DNS and Structure-Based Modeling of Rotated Shear Flows: Implications for Accretion Disks?

1. 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. Motivation Strongly rotating flows are challenging to turbulence models. Surprising lack of modern high resolution simulations of these flows. Most well-known models have been calibrated against 20-year-old LES! 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. Outline Flow configuration Structure-Based Modeling: why is it different (better)? 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 Future steps Discussion

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

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

6. Flow physics: spanwise rotation frame co-rotating frame counter-rotating decay algebraic exponential algebraic decay k k k k k t t t t t ? ? ? ? - +

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

8. 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. Mean shear skews the computational grid, but periodic remeshing allows the simulation to progress to large total shear. Aliasing errors due to periodic remeshing are removed. 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. ReynoldsDecomposition continuity: momentum: averaging

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

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

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

13. What Other Information? Velocity magnitude 5123 DNS of rotated shear flow Most turbulent kinetic energy organized in large structures. The statistical description of the energy-containing structures is another degree of freedom in addition to .

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

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

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

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

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

19. Structure-Based Formulation

20. Differential SBM 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:

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

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

23. Results (preliminary): equilibrium P/e

24. Results (preliminary): equilibrium P/e

25. (h = 0.15 ) Results: evolution histories of structure tensors normalized Reynolds stress normalized dimensionality normalized circulicity solid lines: DNS dashed lines: SBM using large-scale enstrophy. Level of agreement between DNS and SBM typical for other .

26. Results: Reynolds stress tensor at St = 9 vs. h h h

27. (h = 0.4) Results: evolution histories of structure tensors normalized Reynolds stress normalized dimensionality normalized circulicity 22 33 11 11 33 11 12 12 12 solid lines: DNS dashed lines: SBM using large-scale enstrophy. Level of agreement between DNS and SBM typical for other .

28. Results: structure tensors at St = 9 vs. h normalized Reynolds stress normalized dimensionality normalized circulicity h h h symbols: DNS solid lines: SBM using large-scale enstrophy.

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

30. MHD Results (preliminary): equilibrium P/e

31. Evolution of Energies

32. Production over dissipation

33. v-field C221 v-field C221W2-B=0 Results: moderate shear timescale v-field C221W2 horizontal slabs streamwise eddies vertical slabs

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

35. Conclusion DNS seems predicts that turbulence is suppressed for . DNS seems predicts that MHD turbulence with spanwise B can survive . SBM with large-scale enstrophy in excellent agreement with DNS! But… DNS is for low Reynolds number periodic flow in a box. Model predictions are for high-Reynolds number limit. Future Plans Establish if possible Reynolds number dependence 10243 Or bigger DNS would help! (both Re effects and eddy containment issues