Modelling interactions between waves and diffused interfaces

Abstract

When simulating multiphase compressible flows using the diffuse-interface methods, the test cases presented in the literature to validate the modellings with regard to interface problems are always textbook cases: interfaces are sharp and the simulations therefore easily converge to the exact solutions. In real problems, it is rather different because the waves encounter moving interfaces which consequently have already undergone the effects of numerical diffusion. Numerical solutions resulting from the interactions of waves with diffused interfaces have never been precisely investigated and for good reasons, the results obtained are extremely dependent on the model used. Precisely, well-posed models present similar and important issues when such an interaction occurs, coming from the appearance of a wave-trapping phenomenon.To circumvent those issues, we propose to use a thermodynamically-consistent pressure-disequilibrium model with finite, instead of infinite, pressure-relaxation rate to overcome the difficulties inherent in the computation of these interactions. Because the original method to solve this model only enables infinite relaxation, we propose a new numerical method allowing infinite as well as finite relaxation rates. Solutions of the new modelling are examined and compared to literature, in particular we propose the study of a shock on a water-air interface, but also for problems of helium-air and water-air shock tubes, spherical and non-spherical bubble collapses.

A phenomenological analysis of droplet shock-induced cavitation using a multiphase modeling approach

“…The method corrects the total energy before the relaxation procedure, during the flux computation of the hyperbolic step, and therefore allows finite or infinite relaxations. 18 The relaxation terms (system of ordinary differential equations) are integrated with a first-order, explicit, Euler scheme with time step subdivisions. 18 The number of subdivisions is adapted at each time step to verify the volume-fraction and pressure constraints.…”

“…18 The relaxation terms (system of ordinary differential equations) are integrated with a first-order, explicit, Euler scheme with time step subdivisions. 18 The number of subdivisions is adapted at each time step to verify the volume-fraction and pressure constraints. During this procedure, if the pressures are completely relaxed, that is, a unique pressure for all phases, we terminate the Euler scheme and we perform from the initial state an infinite-relaxation procedure 20 to guarantee a unique pressure and better estimate the solution.…”

“…We use herein a slightly modified version of the modeling proposed by Schmidmayer et al 18 to simulate the compression and expansion of each phase within the liquid-gas mixture, while ignoring phase change. The modification is only related to the form of the pressure-relaxation terms (right-hand side) and is detailed in Sec.…”

“…In most if not all the literature, for example, Schmidmayer et al, 18 Saurel et al, 20 Baer and Nunziato, 22 Saurel and Abgrall, 23 the relaxation coefficients are taken as unique and constant for all interactions, that is, l k;j ¼ l.…”