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old-lattice-boltzmann.com/content/1-dimensionless-lbm.md

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+title = "The dimensionless lattice Boltzmann equation"
+description = "How to write the dimensionless lattice Boltzmann equation"
+date = 2024-03-06
+slug = "dimensionless"
+
+[extra]
+katex = true
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+
+## Thed imensionless Navier-Stokes
+
+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 litterature.
+
+The incompressible Navier--Stokes equations reads
+
+££
+\begin{aligned}
+&\bm{\nabla}\cdot\bm{u}=0,\\\\
+&\frac{\partial}{\partial t}\bm{u}+\bm{u}\cdot\bm{\nabla}\bm{u}=-\frac{1}{\rho}\bm{\nabla}p+\nu\bm{\nabla}^2\bm{u},
+\end{aligned}
+££
+where $p$, $\rho$, $\nu$, 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 lengthscales 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{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},\\\\
+&\frac{\partial}{\partial t^\ast}=\frac{L}{U}\frac{\partial}{\partial t}, &\bm{\nabla}^\ast=L\bm{\nabla}
+\end{aligned}
+££
+Replacing these equations into the Navier--Stokes equations one gets
+££
+\begin{aligned}
+&\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+\frac{1}{\mathrm{Re}}\bm{\nabla}^{\ast 2}\bm{u}^\ast,
+\end{aligned}
+££
+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-dimensionnalization 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 chosing 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{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},\\\\
+&\frac{\partial}{\partial t^\ast}=\frac{L^2}{\nu}\frac{\partial}{\partial t}, &\bm{\nabla}^\ast=L\bm{\nabla}
+\end{aligned}
+££
+which in turn give the following dimensionless Navier--Stokes equations
+££
+\begin{aligned}
+&\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{aligned}
+££
+where we see that no dimensionless number is present anymore. This kind of non-dimensionalization is only valid for creeping flows that move only very slowly and will not interest us in this series of tutorials.
+
+## The dimensionless Boltzmann equation
+
+The quantity of interest in the Boltzmann equation is the desity 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[^1]
+££
+\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),
+££
+where $\tau$ is the relaxation time of the fluid, and $f^{eq}$ the equilibrium distribution function which is typically the Maxwell-Boltzmann distribution (more on this later).
+
+The dimensionless density distribution function is then given by
+
+
+[^1]: See reference TODO
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