manus_continuum_granular1

commit 038784f95499ce09c5e32ccbeaba7472ec1f348e
parent 33b45ecbfa70b2cba4b9a06306414cb625a45154
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date:   Mon, 24 Jun 2019 15:54:54 +0200

Add preliminary text on g BCs and add punctuation for Eqs

Diffstat:
M
continuum-granular-manuscript1.tex
 | 
11
+++++++----

1 file changed, 7 insertions(+), 4 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -91,7 +91,7 @@ where
 \label{sub:fluid_pressure_evolution}
 We prescribe the transient evolution of pore-fluid pressure ($p_\text{f}$) by Darcian pressure diffusion \citep[e.g.][]{Goren2010, Goren2011, Damsgaard2017}:
 \begin{equation}
-	\frac{\partial p_\text{f}}{\partial t} = \frac{1}{\phi\mu_\text{f}\beta_\text{f}} \nabla \cdot (k \nabla p_\text{f})
+	\frac{\partial p_\text{f}}{\partial t} = \frac{1}{\phi\mu_\text{f}\beta_\text{f}} \nabla \cdot (k \nabla p_\text{f}),
 	\label{eq:p_f}
 \end{equation}
 where $\mu_\text{f}$ denotes dynamic fluid viscosity [Pa s], $\beta_\text{f}$ is adiabatic fluid compressibility [Pa$^{-1}$], and $k$ is intrinsic permeability [m$^2$].
@@ -115,16 +115,19 @@ We rearrange Eq.~\ref{eq:g} and split the Laplace operator ($\nabla^2$) into a 1
 An iterative scheme is applied to relax the following equation at each grid node $i$:
 \begin{equation}
 	g_i = {\left(1 + \alpha_i\right)}^{-1}
-	\left(\alpha_i g_\text{local}(\sigma_{\text{n},i}', \mu_i)
+	\left(\alpha_i g_\text{local}(\sigma_{\text{n},i}', \mu_i),
 		+ \frac{g_{i+1} + g_{i-1}}{2}
 	\right)
 	\label{eq:g_i}
 \end{equation}
 where
 \begin{equation}
-	\alpha_i = \frac{\Delta z^2}{2\xi^2(\mu_i)}
+	\alpha_i = \frac{\Delta z^2}{2\xi^2(\mu_i)}.
 	\label{eq:alpha}
 \end{equation}
+We apply fixed-value (Dirichlet) boundary conditions for the fluidity field ($g(z=0) = g(z=L_z) = 0$).
+This condition causes the velocity field transition towards a constant value at the domain edges.
+Neumann boundary conditions, on the contrary, create a velocity profile resembling a free surface flow.
 
 The pore-pressure solution (Eq.~\ref{eq:p_f}) 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].
@@ -135,7 +138,7 @@ As for the granular flow solution, we also use operator splitting and finite dif
 		\left(
 			\frac{2 k_{i+1} k_i}{k_{i+1} + k_i} \frac{p_{i+1} - p_i}{\Delta z} -
 			\frac{2 k_{i-1} k_i}{k_{i-1} + k_i} \frac{p_i - p_{i-1}}{\Delta z}
-		\right)
+		\right).
 	\label{eq:p_f_solution}
 \end{equation}
 For each time step $\Delta t$, a solution to Eq.~\ref{eq:p_f_solution} is first found by explicit temporal integration.