Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric...

47
Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering 2 variations on a theme: z =− z 2

Transcript of Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric...

Page 1: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Lagrangian dynamics and statistical geometric structure of turbulence

Charles Meneveau, Yi Li & Laurent Chevillard

Mechanical Engineering

2 variations on a theme: z = −z2

Page 2: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Turbulent flow: multiscale

δu(Δ) = uL (x + ΔeL ) − uL (x)

Characterize velocity field at particular scale(filter out larger-scale advection): use velocity increments

Δ

Δ

All velocity component increments in all directions, at all scales:

Page 3: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Aij =

∂ %u j

∂xi

A11%A12

%A13

%A21%A22

%A23

%A31%A23

%A33

⎜⎜⎜

⎟⎟⎟

Filtered (coarsened) velocity gradient tensor at scale Δ:

E(k)

k

πΔ

FilteredSGS, subfilter

Aij =∂u j

∂xi

Unfiltered full velocity gradient tensor (Δ=0)

Page 4: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Restricted Euler dynamics in (inertial range of) turbulence:

∂ ˜ u j∂ t

+ ˜ u k∂ ˜ u j∂ x k

= −∂ ˜ p ∂ x j

+ ν∇ 2 ˜ u j −∂

∂ x kτ jk

• Filtered Navier-Stokes equations:

Restricted Euler: Vieillefosse, Phys. A, 125, 1985 Cantwell, Phys. Fluids A4, 1992

Filtered turbulence: Borue & Orszag, JFM 366, 1998 Van der Bos et al., Phys Fluids 14, 2002:

E(k)

k

πΔ

Filtered SGS, subfilter

Page 5: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Restricted Euler dynamics in (inertial range of) turbulence:

∂ ˜ u j∂ t

+ ˜ u k∂ ˜ u j∂ x k

= −∂ ˜ p ∂ x j

+ ν∇ 2 ˜ u j −∂

∂ x kτ jk

• Take gradient:

• Filtered Navier-Stokes equations:

Self-interaction Anisotropic Pressure

Hessian Anisotropic subgrid-scale + viscous effects

Restricted Euler: Vieillefosse, Phys. A, 125, 1985 Cantwell, Phys. Fluids A4, 1992

Filtered turbulence: Borue & Orszag, JFM 366, 1998 Van der Bos et al., Phys Fluids 14, 2002:

∂ %Aij

∂t+ %uk

∂ %Aij

∂xk

d %Aij

dt= − %Aik

%Akj −δ ij

3%Amk

%Akm⎛⎝⎜

⎞⎠⎟

+ −∂2 p

∂xi∂x j

+ 13

∂2 p∂xk∂xk

δ ij

⎝⎜

⎠⎟ −

∂2τ kjd

∂xi∂xk

−δ ij

3∂2τ kl

d

∂xl∂xk

⎝⎜⎞

⎠⎟+ ν∇2 %Aij

d %Aij

dt= − %Aik

%Akj −δ ij

3%Amk

%Akm⎛⎝⎜

⎞⎠⎟

+ Hij

Aij

E(k)

k

πΔ

Filtered SGS, subfilter Aij =

∂ %u j

∂xi

Page 6: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Cayley-Hamilton Theorem

˜ A jid ˜ A ijdt

= ˜ A ji( ˜ A ik ˜ A kj − 13

˜ A mk˜ A kmδ ij) →

dQΔ

dt= −3RΔ

˜ A jk ˜ A ki

d ˜ A ijdt

= ˜ A jk ˜ A ki( ˜ A ik ˜ A kj − 13

˜ A mk˜ A kmδij ) →

dRΔ

dt=

23

QΔ2

• Invariants (Cantwell 1992): QΔ ≡ −

12

%Aki%Aik

RΔ ≡ −

13

%Akm%Amn

%Ank

Aik AknAnj + PAik Akj + QAij + Rδ ij = 0

Restricted Euler dynamics in (inertial range of) turbulence:Hij = 0

Page 7: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Cayley-Hamilton Theorem

˜ A jid ˜ A ijdt

= ˜ A ji( ˜ A ik ˜ A kj − 13

˜ A mk˜ A kmδ ij) →

dQΔ

dt= −3RΔ

˜ A jk ˜ A ki

d ˜ A ijdt

= ˜ A jk ˜ A ki( ˜ A ik ˜ A kj − 13

˜ A mk˜ A kmδij ) →

dRΔ

dt=

23

QΔ2

• Invariants (Cantwell 1992): QΔ ≡ −

12

%Aki%Aik

RΔ ≡ −

13

%Akm%Amn

%Ank

Aik AknAnj + PAik Akj + QAij + Rδ ij = 0

Remarkable projection (decoupling)!

Restricted Euler dynamics in (inertial range of) turbulence:Hij = 0

More literature:Equations for all 5 invariants:

Martin, Dopazo & Valiño (Phys. Fluids, 1998)Equations for eigenvalues, and higher-dimensional versions:

Liu & Tadmor (Commun. Math. Phys., 2002)

Page 8: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Cayley-Hamilton Theorem

˜ A jid ˜ A ijdt

= ˜ A ji( ˜ A ik ˜ A kj − 13

˜ A mk˜ A kmδ ij) →

dQΔ

dt= −3RΔ

˜ A jk ˜ A ki

d ˜ A ijdt

= ˜ A jk ˜ A ki( ˜ A ik ˜ A kj − 13

˜ A mk˜ A kmδij ) →

dRΔ

dt=

23

QΔ2

• Invariants (Cantwell 1992):

• Singularity in finite time, but

- Predicts preference for axisymmetric extension

- Predicts alignment of vorticity with intermediate eigenvector of S: βs

Analytical solution:

QΔ ≡ −

12

%Aki%Aik

RΔ ≡ −

13

%Akm%Amn

%Ank

Aik AknAnj + PAik Akj + QAij + Rδ ij = 0

Restricted Euler dynamics in (inertial range of) turbulence:Hij = 0

Remarkable projection (decoupling)!

More literature:Equations for all 5 invariants:

Martin, Dopazo & Valiño (Phys. Fluids, 1998)Equations for eigenvalues, and higher-dimensional versions:

Liu & Tadmor (Commun. Math. Phys., 2002)

From: Cantwell, Phys. Fluids 1992)

Page 9: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Topic 1: Are there other useful “trace-dynamics” simplifications (other than Q & R)

Topic 2: Stochastic Lagrangian model for evolution of Aij

REST OF TALK:

Page 10: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

ui (x + r) − %ui (x) = %Akirk + O(r2 )

δu ≡ %Akirk

ri

rlr

δv2 ≡ δ ij −

rirj

r2

⎛⎝⎜

⎞⎠⎟

%Akjrklr

⎣⎢

⎦⎥

2

Longitudinal

Transverse

See: Yi & Meneveau, Phys. Rev. Lett. 95, 164502, 2005, JFM 2006

Another simplifications: Velocity increments

r(t)

u(x) u(x + r)

ΔFixeddisplacement

δu(t) ≡ %A(t) : (r(t)r(t)) l = %Arrl

Aji =

∂ %ui

∂x j

Page 11: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

δu ≡ %Akirk

ri

rlr

Rate of change of longitudinal velocity increment, following the flow (both end-points in linearized flow):

Velocity increments: Lagrangian evolution

ddt

δu =ddt

%Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

Page 12: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

ddt

δu =ddt

%Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

=d %Aki

dtrkri

rlr

+ %Akidrk

dtri

rlr

+ %Akidri

dtrk

rlr

− 2 %Akirkri

r3

drdt

l

Rate of change of longitudinal velocity increment, following the flow (both end-points in linearized flow):

Velocity increments: Lagrangian evolution

r(t)

r(t + Δt)

δu ≡ %Akirk

ri

rlr

Page 13: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

dri

dt=

∂ %ui

∂xm

rm = %Amirm

d %Aij

dt= −( %Aik

%Akj − 1D

%Amk%Akmδ ij ) +Hij

Hmn =

∂2

∂xm∂xk

pδkn − τ knSGS + 2ν %Skn⎡⎣ ⎤⎦

⎝⎜⎞

⎠⎟

anisotropic

Rate of change of longitudinal velocity increment, following the flow (both end-points in linearized flow):

Velocity increments: Lagrangian evolution

ddt

δu =ddt

%Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

=d %Aki

dtrkri

rlr

+ %Akidrk

dtri

rlr

+ %Akidri

dtrk

rlr

− 2 %Akirkri

r3

drdt

l

δu ≡ %Akirk

ri

rlr

Page 14: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

dri

dt=

∂ %ui

∂xm

rm = %Amirm

d %Aij

dt= −( %Aik

%Akj − 1D

%Amk%Akmδ ij ) +Hij

Hmn =

∂2

∂xm∂xk

pδkn − τ knSGS + 2ν %Skn⎡⎣ ⎤⎦

⎝⎜⎞

⎠⎟

anisotropic

Rate of change of longitudinal velocity increment, following the flow (both end-points in linearized flow):

Velocity increments: Lagrangian evolution

ddt

δu =ddt

%Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

=d %Aki

dtrkri

rlr

+ %Akidrk

dtri

rlr

+ %Akidri

dtrk

rlr

− 2 %Akirkri

r3

drdt

l

δu ≡ %Akirk

ri

rlr

ddt

δu = −( %Akm%Ami − 1

D%Apq

%Aqpδ ki )rkri

rlr

+ %Aki%Amkrm

ri

rlr

+ %Aki%Amirm

rk

rlr

− 2 %Akirkri

r3

rm

rl %Apmrp + Hmn

rmrn

rlr

Page 15: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

dri

dt=

∂ %ui

∂xm

rm = %Amirm

d %Aij

dt= −( %Aik

%Akj − 1D

%Amk%Akmδ ij ) +Hij

Hmn =

∂2

∂xm∂xk

pδkn − τ knSGS + 2ν %Skn⎡⎣ ⎤⎦

⎝⎜⎞

⎠⎟

anisotropic

Rate of change of longitudinal velocity increment, following the flow (both end-points in linearized flow):

Velocity increments: Lagrangian evolution

ddt

δu =ddt

%Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

=d %Aki

dtrkri

rlr

+ %Akidrk

dtri

rlr

+ %Akidri

dtrk

rlr

− 2 %Akirkri

r3

drdt

l

δu ≡ %Akirk

ri

rlr

δv2 ≡ δ ij −rirj

r2

⎛⎝⎜

⎞⎠⎟

%Akjrklr

⎣⎢

⎦⎥

2

ddt

δu = −( %Akm%Ami − 1

D%Apq

%Aqpδ ki )rkri

rlr

+ %Aki%Amkrm

ri

rlr

+ %Aki%Amirm

rk

rlr

− 2 %Akirkri

r3

rm

rl %Apmrp + Hmn

rmrn

rlr

ddt

δu = δ ij −rirj

r2

⎛⎝⎜

⎞⎠⎟

%Akjrklr

⎣⎢

⎦⎥

21l

− %Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

2 1l

+ 1D

%Apq%Aqpl + Hmn

rmrn

r2 l

Page 16: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

dri

dt=

∂ %ui

∂xm

rm = %Amirm

d %Aij

dt= −( %Aik

%Akj − 1D

%Amk%Akmδ ij ) +Hij

Hmn =

∂2

∂xm∂xk

pδkn − τ knSGS + 2ν %Skn⎡⎣ ⎤⎦

⎝⎜⎞

⎠⎟

anisotropic

Rate of change of longitudinal velocity increment, following the flow (both end-points in linearized flow):

Velocity increments: Lagrangian evolution

ddt

δu =ddt

%Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

=d %Aki

dtrkri

rlr

+ %Akidrk

dtri

rlr

+ %Akidri

dtrk

rlr

− 2 %Akirkri

r3

drdt

l

δu ≡ %Akirk

ri

rlr

δv2 ≡ δ ij −rirj

r2

⎛⎝⎜

⎞⎠⎟

%Akjrklr

⎣⎢

⎦⎥

2

ddt

δu = −( %Akm%Ami − 1

D%Apq

%Aqpδ ki )rkri

rlr

+ %Aki%Amkrm

ri

rlr

+ %Aki%Amirm

rk

rlr

− 2 %Akirkri

r3

rm

rl %Apmrp + Hmn

rmrn

rlr

ddt

δu = δ ij −rirj

r2

⎛⎝⎜

⎞⎠⎟

%Akjrklr

⎣⎢

⎦⎥

21l

− %Akirkri

rlr

⎛⎝⎜

⎞⎠⎟

2 1l

+ 1D

%Apq%Aqpl + Hmn

rmrn

r2 l

ddt

δu =1l

δv2 − δu2( )+1D

%Apq%Aqpl + Hmn

rmri

r2 l

Page 17: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Velocity increments: Lagrangian evolution

ddt

δu =1l

δv2 − δu2( )+1D

%Apq%Aqpl + Hmn

rmri

r2 l

Tensor invariant (Q)Write A in frame formed by:

Arr Are Arn

Aer Aee Aen

Anr Ane −(Arr + Eee )

⎢⎢⎢

⎥⎥⎥

=δu δv 0? ? ?? ? −(δu + ?)

⎢⎢⎢

⎥⎥⎥

1l

Apq%Aqpl =

δu δv 0? ? ?? ? −(δu + ?)

⎢⎢⎢

⎥⎥⎥

δu δv 0? ? ?? ? −(δu + ?)

⎢⎢⎢

⎥⎥⎥

1l

= δu2 + [δu + ?]2 + ?+ ...( )1l

= 2δu2 1l

+ ?+ ...

and ? =0ddt

δu = − 1 −2D

⎛⎝⎜

⎞⎠⎟

δu2 + δv2 Hmn =

∂2

∂xm∂xk

pδkn − τ knSGS + 2ν %Skn⎡⎣ ⎤⎦

⎛⎝⎜

⎞⎠⎟

anisotropic

= 0

r =r(t)

r

e : en = δui δ in −rirn

r2⎛⎝⎜

⎞⎠⎟

1δv

n = r × e

Page 18: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

From a similar derivation for δv:

ddt

δv = −2l

δuδv

Velocity increments: Lagrangian evolution

From a similar derivation for δT, neglecting diffusion and SGS fluxes:

ddt

δT = −1l

δuδT

In 2D, vorticity is passive scalar, so for 2D:

ddt

δω z = −1l

δuδω z

Page 19: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Velocity increments: Lagrangian evolution

Set of equations:

ddt

δu = − 1−2D

⎛⎝⎜

⎞⎠⎟

δu2 + δv2

ddt

δv = −2δuδv

ddt

δT = −δuδT

ddt

δω z = −δuδω z (only for D=2)

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪

“Advected delta-vee system”

ddt

δu = −δu2

1-D inviscid Burgers:

ddt

α = −α 2 + χ 2

ddt

χ = −2αχ

Angles & vortex stretching(Galati, Gibbon et al, 1997Gibbon & Holm 2006….):

Page 20: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

ddt

δv = −2l

δu δv

Comparison with DNS,Lagrangian rate of change of velocity increments:

2563 DNS, filtered at 40η, Δ=40 η, evaluated δu, δv, and their Lagrangian rate of change of

velocity increments numericallyρ = 0.51

ρ = 0.61

ddt

δu = −13

δu2 + δv2⎛⎝⎜

⎞⎠⎟

1l

Page 21: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Evolution from Gaussian initial conditions:

Initial condition:

δu = Gaussian zero mean, unit variance

δv = δv12 + δv2

2

δvk = Gaussian zero mean, unit variance, k=1,2

= 1set

δT = Gaussian zero mean, unit variance

Page 22: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Technical point #1: Alignment bias correction factor(thanks to Greg Eyink for pointing out the need for a correction)

See: Yi & Meneveau, Phys. Rev. Lett. 95, 164502,

r(t) r(0), r(0) = l

DdΩ0 = r(t)D dΩ(t)

dΩ(t)dΩ0

=l

r(t)⎛⎝⎜

⎞⎠⎟

D

drdt

= δu(r,t) = δu ⋅rl

⎛⎝⎜

⎞⎠⎟

ddt

ln(r / l ) = δu / l

dΩ(t)dΩ0

= exp −Dl −1 δu(t ')dt '0

t

∫( ) P → P dΩ(t)dΩ0

Can be evaluated from advected delta-vee system

Page 23: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

δu

P u(δu)

-8 -6 -4 -2 0 2 4 6 810-6

10-4

10-2

100

(a)

δvcPc v(δ

v c)-8 -6 -4 -2 0 2 4 6 8

10-6

10-4

10-2

100

(b)

Numerical Results: PDFs in 3D

ddt

δu = −13

δu2 + δv2

ddt

δv = −2δuδv

⎨⎪⎪

⎩⎪⎪

D =3

time time

ddt

δu = − 1−2D

⎛⎝⎜

⎞⎠⎟

δu2 + δv2

ddt

δv = −2δuδv

ddt

δT = −δuδT

ddt

δω z = −δuδω z (only for D=2)

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪

δvc = δv cosθ

Pvc (δvc ) = 1

πPv (δv)

δv2 − δvc2

dδvδvc

Technical point #2:Individual component θ is random,

uniformly distributed in [0, 2π)

Longitudinal velocity: negative skewness Transverse velocity: stretched symmetric tails

Page 24: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Measured intermittency trends:

Longitudinal increment is skewed

Transverse velocity is more intermittent

10243 DNS(Gotoh et al. 2002)

2563 DNS

Page 25: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

δu

P u(δu)

-8 -6 -4 -2 0 2 4 6 810-6

10-4

10-2

100

(a)

δvcPc v(δ

v c)-8 -6 -4 -2 0 2 4 6 8

10-6

10-4

10-2

100

(b)

Numerical Results: PDFs in 2D

ddt

δu = δv2

ddt

δv = −2δuδv

⎨⎪⎪

⎩⎪⎪

D =2

ddt

δu = − 1−2D

⎛⎝⎜

⎞⎠⎟

δu2 + δv2

ddt

δv = −2δuδv

ddt

δT = −δuδT

ddt

δω z = −δuδω z (only for D=2)

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪

2-D turbulence: velocity increments NOT intermittent (near Gaussian)

Page 26: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Measured intermittency trends:

No intermittency in 2-D turbulence for velocity increments

DNS:

Bofetta et al.Phys. Rev. E, 2000

Paret & Tabeling, Phys. Fluids, 1998

Page 27: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Phase portraits in (δu, δv) phase space:

δuδv

-6 -4 -2 0 2 4 60

2

4

6

N = 2(a) D =2

δu

δv

-6 -4 -2 0 2 4 60

2

4

6

N = 3(b) D =3

δu

δv

-6 -4 -2 0 2 4 60

2

4

6

N = 4(c) D =4

δu

δv-6

-6

-4

-4

-2

-2

0

0

2

2

4

4

6

6

0 0

1 1

2 2

3 3

4 4

5 5

6 6

D =∞

U = δu2 +D

D + 2δv2⎛

⎝⎜⎞⎠⎟

δv2 / D−1

Invariant:

ddt

δu = − 1−2D

⎛⎝⎜

⎞⎠⎟

δu2 + δv2

ddt

δv = −2δuδv

⎨⎪⎪

⎩⎪⎪

Self-amplification:Skewness in δu

Cros

s-am

p lifi

c ati o

n :In

ter m

i tte n

cy in

δv

“For small initial δv (particles moving directly towards

each other), gradient can become arbitrarily large at

later times”

Page 28: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

δT, δvcP T

,Pc v

-8 -4 0 4 810-6

10-4

10-2

100

t

(a)

δT

P T(δT

)

-8 -4 0 4 810-6

10-4

10-2

100

Solid lines: δT

Dashed lines: δvc

Passive scalar

D =3

ddt

δu = − 1−2D

⎛⎝⎜

⎞⎠⎟

δu2 + δv2

ddt

δv = −2δuδv

ddt

δT = −δuδT

ddt

δω z = −δuδω z (only for D=2)

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪

ddt

δu = −13

δu2 + δv2

ddt

δv = −2δuδv

ddt

δT = −δuδT

⎪⎪⎪

⎪⎪⎪

Scalar increments MORE intermittent than transverse velocity, after initial transient

Page 29: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Measured intermittency trends:

Passive scalar transport:

Larger intermittency for scalar increments than for velocity increments

δT (l ) = T (x + l eL ) − T (x)

e.g. Antonia et al. Phys. Rev. A 1984, andWatanabe & Gotoh, NJP, 2004:

Page 30: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

δω

P ω(δ

ω)

-8 -4 0 4 810-6

10-4

10-2

100

(b)

Vorticity in 2D

D =2ddt

δu = − 1−2D

⎛⎝⎜

⎞⎠⎟

δu2 + δv2

ddt

δv = −2δuδv

ddt

δT = −δuδT

ddt

δω z = −δuδω z (only for D=2)

⎪⎪⎪⎪⎪⎪

⎪⎪⎪⎪⎪⎪

ddt

δu = δv2

ddt

δv = −2δuδv

ddt

δω z = −δuδω z

⎪⎪⎪

⎪⎪⎪

Recall: velocity in 2D

δu

P u(δu)

-8 -6 -4 -2 0 2 4 6 810-6

10-4

10-2

100

(a)

δvc

Pc v(δv c)

-8 -6 -4 -2 0 2 4 6 810-6

10-4

10-2

100

(b)

Vorticity in 2D is intermittent

Page 31: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Measured intermittency trends:

Intermittency in 2-D turbulence for vorticity increments

Paret & Tabeling, Phys. Rev. Lett., 1999

Page 32: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Conclusions so far:

• Non-Gaussian intermittency trends can be explained simply by “Burgers equation-like” dynamics where instead of 1-D we embed a 1-D direction and follow it in a Lagrangian fashion. Non-Gaussian PDFs evolve very quickly (0.3 turn-over time).

• However, for complete local information, we’d need to follow 3 “perpendicular” lines. The we would need

3 x 2 + 3 angles = 9

variables to be followed.

• Might as well stay with Aij …

Page 33: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Lagrangian Stochastic model for full velocity gradient tensor:

dAij

dt= −AiqAqj −

∂2 p∂xi∂x j

+ ν∂2Aij

∂xq∂xq

+ Wij

Self-stretching unclosed forcing

Aij (t)

Aij =∂u j

∂xi

L. Chevillard & CM, PRL 2006 (in press

Page 34: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering
Page 35: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

∂2 p∂xi∂x j

=∂Xp

∂xi

∂Xq

∂x j

∂2 p

∂Xp∂Xq

+∂

∂xi

∂Xq

∂x j

⎝⎜

⎠⎟

∂p∂Xq

∂2 p∂xi∂x j

≈∂Xp

∂xi

∂Xq

∂x j

∂2 p

∂Xp∂Xq

Focus on Lagrangian pressure field: p(X,t)

Change of variables:

time

ΔX

Δx

L. Chevillard & CM, PRL 2006 (in press):

3 main ingredients

Page 36: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

1. Proposed Pressure Hessian model: Assume that Lagrangian pressure Hessian is isotropic if t-τ is long enough for “memory loss”of dispersion process

Deformation tensor:

Cauchy-Green tensor:

Inverse:

Poisson constraint:

Dij =∂x j

∂Xi

Cij = Dik Djk

(C−1)ij =∂Xk

∂xi

∂Xk

∂x j

∂2 p∂xi∂x j

≈∂Xp

∂xi

∂Xq

∂x j

∂2 p

∂Xp∂Xq

∂2 p∂Xp∂Xq

≈13

∂2 p∂Xk∂Xk

δ pq

∂2 p∂xi∂x j

≈ (C−1)ij13

∂2 p∂Xk∂Xk

dAii

dt= −AiqAqi −

∂2 p∂xi∂xi

= 0

(C−1)ii 13

∂2 p∂Xk∂Xk

= −AiqAqi

∂2 p∂xi∂x j

= −(C−1)ij

(C−1)nn

ApqAqp

???

Equivalent to “tetrad model”(more formally derived)

Page 37: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

2. Proposed viscous Hessian model: Similar approach (Jeong & Girimaji, 2003)

Characteristic Lagrangian displacement after A’s Lagrangian correlation time scale (Kolm time):

ν∂2A

∂xi∂x j

≈∂Xp

∂xi

∂Xq

∂x j

ν∂2A

∂Xp∂Xq

ν∂2A

∂Xp∂Xq

≈ν

(δ X)2 A13

δ pq

δ X ~ (disp veloc) × (correl time) ~ u 'τ K ~ λ

ν∂2Aij

∂xm∂xm

≈ −(C−1)mm

3TAij

δX ~ Taylor microscale !

(ν / δ X 2 ) ~ ν / λ 2 ~ T −1 ν∂2A

∂Xp∂Xq

≈1T

A13

δ pq

Page 38: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

3. Short-time memory material deformation: (Markovianization)

Equation for deformation tensor:

dDdt

= DA

Formal Solution in terms of time-ordered exponential function:

D(t) = D(0) exp[A(t ')dt ']t 0

t

∏ = D(t − τ )dτ (t)

dτ (t) = exp[A(t ')dt ']t −τ

t

∏ ≈ exp[A(t)τ ]

where:

Short-time (Markovian) Cauchy-Green:

cτ (t) = exp[A(t)τ ]exp[AT (t)τ ]

Δt = τ

Two time-scales tested:

• Mean Kolmogorov time

• Local time: strain-rate from A:

τ = τ K = cT Re−1/2

τ = Γ(2SijSij )−1/2

Page 39: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Lagrangian stochastic model for A:

Set of 9 (8) coupled nonlinear stochastic ODE’s:

dA = −A2 +Tr(A2 )Tr(cτ

−1)cτ

−1 −Tr(cτ

−1)3T

A⎛

⎝⎜⎞

⎠⎟dt + dW

dW: white-in-time Gaussian forcing (trace-free-isotropic-covariance structure - unit variance (in units of T)

L. Chevillard & CM, PRL 2006 (in press):

Page 40: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Preferential voricity alignment Preferred strain-state(Tsinober, Ashurst et al..): (Lund & Rogers, 1994)

s* =−3 6αβγ

(α 2 + β 2 + γ 2 )3/2α ≥ β ≥ γ

α

γ β

Recall Phenomenology:

Page 41: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

DNS data: pearl-shape R-Q plane:

Local flow topology (Cantwell, 1992):

Q = −12

ApqAqp R = −13

ApqAqmAmp

Recall Phenomenology:

Page 42: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Statistical Intermittency (stretched PDFs) and anomalous scaling of moments

A11p : ReFL ( p) ⇒ A11

p ~ A112 FL ( p)/FL (2)

A12p : ReFT ( p)

F =A11

4

A112 2

(Sreenivasan & Antonia)

Recall Phenomenology:

Page 43: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Results & Comparison with DNS:

• DNS: 2563: Rλ = 150

• Model: τK/T = 0.1 , consistent with Yeung et al. (JoT 2006) at same Rλ

• Alignment of vorticity w=ε:(A-AT)/2 with S=(A+AT)/2 eigenvectors

Best alignment with intermediate

• PDF of strain-state parameter s*:prevalence of axisymmetric extension

Page 44: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Results & Comparison with DNS: R-Q diagram

• Shows “Viellefosse tail” also seen in Van derBos et al (2002) experimental data

• Good but not “perfect” agreement (better than previous tetrad model)

Page 45: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Intermittency (PDFs as function or Re or parameter Γ)

τ=τK:

• Deforms as funcion of Re (realistic), • Not very realistic at large Re(“Rλ“ > 200)

τ=Γ/(2S:S)1/2 :

• Quite realistic, • But model diverges if Γ < Γcrit

Page 46: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Intermittency (Relative anomalous scaling exponents:)

τ=τK:

• μL = 0.25 • μT = 0.36

• Skewness S ~ -0.35 to -0.5

τ=Γ/(2S:S)1/2 :

•μL = 0.25 • μT = 0.40

• Skewness S ~ -0.35 to -0.5

- - - Kolmogorov (1941), _______ Multifractal scaling (Nelkin 1991)

Page 47: Lagrangian dynamics and statistical geometric …Lagrangian dynamics and statistical geometric structure of turbulence Charles Meneveau, Yi Li & Laurent Chevillard Mechanical Engineering

Conclusions:• It appears that quite a bit (more than previously thought) about turbulence phenomenology may be understood from “local” - “self-stretching” terms “-z2” or “-A2”- here we have:

• (i) “projected” into special directions that simplify things• (ii) “modeled” regularizing terms to get stationary behavior for entire A

• Simple advected delta-vee system “explains” many qualitative intermittency trends from a very low-dimensional system of ODEs - long time behavior wrong….

• Statistically stationary system of 8 forced ODE’s has been proposed - derived from grad(Navier-Stokes) and using physically motivated models for pressure Hessian and viscous Hessian

• Model reproduces structural geometric features of turbulence (RQ, s*, alignments) and statistical intermitency measures such as long tails in PDFs, stronger intermittency in transverse directions, and anomalous relative scaling exponents (in a small range of Re).