| A scheme for giving steady-state solutions of the Navier-Stokes equations on unstructured grids in two-dimensions is presented. The method, called fluctuation splitting, is designed to be independent of the underlying grid geometry and distributes the residual in the cell to the vertices in a manner based upon upwinding ideas. The development starts with a discussion of schemes for the advection equation on triangles which is then extended to include the diffusive terms in the advection-diffusion equation. This scheme is then evaluated on the Smith and Hutton problem, which involves the simultaneous advection and diffusion of a scalar quantity at high and low Peclet numbers around a bend. By integrating continuity on a structured grid, the Blasius boundary layer equations are solved as a scalar advection-diffusion problem. The extension to systems of equations begins with the Euler equations, which involves the development of wave-models to model the local solution within cells of the domain. The vertices of the cells are updated based upon the passage of these simple waves. This is then extended to the Navier-Stokes equations by discretizing the viscous fluxes in a manner similar to that done in the scalar case for advection-diffusion. Results are shown for the Blasius boundary layer and a NACA0012 airfoil. Finally, the parallel aspects of the scheme, which uses a very compact stencil for each vertex, are discussed. It is shown that the algorithm is readily parallelizable and that the performance of the parallel algorithm on an IBM SP/2 is excellent. |
