# Modelling interactions of waves with diffused interfaces

Kevin Schmidmayer, Joris Cazé, Fabien Petitpas, Eric Daniel, Nicolas Favrie

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Kevin Schmidmayer, Joris Cazé, Fabien Petitpas, Eric Daniel, Nicolas Favrie. Modelling interactions of waves with diffused interfaces. MultiMat 2022 - 10th International Conference on Numerical Methods for Multi-Material Fluid Flow, Aug 2022, Zürich, Switzerland. hal-03752096

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Submitted on 16 Aug 2022

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Modelling interactions of waves with diffused interfaces

K. Schmidmayer†*, J. Cazé†‡, F. Petitpas†, É. Daniel† and N. Favrie†

† Aix Marseille Univ, CNRS, IUSTI, Marseille, France (kevin.schmidmayer@univ-amu.fr, fabien.petitpas@univ-amu.fr, eric.daniel@univ-amu.fr, nicolas.favrie@univ-amu.fr)

‡ Centre National d'Etudes Spatiales, Paris, France (joris.caze@univ-amu.fr)

* INRIA Bordeaux Sud-Ouest, project-team CAGIRE, Universite de Pau et des Pays de l'Adour, E2S UPPA, Laboratory of Mathematics and Applied Mathematics (LMAP), Pau, France

**Keywords:** diffuse-interface method, multiphase, compressible, interface, interaction, relaxation.

ABSTRACT

When simulating multiphase compressible flows using the diffuse-interface methods, the test cases pre-
sented 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 interac-
tions 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, such as the model of Kapila et al. [1], 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 [2] 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.

**References**

[1] A. Kapila, R. Menikoff, J. Bdzil, S. Son and D. Stewart, “Two-phase modeling of DDT in granular materials: Reduced equations”, *Phys. Fluids*, **13**, pp. 3002–3024, 2001.

[2] R. Saurel, F. Petitpas and R. Berry, “Simple and efficient relaxation methods for interfaces sepa-
rating compressible fluids, cavitating flows and shocks in multiphase mixtures”, *J. Comp. Phys.*,
228(5), pp. 1678–1712, 2009.

K.S. and N.F. acknowledge support from the A*MIDEX and the ANR under grants ANR-11-LABX-0092, ANR-11-IDEX-0001-02 and ANR-ASTRID project SNIP ANR-19-ASTR-0016-01. J.C., F.P. and E.D. acknowledge support from the CNES under grant RT-CT-2310000-2001-CNES.