Post on 25-Jan-2022
Lagrangian Dynamics and StatisticalGeometry in Turbulence
...and Intermittency
Laurent Chevillard† & Emmanuel Leveque†
laurent.chevillard@ens-lyon.fr & emmanuel.leveque@ens-lyon.fr
†Laboratoire de Physique, ENS Lyon, CNRS, France
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.1/30
3D Fluid Turbulence: Full velocity gradients
Direct Numerical Simulations(picture by Toschi)
kjhkjgfdfgdfsgdfgdfskjhflgjhfA =
∂ux∂x
∂ux∂y
∂ux∂z
∂uy
∂x
∂uy
∂y
∂uy
∂z∂uz∂x
∂uz∂y
∂uz∂z
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.2/30
3D Nature of Turbulence and Non Locality of Pressure
Acceleration︷︸︸︷ai =
dui
dt=
∂ui
∂t+
Recquire 3D︷ ︸︸ ︷
(u.∇)ui
︸ ︷︷ ︸
Anti-Correlated
=
nonlinear︷ ︸︸ ︷
−∇ip︸ ︷︷ ︸
non local
+ν∇2ui
Incompressibility ↔ Poisson equation: ∇2p = −tr(A2)
Velocity Gradient Tensor: A =
∂ux∂x
∂ux∂y
∂ux∂z
∂uy
∂x
∂uy
∂y
∂uy
∂z∂uz∂x
∂uz∂y
∂uz∂z
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.3/30
3D Nature of Turbulence and Non Locality of Pressure
Acceleration︷︸︸︷ai =
dui
dt=
∂ui
∂t+
Recquire 3D︷ ︸︸ ︷
(u.∇)ui
︸ ︷︷ ︸
Anti-Correlated
=
nonlinear︷ ︸︸ ︷
−∇ip︸ ︷︷ ︸
non local
+ν∇2ui
Incompressibility ↔ Poisson equation: ∇2p = −tr(A2)
→ Pressure Gradient: ∇ip(x) =∫
dy
Green fct.︷ ︸︸ ︷
Gi(|y − x|)︸ ︷︷ ︸
∼ 1
|y−x|2
tr(A2(y)
)Non Local!!
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.4/30
3D Nature of Turbulence and Non Locality of Pressure
Acceleration︷︸︸︷ai =
dui
dt=
∂ui
∂t+
Recquire 3D︷ ︸︸ ︷
(u.∇)ui
︸ ︷︷ ︸
Anti-Correlated
=
nonlinear︷ ︸︸ ︷
−∇ip︸ ︷︷ ︸
non local
+ν∇2ui
Incompressibility ↔ Poisson equation: ∇2p = −tr(A2)
→ Pressure Hessian : Pij = ∇ijp(x) =
Local-Isotropic︷ ︸︸ ︷
−tr(A2(x)
) δij
3+
Non Local-Anisotropic︷ ︸︸ ︷
P.V.∫
dy Gij(|y − x|)︸ ︷︷ ︸
∼ 1
|y−x|3
tr(A2(y)
)
See Ohkitani & Kishiba (PoF,95) and Majda, Bertozzi (CUP,01)
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.5/30
3D Nature of Turbulence and Non Locality of Pressure
Acceleration︷︸︸︷ai =
dui
dt=
∂ui
∂t+
Recquire 3D︷ ︸︸ ︷
(u.∇)ui
︸ ︷︷ ︸
Anti-Correlated
=
nonlinear︷ ︸︸ ︷
−∇ip︸ ︷︷ ︸
non local
+ν∇2ui
⇓∂
∂xj⇓
Time evolution of the velocity gradient tensor A =
∂ux∂x
∂ux∂y
∂ux∂z
∂uy
∂x
∂uy
∂y
∂uy
∂z∂uz∂x
∂uz∂y
∂uz∂z
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.6/30
3D Nature of Turbulence and Non Locality of Pressure
Acceleration︷︸︸︷ai =
dui
dt=
∂ui
∂t+
Recquire 3D︷ ︸︸ ︷
(u.∇)ui
︸ ︷︷ ︸
Anti-Correlated
=
nonlinear︷ ︸︸ ︷
−∇ip︸ ︷︷ ︸
non local
+ν∇2ui
⇓∂
∂xj⇓
d
dtAij = −AiqAqj
︸ ︷︷ ︸
self-stretching term
−
Pressure Hessian︷ ︸︸ ︷
∂2p
∂xi∂xj+ ν
∂2Aij
∂xq∂xq︸ ︷︷ ︸
Viscous term︸ ︷︷ ︸
need to be modeled
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.7/30
The Lagrangian evolution of the Eulerian velocity gradient tensor
See the review C. Meneveau, Lagrangian dynamics and models of the velocity gradienttensor in Turbulent flows, Ann. Rev. Fluid Mech. (2011)
Let Aij =∂ui
∂xjand
d
dt≡
∂
∂t+ uq
∂
∂xq
A =
Long11 Trans12 Trans13Trans21 Long22 Trans23Trans31 Trans32 Long33
Then, along a fluid trajectory (Léorat 75, Vieillefosse 82):
∇(Navier-Stokes) ⇒d
dtAij = −AiqAqj
︸ ︷︷ ︸
self-stretching term
−
Pressure Hessian︷ ︸︸ ︷
∂2p
∂xi∂xj+ ν
∂2Aij
∂xq∂xq︸ ︷︷ ︸
Viscous term︸ ︷︷ ︸
need to be modeled
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.8/30
Tracking Velocity Gradients along Lagrangian trajectories
DNS
• Yeung & Pope (89).
• Girimaji & Pope, J.F.M. (90).
• Pope & Chen , Phys. Fluids (90).
Experimental
• Zeff et al., Nature (2003).
• Luthi, Tsinober & Kinzelbach, J.F.M.(2005).
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.9/30
Statistical Intermittency and Geometry in turbulence
DNS Results Rλ = 150
−10 −5 0 5 10
−15
−10
−5
0
Long Trans
∂ju
i
ln P
Intermittency
• Non-Gaussianity
• Skewness• Anomalous scaling with Reynolds
number
−1 0 10
0.5
1
1.5
s*
P(s
*)
0 10
1
2
3
4
Min
Int.
Max
cos(ω,λi)
P[c
os(
ω,λ
i)]
Geometry
• Preferential alignment of vorticity
• Preferential axisymmetric expansion
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.10/30
The RQ plane - Local Topology
See Chong, Perry and Cantwell (90) and Cantwell (93)
λi = f(R,Q) ∈ C: Eigenvalues of A
• Second invariant: Q = − 12
Tr(A2) = 14
Enstrophy︷︸︸︷
|ω|2 − 12
Dissipation︷ ︸︸ ︷
Tr(S2)
• Third invariant: R = − 13
Tr(A3) = − 14
ωiSijωj︸ ︷︷ ︸
Enstrophy Production
− 13
Tr(S3)︸ ︷︷ ︸
Strain Skewness
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.11/30
Review of the Restricted Euler approximation (I)
ddtAij = −AiqAqj − ∂2p
∂xi∂xj+ ν
∂2Aij
∂xq∂xq
• Restricted Euler Dynamics (Léorat 75-Vieillefosse 84-Cantwell 92)∂2p
∂xi∂xj= −
δij3
Tr(A2) and ν = 0
d
dtA = −
(
A2 −δij
3Tr(A2)
)
→ Non stationary
• Evolution equations for Q and R:
dQ
dt= −3Rdhgghfgjh and dhgghfgjh
dR
dt=
2
3Q2
• 274R2(t) +Q3(t) is time invariant
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.12/30
Review of the Restricted Euler approximation (II)
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.13/30
Review of the Restricted Euler approximation (III)
ddtAij = −AiqAqj − ∂2p
∂xi∂xj+ ν
∂2Aij
∂xq∂xq
• Restricted Euler Dynamics (Léorat 75-Vieillefosse 84-Cantwell 92)∂2p
∂xi∂xj= −
δij3
Tr(A2) and ν = 0
d
dtA = −
(
A2 −δij
3Tr(A2)
)
→ Non stationary
• Finite time singularity (t∗) of A for any initial condition
• BUT, at t . t∗
• Second eigenvalue λ of S is positive→ Preferential axisymmetric expansion
• Vorticity gets aligned with the associated eigenvector uλ
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.14/30
Review of various models
ddtAij = −AiqAqj − ∂2p
∂xi∂xj+ ν
∂2Aij
∂xq∂xq
• Restricted Euler Dynamics (Vieillefosse 84-Cantwell 92)∂2p
∂xi∂xj= −
δij3
Tr(A2) and ν = 0 → Finite time singularity
• Lognormality of Pseudo-dissipation ϕ = Tr(AAT ) (Girimaji-Pope 90)→ Strong a-priori assumption
• Linear damping term (Martin et al. 98)∂2p
∂xi∂xj= −
δij3
Tr(A2) and ν ∂2A∂xq∂xq
= − 1τ
A → Finite time singularity
• Delta-vee system (Yi-Meneveau (05))Projection on Longitudinal δℓu and Transverse δℓv increments
Using the material Deformation (Cauchy-Green Tensor C)
• Tetrad’s model (Chertkov-Pumir-Shraiman 99)∂2p
∂xi∂xj= −
Tr(A2)
Tr(C−1)C−1
ij and ν = 0 → Non stationary
• Differential damping term (Jeong-Girimaji 03)∂2p
∂xi∂xj= −
δij3
Tr(A2) and ν ∂2A∂xq∂xq
= −Tr(C−1)
3τA
→ Non stationaryLaurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.15/30
Cauchy-Green Tensor: Tracking the volume deformation
dsfqds
x(t)X=x(t0)
dsfqds
Deformation gradient: Dij(t) =∂xi∂Xj
(t)
Dyn.: ddt
D = AD
↔ D(t) =
t∏
t0
eA(ξ)dξ
︸ ︷︷ ︸
Time ordered exponential
Cauchy-Green Tensor : C = DDT or C−1ij =
∂Xp
∂xi
∂Xp
∂xj
Non Stationary
• Monin-Yaglom 75
• Girimaji-Pope 90: DNS Rλ = 90
• Lüthi, Tsinober, Kinzelbach 05: Exp. Rλ = 50
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.16/30
Re-Interpretation of the Chertkov et al. Tetrad Model
lksdfqsg
Eulerian → Lagrangian
∂2p
∂xi∂xj≈
∂Xp
∂xi
∂Xq
∂xj
∂2p
∂Xp∂Xq
fq
hyp.:
Isotropic︷ ︸︸ ︷
∂2p
∂Xp∂Xq=
δpq3
∂2p∂Xm∂Xm
Poisson=
equation−δpq
Tr(A2)
Tr(C−1)
Chertkov, Pumir and Shraiman (99)
∂2p
∂xi∂xj= −
Tr(A2)
Tr(C−1)C−1
ij
Non Stationary
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.17/30
The Jeong et al. Lagrangian Linear Diffusion Model
Eulerian → Lagrangian
∂2A∂xm∂xm
=∂Xp
∂xm
∂Xq
∂xm
∂2A∂Xp∂Xq
hyp.: Linear damping in the Lagrangian frame ν ∂2A∂Xp∂Xq
= −δpq3
AΘ
ν∂2A
∂xm∂xm= −
Tr(C−1)
3ΘA → Θ ??
Jeong and Girimaji (03) Non Stationary
Chevillard-Meneveau (06)→Self-consistent time-scale estimation:
1
Θ∼ ν
δ2
δX2∼
ν
(distance traveled during τK)2∼
ν
λ2︸︷︷︸
Taylor
∼1
TIntegral time scale −1
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.18/30
Stationary Cauchy-Green Tensor
D(t) =∂x∂X
(t) =
t∏
t0
eA(ξ)dξ = dτ (t)︸ ︷︷ ︸
present
old history︷ ︸︸ ︷
D(t− τ)
Over a dissipative time scale τ =
τK Kolmogorov
1/√
Tr(2S2) Local
Eulerian frame
x(t)
Test Volume
C(t0)
x(
t0︷ ︸︸ ︷
t− τ)
C(t0 + τ)
= cτ (t)
with dτ (t) =
t∏
t−τ
eA(ξ)dξ ≈ e∫ tt−τ A(ξ)dξ
≈ eτA(t)
Recent deformation
Let cτ the stationary “Cauchy-Green" Tensor
cτ = dτdTτ
See Chevillard-Meneveau 06Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.19/30
A Stochastic model for the velocity gradient tensor
Chevillard-Meneveau, Physical Review Letters 97, 174501 (2 006)
dA =
−A2
︸ ︷︷ ︸
Self-stretching
+
Pressure Hessian︷ ︸︸ ︷
c−1τ
Tr(c−1τ )
Tr(A2) −Tr(c−1
τ )
3TA
︸ ︷︷ ︸
Viscous
dt+
Forcing︷︸︸︷
dW
• Simplest white-in-time Gaussian forcing→ Tracefree-Isotropic-Homogeneous-Unit variance
• Explicit Reynolds number Re dependence when τ = τK
• Fluctuating (Local) dissipative time scale τ = Γ(Re)/√
Tr(2S2)
Re effects →
Isotropization of Pressure HessianWeakening Viscous term
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.20/30
Prediction of Intermittency (I): Deformation of PDFs
Chevillard & Meneveau, C.R. Mécanique 335, 187 (2007).
−5 0 5
−2
0
2
4
6
8
Re
A11
log
10P
Longitudinal
−5 0 5A
12
Transverseτ = τK
• Continuous deformation ↔Intermittency
• At high Re → Not realistic
−5 0 5
−2
0
2
4
6
8
Re
A11
log
10P
Longitudinal
−5 0 5A
12
Transverseτ = Γ(Re)/
√
Tr(2S2)
• Continuous deformation ↔Intermittency
• Very realistic but Γ > Γc forregularization
Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.21/30
Prediction of Intermittency (II): Relative scalings
- - (K41 Monofractal ) and — Nelkin’s Multifractal predictions (Lognormal → c2)
0 1 2 3
0
2
4
6
8
10 (a)
p=3
p=4
p=5
p=6
S
ln<(A11
)2>
ln<
|A
11|
p>
1 2 3
2
4
6
8
10
12
14(b)
p=3
p=4
p=5
p=6
ln<(A12
)2>
ln<
|A
12|
p>
τ = τK
• c2Long ≈ 0.025 → µ ≈ 0.22
• c2Trans ≈ 0.040
• Skewness ≈ −0.35 → −0.5
−0.5 −0.4 −0.3 −0.2 −0.1 0
−0.5
0
0.5
1
1.5
2
2.5
3 (a)
p=3
p=4
p=5
p=6
S
ln<(A11
)2>
ln<
|A
11|
p>
0.2 0.3 0.4 0.5 0.6 0.7
1
2
3
4
5
6
(b)
p=3
p=4
p=5
p=6
ln<(A12
)2>
ln<
|A
12|
p>
τ = Γ(Re)/√
Tr(2S2)
• c2Long ≈ 0.025
• c2Trans ≈ 0.045
• Skewness ≈ −0.35 → −0.5
⇒ Robustness (Universality)Laurent Chevillard, Laboratoire de Physique de l’ENS Lyon, France – p.22/30
Prediction of Intermittency (III): Relative scalings of Measures
- - (K41) and — Nelkin-Meneveau’s Multifractal predictions (Lognormal → µ)
A =
Deformation︷︸︸︷
S + Ω︸︷︷︸
Rotation
Dissipation ǫ = Tr(S2) → µǫ
Enstrophy ζ = −Tr(Ω2) → µζ
Pseudodissipation ϕ = Tr(AAT ) → µϕ
2 3 4
5
10
15
20
p=2
p=3
p=4
ln<E>
ln<
Ep>
Dissipation
2 3 4
p=2
p=3
p=4
ln<E>
Enstrophy
2 3 4
p=2
p=3
p=4
ln<E/2>
Pseudodissipation
Conclusions :
Same intermittency︷ ︸︸ ︷
µǫ = µζ = µϕ = µ ≈ 0.25 ≈ cLong2 × 9︸ ︷︷ ︸
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