Artificial Compressibility Approaches in Flux Reconstruction for Incompressible Viscous Flow Simulations

TL;DR

This paper compares artificial compressibility methods for incompressible flow simulations using flux reconstruction, highlighting the strengths and limitations of EDAC and hyperbolised ACM across different flow scenarios.

Contribution

It provides a detailed comparison of EDAC and hyperbolised ACM methods within flux reconstruction, revealing their performance and stability characteristics in various flow regimes.

Findings

EDAC improves divergence reduction linearly as compressibility decreases

EDAC performs well on Taylor-Green vortex but causes early transition in complex flows

Hyperbolised ACM can be beneficial but increases setup complexity

Abstract

Several competing artificial compressibility methods for the incompressible flow equations are examined using the high-order flux reconstruction method. The established artificial compressibility method (ACM) of \citet{Chorin1967} is compared to the alternative entropically damped (EDAC) method of \citet{Clausen2013}, as well as an ACM formulation with hyperbolised diffusion. While the former requires the solution to be converged to a divergence free state at each physical time step through pseudo iterations, the latter can be applied explicitly. We examine the sensitivity of both methods to the parameterisation for a series of test cases over a range of Reynolds numbers. As the compressibility is reduced, EDAC is found to give linear improvements in divergence whereas ACM yields diminishing returns. For the Taylor--Green vortex, EDAC is found to perform well; however on the more challenging circular cylinder at $Re=3900$, EDAC gives rise to early transition of the free shear-layer and over-production of the turbulence kinetic energy. This is attributed to the spatial pressure fluctuations of the method. Similar behaviour is observed for an aerofoil at $Re=60,000$ with an attached transitional boundary layer. It is concluded that hyperbolic diffusion of ACM can be beneficial but at the cost of case setup time, and EDAC can be an efficient method for incompressible flow. However, care must be taken as pressure fluctuations can have a significant impact on physics and the remedy causes the governing equation to become overly stiff.