Abstract:Summary
This paper presents a framework for incorporating arbitrary implicit multistep schemes into the lattice Boltzmann method. While the temporal discretization of the lattice Boltzmann equation is usually derived using a second‐order trapezoidal rule, it appears natural to augment the time discretization by using multistep methods. The effect of incorporating multistep methods into the lattice Boltzmann method is studied in terms of accuracy and stability. Numerical tests for the third‐order accurate Adams… Show more
“…A partial answer can be found in the literature. Indeed, a critical Mach number of approximately 0.73 was observed by Wilde et al [74], by adopting several numerical discretizations of the BGK-DVBE. Hereafter, we will check if this result can be extended to the opposite viewpoint, i.e.…”
Section: Sophisticated Collision Models As a Way To Improve The Linear Stabilitymentioning
confidence: 97%
“…Interestingly, this upper stability limit can be shown to depend on both the lattice and the type of equilibrium (polynomial, entropic, etc.). In addition, it does not seem to be impacted by either the collision model [11,12,76], or the numerical discretization [74]. Figure 4Mach number (Ma) corresponding to the linear stability limit of the D2Q9-BGK-DVBE, for various angles ϕ of the mean flow and ν = 10 −6 .…”
Section: Linear Stability Analyses Of the Discrete Velocity Boltzmann Equationmentioning
The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed.
This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.
“…A partial answer can be found in the literature. Indeed, a critical Mach number of approximately 0.73 was observed by Wilde et al [74], by adopting several numerical discretizations of the BGK-DVBE. Hereafter, we will check if this result can be extended to the opposite viewpoint, i.e.…”
Section: Sophisticated Collision Models As a Way To Improve The Linear Stabilitymentioning
confidence: 97%
“…Interestingly, this upper stability limit can be shown to depend on both the lattice and the type of equilibrium (polynomial, entropic, etc.). In addition, it does not seem to be impacted by either the collision model [11,12,76], or the numerical discretization [74]. Figure 4Mach number (Ma) corresponding to the linear stability limit of the D2Q9-BGK-DVBE, for various angles ϕ of the mean flow and ν = 10 −6 .…”
Section: Linear Stability Analyses Of the Discrete Velocity Boltzmann Equationmentioning
The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed.
This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.
“…with the relaxation time τ = ν/(c 2 s δ t ) + 0.5, where the term of 0.5 is a consequence of the second-order time integration in terms of the trapezoidal rule [48][49][50]. The relaxation time τ Pr = (τ − 0.5)/Pr + 0.5 is related to the Prandtl number Pr.…”
This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method [Dorschner, Bösch, and Karlin, Phys. Rev. Lett. 121, 130602 (2018)], the method operates in a static, nonmoving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent time step size due to the integration along characteristics and for the use of unusual velocity sets, like the D2Q25, which is constructed by the roots of the fifth-order Hermite polynomial. The properties of the present model are shown in diverse example simulations of a two-dimensional Taylor-Green vortex, a Sod shock tube, a two-dimensional Riemann problem, and a shock-vortex interaction. It is shown that the cell-based interpolation and the use of Gauss-Lobatto-Chebyshev support points allow for spatially high-order solutions and minimize the mass loss caused by the interpolation. Transformed grids in the shock-vortex interaction show the general applicability to nonuniform grids.
“…For the Hermite polynomial formulations, one obtains the upper limit Ma max = √ 3 − 1 ≈ 0.73, with Ma max obtained by considering all possible orientations of the mean flow and keeping the minimal value. Interestingly, one cannot increase this critical Mach number by changing the collision model [43,79] or the numerical scheme [80]. In fact, this can only be achieved by modifying the velocity discretization and the equilibrium state [9,16,43].…”
Standard lattice Boltzmann methods (LBMs) are based on a symmetric discretization of the phase space, which amounts to study the evolution of particle distribution functions (PDFs) in a reference frame at rest. This choice induces a number of limitations when the simulated flow speed gets closer to the sound speed, such as velocity-dependent transport coefficients. The latter issue is usually referred to as a Galilean invariance defect. To restore the Galilean invariance of LBMs, it was proposed to study the evolution of PDFs in a comoving reference frame by relying on asymmetric shifted lattices [N. Frapolli, S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 117, 010604 (2016)]. From the numerical viewpoint, this corresponds to overcoming the rather restrictive Courant-Friedrichs-Lewy conditions on standard LBMs and modeling compressible flows while keeping memory consumption and processing costs to a minimum (therefore using the standard first-neighbor stencils). In the present work systematic physical error evaluations and stability analyses are conducted for different discrete equilibrium distribution functions (EDFs) and collision models. Thanks to them, it is possible to (1) better understand the effect of this solution on both physics and stability, (2) assess its viability as a way to extend the validity range of LBMs, and (3) quantify the importance of the reference state as compared to other parameters such as the equilibrium state and equilibration path. The results clearly show that, in theory, the concept of shifted lattices allows the scheme to deal with arbitrarily high values of the nondimensional velocity. Furthermore, just like the zero-Mach flow for the standard stencils, it is observed that setting the shift velocity to the fluid velocity results in optimal physical and numerical properties. In addition, a detailed analysis of the obtained results shows that the properties of different collision models and EDFs remain unchanged under the shift of stencil. In other words, by introducing a velocity shift in the stencil, the optimal operating point, in terms of physics and numerics, will also be shifted by the same vector regardless of the EDF or collision model considered. Eventually, while limited to the D2Q9 stencil with the nine possible first-neighbor shifts, the present study and corresponding conclusions can be extended to other stencils and velocity shifts in a straightforward manner.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
