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Jörg Schumacher Dept. of Mechanical Engineering, Technische Universität
Ilmenau, Germany
Local dissipation scales in turbulence
Collaborators
Katepalli R. Sreenivasan (ICTP Trieste) Victor Yakhot (Boston University)
Outline
How can local dissipation scales be defined and determined?
What is their impact on the physics in the inertial range of turbulence?
Non-premixed turbulent combustion
Fuel
Air
PνOνFν P2OF 2
AirF
F
mm
mZ
ZDZuρt
Zρ
1Z0
stZZ
1Z
0Z
Example: Jet diffusion flame
F=CH4: Zst=0.055
Laser diagnostics(Jeffrey A. Sutton, PhD thesis, U of Michigan 2005)
22Dln Z Z
stZ
Local dissipation scales
Kolmogorov length
Paladin & Vulpiani (1987), Frisch & Vergassola (1991): Intermediate dissipation range (IDR) spanned by (h)
Chevillard et al. (2005): Rapid increase of F(rv) between -
and +
Schumacher et al. (2005):
1/4
3/4
Kxε
νxηη
1/4
3/4
Kε
νη
Why high-resolution DNS?
Spectral resolution larger by a factor of 8 compared to standard case
64R λ
Dynamical definition of dissipation scale
1
ν
xuδxηxRe η
η
xuδ
νxη
η
Re,ηxηQ K
Re,ηxηQ K
Finest local dissipation scales
Finest dissipation scales Energy dissipation maxima
K
K
η3
4η
ηη
ε4ε
ε10ε
Theoretical prediction for Q(Yakhot, Physica D 2006)
udδuδPuδ2uδ0
2p2p
dpuδuδuδπi
2uδP 2p
i
i
2p
Mellin transform
2pξ
2pL
2p
Lσ !!12puδ
uδ
νη
η
dxexπ
2 !!12p
2x
0
2p1p
dx
ηLlnb4
LηRex2lnx-exp
ηLlnbηπ
2ηQ
0
1a22
Saddle point approximation
24bp2ap2pξ
Comparison with DNS
Qualitative agreement between DNS and theoretical model
Local scales and anomalous scaling(Hill, J.Fluid Mech. 2002; Yakhot, J. Fluid Mech. 2003)
2-2nr2,22n2n,0
2n.0 uδa12nSr
1d12nS
r
1-d
r
S
unclosed term 22nr
3η
3η uδuδ
ν
12n
ν
uδa
mr
nrn.m vδuδrS
v(x+r)
u(x+r)
v(x)
u(x)
Local scales and anomalous scaling(Hill, J.Fluid Mech. 2002; Yakhot, J. Fluid Mech. 2003; Gotoh & Nakano, J. Stat. Phys. 2003)
2-2nr2,22n2n,0
2n.0 uδa12nSr
1d12nS
r
1-d
r
S
unclosed term 22nr
3η
3η uδuδ
ν
12n
ν
uδa
12n2n ξ12nL
ξ2nL1,02n2n,0
L
η
ν
σ
L
η
η
σ
ν
ηS
η
ηS
for r→
mr
nrn.m vδuδrS
v(x+r)
u(x+r)
v(x)
u(x)
1ξξ12n
12n2nLReη
Exponents for velocity derivatives(Yakhot & Sreenivasan, J. Stat. Phys 2005)
1ξξ
ξn
n4n4n
n
4nη
2n
ηnn 14n4n
4n
Re~ν
ηS
ν
uδ
η
uδνε
uδ
νη
η
1ξξ12n
12n2nLReη
3/42n2n LReη
3
2nξ
12n LReη:n
1ξξ
ξn
n2n2n
n
2nη
n
η 12n2n
2n
Re~ν
ηS
ν
uδ
η
uδ
x
u
n
Scaling of velocity gradient moments
114nξ4nξ
4nξndn
Theory
0.157
0.489
0.944
13ξ2ξ
2ξ1ρ1
706.0ξ2
0.465
High-Re experiments: 0.71 (Benzi et al., PRE 1993)
Outlook: Far-dissipation range Kraichnan J. Fluid Mech.1959 Chen, Doolen, Herring, Kraichnan, Orszag & She, Phys. Rev. Lett. 1993
Kraichnan (1959): Universal behavior ~(k/kd)3 exp(-11k/kd)
Reynolds number dependence in high-Schmidt number mixing
Summary
Local dissipation scales are defined in a dynamical content.
Velocity gradient statistics on Kolmogorov and sub-Kolmogorov scales leads to asymptotic scaling exponents for velocity increment statistics on super-Kolmogorov scales.
Numerical effort has to go into the correct resolution of finest scales or strongest gradients.
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