### COMPARISON OF NUMERICAL METHODS FOR TERNARY FLUID FLOWS: IMMERSED BOUNDARY, LEVEL-SET, AND PHASE-FIELD METHODS

DOI QR Code

LEE, SEUNGGYU;JEONG, DARAE;CHOI, YONGHO;KIM, JUNSEOK

• 투고 : 2016.03.02
• 심사 : 2016.03.11
• 발행 : 2016.03.25
• 4 1

#### 초록

This paper reviews and compares three different methods for modeling incompressible and immiscible ternary fluid flows: the immersed boundary, level set, and phase-field methods. The immersed boundary method represents the moving interface by tracking the Lagrangian particles. In the level set method, an interface is defined implicitly by using the signed distance function, and its evolution is governed by a transport equation. In the phase-field method, the advective Cahn-Hilliard equation is used as the evolution equation, and its order parameter also implicitly defines an interface. Each method has its merits and demerits. We perform the several simulations under different conditions to examine the merits and demerits of each method. Based on the results, we determine the most suitable method depending on the specific modeling needs of different situations.

#### 키워드

ternary fluid flows;continuum surface force;immersed boundary method;level set method;phase-field method;Navier-Stokes equation

#### 참고문헌

1. J.M. Park and P.D. Anderson, A ternary model for double-emulsion formation in a capillary microfluidic device, Lab Chip, 12(15) (2012), 2672-2677. https://doi.org/10.1039/c2lc21235h
2. L. Szalmas, Viscous velocity, diffusion and thermal slip coefficients for ternary gas mixtures, Euro. J. Mech. B-Fluid., 53 (2015), 264-271. https://doi.org/10.1016/j.euromechflu.2015.06.005
3. A.S. Utada, E. Lorenceau, D.R. Link, P.D. Kaplan, and H.A. Stone, Weitz DA. Monodisperse double emulsions generated from a microcapillary device, Science, 308(5721) (2005), 537-541. https://doi.org/10.1126/science.1109164
4. C.S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25(3) (1977), 220-252. https://doi.org/10.1016/0021-9991(77)90100-0
5. H.C. Kan, H.S. Udaykumar, W. Shyy, and R. Tran-Son-Tay, Hydrodynamics of a compound drop with application to leukocyte modeling, Phys. Fluid., 10(4) (1998), 760-774. https://doi.org/10.1063/1.869601
6. H.C. Kan, W. Shyy, H.S. Udaykumar, P. Vigneron, and R. Tran-Son-Tay, Effects of nuclues on leukocyte recovery, Ann. Biomed. Eng., 27 (1999), 648-655. https://doi.org/10.1114/1.214
7. R. Gautier, S. Laizet, and E. Lamballais, A DNS study of jet control with microjets using an immersed boundary method, Int. J. Comput. Fluid Dyn., 28 (2014), 393-410. https://doi.org/10.1080/10618562.2014.950046
8. P. Ouro, L. Cea, L. Ramirez, and X. Nogueira, An immersed boundary method for unstructured meshes in depth averaged shallow water models, Int. J. Numer. M. Fluid., DOI: 10.1002/fld.4201, 2015. https://doi.org/10.1002/fld.4201
9. C. Yan, W.X. Huang, G.W. Cui, C. Xu, and Z.S. Zhang, A ghost-cell immersed boundary method for large eddy simulation of flows in complex geometries, Int. J. Comput. Fluid Dyn., 29 (2015), 12-25. https://doi.org/10.1080/10618562.2014.1002484
10. H. Hua, J. Shin, and J. Kim, Dynamics of a compound droplet in shear flow, Int. J. Heat Fluid Fl., 50 (2014), 63-71. https://doi.org/10.1016/j.ijheatfluidflow.2014.05.007
11. Y. Kim, M.C. Lai, and C.S. Peskin, Numerical simulations of two-dimensional foam by the immersed boundary method, J. Comput. Phys., 229(13) (2010) 5194-5207. https://doi.org/10.1016/j.jcp.2010.03.035
12. Y. Kim and Y. Seol, Numerical simulations of two-dimensional wet foam by the immersed boundary method, Comput. Struct., 122 (201), 259-269. https://doi.org/10.1016/j.compstruc.2013.03.015
13. Y. Kim, M.C. Lai, C.S. Peskin, and Y. Seol, Numerical simulations of three-dimenisonal foam by the immersed boundary method, J. Comput. Phys., 269 (2014), 1-21. https://doi.org/10.1016/j.jcp.2014.03.016
14. S. Osher and J.A. Sethian, Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79(1) (1988), 12-49. https://doi.org/10.1016/0021-9991(88)90002-2
15. S. Osher and R.P. Fedkiw RP, Level Set Methods and Dynamic Implicit Surfaces. Springer-Verlag, New York, 2003.
16. F. Raees, D.R. Heul, and C. Vuik, A mass-conserving level-set method for simulation of multiphase flow in geometrically complicated domains, Int. J. Numer. M. Fluid, DOI:10.1002/fld.4188, 2005. https://doi.org/10.1002/fld.4188
17. J.A. Sethian and P. Smereka, Level set methods for fluid interfaces, Ann. Rev. Fluid Mech., 35 (2003), 341-372. https://doi.org/10.1146/annurev.fluid.35.101101.161105
18. B. Merriman, J.K. Bence, and S. Osher, Motion of multiple junctions a level set approach, J. Comput. Phys., 12(2) (1994), 334-363.
19. K.A. Smith, F.J. Solis, and D.L. Chopp, A projection method for motion of triple junctions by level sets, Interface. Free Bound., 4 (2002) 263-276.
20. S. Aland and F. Chen, An efficient and energy stable shceme for a phase-field model for the moving contact line problem, Int. J. Numer. M. Fluid., DOI: 10.1002/fld.4200, 2015. https://doi.org/10.1002/fld.4200
21. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. interfacial free energy, J. Chem. Phys. 28(2) (1958), 258-267. https://doi.org/10.1063/1.1744102
22. D. Anderson, G.B. McFadden, and A.A. Wheeler, Diffuse interface methods in fluid mechanics, Ann. Rev. Fluid Mech., 30(1) (1998) 139-165. https://doi.org/10.1146/annurev.fluid.30.1.139
23. D. de Fontaine, A computer simulation of the evolution of coherent composition variations in solid solutions, Ph.D. Thesis, Northwestern University, USA, 1967.
24. D. Eyre, Systems of Cahn-Hilliard equations, SIAM J. Appl. Math., 53 (1993), 1686-1712. https://doi.org/10.1137/0153078
25. J.F. Blowey, M. Copetti, and C.M. Elliott, Numerical analysis of a model for phase separation of a multi-component alloy, IMA J. Numer. Anal. 16 (1996), 111-139. https://doi.org/10.1093/imanum/16.1.111
26. F. Boyer, C. Lapuerta, S. Minjeaud, B. Piar, and M. Quintard, Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows, Transport Porous Med., 82(3) (2010), 463-483. https://doi.org/10.1007/s11242-009-9408-z
27. M. Copetti, Numerical experiments of phase separation in ternary mixtures, Math. Comput. Simul. 52(1) (2000), 41-51. https://doi.org/10.1016/S0378-4754(99)00153-6
28. J.S. Kim, Phase field computations for ternary fluid flows, Comput. M. Appl. Mech. Eng., 196 (2007), 4779-4788. https://doi.org/10.1016/j.cma.2007.06.016
29. J.S. Kim, Phase-field models for multi-component fluid flow Commun. Comput. Phys. 12 (2012) 613-661. https://doi.org/10.4208/cicp.301110.040811a
30. H.G. Lee and J. Kim, Two-dimensional Kelvin-Helmholtz instabilities of multi-component fluids, Euro. J. Mech. B-Fluid., 49 (2015), 77-88. https://doi.org/10.1016/j.euromechflu.2014.08.001
31. T.Y. Hou, Z. Li, S. Osher, and H. Zhao, A hybrid method for moving interface problems with application to the Hele-Shaw flow, J. Comput. Phys. 132(2) (1997), 236-252.
32. F. Losasso, T. Shinar, A. Selle, and R. Fedkiw, Multiple interacting liquids, ACM T. Graphic., 25(3) (2006), 812-819. https://doi.org/10.1145/1141911.1141960
33. T. Oda, N. Satofuka, and H. Nishida, Numerical analysis of particle behavior penetrating into liquid by level set method, in: S.W. Armfield, P. Morgan (Eds.), Compututional Fluid Dynamics 2002, Springer, Berlin Heidelberg, 2003, pp. 529-534.
34. K.A. Smith, F.J. Solis, L. Tao, K. Thornton, and M.O. De La Cruz, Domain growth in ternary fluids: a level set approach, Phys. Rev. Lett., 84 (2000), 91-94. https://doi.org/10.1103/PhysRevLett.84.91
35. J.S. Kim and J. Lowengrub, Phase field modeling and simulation of three-phase flows, Interface. Free Bound., 7 (2005), 435-466.
36. M. Sussman, P. Smereka, and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114(1) (1994), 146-159. https://doi.org/10.1006/jcph.1994.1155
37. H.G. Lee and J. Kim, A second-order accurate non-linear differnce scheme for the Ncomponent Cahn-Hilliard system, Physica A, 387(19) (2008), 4787-4799. https://doi.org/10.1016/j.physa.2008.03.023
38. D. Jacqmin, Contact-line dynamics of a diffuse fluid interface, J. Fluid Mech., 402 (2000), 57-88. https://doi.org/10.1017/S0022112099006874
39. H. Hua, J. Shin, and J. Kim, Level set, phase-field, and immersed boundary methods for two-phase fluid flows, J. Fluid. Eng., 136 (2014), 021301.
40. J.U. Brackbill, D.B. Kothe, and C. Zemach, A continuum method for modelling surface tension, J. Comput. Phys., 100(2) (1992), 335-354. https://doi.org/10.1016/0021-9991(92)90240-Y
41. Y. Li, A. Yun, and J. Kim, An immersed boundary method for simulating a single axisymmetric cell growth and division, J. Math. Bio., 65(4) (2012), 653-675. https://doi.org/10.1007/s00285-011-0476-7
42. F. Boyer and C. Lapuerta, Study of a three component Cahn-Hilliard flow model, ESAIMMath. Model. Numer. Anal., 40(4) (2006), 653-687. https://doi.org/10.1051/m2an:2006028
43. A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22(104) (1968), 745-762. https://doi.org/10.1090/S0025-5718-1968-0242392-2
44. Y. Li, A. Yun, D. Lee, J. Shin, D. Jeong, and J. Kim, Three-dimensional Volum-econserving immersed boundary model for two-phase fluid flows, Compu. Meth. Appl. Mech. Eng., 257 (2013), 36-46. https://doi.org/10.1016/j.cma.2013.01.009
45. H. Hua, Y. Li, J. Shin, H. Song, and J. Kim, Effect of confinement on droplet deformation in shear flow, Int. J. Comput. Fluid Dyn., 27 (2013), 317-331. https://doi.org/10.1080/10618562.2013.857406
46. C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Springer, Berlin Heidelberg, 1998.
47. H.G. Lee, J.W. Choi, and J. Kim, A practically unconditionally gradient stable scheme for the N-component Cahn-Hilliard system, Physica A, 391 (2012), 1009-1019. https://doi.org/10.1016/j.physa.2011.11.032
48. J.J. Eggleston, G.B. McFadden, and P.W. Voorhees, A phase-field model for highly anisotropic interfacial energy, Physica D, 150 (2001), 91-103. https://doi.org/10.1016/S0167-2789(00)00222-0
49. J. Kim, S. Lee, and Y. Choi, A conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier, Int. J. Eng. Sci., 84 (2014), 11-17. https://doi.org/10.1016/j.ijengsci.2014.06.004

#### 과제정보

연구 과제 주관 기관 : National Research Foundation of Korea (NRF)