1. Jörg Schumacher Dept. of Mechanical Engineering, Technische Universität Ilmenau, Germany Local dissipation scales in turbulence

2. Collaborators  Katepalli R. Sreenivasan (ICTP Trieste)  Victor Yakhot (Boston University)

3. Outline  How can local dissipation scales be defined and determined?  What is their impact on the physics in the inertial range of turbulence?

4. Non-premixed turbulent combustion Fuel Air Example: Jet diffusion flame F=CH 4 : Z st =0.055

5. Laser diagnostics (Jeffrey A. Sutton, PhD thesis, U of Michigan 2005)

6. 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(  r v) between  - and  +  Schumacher et al. (2005):

7. Why high-resolution DNS? Spectral resolution larger by a factor of 8 compared to standard case

8. Dynamical definition of dissipation scale

9. Finest local dissipation scales Finest dissipation scalesEnergy dissipation maxima

11. Theoretical prediction for Q  (Yakhot, Physica D 2006) Mellin transform Saddle point approximation

12. Comparison with DNS Qualitative agreement between DNS and theoretical model

13. Local scales and anomalous scaling (Hill, J.Fluid Mech. 2002; Yakhot, J. Fluid Mech. 2003) unclosed term v(x+r) u(x+r) v(x) u(x)

14. Local scales and anomalous scaling (Hill, J.Fluid Mech. 2002; Yakhot, J. Fluid Mech. 2003; Gotoh & Nakano, J. Stat. Phys. 2003) unclosed term for r→  v(x+r) u(x+r) v(x) u(x)

15. Exponents for velocity derivatives (Yakhot & Sreenivasan, J. Stat. Phys 2005)

16. Scaling of velocity gradient moments Theory 0.157 0.489 0.944 0.465 High-Re experiments: 0.71 (Benzi et al., PRE 1993)

17. Outlook: Far-dissipation range Kraichnan J. Fluid Mech.1959 Chen, Doolen, Herring, Kraichnan, Orszag & She, Phys. Rev. Lett. 1993 Kraichnan (1959): Universal behavior ~(k/k d ) 3 exp(-11k/k d ) Reynolds number dependence in high-Schmidt number mixing

18. 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.