A multigrid-accelerated three-dimensional transient-flow code and its application to a new test problem

A multigrid-assisted solver for the three-dimensional time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes is developed. The spatial accuracy is third-order for the convective terms and fourth-order for the viscous terms, and a fractional-step strategy ensures second-order time accuracy. To achieve good time-wise efficiency a multigrid technique is used to solve the highly time-consuming pressure-Poisson equation that requires to be solved at every time step. The speed-up achieved by multigrid is shown in tabular form. The performance and accuracy of the code are first ascertained by computing the flow in a single-sided lid-driven cubic cavity with good grid-economy and comparing the results available in the literature. The code, thus validated, is then applied to a new test problem we propose and various transient and asymptotically obtained steady-state results are presented. Given the care taken to establish the credibility of the code and the good spatio-temporal accuracy of the discretization, these results are accurate and may be used for ascertaining the performance of any computational algorithm applied to this test problem.