Lagrangian and Eulerian velocity structure functions in hydrodynamic turbulence

1. LagrangianandEulerianvelocitystructurefunctionsinhydrodynamicturbulenceLagrangianandEulerianvelocitystructurefunctionsinhydrodynamicturbulence Article from K. P. Zybin and V. A. Sirota Enrico Dammers & Christel Sanders Course 3T220 Chaos

2. Content • Goal • Euler vs. lagrangian • Background • Theory from earlier articles • Structure functions • Bridge relations • Results • Conclusions

3. Goal of the article • Showing eulerian and lagrangian structure formulas are obeying scaling relations • Determine the scaling constants analytical without dimensional analyses

4. Euler vs. Lagrangian Lagrangian Euler • Measured between t and t+τ • Along streamline • Structure function • Measured between r and r+l • Between fixed points • Structure function

5. Structure Functions • Kolmogorov: • She-Leveque:

6. Background • Turbulent flow, • Assumptions: • Stationary • Isotropic • Eddies , which are characterized by velocity scales and time scales(turnover time) • Model: Vortex Filaments • Thin bended tubes with vorticity, ω. • Assumption: • Straight Tubes • Regions with high vorticity make the main contribution to structure functions ω

7. Theory of earlier Articles:Navier-stokes on vortex filament • Dot product with • relation pressure en velocity • Change to Lagrange Frame: • Lagrange: • , • , at r=

8. Theory of earlier Articles:Navier-stokes on vortex filament • Taylor expansion of v’ and P around r= • , • Splitting in sum of symmetric and anti-symmetric term • Vorticity

9. Theory of earlier Articles:Navier-stokes on vortex filament • Combining all terms • = • 15 different values • 10 equations • 5 undefined functions

10. Theory of earlier Articles:Navier-stokes on vortex filament • Assumption: • are random functions, stationary • With: • Where is a function depending on profile • When • For Simplicity:

11. Theory of earlier Articles:Eigenfunctions • Small n, value of order , non-linear function • In real systems for large n: • assumption of article • Where  is maximum possible rate of vorticity growth

12. Eulerian structure function • Assume circular orbit of particle in a filament: • Average over all point pairs: • l must be smaller then R: • This restriction gives a maximum to t for the filament

13. Eulerian structure function • This results in the following condition: • : Eddy Turn over time • : Eddy size • for • Gives:

14. Eulerian structure function • The eulerian structure function now becomes: • With

15. Lagrangian structure function • For the lagrangian function: • : curvature radius of the trajectory • Assume which is the same restriction as in the euler case, • Same steps as with the eulerian function gives:

16. Lagrangian structure Function • The lagrangian structure function now becomes: • With

17. Bridge relation • Now we have • Combination of ’s gives relation: • (n-)=2(n-

18. Results • Compare with numerical simulation

19. Conclusions • Showing eulerian and lagrangian structure formulas are obeying scaling relations • Determine the scaling constants analytical without dimensional analyses • Using Eigen functions: • (n-)=2(n-

20. Questions?

21. Results

22. Results