Extension of the finite volume particle method to viscous flow

The finite volume particle method (FVPM) is a mesh-free method for fluid dynamics which allows simple and accurate implementation of boundary conditions and retains the conservation and consistency properties of classical finite volume methods. In this article, the FVPM is extended to viscous flows using a consistency-corrected smoothed particle hydrodynamics (SPH) approximation to evaluate velocity gradients. The accuracy of the viscous FVPM is improved by a higher-order discretisation of the inviscid flux combined with a second-order temporal discretisation. The higher-order inviscid FVPM is validated for a 1-D shock tube problem, in which it demonstrates an enhanced shock capturing ability. For two-dimensional simulations, a small arbitrary Lagrange-Euler correction to fully Lagrangian particle motion is beneficial in maintaining a favourable particle distribution over long simulation times. The viscous FVPM is validated for two-dimensional Poiseuille, Taylor-Green and lid-driven cavity flows, and good agreement is achieved with analytic or reference numerical solutions. These results establish the viability of FVPM as a tool for mesh-free simulation of viscous flows in engineering.