An approach to the Riemann problem in the light of a reformulation of the state equation for SPH inviscid ideal flows: a highlight on spiral hydrodynamics in accretion discs

TL;DR

This paper introduces a new approach to solving the Riemann problem in inviscid ideal flows by reformulating the equation of state, improving shock capturing in SPH simulations and enhancing the analysis of spiral structures in accretion discs.

Contribution

It proposes an empirical reformulation of the equation of state that simplifies Riemann problem solutions in SPH, reducing dependence on smoothing resolution and improving astrophysical flow modeling.

Findings

Effective Riemann solver for inviscid flows demonstrated on shock tube tests

Enhanced identification of spiral structures in accretion discs

Reduced resolution dependence in SPH shock capturing

Abstract

In physically inviscid 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) are to 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 problems. 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 an effective solution to the Riemann problem excludes any dependence, in the transport phenomena, on particle smoothing resolution length $h$ in non viscous SPH flows. Results on 1D shock tube tests, as well as examples of application for 2D turbulence and 2D shear flows are here shown. As an astrophysical application, a much better identification of spiral structures in accretion discs in a close binary (CB), as a result of this reformulation is also shown here.