zbMATH — the first resource for mathematics

A volume-of-fluid method for incompressible free surface flows. (English) Zbl 1422.76133
Summary: This paper proposes a hybrid volume-of-fluid (VOF) level-set method for simulating incompressible two-phase flows. Motion of the free surface is represented by a VOF algorithm that uses high resolution differencing schemes to algebraically preserve both the sharpness of interface and the boundedness of volume fraction. The VOF method is specifically based on a simple order high resolution scheme lower than that of a comparable method, but still leading to a nearly equivalent order of accuracy. Retaining the mass conservation property, the hybrid algorithm couples the proposed VOF method with a level-set distancing algorithm in an implicit manner when the normal and the curvature of the interface need to be accurate for consideration of surface tension. For practical purposes, it is developed to be efficiently and easily extensible to three-dimensional applications with a minor implementation complexity. The accuracy and convergence properties of the method are verified through a wide range of tests: advection of rigid interfaces of different shapes, a three-dimensional air bubble’s rising in viscous liquids, a two-dimensional dam-break, and a three-dimensional dam-break over an obstacle mounted on the bottom of a tank. The standard advection tests show that the volume advection algorithm is comparable in accuracy with geometric interface reconstruction algorithms of higher accuracy than other interface capturing-based methods found in the literature. The numerical results for the remainder of tests show a good agreement with other numerical solutions or available experimental data.

76M12 Finite volume methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76T99 Multiphase and multicomponent flows
Full Text: DOI
[1] Hirt, Volume of fluid (VOF) method for the dynamics of free boundaries, Journal of Computational Physics 39 pp 201– (1981) · Zbl 0462.76020
[2] Dommermuth DG, Sussman M, Beck RF, O’Shea TT, Wyatt DC, Olson K, MacNeice P. The numerical simulation of ship waves using Cartesian grid methods with adaptive mesh refinement. Proceedings of the 25th Symposium on Naval Hydrodynamics, St. John’s, Newfoundland and Labrador, Canada, August 2004.
[3] Fekken G, Veldman AEP, Buchner B. Simulation of green water loadings using the NavierâStokes equations. Proceedings of the Seventh International Conference on Numerical Ship Hydrodynamics, Nantes, France, 1999.
[4] Gerrits J. Dynamics of liquid-filled spacecraft. Ph.D. Thesis, University of Groningen, The Netherlands, 2001. Available from: www.ub.rug.nl/eldoc/dis/science/j.gerrits.
[5] Kleefsman, A volume-of-fluid based simulation method for wave impact problems, Journal of Computational Physics 206 pp 363– (2005) · Zbl 1087.76539
[6] Loots GE. Fluidâstructure interaction in hemodynamics. Ph.D. Thesis, University of Groningen, The Netherlands, 2003. Available from:www.ub.rug.nl/eldoc/dis/science/g.e.loots.
[7] Loots, The role of hemodynamics in the development of the outflow tract of the heart, Journal of Engineering Mathematics 45 pp 91– (2003) · Zbl 1112.76516
[8] Muzaferija S, PericÌ M, Sames P, Schellin T. A two-fluid NavierâStokes solver to simulate water entry. Proceedings of the 22nd Symposium on Naval Hydrodynamics, Washington, DC, U.S.A., 1998.
[9] Repetto RA. Computation of turbulent free-surface flows around ships and floating bodies. Ph.D. Thesis, Technical University of Hamburg, Harburg, 2001.
[10] Wan, Proceedings of the 14th International Workshop on Water Waves and Floating Bodies pp 163– (1999)
[11] Craft, Engineering Turbulence Modelling and Measurements 4 pp 73– (1999)
[12] Dommermuth DG, Gharib M, Huang H, Innis GE, Maheo P, Novikov EA, Talcott JC, Wyatt DC. Turbulent free-surface flows: a comparison between numerical simulations and experimental measurements. Proceedings of the 21st Symposium on Naval Hydrodynamics, Trondheim, 1996; 249â265.
[13] Sussman, An adaptive level set approach for incompressible two-phase flows, Journal of Computational Physics 148 pp 81– (1999) · Zbl 0930.76068
[14] Sussman M, Dommermuth D. The numerical simulation of ship waves using Cartesian grid methods. Proceedings of the 23rd Symposium on Naval Hydrodynamics, Val-De-Reuel, France, September 2000.
[15] Scardovelli, Direct numerical simulation of free-surface and interfacial flow calculations, Annual Review of Fluid Mechanics 31 pp 567– (1999)
[16] Noh, Proceedings of the Fifth International Conference on Fluid Dynamics pp 330– (1976)
[17] Youngs DL. An interface tracking method for a 3D Eulerian hydrodynamics code. Technical Report AWRE/44/92/35, Atomic Weapons Research Establishment, 1987.
[18] Harvie, A new volume of fluid advection algorithm: the stream scheme, Journal of Computational Physics 162 pp 1– (2000) · Zbl 0964.76068
[19] Harvie, A new volume of fluid advection algorithm: the defined donating region scheme, International Journal for Numerical Methods in Fluids 35 (2) pp 151– (2001) · Zbl 0991.76062
[20] LÃ{\({}^3\)}peza, An improved PLICâVOF method for tracking thin fluid structures in incompressible two-phase flows, Journal of Computational Physics 208 pp 51– (2005)
[21] Renardy, PROST: parabolic reconstruction of surface tension for the volume-of-fluid method, Journal of Computational Physics 183 pp 400– (2002) · Zbl 1057.76569
[22] Rudman, A volume tracking method for interfacial flows with large density variations, International Journal for Numerical Methods in Fluids 28 pp 357– (1998) · Zbl 0915.76060
[23] Fluent, Fluent 6.3 Users Guide (2006)
[24] Kothe, RIPPLE: a new model for incompressible flows with free surfaces, AIAA Journal 30 (11) pp 2694– (1992) · Zbl 0762.76074
[25] Lafaurie, Modelling merging and fragmentation in multiphase flows with SURFER, Journal of Computational Physics 113 pp 134– (1994) · Zbl 0809.76064
[26] Leonard, The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection, Computer Methods in Applied Mechanics and Engineering 88 pp 17– (1991) · Zbl 0746.76067
[27] Maronnier, Numerical simulation of free surface flows, Journal of Computational Physics 155 pp 439– (1999) · Zbl 0952.76070
[28] Ubbink, A method for capturing sharp fluid interfaces on arbitrary meshes, Journal of Computational Physics 153 pp 26– (1999) · Zbl 0955.76058
[29] Zalesak, Fully multi-dimensional flux corrected transport algorithms for fluids, Journal of Computational Physics 31 pp 335– (1979) · Zbl 0416.76002
[30] Bourlioux A. A coupled level-set volume-of-fluid algorithm for tracking material interfaces. Proceedings of the Sixth International Symposium on Computational Fluid Dynamics, Lake Tahoe, CA, 1995.
[31] Son, Efficient implementation of a coupled level-set and volume-of-fluid method for three-dimensional incompressible two-phase flows, Numerical Heat Transfer B 43 pp 549– (2003)
[32] Sussman, A coupled level set and volume of fluid method for computing 3D and axisymmetric incompressible two-phase flows, Journal of Computational Physics 162 pp 301– (2000) · Zbl 0977.76071
[33] Sussman, A sharp interface method for incompressible two-phase flows, Journal of Computational Physics 221 pp 469– (2007) · Zbl 1194.76219
[34] van der Pijl, A mass-conserving level-set method for modeling of multi-phase flows, International Journal for Numerical Methods in Fluids 47 pp 339– (2005) · Zbl 1065.76160
[35] Launder, The numerical computation of turbulent flows, Computer Methods in Applied Mechanics and Engineering 3 pp 269– (1974) · Zbl 0277.76049
[36] Brackbill, A continuum method for modeling surface tension, Journal of Computational Physics 100 pp 335– (1992) · Zbl 0775.76110
[37] Gaskell, Curvature compensated convective transport: SMART, a new boundedness preserving transport algorithm, International Journal for Numerical Methods in Fluids 8 pp 617– (1988) · Zbl 0668.76118
[38] Darwish, Normalized variable and space formulation methodology for high-resolution schemes, Numerical Heat Transfer B 26 pp 79– (1994)
[39] Sussman, An improved level set method for incompressible two-phase flows, Computers and Fluids 27 (5â6) pp 663– (1998) · Zbl 0967.76078
[40] Ferziger, Computational Methods for Fluid Dynamics (1996)
[41] Khosla, A diagonally dominant second-order accurate implicit scheme, Computers and Fluids 2 pp 207– (1974) · Zbl 0335.76009
[42] Van Leer, Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov’s method, Journal of Computational Physics 32 pp 101– (1979) · Zbl 1364.65223
[43] Muzaferija S. Adaptive finite volume method for flow predictions using unstructured meshes and multigrid approach. Ph.D. Thesis, University of London, 1994.
[44] Stone, Iterative solution of implicit approximations of multi-dimensional partial differential equations, SIAM Journal on Numerical Analysis 5 pp 530– (1968) · Zbl 0197.13304
[45] Patankar, A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, International Journal of Heat and Mass Transfer 15 pp 1787– (1972) · Zbl 0246.76080
[46] Jiang, Weighted ENO schemes for HamiltonâJacobi equations, SIAM Journal on Scientific Computing 21 pp 2126– (2000)
[47] Croce R, Griebel M, Schweitzer MA. A parallel level-set approach for two-phase flow problems with surface tension in three space dimensions, Technical Report 157, Sonderforschungsbereich 611, UniversitÃ{\currency}t Bonn, 2004.
[48] Hao, A numerical method for three-dimensional gasâliquid flow computations, Journal of Computational Physics 196 pp 126– (2004)
[49] Unverdi, A front-tracking method for viscous, incompressible, multi-fluid flows, Journal of Computational Physics 100 pp 25– (1992) · Zbl 0758.76047
[50] Zhou ZQ, De Kat JO, Buchner B. A nonlinear 3-D approach to simulate green water dynamics on deck. Proceedings of the Seventh International Conference on Numerical Ship Hydrodynamics, Nantes, France, 1999; 5.1â1, 15.
[51] Martin, An experimental study of the collapsed liquid columns on a rigid horizontal plate, Philosophical Transactions of the Royal Society of London Series AâMathematical Physical and Engineering Sciences 244 pp 312– (1952)
[52] Ferreira, Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations, Mathematics and Computers in Simulation (2007) · Zbl 1157.76033
[53] Koshizuka, Moving particle semi-implicit method for fragmentation of incompressible fluid, Nuclear Science and Engineering 123 pp 421– (1984)
[54] Violeau, Numerical modelling of complex turbulent free-surface flows with the SPH method: an overview, International Journal for Numerical Methods in Fluids 53 pp 277– (2007) · Zbl 1227.76022
[55] Colicchio, Free-surface flow after a dam break: a comparative study, Ship Technology Research 49 (3) pp 95– (2002)
[56] Colagrossi, Numerical simulation of interfacial flows by smoothed particle hydrodynamics, Journal of Computational Physics 191 pp 448– (2003) · Zbl 1028.76039
[57] Nielsen KB. Numerical prediction of green water loads on ships. Ph.D. Thesis, Technical University of Denmark, 2003.
[58] ten Caat M. Numerical simulation of incompressible two-phase flow. Master’s Thesis, University of Groningen, 2002.
[59] Greco M. A two-dimensional study of green-water loading. Ph.D. Thesis, Norwegian University of Science and Technology, 2001.
[60] Greco, Impact flows and loads on ship-deck structures, Journal of Fluids and Structures 19 (3) pp 251– (2004)
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.