t h a t r e p r e s e n t l i t t l e , i f a n y , p h y s i c a l r e a l i t y . T h i s m o t i v a t e s s t u d y i n g t h e

c o n c e p t s o f s t a b i l i t y , c o n v e r g e n c e , a n d c o n s i s t e n c y . O n t h e o t h e r h a n d , e v e n i f t h e

e r r o r s a r e k e p t s m a l l e n o u g h t h a t t h e y c a n b e n e g l e c t e d ( f o r e n g i n e e r i n g p u r p o s e s ) ,

t h e r e s u l t i n g s i m u l a t i o n c a n s t i l l b e o f l i t t l e p r a c t i c a l u s e i f i n e c i e n t o r i n a p p r o p r i a t e

a l g o r i t h m s a r e u s e d . T h i s m o t i v a t e s s t u d y i n g t h e c o n c e p t s o f s t i n e s s , f a c t o r i z a t i o n ,

a n d a l g o r i t h m d e v e l o p m e n t i n g e n e r a l . A l l o f t h e s e c o n c e p t s w e h o p e t o c l a r i f y i n

t h i s b o o k .

1 . 2 B a c k g r o u n d

T h e e l d o f c o m p u t a t i o n a l u i d d y n a m i c s h a s a b r o a d r a n g e o f a p p l i c a b i l i t y . I n d e p e n -

d e n t o f t h e s p e c i c a p p l i c a t i o n u n d e r s t u d y , t h e f o l l o w i n g s e q u e n c e o f s t e p s g e n e r a l l y

m u s t b e f o l l o w e d i n o r d e r t o o b t a i n a s a t i s f a c t o r y s o l u t i o n .

1 . 2 . 1 P r o b l e m S p e c i c a t i o n a n d G e o m e t r y P r e p a r a t i o n

T h e r s t s t e p i n v o l v e s t h e s p e c i c a t i o n o f t h e p r o b l e m , i n c l u d i n g t h e g e o m e t r y , o w

c o n d i t i o n s , a n d t h e r e q u i r e m e n t s o f t h e s i m u l a t i o n . T h e g e o m e t r y m a y r e s u l t f r o m


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1 . 2 . B A C K G R O U N D 3

m e a s u r e m e n t s o f a n e x i s t i n g c o n g u r a t i o n o r m a y b e a s s o c i a t e d w i t h a d e s i g n s t u d y .

A l t e r n a t i v e l y , i n a d e s i g n c o n t e x t , n o g e o m e t r y n e e d b e s u p p l i e d . I n s t e a d , a s e t

o f o b j e c t i v e s a n d c o n s t r a i n t s m u s t b e s p e c i e d . F l o w c o n d i t i o n s m i g h t i n c l u d e , f o r

e x a m p l e , t h e R e y n o l d s n u m b e r a n d M a c h n u m b e r f o r t h e o w o v e r a n a i r f o i l . T h e

r e q u i r e m e n t s o f t h e s i m u l a t i o n i n c l u d e i s s u e s s u c h a s t h e l e v e l o f a c c u r a c y n e e d e d , t h e

t u r n a r o u n d t i m e r e q u i r e d , a n d t h e s o l u t i o n p a r a m e t e r s o f i n t e r e s t . T h e r s t t w o o f

t h e s e r e q u i r e m e n t s a r e o f t e n i n c o n i c t a n d c o m p r o m i s e i s n e c e s s a r y . A s a n e x a m p l e

o f s o l u t i o n p a r a m e t e r s o f i n t e r e s t i n c o m p u t i n g t h e o w e l d a b o u t a n a i r f o i l , o n e m a y

b e i n t e r e s t e d i n i ) t h e l i f t a n d p i t c h i n g m o m e n t o n l y , i i ) t h e d r a g a s w e l l a s t h e l i f t

a n d p i t c h i n g m o m e n t , o r i i i ) t h e d e t a i l s o f t h e o w a t s o m e s p e c i c l o c a t i o n .

1 . 2 . 2 S e l e c t i o n o f G o v e r n i n g E q u a t i o n s a n d B o u n d a r y C o n -

d i t i o n s

O n c e t h e p r o b l e m h a s b e e n s p e c i e d , a n a p p r o p r i a t e s e t o f g o v e r n i n g e q u a t i o n s a n d

b o u n d a r y c o n d i t i o n s m u s t b e s e l e c t e d . I t i s g e n e r a l l y a c c e p t e d t h a t t h e p h e n o m e n a o f

i m p o r t a n c e t o t h e e l d o f c o n t i n u u m u i d d y n a m i c s a r e g o v e r n e d b y t h e c o n s e r v a t i o n

o f m a s s , m o m e n t u m , a n d e n e r g y . T h e p a r t i a l d i e r e n t i a l e q u a t i o n s r e s u l t i n g f r o m

t h e s e c o n s e r v a t i o n l a w s a r e r e f e r r e d t o a s t h e N a v i e r - S t o k e s e q u a t i o n s . H o w e v e r , i n

t h e i n t e r e s t o f e c i e n c y , i t i s a l w a y s p r u d e n t t o c o n s i d e r s o l v i n g s i m p l i e d f o r m s

o f t h e N a v i e r - S t o k e s e q u a t i o n s w h e n t h e s i m p l i c a t i o n s r e t a i n t h e p h y s i c s w h i c h a r e

e s s e n t i a l t o t h e g o a l s o f t h e s i m u l a t i o n . P o s s i b l e s i m p l i e d g o v e r n i n g e q u a t i o n s i n c l u d e

t h e p o t e n t i a l - o w e q u a t i o n s , t h e E u l e r e q u a t i o n s , a n d t h e t h i n - l a y e r N a v i e r - S t o k e s

e q u a t i o n s . T h e s e m a y b e s t e a d y o r u n s t e a d y a n d c o m p r e s s i b l e o r i n c o m p r e s s i b l e .

B o u n d a r y t y p e s w h i c h m a y b e e n c o u n t e r e d i n c l u d e s o l i d w a l l s , i n o w a n d o u t o w

b o u n d a r i e s , p e r i o d i c b o u n d a r i e s , s y m m e t r y b o u n d a r i e s , e t c . T h e b o u n d a r y c o n d i t i o n s

w h i c h m u s t b e s p e c i e d d e p e n d u p o n t h e g o v e r n i n g e q u a t i o n s . F o r e x a m p l e , a t a s o l i d

w a l l , t h e E u l e r e q u a t i o n s r e q u i r e o w t a n g e n c y t o b e e n f o r c e d , w h i l e t h e N a v i e r - S t o k e s

e q u a t i o n s r e q u i r e t h e n o - s l i p c o n d i t i o n . I f n e c e s s a r y , p h y s i c a l m o d e l s m u s t b e c h o s e n

f o r p r o c e s s e s w h i c h c a n n o t b e s i m u l a t e d w i t h i n t h e s p e c i e d c o n s t r a i n t s . T u r b u l e n c e

i s a n e x a m p l e o f a p h y s i c a l p r o c e s s w h i c h i s r a r e l y s i m u l a t e d i n a p r a c t i c a l c o n t e x t ( a t

t h e t i m e o f w r i t i n g ) a n d t h u s i s o f t e n m o d e l l e d . T h e s u c c e s s o f a s i m u l a t i o n d e p e n d s

g r e a t l y o n t h e e n g i n e e r i n g i n s i g h t i n v o l v e d i n s e l e c t i n g t h e g o v e r n i n g e q u a t i o n s a n d

p h y s i c a l m o d e l s b a s e d o n t h e p r o b l e m s p e c i c a t i o n .

1 . 2 . 3 S e l e c t i o n o f G r i d d i n g S t r a t e g y a n d N u m e r i c a l M e t h o d

N e x t a n u m e r i c a l m e t h o d a n d a s t r a t e g y f o r d i v i d i n g t h e o w d o m a i n i n t o c e l l s , o r

e l e m e n t s , m u s t b e s e l e c t e d . W e c o n c e r n o u r s e l v e s h e r e o n l y w i t h n u m e r i c a l m e t h -

o d s r e q u i r i n g s u c h a t e s s e l l a t i o n o f t h e d o m a i n , w h i c h i s k n o w n a s a g r i d , o r m e s h .


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M a n y d i e r e n t g r i d d i n g s t r a t e g i e s e x i s t , i n c l u d i n g s t r u c t u r e d , u n s t r u c t u r e d , h y b r i d ,

c o m p o s i t e , a n d o v e r l a p p i n g g r i d s . F u r t h e r m o r e , t h e g r i d c a n b e a l t e r e d b a s e d o n

t h e s o l u t i o n i n a n a p p r o a c h k n o w n a s s o l u t i o n - a d a p t i v e g r i d d i n g . T h e n u m e r i c a l

m e t h o d s g e n e r a l l y u s e d i n C F D c a n b e c l a s s i e d a s n i t e - d i e r e n c e , n i t e - v o l u m e ,

n i t e - e l e m e n t , o r s p e c t r a l m e t h o d s . T h e c h o i c e s o f a n u m e r i c a l m e t h o d a n d a g r i d -

d i n g s t r a t e g y a r e s t r o n g l y i n t e r d e p e n d e n t . F o r e x a m p l e , t h e u s e o f n i t e - d i e r e n c e

m e t h o d s i s t y p i c a l l y r e s t r i c t e d t o s t r u c t u r e d g r i d s . H e r e a g a i n , t h e s u c c e s s o f a s i m -

u l a t i o n c a n d e p e n d o n a p p r o p r i a t e c h o i c e s f o r t h e p r o b l e m o r c l a s s o f p r o b l e m s o f

i n t e r e s t .

1 . 2 . 4 A s s e s s m e n t a n d I n t e r p r e t a t i o n o f R e s u l t s

F i n a l l y , t h e r e s u l t s o f t h e s i m u l a t i o n m u s t b e a s s e s s e d a n d i n t e r p r e t e d . T h i s s t e p c a n

r e q u i r e p o s t - p r o c e s s i n g o f t h e d a t a , f o r e x a m p l e c a l c u l a t i o n o f f o r c e s a n d m o m e n t s ,

a n d c a n b e a i d e d b y s o p h i s t i c a t e d o w v i s u a l i z a t i o n t o o l s a n d e r r o r e s t i m a t i o n t e c h -

n i q u e s . I t i s c r i t i c a l t h a t t h e m a g n i t u d e o f b o t h n u m e r i c a l a n d p h y s i c a l - m o d e l e r r o r s

b e w e l l u n d e r s t o o d .

1 . 3 O v e r v i e w

I t s h o u l d b e c l e a r t h a t s u c c e s s f u l s i m u l a t i o n o f u i d o w s c a n i n v o l v e a w i d e r a n g e o f

i s s u e s f r o m g r i d g e n e r a t i o n t o t u r b u l e n c e m o d e l l i n g t o t h e a p p l i c a b i l i t y o f v a r i o u s s i m -

p l i e d f o r m s o f t h e N a v i e r - S t o k e s e q u a t i o n s . M a n y o f t h e s e i s s u e s a r e n o t a d d r e s s e d

i n t h i s b o o k . S o m e o f t h e m a r e p r e s e n t e d i n t h e b o o k s b y A n d e r s o n , T a n n e h i l l , a n d

P l e t c h e r 1 ] a n d H i r s c h 2 ] . I n s t e a d w e f o c u s o n n u m e r i c a l m e t h o d s , w i t h e m p h a s i s

o n n i t e - d i e r e n c e a n d n i t e - v o l u m e m e t h o d s f o r t h e E u l e r a n d N a v i e r - S t o k e s e q u a -

t i o n s . R a t h e r t h a n p r e s e n t i n g t h e d e t a i l s o f t h e m o s t a d v a n c e d m e t h o d s , w h i c h a r e

s t i l l e v o l v i n g , w e p r e s e n t a f o u n d a t i o n f o r d e v e l o p i n g , a n a l y z i n g , a n d u n d e r s t a n d i n g

s u c h m e t h o d s .

F o r t u n a t e l y , t o d e v e l o p , a n a l y z e , a n d u n d e r s t a n d m o s t n u m e r i c a l m e t h o d s u s e d t o

n d s o l u t i o n s f o r t h e c o m p l e t e c o m p r e s s i b l e N a v i e r - S t o k e s e q u a t i o n s , w e c a n m a k e u s e

o f m u c h s i m p l e r e x p r e s s i o n s , t h e s o - c a l l e d \ m o d e l " e q u a t i o n s . T h e s e m o d e l e q u a t i o n s

i s o l a t e c e r t a i n a s p e c t s o f t h e p h y s i c s c o n t a i n e d i n t h e c o m p l e t e s e t o f e q u a t i o n s . H e n c e

t h e i r n u m e r i c a l s o l u t i o n c a n i l l u s t r a t e t h e p r o p e r t i e s o f a g i v e n n u m e r i c a l m e t h o d

w h e n a p p l i e d t o a m o r e c o m p l i c a t e d s y s t e m o f e q u a t i o n s w h i c h g o v e r n s s i m i l a r p h y s -

i c a l p h e n o m e n a . A l t h o u g h t h e m o d e l e q u a t i o n s a r e e x t r e m e l y s i m p l e a n d e a s y t o

s o l v e , t h e y h a v e b e e n c a r e f u l l y s e l e c t e d t o b e r e p r e s e n t a t i v e , w h e n u s e d i n t e l l i g e n t l y ,

o f d i c u l t i e s a n d c o m p l e x i t i e s t h a t a r i s e i n r e a l i s t i c t w o - a n d t h r e e - d i m e n s i o n a l u i d

o w s i m u l a t i o n s . W e b e l i e v e t h a t a t h o r o u g h u n d e r s t a n d i n g o f w h a t h a p p e n s w h e n


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1 . 4 . N O T A T I O N 5

n u m e r i c a l a p p r o x i m a t i o n s a r e a p p l i e d t o t h e m o d e l e q u a t i o n s i s a m a j o r r s t s t e p i n

m a k i n g c o n d e n t a n d c o m p e t e n t u s e o f n u m e r i c a l a p p r o x i m a t i o n s t o t h e E u l e r a n d

N a v i e r - S t o k e s e q u a t i o n s . A s a w o r d o f c a u t i o n , h o w e v e r , i t s h o u l d b e n o t e d t h a t ,

a l t h o u g h w e c a n l e a r n a g r e a t d e a l b y s t u d y i n g n u m e r i c a l m e t h o d s a s a p p l i e d t o t h e

m o d e l e q u a t i o n s a n d c a n u s e t h a t i n f o r m a t i o n i n t h e d e s i g n a n d a p p l i c a t i o n o f n u -

m e r i c a l m e t h o d s t o p r a c t i c a l p r o b l e m s , t h e r e a r e m a n y a s p e c t s o f p r a c t i c a l p r o b l e m s

w h i c h c a n o n l y b e u n d e r s t o o d i n t h e c o n t e x t o f t h e c o m p l e t e p h y s i c a l s y s t e m s .

1 . 4 N o t a t i o n

T h e n o t a t i o n i s g e n e r a l l y e x p l a i n e d a s i t i s i n t r o d u c e d . B o l d t y p e i s r e s e r v e d f o r r e a l

p h y s i c a l v e c t o r s , s u c h a s v e l o c i t y . T h e v e c t o r s y m b o l ~ i s u s e d f o r t h e v e c t o r s ( o r

c o l u m n m a t r i c e s ) w h i c h c o n t a i n t h e v a l u e s o f t h e d e p e n d e n t v a r i a b l e a t t h e n o d e s

o f a g r i d . O t h e r w i s e , t h e u s e o f a v e c t o r c o n s i s t i n g o f a c o l l e c t i o n o f s c a l a r s s h o u l d

b e a p p a r e n t f r o m t h e c o n t e x t a n d i s n o t i d e n t i e d b y a n y s p e c i a l n o t a t i o n . F o r

e x a m p l e , t h e v a r i a b l e u c a n d e n o t e a s c a l a r C a r t e s i a n v e l o c i t y c o m p o n e n t i n t h e E u l e r

a n d N a v i e r - S t o k e s e q u a t i o n s , a s c a l a r q u a n t i t y i n t h e l i n e a r c o n v e c t i o n a n d d i u s i o n

e q u a t i o n s , a n d a v e c t o r c o n s i s t i n g o f a c o l l e c t i o n o f s c a l a r s i n o u r p r e s e n t a t i o n o f

h y p e r b o l i c s y s t e m s . S o m e o f t h e a b b r e v i a t i o n s u s e d t h r o u g h o u t t h e t e x t a r e l i s t e d

a n d d e n e d b e l o w .

P D E P a r t i a l d i e r e n t i a l e q u a t i o n

O D E O r d i n a r y d i e r e n t i a l e q u a t i o n

O E O r d i n a r y d i e r e n c e e q u a t i o n

R H S R i g h t - h a n d s i d e

P . S . P a r t i c u l a r s o l u t i o n o f a n O D E o r s y s t e m o f O D E ' s

S . S . F i x e d ( t i m e - i n v a r i a n t ) s t e a d y - s t a t e s o l u t i o n

k - D k - d i m e n s i o n a l s p a c e

~

b c

B o u n d a r y c o n d i t i o n s , u s u a l l y a v e c t o r

O ( ) A t e r m o f o r d e r ( i . e . , p r o p o r t i o n a l t o )


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C h a p t e r 2

C O N S E R V A T I O N L A W S A N D

T H E M O D E L E Q U A T I O N S

W e s t a r t o u t b y c a s t i n g o u r e q u a t i o n s i n t h e m o s t g e n e r a l f o r m , t h e i n t e g r a l c o n s e r v a -

t i o n - l a w f o r m , w h i c h i s u s e f u l i n u n d e r s t a n d i n g t h e c o n c e p t s i n v o l v e d i n n i t e - v o l u m e

s c h e m e s . T h e e q u a t i o n s a r e t h e n r e c a s t i n t o d i v e r g e n c e f o r m , w h i c h i s n a t u r a l f o r

n i t e - d i e r e n c e s c h e m e s . T h e E u l e r a n d N a v i e r - S t o k e s e q u a t i o n s a r e b r i e y d i s c u s s e d

i n t h i s C h a p t e r . T h e m a i n f o c u s , t h o u g h , w i l l b e o n r e p r e s e n t a t i v e m o d e l e q u a t i o n s ,

i n p a r t i c u l a r , t h e c o n v e c t i o n a n d d i u s i o n e q u a t i o n s . T h e s e e q u a t i o n s c o n t a i n m a n y

o f t h e s a l i e n t m a t h e m a t i c a l a n d p h y s i c a l f e a t u r e s o f t h e f u l l N a v i e r - S t o k e s e q u a t i o n s .

T h e c o n c e p t s o f c o n v e c t i o n a n d d i u s i o n a r e p r e v a l e n t i n o u r d e v e l o p m e n t o f n u -

m e r i c a l m e t h o d s f o r c o m p u t a t i o n a l u i d d y n a m i c s , a n d t h e r e c u r r i n g u s e o f t h e s e

m o d e l e q u a t i o n s a l l o w s u s t o d e v e l o p a c o n s i s t e n t f r a m e w o r k o f a n a l y s i s f o r c o n s i s -

t e n c y , a c c u r a c y , s t a b i l i t y , a n d c o n v e r g e n c e . T h e m o d e l e q u a t i o n s w e s t u d y h a v e t w o

p r o p e r t i e s i n c o m m o n . T h e y a r e l i n e a r p a r t i a l d i e r e n t i a l e q u a t i o n s ( P D E ' s ) w i t h

c o e c i e n t s t h a t a r e c o n s t a n t i n b o t h s p a c e a n d t i m e , a n d t h e y r e p r e s e n t p h e n o m e n a

o f i m p o r t a n c e t o t h e a n a l y s i s o f c e r t a i n a s p e c t s o f u i d d y n a m i c p r o b l e m s .

2 . 1 C o n s e r v a t i o n L a w s

C o n s e r v a t i o n l a w s , s u c h a s t h e E u l e r a n d N a v i e r - S t o k e s e q u a t i o n s a n d o u r m o d e l

e q u a t i o n s , c a n b e w r i t t e n i n t h e f o l l o w i n g i n t e g r a l f o r m :

Z

V ( t

2

)

Q d V ;

Z

V ( t

1

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Q d V +

Z

t

2

t

1

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n : F d S d t =

Z

t

2

t

1

Z

V ( t )

P d V d t ( 2 . 1 )

I n t h i s e q u a t i o n , Q i s a v e c t o r c o n t a i n i n g t h e s e t o f v a r i a b l e s w h i c h a r e c o n s e r v e d ,

e . g . , m a s s , m o m e n t u m , a n d e n e r g y , p e r u n i t v o l u m e . T h e e q u a t i o n i s a s t a t e m e n t o f

7


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1 0 C H A P T E R 2 . C O N S E R V A T I O N L A W S A N D T H E M O D E L E Q U A T I O N S

t o d e r i v e t h e u x J a c o b i a n m a t r i c e s , w e m u s t r s t w r i t e t h e u x v e c t o r s E a n d F i n

t e r m s o f t h e c o n s e r v a t i v e v a r i a b l e s , q

1

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, a n d q

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F =

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W e h a v e a s s u m e d t h a t t h e p r e s s u r e s a t i s e s p = ( ; 1 ) e ; ( u

2

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i d e a l g a s l a w , w h e r e i s t h e r a t i o o f s p e c i c h e a t s , c

p

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v

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t h e u x J a c o b i a n o f E c a n b e w r i t t e n i n t e r m s o f t h e c o n s e r v a t i v e v a r i a b l e s a s

A =

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w h e r e

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2 . 6 . P R O B L E M S 1 9

w h e r e ! = 0 f o r i r r o t a t i o n a l o w . F o r i s e n t r o p i c a n d h o m e n t h a l p i c o w , t h e

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P D E ' s , w e h a v e

@ f ( q )

@ x

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w h e r e

q =

u

v

f =

; u

v

g =

; v

; u

O n e a p p r o a c h t o s o l v i n g t h e s e e q u a t i o n s i s t o a d d a t i m e - d e p e n d e n t t e r m a n d

n d t h e s t e a d y s o l u t i o n o f t h e f o l l o w i n g e q u a t i o n :

@ q

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( a ) F i n d t h e u x J a c o b i a n s o f f a n d g w i t h r e s p e c t t o q .

( b ) D e t e r m i n e t h e e i g e n v a l u e s o f t h e u x J a c o b i a n s .

( c ) D e t e r m i n e t h e c o n d i t i o n s ( i n t e r m s o f a n d u ) u n d e r w h i c h t h e s y s t e m i s

h y p e r b o l i c , i . e . , h a s r e a l e i g e n v a l u e s .

( d ) A r e t h e a b o v e u x e s h o m o g e n e o u s ? ( S e e A p p e n d i x C . )


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! " $

% & &

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13 2 ) 4 5 7 7 5 2 @ B D F D F H H P R T D B 5 2 Y ` 5 b H d 2 B 2 ` R 2 Y D H T h i q R B h q 5 @ t 5 R d 7 5 h H y H P R T D B 58 2 p Y

4 5 2 S D T B 2 T d D B T y h H d B b T D B b H Y @ B D F d H Y H 4 D D 5 5 D F Y T 4 H T 2 h D B 7 H 2 I H 4 T 2 T d 5 B 7 T D H

D F H Y H Y B 7 R y D T 2 I H 5 R Y y i T 2 h D F H 2 Y 5 y b H D F H d H Y R y D B 2 ` h B H d H 2 4 H H P R T D B 5 2 Y y D H d 2 T D B b H y i t

5 2 H 4 T 2 T d 5 B 7 T D H D F H Y T D B T y h H d B b T D B b H Y n d Y D t D F H d H i d 5 h R 4 B 2 ` T Y i Y D H 7 5 5 d

h B 2 T d i h B H d H 2 D B T y H P R T D B 5 2 Y F H D B 7 H h H d B b T D B b H Y T d H T d 5 B 7 T D H h# 2 I H D t y H T h B 2 ` D 5

T { } ~ { { } @ F B 4 F d 5 h R 4 H Y T Y H D 5 h B H d H 2 4 H H P R T D B 5 2 Y @ F B 4 F 4 TV 2 r H

Y 5 y b H h F B Y B Y D F H T d 5 T 4 F H 7 F T Y B H h F H d H 1 2 D F B Y 4 F T D H d t D F H 4 5 2 4 H D 5 n 2 B D H

h B H d HQ 2 p 4 H T d 5 B 7 T D B 5 2 Y D 5 T d D B T y h H d B b T D B b H Y B Y d H Y H 2 D H h F H Y H 4 T 2 H T y B H h

H B D F H d D 5 Y T D B T y h H d B b T D B b H Y 5 d D B 7 H h H d B b T D B b H Y R d H 7 F T Y B Y BX 2 D F B Y F T D H d B Y 5 2

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D F H Y H nU 2 I B D H h B H d H 2 4 H T d 5 B 7 T D B 5 2 Y @ B y y H h B Y 4 R Y Y H h BX 2 F T D H d

y y 5 D F H 7 T D H d B T y H y 5 @ B Y d H Y H 2 D H h B 2 T T d D H Y B T 2 Y i Y D H 7 H H 7 F T Y B H D F H

T 4 D D F T D P R B D H ` H 2 H d T y 4 y T Y Y H Y 5 7 H Y F H Y H d H Y Y H h B 2 ` H 2 H d T y 4 R d b B y B 2 H T d 4 5 5 d h B 2 T D H Y

BX 2 e } 4 T 2 H D d T 2 I Y 5 d 7 H h D 5 T RI 2 p B 5 d 7 T d D H Y B T 2 7 H Y F @ B D F H P R B Y T 4 H h

BX 2 o D H d b T y Y B 2 T Y 5 4 T y y H h { } t T Y Y F 5 @C 2 w BE 2 u B ` R d H F H {

} B Y R 2 B 5 d 7 T y y D F H ` H 5 7 H D d B 4 b T d B T D B 5 2 B Y T Y 5 d H h B 2 D 5 b T d B T y H 4 5 H 4 B H 2 D Y 5 D F H

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d T 4 D B 4 T y T y B 4 T D B 58 2

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4 5 7 BX 2 I T D B 5 2 Y 5 T 2 h 5 d e e

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y 5 @ H Y D 5 @ H d ` B b H Y D F H 5 d h H d 5 D F H 7 H D F 5 h d 5 7 P

t @ H Y H H D F T D D F H H d H Y Y B 5 2

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4 6 4 r

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