×

A cascaded phase-field lattice Boltzmann model for the simulation of incompressible, immiscible fluids with high density contrast. (English) Zbl 1443.76173

Summary: In this work, a conservative phase-field model for the simulation of immiscible multiphase flows is developed using an incompressible, velocity-based, cascaded lattice Boltzmann method (CLBM). Extensions are made to the lattice Boltzmann (LB) equations for interface tracking and incompressible hydrodynamics, proposed by A. Fakhari et al. [“Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios”, Phys. Rev. E 96, No. 5, Article ID 053301, 14 p. (2017; doi:10.1103/PhysRevE.96.053301)], by performing relaxation operations in central moment space. This was motivated by the work of L. Fei et al. [“Consistent forcing scheme in the cascaded lattice Boltzmann method”, Phys. Rev. E 96, No. 5, Article ID 053307, 8 p. (2017; doi:10.1103/PhysRevE.96.053307); “Modeling incompressible thermal flows using a central-moments-based lattice Boltzmann method”, Int. J. Heat Mass Transfer 120, 624–634 (2018; doi:10.1016/j.ijheatmasstransfer.2017.12.052)], where promising results from such a transformation were observed. The relaxation of central moments is defined in a reference frame moving with the fluid, while the existing multiple-relaxation time [P. Lallemand and L.-S. Luo, “Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability”, Phys. Rev. E 61, No. 6, 6546–6562 (2000; doi:10.1103/PhysRevE.61.6546); A. Fakhari and T. Lee, “Multiple-relaxation-time lattice Boltzmann method for immiscible fluids at high Reynolds numbers”, Phys. Rev. E 87, No. 2, Article ID 023304, 8 p. (2013; doi:10.1103/PhysRevE.87.023304)] scheme performs collision in a fixed frame of reference. Moreover, the derivations make use of continuous, Maxwellian distribution functions. As a result, the CLBM enhances the Galilean invariance and stability of the method when high lattice Mach numbers are evident. The cascaded scheme has been previously used in the literature to simulate multiphase flows based on the pseudo-potential model, where it allowed for high density and viscosity contrasts to be captured [D. Lycett-Brown and K. H. Luo, Comput. Math. Appl. 67, No. 2, 350–362 (2014; Zbl 1381.76290); D. Lycett-Brown and K. H. Luo, “Cascaded lattice Boltzmann method with improved forcing scheme for large-density-ratio multiphase flow at high Reynolds and Weber numbers”, Phys. Rev. E 94, No. 5, Article ID 053313, 20 p. (2016; doi:10.1103/PhysRevE.94.053313)]. Here, the CLBM is implemented within the phase-field framework and is verified through the analysis of a layered Poiseuille flow. The performance of the CLBM is then investigated in terms of spurious currents, Galilean invariance and computational efficiency. Finally, the work of Fakhari et al. [loc. cit.] is extended by validating the model’s ability to capture the relation between surface tension and the rise velocity of a planar Taylor bubble, in both stagnant and flowing fluids. New counter-current results indicate that the rise velocity model of H. Ha-Ngoc and J. Fabre [“Test-case no 29A: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN). I: In a stagnant liquid”, Multiphase Sci. Techn. 16, No. 1, 175–188 (2004; doi:10.1615/MultScienTechn.v16.i1-3.270)] also applies in this regime.

MSC:

76M28 Particle methods and lattice-gas methods
65Z05 Applications to the sciences

Citations:

Zbl 1381.76290
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fakhari, A.; Mitchell, T.; Leonardi, C.; Bolster, D., Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios, Phys. Rev. E, 96, 5, 1-14 (2017)
[2] Fei, L.; Luo, K. H., Consistent forcing scheme in the cascaded lattice Boltzmann method, Phys. Rev. E, 96, 5 (2017)
[3] Fei, L.; Luo, K. H.; Lin, C.; Li, Q., Modeling incompressible thermal flows using a central-moments-based lattice Boltzmann method, Int. J. Heat Mass Transfer, 120, 624-634 (2018)
[4] Lallemand, P.; Luo, L.-S., Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 61, 6, 6546 (2000)
[5] Fakhari, A.; Lee, T., Multiple-relaxation-time lattice Boltzmann method for immiscible fluids at high Reynolds numbers, Phys. Rev. E, 87, 2, 1-8 (2013)
[6] Lycett-Brown, D.; Luo, K. H., Multiphase cascaded lattice Boltzmann method, Comput. Math. Appl., 67, 2, 350-362 (2014) · Zbl 1381.76290
[7] Lycett-Brown, D.; Luo, K. H., Cascaded lattice Boltzmann method with improved forcing scheme for large-density-ratio multiphase flow at high Reynolds and Weber numbers, Phys. Rev. E, 94, 5, 1-20 (2016)
[8] Ha-Ngoc, H.; Fabre, J., Test-case no 29A: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA, PN ) Part I: In a stagnant liquid, Multiph. Sci. Technol., 16, 1-3, 175-188 (2004)
[9] Succi, S., The Lattice Boltzmann Equation: for Fluid Dynamics and Beyond (2001), Oxford University Press · Zbl 0990.76001
[10] Krüger, T.; Kusumaatmaja, H.; Kuzmin, A.; Shardt, O.; Silva, G.; Viggen, E. M., The Lattice Boltzmann Method - Principles and Practice (2016)
[11] Li, Q.; Luo, K. H.; Kang, Q. J.; He, Y. L.; Chen, Q.; Liu, Q., Lattice Boltzmann methods for multiphase flow and phase-change heat transfer, Prog. Energy Combust. Sci., 52, 62-105 (2016)
[12] Huang, H.; Sukop, M.; Lu, X., Multiphase Lattice Boltzmann Methods: Theory and Application (2015), John Wiley & Sons
[13] Wacławczyk, T., On a relation between the volume of fluid, level-set and phase field interface models, Int. J. Multiph. Flow., 97, 60-77 (2017)
[14] Mirjalili, B. S.; Jain, S.; Dodd, M., Interface-capturing methods for two-phase flows : An overview and recent developments, Annu. Res. Briefs, 1, 117-135 (2017)
[15] Hirt, C.; Nichols, B., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., 39, 1, 201-225 (1981) · Zbl 0462.76020
[16] Wang, Z.; Yang, J.; Koo, B.; Stern, F., A coupled level set and volume-of-fluid method for sharp interface simulation of plunging breaking waves, Int. J. Multiph. Flow., 35, 3, 227-246 (2009)
[17] Wacławczyk, T., A consistent solution of the re-initialization equation in the conservative level-set method, J. Comput. Phys., 299, 487-525 (2015) · Zbl 1351.76225
[18] Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517 (2002) · Zbl 1123.74309
[19] Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, N.; Tauber, W.; Han, J.; Nas, S.; Jan, Y.-J., A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169, 2, 708-759 (2001) · Zbl 1047.76574
[20] Anderson, D. M.; McFadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 1, 139-165 (1998) · Zbl 1398.76051
[21] Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling, J. Comput. Phys., 155, 1, 96-127 (1999) · Zbl 0966.76060
[22] Prosperetti, A.; Tryggvason, G., Computational Methods for Multiphase Flow (2009), Cambridge University Press
[23] Shan, X.; Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, 47, 3, 1815-1819 (1993)
[24] Shan, X.; Chen, H., Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation, Phys. Rev. E, 49, 4, 2941-2948 (1994)
[25] Cui, J.; Li, W.; Lam, W. H., Numerical investigation on drag reduction with superhydrophobic surfaces by Lattice-Boltzmann method, Comput. Math. Appl., 61, 12, 3678-3689 (2011)
[26] Swift, M. R.; Osborn, W. R.; Yeomans, J. M., Lattice Boltzmann simulation of nonideal fluids, Phys. Rev. Lett., 75, 830-833 (1995)
[27] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28, 2, 258-267 (1958) · Zbl 1431.35066
[28] Allen, S. M.; Cahn, J. W., Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys, Acta Metall., 24, 5, 425-437 (1976)
[29] Geier, M.; Fakhari, A.; Lee, T., Conservative phase-field lattice Boltzmann model for interface tracking equation, Phy. Rev. E, 91, 6 (2015)
[30] Mitchell, T.; Leonardi, C.; Fakhari, A., Development of a three-dimensional phase-field lattice Boltzmann method for the study of immiscible fluids at high density ratios, Int. J. Multiph. Flow., 107, 1-15 (2018)
[31] Chiu, P. H.; Lin, Y. T., A conservative phase field method for solving incompressible two-phase flows, J. Comput. Phys., 230, 1, 185-204 (2011) · Zbl 1427.76201
[32] Prasianakis, N. I.; Karlin, I. V., Lattice Boltzmann method for thermal flow simulation on standard lattices, Phys. Rev. E, 76 (2007)
[33] Geier, M.; Schonherr̈, M.; Pasquali, A.; Krafczyk, M., The cumulant lattice Boltzmann equation in three dimensions: Theory and validation, Comput. Math. Appl., 70, 4, 507-547 (2015) · Zbl 1443.76172
[34] Nie, X.; Shan, X.; Chen, H., Galilean invariance of lattice Boltzmann models, Europhys. Lett., 81, 3, 34005 (2008)
[35] Geier, M.; Greiner, A.; Korvink, J. G., Cascaded digital lattice Boltzmann automata for high Reynolds number flow, Phys. Rev. E, 73, 6, 1-10 (2006)
[36] Feng, Y.; Sagaut, P.; Tao, W., A three dimensional lattice model for thermal compressible flow on standard lattices, J. Comput. Phys., 303, 514-529 (2015) · Zbl 1349.76680
[37] Fei, L.; Luo, K. H.; Li, Q., Three-dimensional cascaded lattice Boltzmann method: Improved implementation and consistent forcing scheme, Phys. Rev. E, 97, 5, 053309 (2018)
[38] De Rosis, A.; Luo, K. H., Role of higher-order Hermite polynomials in the central-moments-based lattice Boltzmann framework, Phys. Rev. E, 99, 1 (2019)
[39] Kupershtokh, A. L.; Medvedev, D. A.; Karpov, D. I., On equations of state in a lattice Boltzmann method, Comput. Math. Appl., 58, 5, 965-974 (2009) · Zbl 1189.76413
[40] He, X.; Shan, X.; Doolen, G. D., Discrete Boltzmann equation model for nonideal gases, Phys. Rev. E, 57, 1, R13-R16 (1998)
[41] Premnath, K. N.; Banerjee, S., Incorporating forcing terms in cascaded lattice Boltzmann approach by method of central moments, Phys. Rev. E, 80, 3 (2009)
[42] Huang, R.; Wu, H.; Adams, N. A., Eliminating cubic terms in the pseudopotential lattice Boltzmann model for multiphase flow, Phys. Rev. E, 97, 5, 53308 (2018)
[43] Guo, Z.; Zheng, C.; Shi, B., Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65, 4, 6 (2002) · Zbl 1244.76102
[44] Dellar, P., Nonhydrodynamic modes and a priori construction of shallow water lattice Boltzmann equations, Phys. Rev. E, 65, Article 036309 pp. (2002)
[45] Fakhari, A.; Bolster, D.; Luo, L., A weighted multiple-relaxation-time lattice Boltzmann method for multiphase flows and its application to partial coalescence cascades, J. Comput. Phys., 341, 22-43 (2017) · Zbl 1376.76047
[46] Chen, H.; Gopalakrishnan, P.; Zhang, R., Recovery of galilean invariance in thermal lattice Boltzmann models for arbitrary prandtl number (2014)
[47] Latt, J.; Chopard, B., Lattice Boltzmann method with regularized pre-collision distribution functions, Math. Comput. Simulation, 72, 2-6, 165-168 (2006) · Zbl 1102.76056
[48] Wang, H.; Chai, Z.; Shi, B.; Liang, H., Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations, Phys. Rev. E, 94, 3, Article 033304 pp. (2016)
[49] Chai, Z.; Shi, B.; Guo, Z., A multiple-relaxation-time lattice Boltzmann model for general nonlinear anisotropic Convection-Diffusion equations, J. Sci. Comput., 69, 1, 355-390 (2016) · Zbl 1356.82032
[50] Kusumaatmaja, H.; Hemingway, E. J.; Fielding, S. M., Moving contact line dynamics: from diffuse to sharp interfaces, J. Fluid Mech., 788, 209-227 (2016) · Zbl 1381.76264
[51] Zu, Y. Q.; He, S., Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts, Phy. Rev. E, 87, 4, 1-23 (2013)
[52] Kumar, A., Isotropic finite-differences, J. Comput. Phys., 201, 1, 109-118 (2004) · Zbl 1059.65075
[53] Mattila, K. K.; Hegele Júnior, L. A.; Philippi, P. C., High-accuracy approximation of high-rank derivatives: Isotropic finite differences based on lattice-Boltzmann stencils, Sci. World J., 2014 (2014)
[54] Banari, A., Lattice Boltzmann Simulation of Multiphase Flows; Application to Wave Breaking and Sea Spray Generation (2014), University of Rhode Island, (Ph.D. thesis)
[55] Liang, H.; Xu, J.; Chen, J.; Wang, H.; Chai, Z.; Shi, B., Phase-field-based lattice Boltzmann modeling of large-density-ratio two-phase flows, Phys. Rev. E, 97, 3, Article 33309 pp. (2018)
[56] Ren, F.; Song, B.; Sukop, M. C.; Hu, H., Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation, Phys. Rev. E, 94, 2, 1-12 (2016)
[57] Dzikowski, M.; Łaniewski-Wołłk, Ł.; Rokicki, J., Single component multiphase lattice Boltzmann method for Taylor/Bretherton bubble train flow simulations, Commun. Comput. Phys., 19, 4, 1042-1066 (2016) · Zbl 1373.76318
[58] Chopard, B.; Falcone, J. L.; Latt, J., The lattice Boltzmann advection-diffusion model revisited, Eur. Phys. J. Spec. Top., 171, 1, 245-249 (2009)
[59] Nicklin, D.; Wilkes, J.; Davidson, J., Two-phase flow in vertical tubes, Trans. Inst. Chem. Eng., 40, 61-68 (1962)
[60] Ha-Ngoc, H.; Fabre, J., Test-case no 29B: The velocity and shape of 2D long bubbles in inclined channels or in vertical tubes (PA,PN) part II: In a flowing liquid, Multiphas. Sci. Technol., 16, 1-3, 189-204 (2004)
[61] Łaniewski-Wołłk, Ł.; Rokicki, J., Adjoint Lattice Boltzmann for topology optimization on multi-GPU architecture, Comput. Math. Appl., 71, 833-848 (2016) · Zbl 1359.76231
[62] Asinari, P., Generalized local equilibrium in the cascaded lattice Boltzmann method, Phys. Rev. E, 78, 1, 1-5 (2008)
[63] De Rosis, A., Nonorthogonal central-moments-based lattice Boltzmann scheme in three dimensions, Phys. Rev. E, 95, 1, 13310 (2017)
[64] He, X.; Luo, L.-S., Lattice Boltzmann model for the incompressible Navier-Stokes equation, J. Stat. Phys., 88, 3/4, 927-944 (1997) · Zbl 0939.82042
[65] Guo, Z.; Zheng, C.; Shi, B., Force imbalance in lattice Boltzmann equation for two-phase flows, Phys. Rev. E, 83, 3, 1-8 (2011)
[66] De Rosis, A., Non-orthogonal central moments relaxing to a discrete equilibrium: A D2Q9 lattice Boltzmann model, Europhys. Lett., 116, 4 (2016)
[67] De Rosis, A., Central-moments-based lattice Boltzmann schemes with force-enriched equilibria, Europhys. Lett., 117, 3 (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.