Seven-velocity three-dimensional vectorial lattice Boltzmann method including various types of approximations to the pressure and two-parameterized second-order boundary treatments

Abstract In this paper we present a seven-velocity three-dimensional (D3N7) vectorial lattice Boltzmann method (LBM) including various types of approximations to the pressure and propose a family of two-parameterized second-order boundary schemes with accuracy independent of the boundary location. In order to show the numerical stability of the D3N7 model, we construct a symmetrizer to handle the nonlinear approximations to the pressure. In the meantime, we relate the stability based on the vectorial model to that based on the conventional scalar model through an orthogonal similarity transformation. Finally, two 3-D examples with straight and curved boundaries numerically validate the D3N7 model, including linear and nonlinear approximations to the pressure, together with the proposed boundary schemes.

[1]  Franccois Dubois.  Simulation of strong nonlinear waves with vectorial lattice Boltzmann schemes , 2014 .

[2]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation , 1993, Journal of Fluid Mechanics.

[3]  D. d'Humières,et al.  Thirteen-velocity three-dimensional lattice Boltzmann model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  A. Ladd.  Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results , 1993, Journal of Fluid Mechanics.

[5]  R. Natalini,et al.  A discrete kinetic approximation for the incompressible Navier-Stokes equations , 2008 .

[6]  Zhaoxia Yang,et al.  Convergence of lattice Boltzmann methods for Stokes flows in periodic and bounded domains , 2008, Comput. Math. Appl..

[7]  R. Occelli,et al.  Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations , 2018 .

[8]  Juntao Huang,et al.  Boundary Conditions for Kinetic Theory Based Models I: Lattice Boltzmann Models , 2019, Multiscale Model. Simul..

[9]  Jin Zhao,et al.  Vector-Type Boundary Schemes for the Lattice Boltzmann Method Based on Vector-BGK Models , 2020, SIAM J. Sci. Comput..

[10]  P. Dellar.  Lattice Kinetic Schemes for Magnetohydrodynamics , 2002 .

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

[12]  C. Ross Ethier,et al.  Exact fully 3D Navier–Stokes solutions for benchmarking , 1994 .

[13]  Zhaoxia Yang,et al.  Convergence of lattice Boltzmann methods for Navier–Stokes flows in periodic and bounded domains , 2009, Numerische Mathematik.

[14]  Zhimin Zhang,et al.  Approximation of the multi-dimensional incompressible Navier-Stokes equations by discrete-velocity vector-BGK models , 2020, Journal of Mathematical Analysis and Applications.

[15]  François Bouchut,et al.  Entropy satisfying flux vector splittings and kinetic BGK models , 2003, Numerische Mathematik.

[16]  F. Bouchut.  Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws , 1999 .

[17]  Wen-An Yong,et al.  Weighted L2-Stability of the Lattice Boltzmann Method , 2009, SIAM J. Numer. Anal..

[18]  Jin Zhao.  Discrete-Velocity Vector-BGK Models Based Numerical Methods for the Incompressible Navier-Stokes Equations , 2021, Communications in Computational Physics.

[19]  R. Natalini,et al.  Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations , 2017, Kinetic & Related Models.

[20]  F. Topin,et al.  Determining permeability tensors of porous media: A novel ‘vector kinetic’ numerical approach , 2019, International Journal of Multiphase Flow.

[21]  J. Boon.  The Lattice Boltzmann Equation for Fluid Dynamics and Beyond , 2003 .