The dimensionless lattice Boltzmann equation

The non-dimensional Navier-Stokes equations

In order to write the dimensionless lattice Boltzmann equation, we start by reminding the basis of dimensional analysis for fluid flows by writing the dimensionless incompressible Navier--Stokes which are more common in the literature.

The incompressible Navier--Stokes equations reads

$$ \begin{align} &\bm{\nabla}\cdot\bm{u}=0,\\ &\partial_t\bm{u}+\bm{u}\cdot\bm{\nabla}\bm{u}=-\frac{1}{\rho}\bm{\nabla}p+\nu\bm{\nabla}^2\bm{u}, \end{align} $$ where $p$, $\rho$ (which is a constant in the incompressible model), $\nu$, and $\bm{u}$ are respectively the pressure, density, kinematic viscosity, and velocity of the flow.

In order to transform this equation into its dimensionless form a certain amount of characteristic lengths calves must be chosen. Here we will define $U$ as the characteristic velocity of the flow and $L$ its characteristic length.

We can the write all the above quantities dimensionless form $$ \begin{equation} \begin{aligned} &\bm{u}^\ast=\frac{\bm{u}}{U}, &p^\ast=\frac{p}{\rho U^2},\\ &t^\ast=\frac{U}{L}t, &\bm{x}^\ast=\frac{\bm{x}}{L},\\ &\partial_t^\ast=\frac{L}{U}\partial_t, &\bm{\nabla}^\ast=L\bm{\nabla} \end{aligned} \end{equation} $$ Replacing these equations into the Navier--Stokes equations one gets $$ \begin{align} &\bm{\nabla}^\ast\cdot\bm{u}^\ast=0,\\ &\partial_t^\ast\bm{u}^\ast+\bm{u}^\ast\cdot\bm{\nabla}^\ast\bm{u}^\ast=-\bm{\nabla}^\ast p^\ast+\frac{1}{\mathrm{Re}}\bm{\nabla}^{\ast 2}\bm{u}^\ast, \end{align} $$ where $\mathrm{Re}=U\cdot L/\nu$ is the famous Reynolds number which represents the ratio of the inertial over the viscous forces.

It must me noted that this non dimensionalization procedure heavily relies on the choice of the characteristic timescale of the flow which is this case can be constructed through the characteristic velocity of the flow, $T=L/U$. By choosing the kinematic viscosity instead of the velocity, the characteristic time of the flow becomes, $T=L^2/\nu$. With this new characteristic time the dimensionless quantities become $$ \begin{equation} \begin{aligned} &\bm{u}^\ast=\frac{L}{\nu}\bm{u}, &p^\ast=\frac{L^2}{\nu^2\rho}p,\\ &t^\ast=\frac{\nu}{L^2}t, &\bm{x}^\ast=\frac{\bm{x}}{L},\\ &\partial_t^\ast=\frac{L^2}{\nu}\partial_t, &\bm{\nabla}^\ast=L\bm{\nabla} \end{aligned} \end{equation} $$ which in turn give the following dimensionless Navier--Stokes equations $$ \begin{align} &\bm{\nabla}^\ast\cdot\bm{u}^\ast=0,\\ &\frac{\partial}{\partial t^\ast}\bm{u}^\ast+\bm{u}^\ast\cdot\bm{\nabla}^\ast\bm{u}^\ast=-\bm{\nabla}^\ast p^\ast+\bm{\nabla}^{\ast 2}\bm{u}^\ast, \end{align} $$ where we see that no dimensionless number is present anymore. This case represents creeping flows that move only very slowly and will not interest us in this series of tutorials.

The non-dimensional Boltzmann equation

The quantity of interest in the Boltzmann equation is the density probability distribution function $$ f(\bm{x},\bm{\xi}, t) $$ which represents the probability of finding a particle at position $\bm{x}$ with velocity $\bm{\xi}$ at time $t$.

In this tutorial we will only be interested in the Boltzmann-BGK equation, which reads $$ \begin{equation} \partial_t f(\bm{x}, \bm{\xi}, t)+\bm{\xi}\cdot \bm{\nabla}f(\bm{x},\bm{\xi}, t)=-\frac{1}{\tau}\left(f(\bm{x}, \bm{\xi}, t)-f^{eq}(\bm{x}, \bm{\xi}, t)\right), \end{equation} $$ where $\tau$ is the relaxation time of the fluid, and $f^{eq}$ the equilibrium distribution function which is typically the Maxwell-Boltzmann density distribution function. Considering our fluid is made of particles of mass $m$, the Boltzmann distribution reads $$ \begin{equation} f^{eq}(\bm{x}, \bm{\xi}, t) = \rho(\bm{x}, t)\left(\frac{m}{2\pi k_B T(\bm{x}, t)}\right)^{D/2}\exp\left(-\frac{m(\bm{\xi}-\bm{u}(\bm{x}, t))^2}{2k_B T(\bm{x}, t)}\right), \end{equation} $$ where $\rho$, $\bm{u}$, $T$ are respectively the density, velocity and temperature of the fluid, $k_B$ the Boltzmann constant, and $D$ the dimension of the velocity.

The dimensions of the density distribution function are therefore $$ \left[M \cdot L^{-2D}\cdot T^D\right] $$ By defining the characteristic velocity of our particles by $\xi_0$ (due to thermal agitation) $$ \begin{equation} \xi_0^2=\frac{k_BT}{m}, \end{equation} $$ the non-dimensional quantities of interest become $$ \begin{equation} \begin{aligned} &\bm{\xi}^\ast=\frac{1}{\xi_0}\bm{\xi},&\quad\rho^\ast=\frac{1}{\rho_0}\rho,\\ &t^\ast=\frac{\xi_0}{L}t,&\quad\bm{x}^\ast=\frac{\bm{x}}{L},\\ &\partial_t^\ast=\frac{L}{\xi_0}\partial_t,&\quad\bm{\nabla}^\ast=L\bm{\nabla},\\ &f^\ast=\frac{\xi_0^D}{\rho_0}f,&\quad{f^{eq}}^\ast=\frac{\xi_0^D}{\rho_0}f^{eq}, \end{aligned} \end{equation} $$ With these relations, we obtains the non-dimensional BGK equation $$ \begin{equation} \partial_t^\ast f^\ast+\bm{\xi}^\ast\cdot \bm{\nabla}^\ast f^\ast=-\frac{1}{\mathrm{Kn}}\left(f^\ast-{f^{eq}}^\ast\right), \end{equation} $$ where the space, time, and microscopic velocity dependence is omitted and where $$ \begin{equation} \mathrm{Kn}=\frac{\tau\xi_0}{L}, \end{equation} $$ is the Knudsen number. As for the Navier-Stokes equations we see that the non-dimensional BGK equation is parametrized with an unique transport coefficient which is the Knudsen number in this case. In the next episodes of this series we will make the link between the BGK and the Navier--Stokes equations and show how we discretize the BGK equation in order to simulate weakly compressible fluid flows.

Finally we want to express the non-dimensional Maxwell-Boltzmann distribution as a function of only non-dimensional quantities and it reads $$ \begin{equation} {f^{eq}}^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast) = \frac{\rho^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)}{(2\pi\theta^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast))^{D/2}}\exp\left(-\frac{\left(\bm{\xi}^\ast-\bm{u}^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)\right)^2}{2\theta^\ast(\bm{x}^\ast, \bm{\xi}^\ast, t^\ast)}\right), \end{equation} $$ where $\theta^\ast$ is given by $$ \begin{equation} \theta^\ast = \frac{k_B T}{m\cdot \xi_0^2} \end{equation} $$ In the next episode we will discuss in more details, the link between the description of a fluid in the Boltzmann framework and in the Navier--Stokes framework.