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

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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.

Introduction

Recently the vector-BGK models and the corresponding numerical methods have attracted great attention [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. It was proved in [8], [9], [10], from the viewpoint of partial differential equations, that the vector-BGK models can be regarded as good approximations to the incompressible Navier–Stokes equations. In [6], a vectorial kinetic method was proposed to solve the incompressible Navier–Stokes equations at pore scale and excellent results were obtained. F. Bouchut et al. in [7] proposed second-order entropy satisfying BGK–FVS schemes for the incompressible Navier–Stokes equations. Furthermore, we showed in [11] that the lattice Boltzmann method (LBM) of the vector-BGK models provides a good numerical scheme.

For the stability of the LBM based on the vector-BGK models in [10], [12], we utilized a symmetrizer, which depends on the linear approximation to the pressure. It seems that the nonlinear approximation to the pressure will destroy the structure of the symmetrizer, and then the corresponding stability analysis is no longer valid. But our recent work finds that this is not the case. As one of the goals of this paper, we extend the linear approximation to more general situations (nonlinear). We show the stability by a small adaptation of the symmetrizer, and then get the stability condition of the vectorial model including various types of approximations to the pressure. In addition, we relate the numerical stability based on the vectorial model to that based on the conventional scalar model through an orthogonal similarity transformation. It is numerically validated that the linear and nonlinear approximations play the similar effect for the fluid velocity in the incompressible Navier–Stokes equations.

This paper also aims to present a seven-velocity three-dimensional (D3N7) vectorial LBM with a family of two-parameterized second-order boundary schemes for the incompressible Navier–Stokes equations. It is well-known that the LBM with fewer discrete velocities would show an enormous advantage to handle the problems in complex geometries, especially the domain including corners. D. d’Humières et al. in [13] pointed out that the conventional scalar LBM seems to need no less than thirteen velocities in three dimensions to get the correct Navier–Stokes equations. Here the vectorial LBM in three dimensions just requires seven velocities. On the other hand, we proposed a vector-type bounce-back boundary scheme to accompany the vectorial model in [12], which is shown second-order accurate when the boundary is located at the middle of two neighboring lattice nodes. Furthermore, inspired by [14], we based on the vector-type boundary scheme to construct a family of one-parameterized second-order boundary schemes with accuracy independent of the boundary location. In this paper, through a combination we give a family of two-parameterized second-order boundary schemes, and examine them numerically with linear and nonlinear approximations to the pressure.

This paper is organized as follows. In Section 2 we introduce the D3N7 model, and give the vector-type bounce-back boundary scheme and the parameterized second-order boundary schemes. We show the stability of the D3N7 model with the various types of approximations to the pressure on the bounded domain in Section 3. Numerical experiments are reported in Section 4. Finally, some conclusions and remarks are given in Section 5.

Section snippets

D3N7

In this section, we introduce a seven-velocity three-dimensional (D3N7) vectorial lattice Boltzmann method (LBM) (see [11] for more details) and give its consistency to the incompressible Navier–Stokes equations.

The D3N7 model reads as fi(t+Δt,x+hei)fi(t,x)=Mifiτ(t,x)+hΔtFi(t,x),i=1,2,,7.Here fi=fit,xR4 is a vector-valued distribution function along the discrete velocity ei; the discrete velocities are e1=e4=(1,0,0),e2=e5=(0,1,0),e3=e6=(0,0,1) and e7=(0,0,0); Δt and h are the time step

Stability analysis

In this section, we are concerned about the numerical stability, with respect to disturbances of initial data, of the vectorial LBM (2.1) together with its vector-type bounce-back boundary scheme (2.13). The convergence analyses in [15], [16] told us that it is sufficient to consider a linearization of the vectorial LBM at the quiescent state (ρ,u)=(1,0).

Here we set f=(f1,,f7), M=(M1,,M7), and Q=(Mf)τ, where Mi is the ith equilibrium distribution (Maxwellian) defined in (2.2) and can be

Numerical experiments

In this section, we will report several numerical results to validate the D3N7 model (2.1) including various types of P(ρ) for the incompressible Navier–Stokes equations with the Dirichlet boundary condition (2.11). Specifically, we consider the following four cases: P(ρ)=ρ,ρ43,ρ2,ρ3. For the boundary treatment we adopt the vector-type bounce-back boundary scheme (2.13) and the parameterized second-order boundary scheme (2.16). To be concrete, we take b=0.5,0.7,1 and l=γ,2γ, which satisfy the

Conclusions and remarks

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 the two-parameterized second-order boundary schemes with accuracy independent of the boundary location. Utilizing the vector nature of the equilibrium distribution function (Maxwellian), we construct a symmetrizer to show the numerical stability of the D3N7 model including the linear and nonlinear

Acknowledgments

Jin Zhao would like to thank Prof. Wen-An Yong for helpful discussions. This was supported by the National Natural Science Foundation of China (NSFC) with Grant No. U1930402.

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