A brief review of the phase-field-based lattice Boltzmann method for multiphase flows
Abstract view|2279|times PDF download|675|times
Abstract
In this paper, we present a brief overview of the phase-field-based lattice Boltzmann method (LBM) that is a distinct and efficient numerical algorithm for multiphase flow problems. We first give an introduction to the mathematical theory of phase-field models for multiphase flows, and then present some recent progress on the LBM for the phase-field models which are composed of the classic Navier-Stokes equations and the Cahn-Hilliard or Allen-Cahn equation. Finally, some applications of the phase-field-based LBM are also discussed.
Cited as: Wang, H., Yuan, X., Liang, H., Chai, Z., Shi, B. A brief review of the phase-field-based lattice Boltzmann method for multiphase flows. Capillarity, 2019, 2(3): 33-52, doi: 10.26804/capi.2019.03.01
Keywords
Full Text:
PDFReferences
Abadi, R.H., Rahimian, M.H., Fakhari, A. Conservative phase-field lattice-Boltzmann model for ternary fluids. J. Comput. Phys. 2018, 374: 668-691.
Aidun, C.K., Clausen, J.R. Lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 2010, 42: 439- 472.
Allen, S., Cahn, J. Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys. Acta. Metall. 1976, 24(5): 425-437.
Amaya-Bower, L., Lee, T. Single bubble rising dynamics for moderate Reynolds number using lattice Boltzmann method. Comput. Fluids 2010, 39(7): 1191-1207.
Anderson, D.M., McFadden, G.B., Wheeler, A.A. Diffuse- interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 1998, 30(1): 139-165.
Anna, S. Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 2016, 48: 285-309.
Bandalassi, V.E., Ceniceros, H.D., Banerjee, S. Computation of multiphase systems with phase field models, J. Comput. Phys. 2003, 190: 371-397.
Bhaga, D., Weber, M.E. Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 1981, 105: 61-85.
Bonhomme, R., Magnaudet, J., Duval, F., et al. Inertial dynamics of air bubbles crossing a horizontal fluid-fluid interface. J. Fluid Mech. 2012, 707: 405-443.
Brassel, M., Bretin, E. A modified phase field approximation for mean curvature flow with conservation of the volume. Math. Meth. Appl. Sci. 2011, 34: 1157-1180.
Brennen, C.E. Fundamentals of multiphase flow. Cambridge university press, 2005.
Cahn, J.W., Hilliard, J.E. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 1958, 28(2): 258-267.
Cahn, J.W., Hilliard, J.E. Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 1959, 31(3): 688-699.
Chai, Z., He, N., Guo, Z., et al. Lattice Boltzmann model for high-order nonlinear partial differential equations. Phys. Rev. E 2018a, 97: 013304.
Chai, Z., Liang, H., Du, R., et al. A lattice Boltzmann model for two-phase flow in porous media. SIAM J. Sci. Comput. 2018, 41: B746-B772.
Chai, Z., Shi, B., Guo, Z. A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic convection-diffusion equations. J. Sci. Comput. 2016, 69(1): 355-390.
Chai, Z., Sun, D., Wang, H., et al. A comparative study of local and nonlocal Allen-Cahn equations with mass conservation. Int. J. Heat Mass Transf. 2018b, 122: 631- 642.
Chai, Z., Zhao, T. Effect of the forcing term in the multiple- relaxation-time lattice Boltzmann equation on the shear stress or the strain rate tensor. Phys. Rev. E 2002, 86: 016705.
Chen, L., Kang, Q., Mu, Y., et al. A critical review of the pseudopotential multiphase lattice Boltzmann model: Methods and applications. Int. J. Heat Mass Transf. 2014, 76: 210-236.
Chen, X., Hu G. Multiphase flow in mircrofluidic devices. Adv. Appl. Mech. 2015, 45: 201503.
Chiu, P.H., Lin, Y.T. A conservative phase field method for solving incompressible two-phase flows. J. Comput. Phys. 2011, 230(1): 185-204.
Chopard, B., Falcone, J.L., Latt, J. The lattice Boltzmann advection-diffusion model revisited. Eur. Phys. J. Spec. Top. 2009, 171(1): 245-249.
Cristini, V., Tan, Y. Theory and numerical simulation of droplet dynamics in complex flows-a review. Lab Chip 2004, 4(4): 257-264.
d’Humieres, D. Generalized lattice-Boltzmann equations. in: B.D. Shizgal, D.P. Weave (Eds.), Rarefied Gas Dynamics: Theory and Simulations, Prog. Astronaut. Aeronaut, vol. 159, AIAA, Washington, DC, 1992, pp. 450-458.
Ding, H., Spelt, P.D., Shu, C. Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 2007, 226(2): 2078-2095.
Ding, H., Spelt, P.D.M. Wetting condition in diffuse interface simulations of contact line motion. Phys. Rev. E 2007, 75(4): 046708.
Dong, H., Carr, W.W., Bucknall, D.G., et al. Temporally- resolved inkjet drop impaction on surfaces. AIChE J. 2007, 53(10): 2606-2617.
Fakhari, A., Geier, M., Bolster, D. A simple phase-field model for interface tracking in three dimensions. Comput. Math. Appl. 2016a, 78(4): 1154-1165.
Fakhari, A., Geier, M., Lee, T. A mass-conserving lattice Boltzmann method with dynamic grid refinement for immiscible two-phase flows. J. Comput. Phys. 2016b, 315: 434-457.
Fakhari, A., Rahimian, M.H. Phase-field modeling by the method of lattice Boltzmann equations. Phys. Rev. E 2010, 81(3): 036707.
Folch, R., Casademunt, J., Hernández-Machado, A. Phase- field model for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach. Phys. Rev. E 1999, 60: 1724-1733.
Gan, H., Shan, X., Eriksson, T., et al. Reduction of droplet volume by controlling actuating waveforms in inkjet printing for micro-pattern formation. J. Micromech. Microeng. 2009, 19(5): 055010.
Geier, M., Fakhari, A., Lee, T. Conservative phase-field lattice Boltzmann model for interface tracking equation. Phys. Rev. E 2015, 91(6): 063309.
Goncharov, V.N. Analytical model of nonlinear, single-mode, classical Rayleigh-Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 2002, 88(13): 134502.
Gunstensen, A.K., Rothman, D.H., Zaleski, S., et al. Lattice Boltzmann model of immiscible fluids. Phys. Rev. A 1991, 43(8): 4320.
Guo, Z., Shi, B., Wang, N. Lattice BGK model for incompressible Navier-Stokes equation. J. Comput. Phys. 2000, 165(1): 288-306.
Guo, Z., Shu, C. Lattice Boltzmann method and its applica- tions in engineering. Singapore: World Scientific, 2013.
Guo, Z., Zheng, C., Shi, B. Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys. Rev. E 2008, 77: 036707.
Guo, Z., Zheng, C., Shi, B. Force imbalance in lattice Boltzmann equation for two-phase flows. Phys. Rev. E 2011, 83(3): 036707.
He, X., Chen, S., Zhang, R. A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability. J. Comput. Phys. 1999a, 152(2): 642-663.
He, X., Zhang, R., Chen, S., et al. On the three-dimensional Rayleigh-Taylor instability. Phys. Fluids 1999b, 11(5): 1143-1152.
Hirt, C.W., Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 1981, 39(1): 201-225.
Hohenberg, P.C., Halperin, B.I. Theory of dynamic critical phenomena. Rev. Mod. Phys. 1977, 49(3): 435.
Hu, Y., Li, D., Jin, L., et al. Hybrid Allen-Cahn-based lattice Boltzmann model for incompressible two-phase flows: The reduction of numerical dispersion. Phys. Rev. E 2019a, 99(2): 023302.
Hu, Y., Li, D., Niu, X., et al. A diffuse interface lattice Boltzmann model for thermocapillary flows with large density ratio and thermophysical parameters contrasts. Int. J. Heat Mass Transf. 2019b, 138: 809-824.
Hua, J., Lin, P., Liu, C., et al. Energy law preserving C0 finite element schemes for phase field models in two-phase flow computations. J. Comput. Phys. 2011, 230(19): 7115- 7131.
Hua, J., Lou, J. Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 2007, 222: 769-795.
Huang, H., Huang, J., Lu, Y. A mass-conserving axisymmetric multiphase lattice Boltzmann method and its application in simulation of bubble rising, J. Comput. Phys. 2014, 269: 386-402.
Huang, H., Sukop, M., Lu, X. Multiphase lattice Boltzmann methods: Theory and application. John Wiley & Sons, 2015.
Inamuro, T., Konishi, N., Ogino, F. A Galilean invariant model of the lattice Boltzmann method for multiphase fluid flows using free-energy approach. Comput. Phys. Commun. 2000, 129(1-3): 32-45.
Jacqmin, D. Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 1999, 155(1): 96-127.
Jullien, M., Tsang, M., Cohen, C., et al. Droplet breakup in microfluidic T-junctions at small capillary numbers. Phys. Fluids 2009, 21(7): 072001.
Kendon, V.M., Cates, M.E., Pagonabarraga, I., et al., Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: A lattice Boltzmann study, J. Fluid Mech. 2001, 440: 147-203.
Kim, J., Lee, S., Choi, Y. A conservative Allen-Cahn equation with a spaceCtime dependent Lagrange multiplier. Int. J. Eng. Sci. 2014, 84: 11-17.
Kim, J., Jeong, D., Yang, S.D., et al. A finite difference method for a conservative Allen-Cahn equation on non- flat surfaces. J. Comput. Phys. 2017, 334: 170-181.
Kintses, B., Van Vliet, L., Devenish, S., et al. Microfluidic droplets: New integrated workflows for biological experiments. Curr. Opin. Chem. Biol. 2010, 14(5): 548- 555.
Krüger, T., Kusumaatmaja, H., Kuzmin, A., et al. The lattice Boltzmann method: Principles and practice. Springer, Switzerland, 2017.
Kurtoglu, I.O., Lin, C.L. Lattice Boltzmann study of bubble dynamics. Numer. Heat Transf. B 2006, 50(4): 333-351.
Lallemand, P., Luo, L.S. Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 2000, 61(6): 6546.
Lee, D., Kim, J. Comparison study of the conservative Allen- Cahn and the Cahn-Hilliard equations. Math. Comput. Simul. 2016, 119: 35-56.
Lee, H., Kim, J. An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations. Comput. Phys. Commun. 2012, 183: 2107-2115.
Lee, T., Lin, C.L. A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. J. Comput. Phys. 2005, 206(1): 16-47.
Leshansky, A., Afkhami, S., Jullien, M., et al. Obstructed breakup of slender drops in a microfluidic T junction. Phys. Rev. Lett. 2012, 108(26): 264502.
Li, Q., Luo, K., Gao, Y., et al. Additional interfacial force in lattice Boltzmann models for incompressible multiphase flows. Phys. Rev. E 2012, 85(2): 026704.
Li, Q., Luo, K., Kang, Q., et al. Lattice Boltzmann methods for multiphase flow and phase-change heat transfer. Prog. Energy Combust. Sci. 2016, 52: 62-105.
Li, W., Vigil R.D., Beresnev I.A., et al. Vibration-induced mobilization of trapped oil ganglia in porous media: Ex- perimental validation of a capillary-physics mechanism. J. Colloid Interface Sci. 2005, 289(1): 193-199.
Liang H, Liu H, Chai Z, et al. Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio. Phys. Rev. E 2019b, 99(6): 063306.
Liang, H., Li, Q., Shi, B., et al. Lattice Boltzmann simulation of three-dimensional Rayleigh-Taylor instability. Phys. Rev. E 2016b, 93(3): 033113.
Liang, H., Li, Y., Chen, J., et al. Axisymmetric lattice Boltzmann model for multiphase flows with large density ratio. Int. J. Heat Mass Tran. 2019a, 130: 1189-1205.
Liang, H., Shi, B., Chai, Z. Lattice Boltzmann modeling of three-phase incompressible flows. Phys. Rev. E 2016a, 93(1): 013308.
Liang, H., Shi, B., Chai, Z. An efficient phase-field-based multiple-relaxation-time lattice Boltzmann model for three-dimensional multiphase flows. Comput. Math. Appl. 2017, 73(7): 1524-1538.
Liang, H., Shi, B., Guo Z., et al. Phase-field-based multiple- relaxation-time lattice Boltzmann model for incompress- ible multiphase flows, Phys. Rev. E 2014, 89: 053320.
Liang, H., Xu, J., Chen, J., et al. Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows. Phys. Rev. E 2018, 97(3): 033309.
Link, D., Anna, S., Weitz, D., et al. Geometrically mediated breakup of drops in microfluidic devices. Phys. Rev. Lett. 2004, 92(5): 054503.
Liu, C., Shen, J. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier- spectral method. Physica D 2003, 179(3-4): 211-228.
Liu, H., Kang, Q., Leonardi C.R., et al. Multiphase lattice Boltzmann simulations for porous media applications. Comput. Geosci. 2016, 20(4): 777-805.
Liu, H., Valocchi, A.J., Zhang, Y., et al. Phase-field-based lattice Boltzmann finite-difference model for simulating thermocapillary flows. Phys. Rev. E 2013, 87(1): 013010.
Liu, H., Valocchi, A.J., Zhang, Y., et al. Lattice Boltzmann phase-field modeling of thermocapillary flows in a confined microchannel. J. Comput. Phys. 2014, 256: 334-356.
Liu, H., Zhang, Y. Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 2010, 229(24): 9166-9187.
Liu, X., Chai, Z., Shi, B. A phase-field-based lattice Boltzmann modeling of two-phase electro-hydrodynamic flows. Phys. Fluids 2019, 31: 092103.
Lou, Q., Guo, Z., Shi, B. Effects of force discretization on mass conservation in lattice Boltzmann equation for two- phase flows. Europhys. Let. 2012, 99(6): 64005.
Lowengrub, J., Truskinovsky, L. Quasi-incompressible Cahn- CHilliard fluids and topological transitions. P. Roy. Soc. A-Math. Phy. 1998, 454(1978): 2617-2654.
Muradoglu, M., Tasoglu, S. A front-tracking method for computational modeling of impact and spreading of viscous droplets on solid walls. Comput. Fluids 2010, 39(4): 615-625.
Qian, Y.H., d’Humires, D., Lallemand, P. Lattice BGK models for Navier-Stokes equation. Europhys. Let. 1992, 17(6): 479.
Ren, F., Song, B., Sukop, M.C., et al. Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation. Phys. Rev. E 2016a, 94(2): 023311.
Ren, F., Song, B., Sukop, M.C. Terminal shape and velocity of a rising bubble by phase-field-based incompressible lattice Boltzmann model. Water Resour. Res. 2016b, 97: 100-109.
Rubinstein, J., Sternberg, P. Nonlocal reactionłdiffusion equations and nucleation. IMA J. Appl. Math. 1992, 48(3): 249-264.
Scarbolo, L., Molin, D., Perlekar, P., et al. Unified framework for a side-by-side comparison of different multicompo- nent algorithms: Lattice Boltzmann vs. phase field model. J. Comput. Phys. 2013, 234: 263-279.
Shan, X., Chen, H. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E 1993, 47(3): 1815.
Shan, X., Chen, H. Simulation of nonideal gases and liquid- gas phase transitions by the lattice Boltzmann equation. Phys. Rev. E 1994, 49(4): 2941.
Shao, J.Y., Shu, C., Huang, H.B., et al. Free-energy-based lattice Boltzmann model for the simulation of multiphase flows with density contrast. Phys. Rev. E 2014, 89(3): 033309.
Shen, J. Modeling and numerical approximation of two- phase incompressible flows by a phase-field approach. Multiscale Modeling And Analysis For Materials Simulation, 2012.
Shi, B., Deng, B., Du, R., et al. A new scheme for source term in LBGK model for convection- diffusion equation. Comput. Math. Appl. 2008, 55: 1568-1575.
Shi, B., Guo, Z. Lattice Boltzmann model for nonlinear convection-diffusion equations. Phys. Rev. E 2009, 79(1): 016701.
Smith, K.A., Solis, F.J., Chopp, D. A projection method for motion of triple junctions by level sets. Interface. Free Bound. 2002, 4(3): 263-276.
Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford: Clarendon Press, 2001. Sun, Y., Beckermann, C. Sharp interface tracking using the phase-field equation. J. Comput. Phys. 2007, 220(2): 626- 653.
Sussman, M., Smereka, P., Osher, S. A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 1994, 114(1): 146-159.
Sussman, M., Smith, K.M., Hussaini, M.Y., et al. A sharp interface method for incompressible two-phase flows. J. Comput. Phys. 2007, 221(2): 469-505.
Su, T., Li, Y., Liang, H., et al. Numerical study of single bubble rising dynamics using the phase field lattice Boltzmann method. Int. J. Mod. Phys. C 2018, 29(11): 1-27.
Swift, M.R., Osborn, W., Yeomans, J. Lattice Boltzmann simulation of nonideal fluids. Phys. Rev. Lett. 1995, 75(5): 830.
Teh, S.Y., Lin, R., Hung, L.H., et al. Droplet microfluidics. Lab Chip 2008, 8(2): 198-220.
Unverdi, S.O., Tryggvason, G. A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 1992, 100(1): 25-37.
Wang, H., Chai, Z., Shi, B., et al. Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn- Hilliard equations. Phys. Rev. E 2016, 94(3): 033304.
Wang, H. Phase-field lattice Boltzmann method for flow and heat transfer of multiphase fluid. Ph.D Thesis, Huazhong University of Science and Technology, 2018.
Wang, Y., Shu, C., Shao, J.Y., et al. A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio. J. Comput. Phys. 2015, 290: 336-351.
Wörner, M. Numerical modeling of multiphase flows in microfluidics and microprocess engineering: A review of methods and applications. Microfluid. Nanofluid. 2012, 12(6): 841-886.
Yang, K., Guo, Z. Lattice Boltzmann method for binary fluids based on mass-conserving quasi-incompressible phase- field theory. Phys. Rev. E 2016, 93(4): 043303.
Yang, X., Forest, M.G., Li, H., et al. Modeling and simulations of drop pinch-off from liquid crystal filaments and the leaky liquid crystal faucet immersed in viscous fluids. J. Comput. Phys. 2013, 236: 1-14.
Yuan, H.Z., Chen, Z., Shu, C, et al. A free energy-based surface tension force model for simulation of multiphase flows by level-set method. J. Comput. Phys. 2017, 345: 404-426.
Yue, P., Feng, J., Liu, C., et al. A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 2004, 515: 293-317.
Yue. P., Zhou. C., Feng. J. Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 2007, 223(1): 1-9.
Zhai, S., Weng, Z., Feng, X. Investigations on several numer- ical methods for the non-local Allen-Cahn equation. Int. J. Heat Mass Transf. 2015, 87: 111-118.
Zhang, S., Wang, M. A nonconforming finite element method for the Cahn-Hilliard equation. J. Comput. Phys. 2010, 229(19): 7361-7372.
Zheng, L., Zheng, S., Zhai, Q. Lattice Boltzmann equation method for the Cahn-Hilliard equation. Phys. Rev. E 2015, 91(1): 013309.
Zheng, L., Zheng, S., Zhai, Q.. Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow. Phys. Lett. A 2016, 380(4): 596- 603.
Zheng, L., Zheng, S. Phase-field-theory-based lattice Boltz- mann equation method for N immiscible incompressible fluids. Phys. Rev. E 2019, 99(6): 063310.
Zheng, H., Shu, C., Chew, Y.T. Lattice Boltzmann interface capturing method for incompressible flows. Phys. Rev. E 2005, 72(5): 056705.
Zheng, H., Shu, C., Chew, Y.T. A lattice Boltzmann model for multiphase flows with large density ratio. J. Comput. Phys. 2006, 218(1): 353-371.
Zheng, H., Shu, C., Chew, Y.T., et al. Three-dimensional lattice Boltzmann interface capturing method for incompressible flows. Int. J. Numer. Methods Fluids 2008, 56(9): 1653- 1671.
Zhou, Y. Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 2017a, 720: 1-136.
Zhou, Y. Rayleigh-Taylor and Richtmyer-Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 2017b, 723: 1-160.
Zu, Y., He, S. Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts. Phys. Rev. E 2013, 87(4): 043301.
Refbacks
- There are currently no refbacks.
Copyright (c) 2019 The Author(s)
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.