Introduction to the Lattice Boltzmann Method for Fluid Mechanics

The Lattice Boltzmann Method (LBM) is a CFD (Computational Fluid Dynamics) method that has undergone considerable development since the early 2000s. Unlike traditional CFD methods, which use macroscopic quantities such as velocity, pressure or density as fundamental variables, LBM is based on calculating the velocity distribution of molecules. The usual quantities are then obtained by calculating the moments of the velocity distribution.

This approach can be seen as a discretization of the Boltzmann equation, which corresponds to the balance over an infinitesimal volume of the probability density of molecular velocities in a dilute gas.

Boltzmann showed that this function converged to an equilibrium known as the Maxwell-Boltzmann distribution, which takes the form of a Gaussian whose mean corresponds to fluid velocity and whose standard deviation is related to temperature.

Based on this equation, the Bhatnagar-Gross-Krook (BGK) model proposes to represent time evolution as a linear relaxation towards equilibrium. This assumption is valid for quasi-incompressible flows. The characteristic relaxation time is then linked to the fluid's viscosity.

If Boltzmann's work was developed within the strict framework of the kinetic theory of gases, it is possible to demonstrate that the discretized Boltzmann equation converges to the Navier-Stokes equations as long as the assumption of quasi-incompressible flow remains valid.

This result makes it possible to use the LBM for viscous fluids well beyond simple dilute gases. Extrapolating further, the discrete BGK equation can be seen simply as an original way of representing transport equations. For example, the heat equation can be solved by a similar approach to realize the coupling of fluid mechanics with thermics.

The advantages of this method are as follows:

• the simplicity of the algorithm, enabling CFD code to be implemented in just a few dozen lines for high-level languages (python, matlab...);

• an Eulerian approach that allows complex geometries to be represented using a simple structured grid;

• the purely explicit approach, which facilitates massive parallelization of calculation codes;

• the possibility of multiphysics coupling to simulate complex flows, including thermals, phase changes or multi-component flows.

These advantages have led to strong development of the LBM and its increasingly widespread use in the scientific community.

In this article, the basics of LBM will be presented, starting with an introduction...