A class of high-resolution algorithms for incompressible flows

We present a class of a high-resolution Godunov-type algorithms for solving flow problems governed by the incompressible Navier–Stokes equations. The algorithms use high-resolution finite volume methods developed in LeVeque (SIAM J Numer Anal 1996;33:627–665) for the advective terms and finite difference methods for the diffusion and the Poisson pressure equation. The high-resolution algorithm advects the cell-centered velocities using the divergence-free cell-edge velocities. The resulting cell-centered velocity is then updated by the solution of the Poisson equation. The algorithms are proven to be robust for constant-density flows at high Reynolds numbers via an example of lid-driven cavity flow. With a slight modification for the projection operator in the constant-density solvers, the algorithms also solve incompressible flows with finite-amplitude density variation. The strength of such algorithms is illustrated through problems like Rayleigh–Taylor instability and the Boussinesq equations for Rayleigh–Bénard convection. Numerical studies of the convergence and order of accuracy for the velocity field are provided. While simulations for two-dimensional regular-geometry problems are presented in this study, in principle, extension of the algorithms to three dimensions with complex geometry is feasible.