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