Non-uniform mesh based FDM simulation of lid-driven cavity problem governed by N-S equations in stream function-vorticity formulation

TL;DR

This paper presents a finite difference method using non-uniform meshes for simulating the lid-driven cavity problem governed by Navier-Stokes equations in stream function-vorticity form, demonstrating improved accuracy and efficiency.

Contribution

It introduces a variable grid finite difference scheme tailored for the cavity problem, enhancing detail capture and solution accuracy over traditional uniform grid methods.

Findings

Variable grids better capture flow separation and reversed flow.

Symmetric dense grids yield more accurate results.

Vorticity and stream function are more accurate than velocity.

Abstract

In this paper, the driven cavity problem was solved using finite difference scheme in stream function-vorticity formulation. A variable grid is adopted to capture more details and information in the area nearby the wall. The Navier-Stokes equation is rewritten as a particular form suitable to the variable grids. In simulation, Reynolds number is set 100 as an example. The velocity, vorticity and streamline contour are produced by the CFD scheme developed in this paper and then are compared with those by Ghia et. al. (1982) to validate this numerical scheme. It shows that the numerical CFD scheme developed in this paper works very well for both uniform grids and variable grids. The numerical tests with different grids setting show that a) the variable grids have advantages in capturing the reversed flow and separation bubbles produced in the corners associated with a good efficiency, b) the numerical schemes with symmetric and dense grids gives a more accurate solution than those with non-symmetric and sparse grids, and c) both the vorticity and stream function have a better accuracy than velocity.