Development of a Compact and Accurate Discretization for Incompressible Navier-Stokes Equations Based on an Equation-Solving Solution Gradient, Part I: Kernel Scheme Fundamentals

Abstract

This is the first part of our attempt to develop a compact and accurate numerical scheme for incompressible Navier-Stokes equations in complicated domains. The objective of this article is to unveil the kernel scheme fundamentals in our formulation, where the solution gradient required for an accurate discretization is computed directly as an additional variable rather than interpolated from solution values at neighboring computational nodes. To achieve this goal, a supplementary equation and its associated control volume are introduced to retain a compact and accurate discretization. Scheme essentials are exposed by numerical analyses on simple one-dimensional modeled problems to reveal its formal accuracy. Due to its highly comprehensible and practical features, this formulation can be easily extended to solve problems in two-dimensional rectangular grid systems. Several one- and two-dimensional problems are solved to verify its simulation accuracy. From the numerical analyses and computational results of test problems, it is found that the present formulation is a useful tool to solve convection-diffusion equations and can be employed as the kernel scheme for fluid flow simulations.