A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows

Journal of Computational Physics

Volumn 219, Issue 2, 2006, Pages 780-800

  Marchandise, Emilie ^a,b   Remacle, Jean François ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^b FNRS   (Belgium)

Author keywords

3D incompressible Navier Stokes; Discontinuous Galerkin; Free surface; Level set; Two phase flow model

Indexed keywords

FINITE ELEMENT METHOD; GALERKIN METHODS; INCOMPRESSIBLE FLOW; NAVIER STOKES EQUATIONS; NUMERICAL METHODS;

3D INCOMPRESSIBLE NAVI-STOKE; DISCONTINOUS GALERKIN METHODS; DISCONTINUOUS GALERKIN; FREE SURFACES; INCOMPRESSIBLE NAVIER-STOKES; LEVEL SET; STABILIZED FINITE ELEMENT; TWO PHASE; TWO PHASES FLOW; TWO-PHASE FLOW MODELLING;

TWO PHASE FLOW;

EID: 33750618189     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2006.04.015     Document Type: Article

References (62)

1. Marchandise E., Remacle J.-F., and Chevaugeon N. A quadrature free discontinuous Galerkin method for the level set equation. Journal of Computational Physics 212 (2006) 338-357
2. Remacle J.-F., Chevaugeon N., Marchandise E., and Geuzaine C. Efficient visualization of high order finite elements. International Journal for Numerical Methods in Engineering (2006)
3. C.W. Hirt, A.A. Amsden, J.L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, Journal of Computational Physics 135 (2) (1997), in press.
4. Hughes T., Liu W., and Zimmermann T. Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Computer Methods in Applied Mechanics and Engineering 29 (1981) 239-349
5. Donea J. Arbitrary Lagrangian-Eulerian finite element methods. Computational Methods for Transient Analysis 1 (1983) 473-516
6. Dettmer W., Saksono P., and Peric D. On a finite element formulation for incompressible Newtonian fluid flows on moving domains in the presence of surface tension. Computer Methods in Applied Mechanics and Engineering 19 (2003) 659-668
7. Harlow F., and Welch J. Volume tracking methods for interfacial flow calculations. Physics of Fluids 8 (1965) 21-82
8. Hirt C., and Nichols B. Volume of fluid method (VOF) for the dynamics of free boundaries. Journal of Computational Physics 39 (1981) 201-225
9. Pilliot J., and Puckett E. Second order accurate volume-of-fluid algorithms for tracking material interfaces. Journal of Computational Physics 199 (2004) 465-502
10. Osher S., and Sethian J.A. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79 (1988) 12-49
11. Sussman M., and Fatemi E. An efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow. SIAM Journal on Scientific Computing 20 (1999) 1165-1191
12. Sussman M., Smereka P., and Osher S. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics 114 (1994) 146-159
13. Sussman M., Fatemi E., Smereka P., and Osher S. An improved level set method for incompressible two-fluid flows. Computers and Fluids 27 (1998) 663-680
14. Chessa J., and Belytschko T. An extended finite element method for two-phase fluids. Journal of Applied Mechanics (Transactions of the ASME) 70 (2003) 10-17
15. Chessa J., and Belytschko T. An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension. International Journal for Numerical Methods in Engineering 58 (2003) 2041-2064
16. Norman C., and Miksis M. Non-linear dynamics of thin films and fluid interfaces. Physica D: Nonlinear Phenomena 209 (2005) 191-204
17. Nagrath S., Jansen K., and Lahey R. Computation of incompressible bubble dynamics with a stabilized finite element level set method. Computer Methods in Applied Mechanics and Engineering 194 42-44 (2005) 4565-4587
18. Osher S., and Shu C.-W. High order essentially non-oscillatory schemes for Hamilton-Jacobi equations. SIAM Journal on Numerical Analysis 728 (1991) 902-921
19. Hu C., and Shu C.-W. Weighted ENO schemes for Hamilton-Jacobi equations. SIAM Journal on Scientific Computing 21 6 (1999) 2126-2143
20. Enright D., Fedkiw R., Ferziger J., and Mitchell I. A hybrid particle level set method for improved interface capturing. Journal of Computational Physics 183 (2002) 83-116
21. Nagrath S., Jansen K., and Lahey R. Computation of incompressible bubble dynamics with a stabilized finite element level set method. Computer Methods in Applied Mechanics and Engineering 194 (2005) 4565-4587
22. Tornberg A.-K., and Engquist B. A finite element based level set method for multiphase flow applications. Computing Visualization in Science 3 (2000) 93-101
23. van der Pijl S., Segal A., Vuik C., and Wesseling P. A mass-conserving levelset method for modeling of multiphase flows. International Journal for Numerical Methods in Fluids 47 (2004) 339-361
24. Ferziger J. Interfacial transfer in Tryggvason's method. International Journal for Numerical Methods in Fluids 41 5 (2003) 551-560
25. A. Smoliansky, Numerical modeling of two-fluid interfacial flows, Ph.D. Thesis, University of Jyvaskyla, Finland, 2001.
26. A.-K. Tornberg, Interface tracking methods with applications to multiphase flows, Ph.D. Thesis, Royal Institute of Technology (KTH), Finland, 2000.
27. Chorin A. Numerical solution of the Navier-Stokes equations. Mathematical Computation 22 (1968) 745-762
28. Y. Kallinderis, K. Nakajima, Finite element method for incompressible viscous flow with adaptative grids, AIAA Journal 8 (32) (1994), in press.
29. Kallinderis Y., and Schulz W. Unsteady flow structure interaction for incompressible flows using deformable grids. Journal of Computational Physics 143 2 (1998) 569-597
30. Hughes T.J.R., and Franca L.P. A new finite element formulation for computational fluid dynamics: V. circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations. Computer Methods in Applied Mechanics and Engineering 59 (1986) 85-99
31. T. De Mulder, Stabilized finite elements methods for turbulent incompressible single-phase and dispersed two-phase flows, Ph.D. Thesis, von Karman Institute For Fluid Dynamics, Belgium, 1997.
32. Chorin A. A numerical method for solving incompressible viscous flow problems. Journal of Computational Physics 135 2 (1997) 118-125
33. Rogers S., Kwak D., and Kiris C. Steady and unsteady solutions of the incompressible Navier-Stokes equations. AIAA Journal 29 (1991) 306-610
34. Hughes T., Franca L., and Hulbert G. A new finite element formulation for fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations. Computer Methods in Applied Mechanics and Engineering 73 (1989) 173-189
35. Franca L., and Frey S. Stabilized finite element methods: II. The incompressible Navier Stokes equations. Computer Methods in Applied Mechanics and Engineering 59 (1992) 209-233
36. Ladyshenskaya O. The Mathematical Theory of Viscous Incompressible Flow (1969), Gordon and Breach, New York
37. I. Babuska, The finite element method with Lagrangian multipliers, Numerical Mathematics 179 (1971), in press.
38. Brezzi F., and Fortin M. Mixed and Hybrid Finite Element Methods (1991), Springer, New York
39. Remacle J.-F., Flaherty J., and Shephard M. An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Review 45 (2003) 53-72
40. Remacle J.-F., and Shephard M.S. An algorithm oriented mesh database. International Journal for Numerical Methods in Engineering 58 2 (2003) 349-374
41. Babuska I. The finite element method with Lagrangian multipliers. Numerische Mathematik 20 (1971) 179-192
42. Brezzi F. On the existence uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. R.A.I.R.O., Série Rouge Analyse Numérique 8 (1974) 129-151
43. Tezduyar T., and Osawa Y. Finite element stabilization parameters computed from element matrices and vectors. Computer Methods in Applied Mechanics and Engineering 190 (2000) 411-430
44. Shakib F., Hughes T., and Johan Z. A new finite element formulation for computational fluid dynamics: the compressible Euler and Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 89 (1991) 141-219
45. T.J. Barth, Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations, Special course on unstructured grid methods for advection dominated flows, AGARD R-787, 1992 (Chapter 4).
46. Selmin V., and Formaggia L. Unified construction of finite element and finite volume discretizations for compressible flows. International Journal for Numerical Methods in Fluids 39 (1996) 1-32
47. P. Geuzaine, An implicit upwind finite volume method for compressible turbulent flows on unstructured meshes, Ph.D. Thesis, Université de Liège, Belgium, 1999.
48. M. Mount, N.S. Arya, R. Netanyahu, Silverman, An optimal algorithm for approximate nearest neighbor searching, Journal of the ACM 45 (1998), in press.
49. ANN library. Available from: URL .
50. Geuzaine P. Newton-Krylov strategy for compressible turbulent flows on unstructured meshes. AIAA Journal 39 (2001) 528-531
51. Cai X.C., Farhat C., and Sarkis M. A minimum overlap restricted additive Schwarz preconditioner and applications in 3D flow simulations. Contemporary Mathematics 218 (1998) 478-484
52. Prosperetti A. Motion of two superimposed viscous fluids. Physics of Fluids 24 (1981) 1217-1223
53. Wu G., Eatock Taylor R., and Greaves D. The effect of viscosity on the transient free-surface waves in a two-dimensional tank. Journal of Engineering Mathematics 40 (2001) 77-90
54. Martin J., and Moyce W. An experimental study of the collapse of liquid columns on a rigid horizontal plane. Philosophical Transactions A 244 (1952) 312-324
55. A. Ritter, Die fortpflanzung der wasserwellen., Z. Ver. deut. Ing. 36 (1982), in press.
56. M. Quecedo, M. Pastor, M. Herreros, Comparison of two mathematical models for solving the dam break problem using the FEM method, Computer Methods in Applied Mechanics and Engineering (in press).
57. Shigematsu T., Liu P., and Oda K. Numerical modeling of the initial stages of dam-break waves. Journal of Hydraulic Research 42 2 (2004) 183-195
58. Puckett E., Almgren A., Bell J., Marcus D., and Rider W. A high-order projection method for tracking fluid interfaces in variable density incompressible flows. Journal of Computational Physics 130 (1997) 269-282
59. Popinet S., and Zaleski S. A front-tracking algorithm for accurate representation of surface tension. International Journal for Numerical Methods in Fluids 30 (1999) 775-793
60. Nourgaliev R.R., Dinh T.N., and Theofanous T.G. A pseudocompressibility method for the numerical simulation of incompressible multifluid flows. International Journal of Multiphase Flow 30 (2004) 6001-6937
61. Mount M., Arya N.S., Netanyahu R., and Silverman. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh. Journal of Computational Physics 152 2 (1999) 642-663
62. Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability (1961), Clarendon Press, Oxford