A compact streamfunction-velocity scheme on nonuniform grids for the 2D steady incompressible Navier-Stokes equations

In this paper, an effective compact finite difference approximation which carries streamfunction and its first derivatives (velocities) as the unknown variables for the streamfunction-velocity formulation of the steady two dimensional incompressible Navier-Stokes equation is developed on non-uniform orthogonal Cartesian grids. To solve the resulting system of equations, a multigrid iterative strategy on nonuniform grids is introduced by using the interpolation techniques. Numerical experiments, involving two test problems with analytical solutions and the lid-driven square cavity flow problem are carried out to display the superiority of the currently developed method on nonuniform grid. Numerical results show that the present method on nonuniform grids gets as similarly efficient convergence rate as on uniform grids, viz., second order accuracy and the resolution of the computed solutions for the problems with the sharp changes can be significantly improved when the nonuniform grid strategy is utilized. The backward-facing step flow is also calculated by the present method to exhibit the capability to simulate the distant field using fewer grid points. The solution for the natural convection problem reveals further the wide applications of the present method not only in the flow problems but also in the heat transfer problems. All of these numerical results demonstrate the accuracy and efficiency of the currently proposed schemes. (C) 2013 Elsevier Ltd. All rights reserved.