Multi-Dimensional, Compressible Viscous Flow on a Moving Voronoi Mesh

TL;DR

This paper introduces a novel finite volume scheme for viscous flow on a moving Voronoi mesh, enabling dynamic mesh adjustment with flow, reducing errors, and improving accuracy in complex geometries.

Contribution

It presents a new formulation of viscous hydrodynamics on a moving Voronoi mesh, including a method for accurate viscous flux computation, implemented in the AREPO code for 2D and 3D simulations.

Findings

The scheme achieves good accuracy in test problems.

It offers high geometric flexibility for complex flows.

It is suitable for astrophysical applications involving viscosity.

Abstract

Numerous formulations of finite volume schemes for the Euler and Navier-Stokes equations exist, but in the majority of cases they have been developed for structured and stationary meshes. In many applications, more flexible mesh geometries that can dynamically adjust to the problem at hand and move with the flow in a (quasi) Lagrangian fashion would, however, be highly desirable, as this can allow a significant reduction of advection errors and an accurate realization of curved and moving boundary conditions. Here we describe a novel formulation of viscous continuum hydrodynamics that solves the equations of motion on a Voronoi mesh created by a set of mesh-generating points. The points can move in an arbitrary manner, but the most natural motion is that given by the fluid velocity itself, such that the mesh dynamically adjusts to the flow. Owing to the mathematical properties of the Voronoi tessellation, pathological mesh-twisting effects are avoided. Our implementation considers the full Navier-Stokes equations and has been realized in the AREPO code both in 2D and 3D. We propose a new approach to compute accurate viscous fluxes for a dynamic Voronoi mesh, and use this to formulate a finite volume solver of the Navier-Stokes equations. Through a number of test problems, including circular Couette flow and flow past a cylindrical obstacle, we show that our new scheme combines good accuracy with geometric flexibility, and hence promises to be competitive with other highly refined Eulerian methods. This will in particular allow astrophysical applications of the AREPO code where physical viscosity is important, such as in the hot plasma in galaxy clusters, or for viscous accretion disk models.