×

zbMATH — the first resource for mathematics

An adaptive level set approach for incompressible two-phase flows. (English) Zbl 0930.76068
Summary: We present a numerical method using the level set approach for solving incompressible two-phase flow with surface tension. In the level set approach, the free surface is represented as the zero level set of a smooth function; this has the effect of replacing the advection of density, which has steep gradients at the free surface, with the advection of the level set function, which is smooth. In addition, the free surface can merge or break up with no special treatment. We maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects. In this work, we couple the level set scheme to an adaptive projection method for the incompressible Navier-Stokes equations, in order to achieve higher resolution of the free surface with a minimum of additional expense. We present two-dimensional axisymmetric and fully three-dimensional results of air bubble and water drop computations.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76T10 Liquid-gas two-phase flows, bubbly flows
76D45 Capillarity (surface tension) for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
Software:
Wesseling
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alcouffe, R.E; Brandt, A; Dendy, J.E; Painter, J.W, The multi-grid method for the diffusion equation with strongly discontinuous coefficients, SIAM J. sci. stat. comput., 2, 430, (1981) · Zbl 0474.76082
[2] Allen, R.R; Meyer, J.D; Knight, W.R, Thermodynamics and hydrodynamics of thermal ink jets, Hewlett-packard J., (May 1985)
[3] A. S. Almgren, J. B. Bell, P. Colella, L. H. Howell, M. Welcome, A conservative adaptive projection method for the incompressible Navier-Stokes equations in three dimensions, J. Comput. Phys. · Zbl 0933.76055
[4] Almgren, A.S; Bell, J.B; Colella, P; Marthaler, T, A Cartesian grid projection method for the incompressible Euler equations in complex geometries, SIAM J. sci. comput., 18, 1289, (1997) · Zbl 0910.76040
[5] Almgren, A.S; Bell, J.B; Szymczak, W.G, A numerical method for the incompressible navier – stokes equations based on an approximate projection, SIAM J. sci. comput., 17, (1996) · Zbl 0845.76055
[6] Bell, J.B; Berger, M.J; Saltzman, J.S; Welcome, M.L, Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM J. sci. comput., 15, 1277, (1994)
[7] Bell, J.B; Colella, P; Glaz, H.M, A second-order projection method for the incompressible navier – stokes equations, J. comput. phys., 85, 257, (1989) · Zbl 0681.76030
[8] J. B. Bell, P. Colella, L. H. Howell, An efficient second-order projection method for viscous incompressible flow, 10th AIAA Computational Fluid Dynamics Conference, Honolulu, June 24-27, 1991
[9] Bell, J.B; Marcus, D.L, A second-order projection method for variable-density flows, J. comput. phys., 101, 334, (1992) · Zbl 0759.76045
[10] Berger, M.J; Colella, P, Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1989) · Zbl 0665.76070
[11] Berger, M.J; Oliger, J, Adaptive mesh refinement for hyperbolic partial differential equations, J. comput. phys., 53, 484, (1984) · Zbl 0536.65071
[12] Berger, M.J; Rigoustsos, I, An algorithm for point clustering and grid generation, (1991)
[13] Best, J.P, The formation of toroidal bubbles upon the collapse of transient cavities, J. fluid mech., 251, 79, (1993) · Zbl 0784.76011
[14] Boulton-Stone, J.M; Blake, Gas bubbles bursting at a free surface, J. fluid mech., 254, 437, (1993) · Zbl 0780.76011
[15] Brackbill, J.U; Kothe, D.B; Zemach, C, A continuum method for modeling surface tension, J. comput. phys., 100, 335, (1992) · Zbl 0775.76110
[16] D. Chambers, D. Marcus, M. Sussman, Relaxation spectra of surface waves, Proceedings of the 1995 International Mechanical Engineering Congress and Exposition, November 1995
[17] Chang, Y.C; Hou, T.Y; Merriman, B; Osher, S, Eulerian capturing methods based on a level set formulation for incompressible fluid interfaces, J. comput. phys., 124, 449, (1996) · Zbl 0847.76048
[18] Colella, P, A direct Eulerian MUSCL scheme for gas dynamics, SIAM. J. comput., 6, 104, (1985) · Zbl 0562.76072
[19] Colella, P, A multidimensional second-order Godunov scheme for conservation laws, J. comput. phys., 87, 171, (1990)
[20] N. V. Deshpande, Fluid mechanics of bubble growth and collapse in a thermal ink-jet printer, SPSE/SPIES Electronic Imaging Devices and Systems Symposium, January 1989
[21] Esmaeeli, A; Tryggvason, G, An inverse energy cascade in two-dimensional low Reynolds number bubbly flows, J. fluid mech., 314, (1996) · Zbl 0875.76636
[22] Haj-Hariri, H; Shi, Q; Borhan, A, Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers, Phys. fluids, 9, 845, (1997)
[23] Harten, A, J. comput. phys., 83, 148, (1989)
[24] M. Hinatsu, H. Takeshi, Breaking waves in front of a box-shaped ship, 2nd Japan-Korea Workshop on Ship and Marine Hydrodynamics, Osaka, Japan, 1993
[25] Hnat, J.G; Buckmaster, J.D, Spherical cap bubbles and skirt formation, Phys. fluids, 19, 182, (1976) · Zbl 0319.76072
[26] L. H. Howell, A multilevel adaptive projection method for unsteady incompressible flow, 6th Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, April 4-9 1993
[27] Jacqmin, D, An energy approach to the continuum surface method, (1996)
[28] Juric, D; Tryggvason, G, Computations of boiling flows, Int. J. multiphase flow, 24, 387, (1998) · Zbl 1121.76455
[29] Lamb, H, Hydrodynamics, (1932)
[30] Longuet-Higgins, M.S; Cokelet, E.D, Deformation of steep surface waves on water I: A numerical method of computation, Proc. R. soc. lond. A., 350, 1, (1975) · Zbl 0346.76006
[31] Lundgren, T.S; Mansour, N.N, Vortex ring bubbles, J. fluid mech., 224, 177, (1991) · Zbl 0717.76119
[32] R. B. Milne, An adaptive level set method, U. C. Berkeley Department of Mathematics, 1995
[33] Minion, M.L, A projection method for locally refined grids, J. comput. phys., 127, (1996) · Zbl 0859.76047
[34] M. L. Minion, On the stability of Godunov-projection methods for incompressible flow, J. Comput. Phys.
[35] Nobari, M.R; Jan, Y.J; Tryggvason, G, Head on collision of drops; A numerical investigation, (November 1993)
[36] Oguz, H.N; Prosperetti, A, Bubble entrainment by the impact of drops on liquid surfaces, J. fluid mech., 203, 143, (1990)
[37] Osher, S; Sethian, J.A, Fronts propagating with curvature-dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12, (1988) · Zbl 0659.65132
[38] Pember, R.B; Bell, J.B; Colella, P; Crutchfield, W.Y; Welcome, M.L, An adaptive Cartesian grid method for unsteady compressible flow in irregular regions, J. comput. phys., 120, 278, (1995) · Zbl 0842.76056
[39] Puckett, E.G; Almgren, A.S; Bell, J.B; Marcus, D.L; Rider, W.G, A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. comput. phys., 130, 269, (1997) · Zbl 0872.76065
[40] Quirk, J; Karni, S, On the dynamics of a shock bubble interaction, J. fluid mech., 318, 129, (1996) · Zbl 0877.76046
[41] Ryskin, G; Leal, L.G, Numerical solution of free boundary problems in fluid mechanics. part 1. the finite-difference technique, J. fluid mech., 148, 1, (1984) · Zbl 0548.76031
[42] Shu, C.W; Osher, S, Efficient implementation of essentially non-oscillatory shock capturing schemes, II, J. comput. phys., 83, 32, (1989) · Zbl 0674.65061
[43] M. Sussman, E. Fatemi, An efficient, interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. Sci. Comput. · Zbl 0958.76070
[44] Sussman, M; Smereka, P, Axisymmetric free boundary problems, J. fluid mech., 341, 269, (1997) · Zbl 0892.76090
[45] Sussman, M; Smereka, P; Osher, S.J, A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146, (1994) · Zbl 0808.76077
[46] Sussman, M; Uto, S, Computing oil spreading underneath a sheet of ice, (July 1998)
[47] Szymczak, W.G; Rogers, J; Solomon, J.M; Berger, A.E, A numerical algorithm for hydrodynamic free boundary problems, J. comput. phys., 106, 319, (1993) · Zbl 0770.76047
[48] O. Tatebe, The multigrid preconditioned conjugate gradient method, 6th Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, April 4-9 1993
[49] Udaykumar, H.S; Rao, M.M; Shyy, W, Elafint—A mixed eulerian – lagrangian method for fluid flows with complex and moving boundaries, Int. J. numer. methods fluids., 22, 691, (1996) · Zbl 0887.76059
[50] Unverdi, S.O; Tryggvason, G, A front-tracking method for viscous, incompressible, multi-fluid flows, J. comput. phys., 100, 25, (1992) · Zbl 0758.76047
[51] Wesseling, P, An introduction to multigrid methods, (1992) · Zbl 0760.65092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.