Analytical solutions to shock and expansion waves for non-ideal equations of state

Abstract

We present analytical solutions to the stationary normal shock and centered rarefaction waves, which are valid for arbitrary non-ideal equations of state (EOS). Generalized shock functions are defined, which are shown to be well-behaved and locally convex, facilitating rapid and exact computation of shock ratios. For rarefactions, a novel domain mapping is used to derive flow variables as closed-form analytical functions in space and time, independent of the EOS. Results are discussed for transcritical and supercritical CO2. The solutions enable researchers to test shock-capturing codes designed for non-ideal flows, and the derivation strategy opens possibilities to revisit nonlinear hyperbolic conservation problems that traditionally lack analytical solutions.

An adaptive primitive-conservative scheme for high speed transcritical flow with an arbitrary equation of state

“…Previous studies have shown that the fluxes of the Euler equation are convex, hence this discontinuity should not exist in rarefaction. Wang et.al 35,36 recently suggested a structurally complete approximate Riemann solution (StARS) for transcritical flow in the context of cubic state equations in order to restore the expansion wave. Some complex EoS may affect the convexity of isentropes [37][38][39] , re-sulting in anomalous wave structures such as composite or split waves.…”