Flash evaporation Riemann Problem: Formulation and its Exact Solution

Flash evaporation, a liquid-to-gas phase transition phenomenon in real fluids, is prevalent in aerospace propulsion systems. To elucidate the physical mechanisms of such complex flows and provide theoretical benchmarks for Computational Fluid Dynamics simulations, this paper formalizes the Flash evaporation Riemann problem (FeRP) characterized by the expansion branch crossing the saturation line, within the framework of Homogeneous Equilibrium and Vapor-Liquid Equilibrium assumptions. An exact solution framework that analytically resolves all thermodynamic derivatives of equilibrium two-phase fluids is established for arbitrary two-parameter equations of state. By evaluating the Landau fundamental derivative, the non-classical wave structures arising in the FeRP are analyzed, for which a stable iterative solution strategy incorporating the Chapman-Jouguet condition as an outer constraint is proposed. Furthermore, an exact solution for the FeRP based on Wood’s mechanical equilibrium speed of sound is developed, enabling a comprehensive evaluation of its thermodynamic implications. Results indicate that Wood’s model alters the definition of the two-phase mixture entropy in the Euler equations, introducing an isentropic path characterized by a “density lag” effect and non-physical entropy decrease. Comparative analysis of the FeRP under typical scramjet fuel injection conditions reveals that, although Wood’s model captures the general trend of the Riemann solution curve, it significantly underestimates intermediate pressure, velocity, and the extent of vaporization relative to the complete equilibrium model.

💡 Research Summary

The paper addresses the phenomenon of flash evaporation—an abrupt liquid‑to‑gas phase transition triggered by rapid depressurization—in aerospace propulsion systems and formulates it as a Riemann problem (FeRP). By adopting the Homogeneous Equilibrium Model (HEM) together with a Vapor‑Liquid Equilibrium (VLE) model, the authors assume that the phase change occurs instantaneously on the saturation line, with liquid and vapor sharing the same pressure, temperature (equal to the saturation temperature), and velocity. This eliminates metastable effects and restores hyperbolicity to the Euler equations, allowing a self‑similar solution.

A major contribution is the derivation of complete thermodynamic relations for both single‑phase and equilibrium two‑phase mixtures from a generic two‑parameter equation of state (P=P(\rho,T)) and a caloric equation (C_v=C_v(\rho,T)). Internal energy, entropy, and sound speed are expressed analytically for arbitrary fluids, and the equilibrium sound speed (c_{eq}) is obtained from the ratio of entropy and pressure derivatives of the mixture. The authors also present the fundamental Landau derivative (\Gamma) and show that for typical propulsion fluids such as n‑dodecane and CO₂, (\Gamma>0) throughout both single‑phase and two‑phase regions, guaranteeing a classical wave structure (shocks, rarefactions, contacts) in most of the domain. However, when the expansion fan crosses the saturation line, rarefaction splitting occurs, producing composite waves that contain expansion shocks and compression fans. These non‑classical waves violate the Lax entropy condition but satisfy the more general Liu entropy condition, making them physically admissible.

To solve the FeRP exactly, the paper extends a Newton‑iteration framework previously developed for real‑fluid Riemann problems. The unknowns are the post‑wave states that satisfy the Rankine‑Hugoniot relations for shocks, continuity of pressure and velocity across contacts, and isentropic relations for rarefactions. The Chapman‑Jouguet (CJ) condition is imposed as an outer constraint, ensuring that the wave speeds remain physically admissible and that the solution converges even when composite waves are present. Careful initialization using the saturation line and entropy curves, together with appropriate scaling, yields robust convergence for a wide range of fluids and initial conditions.

The paper also scrutinizes the widely used Wood model, which defines the two‑phase sound speed as a simple mass‑fraction weighted average of the liquid and vapor sound speeds. By comparing Wood’s formulation with the full equilibrium thermodynamics, the authors demonstrate that Wood’s model effectively modifies the mixture entropy definition, leading to a “density lag” effect along isentropic paths and, paradoxically, a decrease in entropy—an unphysical result. Consequently, while Wood’s model may be computationally convenient, it lacks thermodynamic consistency.

A practical assessment is performed for scramjet fuel injection conditions, where n‑dodecane is subjected to high‑pressure, high‑temperature expansion. The exact equilibrium solution predicts significantly higher intermediate pressures, flow velocities, and vapor fractions than the Wood‑based solution. The discrepancy highlights the risk of under‑predicting flash evaporation intensity if Wood’s model is employed in design calculations.

In summary, the authors provide: (1) a rigorous formulation of the flash evaporation Riemann problem under HEM‑VLE assumptions; (2) analytical expressions for all required thermodynamic derivatives for arbitrary equations of state; (3) a stable Newton‑iteration algorithm augmented by the CJ condition to obtain exact solutions, including non‑classical composite waves; (4) a critical evaluation of Wood’s mechanical equilibrium sound speed, exposing its entropy‑consistency flaws; and (5) quantitative evidence that the full equilibrium model is essential for accurate prediction of flash evaporation in high‑speed propulsion applications. This work establishes a solid theoretical benchmark for CFD validation and offers clear guidance on the limitations of simplified sound‑speed models in two‑phase flow simulations.