C o e x i s t e n c e o f t w o d i s s i p a - PowerPoint PPT Presentation

C o e x i s t e n c e o f t w o d i s s i p a t i v e me c h a n i s ms i n t w o - d i me n s i o n a l t u r b u l e n t f l o w s Romain Nguyen van yen 1 , Marie Farge 2 , Kai Schneider 3 1 FB Mathematik, Freie Universität Berlin and Humboldt Foundation, Berlin, Germany 2 LMD-CNRS-IPSL, École Normale Supérieure, Paris, France 3 M2P2-CNRS and CMI, Université de Provence, Marseille, France European Turbulence Conference, Warszawa, 14.09.2011

̶ ̶ ̶ ̶ Wh a t a r e w e l o o k i n g f o r ? ● T o s t u d y s o m e g e n e r i c r e l a x a t i o n p r o c e s s e s i n t u r b u l e n t f l o w s . ● T o y m o d e l : t h e 2 D p l a n a r C o u e t t e f l o w n o i n s t a b i l i t y , n o n o n - n o r m a l g r o w t h ( c o n s e r v e d v o r t i c i t y ) , c a n n o t s t u d y t r a n s i t i o n ( n o t e v e n b y p a s s ) , v o r t i c i t y p r o d u c t i o n o n l y n o n l i n e a r l y a t t h e b o u n d a r i e s , f e w r e s u l t s i n l i t e r a t u r e ( f o c u s h a s b e e n 3 D ) . Q u e s t i o n s : ● S t a t i s t i c a l e q u i l i b r i a ? ● Mo 2L m e n t u m t r a n s p o r t ? U -U ● T u r b u l e n t e n e r g y d i s s i p a t i o n ? L y M. Couette, Ann. Chim. Phys. ser 6 23 , p. 433 (1890)

̶ ̶ ̶ Wh a t a r e t h e e x p e c t e d s t a t e s ? ● L a m i n a r s t e a d y s t a t e : U -U ● A p p r o a c h t o K e l v i n ' s « C a t ' s e y e s » ? ( i n v i s c i d s t e a d y s t a t e s ) ● T u r b u l e n t s t e a d y s t a t e ( s ) ? ? N o l i n e a r g r o w t h m e c h a n i s m ! ! S o m m e r f e l d p a r a d o x . L o n g l i v e d t r a n s i e n t s ? U. Ehrenstein, M. Nagata, & F. Rincon, Phys. Fluids 20 , 064103 (2008) F. Rincon, Phys. Fluids 19 , 074105 (2007) Lord Kelvin, Nature 23 , p. 45 (1880) D. Holm, J. Marsden & T. Ratiu (1986)

N u me r i c a l mo d e l NSE(ν) → p e n a l i z a t i o n + v o r t i c i t y f o r m u l a t i o n + F o u r i e r d i s c r e t i z a t i o n PNSE(ν,η) → approximates Navier BC C. Navier, Mém. Acad. Sci. Inst. France 6 p. 389 (1823) P. Angot, C.-H. Bruneau, P. Fabrie, Numer. Math. 81 , p.497 (1999) KS & MF, PRL 95 , 244502 (2005) with slip length RNVY, MF, KS, PRL 106, 184502 (2011) at a precision O(η). RNVY, D. Kolomenskiy, KS, preprint (2011)

̶ ̶ Me t h o d ● K e e p i n i t i a l c o n d i t i o n f i x e d . ● T w o R e y n o l d s n u m b e r s : U b a s e d o n e x t e r n a l s h e a r v e l o c i t y U Re 0 b a s e d o n i n i t i a l R MS v e l o c i t y Re ● I n t h e f u t u r e : v a r y v i s c o s i t y . ● I n t h i s p r e s e n t a t i o n : a f e w e x a m p l e s . ● A l w a y s s u b s t r a c t l a m i n a r s h e a r p r o f i l e w h e n p r e s e n t i n g r e s u l t s . ( ! ! c a r e f u l l y c o r r e c t f o r p e n a l i z a t i o n e f f e c t t o h a v e s t a t i o n n a r y s o l u t i o n o f P N S E ) I n i t i a l v o r t i c i t y f l u c t u a t i o n s

R e s u l t s o v e r v i e w → d e c a y t o « c a t ' s e y e s » → s o l u t i o n r e m a i n s t u r b u l e n t , a n d t h e n ? ? t = 600

V o r t e x - w a l l i n t e r a c t i o n ● A t v a n i s h i n g v i s c o s i t y , v o r t e x - w a l l i n t e r a c t i o n s r e s u l t i n t h e p r o d u c t i o n o f i n t e n s e v o r t e x s h e e t s a n d s u b s e q u e n t l y e n e r g y d i s s i p a t i n g s t r u c t u r e s . ● T h i s i s b e t t e r s e e n w h e n c o n s i d e r i n g a s i m p l i f i e d d i p o l e - w a l l c o l l i s i o n : t = 0.36 Re = 985 Re = 1970 Re = 3940 Re = 7880 Z t = 0 t = 0.36 t → a t v a n i s h i n g v i s c o s i t y t h e p e r f e c t f l u i d m o d e l i s i n v a l i d ! A f i r s t me c h a n i s m o f d i s s i p a t i o n : mo l e c u l a r d i s s i p a t i o n a t s i n g u l a r i t i e s . L. Prandtl, ZAMM 1 , p.431 (1921) T. Kato in Sem. Nonlin. Par. Diff. Eq. (1984) RNVY, MF, KS, PRL 106, 184502 (2011) J.M. Burgers, Proc. KNAW 26 , p.582 (1923)

P r o d u c t i o n o f r a n d o mn e s s ● T h e w a l l p r o d u c e s l o c a l i z e d v o r t i c e s a s w e h a v e s e e n , b u t n e v e r t h e l e s s t h e f l o w m a i n t a i n s a r a n d o m a s p e c t . ● T h i s i s d u e t o n o n l i n e a r m i x i n g i n t h e b u l k . ● S a m e e f f e c t c a n b e s e e n i n 2 D h o m o g e n e o u s i s o t r o p i c - 3 t u r b u l e n c e a n d l e a d s t o k s p e c t r u m . ● H o w t o q u a n t i f y i t ?

H o w t o q u a n t i f y i t ? ● Wa v e l e t t o o l t o q u a n t i f y f l o w r a n d o m i z a t i o n : S c a l e - w i s e c o h e r e n t v o r t i c i t y e x t r a c t i o n ( S C V E ) INCOHERENT COHERENT ● S p l i t t h e w a v e l e t c o e f f i c i e n t s o f v o r t i c i t y COHERENT COHERENT i n t o t w o s e t s . ● T h e s e t s a r e d e f i n e d b y a t h r e s h o l d d e p e n d i n g o n s c a l e a n d d i r e c t i o n . INCOHERENT ● T h e t h r e s h o l d s a r e d e t e r m i n e d s e l f - c o n s i s t e n t l y b y : M. Farge, K. Schneider & N. Kevlahan, Phys. Fluids 11 , p. 2187 (1999) RNVY, MF, KS, Physica D doi:10.1016/j.physd.2011.05.022, in press.

R e s u l t s f o r 2 D H I T I n c o h e r e n t v o r t i c i t y C o h e r e n t v o r t i c i t y T o t a l v o r t i c i t y = + = +

R e s u l t s f o r 2 D C o u e t t e

P e r s p e c t i v e s

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P e r s p e c t i v e : mo d e l l i n g ● E x t e n s i v e s t u d y f o r v a r y i n g R e y n o l d s n u m b e r . ● D e s i g n a n d v a l i d a t e s t a t i s t i c a l m o d e l s f o r i n c o h e r e n t p a r t . S e l f - o r g a n i s a t i o n C o h e r e n t I n c o h e r e n t m o t i o n m o t i o n → d e t e r m i n i s t i c → s t a t i s t i c a l m o d e l m o d e l Mi x i n g « C a s c a d e » F o r c i n g b y w a l l s o r b o d y f o r c e s S i n g u l a r i t i e s H e a t

T h a n k y o u ! A n d t h a n k s t o R u p e r t K l e i n , D m i t r y K o l o m e n s k i y , C l a u d e B a r d o s , J e a n - Ma r c e l R a x a n d o t h e r s f o r s t i m u l a t i n g d i s c u s s i o n s ! C++ code download : http://justpmf.com/romain Papers : http://wavelets.ens.fr