hiroaki nishikawa national institute of aerospace
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Towards Future Navier-Stokes Schemes Uniform Accuracy, O(h) Time Step, Accurate Viscous/Heat Fluxes. Hiroaki Nishikawa National Institute of Aerospace. Future Navier-Stokes Schemes. One Scheme for the Navier-Stokes System Uniform Accuracy for ALL Reynolds Numbers - PowerPoint PPT PresentationTRANSCRIPT
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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
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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…
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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
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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:
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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).
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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:
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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.
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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.
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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:
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2D Finite-Difference Scheme
Simply apply the 1D scheme in each dimension.
Dimension by dimension decomposition:
2D Advection-Diffusion System:
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2D Fast Laplace Solver
In the diffusion limit, the 2D FD scheme reduces to
Jacobi iteration scheme with convergence
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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
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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.
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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.
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1D Convergence ResultsComparison with a scalar scheme:
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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
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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.
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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
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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)
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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)
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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.
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First-Order Navier-Stokes System
Finite-volume, Finite-element, Residual-distribution, etc.