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
SMART_READER_LITE
LIVE PREVIEW

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

Mar 22, 2023 299 likes •466 views

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

slide-1
SLIDE 1

C

  • e

x i s t e n c e

  • f

t w

  • d

i s s i p a t i v e me c h a n i s ms i n t w

  • d

i me n s i

  • n

a l t u r b u l e n t f l

  • w

s

Romain Nguyen van yen1, Marie Farge2, Kai Schneider3

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

slide-2
SLIDE 2

Wh a t a r e w e l

  • k

i n g f

  • r

?

  • T
  • s

t u d y s

  • m

e g e n e r i c r e l a x a t i

  • n

p r

  • c

e s s e s i n t u r b u l e n t f l

  • w

s .

  • T
  • y

m

  • d

e l : t h e 2 D p l a n a r C

  • u

e t t e f l

  • w

̶

n

  • i

n s t a b i l i t y , n

  • n
  • n
  • n
  • r

m a l g r

  • w

t h ( c

  • n

s e r v e d v

  • r

t i c i t y ) ,

̶

c a n n

  • t

s t u d y t r a n s i t i

  • n

( n

  • t

e v e n b y p a s s ) ,

̶

v

  • r

t i c i t y p r

  • d

u c t i

  • n
  • n

l y n

  • n

l i n e a r l y a t t h e b

  • 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

  • c

u s h a s b e e n 3 D ) .

U

  • U

2L Ly

Q u e s t i

  • n

s :

  • S

t a t i s t i c a l e q u i l i b r i a ?

  • Mo

m e n t u m t r a n s p

  • r

t ?

  • T

u r b u l e n t e n e r g y d i s s i p a t i

  • n

?

  • M. Couette, Ann. Chim. Phys. ser 6 23, p. 433 (1890)
slide-3
SLIDE 3

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 :

  • A

p p r

  • a

c h t

  • 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

  • l

i n e a r g r

  • w

t h m e c h a n i s m ! !

̶

S

  • m

m e r f e l d p a r a d

  • x

.

̶

L

  • n

g l i v e d t r a n s i e n t s ?

U

  • U
  • 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)
slide-4
SLIDE 4

N u me r i c a l mo d e l

  • 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) RNVY, MF, KS, PRL 106, 184502 (2011) RNVY, D. Kolomenskiy, KS, preprint (2011)

NSE(ν) PNSE(ν,η)

→ approximates Navier BC with slip length at a precision O(η).

→ p e n a l i z a t i

  • n

+ v

  • r

t i c i t y f

  • r

m u l a t i

  • n

+ F

  • u

r i e r d i s c r e t i z a t i

  • n
slide-5
SLIDE 5

Me t h

  • d
  • K

e e p i n i t i a l c

  • n

d i t i

  • n

f i x e d .

  • T

w

  • R

e y n

  • l

d s n u m b e r s :

̶

Re

U

b a s e d

  • n

e x t e r n a l s h e a r v e l

  • c

i t y U

̶

Re b a s e d

  • n

i n i t i a l R MS v e l

  • c

i t y

  • I

n t h e f u t u r e : v a r y v i s c

  • s

i t y .

  • I

n t h i s p r e s e n t a t i

  • 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

  • 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

  • r

r e c t f

  • r

p e n a l i z a t i

  • n

e f f e c t t

  • h

a v e s t a t i

  • n

n a r y s

  • l

u t i

  • n
  • f

P N S E )

I n i t i a l v

  • r

t i c i t y f l u c t u a t i

  • n

s

slide-6
SLIDE 6

R e s u l t s

  • v

e r v i e w

→ d e c a y t

  • «

c a t ' s e y e s » → s

  • l

u t i

  • 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

?

slide-7
SLIDE 7

Z

Re = 985 Re = 1970 Re = 3940 Re = 7880

t = 0.36

t

V

  • r

t e x

  • w

a l l i n t e r a c t i

  • n
  • A

t v a n i s h i n g v i s c

  • s

i t y , v

  • r

t e x

  • w

a l l i n t e r a c t i

  • n

s r e s u l t i n t h e p r

  • d

u c t i

  • n
  • f

i n t e n s e v

  • 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

  • n

s i d e r i n g a s i m p l i f i e d d i p

  • l

e

  • w

a l l c

  • l

l i s i

  • n

:

t = 0 t = 0.36

  • L. Prandtl, ZAMM 1, p.431 (1921)

J.M. Burgers, Proc. KNAW 26, p.582 (1923)

→ a t v a n i s h i n g v i s c

  • s

i t y t h e p e r f e c t f l u i d m

  • d

e l i s i n v a l i d !

  • T. Kato in Sem. Nonlin. Par. Diff. Eq. (1984)

RNVY, MF, KS, PRL 106, 184502 (2011)

A f i r s t me c h a n i s m

  • f

d i s s i p a t i

  • n

: mo l e c u l a r d i s s i p a t i

  • n

a t s i n g u l a r i t i e s .

slide-8
SLIDE 8

P r

  • d

u c t i

  • n
  • f

r a n d

  • mn

e s s

  • T

h e w a l l p r

  • d

u c e s l

  • c

a l i z e d v

  • 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

  • w

m a i n t a i n s a r a n d

  • m

a s p e c t .

  • T

h i s i s d u e t

  • n
  • 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

  • m
  • g

e n e

  • u

s i s

  • t

r

  • p

i c t u r b u l e n c e a n d l e a d s t

  • k
  • 3

s p e c t r u m .

  • H
  • w

t

  • q

u a n t i f y i t ?

slide-9
SLIDE 9

H

  • w

t

  • q

u a n t i f y i t ?

  • Wa

v e l e t t

  • l

t

  • q

u a n t i f y f l

  • w

r a n d

  • m

i z a t i

  • n

: S c a l e

  • w

i s e c

  • h

e r e n t v

  • r

t i c i t y e x t r a c t i

  • n

( S C V E )

  • 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.

INCOHERENT COHERENT COHERENT COHERENT INCOHERENT

  • S

p l i t t h e w a v e l e t c

  • e

f f i c i e n t s

  • f

v

  • r

t i c i t y i n t

  • t

w

  • 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

  • l

d d e p e n d i n g

  • n

s c a l e a n d d i r e c t i

  • n

.

  • T

h e t h r e s h

  • l

d s a r e d e t e r m i n e d s e l f

  • c
  • n

s i s t e n t l y b y :

slide-10
SLIDE 10

R e s u l t s f

  • r

2 D H I T

= = + +

T

  • t

a l v

  • r

t i c i t y C

  • h

e r e n t v

  • r

t i c i t y I n c

  • h

e r e n t v

  • r

t i c i t y

slide-11
SLIDE 11

R e s u l t s f

  • r

2 D C

  • u

e t t e

slide-12
SLIDE 12

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

slide-13
SLIDE 13

Mo me n t u m t r a n s p

  • r

t

slide-14
SLIDE 14

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

  • r

v a r y i n g R e y n

  • 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

  • d

e l s f

  • r

i n c

  • h

e r e n t p a r t .

C

  • h

e r e n t m

  • t

i

  • n

→ d e t e r m i n i s t i c m

  • d

e l I n c

  • h

e r e n t m

  • t

i

  • n

→ s t a t i s t i c a l m

  • d

e l

H e a t

S i n g u l a r i t i e s « C a s c a d e » Mi x i n g S e l f

  • r

g a n i s a t i

  • n

F

  • r

c i n g b y w a l l s

  • r

b

  • d

y f

  • r

c e s

slide-15
SLIDE 15

T h a n k y

  • u

!

A n d t h a n k s t

  • R

u p e r t K l e i n , D m i t r y K

  • l
  • m

e n s k i y , C l a u d e B a r d

  • s

, J e a n

  • Ma

r c e l R a x a n d

  • t

h e r s f

  • r

s t i m u l a t i n g d i s c u s s i

  • n

s !

C++ code download : http://justpmf.com/romain Papers : http://wavelets.ens.fr