Proper Finite Difference Schemes for Simulations of Incompressible Turbulent Flow in Generalized Curvilinear Coordinates : (1st Report, Analytical Conservation Properties of Coordinate Transformed Basic Equations and Extension of the Proper Finite Difference Schemes to Generalized Curvilinear Coordinates)

Abstract

In order to conduct direct numerical simulation (DNS) or large eddy simulation (LES) of turbulent flows in complicated flow geometry, accurate finite difference methods are needed in generalized curvilinear coordinate system. Recently, it was shown that the analytical conservation properties of the set of basic equations are needed to be satisfied properly even in discretized basic equations in order to obtain accurate and stable solutions in simulations of incompressible turbulent flow using finite difference method. The basic equations treated here are the continuity equation, the Navier-Stokes equation and the transport equations of the square values of velocity components and the kinetic energy. In this paper, finite difference schemes in generalized curvilinear coordinate system that are suitable for simulations of incompressible turbulent flow are constructed from relatively simple extension of the proper finite difference schemes derived in equidistant Cartesian coordinate system. The extension of finite difference scheme into generalized curvilinear coordinate system is based on the fact that the analytical conservation properties of the coordinate transformed basic equations for incompressible viscous flows are identical with that in Cartesian coordinates.