Recursive finite-difference Lattice Boltzmann schemes

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Abstract

The motivation of this study is twofold. First, a recursive mathematical formulation of the discrete-velocity Boltzmann equation (DVBE) under the Bhatnagar-Gross-Krook (BGK) approximation is introduced. This formulation allows us to formally express the solution of the DVBE as an infinite sum over successive particle derivatives of the distributions associated with local equilibrium. A Chapman-Enskog multiple-scales expansion shows that this sum can be safely truncated beyond the second order if the Navier-Stokes level of description is requested. Therefore, the distribution functions depend only on the first and second-order derivatives of the related equilibrium distributions. This alternative equation to the DVBE defines a basis to design kinetic schemes for the evolution of the distribution functions based solely on flow variables that are sufficient to define local equilibrium. Second, a family of mass-conserving numerical schemes is introduced from this kinetic equation by discretizing the particle derivatives by backward finite differences. Interestingly, a so-called “simplified Lattice Boltzmann method” introduced by Chen et al. in 2017 can be recast in this family. Numerical simulations highlight a level of numerical dissipation that is generally higher than the level obtained with a standard Lattice Boltzmann scheme, as expected by approximating derivatives by finite differences. Nevertheless, we show by using a von Neumann analysis that it is possible to parametrize our scheme, according to the relaxation coefficient of the DVBE, to reduce significantly its numerical dissipation and improve its spectral properties.

We believe that this modeling can also be of interest to connect macroscopic and kinetic representations, e.g. when dealing with initial and boundary conditions or in hybrid simulations matching Navier-Stokes and Lattice Boltzmann schemes.

Introduction

The Lattice Boltzmann (LB) method follows from a discretization of the Boltzmann equation to solve weakly-compressible fluid dynamics [1], [2]. Using a kinetic approach to simulate continuum flows seems at first glance unreasonable but, nevertheless, has certain advantages. At first, the velocity space is amenable to a radical decimation so that only a small set of microscopic velocities, e.g. nine in two dimensions, is sufficient to reconstruct the Navier-Stokes dynamics at the macroscopic level. In addition, non-locality and non-linearity are disentangled in the kinetic equations, which facilitates numerical integration [3]. As a result, the so-called stream-and-collide LB algorithm is simple, accurate and formidably efficient in terms of computations [4].

In the LB approach, the degrees of freedom, or nodal values, refer to the distribution functions (f0,,fN1) of particles with the microscopic velocities (c0,,cN1). The scheme governs the evolution in time of these distributions at each lattice node. At the macroscopic level, flow variables are recovered by summing the contributions from the different (microscopic) velocities so thatρ=α=0N1fαandρu=α=0N1fαcα, where ρ and u denote respectively the mass density and the velocity of the fluid. Even if the decimation in velocity space is drastic, the number of nodal values remains significantly higher that the number of reconstructed flow variables, e.g. in two dimensions nine distribution functions are required to reconstruct locally the mass density and the two components of the velocity. The difference is even more pronounced in three-dimensions where nineteen or twenty seven densities are required. This overload of information is problematic when a given macroscopic solution needs to be prescribed at the mesoscopic level, e.g. for initial or boundary conditions. In that case, the kinetic solution is indeterminate and ad-hoc constraints must be invoked [2]. From a computational viewpoint, the LB algorithm is data intensive and memory bound, which can be detrimental to its portability on accelerators such as Graphical Processing Units (GPUs). In this article, a LB algorithm that relies on flow variables only, and therefore requires less degrees of freedom, is introduced. It stems from the truncation and the discretization of a recursive formulation of the discrete-velocity Boltzmann equation (DVBE).

Some variants of the original stream-and-collide algorithm, depending on flow variables only, have already been proposed. In particular, Inamuro suggested a Bhatnagar-Gross-Krook (BGK) collision with a relaxation time (towards statistical equilibrium) equal to the time step of the algorithm. Therefore, distributions are considered at equilibrium and depend only on flow variables (ρ,u) [5]. This strong assumption gives a non-physical value to the fluid viscosity that can then be corrected by adding a contribution of the viscous stress tensor estimated by finite differences. Asinari et al. derived a link-wise artificial compressibility scheme that can be formulated in the LB framework, with a collision operator that is non-local but depends on distributions at equilibrium only [6]. Let us mention that the scheme developed by Inamuro can be viewed as a special case of the link-wise artificial compressibility scheme. More recently, Chen et al. suggested to consider the equilibrium state at the previous time step to build a prediction of the solution, which is afterwards corrected [7] by considering a first-order approximation (in Knudsen number) of the DVBE. This prediction-correction scheme only involves macroscopic quantities. We shall see that this last scheme fits into the general framework developed in the present study.

A key ingredient behind formulating a LB scheme that relies only on flow variables is to express the non-equilibrium component of the distributions, fineqfifieq, as a function of the space-and-time derivatives of the equilibrium component fieq(ρ,u). In this regard, Holdych et al. [8] showed that fineq could be rewritten in a recursive manner asfαneq=m=1(τ)m(t+cα)mfαeq. Their motivation was to derive the truncation errors of this expansion and to show that some errors cancel out for specific values of the relaxation time τ. This iterative approach is similar to the so-called “Maxwell iteration” for the LB method [9], in which the macroscopic equations are derived without explicitly resorting to a multiple-scale Chapman-Enskog expansion [2]. Finally, it has also been used to tailor the equilibrium function of the standard LB scheme to solve non-linear equations such as the Burgers, Korteweg-de-Vries or Kuramoto-Sivashinsky equations by Otomo et al. [10]. In the present article, our contribution is to use Eq. (2) to derive original LB schemes which rely on flow variables only. This is made possible by first showing that the sum can actually be truncated at m=2 to comply with the Navier-Stokes dynamics at the macroscopic level. Then, numerical schemes can be designed by finite-difference discretization of the first and second-order particle derivatives.

The remainder of the paper is organized as follows. Section 2 introduces the recursive formulation of the discrete-velocity Boltzmann equation and its discretization by finite differences. An original finite-difference LB scheme is proposed. It is also shown that the aforementioned prediction-correction scheme by Chen et al. [7] can be derived in this framework. In section 3, a detailed comparison with the classical LB scheme is performed on the double shear layer and Taylor-Green vortex two-dimensional flows. Section 4 gives further insight on the proposed scheme by exploring spectral properties with a von-Neumann analysis. Finally, concluding remarks and perspectives are drawn in Section 5.

Section snippets

Context and derivation of the recursive formulation

In the LB approach, the sums replace the integrals in the statistical moments, as expressed in Eq. (1). This discretization of the velocity space stems from expanding and truncating the solution of the continuum Boltzmann equation onto a finite basis of Hermite polynomials in velocity, and resorting to a Gaussian quadrature formula to express the statistical moments [1], [11]. Therefore, the discrete set of velocities may be thought of as the nodes in the Gaussian quadrature formula. It can

Numerical simulations

In this section, two flow configurations are considered to assess the validity of our recursive finite-difference scheme (abbreviated as rfd, Eqs. (26), (27), and (28)) against the prediction-correction scheme (precorr, Eqs. (41), (42), (43), and (44)) and the standard stream-and-collide LB algorithm (lbm, Eq. (45)). This latter expresses asgα(x+cαδt,t+δt)=gα(x,t)1τ¯g(gα(x,t)gαeq(x,t)) with τ¯g=τ¯+1/2 and gαeq=fαeq. The rfd algorithm is initialized by performing the first two steps with the

Comparisons

The so-called von Neumann (stability) analysis is a procedure that is commonly used to investigate the stability and Fourier spectral properties of finite-difference schemes [31]. It is here used to gain insight about the calibration of the parameter γ in the rfd scheme. At first, the distributions fα(x,t) are considered as small perturbations around a stationary and uniform reference state (ρ, U): fα=f¯α+fα with f¯α=fαeq(ρ,U). The fluctuating components are expressed as complex monochromatic

Conclusion and perspectives

This work introduces on a rigorous basis an alternative equation to the discrete-velocity Boltzmann equation to address fluid dynamics at a kinetic level, while conforming to the Navier-Stokes dynamics at the macroscopic level (in the low-Mach-number limit). This kinetic equation expresses the full distribution functions, i.e. including the equilibrium and non-equilibrium components, in terms of the flow variables (ρ,u) only. However, this comes at the cost of considering a second-order

Funding

The authors acknowledge the funding of the french DGAC under the project OMEGA3 (DGAC/DTA/SDC no 2018-16). This work is part of a scientific collaboration including CS-group, Renault, Airbus, Ecole Centrale de Lyon, CNRS and Aix-Marseille University.

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