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

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.

💡 Research Summary

The paper presents a novel finite‑volume formulation for solving the full Navier‑Stokes equations on a moving, unstructured Voronoi mesh. Traditional high‑order schemes for compressible flow have largely been confined to structured, stationary grids, which suffer from large advection errors when dealing with complex geometries, moving boundaries, or strongly sheared flows. By allowing the mesh‑generating points to move arbitrarily—most naturally with the local fluid velocity—the authors create a quasi‑Lagrangian framework in which the mesh automatically adapts to the flow while preserving the topological regularity inherent to Voronoi tessellations. This property eliminates pathological cell twisting and avoids the need for frequent mesh reconnection or remeshing operations.

A central technical contribution is the construction of accurate viscous fluxes on a dynamically evolving mesh. The authors first compute spatial gradients of primitive variables (velocity, temperature, etc.) inside each cell using a least‑squares reconstruction that incorporates information from all neighboring cells. These gradients are then used to evaluate the viscous stress tensor at each cell face. Because the Voronoi faces change area and orientation as the mesh moves, the flux formulation includes additional terms that account for the time‑derivative of the face area and the motion of the face itself. This ensures strict conservation of mass, momentum, and energy even when the control volumes deform.

Implementation is carried out within the AREPO code, which already supports a moving‑mesh hydrodynamics solver for the Euler equations. The new viscous module extends AREPO to 2‑D and 3‑D Navier‑Stokes calculations. The authors validate the method with a suite of benchmark problems. In the circular Couette flow test, the numerical velocity profile matches the analytical solution to within 1 % across a range of Reynolds numbers, and the convergence rate is second‑order in space. In the flow past a cylindrical obstacle, the simulated boundary‑layer thickness, separation point, and drag coefficient agree closely with experimental data and high‑resolution fixed‑grid simulations, demonstrating that the method captures both shear‑driven and pressure‑driven viscous effects accurately. Additional tests involving high‑Mach number advection show that the moving mesh dramatically reduces advection errors compared with a static grid, while the viscous terms remain correctly represented.

The authors argue that this capability opens up a new class of astrophysical applications where physical viscosity cannot be ignored. Examples include the hot intracluster medium of galaxy clusters, where Braginskii viscosity influences turbulence and heat transport, and viscous accretion disks around compact objects, where angular momentum transport is governed by a physical shear viscosity rather than an artificial prescription. Because the mesh follows the flow, the method naturally resolves thin shear layers and reduces numerical diffusion, allowing the physical viscosity to dominate the dynamics.

In the discussion, the paper outlines future extensions: coupling the viscous solver with magnetohydrodynamics (MHD) to treat anisotropic Braginskii viscosity, incorporating multi‑phase or reactive flows, and developing adaptive mesh refinement (AMR) strategies that preserve the Voronoi structure while concentrating resolution where needed. The authors also note that the underlying mathematical framework could be adapted to other moving‑mesh codes, suggesting a broader impact beyond AREPO.

In summary, the work delivers a robust, geometrically flexible, and highly accurate scheme for compressible, viscous fluid dynamics on a moving Voronoi mesh. By integrating a rigorous viscous flux calculation with the inherent advantages of a quasi‑Lagrangian mesh, the authors achieve low advection error, strict conservation, and excellent agreement with analytical and experimental benchmarks. This positions the method as a competitive alternative to refined Eulerian solvers and paves the way for realistic, viscosity‑driven astrophysical simulations.