Document Type : Regular Article

**Authors**

Faculty of New Technologies and Aerospace Engineering, Shahid Beheshti University, Tehran, Iran

10.47176/jafm.15.06.33536

**Abstract**

In this work, the accuracy of the multiphase lattice Boltzmann method (LBM) based on the phase-field models, namely the Cahn-Hilliard (C-H) and Allen-Cahn (A-C) equations, are evaluated for simulation of two-phase flow systems with high-density ratios. The mathematical formulation and the schemes used for discretization of the derivatives in the C-H LBM and A-C LBM are presented in a similar notation that makes it easy to implement and compare these two phase-field models. The capability and performance of the C-H LBM and A-C LBM are investigated, specifically at the interface region between the phases, for simulation of flow problems in the two-dimensional (2D) and three-dimensional (3D) frameworks. Herein, the equilibrium state of a droplet and the practical two-phase flow problem of the rising bubble are considered to evaluate the mass conservation capability of the phase-filed models employed at different flow conditions and the obtained results are compared with available numerical and experimental data. The effect of employing different equations proposed in the literature for calculating the relaxation time on the accuracy of the implemented phase-field LBMs in the interfacial region is also studied. The present study shows that the LBM based on the A-C equation (A-C LBM) is advantageous over that based on the C-H equation in dealing with the conservation of the total mass of a two-phase flow system. Also, the results obtained by the A-C LBM is more accurate than those obtained using the C-H LBM in comparison with other numerical results and experimental observations. The present study suggests the A-C LBM as a sufficiently accurate and computationally efficient phase-field model for the simulation of practical two-phase flows to resolve their structures and properties even at high-density ratios.

**Keywords**

Allen, S. M. and J. W. Cahn (1976). Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys. *Acta Metallurgica* 24(5), 425-437.##

Amaya-Bower, L. and T. Lee (2010). Single bubble rising dynamics for moderate Reynolds number using Lattice Boltzmann Method. *Computers & Fluids* 39(7), 1191-1207.##

Amaya-Bower, L. and T. Lee (2011). Numerical simulation of single bubble rising in vertical and inclined square channel using lattice Boltzmann method. *Chemical Engineering Science* 66(5), 935-952.##

Bao, J. and L. Schaefer (2013). Lattice Boltzmann equation model for multi-component multi-phase flow with high density ratios. *Applied Mathematical Modelling* 37(4), 1860-1871.##

Bhaga, D. and M. E. Weber (2006). Bubbles in viscous liquids: shapes, wakes and velocities. *Journal of Fluid Mechanics* 105, 61-85.##

Cahn, J. W. and J. E. Hilliard (1958). Free Energy of a Nonuniform System. I. Interfacial Free Energy. *The Journal of Chemical Physics *28(2), 258-267.##

Chiu, P. H. and Y. T. Lin (2011). A conservative phase field method for solving incompressible two-phase flows. Journal of Computational Physics 230(1), 185-204.##

Clift, R., J. R. G., M. E. Weber. (1978). *Bubbles, drops, and particles*. New York ; London, Academic Press.##

Ding, H., P. D. M. Spelt and C. Shu (2007). Diffuse interface model for incompressible two-phase flows with large density ratios. *Journal of Computational Physics* 226(2), 2078-2095.##

Ezzatneshan, E. (2017). Study of surface wettability effect on cavitation inception by implementation of the lattice Boltzmann method. *Physics of Fluids *29(11), 113304.##

Ezzatneshan, E. (2019). Comparative study of the lattice Boltzmann collision models for simulation of incompressible fluid flows. *Mathematics and Computers in Simulation *156, 158-177##

Ezzatneshan, E. and H. Vaseghnia (2020). Evaluation of equations of state in multiphase lattice Boltzmann method with considering surface wettability effects. *Physica A: Statistical Mechanics and its Applications* 541, 123258.##

Fakhari, A., D. Bolster and L. S. Luo (2017). A weighted multiple-relaxation-time lattice Boltzmann method for multiphase flows and its application to partial coalescence cascades. *Journal of Computational Physics* 341, 22-43.##

Fakhari, A., M. Geier and D. Bolster (2019). A simple phase-field model for interface tracking in three dimensions. *Computers & Mathematics with Applications *78(4), 1154-1165.##

Fakhari, A., M. Geier and T. Lee (2016). A mass-conserving lattice Boltzmann method with dynamic grid refinement for immiscible two-phase flows. *Journal of Computational Physics* 315, 434-457.##

Fakhari, A., Y. Li, D. Bolster and K. T. Christensen (2018). A phase-field lattice Boltzmann model for simulating multiphase flows in porous media: Application and comparison to experiments of CO2 sequestration at pore scale. *Advances in Water Resources* 114, 119-134.##

Fakhari, A., T. Mitchell, C. Leonardi and D. Bolster (2017). Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios. *Physical Review E *96(5-1), 053301.##

Fakhari, A. and M. H. Rahimian (2010). Phase-field modeling by the method of lattice Boltzmann equations.* Physical Review E *81(3 Pt 2), 036707.##

Geier, M., A. Fakhari and T. Lee (2015). Conservative phase-field lattice Boltzmann model for interface tracking equation. *Physical Review E *91(6), 063309.##

Hejranfar, K. and E. Ezzatneshan (2015). Simulation of two-phase liquid-vapor flows using a high-order compact finite-difference lattice Boltzmann method. *Phys Rev E Stat Nonlin Soft Matter Phys* 92(5), 053305.##

Holdych, D. J., D. Rovas, J. G. Georgiadis and R. O. Buckius (2011). An Improved Hydrodynamics Formulation for Multiphase Flow Lattice-Boltzmann Models. *International Journal of Modern Physics C* 09(08), 1393-1404.##

Inamuro, T., T. Ogata, S. Tajima and N. Konishi (2004). A lattice Boltzmann method for incompressible two-phase flows with large density differences. Journal of Computational Physics 198(2), 628-644.##

Lee, T. (2009). Effects of incompressibility on the elimination of parasitic currents in the lattice Boltzmann equation method for binary fluids. *Computers & Mathematics with Applications* 58(5), 987-994.##

Lee, T. (2019). Fully implicit force splitting scheme to two-phase lattice Boltzmann equation in pressure-velocity formulation. *72nd Annual Meeting of the APS Division of Fluid Dynamics *64(13), November 23–26, Seattle, Washington.##

Lee, T. and C. L. Lin (2005). A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. *Journal of Computational Physics* 206(1), 16-47.##

Lee, T. and L. Liu (2010). Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces. *Journal of Computational Physics* 229(20), 8045-8063.##

Liang, H., B. C. Shi, Z. L. Guo and Z. H. Chai (2014). Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows. *Physical Review E* 89(5), 053320.##

Liang, H., J. Xu, J. Chen, H. Wang, Z. Chai and B. Shi (2018). Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows. *Physical Review E* 97(3), 033309.##

Lou, Q. and Z. Guo (2015). Interface-capturing lattice Boltzmann equation model for two-phase flows. *Physical Review E *91(1), 013302.##

Lou, Q., Z. L. Guo and B. C. Shi (2012). Effects of force discretization on mass conservation in lattice Boltzmann equation for two-phase flows. *EPL (Europhysics Letters) *99(6), 64005.##

Lycett-Brown, D. and K. H. Luo (2015). Improved forcing scheme in pseudopotential lattice Boltzmann methods for multiphase flow at arbitrarily high density ratios. *Physical Review E* 91(2), 023305.##

Mattila, K. K., D. N. Siebert, L. A. Hegele and P. C. Philippi (2013). High-Order Lattice-Boltzmann Equations and Stencils for Multiphase Models. *International Journal of Modern Physics C* 24(12), 1340006 .##

Otomo, H., R. Zhang and H. Chen (2019). Improved phase-field-based lattice Boltzmann models with a filtered collision operator. *International Journal of Modern Physics C* 30(10), 1941009.##

Ren, F., B. Song, M. C. Sukop and H. Hu (2016). Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation. *Physical Review E *94(2-1), 023311.##

Spencer, T. J., I. Halliday and C. M. Care (2011). A local lattice Boltzmann method for multiple immiscible fluids and dense suspensions of drops. *Philosophical Transactions A: Mathematical, Physical and Engineering Sciences* 369(1944), 2255-2263.##

Su, T., Y. Li, H. Liang and J. Xu (2018). Numerical study of single bubble rising dynamics using the phase field lattice Boltzmann method. *International Journal of Modern Physics C* 29(11), 1850111.##

Sun, Y. and C. Beckermann (2007). Sharp interface tracking using the phase-field equation. *Journal of Computational Physics* 220(2), 626-653.##

Tölke, J., G. D. Prisco and Y. Mu (2013). A lattice Boltzmann method for immiscible two-phase Stokes flow with a local collision operator. *Computers & Mathematics with Applications *65(6), 864-881.##

Wang, H., X. Yuan, H. Liang, Z. Chai and B. Shi (2019). A brief review of the phase-field-based lattice Boltzmann method for multiphase flows. *Capillarity* 2(3), 33-52.##

Wang, H. L., Z. H. Chai, B. C. Shi and H. Liang (2016). Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations. *Physical Review E* 94(3-1): 033304.##

Wang, Y., C. Shu, J. Y. Shao, J. Wu and X. D. Niu (2015). A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio. *Journal of Computational Physics *290, 336-351.##

Weil, K. G. (1984). J. S. Rowlinson and B. Widom: Molecular Theory of Capillarity, Clarendon Press, Oxford 1982. 327 Seiten. Berichte der Bunsengesellschaft für physikalische Chemie 88(6), 586-586.##

Yan, X., Y. Ye, J. Chen, X. Wang and R. Du (2021). Improved multiple-relaxation-time lattice Boltzmann model for Allen–Cahn equation. *International Journal of Modern Physics C,* 32(7), 2150086*. ##*

Yang, J. and E. S. Boek (2013). A comparison study of multi-component Lattice Boltzmann models for flow in porous media applications. *Computers & Mathematics with Applications *65(6), 882-890.##

Zhao, W., Y. Zhang and B. Xu (2019). An improved pseudopotential multi-relaxation-time lattice Boltzmann model for binary droplet collision with large density ratio. *Fluid Dynamics Research* 51(2), 025510.##

Zu, Y. Q. and S. He (2013). Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts. *Physical Review E *87(4), 043301.##

November and December 2022

Pages 1771-1787

**Received:**07 November 2021**Revised:**07 June 2022**Accepted:**16 June 2022**First Publish Date:**07 September 2022