Changeset 8722

Timestamp:

Mar 5, 2013, 1:41:47 AM (12 years ago)

Author:

mungkasi

Message:

Updating code and adding report for avalanche dry.

Location:

trunk/anuga_core/validation_tests/Tests/Analytical_exact/avalanche_dry

Files:

• numerical_avalanche_dry.py (modified) (1 diff)
• report.tex (added)
• results.tex (modified) (1 diff)

trunk/anuga_core/validation_tests/Tests/Analytical_exact/avalanche_dry/numerical_avalanche_dry.py

@@ -149 +149 @@
 
 #------------------------------------------------------------------------------
+# Produce a documentation of parameters
+#------------------------------------------------------------------------------
+parameter_file=open('parameters.tex', 'w')
+parameter_file.write('\\begin{verbatim}\n')
+from pprint import pprint
+pprint(domain.get_algorithm_parameters(),parameter_file,indent=4)
+parameter_file.write('\\end{verbatim}\n')
+parameter_file.close()
+
+#------------------------------------------------------------------------------
 # Evolve system through time
 #------------------------------------------------------------------------------

trunk/anuga_core/validation_tests/Tests/Analytical_exact/avalanche_dry/results.tex

@@ -2 +2 @@
 \section{Avalanche involving a dry area}
 
-An avalanche test problem involving a dry area. Here debris is approximated as water. The simulation should show a rarefaction fan and wetting process. 
+An avalanche problem involving a dry area is solved using shallow water approach. This problem is very similar to the dry dam break, but it is on a sloping topography. The debris could be snow, sand, or even rock. The simulation should show a rarefaction and wetting process, just like the dry dam break problem. The analytical solution of this problem was derived by Mungkasi and Roberts~\cite{MR2011DA}. This shallow water approach to solve debris avalanche problems was also implemented by a number of researchers, such as Mangeney et al.~\cite{MHR2000} and Naaim et al.~\cite{NVC1997}. 
+
+The initial condition is
+\begin{equation} \label{eq:dap_init}
+u(x,0)=0, ~~v(x,y)=0, ~~\textrm{and}~~
+h(x,0) = \left\{ \begin{array}{ll}
+h_1 & \textrm{if $x < 0$}\\
+0 & \textrm{if $x > 0$}\\
+\end{array} \right.
+\end{equation}
+where $h_1>0$. The topography is a flat bed with positive slope.
+
+The analytical solution~\cite{MR2011DA} at time $t>0$ is
+\begin{equation} 
+h(x) = \left\{ \begin{array}{ll}
+0 & \textrm{if $x \leq -2 c_0 t + \frac12 mt^2$}\\
+h_R=\frac{1}{9g} \left( \frac{x}{t} + 2c_0 - \frac12 mt \right)^2 & \textrm{if $-2 c_0 t + \frac12 mt^2 \leq x \leq c_0 t + \frac12 mt^2$}\\
+h_0 & \textrm{if $x \geq c_0 t + \frac12 mt^2$}\\
+\end{array} \right.
+\end{equation}
+which is the free surface and
+\begin{equation} 
+u(x) = \left\{ \begin{array}{ll}
+0 & \textrm{if $x \leq -2 c_0 t + \frac12 mt^2$}\\
+u_R=\frac23 \left( \frac{x}{t} - c_0 + mt \right) & \textrm{if $-2 c_0 t + \frac12 mt^2 \leq x \leq c_0 t + \frac12 mt^2$}\\
+mt & \textrm{if $x \geq c_0 t + \frac12 mt^2$}\\
+\end{array} \right.
+\end{equation}
+which is the velocity. Here $m=-g\tan{\theta}+F$, where $\tan{\theta}$ is the slope of the topography. Variable $F$ is the Coulomb-type friction given by 
+\begin{equation}
+F=g \cos^2{\theta} \tan{\delta},
+\end{equation}
+in which $\tan{\delta}$ is a given value of friction slope such that $\tan{\delta} \leq \tan{\theta}$.
+
 
 \subsection{Results}
 
-
-We should see excellent agreement between the analytical and numerical solutions.
-
+For our test, we consider $h_0=20$ in (\ref{eq:dap_init}).
+The following figures show the stage, $x$-momentum, and $x$-velocity at several instants of time. We should see excellent agreement between the analytical and numerical solutions. The wet/dry interface is difficult to resolve and it usually produces large errors, similar to the dry dam break problem.
 \begin{figure}[h]
 \begin{center}