Lagrangian Dynamics and Statistical Geometry in Turbulence ...and Intermittency Laurent Chevillard † & Emmanuel L ´ ev ˆ eque † [email protected]& [email protected]† Laboratoire de Physique, ENS Lyon, CNRS, France Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.1/30
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Lagrangian Dynamics and Statistical Geometry in Turbulence
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Lagrangian Dynamics and StatisticalGeometry in Turbulence
Refined Similarity HypothesisLaurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.23/30
DNS comparisons (I)
• DNS 2563: Rλ = 150
• Model : τK/T = 0.1 (Consistent with Yeung et al. JoT 06)
0 10
1
2
3
4
MinIntMax
cos(ω,λi)
P
DNS
0 1
cos(ω,λi)
Model sgfqdgfdgfdgdsgfqdgfdgfdgd
Alignment of vorticity with eigenvectors ofstrain S
→ Preferential alignment
−1 0 10
1
2
s*
P
DNS
−1 0 1
s*
Model sgfqdgfdgfdgdsgfqdgfdgfdgd
PDF of rate of Strain s∗:
s∗ = −3√
6αβγ
(α2+β2+γ2)3/2
→ Preferential axisymmetric expansion
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.24/30
DNS comparisons (II)
Joint probability of R and Q
−1 0 1−1
0
1
R*
Q*
DNS
(a)
−1 0 1
R*
Model
(b)
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.25/30
DNS comparisons (III)
Focussing on Enstrophy-Dissipation dominated regions (see Chertkov et al. 99):
Q = − 12
Tr(A2) = 14
Enstrophy︷︸︸︷
|ω|2 − 12
Dissipation︷ ︸︸ ︷
Tr(S2)
−1
0
1
Q*
DNS
(a)
Model
En
strop
hy
(b)
−1 0 1−1
0
1
R*
Q*
(c)
−1 0 1
R*
Dissip
atio
n(d)
Conditional average:
〈|ω|2|R,Q〉P(R,Q)
〈Tr(S2)|R,Q〉P(R,Q)
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.26/30
DNS comparisons (IV)
R = − 13
Tr(A3) = − 14
Enstrophy Production︷ ︸︸ ︷
ωiSijωj − 13
Strain Skewness︷ ︸︸ ︷
Tr(S3)
−1
0
100.1
0.3
1
Q*
DNS
(a)
Model
Stra
in S
kew
ness
(b)
−1
0
100.3
0.6−0.02
Q*
(c)
En
strop
hy P
rod
uctio
n
(d)
−1 0 1−1
0
10
0.3
0.6
−0.02
R*
Q*
(e)
−1 0 1
R*
Tra
nsfe
r
(f)
Conditional average (Chertkov et al. 99):
〈−Tr(S3)|R,Q〉P(R,Q)
〈ωiSijωj |R,Q〉P(R,Q)
〈−Tr(AT A2)|R,Q〉P(R,Q)
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.27/30
DNS comparisons (V): Focussing on Pressure Hessian and Viscous effects
−1
−0.5
0
0.5
1Q
*DNS
(a)
Model
Restricte
d E
ule
r
(b)
−1
−0.5
0
0.5
1
Q*
(c) Pre
ssure
Hessia
n
(d)
−1
−0.5
0
0.5
1
Q*
(e)
Visco
us
(f)
−1 −0.5 0 0.5 1−1
−0.5
0
0.5
1
R*
Q*
(g)
−1 −0.5 0 0.5 1
↑ 0.2R*
Tota
l
(h)
Cond. average (van der Bos et al. 02):
⟨(
dR/dt
dQ/dt
)
RE
∣∣∣∣∣R,Q
⟩
P(R,Q)
⟨(
dR/dt
dQ/dt
)
PH
∣∣∣∣∣R,Q
⟩
P(R,Q)
⟨(
dR/dt
dQ/dt
)
Viscous
∣∣∣∣∣R,Q
⟩
P(R,Q)
⟨(
dR/dt
dQ/dt
)
Total
∣∣∣∣∣R,Q
⟩
P(R,Q)
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.28/30
Alignment of Vorticity with Pressure Hessian eigenvectors
Euler equations︷ ︸︸ ︷
Ohkitani (93), Gibbon et al. (97, 06, 07) ⇒
dωidt
= Sijωj Vorticity strechingd2ωidt2
= −P ijωj Ertel’s Theorem (42)
0 10
1
2
3
cos θ
PDNS
(a)
0 1
cos θ
Model
(b)
Alignments with "intermediate" eigenvector reproducedAlignments with "smallest" eigenvector NOT reproduced
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.29/30
Conclusions
• A
8 independent ODEs dA/dt=−A2−P+ν∆A︷ ︸︸ ︷
new stationary stochastic model for A including closures for• Pressure Hessian P• Velocity gradient Laplacian ν∆A
→ Physics of Recent deformation
• Well-known properties of turbulence (vorticity alignments, RQ-plane, skewness)well reproduced. Discrepancies in Enstrophy dominated regions.
• Prediction of Intermittency• quantitative agreement with
standard data• Transverse more intermittent
than Longitudinal• Dissipation and Enstrophy
scale the same0 2 4 6
0
0.5
1
1.5
2
Exp. Long.
Exp. Trans.
Mod. Pred. Long.
Mod. Pred. Trans.
She−Leveque
pζ p
Perspectives
Improving rotation -vorticity stretching dominated regionsReaching very high Re (see Biferale et al., PRL 98, 214501 (2007).)Modeling Subgrid -scale stress tensor (See Chevillard, Li, Eyink, Meneveau 07)
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.30/30