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.