commit 574096d99e66aa06c735706ec9fc6504c57c2014
parent 7d46adf24841dcf808e8d2dca6a27de2f6120a1e
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 24 Jun 2019 13:30:20 +0200
Use same order for pressure-solution terms
Diffstat:
1 file changed, 4 insertions(+), 4 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -127,11 +127,11 @@ where
\label{eq:alpha}
\end{equation}
-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 ($p_\text{f}(z = L_z) = A \sin(2\pi f t) + p_{\text{f},0}$).
+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].
-We also use operator splitting and finite differences to solve the equation for pore-pressure diffusion (Eq.~\ref{eq:p_f}):
+As for the granular flow solution, we also use operator splitting and finite differences to solve the equation for pore-pressure diffusion (Eq.~\ref{eq:p_f}):
\begin{equation}
- \Delta p_{\text{f},i} = \frac{\Delta t}{\beta_\text{f} \phi_i \mu_\text{f}}
+ \Delta p_{\text{f},i} = \frac{\Delta t}{\phi_i \mu_\text{f} \beta_\text{f}}
\frac{1}{\Delta z}
\left(
\frac{2 k_{i+1} k_i}{k_{i+1} + k_i} \frac{p_{i+1} - p_i}{\Delta z} -
@@ -140,7 +140,7 @@ We also use operator splitting and finite differences to solve the equation for
\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.
-We then use Jacobian iterations to find an implicit solution to the same equation through underrelaxation.
+We then use Jacobian iterations to find an implicit solution to the same equation using underrelaxation.
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.
