Numerical analysis/Mathematical problems in mechanics
A kinematic vector penalty–projection method for incompressible flow with variable density
Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1124-1131.

In this Note, we present a new version of the vector penalty–projection splitting method described in [1] for the fast numerical computation of incompressible flows with variable density and viscosity. We show that the velocity correction can be made completely independent of the mass density ρ. Hence, this step is purely kinematic using the fast Helmholtz–Hodge decompositions proposed in [2]. Then, it is shown that the dynamic step of pressure gradient correction can be fast and locally consistent on edge-based generalized MAC-type unstructured meshes that naturally verify the compatibility condition in the proposed discrete setting. By the way, a new accurate front-tracking Lagrangian-advection technique is also introduced for multiphase flows.

This new method preserves the fully vector formulation of both the prediction and correction steps of the original scheme, the primary unknowns being (v,p) and ρ by advection, since the pressure Neumann–Poisson problem remains eliminated. The efficiency of the present method is demonstrated through numerical results on sharp test cases.

On présente dans cette Note une nouvelle version de la méthode de splitting par pénalité–projection vectorielle décrite dans [1] pour le calcul des écoulements incompressibles à masse volumique et viscosité variables. Le principal résultat est de rendre la correction vectorielle de vitesse complètement indépendante de la masse volumique ρ. Cette étape devient donc purement cinématique et correspond à une décomposition rapide de Helmholtz–Hodge proposée dans [2]. On montre que l'étape dynamique de correction du gradient de pression peut être rapide et localement consistante sur des maillages généralisés de type MAC non structurés.

Published online:
DOI: 10.1016/j.crma.2016.06.007
Angot, Philippe 1; Caltagirone, Jean-Paul 2; Fabrie, Pierre 3

1 Aix-Marseille Université, Institut de mathématiques de Marseille – CNRS UMR 7373, Centrale Marseille, 39, rue Frédéric-Joliot-Curie, 13453 Marseille cedex 13, France
2 Université de Bordeaux & IPB, Institut de mécanique et d'ingénierie de Bordeaux – CNRS UMR 5295, 16, avenue Pey-Berland, 33607 Pessac, France
3 Université de Bordeaux & IPB, Institut de mathématiques de Bordeaux – CNRS UMR 5251, ENSEIRB–MATMECA, 33400 Talence, France
     author = {Angot, Philippe and Caltagirone, Jean-Paul and Fabrie, Pierre},
     title = {A kinematic vector penalty{\textendash}projection method for incompressible flow with variable density},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1124--1131},
     publisher = {Elsevier},
     volume = {354},
     number = {11},
     year = {2016},
     doi = {10.1016/j.crma.2016.06.007},
     language = {en},
     url = {}
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PB  - Elsevier
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DO  - 10.1016/j.crma.2016.06.007
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%A Caltagirone, Jean-Paul
%A Fabrie, Pierre
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Angot, Philippe; Caltagirone, Jean-Paul; Fabrie, Pierre. A kinematic vector penalty–projection method for incompressible flow with variable density. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1124-1131. doi : 10.1016/j.crma.2016.06.007.

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