Investigation of an entropic stabilizer for the lattice-Boltzmann method

Discrete-velocity Boltzmann model: Regularization and linear stability

A discrete-velocity Boltzmann model for a nine-velocity lattice is considered. Compared to the conventional lattice Boltzmann (LB) schemes the collisions for the model are defined explicitly. Space and time discretization of the model is based on the collide and stream method; in addition, the regularization of the collision term is proposed. It is demonstrated that the regularized model can be represented as a two-relaxation-time LB model of a special type. The scheme is compared to the Onsager regularized (a specific filtered Galilean invariant model) and recursively regularized LB equations in terms of stability and dissipation properties, and linear stability analysis is performed. Several numerical experiments are carried out: double shear layer, lid-driven cavity flow, and propagation of acoustic and shear waves are considered for different grid resolutions, Mach and Reynolds numbers. It is shown that free parameters in the model corresponding to collision cross sections can be adjusted in such a way that the dissipation and stability properties can be controlled.

Linear stability and isotropy properties of athermal regularized lattice Boltzmann methods

The present work proposes a general methodology to study stability and isotropy properties of lattice Boltzmann (LB) schemes. As a first investigation, such a methodology is applied to better understand these properties in the context of regularized approaches. To this extent, linear stability analyses of two-dimensional models are proposed: the standard Bhatnagar-Gross-Krook collision model, the original precollision regularization, and the recursive regularized model, where off-equilibrium distributions are partially computed thanks to a recursive formula. A systematic identification of the physical content carried by each LB mode is done by analyzing the eigenvectors of the linear systems. Stability results are then numerically confirmed by performing simulations of shear and acoustic waves. This work allows drawing fair conclusions on the stability properties of each model. In particular, among the aforementioned models, recursive regularization turns out to be the most stable one for the D2Q9 lattice, especially in the zero-viscosity limit. Two major properties shared by every regularized model are highlighted: (1) a mode filtering property and (2) an incorrect, and broadly anisotropic, dissipation rate of the modes carrying physical waves in under-resolved conditions. The first property is the main source of increased stability, especially for the recursive regularization. It is a direct consequence of the reconstruction of off-equilibrium populations before each collision process, decreasing the rank of the system of discrete equations. The second property seems to be related to numerical errors directly induced by the equilibration of high-order moments. In such a case, this property is likely to occur with any collision model that follows such a stabilization methodology.

Consistent vortex initialization for the athermal lattice Boltzmann method

A barotropic counterpart of the well-known convected vortex test case is rigorously derived from the Euler equations along with an athermal equation of state. Starting from a given velocity distribution corresponding to an intended flow recirculation, the athermal counterpart of the Euler equations are solved to obtain a consistent density field. The present initialization is assessed on a standard lattice Boltzmann solver based on the D2Q9 lattice. Compared to the usual isentropic initialization, a much lower spurious relaxation toward the targeted solution is observed, which is due to the spatial resolution rather than approximated macroscopic quantities. The amplitude of the spurious waves can be further reduced by including an off-equilibrium part in the initial distribution functions.

Central-moment-based Galilean-invariant multiple-relaxation-time collision model

Aiming at systematically correcting the non-Galilean-invariant thermal diffusivity in the previous multiple-relaxation-time Boltzmann equation collision model [Shan and Chen, Int. J. Mod. Phys. C 18, 635 (2007)IJMPEO0129-183110.1142/S0129183107010887], we show that by separately relaxing the central moments of the distribution function, Chapman-Enskog calculation leads to the correct hydrodynamic equations with mutually independent and Galilean invariant viscosity and thermal diffusivity, provided the velocity-space discretization preserves moments up to the fourth order. By transforming the central moments back to the absolute reference frame and evaluating using fixed discrete velocities, the efficient and accurate streaming-collision time-stepping algorithm is preserved. The lattice Boltzmann model is found to have excellent numerical stability in high-Reynolds-number simulations.

Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. This article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant, or regularized form. In parallel with that, several bases (nonorthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post-collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. This work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of this article, general instructions are provided to help the reader with the implementation of the most complicated collision models.

Stability of the lattice kinetic scheme and choice of the free relaxation parameter

The lattice kinetic scheme (LKS), a modified version of the classical single relaxation time (SRT) lattice Boltzmann method, was initially developed as a suitable numerical approach for non-Newtonian flow simulations and a way to reduce memory consumption of the original SRT approach. The better performances observed for non-Newtonian flows are mainly due to the additional degree of freedom allowing an independent control over the relaxation of higher-order moments, independently from the fluid viscosity. Although widely applied to fluid flow simulations, no theoretical analysis of LKS has been performed. The present work focuses on a systematic von Neumann analysis of the linearized collision operator. Thanks to this analysis, the effects of the modified collision operator on the stability domain and spectral behavior of the scheme are clarified. Results obtained in this study show that correct choices of the "second relaxation coefficient" lead, to a certain extent, to a more consistent dispersion and dissipation for large values of the first relaxation coefficient. Furthermore, appropriate values of this parameter can lead to a larger linear stability domain. At moderate and low values of viscosity, larger values of the free parameter are observed to increase dissipation of kinetic modes, while leaving the acoustic modes untouched and having a less pronounced effect on the convective mode. This increased dissipation leads in general to less pronounced sources of nonlinear instability, thus improving the stability of the LKS.

Theoretical and numerical analysis of the lattice kinetic scheme for complex-flow simulations

The lattice kinetic scheme (LKS) is a modified version of the classical single relaxation time lattice Boltzmann method. Although used for many applications, especially when large variations in viscosity are involved, a thorough analysis of the scheme has not been provided yet. In the context of this work, the macroscopic behavior of this scheme is evaluated through the Chapman-Enskog analysis. It is shown that the additional degree of freedom provided in the scheme allows for an independent control of higher-order moments. These results are further corroborated by numerical simulations. The behavior of this numerical scheme is studied for selected external and internal flows to clarify the effect of the free parameter on the different moments of the distribution function. It is shown that it is more stable than SRT (single relaxation time) when confronted to fully periodic under-resolved simulations (especially for λ≈1). It can also help minimize the error coming from the viscosity-dependence of the wall position when combined with the bounce-back approach; although still present, viscosity-dependence of the wall position is reduced. Furthermore, as shown through the multiscale analysis, specific choices of the free parameter can cancel out the leading-order error. Overall, the LKS is shown to be a useful and efficient alternative to the SRT method for simulating numerically complex flows.

Recursive regularization step for high-order lattice Boltzmann methods

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of nonequilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second- (and higher-) order nonhydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from 10^{4} to 10^{6}, and where a thorough analysis of the case at Re=3×10^{4} is conducted. In the latter, results obtained using both regularization steps are compared against the Bhatnagar-Gross-Krook LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.

Lattice Boltzmann simulation of the gas-solid adsorption process in reconstructed random porous media

The gas-solid adsorption process in reconstructed random porous media is numerically studied with the lattice Boltzmann (LB) method at the pore scale with consideration of interparticle, interfacial, and intraparticle mass transfer performances. Adsorbent structures are reconstructed in two dimensions by employing the quartet structure generation set approach. To implement boundary conditions accurately, all the porous interfacial nodes are recognized and classified into 14 types using a proposed universal program called the boundary recognition and classification program. The multiple-relaxation-time LB model and single-relaxation-time LB model are adopted to simulate flow and mass transport, respectively. The interparticle, interfacial, and intraparticle mass transfer capacities are evaluated with the permeability factor and interparticle transfer coefficient, Langmuir adsorption kinetics, and the solid diffusion model, respectively. Adsorption processes are performed in two groups of adsorbent media with different porosities and particle sizes. External and internal mass transfer resistances govern the adsorption system. A large porosity leads to an early time for adsorption equilibrium because of the controlling factor of external resistance. External and internal resistances are dominant at small and large particle sizes, respectively. Particle size, under which the total resistance is minimum, ranges from 3 to 7 μm with the preset parameters. Pore-scale simulation clearly explains the effect of both external and internal mass transfer resistances. The present paper provides both theoretical and practical guidance for the design and optimization of adsorption systems.