Two-phase flows on interface refined grids modeled with VOF, staggered finite volumes, and spline interpolants

Journal of Computational Physics

Volumn 166, Issue 2, 2001, Pages 302-335

  Ginzburg, I ^a   Wittum, G ^a  

^a UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

Buoyant bubbles; Cubic splines; Finite volume method; Laplace law; Linear interface interpolant; Multigrid; Navier Stokes equation; Rotated elements; Saddle point problem; Two phase flow; Volume of fluid method

Indexed keywords

ADVECTION; COMPUTATIONAL FLUID DYNAMICS; DEGREES OF FREEDOM (MECHANICS); FINITE VOLUME METHOD; INTERPOLATION; MESH GENERATION; NEWTON-RAPHSON METHOD; REFINING; TWO PHASE FLOW; VISCOUS FLOW;

BUOYANT BUBBLES; CUBIC SPLINE; FINITE-VOLUME METHOD; INTERPOLANTS; LAPLACE LAW; LINEAR INTERFACE INTERPOLANT; MULTIGRIDS; NAVIER-STOKES EQUATION; ROTATED ELEMENT; SADDLE POINT PROBLEMS; TWO PHASES FLOW; VOLUME-OF-FLUID METHOD;

NAVIER STOKES EQUATIONS;

EID: 0001516397     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcph.2000.6655     Document Type: Article

References (72)

1. N. Ashgriz and J. Y. Poo, A computational method for determining curvatures, J. Comput. Phys. 84, 483 (1989)
2. I. Aleinov and E. G. Puckett, Computing surface tension with high order kernels, in Proc. 6th International Symposium on Computational Dynamics, edited by K. Oshima, Lake Tahoe. CA, September 4-8, 1995
3. N. Ashgriz and J. Y. Poo, FLAIR: Flux line-segment model for advection and interface reconstruction, J. Comput. Phys. 93, 449 (1991)
4. R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst, Templates for the solution of linear systems: Building blocks for iterative methods, SIAM (1994)
5. P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuss, H. Rentz-Reichert, and C. Wieners, UG - A flexible software toolbox for solving partial differential equations, Computing and Visualization in Science 1, 27 (1997)
6. J. B. Bell, P. Colella, and H. M. Glaz, A second-order projection method of the incompressible Navier-Stokes equation. J. Comput. Phys. 85, 257 (1989)
7. J. H. Bramble and J. E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems, Math. Comput. 50, 1 (1988)
8. J. H. Bramble, Multigrid Methods, Pitman Res. Notes in Math. Sci., ISSN 0296-3674, 1993
9. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Series in Comp. Math. 15 (Springer-Verlag, 1991)
10. D. Bhaga and M. E. Weber, Bubbles in viscous liquids: Shapes, wakes, and velocities, J. Fluid Mech. 105, 61 (1992)
11. T. J. Barth, Aspects of Unstructured Grids and Finite-Volume Solvers for Euler and Navier-Stokes Equations,** VKI/NASA/AGARD Special Course on Unstructured Grid Methods for Advection Dominated Flows, AGARD Publication R-787, 1995
12. T. J. Barth and J. A. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulate domains, J. Comput. Phys. 145, 1 (1998)
13. J. U. Brackbill, D. B. Kothe, and C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100, 335 (1992)
14. R. Clift, J. R. Grace, and M. E. Weber, Bubbles, Drops, and Particles (Academic Press, San Diego, 1978)
15. D. L. Chopp, Computing minimal surfaces via level set curvature flow, J. Comput. Phys. 106, 77 (1993)
16. R. Collins, A simple model of the plane gas bubble in a finite liquid, J. Fluid Mech. 22, 763 (1965)
17. A. V. Coward, Y. Y. Renardy, M. Renardy, and J. R. Richards, Temporal evolution of periodic disturbances in two-layer Couette flow, J. Comput. Phys. 132, 346 (1997)
18. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary stokes equations. Revue Francaise d’Automatique, Informatique et Recherche Opérationnele R-3, 33 (1973)
19. B. J. Daly, A technique for including surface tension effects in hydrodynamic calculations, J. Comput. Phys. 4, 97 (1969)
20. R. DeBar, Fundamentals of the Kraken Code, Technical Report UCIR-760, LLNL, 1974
21. D. E. Fyfe, E. S. Oran, and M. J. Fritts, Surface tension and viscosity with Lagrangian hydrodynamics on a triangular mesh. J. Comput. Phys. 76, 394 (1988)
22. D. M. Gao, A three-dimensional hybrid finite element-volume tracking model for mould filing in casting processes, Int. J. Numer. Methods Fluids 29, 877 (1999)
23. O. S. Galaktionov, P. D. Anderson, G. W. M. Peters, and F. N. Van de Vosse, An adaptive front tracking technique for three-dimensional transient flows, Int. J. Numer. Methods Fluids 32, 201 (2000)
24. D. M. Greaves and A. G. L. Borthwick, On the use of adaptive hierarchical meshes for numerical simulation of separated flows, Int. J. Numer. Methods Fluids 26, 303 (1998)
25. J. Glimm, J. Grove, B. Lindquist, O. McBryan, and G. Tryggvason, The bifurcation of tracked scalar waves, SIAM J. Sci. Stat. Comput. 9, 61 (1988)
26. D. Gueyffier, J. Lie, R. Scardovelli, and S. Zaleski, Volume of fluid interface tracking with smoothed surface stress methods for three-dimensional flows, unpublished (1998)
27. W. Hackbusch, Multi-Grid Methods and Applications (Springer-Verlag, New York, 1985)
28. F. H. Harlow and J. E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8, 2182 (1965)
29. C. W. Hirt and B. D. Nicholls, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39, 201 (1981)
30. J. G. Hnat and J. D. Buckmaster, Spherical cap bubbles and skirt formation, Phys. Fluids 19, 182 (1976)
31. J. M. Hyman, Numerical methods for tracking interfaces, Physica D 12, 396 (1984)
32. J. H. Jeong and D. Y. Yang, Finite element analysis of transient fluid flow with free surface using VOF (VOLUME-OF-FLUID) method and adaptive grid. Int. J. Numer. Methods Fluids 26, 1127(1998)
33. D. B. Kothe, Perspective on Eulerian finite volume methods for incompressible interfacial flows, lecture notes presented at Free Surface Flow Workshop, International Centre for Mechanical Sciences, Udine, Italy (September 1-5, 1997). [In Free Surface Flows, edited by H. C. Kuhlmann and H.-J. Rath (Springer-Verlag, New York), p. 267]
34. B. Lafaurie, C. Nardone, R. Scardovelli, and S. Zaleski, Modeling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys. 113, 134 (1994)
35. Les Piegl and Wayane Tiller, The NURBS Book (Springer-Verlag, New York, 1997)
36. R. J. Leveque, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal. 33, 627 (1996)
37. J. Li, Calcul d’Interface Affine par Morceaux, C. R. Acad. Sci. Paris 320, série IIb, 391 (1995)
38. N. Lock, M. Jaeger, M. Medale, and R. Occelli, Local mesh adaptation technique for front tracking problems, Int. J. Numer. Methods Fluids 28, 719 (1998)
39. T. S. Ludgren and Mansour, Vortex ring bubbles, J. Fluid Mech. 224, 177 (1991)
40. F. Mashayek and N. Ashgriz, A hybrid finite-element-volume-of-fluid method for simulating free surface flows and interfaces, Int. J. Numer. Methods Fluids 20, 1363 (1995)
41. S. J. Mosso, B. K. Swartz, D. B. Kothe, and S. P. Clancy, Recent enhancements of volume tracking algorithm for irregular grids (Los Alamos National Laboratory, Los Alamos, NM, LA_UR_96_277, 1996), presented at the Parallel CFD Conference, Capri, Italy (March 20-23, 1996)
42. S. J. Mosso, B. K. Swartz, D. B. Kothe, and R. C. Ferrell, A Parallel, Volume-Tracking Algorithm for Unstructured Meshes (Los Alamos National Laboratory, Los Alamos, NM, LA_UR_96_2420, 1996)
43. W. F. Noh and P. R. Woodward, SLIC (simple line interface method), in Lecture Notes in Phys. 59, edited by A. I. van de Vooren and P. J. Zandbergen (Springer-Verlag, Berlin/New York, 1976), p. 330
44. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 130, 269 (1997)
45. S. Popinet and S. Zaleski, A front-tracking algorithm for the accurate representation of surface tension, submitted for publication
46. W. H. Press, S. A. Teukolsky, W. T. Wetterling, and B. P. Flannnery, Numerical Recipes in C. (Cambridge Univ. Press, Cambridge, UK, 1992)
47. E. G. Puckett, A. S. Almgren, J. B. Bell, D. L. Marcus, and W. J. Rider, A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. Comput. Phys. 79, 12 (1988)
48. R. Rannacher and S.Turek, Simple nonconforming quadrilateral stokes element, Numer. Methods Partial Diff. Equations 8, 97 (1992)
49. A. Rappoport, A. Sheffer, and M. Bercovier, Volume preserving free form solids, IEEE Trans. Visualiz. Comput. Graphics 2(1), 19 (1996)
50. H. Rentz-Reichert, Robuste Mehrgitterverfahren zur Lösung der inkompressiblen Navier-Stokes Gleichung: Ein Vergleich, PH.D. thesis (Stuttgart University, 1996)
51. S.-O. Kim and H. C. No, Second order model for free surface convection and interface reconstruction, Int. J. Numer. Methods Fluids 26, 79 (1998)
52. J. E. Jr. Pilliod and E. G. Puckett, Second Order Accurate Volume-of-Fluid Algorithms for Tracking Material Interfaces, Technical Report No. LBNL-40744 (Lawrence Berkley National Laboratory)
53. G. R. Price, G. T. Reader, R. D. Rowe, and J. D. Bugg, A piecewise parabolic interface calculation for volume tracking, Proceedings of the Sixth Annual Conference of the Computational Fluid/Dynamics Society of Canada (University of Victoria, Victoria, British Columbia, 1998)
54. W. J. Rider and D. B. Kothe, Reconstructing volume tracking, J. Comput. Phys. 141, 112 (1998)
55. G. Ryskin and L. G. Leal, Numerical solution of free-boundary problems in fluid mechanics, Part 2, Buoyancy-driven motion of a gas bubble through a quiescent liquid, J. Fluid Mech. 148, 19 (1984)
56. T. Sato and S. M. Richardson, Numerical simulation method for viscoelastic flow with free surfaces-fringe element generation method. Int. J. Numer. Methods Fluids 26, 555 (1994)
57. R. Scardovelli and S. Zaleski, Direct numerical simulation of free-surface and interfacial flow, Annu. Rev. Fluid Mech 31, 567 (1999)
58. G. L. Sleijpen, H. A. van der Vorst, and D. R. Fokkema, BiCGstab(1) and other hybrid BI-CG methods, Numer. Algorithms 7, 75 (1994)
59. G. E. Schneider and M. J. Raw, Control volume finite element method for heat transfer and fluid flow using collocated variables, Numer. Heat Transfer 11, 363 (1987)
60. M. Rudman, Volume tracking methods for interfacial flow calculations. Int. J. Numer. Methods Fluids 24, 671 (1997)
61. M. Rudman, Volume tracking methods for incompressible multifluid flows with large density variations, Int. J. Numer. Methods Fluids 28, 357-378 (1998)
62. M. Sussmann, P. Smereka, and S. Osher, A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146 (1994)
63. M. Sussmann and P. Smereka, Axisymmetric free boundary problems, J. Fluid Mech. 341, 269 (1997)
64. M. Sussmann, A. S. Almgren, J. B. Bell, P. Collela, L. Howell, and M. Welcome, An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys. 148, 81 (1994)
65. B. Swartz, The second order sharpening of blurred smooth borders, Math. Computat. 52, 675 (1989)
66. S. H. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys. 100, 25 (1992)
67. S. P. Vanka, Block-implicit multigrid solution of Navier-Stokes equations in primitive variables, J. Comput. Phys. 65, 138 (1986)
68. P. Wesseling, An Introduction to Multi-Grid Methods (Wiley, 1992)
69. M. W. Williams, D. B. Kothe, and E. G. Puckett, Convergence and accuracy of kernel-based continuum surface tension models, LANL Report LA-UR-98-2268, invited paper of the Chia-Shuh Yih Memorial Symposium, in Proceedings of the Thirteenth U.S. National Congress of Applied Mechanics, Gainesville, FL 1998
70. G. Wittum, Multi-grid methods for Stokes and Navier-Stokes Equations, Transforming smoothers: Algorithms and numerical results, Numer. Math. 54, 543 (1989)
71. D. L. Youngs, Time-dependent multi-material flow with large fluid distortion, in Numerical Methods for Fluid Dynamics, edited by K. W. Morton and M. L. Norman, 1986
72. D. L. Youngs, An Interface Tracking Method for a 3D Eulerian Hydrodynamics Code, Technical Report 44/92/35 (AWRE, 1984)