commit 3cabd0bed45a1f2f191cd53c3cc8ae807c34b304
parent ca0f8b4895f832ea82a675f340ba72be6889a4e2
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 24 Jun 2019 13:03:13 +0200
Add info on fluid solver and BCs
Diffstat:
1 file changed, 15 insertions(+), 5 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -55,11 +55,11 @@ We generalize these previous models to a water-saturated granular flow, by compa
\section{Methods}%
\label{sec:methods}
+We expand a steady-state continuum model for granular flow by \citet{Henann2013} with a coupling to pore-pressure diffusion.
\subsection{Granular flow}%
\label{sub:granular_flow}
In our model, the sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit for a cohesionless material.
-We expand a steady-state continuum model for granular flow by \citet{Henann2013} with a coupling to pore water.
We assume that the elasticity is negligible and set the total shear rate $\dot{\gamma}$ to consist of a plastic contribution $\dot{\gamma}^\text{p}$:
\begin{equation}
\dot{\gamma} \approx \dot{\gamma}^\text{p} = g(\mu_\text{c}, \sigma_\text{n}') \mu,
@@ -128,12 +128,22 @@ where
\label{eq:alpha}
\end{equation}
-\textbf{TODO: Add info on fluid solver}.
+Similarly, we also use operator splitting and finite differences to solve the equation for pore-pressure diffusion (Eq.~\ref{eq:p_f}).
+The pore-pressure solution is constrained by a zero pressure gradient at the bottom ($dp_\text{f}/dz (z=0) = 0$), and a sinusoidal pressure forcing at the top ($p_\text{f}(z = L_z) = A \sin(2\pi f t) + p_{\text{f},0}$).
+Here, $A$ is the forcing amplitude [Pa], $f$ is the forcing frequency [1/s], and $p_{\text{f},0}$ is the mean pore pressure over time [Pa].
+
+For each time step $\Delta t$, a pore-pressure solution is found by explicit temporal integration.
+We then use Jacobian iterations to find an implicit solution through underrelaxation over the same time step.
+For the final pressure field at $t + \Delta t$ we mix the explicit and implicit solutions with equal weight, which is known as the Crank-Nicholson method \citep[e.g.][]{Patankar1980, Ferziger2002, Press2007}.
+The method is unconditionally stable and second-order accurate in time and space.
+
+
+
+\section*{Appendix}%
+\label{sec:appendix}
+The source code for the model is available at \url{https://gitlab.com/admesg/1d_fd_simple_shear}.
%% Bibliography
-%\newpage
-%\addcontentsline{toc}{part}{Bibliography}
-%\bibliography{/Users/ad/articles/own/BIBnew.bib}
\printbibliography{}
\end{document}
