Lattice Boltzmann Method

The Lattice Boltzmann Method (LBM) is a computational fluid dynamics (CFD) technique used to simulate fluid flow and other complex physical phenomena. Unlike traditional CFD methods that solve the Navier-Stokes equations directly, LBM operates at a mesoscopic level, modeling fluid as a collection of particles that propagate and collide on a discrete lattice grid.

Distribution Function

The LBM method uses a distribution function $f_i(\mathbf{x}, t)$ that describes the probability of finding a particle at a given lattice site $\mathbf{x}$ and time $t$ with a discrete velocity $\mathbf{c}_i$. The distribution function evolves according to the following equation:

\[f_i(\mathbf{x} + \mathbf{c}_i \Delta t, t + \Delta t) = f_i(\mathbf{x}, t) + \frac{1}{\tau} \left( f_i^{\text{eq}}(\mathbf{x}, t) - f_i(\mathbf{x}, t) \right)\]

Where:

$f_i(\mathbf{x}, t)$ is the distribution function at lattice site $\mathbf{x}$ at time $t$.
$\mathbf{c}_i$ are the discrete lattice velocities.
$\Delta t$ is the time step.
$\tau$ is the relaxation time.
$f_i^{\text{eq}}(\mathbf{x}, t)$ is the equilibrium distribution function, typically computed using the Maxwell-Boltzmann distribution.

Equilibrium Distribution Function

The equilibrium distribution function $f_i^{\text{eq}}(\mathbf{x}, t)$ is given by:

\[f_i^{\text{eq}} = w_i \left( \rho + \frac{\rho}{c_s^2} \mathbf{c}_i \cdot \mathbf{u} + \frac{\rho}{2 c_s^4} \left( (\mathbf{c}_i \cdot \mathbf{u})^2 - u^2 \right) \right)\]

Where:

$w_i$ are the weights associated with the lattice velocities.
$\rho$ is the fluid density.
$\mathbf{u}$ is the macroscopic velocity of the fluid.
$c_s$ is the speed of sound in the lattice.
$$ u = \mathbf{u} $$ is the magnitude of the macroscopic velocity.

Macroscopic Variables

The macroscopic fluid variables, such as density $\rho$ and velocity $\mathbf{u}$, are obtained by moments of the distribution function. The density $\rho$ is the zeroth moment, and the velocity $\mathbf{u}$ is the first moment:

\[\rho(\mathbf{x}, t) = \sum_i f_i(\mathbf{x}, t)\] \[\rho(\mathbf{x}, t) \mathbf{u}(\mathbf{x}, t) = \sum_i \mathbf{c}_i f_i(\mathbf{x}, t)\]

Where:

$\rho(\mathbf{x}, t)$ is the density at position $\mathbf{x}$ and time $t$.
$\mathbf{u}(\mathbf{x}, t)$ is the velocity field.

Relaxation Time and Kinematic Viscosity

The relaxation time $\tau$ is related to the kinematic viscosity $\nu$ through the following relation:

\[\nu = \frac{c_s^2 (\tau - \frac{1}{2})}{6}\]

Where $c_s$ is the speed of sound in the lattice.

Boundary Conditions

LBM requires the specification of boundary conditions to handle fluid interactions with solid boundaries. Common boundary conditions include:

Bounce-back boundary condition for no-slip walls.
Periodic boundary condition for simulations of periodic flows.

The three main components of the method are firstly the streaming part, where the distribution functions from neighboring lattice sites are pulled, the calculation of macroscopic variables, and lastly the collision step, wherein an operator is executed, calculating the new distribution function and storing the new grid.

In every timestep, the lattice grid is traversed, and the new distribution function values are then calculated only depend on old grid values. The means the algorithm is embarrassingly parallel and suited for distributed HPC systems.

Related program(s): Lattice Boltzmann kernel (LBC)
Related discipline(s): Computational Fluid Dynamics

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreements No 676553 (POP1) and 824080 (POP2).

Currently, the project receives funding from the European High-Performance Computing Joint Undertaking (JU) under grant agreement No 101143931 (POP3). The JU receives support from the European Union's Horizon Europe research and innovation programme and Spain, Germany, France, Portugal and the Czech Republic.