diff --git a/old-lattice-boltzmann.com/config.toml b/old-lattice-boltzmann.com/config.toml deleted file mode 100644 index fca84181c5547d4ec3a487607d0ed65c9d886c2e..0000000000000000000000000000000000000000 --- a/old-lattice-boltzmann.com/config.toml +++ /dev/null @@ -1,28 +0,0 @@ -# The URL the site will be built for -base_url = "https://www.lattice-boltzmann.com" - -# Whether to automatically compile all Sass files in the sass directory -compile_sass = true - -# Whether to build a search index to be used later on by a JavaScript library -build_search_index = true - -theme = "toucan" - -transparent = "true" - -[markdown] -# Whether to do syntax highlighting -# Theme can be customised by setting the `highlight_theme` variable to a theme supported by Zola -highlight_code = true -highlight_theme = "base16-ocean-light" - -[extra] -# Put all your custom variables here - -# Whether to do syntax highlighting -# Theme can be customised by setting the `highlight_theme` variable to a theme supported by Zola -title = "High performance lattice Boltzmann method" - -[extra.theme] -read_more = "Read more" diff --git a/old-lattice-boltzmann.com/content/1-dimensionless-lbm.md b/old-lattice-boltzmann.com/content/1-dimensionless-lbm.md deleted file mode 100644 index e86b25eca1bd3eac45fbb36947303e806bcc725e..0000000000000000000000000000000000000000 --- a/old-lattice-boltzmann.com/content/1-dimensionless-lbm.md +++ /dev/null @@ -1,101 +0,0 @@ -+++ -title = "The dimensionless lattice Boltzmann equation" -description = "How to write the dimensionless lattice Boltzmann equation" -date = 2024-03-06 -slug = "dimensionless" - -[extra] -katex = true -+++ - -## 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 - - - - - - - - - - - - - - - - - - - - - diff --git a/old-lattice-boltzmann.com/content/_index.md b/old-lattice-boltzmann.com/content/_index.md deleted file mode 100644 index eb8603506fab731edde9976d0847cd425d624b72..0000000000000000000000000000000000000000 --- a/old-lattice-boltzmann.com/content/_index.md +++ /dev/null @@ -1,4 +0,0 @@ -+++ -sort_by = "date" -template = "section.html" -+++