An unfitted interior penalty discontinuous Galerkin method for incompressible Navier–Stokes two‐phase flow |
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Authors: | F Heimann C Engwer O Ippisch P Bastian |
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Institution: | 1. MAPMO UMR CNRS 7349, Université d'Orléans, UFR Sciences, Batiment de mathématiques, B.P. 6759, F‐45067 Orléans cedex 2, France;2. Institut Jean Le Rond d'Alembert, CNRS & UPMC Université Paris 06, UMR 7190, 4 place Jussieu, Bo?te 162, F‐75005 Paris, France;3. Correspondence to: Olivier Delestre, Laboratoire de Mathématiques J.A. Dieudonné & Polytech Nice‐Sophia, Université de Nice ‐ Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France.;4. E‐mail: Delestre@unice.fr;5. INRA, UR0272, UR Science du sol, Centre de recherche d'Orléans, CS 40001, F‐45075 Orléans Cedex 2, France;6. Presently at: Department of Applied Mathematics, National University of Ireland, Galway, Republic of Ireland. |
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Abstract: | A discontinuous Galerkin method for the solution of the immiscible and incompressible two‐phase flow problem based on the nonsymmetric interior penalty method is presented. Therefore, the incompressible Navier–Stokes equation is solved for a domain decomposed into two subdomains with different values of viscosity and density as well as a singular surface tension force. On the basis of a piecewise linear approximation of the interface, meshes for both phases are cut out of a structured mesh. The discontinuous finite elements are defined on the resulting Cartesian cut‐cell mesh and may therefore approximate the discontinuities of the pressure and the velocity derivatives across the interface with high accuracy. As the mesh resolves the interface, regularization of the density and viscosity jumps across the interface is not required. This preserves the local conservation property of the velocity field even in the vicinity of the interface and constitutes a significant advantage compared with standard methods that require regularization of these discontinuities and cannot represent the jumps and kinks in pressure and velocity. A powerful subtessellation algorithm is incorporated to allow the usage of standard time integrators (such as Crank–Nicholson) on the time‐dependent mesh. The presented discretization is applicable to both the two‐dimensional and three‐dimensional cases. The performance of our approach is demonstrated by application to a two‐dimensional benchmark problem, allowing for a thorough comparison with other numerical methods. Copyright © 2012 John Wiley & Sons, Ltd. |
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Keywords: | discontinuous Galerkin unfitted finite elements Navier– Stokes two‐phase flow Cartesian cut‐cell methods level set methods |
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