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Local Dissipation Scales in Turbulence: Definition, Impact, and Dynamics

Explore the definition, impact, and dynamics of local dissipation scales in turbulence, including examples and theoretical predictions. Learn about Kolmogorov length, energy dissipation maxima, and anomalous scaling in velocity gradients.

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Local Dissipation Scales in Turbulence: Definition, Impact, and Dynamics

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

  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 Example: Jet diffusion flame F=CH4: Zst=0.055 Air Fuel

  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(h) • Chevillard et al. (2005): Rapid increase of F(drv) between h- and h+ • 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 scales Energy dissipation maxima

  10. Theoretical prediction for Q(h) (Yakhot, Physica D 2006) Mellin transform Saddle point approximation

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

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

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

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

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

  16. 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

  17. Summary • Local dissipation scales are defined in a dynamical content. • Velocity gradient statistics onKolmogorov andsub-Kolmogorov scales leads to asymptotic scaling exponents for velocity increment statistics onsuper-Kolmogorov scales. • Numerical effort has to go into the correct resolution of finest scales or strongest gradients.

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