Numerical Simulation of Phase Transition with the Hyperbolic Godunov-Peshkov-Romenski Model

Abstract

In this paper, a thermodynamically consistent solution of the interfacial Riemann problem for the first-order hyperbolic continuum model of Godunov, Peshkov and Romenski (GPR model) is presented. In the presence of phase transition, interfacial physics are governed by molecular interaction on a microscopic scale, beyond the scope of the macroscopic continuum model in the bulk phases. The developed two-phase Riemann solvers tackle this multi-scale problem, by incorporating a local thermodynamic model to predict the interfacial entropy production. Using phenomenological relations of non-equilibrium thermodynamics, interfacial mass and heat fluxes are derived from the entropy production and provide closure at the phase boundary. We employ the proposed Riemann solvers in an efficient sharp interface level-set Ghost-Fluid framework to provide coupling conditions at phase interfaces under phase transition. As a single-phase benchmark, a Rayleigh-Bénard convection is studied to compare the hyperbolic thermal relaxation formulation of the GPR model against the hyperbolic-parabolic Euler-Fourier system. The novel interfacial Riemann solvers are validated against molecular dynamics simulations of evaporating shock tubes with the Lennard-Jones shifted and truncated potential. On a macroscopic scale, evaporating shock tubes are computed for the material n-Dodecane and compared against Euler-Fourier results. Finally, the efficiency and robustness of the scheme is demonstrated with shock-droplet interaction simulations that involve both phase transfer and surface tension, while featuring severe interface deformations.

Figures (19)

• Figure 1: Exact and approximate wave pattern for a GPR two-phase Riemann problem with phase transition. The non-classical wave of the phase interface, associated with the velocity of the phase boundary $s^{\#}$, is highlighted in pink. The remaining waves are shock or rarefaction waves.
• Figure 2: Flowchart, illustrating the iterative solution procedure of the $\text{HLLP}_{mq}$ Riemann solver.
• Figure 3: Flowchart, illustrating the iterative solution procedures of the $\text{HLLP}_{m}$ Riemann solver.
• Figure 4: Computational setup for the one-dimensional heat conduction test case. Heat fluxes are imposed on the left and right boundaries.
• Figure 5: Temperature profile of the one-dimensional heat conduction test case at different time instances. The heat conductivity is increased from left to right and the GPR model is compared against a Euler-Fourier reference solution.