1. High-Order Spatial and Temporal Methods for Simulation and Sensitivity Analysis of High-Speed Flows PI Dimitri J. Mavriplis University of Wyoming Co-PI Luigi Martinelli Princeton University

2. Project Scope and Relevance Develop novel approaches for improving simulation capabilities for high-speed flows –Emerging consensus about higher-order methods May be only way to get desired accuracy –Asymptotic arguments Superior scalability –Sensitivity analysis and adjoint methods Now seen as indispensible component of new emerging class of simulation tools Automated (adaptive) solution process with certifiable accuracy –Other novel approaches: BGK methods

3. Advantages of DG Discretizations Superior Asymptotic Properties Smaller meshes –Easier to generate/manage Superior Scalability: small meshes on many cores Dense kernels, well suited for GPUs, Cell processors 2.5 million cell DG (h-p Multigrid)

4. Disadvantages of DG Discretizations High-Risk, Revolutionary –Still no production level DG code for subsonics Relies on smooth solution behavior to achieve favorable asymptotic accuracy –Difficulties for strong shocks –Robustness issues

5. Overview of Current Work 1.Viscous discretizations and solvers for DG 2.ALE Formulation for moving meshes 3.BGK Flux flunction implementation/results 4.Shock capturing - Artificial dissipation - High-order filtering/limiting 5.Adjoint-based h-p refinement - Shocks captured with no limiting/added dissipation 6.Conclusions

6. Extension to Viscous Flows DG methods developed initially for hyperbolic problems –Diffusion terms for DG non-trivial Interior Penalty (IP) method –Simplest approach, compact stencil –Explicit expression for penalty parameter derived (JCP) IP method derived and implemented for compressible Navier-Stokes formulation up to p=5 –Studied symmetric and non-symmetric forms for IP –h and p independent convergence observed for Poisson and Navier- Stokes problems

7. DG Navier-Stokes Solutions Mach =0.5, Re =5000 2000 mesh elements Non-symmetric grid

8. DG Navier-Stokes Solutions h-p multigrid convergence maintained (50 – 80 cycles) Accuracy validated by comparison with high-resolution finite-volume results –Separation location ~ 81% chord (p=3) p=1: second-order accuracyp=3: fourth-order accuracy

9. Solution of DG Discretization for NS Equations h-p multigrid solver: h and p independent convergence rates Used as preconditioner to GMRES for further efficiency improvements

10. Kinetic Based Flux Formulations (BGK) L. Martinelli Princeton University Alternative for extension to Navier-Stokes: –It is not necessary to compute the rate of strain tensor in order to calculate viscous fluxes Automatic upwinding via the kinetic model. Satisfy Entropy Condition (H-Theorem) at the discrete level. Implemented in 2D Unstructured Finite-Volume code by Martinelli Extension to 2D DG code under development

11. BGK Finite Volume Solver Mach 10 Cylinder Robust 2 nd order accurate solution BGK –DG solutions obtained for low speed flows –BGK-DG cases with strong shocks initiated

12. Treatment of Shock Waves High-order (DG) methods based on smooth solution behavior 3 approaches investigated for high-order shock wave simulation –Smoothing out shock: Artificial viscosity Use IP method discussed previously Sub-cell shock resolution possible –Limiting or Filtering High Order Solution Remove spurious oscillations Sub-cell shock resolution possible –h-p adaption Start with p=0 (1 st order) solution Raise p (order) only were solution is smooth Refine mesh (h) where solution is non-smooth (shock) No limiting required!

13. Shock Capturing with Artificial Dissipation (p=4) IP Method used for artificial viscosity terms (Laplacian) Artificial Viscosity scales as ~ h/p An alternative to limiting or reducing accuracy in vicinity of non- smooth solutions (Persson and Peraire 2006)

14. Shock Capturing with Artificial Dissipation Sub-cell shock capturing resolution (p=4)

15. Mach 6 Flow over Cylinder Third order accurate (p=2) Relatively coarse grid Sub-cell shock resolution captured with artificial dissipation Principal issue: Convergence/Robustness

16.  Euler-Lagrange equation (1 st variation) Nonlinear partial differential equations (PDE) based  Pseudo-time stepping (Rudin, Osher and Fatemi 1992)  Solved locally in each element Total Variation based nonlinear Filtering Formulation  Minimization where,

17.  Euler-Lagrange equation (1 st variation) Nonlinear partial differential equations (PDE) based  Pseudo-time stepping (Rudin, Osher and Fatemi 1992)  Solved locally in each element Total Variation based nonlinear Filtering Formulation  Minimization where, Controls amount of filtering

18. Shock Capturing with Filtering p=3 (4 th order accuracy) Weak (transonic) shock captured with sub-cell resolution using filtering/limiting Enables highest order polynomial without oscillations

19. DG Filtering for High Speed Flows Mach 6 flow over cylinder at p=2 (3 rd order) –Lax Friedrichs flux Relatively robust Shock spread over more than one element

20. DG Filtering for High Speed Flows Mach 6 flow over cylinder at p=2 (3 rd order) –Van-Leer Flux Relatively robust Thinner Shock spread over approximately one element

21. DG Filtering for Strong Shocks Shock resolution determined by convergence robustness –(not necessarily property of flux function) –Van Leer flux could be run with larger filter  value –Higher order solutions should deliver higher resolution shocks Convergence issues remain above p=2 Lax-Friedrichs Van Leer

22. Formulation –Key objective functionals with engineering applications Surface integrals of the flow-field variables Lift, drag, integrated temperature, surface heat flux A single objective, expressed as –Current mesh (coarse mesh, H) Coarse flow solution, Objective on the coarse mesh, –Globally refined mesh (fine mesh, h) Fine flow solution, Objective on the fine mesh, Goal : find an approximate for without solving on the fine mesh ADJOINT-BASED ERROR ESTIMATION 22 NOT DESIRED!

23. Formulation –Coarse grid solution projected onto fine grid gives non-zero residual –Change in objective calculated on fine grid: = inner product of residual with adjoint Procedure –Compute coarse grid solution and adjoint –Project solution and adjoint to fine grid –Form inner product of residual and adjoint on fine grid Global Error estimate of objective Local error estimate (in each cell) –Use to drive adaptive refinement –Smoothness indicator used to choose between h and p refinement –Naturally maintains p=0 in shock region ADJOINT-BASED ERROR ESTIMATION 23

24. High-speed flow over a half circular-cylinder (M ∞ =6) Combined h-p Refinement for Hypersonic Cases Target function of integrated temperature hp -refinement starting discretization order p = 0 (first-order accurate) 24 initial mesh: 17,072 elements

25. High-speed flow over a half circular-cylinder (M ∞ =6) h-p Refinement for High-Speed Flows 25 adapted mesh: 42,234 elements, discretization orders p=0~3 No shock refinement in regions not affecting surface temperature

26. h-p Refinement Objective=Surface T Mach 6 26 Pressure Mach Number Shock captured without limiting or dissipation Naturally remains at p=0 in shock region

27. h-p Refinement for Mach 10 Case High-speed flow over a half circular-cylinder (M ∞ =10) Target function of drag 27

28. H-p Refinement: Functional Convergence 28 M ∞ =6, functional: integrated temperatureM ∞ =10, functional: drag

29. Conclusions and Future Work DG methods hold promise for advancing state-of- the-art for difficult problems such as Hypersonics Recent advances in: –Viscous discretizations –Flux functions (BGK) –ALE formulations –Solver technology (h-p multigrid) –Shock capturing Extend into 3D DG parallel code –Diffusion terms –Shock capturing –h-p adaptivity (adjoint based) Real gas effects –5 species, 2 temperature model for DG code

30. Remaining Difficulties DG Methods need to be robust –Often requires accuracy reduction (limiting) Shock capturing with artificial viscosity becomes very non- linear/difficult to converge for high p and high Mach Limiting is very robust initially, but convergence to machine zero stalls –Other limiter formulations are possible Adjoint h-p refinement is promising but will likely require use with limiter for necessary robustness –Linearization of limiter/filter