Abstract
We consider two-fluid flow problems, where each fluid is governed by the Navier-Stokes equations and the surface tension proportional to the curvature acts on the interface. The domain which each fluid occupies is unknown, and the interface moves with the velocity of the particle on it. We have developed an energy-stable Lagrange-Galerkin finite element scheme for the two-fluid flow problems. It maintains not only the advantages of Lagrange-Galerkin method of the robustness to high-Reynolds numbers and of the symmetry of the resultant matrix but also the property of energy-stability under the condition of the smoothness of the interface. Here we perform numerical simulation of the behavior of a rising bubble by the scheme.
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References
Bänsch, C.: Finite element discretization of the Navier–Stokes equations with a free capillary surface. Numer. Math. 88, 203–235 (2001)
Notsu, H., Tabata, M.: Error estimates of a pressure-stabilized characteristics finite element scheme for the Oseen equations. J. Sci. Comput. 65, 940–955 (2015)
Notsu, H., Tabata, M.: Error estimates of a stabilized Lagrange–Galerkin scheme for the Navier–Stokes equations. ESAIM Math. Model. Numer. Anal. 50, 361–380 (2016)
Pironneau, O.: Finite Element Methods for Fluids. Wiley, Chichester (1989)
Pironneau, O., M. Tabata, M.: Stability and convergence of a Galerkin-characteristics finite element scheme of lumped mass type. Int. J. Numer. Methods Fluids 64, 1240–1253 (2010)
Prosperetti, A., Tryggvason, G.: Computational Methods for Multiphase Flow. Cambridge University Press, Cambridge (2009)
Rui, H., Tabata, M.: A mass-conservative characteristic finite element scheme for convection-diffusion problems. J. Sci. Comput. 43, 416–432 (2010)
Süli, E.: Convergence and nonlinear stability of the Lagrange–Galerkin method for the Navier–Stokes equations. Numer. Math. 53, 459–483 (1988)
Tabata, M.: Finite element schemes based on energy-stable approximation for two-fluid flow problems with surface tension. Hokkaido Math. J. 36, 875–890 (2007)
Tabata, M.: Numerical simulation of fluid movement in an hourglass by an energy-stable finite element scheme. In: Hafez, M.N., Oshima, K., Kwak, D. (eds.) Computational Fluid Dynamics Review 2010, pp. 29–50. World Scientific, Singapore (2010)
Tabata, M.: Energy-stable Lagrange-Galerkin schemes for two-fluid flow problems (to appear)
Tezduyar, T.E., Behr, M., Liou, J.: A new strategy for finite element computations involving boundaries and interfaces - the deforming-spatial-domain /space-time procedure: I. Comput. Methods Appl. Mech. Eng. 94, 339–351 (1992)
Acknowledgements
The author was supported by JSPS (Japan Society for the Promotion of Science) under Grants-in-Aid for Scientific Research (C), No. 25400212 and (S), No. 24224004 and under the Japanese-German Graduate Externship (Mathematical Fluid Dynamics) and by Waseda University under Project research, Spectral analysis and its application to the stability theory of the Navier-Stokes equations of Research Institute for Science and Engineering.
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Tabata, M. (2016). Numerical Simulation of the Behavior of a Rising Bubble by an Energy-Stable Lagrange-Galerkin Scheme. In: Bazilevs, Y., Takizawa, K. (eds) Advances in Computational Fluid-Structure Interaction and Flow Simulation. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-40827-9_10
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DOI: https://doi.org/10.1007/978-3-319-40827-9_10
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