Comparisons of weakly-compressible and truly incompressible approaches for viscous flow into a high-order Cartesian-grid finite volume framework

Abstract

An original strategy to address hydrodynamic flow was recently proposed through a high-order weakly-compressible Cartesian grid approach [1]. The method, named Weakly-Compressible Cartesian hydrodynamics (WCCH), is based on a fully-explicit temporal scheme for solving the Navier-Stokes equations while implicit incompressible schemes are usually preferred in the literature to address such flows. The present study aims to position and compare the WCCH method with a standard incompressible formulation. To this end, an incompressible scheme has been implemented in the same numerical framework. As far as possible, the algorithm used in the incompressible approach has been designed to be the same as (or close to) the one used in the weakly-compressible approach. In particular, high-order schemes for spatial and time discretization are employed. Pros and cons for each formulation are discussed in conjunction with a series of test cases on extensive criteria including implementation convenience, easy use of mesh refinement, convergence order and accuracy, numerical diffusion, parallel CPU scaling for high performance computing, etc. These comparisons demonstrate the relevance of the incompressible approach, at least for the selected test cases.