Development of a Compact and Accurate Discretization for Incompressible Navier-Stokes Equations Based on an Equation-Solving Solution Gradient, Part II: Formulation on Unstructured Polygonal Grids

Abstract

In this article, we develop a feasible solution procedure for solving convection-diffusion equations on unstructured grids as our second step to simulate incompressible fluid flow problems in complicated domains. Fundamentals of this procedure have been introduced in our previous work and verified in the one-dimensional situation as well as on a two-dimensional rectangular grid. All essential aspects of the discretization process on unstructured grids are detailed in the present work. Since the solution gradients required for a second-order accurate discretization are solved directly by their corresponding equations, it will result in a compact scheme which directly involves minimal computational nodes. Test problems are solved numerically with various grid arrangements to verify this proposition. From the numerical results, it is clearly shown that the present formulation can provide accurate difference expressions for the steady convection-diffusion equation on unstructured grids regardless of the grid arrangement. Therefore, the present formulation will serve as a kernel scheme to simulate fluid flow phenomena in complicated domains.