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A high-order projection method for tracking fluid interfaces in variable density incompressible flows. (English) Zbl 0872.76065

Summary: We present a numerical method for computing solutions of the incompressible Euler or Navier-Stokes equations in the presence of an interface between two fluids with different fluid properties. The method is based on a second-order projection method for variable density flows using an “approximate projection” formulation. The boundary between the fluids is tracked with a second-order, volume-of-fluid interface tracking algorithm. We present results for viscious Rayleigh-Taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm. We also present computational results for the Rayleigh-Taylor instability in air-helium and for bubbles and drops in an air-water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76V05 Reaction effects in flows
76B47 Vortex flows for incompressible inviscid fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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[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] 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. Stat. Comput., 17 (1996) · Zbl 0845.76055
[3] Baker, G. R.; Meiron, D. I.; Orszag, S. A., Boundary integral methods for axisymmetric and three-dimensional Rayleigh-Taylor instability, Physica D, 12, 19 (1984) · Zbl 0585.76063
[4] J. B. Bell, P. Colella, L. H. Howell, 1991, A Multilevel Adaptive Projection Method for Unsteady Incompressible Flow, Proceedings, 10th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, June 24-27, 1991; J. B. Bell, P. Colella, L. H. Howell, 1991, A Multilevel Adaptive Projection Method for Unsteady Incompressible Flow, Proceedings, 10th AIAA Computational Fluid Dynamics Conference, Honolulu, Hawaii, June 24-27, 1991
[5] J. B. Bell, P. Colella, M. L. Welcome, 1991, Conservative Front Tracking for Inviscid Compressible Flow, Proceedings, 22nd Annual Fluid Dynamics, Plasma Dynamics, and Lasers Conference, Honolulu, HI, 1991, 24, 26; J. B. Bell, P. Colella, M. L. Welcome, 1991, Conservative Front Tracking for Inviscid Compressible Flow, Proceedings, 22nd Annual Fluid Dynamics, Plasma Dynamics, and Lasers Conference, Honolulu, HI, 1991, 24, 26
[6] Bell, J. B.; Dawson, C. N.; Shubin, G. R., An unsplit, higher order godunov method for scalar conservation laws in multiple dimensions, J. Comput. Phys., 74, 1 (1988) · Zbl 0684.65088
[7] Bell, J. B.; Marcus, D. L., A second-order projection method for variable density flows, J. Comput. Phys., 101, 334 (1992) · Zbl 0759.76045
[8] Briggs, W. L., A Multigrid Tutorial (1987), SIAM: SIAM Philadelphia
[9] Chan, C.; Mazumder, J.; Chen, M. M., A two-dimensional transient model for convection in a laser melted pool, Metallurg. Trans. A, 15, 2175 (1984)
[10] Chang, Y. C.; Hou, T. Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. Comput. Phys., 124, 449 (1996) · Zbl 0847.76048
[11] I. L. Chern, P. Colella, July 1987, A Conservative Front Tracking Method for Hyperbolic Conservation Laws, Lawrence Livermore National Laboratory; I. L. Chern, P. Colella, July 1987, A Conservative Front Tracking Method for Hyperbolic Conservation Laws, Lawrence Livermore National Laboratory
[12] Chorin, A. J., Flame advection and propagation algorithms, J. Comput. Phys., 35, 1 (1980) · Zbl 0425.76086
[13] Colella, P., A direct Eulerian MUSCL scheme for gas dynamics, SIAM J. Sci. Stat. Comput., 6, 104 (1985) · Zbl 0562.76072
[14] Colella, P., Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys., 87, 71 (1990) · Zbl 0694.65041
[15] N. V. Deshpande, 1989, Fluid Mechanics of Bubble Growth and Collapse in a Therml Ink-Jet Printhead, SPSE/SPIES Electronic Imaging Devices and Systems Symposium, January 1989; N. V. Deshpande, 1989, Fluid Mechanics of Bubble Growth and Collapse in a Therml Ink-Jet Printhead, SPSE/SPIES Electronic Imaging Devices and Systems Symposium, January 1989
[16] Ghoniem, A. F.; Chorin, A. J.; Oppenheim, A. K., Numerical modeling of turbulent flow in a combustion tunnel, Phil. Trans. R. Soc. London A, 304, 303 (1982) · Zbl 0478.76066
[17] Glimm, J.; McBryan, O., A computational model for interfaces, Adv. Appl. Math., 6, 422 (1985)
[18] Grove, J. W., The interaction of shocks with fluid interfaces, Adv. Appl. Math., 10, 201 (1989) · Zbl 0669.76085
[19] Haj-Hariri, H.; Shi, Q.; Borhan, A., Effect of local property smearing on global variables, implications for numerical simulation of multiphase flows, Phys. Fluids A, 6, 2555 (1994) · Zbl 0825.76865
[20] J. J. Helmsen, 1994, A Comparison of Three-Dimensional Photolithography Simulators, U. C. Berkeley; J. J. Helmsen, 1994, A Comparison of Three-Dimensional Photolithography Simulators, U. C. Berkeley
[21] L. F. Henderson, E. G. Puckett, Anomalous refraction of shock waves in materials with general equations of state. Part I. The shock pair system, Trans. R. Soc. London A; L. F. Henderson, E. G. Puckett, Anomalous refraction of shock waves in materials with general equations of state. Part I. The shock pair system, Trans. R. Soc. London A
[22] L. F. Henderson, E. G. Puckett, P. Colella, 1990, On the Anomalous Refraction of Shock Waves, Proceedings, Second Japan-Soviet Union Symposium on Computational Fluid Dynamics, Tsukuba, Japan, 1990, 144; L. F. Henderson, E. G. Puckett, P. Colella, 1990, On the Anomalous Refraction of Shock Waves, Proceedings, Second Japan-Soviet Union Symposium on Computational Fluid Dynamics, Tsukuba, Japan, 1990, 144 · Zbl 0960.76520
[23] Henderson, L. F.; Puckett, E. G.; Colella, P., Anomalous refraction of shock waves, Shock Waves (1992), Springer-Verlag: Springer-Verlag New York/Berlin, p. 283- · Zbl 0960.76520
[24] Hirt, C. W., Flow-3D Users Manual (1988), Flow Sciences, Inc
[25] Hirt, C. W.; Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, 201 (1981) · Zbl 0462.76020
[26] D. B. Kothe, J. R. Baumgardner, S. T. Bennion, J. H. Cerutti, B. J. Daly, K. S. Holian, E. M. Kober, S. J. Mosso, J. W. Painter, R. D. Smith, M. D. Torrey, 1992, PAGOSA: A Massively-Parallel, Multi-Material Hydro-Dynamics Model for Three-Dimensional High-Speed Flow and High-Rate Deformation, Los Alamos National Laboratory; D. B. Kothe, J. R. Baumgardner, S. T. Bennion, J. H. Cerutti, B. J. Daly, K. S. Holian, E. M. Kober, S. J. Mosso, J. W. Painter, R. D. Smith, M. D. Torrey, 1992, PAGOSA: A Massively-Parallel, Multi-Material Hydro-Dynamics Model for Three-Dimensional High-Speed Flow and High-Rate Deformation, Los Alamos National Laboratory
[27] Kothe, D. B.; Mjolsness, R. C., RIPPLE: A new model for incompressible flows with free surfaces, AIAA J., 30, 2694 (1992) · Zbl 0762.76074
[28] D. B. Kothe, R. C. Mjolsness, M. D. Torrey, April 1991, RIPPLE: A Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory; D. B. Kothe, R. C. Mjolsness, M. D. Torrey, April 1991, RIPPLE: A Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory
[29] LaFaurie, B.; Nardone, C.; Scardovelli, R.; Zaleski, S.; Zanetti, G., Modelling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys., 113, 134 (1994) · Zbl 0809.76064
[30] LeVeque, R. J., High-Resolution Conservative Algorithms for Advection in Incompressible Flow, SIAM J. Numer. Anal., 33, 627 (1996) · Zbl 0852.76057
[31] Liu, H.; Lavernia, E. J.; Rangel, R. H., Numerical investigation of micropore formation during substrate impact of molten droplets in plasma spray processes, Atomization Sprays, 4, 369 (1994)
[32] Marcus, D. L.; Bell, J. B., Numerical simulation of a viscious vortex ring interaction with a density interface, Phys. Fluids A, 6, 1505 (1994) · Zbl 0828.76018
[33] D. L. Marcus, E. G. Puckett, J. B. Bell, J. S. Saltzmann, 1991, Numerical Simulation of Accelerated Interfaces, 3rd International Workshop on the Physics of Compressible Turbulent Mixing, R. Dautray, 63, CEA DAM; D. L. Marcus, E. G. Puckett, J. B. Bell, J. S. Saltzmann, 1991, Numerical Simulation of Accelerated Interfaces, 3rd International Workshop on the Physics of Compressible Turbulent Mixing, R. Dautray, 63, CEA DAM
[34] Mulder, W.; Osher, S.; Sethian, J. A., Computing interface motion in compressible gas dynamics, J. Comput. Phys., 100, 209 (1992) · Zbl 0758.76044
[35] B. D. Nichols, C. W. Hirt, R. S. Hotchkiss, August 1980, SOLA-VOF: A Solution Algorithm for Transient Fluid Flow with Multiple Free Boundaries, Los Alamos National Laboratory; B. D. Nichols, C. W. Hirt, R. S. Hotchkiss, August 1980, SOLA-VOF: A Solution Algorithm for Transient Fluid Flow with Multiple Free Boundaries, Los Alamos National Laboratory
[36] Noh, W. F.; Woodward, P. R., SLIC (Simple Line Interface Calculation), (van der Vooren, A. I.; Zandbergen, P. J., Lecture Notes in Physics (1976), Springer-Verlag: Springer-Verlag New York/Berlin), 330 · Zbl 0382.76084
[37] Oguz, H. N.; Prosperetti, A., Dynamics of bubble growth and detachment from a needle, J. Fluid Mech., 257, 111 (1993)
[38] 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
[39] B. J. Parker, D. L. Youngs, Feb. 1992, Two and Three Dimensional Eulerian Simulation of Fluid Flow with Material Interfaces, UK Atomic Weapons Establishment, Aldermaston, Berkshire; B. J. Parker, D. L. Youngs, Feb. 1992, Two and Three Dimensional Eulerian Simulation of Fluid Flow with Material Interfaces, UK Atomic Weapons Establishment, Aldermaston, Berkshire
[40] J. E. Pilliod, September 1992, An Analysis of Piecewise Linear Interface Reconstruction Algorithms for Volume-Of-Fluid Methods, U.C. Davis; J. E. Pilliod, September 1992, An Analysis of Piecewise Linear Interface Reconstruction Algorithms for Volume-Of-Fluid Methods, U.C. Davis
[41] J. E. Pilliod, E. G. Puckett, Second-order volume-of-fluid interface tracking algorithms, J. Comput. Phys.; J. E. Pilliod, E. G. Puckett, Second-order volume-of-fluid interface tracking algorithms, J. Comput. Phys. · Zbl 1126.76347
[42] E. G. Puckett, 1991, A Volume-of-Fluid Interface Tracking Algorithm with Applications to Computing Shock Wave Refraction, Proceedings, 4th International Symposium on Computational Fluid Dynamics, Davis, CA, 1991, H. Dwyer, 933; E. G. Puckett, 1991, A Volume-of-Fluid Interface Tracking Algorithm with Applications to Computing Shock Wave Refraction, Proceedings, 4th International Symposium on Computational Fluid Dynamics, Davis, CA, 1991, H. Dwyer, 933
[43] E. G. Puckett, L. F. Henderson, P. Colella, The anomalous refraction of shock waves in materials with general equations of state. Part II. Anomalous refraction wave systems, Trans. R. Soc. London A; E. G. Puckett, L. F. Henderson, P. Colella, The anomalous refraction of shock waves in materials with general equations of state. Part II. Anomalous refraction wave systems, Trans. R. Soc. London A · Zbl 0960.76520
[44] Puckett, E. G.; Henderson, L. F.; Colella, P., A General Theory of Anomalous Refraction, (Brun, R.; Dumitrescu, L. Z., Shock Waves @ Marseilles (1995), Springer-Verlag: Springer-Verlag New York/Berlin), 139 · Zbl 0960.76520
[45] Puckett, E. G.; Saltzman, J. S., A 3-D adaptive mesh refinement algorithm for multimaterial gas dynamics, Physica D, 60, 84 (1992) · Zbl 0779.76059
[46] W. J. Rider, D. B. Kothe, S. J. Mosso, J. H. Cerutti, J. I. Hochstein, 1995, Accurate Solution Algorithms for Incompressible Multiphase Flows, Proceedings, 33rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995; W. J. Rider, D. B. Kothe, S. J. Mosso, J. H. Cerutti, J. I. Hochstein, 1995, Accurate Solution Algorithms for Incompressible Multiphase Flows, Proceedings, 33rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 1995
[47] Sethian, J. A., Turbulent combustion in open and closed vessels, J. Comput. Phys., 54, 425 (1984) · Zbl 0594.76047
[48] Sethian, J. A., Numerical algorithms for propagating interfaces: Hamilton-Jacobi equations and conservation laws, J. Differential Geom., 31, 131 (1990) · Zbl 0691.65082
[49] Sethian, J. A.; Strain, J., Crystal growth and dendritic solidification, J. Comput. Phys., 98, 231 (1992) · Zbl 0752.65088
[50] Strain, J., A boundary integral approach to unstable solidification, J. Comput. Phys., 85, 342 (1989) · Zbl 0685.65109
[51] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 506 (1968) · Zbl 0184.38503
[52] M. Sussman, E. Fatemi, P. Smereka, S. Osher, 1995, A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow II, Proceedings, 6th International Symposium on Computational Fluid Dynamics, Lake Tahoe, NV, September 1995, M. Hafez; M. Sussman, E. Fatemi, P. Smereka, S. Osher, 1995, A Level Set Approach for Computing Solutions to Incompressible Two-Phase Flow II, Proceedings, 6th International Symposium on Computational Fluid Dynamics, Lake Tahoe, NV, September 1995, M. Hafez · Zbl 0967.76078
[53] M. Sussman, E. Fatemi, P. Smereka, S. Osher, An improved level set method for incompressible two-phase flow, J. Comput. & Fluids; M. Sussman, E. Fatemi, P. Smereka, S. Osher, An improved level set method for incompressible two-phase flow, J. Comput. & Fluids · Zbl 0967.76078
[54] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146 (1994) · Zbl 0808.76077
[55] O. Tatebe, 1993, The Multigrid Preconditioned Conjugate Gradient Method, Proceedings, Sixth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, April 4-9, 1993, N. D. MelsonT. A. ManteuffelS. F. McCormick, CP-3224, 621; O. Tatebe, 1993, The Multigrid Preconditioned Conjugate Gradient Method, Proceedings, Sixth Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO, April 4-9, 1993, N. D. MelsonT. A. ManteuffelS. F. McCormick, CP-3224, 621
[56] P. A. Torpey, 1988, Prevention of Air Ingestion in a Thermal Ink-Jet Device, Proceedings, 4th International Congress on Advances in Non-Impact Print Technologies, March 1988; P. A. Torpey, 1988, Prevention of Air Ingestion in a Thermal Ink-Jet Device, Proceedings, 4th International Congress on Advances in Non-Impact Print Technologies, March 1988
[57] M. D. Torrey, L. D. Cloutman, R. C. Mjolsness, C. W. Hirt, December 1985, NASA-VOF2D: A Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory; M. D. Torrey, L. D. Cloutman, R. C. Mjolsness, C. W. Hirt, December 1985, NASA-VOF2D: A Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory
[58] M. D. Torrey, R. C. Mjolsness, L. R. Stein, July 1987, NASA-VOF3D: A Three-Dimensional Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory; M. D. Torrey, R. C. Mjolsness, L. R. Stein, July 1987, NASA-VOF3D: A Three-Dimensional Computer Program for Incompressible Flows with Free Surfaces, Los Alamos National Laboratory
[59] Trapaga, G.; Matthys, E. F.; Valencia, J. J.; Szekely, J., Fluid flow, heat transfer, and solidification of molten metal droplets impinging on substrates—Comparison of numerical and experimental results, Metall. Trans. B., 23, 701 (1992)
[60] G. Tryggvason, 1994; G. Tryggvason, 1994
[61] Unverdi, S. O.; Tryggvason, G., Computations of multi-fluid flows, Physica D, 60, 70 (1992) · Zbl 0779.76101
[62] Walters, J. K.; Davidson, J. F., The initial motion of a gas bubble formed in an inviscid liquid. Part 1. The two-dimensional bubble, J. Fluid Mech., 12, 408 (1962) · Zbl 0125.16203
[63] Youngs, D. L., Time-dependent multi-material flow with large fluid distortion, (Morton, K. W.; Baines, M. J., Numerical Methods for Fluid Dynamics (1982), Academic Press: Academic Press New York) · Zbl 0537.76071
[64] Youngs, D. L., Numerical Simulation of Turbulent Mixing by Rayleigh-Taylor Instability, (Bishop, A. R.; Campbell, L. J.; Channell, P. J., Fronts, Interfaces and Patterns (1984), North-Holland: North-Holland Amsterdam), 32
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