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Lagrangian dynamics and statistical geometric structure of turbulence
Charles Meneveau, Yi Li & Laurent Chevillard
Mechanical Engineering
2 variations on a theme: z = −z2
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:
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)
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
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
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
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)
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)
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:
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
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
⎛⎝⎜
⎞⎠⎟
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
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
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
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
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
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
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
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….):
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
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
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
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
Measured intermittency trends:
Longitudinal increment is skewed
Transverse velocity is more intermittent
10243 DNS(Gotoh et al. 2002)
2563 DNS
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)
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
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”
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
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:
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
Measured intermittency trends:
Intermittency in 2-D turbulence for vorticity increments
Paret & Tabeling, Phys. Rev. Lett., 1999
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 …
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
∂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
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)
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
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
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):
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:
DNS data: pearl-shape R-Q plane:
Local flow topology (Cantwell, 1992):
Q = −12
ApqAqp R = −13
ApqAqmAmp
Recall Phenomenology:
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:
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
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)
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
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)
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).