1. University of Cyprus DNS and Structure-Based Modeling of Rotated Shear Flows: Implications for Accretion Disks? S. C. Kassinos Stanford University/University of Cyprus ESF PESC Exploratory Workshop: Frontiers for Computational Astrophysics Wengen, Switzerland 26-30 September 2004 Also supported by AFOSR Grant No. F49620-99-0138

2. University of Cyprus Motivation Strongly rotating flows are challenging to turbulence models. Most well-known models have been calibrated against 20-year-old LES! Surprising lack of modern high resolution simulations of these flows. Objectives Create a modern high resolution DNS database of homogeneous turbulence that is sheared or strained in rotating frames. Modeling DNS results are used to validate a new type of model that was developed before the results were available.

3. University of Cyprus Outline Flow configuration Results that could be of relevance to accretion Direct Numerical Simulation (DNS): what are the open issues Structure-Based Modeling: what are the open issues Structure-Based Modeling: why is it different (better)? Future steps Discussion

4. University of Cyprus Discussion focus DNS: +more accurate physics – limited to low Reynolds numbers We discuss results from Direct Numerical Simulations (DNS) and one-point turbulence modeling based on RANS Turbulence models: +calibrated for high Reynolds numbers –often questionable physics

5. University of Cyprus Flow Configurations DNS configurations - + counter-rotating frame co-rotating frame

6. University of Cyprus frame co-rotatingframe counter-rotating ???? Flow physics: spanwise rotation tt k k k kk ttt decay exponential algebraic - +

7. University of Cyprus How does equilibrium vary if at all with ? 1 turbulence thrives turbulence dies Flow physics: basic question

8. University of Cyprus DNS Code Description Governing equations solved in coords deforming with the mean flow to allow Fourier pseudo-spectral methods with periodic B.C.’s. Time advance is based on a third-order Runge-Kutta method. Aliasing errors due to periodic remeshing are removed. Mean shear skews the computational grid, but periodic remeshing allows the simulation to progress to large total shear. The code is implemented in Vectoral using MPI and has been ported to the ASCI Red and a 48-node Linux cluster. Accuracy, grid independence and scalability have been tested.

9. University of Cyprus Reynolds Decomposition continuity: momentum: averaging

10. University of Cyprus one-point model mean deformation rate turbulence scales Directional intensity of velocity fluctuations Standard Assumption (RST)

11. University of Cyprus one-point model mean deformation rate turbulence scales Directional intensity of velocity fluctuations Standard Assumption (RST) Is this enough information for consistent accuracy?

12. University of Cyprus one-point model mean deformation rate turbulence scales Directional intensity of velocity fluctuations Standard Assumption (RST) Is this enough information for consistent accuracy? ONLY FOR SIMPLE CASES!!

13. University of Cyprus What Other Information? Most turbulent kinetic energy organized in large structures. The statistical description of the energy-containing structures is another degree of freedom in addition to. Velocity magnitude 512 3 DNS of rotated shear flow

14. University of Cyprus One-point turbulence structure tensors means eddy-alignment in the x  direction. means all velocity fluctuations organized in jetal motion in x  direction. means all large-scale circulation organized in vortical motion around x  direction.

15. University of Cyprus Importance of Structure in Dynamics Two fields with same, but different structure have different dynamics.   No dynamical effect of rapid frame rotation. Rapid frame rotation modifies one-point state of the turbulence.

16. University of Cyprus One-point turbulence structure tensors Turbulent streamfunction: dimensionality circulicity stropholysis describes the elongation and orientation of energy-containing eddies. describes the distribution of large-scale circulation in the turbulence field. contains information about the breaking of reflectional symmetry by mean/frame rotation. Like the pressure,  carries non-local information

17. University of Cyprus One-point turbulence structure tensors Near-wall streaks in fully-developed channel flow skin friction

18. University of Cyprus one-point model mean deformation rate turbulence scales Results support this as a more fundamentally based approach Directional intensity of velocity fluctuations and morphology of large eddies Structure-Based Modeling (SBM) Assumption

19. University of Cyprus Structure-Based Formulation

20. University of Cyprus Coefficients set by matching standard homogeneous flows with mean and frame rotation (shear, elliptic, axisymmetric strain+rotation, plane strain+rotation). Then validated in fully developed channel flow and rotating pipe flow. Relative narrow range supported by match: Differential SBM

21. University of Cyprus (For details see Phys. Fluids, 14(7), April 2002) At high Re, LSE model constants are evaluated by an asymptotic analysis for decaying turbulence in stationary and rotating frames. The High Re Large-Scale Enstrophy (LSE) Equation

22. University of Cyprus Results (preliminary): equilibrium P/  DNS SBM predictions using standard  model eqn. (SSG, v2f, …) SBM using the large-scale enstrophy equation agrees with DNS. standard  model seriously in error! From a practical point of view, the most important info are the values of where crosses 1 (that we can answer).

23. University of Cyprus Results (preliminary): equilibrium P/ 

24. University of Cyprus Results (preliminary): equilibrium P/ 

25. University of Cyprus Results: evolution histories of structure tensors normalized Reynolds stressnormalized dimensionalitynormalized circulicity  Level of agreement between DNS and SBM typical for other. solid lines: DNS dashed lines: SBM using large-scale enstrophy.

26. University of Cyprus Results: Reynolds stress tensor at St = 9 vs.   

27. University of Cyprus Results: evolution histories of structure tensors normalized Reynolds stressnormalized dimensionalitynormalized circulicity  Level of agreement between DNS and SBM typical for other. solid lines: DNS dashed lines: SBM using large-scale enstrophy. 11 22 12 33 11 12 11 12 33

28. University of Cyprus Results: structure tensors at St = 9 vs.    symbols: DNS solid lines: SBM using large-scale enstrophy. normalized Reynolds stressnormalized dimensionalitynormalized circulicity

29. University of Cyprus DNS configurations for MHD turbulence B B Stuart Number Magnetic Reynolds No.

30. University of Cyprus MHD Results (preliminary): equilibrium P/ 

31. University of Cyprus Evolution of Energies

32. University of Cyprus Production over dissipation

33. University of Cyprus v-field C221 v-field C221W2 v-field C221W2-B=0 Results: moderate shear timescale horizontal slabsvertical slabsstreamwise eddies

34. University of Cyprus Results: scale-dependent anisotropy Reminiscent of the observations of Cho and Lazarian (2003) in high Rm compressible MHD turbulence

35. University of Cyprus But… DNS is for low Reynolds number periodic flow in a box. Model predictions are for high-Reynolds number limit. Establish if possible Reynolds number dependence Conclusion SBM with large-scale enstrophy in excellent agreement with DNS! Future Plans DNS seems predicts that turbulence is suppressed for. DNS seems predicts that MHD turbulence with spanwise B can survive. 1024 3 Or bigger DNS would help! (both Re effects and eddy containment issues