Extensive analysis of the lattice Boltzmann method on shifted stencils

S. A. Hosseini, C. Coreixas, N. Darabiha, and D. Thévenin
Phys. Rev. E 100, 063301 – Published 4 December 2019

Abstract

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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  • Received 21 September 2019

DOI:https://doi.org/10.1103/PhysRevE.100.063301

©2019 American Physical Society

Physics Subject Headings (PhySH)

Fluid DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

S. A. Hosseini1,2,3, C. Coreixas4, N. Darabiha2, and D. Thévenin1

  • 1Laboratory of Fluid Dynamics and Technical Flows, University of Magdeburg “Otto von Guericke,” D-39106 Magdeburg, Germany
  • 2Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 91192 Gif-sur-Yvette Cedex, France
  • 3International Max Planck Research School (IMPRS) for Advanced Methods in Process and Systems Engineering, Magdeburg, Germany
  • 4Department of Computer Science, University of Geneva, 1204 Geneva, Switzerland

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Issue

Vol. 100, Iss. 6 — December 2019

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