1. DNS, LES & High Accuracy Computing High Performance Computing Laboratory, Department of Aerospace Engineering Indian Institute of Technology- Kanpur Prof. T. K. Sengupta, Swagata Bhaumik and Yogesh G. Bhumkar tksen@iitk.ac.intksen@iitk.ac.in, swagata@iitk.ac.in and bhumkar@iitk.ac.in

2. Presentation Outline Spectral theory of computing: high-accuracy DNS Dispersion error: Correcting von Neumann error- propagation equation. Parallelization strategy of high-accuracy compact schemes New Upwind & Multi-dimensional filters: Tools to control instability; aliasing; q-waves; LES & DES 3D DNS: A Search for error metrics. – Computations using velocity-vorticity formulation: A panacea?

3. 2 D Navier – Stokes equation in stream function – vorticity formulation 2D Navier - Stokes equation in stream function – vorticity formulation is solved using orthogonal grid. This is the most accurate equation to simulate 2D Navier-Stokes eauation Stream Function Equation (SFE): Vorticity Transport Equation (VTE): Where and are the scale factors.

4. General Solution Algorithm for Solving 2D Navier- Stokes Equation Initialise the unknown, and Solve SFE at time t Obtain the vorticity that satisfies no-slip condition at Solve the VTE for at Is Stop No Yes

5. Boundary conditions for LDC flow Vorticity contours Re = 10,000

6. Numerical Amplification Factor: A New Spectral Theory of Computations Most of the fluid-dynamical problems are convection dominated. Hence, the simplest model equation to test any numerical scheme for CFD is the following 1D convection equation This equation is non-dissipative and non-dispersive. The solution propagates towards right with a constant speed c.

7. Thus, a general numerical solution of (1) is, Numerical Phase Speed & Group Velocity

8. High Accuracy Computing: Optimized Upwind Compact Schemes For the n th derivative of the function u j at node j is defined as, We have obtained optimised compact schemes (OUCS3) by optimising them in spectral planes. Analysis of central and upwind compact schemes, T. K. Sengupta, G. Ganeriwal and S. De, J. Comput. Phys., 192(2), 677-694 (2003).

9. Numerical properties of the new method Optimal time advancing dispersion relation preserving schemes, M. K. Rajpoot, T. K. Sengupta and P. Dutt, J. Comput. Phys., 229(10), 3623-3651 (2010).

10. q waves of Different Schemes

11. Error Propagation Equation: Beoynd von Neumann Analysis Error dynamics: Beyond von Neumann analysis; T. K. Sengupta, A. Dipankar and P. Sagaut; J. Comput. Phys.; Vol. 226, 1211-1218 (2007).

12. Boundary closure of compact schemes One-sided schemes are need at the boundary for the evaluation of the compact schemes. We use =-0.09 for =2 and =0.12 for =-1 Analysis of central and upwind compact schemes, T. K. Sengupta, G. Ganeriwal and S. De J. Comput. Phys., 192(2), 677-694 (2003).

13. Parallelization of the Compact Schemes for Domain Decomposition Technique Spectral analysis [Sengupta et al. (2003)] shows problematic bias of compact schemes near inflow & outflow: over six to seven nodes. This has been cured by symmetrized S − OUCS3 scheme given in: Symmetrized compact scheme for receptivity study of 2D transitional channel flow; A. Dipankar and T.K. Sengupta, J. Comput. Phys.; Volume 215, Issue 1, Pages 245-273 (2006). Extension to parallel problem is developed in: A new compact scheme for parallel computing using domain decomposition T. K. Sengupta, A. Dipankar and A. K. Rao, J. Comput. Phys., Vol 220(2), 645-677 (2007)

14. Validation of a Symmetrized Compact Scheme for Parallel Computing: Compressible Navier-Stokes Equation (M = 1.6) “Receptivity Study of a Hypersonic Boundary Layer to Free-Stream Vortical Excitation” (M.Tech. Thesis), Naresh Kumar, Aerospace Engg. Dept., IIT Kanpur, 2007

15. Receptivity Study of a Hypersonic Boundary Layer : NS Solution at M = 4 Receptivity Study of a Hypersonic Boundary Layer to Free-Stream Vortical Excitation; (M.Tech. Thesis), Naresh Kumar, Aerospace Engg. Dept., IIT Kanpur, 2007

16. Scalability Results All our parallel codes are programmed in MPI employing parallel compact schemes.

17. High Accuracy Filters: Applications for Linearized Rotating Shallow Water Equation (LRSWE) Horizontal velocity in - direction coriolis parameter Acceleration due to gravity Surface elevation w.r.t. the mean level H Mean depth Horizontal velocity in - direction coriolis parameter Acceleration due to gravity Surface elevation w.r.t. the mean level H Mean depth Analytical dispersion relation : Here, and

18. Solving Focusing of LRSWE by Filters A Linear Focusing Mechanism for Dispersive and Non-Dispersive Wave Problems, Y. G. Bhumkar, M. K. Rajpoot and T. K. Sengupta, J. Comput. Phys., (2010) Link : http://dx.doi.org/10.1016/j.jcp.2010.11.026http://dx.doi.org/10.1016/j.jcp.2010.11.026 Unfiltered solution Filtered solution

19. Introduction to High Accuracy Explicit Filters D. V. Gaitonde, J. S. Shang and J. L. Young, Practical aspects of higher-order numerical schemes for wave propagation phenomena. Int. J. Numer. Meth. Eng. 45 (1999), p. 1849

20. Upwind Filters: Removing q waves Where, Filtered variable Unfiltered variable M Order of the filter Free parameter with Upwind coefficient Sixth order dissipation term T.K. Sengupta, Y. G. Bhumkar and V. Lakshmanan, Design and analysis of a new filter for LES and DES, Comput. Struct. 87, 735–750 (2009)

21. Altered Properties by Filtering: Numerical Amplification Factor with & without Upwind Filters Numerical amplification contours for the solution of 1D wave equation, when the spatial discretization is carried out by OUCS3 and time discretization is obtained by RK4.

22. Scaled numerical group velocity contours for the solution of 1D wave equation, when the spatial discretization is carried out by OUCS3 and time discretization is obtained by RK4. Effects of Filters: Numerical Group Velocity with & without Upwind Filter

23. Multi-Dimensional Filters: A Tool for Aliasing, LES & DES Filter stencil in 2D: Filter coefficient : Filter transfer function : New Explicit Two-Dimensional Higher Order Filters, T. K. Sengupta and Y. G. Bhumkar, Comput. Fluids, Volume 39, 1848-1863, 2010

24. Transfer Function Variation of 1D and 2D Filters in k i h i -k j h j Plane Aliased Calculations De-aliased Calculations

25. Application of Adaptive Explicit Two-Dimensional Filters: Flow Past Circular Cylinder Performing Rotary Oscillations Unfiltered Case Unfiltered Case Adaptive Filtered Case Adaptive Filtered Case Re = 150, A= 5, F f /F o = 0.5

28. We examin the effect of error in 2D lid-driven cavity (LDC) for Re=9000 and Re=10,000. Both these Reynolds numbers are post-critical. The non-zero value of can excite flow by spurious sources and sinks. The strategies are (1) Scheme A : Solving two Poisson equation for u- and v- velocity with BiCGSTAB iterative method. (2) Solving one Poisson equation for u- velocity by BiCGSTAB and v- velocity directly from continuity equation. (1) Scheme A : Solving two Poisson equation for u- and v- velocity with BiCGSTAB iterative method. (2) Solving one Poisson equation for u- velocity by BiCGSTAB and v- velocity directly from continuity equation. Can Spurious Mass Excite Flow Instabilities?

29. Flow inside LDC at Re=9000 Scheme A with Scheme B with Max error plotted as a function of time for scheme A and scheme B. Vorticity at (0.95,0.95) plotted as a function of time for scheme A and scheme B.

30. 2D LDC Flow at Re= 10,000 Here, scheme A with varying levels of (10 -4, 10 -6 and 10 -9 ) and scheme B are used for the simulations.

31. Re=10,000 Contd… Scheme A with Scheme B with

32. 3D calculation of flow inside LDC

33. Other Activities Nonlinear instability / POD of bypass transition Chimera / overset grid method to solve flow past complex geometries Dynamical system approach to instability of flow past a circular cylinder, T. K. Sengupta, Neelu Singh & V. K. Suman, J. Fluid Mech., vol. 656, 82-115, (2010) Universal Instability Modes in Internal and External Flows, T. K. Sengupta, V. V. S. N. Vijay and Neelu Singh, Comput. Fluids, Volume 40(1), 221-235 (2011) Solving Navier-Stokes equation for flow past cylinders using single- block structured and overset grids, T. K Sengupta, V. K. Suman and Neelu Singh, J. Comput. Phys., vol. 229(1), pp 178-199 (2010)

34. People@HPCL Unnikrishnan SS. SaurabhSreejith N A Prof. Tapan K. Sengupta Y. G. BhumkarM. K. RajpootS. Bhaumik R. Roy Chowdhury Peeyush Singh K. Asthana Mubashir Ali Omer Farooq