A fourth-order accurate compact scheme for the solution of steady Navier–Stokes equations on non-uniform grids

Abstract

This paper deals with the formulation of a higher-order compact (HOC) scheme on non-uniform grids in complex geometries to simulate two-dimensional (2D) steady incompressible viscous flows governed by the Navier–Stokes (N–S) equations. The proposed scheme which is spatially fourth-order accurate is then tested on three nonlinear problems, namely (i) a problem governed by N–S equations with a constructed analytical solution, (ii) lid-driven cavity flow problem, and (iii) constricted channel flow problem. In the process, we have also expanded the scope of fourth-order 9-point compact schemes to geometries beyond rectangular. It is seen to efficiently capture steady-state solutions of the N–S equations with Dirichlet as well as Neumann boundary conditions. In addition to this, it captures viscous flows involving free and wall bounded shear layers which invariably contain spatial scale variations. Our results are in excellent agreement with analytical and numerical results whenever available and they clearly demonstrate the superior scale resolution of the proposed scheme.