Detailed analysis of the lattice Boltzmann method on unstructured grids

Abstract

The lattice Boltzmann method has become a standard for efficiently solving problems in fluid dynamics. While unstructured grids allow for a more efficient geometrical representation of complex boundaries, the lattice Boltzmann methods is often implemented using regular grids. Here we analyze two implementations of the lattice Boltzmann method on unstructured grids, the standard forward Euler method and the operator splitting method. We derive the evolution of the macroscopic variables by means of the Chapman-Enskog expansion, and we prove that it yields the Navier-Stokes equation and is first order accurate in terms of the temporal discretization and second order in terms of the spatial discretization. Relations between the kinetic viscosity and the integration time step are derived for both the Euler method and the operator splitting method. Finally we suggest an improved version of the bounce-back boundary condition. We test our implementations in both standard benchmark geometries and in the pore network of a real sample of a porous rock.

Evaluation of the finite element lattice Boltzmann method for binary fluid flows

“…The validity and grid convergence is This work establishes a promising venue for simulations of multiphase flows in non-trivial geometries such as real porous media, e.g., by invoking the boundary treatment developed in [18,25].…”

Simulating anomalous dispersion in porous media using the unstructured lattice Boltzmann method

“…We chose this method over the finite volume approach [10,24] due to its improved stability. The characteristic based, off-lattice Boltzmann method has been specifically designed to remove the coupling between the positions of the computational grid's nodes and the discrete set of velocities, while retaining the stability of the standard, regular grid-based approach.…”

“…Similarly to the popular bounce back method, the first step is to determine the subsetD of the discrete directions that are corresponding to the "missing" populations (i.e., point away from the solid phase and into the fluid phase). The procedure for finding such subset in an unstructured grid setting is described in detail in Misztal et al [10]. By setting the target values of the density ρ tg and velocity u tg at the boundary node, the local populations are recovered using the following rules…”

“…In order to faithfully compute the broadly distributed fluid velocities, it is required to represent the pore boundaries with a high resolution. Compared to classical lattice Boltzmann methods on regular grids, the unstructured grids offer a superior resolution and adaptivity [10]. We further analyze the impact of the complex pore geometry on pollutant removal from the pore space.…”

A simplified finite volume lattice Boltzmann method for simulations of fluid flows from laminar to turbulent regime, Part I: Numerical framework and its application to laminar flow simulation

“…The relationship between the viscosity of fluid and the relaxation time are the reflection of flow features. The detail analysis about this relationship at different temporal and spatial discretized schemes through Chapman-Enskog expansion [26,27] are also scarce and need further research.…”