Hiroaki Nishikawa National Institute of Aerospace

1. Towards Future Navier-Stokes SchemesUniform Accuracy, O(h) Time Step, Accurate Viscous/Heat Fluxes Hiroaki Nishikawa National Institute of Aerospace

2. Future Navier-Stokes Schemes • One Scheme for the Navier-Stokes System • Uniform Accuracy for ALL Reynolds Numbers • O(h) Time Step for ALL Reynolds Numbers • Accurate Viscous Stresses and Heat Fluxes • More…

3. Also equivalent at a steady state: Equivalent for any Tr Stiffness is NOT an issue for steady computation. The system can stay strongly hyperbolic toward a steady state. • JCP 2007 vol.227, pp315-352 New Approach for Diffusion Diffusion Equation Hyperbolic Heat Equations Advection scheme for diffusion

4. Eigenvalues are real: Waves travelling to the left and right at the same finite speed. E.g., Upwind scheme for diffusion: CFL condition: Diffusion System Time step is O(h) for Tr = O(1).

5. Tr is derived by requiring: Tr = Lr / eigenvalue: Eigenvalues: Lr is derived by optimizing the condition number of the system: Advection-Diffusion System The FOS advection-diffusion system is completely defined, in the differential level with Tr = (1).

6. Time step restriction for a common scheme: O(h) Time Step Time step restriction for FOS-based schemes(CFL Condition): O(h) Time Step for all Reynolds numbers.

7. O(h) time step also for two and three dimensions. Two Dimensions: Three Dimensions: O(h) Time Step and Iterations Number of time steps to reach a steady state: ----- > Iterative solver with O(N) convergence. Jacobi iteration is as fast as Krylov-subspace methods.

8. Whatever the discretization • One Scheme for Advection-Diffusion System: Hyperbolic scheme for the whole advection-diffusion system. • Uniform Accuracy for ALL Reynolds Numbers:No need to combine two different schemes (advection and diffusion). • O(h) Time Step for ALL Reynolds Numbers Rapid convergence to a steady state by explicit schemes. • Accurate Solution Gradient (Diffusive Fluxes): Same order of accuracy as the main variables. Boundary conditions made simple (Neumann -> Dirichlet). • Various Other Techniques Available:Techniques for advection apply directly to advection-diffusion.

9. Upwind Finite-Volume Scheme: Advection-Diffusion Scheme Upwind Residual-Distribution Scheme: These schemes gives an identical 3-point finite-difference formula, and 2nd-order accurate at a steady state.

10. 2D Finite-Difference Scheme 2D Advection-Diffusion System: Dimension by dimension decomposition: Simply apply the 1D scheme in each dimension.

11. Jacobi iteration scheme with convergence 2D Fast Laplace Solver In the diffusion limit, the 2D FD scheme reduces to

12. Upwind Scheme for Triangular Grid Upwind Residual-Distribution Scheme: LDA scheme Upwind Finite-Volume Scheme: Not implemented in this work. But it is straightforward to apply any discretization scheme to the 2D system. Just make sure that accuracy is obtained at a steady state.

13. 1D Test Problem Problem: with u(0)=0 and u(1)=1, and the source term, Stretched Grids: 33, 65, 129, 257 points. Re = 10^k, k = -3, -2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 2, 3. CFL = 0.99, Forward Euler time-stepping. Residual reduction by 5 orders of magnitude p is NOT given but computed on the boundary.

14. 1D Convergence Results The number of iterations (time steps) to reach the steady state: The number of iteration is nearly independent of the Reynolds number.

15. 1D Convergence Results Comparison with a scalar scheme:

16. 1D Accuracy Results L_infinity norm of the errors: Error of the solution, u Error of the gradient, p 2nd-Order accurate for both u and p for all Reynolds numbers.

17. 2D Problem Problem: with u (and either p or q) given on the boundary. 17x17, 33x33, 65x65 17x17, 24x24, 33x33, 41x41, 49x49, 57x57, 65x65 10 orders of magnitude reduction in residuals.

18. 2D Convergence Results Number of iterations (time steps) to reach the steady state: Structured Grids Unstructured Grids Number of iterations is almost independent of the Reynolds numbers.

19. 2D Accuracy Results L_infinity norm of the errors for Structured grid case: Error of the solution, u Error of p (=ux) Error of q (=uy) 2nd-Order accurate for both u and p for all Reynolds numbers.

20. 2D Accuracy Results L_1 norm of the errors for Unstructured grid case: Error of the solution, u Error of p (=ux) Error of q (=uy) 2nd-Order accurate for both u and p for all Reynolds numbers.

21. Future Navier-Stokes Schemes • One Scheme for the Navier-Stokes System: Hyperbolic scheme for the whole Navier-Stokes system. • Uniform Accuracy for ALL Reynolds Numbers:No need to combine two different (inviscid and viscous) methods. • O(h) Time Step for ALL Reynolds NumbersRapid convergence to a steady state by explicit schemes. • Accurate Viscous Stresses/ Heat Fluxes: Same order of accuracy as the main variables. Boundary conditions made simple (Neumann -> Dirichlet). • Various Techniques Directly Applicable to NS:Techniques for the Euler apply directly to the Navier-Stokes.

22. First-Order Navier-Stokes System Finite-volume, Finite-element, Residual-distribution, etc.