Towards Future Navier-Stokes SchemesTowards Future Navier-Stokes SchemesUniform Accuracy, O(h) Time Step, Accurate Viscous/Heat FluxesUniform Accuracy, O(h) Time Step, Accurate Viscous/Heat Fluxes

Hiroaki NishikawaHiroaki NishikawaNational Institute of AerospaceNational Institute of Aerospace

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…

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

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:

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).

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:

O(h) Time Step and IterationsNumber 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.

Whatever the discretization1. 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 diffusio

n).

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.

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.

Upwind Finite-Volume Scheme:

2D Finite-Difference Scheme

Simply apply the 1D scheme in each dimension.

Dimension by dimension decomposition:

2D Advection-Diffusion System:

2D Fast Laplace Solver

In the diffusion limit, the 2D FD scheme reduces to

Jacobi iteration scheme with convergence

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

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.

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.

1D Convergence ResultsComparison with a scalar scheme:

1D Accuracy Results

L_infinity norm of the errors:

2nd-Order accurate for both u and p for all Reynolds numbers.

Error of the solution, u Error of the gradient, p

2D Problem

10 orders of magnitude reduction in residuals.

17x17, 33x33, 65x65 17x17, 24x24, 33x33, 41x41, 49x49, 57x57, 65x65

Problem:with u (and either p or q) given on the boundary.

2D Convergence Results

Number of iterations (time steps) to reach the steady state:

Number of iterations is almost independent of the Reynolds numbers.

Structured Grids Unstructured Grids

2D Accuracy ResultsL_infinity norm of the errors for Structured grid case:

2nd-Order accurate for both u and p for all Reynolds numbers.

Error of the solution, u Error of p (=ux) Error of q (=uy)

2D Accuracy ResultsL_1 norm of the errors for Unstructured grid case:

2nd-Order accurate for both u and p for all Reynolds numbers.

Error of the solution, u Error of p (=ux) Error of q (=uy)

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.

First-Order Navier-Stokes System

Finite-volume, Finite-element, Residual-distribution, etc.