Commit 895e75b8 authored by Jonathan Lambrechts's avatar Jonathan Lambrechts

doc : add todo :-)

parent f1ba7f03
\section{Fluid equations}
\newcommand{\dpartial}[2]{\frac{\partial #1}{\partial #2}}
\section{Averaging of variables and operators}
\subsection{Averaging of variables and operators}
Distribution function of the $\alpha$ phase:
\begin{equation}
\xi_{\alpha}(\vc{r},t) = \begin{cases} 1 \quad \text{si} \quad \vc{r} \in dv_{\alpha} \\
......@@ -78,7 +78,7 @@ c_{\alpha} \nabla \langle f \rangle^{\alpha} +
\nabla c_{\alpha} = - \dfrac{1}{dv} \int_{s_{\alpha\beta}} n_{\alpha\beta} ds_{m}
\end{equation}
\section{Mass balance}
\subsection{Mass balance}
Micro~:
\begin{equation}
\dfrac{\partial \rho}{\partial t} + \nabla \cdot \rho \vc{v} = 0
......@@ -108,7 +108,7 @@ assuming a constant density in the fluid ($\langle \rho
\end{equation}
\section{Momentum Balance}
\subsection{Momentum Balance}
Momentum equation (Incompressible Navier Stokes, $ρ=\mbox{cst}$) integrated over the fluid domain:
\begin{equation}
ρ\,\Big(\frac{∂v}{∂t}\Big)_{α} + ρ\,(vv)_{α} + (∇·τ)_α -(∇p)_α = 0 \label{eq:stokes_momentum}
......
......@@ -52,12 +52,14 @@
\begin{document}
\bibliographystyle{plainnat}
\input{todo.tex}
\newpage
\input{equations.tex}
\input{scontact.tex}
\input{particle-fluid-interaction.tex}
\input{stability.tex}
\input{stability2.tex}
\input{imex.tex}
\input{scontact.tex}
\newpage
\bibliography{zotero}
\end{document}
\noindent
\section{Particle-fluid interaction forces}
NB : all those papers deal with gaz-particle mixture in general and fluidized bed in particular. I'm not sure this can be applied to subsurface water.
\section*{Particle-fluid interaction forces}
While other forces can be seen in the literature \citep{zhu_discrete_2007}, the principal forces present in the gas-particle interactions are the drag force $G$ and the pressure gradient force $F$.
\subsection*{Pressure gradient force}
\subsubsection*{Pressure gradient force}
The pressure gradient force can be easily computed by
\begin{equation}
F = -V_p∇p
\end{equation}
where $p$ is the fluid pressure field and $V_p$ the volume of the particle \citep{anderson_fluid_1967}.
\subsection*{Drag force}
\subsubsection*{Drag force}
There exists many parameterization of the drag force, \citet{li_gas-particle_2003} did a systematic study that indicates that they all possess similar predictive capability.
\citet{di_felice_fluid-particle_2012} uses the following simple formulation:
\begin{equation} \label{eq:fdp}
......@@ -28,7 +27,7 @@ dependency on the Reynolds number: for example, that proposed by \citet {di_feli
\begin{equation}
β = 1.8 - 0.65\exp\big(-0.5(1.5 - \log \re_p)²\big).
\end{equation}
\section*{Discretization}
\subsubsection*{Discretization}
\begin{enumerate}
\item $ε$ is computed as the projection of the particles volume on the mesh nodes
\item[] This is exactly like a quadrature rule where the particles position would be the quadrature points and the particle volume the corresponding weights.
......@@ -41,7 +40,7 @@ F_i &= ∑_pF_p(P_p, (∇P)_p)φ_i(x_p)\\
&=∑_p\left(\frac{∂F_p}{∂P_p}φ_j(x_p) + \frac{∂F_p}{∂(∇P)_p}{·}∇φ_j(x_p)\right_i(x_p)
\end{align*}
\end{enumerate}
\section{Comparison with Darcy}
\subsubsection*{Comparison with Darcy}
In our current implementation (Brinkman law), we have
\[
F = -ρ\frac{με²}{K₁}(u_p - u_f) + \dots\text{, with }K₁ = \frac{k₁ε³}{(1-ε)²}
......
\section*{TODO}
\subsection*{For the Hydro-Quebec report}
\begin{itemize}
\item coupling with lmgc
\item doc python
\item build with python instead of (or in addition of*) cmake
\item doc scontactplot ?
\end{itemize}
\subsection*{For the model}
\begin{enumerate}
\item imex : include d/dt in the fluid equations, increase the order, find why there is a delay when we increase dt
\item friction in scontact
\item free surface/ale
\end{enumerate}
\subsection*{For the paper}
\begin{itemize}
\item introduction
\item conclusion
\item result section
\item references (some can be found in my FNRS proposal)
\item polish/structure
\end{itemize}
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