An efficient phase-field-based multiple-relaxation-time lattice Boltzmann model for three-dimensional multiphase flows

In this paper, an efficient phase-field-based lattice Boltzmann (LB) model with multiple-relaxation-time (MRT) collision operator is developed for the simulation of three-dimensional multiphase flows. This model is an extension of our previous two-dimensional model (Liang etal., 2014) to the three dimensions using the D3Q7 (seven discrete velocities in three dimensions) lattice for the CahnHilliard equation and the D3Q15 lattice for the NavierStokes equations. Due to the less lattice-velocity directions used, the computational efficiency can be significantly improved in the study of three-dimensional multiphase flows, and simultaneously the present model can recover the CahnHilliard equation and the NavierStokes equations correctly through the ChapmanEnskog procedure. We compare the present MRT model with its single-relaxation-time version and the previous LB model by using two benchmark interface-tracking problems, and numerical results show that the present MRT model can achieve a significant improvement in the accuracy and stability of the interface capturing. The developed model can also be able to deal with multiphase fluids with high viscosity ratio, which is demonstrated by the simulation of the layered Poiseuille flow and RayleighTaylor instability at various viscosity ratios. The numerical results are found to be in good agreement with the analytical solutions or some available results. In addition, it is also found that the instability induces a more complex structure of the interface at a low viscosity.

[1]  Junseok Kim,et al.  Numerical simulation of the three-dimensional Rayleigh-Taylor instability , 2013, Comput. Math. Appl..

[2]  Chang Shu,et al.  Diffuse interface model for incompressible two-phase flows with large density ratios , 2007, J. Comput. Phys..

[3]  Aiguo Xu,et al.  Complex fields in heterogeneous materials under shock: modeling, simulation and analysis , 2015, 1512.08568.

[4]  Chang Shu,et al.  Three‐dimensional lattice Boltzmann interface capturing method for incompressible flows , 2008 .

[5]  J. Jacobs,et al.  Experimental study of the single-mode three-dimensional Rayleigh-Taylor instability , 2007 .

[6]  Jean-Yves Trépanier,et al.  Numerical evaluation of two recoloring operators for an immiscible two-phase flow lattice Boltzmann model , 2012 .

[7]  Zhenhua Chai,et al.  A Multiple-Relaxation-Time Lattice Boltzmann Model for General Nonlinear Anisotropic Convection–Diffusion Equations , 2016, J. Sci. Comput..

[8]  A. Xu,et al.  Lattice Boltzmann study of thermal phase separation: Effects of heat conduction, viscosity and Prandtl number , 2012 .

[9]  Z. Chai,et al.  Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Hiroaki Yoshida,et al.  Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation , 2010, J. Comput. Phys..

[11]  Hongwei Zheng,et al.  A lattice Boltzmann model for multiphase flows with large density ratio , 2006, J. Comput. Phys..

[12]  Z. Chai,et al.  Phase-field-based lattice Boltzmann model for axisymmetric multiphase flows. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Hua Li,et al.  Phase separation in thermal systems: a lattice Boltzmann study and morphological characterization. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[14]  Aiguo Xu,et al.  Phase-separating binary fluids under oscillatory shear. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Baochang Shi,et al.  Lattice Boltzmann model for nonlinear convection-diffusion equations. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Yeomans,et al.  Lattice Boltzmann simulation of nonideal fluids. , 1995, Physical review letters.

[17]  H Liang,et al.  Lattice Boltzmann simulation of three-dimensional Rayleigh-Taylor instability. , 2016, Physical review. E.

[18]  M. Rudman.  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 24, 671–691 (1997) VOLUME-TRACKING METHODS FOR INTERFACIAL FLOW CALCULATIONS , 2022 .

[19]  D. d'Humières,et al.  Multiple–relaxation–time lattice Boltzmann models in three dimensions , 2002, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[20]  Lin Liu,et al.  Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces , 2010, J. Comput. Phys..

[21]  Raoyang Zhang,et al.  A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability , 1998 .

[22]  Z. Chai,et al.  A comparative study on the lattice Boltzmann models for predicting effective diffusivity of porous media , 2016 .

[23]  Shiyi Chen,et al.  On the three-dimensional Rayleigh–Taylor instability , 1999 .

[24]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[25]  S. Zaleski,et al.  Lattice Boltzmann model of immiscible fluids. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[26]  C. Shu,et al.  Lattice Boltzmann interface capturing method for incompressible flows. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[28]  S. He,et al.  Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  C. Shu,et al.  Lattice Boltzmann Method and Its Applications in Engineering , 2013 .

[30]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[31]  Chuguang Zheng,et al.  Analysis of lattice Boltzmann equation for microscale gas flows: Relaxation times, boundary conditions and the Knudsen layer , 2008 .

[32]  Aiguo Xu,et al.  Morphologies and flow patterns in quenching of lamellar systems with shear. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Ian M. Mitchell,et al.  A hybrid particle level set method for improved interface capturing , 2002 .

[34]  Bin Deng,et al.  A new scheme for source term in LBGK model for convection-diffusion equation , 2008, Comput. Math. Appl..

[35]  B. Shi,et al.  Effects of force discretization on mass conservation in lattice Boltzmann equation for two-phase flows , 2012 .

[36]  F. Massaioli,et al.  Scaling and hydrodynamic effects in lamellar ordering , 2004, cond-mat/0404205.

[37]  Andrea Mazzino,et al.  Phase-field model for the Rayleigh–Taylor instability of immiscible fluids , 2008, Journal of Fluid Mechanics.

[38]  Gretar Tryggvason,et al.  Computations of three‐dimensional Rayleigh–Taylor instability , 1990 .

[39]  P. Lallemand,et al.  Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[40]  Sauro Succi,et al.  Discrete Boltzmann modeling of multiphase flows: hydrodynamic and thermodynamic non-equilibrium effects. , 2015, Soft matter.

[41]  A. Xu,et al.  Multiple-distribution-function lattice Boltzmann kinetic model for combustion phenomena , 2014 .

[42]  Song Zheng,et al.  Lattice Boltzmann equation method for the Cahn-Hilliard equation. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  L. Quartapelle,et al.  A projection FEM for variable density incompressible flows , 2000 .

[44]  Shan,et al.  Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[45]  Z. Chai,et al.  Lattice Boltzmann modeling of three-phase incompressible flows. , 2016, Physical review. E.

[46]  Shan,et al.  Lattice Boltzmann model for simulating flows with multiple phases and components. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[47]  A Lamura,et al.  Lattice Boltzmann simulation of thermal nonideal fluids. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.