🫗 Lattice Boltzmann

Simulation

🧠 Inspired by the famous Fluid Dynamics simulation by Daniel V. Schroeder (Weber State University)
👉 Check out the accompanying Lattice-Boltzmann Fluid Dynamics PDF!

From Boltzmann Statistics to Fluid Dynamics

🚨 This section is a reproduction of From Boltzmann Statistics to Fluid Dynamics

For simplicity, this simulation works in two dimensions, and discretizes space into a square grid or lattice. Each lattice site is big enough to hold a large number of gas molecules, which will be moving in various directions due to their thermal motion plus any large-scale fluid flow.

We approximate this rich complexity by allowing only nine different velocity vectors only, corresponding to nine elementary displacements during a unit of time: up, down, right, left, along any 45° diagonal,or zero (staying in place). The nine elementary displacements are illustrated and listed below, using a unit system in which the lattice spacing is one unit.

$\overrightarrow{ e_0} = 0$
$\overrightarrow{ e_1} = (1,0)$
$\overrightarrow{ e_2} = (0,1)$
$\overrightarrow{ e_3} = (-1,0)$
$\overrightarrow{ e_4} = (0,-1)$
$\overrightarrow{ e_5} = (1,1)$
$\overrightarrow{ e_6} = (-1,1)$
$\overrightarrow{ e_7} = (-1,-1)$
$\overrightarrow{ e_8} = (1,-1)$

To keep track of how many gas molecules at a site are moving in each of these nine directions, we’ll need nine variables. I’ll call these variables $n_i$, for $i=0\ldots8$, and define them as the respective densities of molecules with the corresponding velocities. Then the total density $\rho$ of all molecules at any given site is their sum:

$$\rho = \sum_{i=0}^8 n_i.$$

We can also calculate the macroscopic fluid flow velocity, by taking a weighted average of the nine elementary velocities. I’ll call the flow velocity $\overrightarrow{ u}$, and use the symbol $c$ for the natural unit of velocity, one grid site per unit time. Then the flow velocity at any given site is

$$\overrightarrow{ u} = \sum_{i=0}^8 \frac{n_i}{\rho} \overrightarrow{ e_i} c,$$

which we can write more explicitly as

$$\begin{array}{ll} u_x &=& \frac{c}{\rho}(n_1 + n_5 + n_8 - n_3 - n_6 - n_7) \\ u_y &=& \frac{c}{\rho}(n_2 + n_5 + n_6 - n_4 - n_7 - n_8) \end{array}$$

taking the $x$ and $y$ directions to point rightward and upward, respectively.

Thermal Velocities

The thermal velocities of ideal gas molecules are described by the Boltzmann distribution, which in two dimensions is

$$D(\overrightarrow{ v}) = \frac{m}{2\pi kT} e^{-m|\overrightarrow{ v}|^2/2kT}.$$

This is the function that, when integrated over any range of $v_x$ and $v_y$ values, gives the probability of a molecule’s velocity being in that range, where $m$ is the molecule’s mass, $k$ is Boltzmann’s constant, and $T$ is the temperature. The exponent in the last equation is the molecule’s kinetic energy, and the prefactor $m/2\pi kT$ ensures that the integral of $D(\overrightarrow{ v})$ over all velocity vectors equals $1$ (which you can easily check).