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

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 proposed here, based on a reformulation of the equation of state (EoS) in the Euler equations in fluid dynamics, the limit of which 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 for one‐dimensional shock‐tube tests, as well as examples of application for two‐dimensional shear flows, are shown here. 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.

Molecular dissipation in the nonlinear eddy viscosity in the Navier-Stokes equations: modelling of accretion discs

“…Nevertheless, this is not the conclusion of the full story because Lanzafame (2008Lanzafame ( , 2009 showed that a well bound viscous accretion disc structure modelling strongly depends on several conditions: the kinematic of the mass transfer, γ, αSS and so on. For isothermal or for a quasi-isothermal thermodynamics, a disc is structurally bound even for αSS = 0 because numerical dissipation only is enough to produce a disc in shocks events, a result also discussed by Sawada et al (1987); Spruit et al (1987) and, more recently, by Lanzafame (2010a) in its physical sense.…”