For weak, nearly transonic shock reflection in dense gases (fluids with large specific heats near the thermodynamic critical region, where the fundamental derivative of gas dynamics can become small or negative), existing theory is local: transonic small-disturbance similarity analyses and shock-polar constructions determine the admissible local reflection configurations, including the carry-over of Guderley reflection, and predict that in certain parameter ranges the Mach stem disintegrates into a compound shock–wave fan structure that generates a secondary upstream shock. The global structure of the irregular-reflection pattern — the curved Mach stem of finite extent and the full cascade of supersonic patches downstream of the triple point — has not been captured analytically or numerically for any dense gas, and the compound-disintegration prediction remains unconfirmed by computation or experiment. The problem is to determine this global structure as a function of the dense-gas similarity parameters (the fundamental derivative Γ\Gamma, its density derivative Λ\Lambda, and the upstream Mach number) and to confirm or refute the predicted Mach-stem disintegration.

Background

When a weak planar shock reflects off a rigid wall or wedge at a small incidence angle, inviscid shock-reflection theory predicts a competition between regular reflection (a single reflected shock meeting the incident shock on the wall) and Mach reflection (a three-shock configuration with a Mach stem connecting the triple point to the wall). Von Neumann's two- and three-shock theories, built from oblique shock-polar analysis (a shock polar being the locus of post-wave states in the pressure–flow-deflection-angle plane attainable behind oblique waves), fail to explain the observed existence of a Mach-reflection configuration for very weak incident shocks where no three-shock solution exists — the von Neumann paradox. Guderley resolved the impasse kinematically by proposing that the triple point of weak Mach reflection is accompanied by a centered expansion fan and a sequence of supersonic flow patches embedded in subsonic flow; this Guderley reflection is now accepted as the correct weak-shock limit. Tabak and Rosales (1994) showed that in the weak-shock limit the reflection problem reduces to a scalar transonic small-disturbance equation (a nonlinear equation of mixed elliptic–hyperbolic type for a velocity potential, in which subsonic and supersonic regions coexist), and numerical investigations of this equation by Hunter and Brio (2000), Zakharian, Brio, Hunter and Webb (2000), Tesdall and Hunter (2002), and Tesdall, Sanders and Popivanov (2015) exhibit the full global structure: a slightly curved Mach stem of finite extent and an expanding cascade of supersonic patches separated by weak shocklets downstream of the triple point. This cascade has been observed experimentally in a shock tube (Cachucho and Skews 2012); no closed-form analytical global theory exists even for perfect gases. Dense gases are single-phase vapours of complex molecules (e.g. siloxanes and heavy fluorocarbons) operating near the thermodynamic critical or saturation region, where the equation of state is non-ideal. Their gas dynamics is governed by the fundamental derivative Γ=1+(ρ/2c2) ∂2p/∂ρ2∣s\Gamma = 1 + (\rho/2c^{2})\,\partial^{2}p/\partial\rho^{2}|_{s} (with ρ\rho the density, cc the sound speed and ss the entropy), which measures the nonlinearity of isentropes; in Bethe–Zel'dovich–Thompson (BZT) fluids Γ\Gamma takes negative values over a finite thermodynamic region, admitting rarefaction (expansion) shocks and composite waves impossible in perfect gases. Kluwick and Cox (2018) developed the steady small-disturbance theory of transonic dense-gas flow past compression and expansion ramps, and, with the oblique-wave and shock-polar theory for BZT fluids of Vimercati, Kluwick and Guardone (2018), Kluwick and Cox (2019) extended the canonical weak-shock channel-reflection problem to dense gases. Their analysis, expressed through the similarity parameters Γ\Gamma (fundamental derivative), Λ=ρ ∂Γ/∂ρ∣s\Lambda=\rho\,\partial\Gamma/\partial\rho|_{s} and the upstream Mach number MM, shows that the richer thermodynamics permits compound shock–wave fan reflections and pure wave-fan reflections near normal incidence, that Guderley reflection carries over to dense gases, and — the key novel prediction — that the Mach stem formed in irregular reflection may itself disintegrate into a compound shock–wave fan structure generating an additional secondary upstream shock. Their motivation is partly experimental: wind-tunnel realisations of non-classical gas-dynamic effects such as rarefaction shocks, predicted since Thompson (1971), have withstood detection in shock tubes, and the compound structures predicted here would provide distinctive signatures. The 2019 treatment is local and small-disturbance in character (a steady transonic channel-flow analysis of oblique-shock reflection, unlike the unsteady, self-similar perfect-gas theory before the numerical work of 2000–2015, whose reflection pattern depends on the time-scaled variables x/tx/t and y/ty/t rather than on space and time separately): it determines admissible configurations from local polar arguments but does not capture the global pattern — the finite-domain curved Mach stem and the downstream cascade of supersonic patches — nor does it settle in which regions of the (Γ,Λ,M)(\Gamma,\Lambda,M) parameter space the Mach stem actually disintegrates.

Problem Statement

Determine the global structure of weak, nearly transonic irregular shock reflection in dense gases, and settle the predicted disintegration of the Mach stem, as follows. Consider inviscid reflection of a weak shock from a plane rigid boundary in the steady dense-gas channel-flow configuration of Kluwick and Cox (2019), in a dense gas whose thermodynamic state is characterized by the similarity parameters of that analysis: the fundamental derivative of gas dynamics Γ\Gamma, its density derivative at constant entropy Λ=ρ ∂Γ/∂ρ∣s\Lambda=\rho\,\partial\Gamma/\partial\rho|_{s}, and the upstream Mach number MM, with the parameter class kept fully general (dense gases at large, order-one and negative Γ\Gamma, including but not restricted to Bethe–Zel'dovich–Thompson fluids). Two jointly posed results constitute the common objective: (1) capture the full global wave pattern of irregular (Guderley-type) reflection — the incident shock, reflected shock, centered fan, the curved Mach stem of finite extent attached to the reflecting boundary, and the cascade of supersonic patches and shocklets downstream of the triple point; and (2) determine for which values of the similarity parameters the Mach stem remains a classical shock and for which it disintegrates into a compound shock–wave fan structure generating a secondary upstream shock, thereby confirming, refining or refuting the Kluwick–Cox prediction, which currently rests on local polar analysis alone. The parameter map of the disintegration transition is read from the global structure.

References

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