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

Hosseini, Seyed Ali; Coreixas, Christophe; Darabiha, Nasser; Thévenin, Dominique

doi:10.1103/physreve.99.063305

Cited by 39 publications

(57 citation statements)

References 106 publications

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“…As demonstrated in Ref. [29], the LKS is a minimalist MRT collision model (with two relaxation coefficients) based on Hermite moments. It further reduces to the original regularized collision operator [34,35] when λ = 1.…”

Section: Lattice Kinetic Schemementioning

confidence: 99%

“…In the context of the present study, parameters maximizing the stability domain, as derived in Ref. [29], are used. For the sake of readability these parameters are listed in Table I.…”

Section: Lattice Kinetic Schemementioning

confidence: 99%

“…The von Neumann (VN) stability analysis [9,27,[29][30][31]42,43,[67][68][69][70][71][72][73][74][75][76][77][78] is used to study the time evolution of a perturbation f α injected into the shifted lattice Boltzmann equation. The perturbation is assumed to be a monochromatic plane wave.…”

Section: A Von Neumann Stability Analysis Formalismmentioning

confidence: 99%

“…Looking at the overall structure of the MRT, it is observed that better numerical properties can be achieved by adjusting error terms tied to higher-order moments [28]. As such, closure for the free parameters can only be provided a posteriori through asymptotic and/or spectral analyses of the discrete scheme [29][30][31]. Attempts at further reducing velocity-dependent higher-order error terms later resulted in approaches such as the cascaded method based on relaxation of central moments (evaluated in the local fluid velocity frame) instead of raw moments, and the cumulants LBM [32,33].…”

Section: Introductionmentioning

confidence: 99%

“…For the D2Q9 stencil, for example, keeping up to third-order terms of the expansion is sufficient to correctly recover the off-diagonal components. Keeping higher-order terms, however, while not affecting the NS level terms directly, has been observed to improve the stability of the scheme in the small viscosity region [9,29,43].…”

Section: Introductionmentioning

confidence: 99%

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Extensive analysis of the lattice Boltzmann method on shifted stencils

et al. 2019

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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.

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Section: Lattice Kinetic Schemementioning

confidence: 99%

“…In the context of the present study, parameters maximizing the stability domain, as derived in Ref. [29], are used. For the sake of readability these parameters are listed in Table I.…”

Section: Lattice Kinetic Schemementioning

confidence: 99%

Section: A Von Neumann Stability Analysis Formalismmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extensive analysis of the lattice Boltzmann method on shifted stencils

et al. 2019

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Under-resolved and large eddy simulations of a decaying Taylor–Green vortex with the cumulant lattice Boltzmann method

Geier

Lenz

Schönherr

et al. 2020

Theor. Comput. Fluid Dyn.

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We present a comprehensive analysis of the cumulant lattice Boltzmann model with the three-dimensional Taylor–Green vortex benchmark at Reynolds number 1600. The cumulant model is investigated in several different variants, using regularization, fourth-order convergent diffusion and fourth-order convergent advection with and without limiters. In addition, a cumulant model combined with a WALE sub-grid scale model is being evaluated. The turbulence model is found to filter out the high wave number contributions from the energy spectrum and the enstrophy, while the non-filtered cumulant methods show good correspondence to spectral simulations even for the high wave numbers. The application of the WALE turbulence model appears to be counter productive for the Taylor–Green vortex at a Reynolds number of 1600. At much higher Reynolds numbers ($${\hbox {Re}}=160{,}000$$ Re = 160 , 000 ) a deviation from the ideal Kolmogorov theory can be observed in the absence of an explicit turbulence model. Cumulant models with fourth-order convergent diffusion show much better results than single relaxation time methods.

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