A Multigrid-Accelerated Code on Graded Cartesian Meshes for 2D Time-Dependent Incompressible Viscous Flows

Abstract

The present work deals with the development of a multigrid-assisted solver for the 2D time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes. As finite difference method is used to discretize the governing equations on nonuniform staggered grids, a transformation of the governing equations from the physical space to the computational space is performed. To obtain second-order time accuracy a fractional-step method is employed. The convective terms are discretized using a third-order accurate upwind scheme and the viscous terms are centrally differenced to fourth-order accuracy. To improve the time-wise efficiency of the code a multigrid technique is employed to solve the pressure-Poisson equation that is required to be solved at every time-step. To establish the accuracy and performance of the code the standard 2D lid-driven cavity flow is computed for unsteady, periodic and asymptotically obtained steady solution for a wide range of Reynolds numbers (Re). The code is then used to compute the transient and asymptotically approached steady flows in a hitherto unexamined problem of two-sided lid-driven square cavity, which involves gradual development of a free shear layer and accompanying off-corner vortices. The computations also show that for this configuration, at Re=2000, there exists a steady solution, about which there was some doubt earlier. The reliability of all known and unknown results in the paper is carefully established and efficiency of the method in respect of grid economy is demonstrated.