The Riemann problem for fluid flow of real materials is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. The properties of the isentropes and the shock Hugoniot loci that follow from conditions imposed on the equation of state are reviewed systematically. Important properties of these wave curves are determined by three dimensionless variables characterizing the equation of state: the adiabatic exponent $\ensuremath{\gamma}$, the Gr\"uneisen coefficient $\ensuremath{\Gamma}$, and the fundamental derivative $\mathcal{G}$. Standard assumptions on these variables break down near phase transitions. The result is an anomalous wave structure: either shock waves split into multiple waves, or composite waves form. Additional questions related to shock stability and nonuniqueness of the solution of the Riemann problem are discussed.

The Riemann problem for fluid flow of real materials

A hyperbolic two-phase flow model involving five partial differential equations is constructed for liquid-gas interface modelling. The model is able to deal with interfaces of simple contact where normal velocity and pressure are continuous as well as transition fronts where heat and mass transfer occur, involving pressure and velocity jumps. These fronts correspond to extra waves in the system. The model involves two temperatures and entropies but a single pressure and a single velocity. The closure is achieved by two equations of state that reproduce the phase diagram when equilibrium is reached. Relaxation toward equilibrium is achieved by temperature and chemical potential relaxation terms whose kinetics is considered infinitely fast at specific locations only, typically at evaporation fronts. Thus, metastable states are involved for locations far from these fronts. Computational results are compared to the experimental ones. Computed and measured front speeds are of the same order of magnitude and the same tendency of increasing front speed with initial temperature is reported. Moreover, the limit case of evaporation fronts propagating in highly metastable liquids with the Chapman-Jouguet speed is recovered as an expansion wave of the present model in the limit of stiff thermal and chemical relaxation.

/pdf/modelling-phase-transition-in-metastable-liquids-application-3q040zs8m5.pdf

Modelling phase transition in metastable liquids: Application to cavitating and flashing flows.

Dynamics of first order phase transitions

https://www.ldeo.columbia.edu/~ruprecht/Site/Classes_files/Week11_Toramaru.pdf

BND (bubble number density) decompression rate meter for explosive volcanic eruptions

Localized phase transitions as well as shock waves can often be modeled by material discontinuities satisfying Rankine-Hugoniot (RH) jump conditions. The use of Maxwell, Gibbs-Thompson, Hertz-Knudsen, and similar (supplementary to RH) relations in the theory of dynamic phase changes suggest that the classical system of jump conditions is at least incomplete in the case of phase transitions. While the propagation of a shock wave is completely determined by the conservations laws, the boundary conditions of the problem and the condition that the entropy increases in the process, the same is not true for the propagation of phase boundaries. Additional condition must be added to the RH conditions in order to provide sufficient data for the unique determination of the transformation process. The necessity was tacitly assumed by those who attacked the calculation of the phase boundary velocity without even trying to determine this parameter from the conservation laws and boundary conditions alone.

Kinks versus Shocks

Negative or rarefaction shock waves may exist in single-phase fluids under certain conditions. It is necessary that a particular fluid thermodynamic quantity Γ ≡ −½δ In (δP/δν)s/δ In ν be negative: this condition appears to be met for sufficiently large specific heat, corresponding to a sufficient level of molecular complexity. The dynamic formation and evolution of a negative shock is treated, as well as its properties. Such shocks satisfy stability conditions and have a positive, though small, entropy jump. The viscous shock structure is found from an approximate continuum model. Possible experimental difficulties in the laboratory production of negative shocks are briefly discussed.

Negative shock waves

The emergence of a shockwave from the open end of a shock tube is studied, with special emphasis on test fluids of high molar heat capacity, i.e. retrograde fluids. A variety of wavelike vapour-liquid phase changes are observed in such fluids, including the liquefaction shock, mixture-evaporation shock, condensation waves associated with shock splitting and liquid-evaporation waves (these phenomena have analogues in the polymorphic phase changes of solids; only the first two are treated in this paper). The open end of the shock-tube test section discharges into an observation chamber where photographs of the emerging flow are taken. Calculations were performed with the Benedict-Webb-Rubin, van der Waals and other equations of state. Numerical (finite-difference) predictions of the flow were made for single-phase and two-phase flows: solutions were tested against the experimental shock diffraction and vortex data of Skews. The phase-change properties of the test fluid can be quantified by the ‘retrogradicity’ r(T), measuring the difference in slope between the P, T isentrope and the vapour-pressure curve, and the ‘kink’ k(T), measuring the difference between the single-phase and mixture sound speeds. Mixture-evaporation (i.e. rarefaction) shocks appear to have a sonic-sonic or double Chapman-Jouguet structure and show agreement with amplitude predictions based on k(T). Liquefaction shocks are found to show a reproducible transition from regular, smooth shock fronts to irregular, chaotic shock fronts with increasing shock Mach number. This transition can be correlated with published stability limits.

Shock waves and phase changes in a large-heat-capacity fluid emerging from a tube

Adiabatic Waves in Liquid-Vapor Systems

The splitting of a single pressure discontinuity into a propagating two-wave system is studied for the case of saturated-liquid expansion (liquid-evaporation wave splitting) and vapour compression (vapour-condensation wave splitting). Experimental results from the Max-Planck-Institut fur Stromungsforschung and from Rensselaer Polytechnic Institute show that splitting occurs in test fluids of large molar heat capacity, such as iso-octane (Cv0/R ≈ 37). Each of the two forms of splitting results in a single-phase forerunner wave carrying a pressure discontinuity followed by a phase-change wave, also with a pressure discontinuity. The thermodynamic state between the forerunner wave and the phase-change wave is metastable (supersaturated liquid or vapour). The waves are quantitatively described by systems of adiabats, e.g. shock adiabats. It appears that nucleation processes are predominantly homogeneous.In vapour-compression shock-wave splitting, a combined wave (liquefaction shock) splits into discrete forerunner and condensation waves at a triple point, the intersection of a liquefaction shockfront, forerunner shock and condensation discontinuity: such a point occurs just at critical supersaturation (i.e. the Wilson-line state), where condensation is spontaneous and immediate. For shock waves that produce a metastable state of subcritical supersaturation, condensation is delayed, that is, the condensation discontinuity propagates more slowly; for a split-shock system, the condensation discontinuity propagates subsonically. The pressure amplitude of a real split-shock system is much larger than that predicted by an equilibrium model.In liquid-evaporation wave splitting, the forerunner wave is an acoustic expansion wave and the second wave an evaporation wave with a propagation velocity approximately determined by the Chapman-Jouguet condition for deflagration. Such evaporation wavefronts are increasingly distinct as the temperature approaches the critical-point value. The evaporation rates across the wavefront are comparable to those found in vapour explosions.

Wave splitting in a fluid of large heat capacity

Shock discontinuities for which the upstream state is vapour and the downstream state is liquid are considered. The possibility of such shock waves is associated with a large number of molecular degrees of freedom: it is necessary that the ideal-gas specific heat cv [ges ] 24R at the critical temperature, a condition which is met by several common fluids. Shock properties are found from a corresponding-states thermodynamic model and from several calculations based on data for particular fluids. Condensation shock waves satisfy the usual stability conditions and should be found in laboratory experiments.

Philip A. Thompson

Papers

Negative shock waves

Shock waves and phase changes in a large-heat-capacity fluid emerging from a tube

Adiabatic Waves in Liquid-Vapor Systems

Wave splitting in a fluid of large heat capacity

On the possibility of complete condensation shock waves in retrograde fluids