An approach to the Riemann problem for SPH inviscid ideal flows

TL;DR

This paper proposes a new empirical approach to solving the Riemann problem in inviscid ideal flows within SPH methods, eliminating resolution dependence and demonstrating effectiveness through shock tube tests.

Contribution

It introduces an empirical reformulation of the equation of state for the Riemann problem in SPH, removing resolution dependence in non-viscous flows.

Findings

Effective solution for 1D shock tube tests

Eliminates dependence on particle resolution length

Compatible with inviscid ideal gas flows

Abstract

In the physically non viscous fluid dynamics, "shock capturing" methods adopt either an artificial viscosity contribution or an appropriate Riemann solver algorithm. These techniques are necessary to solve the strictly hyperbolic Euler equations if flow discontinuities (the Riemann problem) must be solved. A necessary dissipation is normally used in such cases. An explicit artificial viscosity contribution is normally adopted to smooth out spurious heating and to treat transport phenomena. Such a treatment of inviscid flows is also widely adopted in the Smooth Particle Hydrodynamics (SPH) finite volume free Lagrangian scheme. In other cases, the intrinsic dissipation of Godunov - type methods is implicitly useful. Instead "shock tracking" methods normally use the Rankine - Hugoniot jump conditions to solve such problem. A simple, effective solution of the Riemann problem in inviscid ideal gases is here proposed, based on an empirical reformulation of the equation of state (EoS) in the Euler equations in fluid dynamics, whose limit for a motionless gas coincides with the classical EoS of ideal gases. The application of such effective solution of the Riemann problem excludes any dependence, in the transport phenomena, on particle resolution length $h$ in non viscous SPH flows. Results on 1D shock tube tests are here shown.