Commun. Comput. Phys., 8 (2010), pp. 51-94.

A Robust, Fully Adaptive Hybrid Level-Set/Front-Tracking Method for Two-Phase Flows with an Accurate Surface Tension Computation

Hector D. Ceniceros 1, Alexandre M. Roma 2, Aristeu Silveira-Neto 3, Millena M. Villar 3*

1 Department of Mathematics, University of California Santa Barbara, CA 93106, USA.
2 Departamento de Matematica Aplicada, Universidade de Sao Paulo, Caixa Postal 66281, CEP 05311-970, Sao Paulo-SP, Brasil.
3 Faculdade de Engenharia Mecanica, Universidade Federal de Uberlandia, CEP 38400-902, Uberlandia-MG, Brasil.

Received 5 May 2009; Accepted (in revised version) 14 October 2009
Available online 12 February 2010


We present a variable time step, fully adaptive in space, hybrid method for the accurate simulation of incompressible two-phase flows in the presence of surface tension in two dimensions. The method is based on the hybrid level set/front-tracking approach proposed in [H. D. Ceniceros and A. M. Roma, J. Comput. Phys., 205, 391-400, 2005]. Geometric, interfacial quantities are computed from front-tracking via the immersed-boundary setting while the signed distance (level set) function, which is evaluated fast and to machine precision, is used as a fluid indicator. The surface tension force is obtained by employing the mixed Eulerian/Lagrangian representation introduced in [S. Shin, S. I. Abdel-Khalik, V. Daru and D. Juric, J. Comput. Phys., 203, 493-516, 2005] whose success for greatly reducing parasitic currents has been demonstrated. The use of our accurate fluid indicator together with effective Lagrangian marker control enhance this parasitic current reduction by several orders of magnitude. To resolve accurately and efficiently sharp gradients and salient flow features we employ dynamic, adaptive mesh refinements. This spatial adaption is used in concert with a dynamic control of the distribution of the Lagrangian nodes along the fluid interface and a variable time step, linearly implicit time integration scheme. We present numerical examples designed to test the capabilities and performance of the proposed approach as well as three applications: the long-time evolution of a fluid interface undergoing Rayleigh-Taylor instability, an example of bubble ascending dynamics, and a drop impacting on a free interface whose dynamics we compare with both existing numerical and experimental data.

AMS subject classifications: 35R35, 65M50, 68U05, 74S20, 76D45, 76T99

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Key words: Front-tracking, immersed boundary method, level set method, adaptive mesh refinements, semi-implicit methods, multilevel multigrid, closest point transform, semi-backward difference formula.

*Corresponding author.
Email: (H. D. Ceniceros), (A. M. Roma), (A. Silveira-Neto), (M. M. Villar)

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