Transcript Document 7884638

A LES-LANGEVIN MODEL
B. Dubrulle
Groupe Instabilite et Turbulence
CEA Saclay
Colls: R. Dolganov and J-P Laval
N. Kevlahan
E.-J. Kim
F. Hersant
J. Mc Williams
S. Nazarenko
P. Sullivan
J. Werne
IS IT SUFFICIENT TO KNOW BASIC
EQUATIONS?
Dissipation
scale
0.1 km
Granule
103 km
Waste of computational resources
Time-scale problem
Necessity of small scale parametrization
Solar
spot
3 10 4 km
Giant
convection
cell
2 10 5 km
Influence of decimated scales
Typical time at scale l:
2
l
t   l 3
u
Decimated scales (small scales) vary very rapidly
We may replace them by a noise with short time scale
u  u  u'
Dtu' i  Aiju' j j
i x,t  j x' ,t'    ij x, x'  t  t' 
Generalized Langevin equation
Obukhov Model
Simplest case
u0
Aij   ij ,   t
 ij x,x'   ij
No mean flow
Large isotropic friction
No spatial correlations
 3   3x 2 3x  u u2 
  3 
P(x ,u,t)  
 


2 exp
2
t
t 
2t   t
u  t
Gaussian velocities
x   2 / 3t 3/ 2
Richardson’s law
u  x 1/ 3
Kolmogorov’s spectra
LES: Langevin
Influence of decimated scales:
transport

x  u  u'
  u

 ( )u  ( )u'
Stochastic computation
kl  u'k u'l
ijk  u'i ku'j
t i  uk k i  k k ui   k kl l i  2 kil k l
Turbulent viscosity
AKA effect
Refined comparison
uÝ u  
Additive noise
Gaussianity
Weak intermittency
Iso-vorticity
True turbulence
Non-Gaussianité
Forte intermittence
Spectrum
PDF of increments
LES: Langevin
LOCAL VS NON-LOCAL INTERACTIONS
• Navier-Stokes equations : two types of triades
tu  u   u    p    u  f
NON-LOCAL
LOCAL
L
L
l
LOCAL VS NON-LOCAL TURBULENCE
NON-LOCAL TURBULENCE
E
tU  (U  )U  p  u    U
t    U     
U

Analogy with MHD equations:
small scale grow via « dynamo » effect
Conservation laws
In inviscid case
E
2
2
U

u

dx
 u   dx
  U   dx
Hm 
Hc
k
A PRIORI TESTS IN NUMERICAL SIMULATIONS
2D TURBULENCE
<<
Uu
uU
uu
Local
small/small
scales
Non-local
U U
Local large/ large scales
3D TURBULENCE
DYNAMICAL TESTS IN NUMERICAL
SIMULATIONS
2D
DNS
3D
DNS
2D
RDT
3D
RDT
THE RDT MODEL
Equation for large-scale velocity
t Ui  U j  j Ui   i P    Ui   j uiU j  ujUi  ui uj 
Linear stochastic inhomogeneous equation
(RDT)
Reynolds stresses
Equation for small scale velocity
t ui  U j  j ui  u j  jU i   i p   t  ui  f i
Turbulent viscosity
Forcing (energy cascade)
Computed (numerics) or prescribed (analytics)
THE FORCING
Iso-vorticity
Iso-force
1.2
10 7
PDF of
increments
Correlations
0.8
< F( t )F( t 0 ) >
10
1
6
10 5
P(x)
10 4
1000
0.6
0.4
0.2
100
0
10
-0.2
-0.4
1
-100
-50
0
x
50
100
0
0.05
0.1
t-t0
0.15
0.2
TURBULENT VISCOSITY
SES
DNS
2  2
t  Cv  q E(q)dq
5k
RDT
LANGEVIN EQUATION AND LAGRANGIAN SCHEME
t ui  U j  j ui   i p   t  ui  fi
Décomposition into wave packets
GT u   x,k    dx' f x  x' e ik ( x x') u x' 
k
x
Dt x  U
Dt k  U  k 
 k

Dt u   T k u  u  2 2 U  k  U  f
 k

2
The wave packet moves with the fluid
Its wave number is changed by shear
Its amplitude depends on forces
coupling (cascade)
friction
“multiplicative noise”
“additive noise”
COMPARISON DNS/SES
Fast numerical 2D simulation
Shear flow
Computational time
10 days
2 hours
QuickTime™ and a
BMP decompressor
are needed to see this picture.
QuickTime™ and a
GIF decompressor
are needed to see this picture.
DNS
Lagrangian
model
(Laval, Dubrulle, Nazarenko, 2000)
Hersant, Dubrulle, 2002
SES SIMULATIONS
Experiment
SES
Hersant, 2003
DNS
LANGEVIN MODEL: derivation
Equation for small scale velocity
t ui  U j  j ui  u j  jU i   i p   t  ui  f i
Turbulent viscosity
Forcing
10 7
10 6
PDF
10 5
10 4
P(x)
Isoforce
 u u  u u 
1000
100
10
1
-100
-50
0
x
50
100
LES: Langevin
Equation for Reynolds stress
 ij  ui uj  uiu j  ui u' j  u' i u j  u' i u' j
 uiu j  uiu j 
 j Lij 
 2 T Sij
with
i
   u   
    f   f  u 
t     
Advection
Distorsion
By non-local
interactions
Lij

t
Generalized Langevin equation
 
Forcing due
To cascade
LES: Langevin
Performances
Comparaison DNS: 384*384*384 et LES: 21*21*21
Spectrum
Intermittency
LES: Langevin
Performances (2)
1
Q  Sij S ji
2
1
R  Sij S jk Ski
3
Q vs R
s probability

s  3 6 2 2 2
   
LES: Langevin
THE MODEL IN SHEARED GEOMETRY
Basic equations
1
 tU   2  r r 2 ur u  U
r
Equation for mean profile
RDT equations
for fluctuations
with stochastic
forcing
 kr2 
kr k
Dtur  2 2   S ur  2u 1 2   T k 2 u  Fr
 k 
k
2k2
kr k
Dtu  2   S ur  2 2u  2  S ur   T k 2 u  F
k
k
2k kz
kr kz
Dtuz  2   Sur  2 2u   T k 2 uz  Fz
k
k
ANALYTICAL PREDICTIONS
Mean flow dominates
Fluctuations dominates
Low Re
2
3/ 2
G  1.46
Re
1  7/ 4
0.5 2
Re 2
G
1  3/ 2 ln(Re 2 )3/ 2
10
TORQUE IN TAYLOR-COUETTE
11
10 10
10 9
G
10 8
10 7
10 10
10 6
10 9
10 5
1000
Re
10 4
10 5
108
10
6
G
100
No adjustable parameter
10 7

10 6

10 5
10 4
Dubrulle and Hersant, 2002

100
1000
R
10 4
10 5