Two-phase flow in complex geometries: A diffuse domain approach

CMES - Computer Modeling in Engineering and Sciences

Volumn 57, Issue 1, 2010, Pages 77-106

  Aland, S ^a   Lowengrub, J ^b   Voigt, A ^a  

^a DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Cahn Hilliard; Chaotic mixing; Complex domain; Contact angle; Diffuse domain; Diffuse interface; Finite element; Fluid structure interaction; Navier Stokes; Phase field

Indexed keywords

CHAOTIC MIXING; COMPLEX DOMAINS; DIFFUSE INTERFACE; NAVIER STOKES; PHASE FIELDS;

BOUNDARY CONDITIONS; CONTACT ANGLE; DROP FORMATION; FINITE ELEMENT METHOD; FLUID STRUCTURE INTERACTION; GEOMETRY; MIXING; NAVIER STOKES EQUATIONS; SERPENTINE; SILICATE MINERALS; TWO PHASE FLOW; WATER WAVE EFFECTS;

PHASE INTERFACES;

EID: 77951946598     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (73)

1. Al-Rawahi, N.; Tryggvason, G. (2002): Numerical simulation of dendritic solidification with convection: Two-dimensional flow. J. Comput. Phys., vol.180, pp. 471-496.
2. Al-Rawahi, N.; Tryggvason, G. (2004): Numerical simulation of dendritic solidification with convection: Three-dimensional flow. J. Comput. Phys., vol.194, pp. 677-696.
3. Anderson, D.; McFadden, G.; Wheeler, A. (1998): Diffuse interface methods in fluid mechanics. Ann. Rev. Fluid Mech., vol.30, pp. 139-165.
4. Balaras, E. (2004): Modeling complex boundaries using an external force on fixed cartesian grids in large-eddy simulations. Comput. Fluids, vol.33, pp. 375-1-04.
5. Bertozzi, A. (1998): The mathematics of moving contact lines in thin liquid films. AMS Notices, vol.45, pp. 689-697.
6. Biros, G.; Ying, L.; Zorin, D. (2003): A fast solver for the Stokes equations with distributed forces in complex geometries. J. Comput. Phys., vol.193, pp. 317-348.
7. Borazjani, I.; Ge, L.; Sotiropoulos, F. (2008): Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies. J. Comput. Phys., vol.227, pp. 7587-7620.
8. Boyer, F. (2002): A theoretical and numerical model for the study of incompressible mixture flows. Computers and Fluids, vol.31, no.1, pp. 41 - 68.
9. Bueno-Orovio, A.; Perez-Garcia, V. (2006): Spectral smoothed boundary methods: The role of external boundary conditions. Numer. Meth. Partial Diff. Eqns., vol.22, pp. 435-448.
10. Bueno-Orovio, A.; Perez-Garcia, V.; Fenton, F. (2006): Spectral methods for partial differential equations in irregular domains: The spectral smoothed boundary method. SIAM J. Sci. Comput., vol.28, pp. 886-900.
11. Choi, J.-I.; Oberoi, R.; Edwards, J.; Rosati, J. (2007): An immersed boundary method for complex incompressible flows. J. Comput. Phys., vol.224, pp. 757-784.
12. Chorin, A. J. (1968): Numerical solution of the Navier-Stokes equations. Math. Comp., vol.22.
13. Chung, M.-H. (2006): Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape. Computers & Fluids, vol.35, pp. 607-623.
14. Davis, S. (2002): Interfacial fluid dynamics. In Batchelor, G.; Moffatt, H.; Worster, M.(Eds): Perspectives in Fluid Dynamics. Cambridge University Press.
15. Davis, T. A. (2004): Algorithm 832: Umfpack v4.3-an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw., vol.30, no.2, pp. 196-199.
16. de Gennes, P. (1985): Wetting: statics and dynamics. Rev. Mod. Phys., vol.57, pp. 827-863.
17. de Zelicourt, D.; Ge, L.; Wang, C.; Sotiropoulos, F.; Gilmanov, A.; Yo- ganathan, A. (2009): Flow simulations in arbitrarily complex cardiovascular anatomies- an unstructured Cartesian grid approach. Computers & Fluids, vol.38, pp. 1749-1762.
18. Deng, J.; Shao, X.-M.; Ren, A.-L. (2006): A new modification of the immersed- boundary method for simulating flows with complex moving boundaries. Int. J. Num. Meth. Fluids, vol.52, pp. 1195-1213.
19. Do-Quang, M.; Amberg, G. (2009): The splash of a solid sphere impacting on a liquid surface: Numerical simulation of the influence of wetting. Physics of Fluids, vol.21, no.2.
20. Duarte, C.; Babuska, I.; Oden, J. (2000): Generalized finite element methods for three-dimensional structural mechanics problems. Comp. Struct., vol.77, pp. 215-232.
21. Emmerich, H. (2008): Advances of and by phase-field modeling in condensed- matter physics. Adv. Phys., vol.57, pp. 1-87.
22. Feng, X. (2006): Fully discrete finite element approximations of the Navier- Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal., vol.44, no.3, pp. 1049-1072.
23. Fenton, F.; Cherry, E.; Karma, A.; Rappel, W.-J. (2005): Modeling wave propagation in realistic heart geometries using the phase-field method. Chaos, vol.15,pp. 013502.
24. Gao, F.; Ingram, D.; Causon, D.; Mingham, C. (2007): The development of a Cartesian cut cell method for incompressible viscous flows. Int. J. Num. Meth. Fluids, vol.54, pp. 1033-1053.
25. Gilmanov, A.; Acharya, S. (2008): A hybrid immersed boundary and material point method for simulating 3d fluid-structure interaction problems. Int. J. Num. Meth. Fluids, vol.56, pp. 2151-2177.
26. Gilmanov, A.; Sotiropoulos, F. (2005): A hybrid Cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies. J. Comput. Phys., vol.207, pp. 457-492.
27. Glowinksi, R.; Pan, T.; Periaux, J. (2006): Numerical simulation of a multi- store separation phenomenon: A fictitious domain approach. Comp. Meth. Appl. Mech. Eng., vol.195, pp. 5566-5581.
28. Glowinsky, R.; Pan, T.; Hesla, T.; Joseph, D.; Periaux, J. (2001): A fictitious domain approach to the direct numerical simluation of incompessible viscous flow past moving rigid bodies: Application to particulate flow. J. Comput. Phys, vol.169, pp. 363-426.
29. Gong, Y.; Li, B.; Li, Z. (2008): Immersed-interface finite-element method for elliptic interface problems with non-homogeneous jump conditions. SIAM J. Numer. Anal., vol.26, pp. 472-495.
30. Gránásy, L.; Pusztai, T.; Saylor, D.; Warren, J. A. (2007): Phase field theory of heterogeneous crystal nucleation. Physical Review Letters, vol.98, no.3, pp. 035703.
31. Hohenberg, P.; Halperin, B. (1977): Theory of dynamic critical phenomena. Rev. Mod. Phys., vol.49, pp. 435-479.
32. Hu, H.; Patankar, N.; Zhu, M. (2001): Direct numerical simulations of fluid- solid systems using the arbitrary Lagrangian-Eulerian technique. J. Comput. Phys., vol.169, pp. 427-462.
33. Jacqmin, D. (1999): Calculation of two-phase navier-stokes flows using phase- field modeling. Journal of Computational Physics, pp. 96-127.
34. Kalliadasis, S.; Chang, H.-C. (1994): Apparent dynamic contact angle of an advancing gas-liquid meniscus. Phys. Fluids, vol.6, pp. 12-23.
35. Kay, D.; Welford, R. (2007): Efficient numerical solution of Cahn-Hilliard- Navier-Stokes fluids in 2d. SIAM J. Sci. Comput., vol.29, no.6, pp. 2241-2257.
36. Kirkpatrick, M.; Armfield, S.; Kent, J. (2003): A representation of curved boundaries for the solution of the Navier-Stokes equations on a staggered three- dimensional Cartesian grid. J. Comput. Phys., vol.184, pp. 1-36.
37. Kockelkoren, J.; Levine, H.; Rappel, W.-J. (2003): Computational approach for modeling intra- and extracellular dynamics. Phys. Rev. E, vol.68, pp. 037702.
38. Koza, J.; Uhlemann, M.; Gebert, A.; Schultz, L. (2008): The effect ofmagnetic fields on electrodeposition of CoFe alloys. Electrochim. Acta, vol.53, pp. 5344-5353.
39. Kropinski, M. (1999): Integral equation methods for particle simulations in creeping flows. Comp. Math. Appl., vol.38, pp. 67-87.
40. Levine, H.; Rappel, W.-J. (2005): Membrane-bound Turing patterns. Phys. rev. E, vol.72, pp. 061912.
41. Li, X.; Lowengrub, J.; Rätz, A.; Voigt, A. (2009): Solving pdes in complex geometries: A diffuse domain approach. Commun. Math. Sci., vol.7, pp. 81-107.
42. Li, Z.; Lin, T.; Wu, X. (2003): New Cartesian grid methods for interface problems using the finite element formulation. Num. Math., vol.96, pp. 61-98.
43. Lin, P. (2007): A fixed-grid model for simulation of a moving body in free surface flows. Computers & Fluids, vol.36, pp. 549-561.
44. Lowengrub, J.; Truskinovsky, L. (1998): Quasi-incompressible Cahn-Hilliard fluids and topological changes. Proc. Roy. Soc. London A, vol.454, pp. 2617-2654.
45. Mayo, A. (1984): The fast solution of Poisson's and the biharmonic equations on irregular regions. SIAM J. Numer. Anal., vol.21, pp. 285-299.
46. Melenk, J.; Babuska, I. (1996): The partition of unity finite element method: Basic theory and applications. Comp. Meth. Appl. Mech. Eng., vol.139, pp. 289-314.
47. Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F.; Vargas, A.; von Loebbecke, A. (2007): A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys., vol.227, pp. 4825-4852.
48. Muradoglu, M.; Kayaalp, A. (2006): An auxilliary grid method for computations of multiphase flows in complex geometries. J. Comput. Phys., vol.214, pp. 858-877.
49. Muradoglu, M.; Stone, H. A. (2005): Mixing in a drop moving through a serpentine channel: A computational study. Physics ofFluids, vol.17, no.7.
50. Oden, J.; Duarte, C.; Zienkiewicz, O. (1998): A new cloud-based hp finite element method. Comp. Meth. Appl. Mech. Eng., vol.153, pp. 117-126.
51. Oron, A.; Davis, S.; Bankoff, S. (1997): Long-scale evolution of thin liquid films. Rev. Mod. Phys., vol.69, pp. 931-980.
52. Pao, J.; Pan, T.; Glowinski, R.; Joseph, D. (2009): A fictitious domain/distributed Lagrange multiplier method for the particulate flow for Oldroyd- B fluids: A positive definiteness preserving approach. J. Non-Newtonian Fluid Mech., vol.156, pp. 95-111.
53. Ratz, A.; Voigt, A. (2006): PDE's on surfaces - a diffuse interface approach. Comm. Math. Sci., vol.4, pp. 575-590.
54. Shinn, A.; Goodwin, M.; Vanka, S. (2009): Immersed boundary computations of shear- and buoyancy-driven flows in complex enclosures. Int. J. Heat Mass Transfer, vol.52, pp. 4082-4089.
55. Singer-Loginova, I.; Singer, H. (2008): The phase field technique for modeling multiphase materials. Rep. Prog. Phys., vol.71, pp. 106501.
56. Teigen, K.; Li, X.; Lowengrub, J.; Wang, F.; Voigt, A. (2010): A diffuse- interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Comm. Math. Sci. in press.
57. Teigen, K.; Song, P.; Lowengrub, J.; Voigt, A. (2010): A diffuse interface method for two-phase flows with soluble surfactants. J. Comput. Phys. in review.
58. Tryggvason, G.; Buner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J. (2001): A front-tracking method for the computation of multiphase flow. J. Comput. Phys., vol.169, pp. 708-759.
59. Tseng, Y.-H.; Ferziger, J. (2003): A ghost-cell immersed boundary method for flow in complex geometry. J. Comput. Phys., vol.192, pp. 593-623.
60. Udaykumar, H.; Kan, H.-C.; Shyy, W.; Tran-Son-Tay, R. (1997): Multiphase dynamics in arbitrary geometries on fixed Cartesian grids. J. Comput. Phys., vol.137, pp. 366-405.
61. Udaykumar, H.; Mittal, R.; Rampunggoon, P.; Khanna, A. (2001): A sharp interface Cartesian grid method for simulating flows with complex moving boundaries. J. Comput. Phys., vol.174, pp. 345-380.
62. Uzgoren, E.; Sim, J.; Shyy, W. (2009): Marker-based, 3-d adaptive Cartesian grid method for multiphase flows around irregular geometries. Commun. Comput. Phys., vol.5, pp. 1-41.
63. Verfürth, R. (1996): A review of a posteriori error estimation and adaptive mesh- refinement techniques. Wiley-Teubner Series Advances in Numerical Mathematics. Chichester: John Wiley & Sons. Stuttgart: B. G. Teubner. vi, 127 p.
64. Vey, S.; Voigt, A. (2007): Amdis: adaptive multidimensional simulations. Computing and Visualization in Science, vol.10, no.1, pp. 57-67.
65. Villanueva, W.; Amberg, G. (2006): Some generic capillary-driven flows. Int. J. Multiphase Flow, vol.32, no.9, pp. 1072-1086.
66. Wang, M. Y.; Zhou, S. (2004): Phase field: A variational method for structural topology optimization. CMES: Computer Modelling in Engineering & Sciences, vol.6, pp. 547-566. (Pubitemid 40020857)
67. Xie, H.; Koshizuka, S.; Oka, Y. (2007): Modeling the wetting effects in droplet impingement using particle method. CMES: Computer Modelling in Engineering & Sciences, vol.18, pp. 1-16.
68. Xu, S.; Wang, Z. (2006): An immersed interface method for simulating the interaction of a fluid with moving boundaries. J. Comput. Phys., vol.216, pp. 454-493.
69. Yu, H. C.; Chen, H. Y.; Thornton, K. (submitted 2009): Smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries.
70. Yu, Z.; Shao, X. (2007): A direct-forcing fictitious domain method for particulate forces. J. Comput. Phys., vol.227, pp. 292-314.
71. Yu, Z.; Thien-Phan, N.; Fan, Y.; Tanner, R. (2002): Viscoelastic mobility problem of a system of particles. J. Non-Newtonian Fluid Mech., vol.104, pp. 87-124.
72. Zhao, H.-K.; Merriman, B.; Osher, S.; Wang, L. (1998): Capturing the behavior of bubbles and drops using the variational level set approach. J. Comput. Phys., vol.143, pp. 495-U8.
73. Zhou, C.; Yue, P.; Feng, J.; Ollivier-Gooch, C.; Hu, H. (2010): 3D phase- field simulations of interfacial dynamics in Newtonian and viscoelastic fluids. J. Comput. Phys., vol.229, pp. 498-511.