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

2. Future Navier-Stokes Schemes 1.One Scheme for the Navier-Stokes System 2.Uniform Accuracy for ALL Reynolds Numbers 3.O(h) Time Step for ALL Reynolds Numbers 4.Accurate Viscous Stresses and Heat Fluxes 5.More…

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

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

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

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

7. 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. Two Dimensions: Three Dimensions: O(h) time step also for two and three dimensions.

8. Whatever the discretization 1.One Scheme for Advection-Diffusion System: Hyperbolic scheme for the whole advection-diffusion system. 2.Uniform Accuracy for ALL Reynolds Numbers: No need to combine two different schemes (advection and diffusion). 3.O(h) Time Step for ALL Reynolds Numbers Rapid convergence to a steady state by explicit schemes. 4.Accurate Solution Gradient (Diffusive Fluxes): Same order of accuracy as the main variables. Boundary conditions made simple (Neumann -> Dirichlet). 5.Various Other Techniques Available: Techniques for advection apply directly to advection-diffusion.

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

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

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

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

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: 2 nd -Order accurate for both u and p for all Reynolds numbers. Error of the solution, uError of the gradient, p

17. 2D Problem 10 orders of magnitude reduction in residuals. 17x17, 33x33, 65x6517x17, 24x24, 33x33, 41x41, 49x49, 57x57, 65x65 Problem: with u (and either p or q) given on the boundary.

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

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

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

21. Future Navier-Stokes Schemes 1.One Scheme for the Navier-Stokes System: Hyperbolic scheme for the whole Navier-Stokes system. 2.Uniform Accuracy for ALL Reynolds Numbers: No need to combine two different (inviscid and viscous) methods. 3.O(h) Time Step for ALL Reynolds Numbers Rapid convergence to a steady state by explicit schemes. 4.Accurate Viscous Stresses/ Heat Fluxes: Same order of accuracy as the main variables. Boundary conditions made simple (Neumann -> Dirichlet). 5.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.