Changeset 8723

Timestamp:

Mar 5, 2013, 1:42:57 AM (12 years ago)

Author:

mungkasi

Message:

Updating code and adding report for avalanche wet.

Location:

trunk/anuga_core/validation_tests/Tests/Analytical_exact/avalanche_wet

Files:

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

trunk/anuga_core/validation_tests/Tests/Analytical_exact/avalanche_wet/numerical_avalanche_wet.py

@@ -157 +157 @@
 
 
+
+#------------------------------------------------------------------------------
+# 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_wet/results.tex

@@ -2 +2 @@
 \section{Avalanche involving a shock wave}
 
-An avalanche test problem involving a shock wave. Here debris is approximated as water. The simulation should show a rarefaction fan and a shock wave. 
+We consider an avalanche involving a shock wave.
+This problem is similar to dam break on wet areas, and so, it involves a shock. We consider a flat topography with positive slope. Shallow water approach is used to solve the problem. The analytical solution of this problem was derived by Mungkasi and Roberts~\cite{MR2012PAAG}. This shallow water approach was also implemented by a number of researchers, such as Mangeney et al.~\cite{MHR2000} and Naaim et al.~\cite{NVC1997}. The simulation should show a rarefaction and a shock. 
+
+
+
+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 $x < 0$}\\
+h_0 & \textrm{if $x > 0$}\\
+\end{array} \right.
+\end{equation}
+where $h_0>h_1>0$. The topography is a flat bed with positive slope. Note that when $h_1=0$, the problem becomes avalanche involving a dry area~\cite{MR2011DA}.
+
+The analytical solution~\cite{MR2012PAAG} at time $t>0$ is
+\begin{equation}
+h(x,t) = \left\{ \begin{array}{ll}
+h_1 & \textrm{if $x < \sigma t + \frac12 mt^2$}\,,\\
+h_2 & \textrm{if $ \sigma t + \frac12 mt^2 \leq x < (u_2+c_2)t + \frac12 mt^2$}\,,\\
+\frac{1}{9g} \left( \frac{x}{t} + 2c_0 - \frac12 mt \right)^2 & \textrm{if $(u_2+c_2)t + \frac12 mt^2 \leq x < c_0 t + \frac12 mt^2$}\,,\\
+h_0 & \textrm{if $x \geq c_0 t + \frac12 mt^2$}\,, \\
+\end{array} \right.
+\end{equation}
+and
+\begin{equation}
+u(x,t) = \left\{ \begin{array}{ll}
+mt & \textrm{if $x < \sigma t + \frac12 mt^2$}\,,\\
+u_2+mt & \textrm{if $ \sigma t + \frac12 mt^2 \leq x < (u_2+c_2)t + \frac12 mt^2$}\,,\\
+\frac23 \left( \frac{x}{t} - c_0 + mt \right) & \textrm{if $(u_2+c_2)t + \frac12 mt^2 \leq x < c_0 t + \frac12 mt^2$}\,,\\
+mt & \textrm{if $x \geq c_0 t + \frac12 mt^2$}\,, \\ \end{array} \right.
+\end{equation}
+for time $t>0$\,. Here $u_2$\,, $c_2$\,, and $\sigma$ are the solutions of the three simultaneous equations 
+\begin{equation}
+u_2  = \sigma - \frac{c_1^2}{4\sigma} \left(1+\sqrt{1+8\left( \frac{\sigma}{c_1} \right)^2}     \right),
+\end{equation}
+\begin{equation}
+c_2 = c_1 \sqrt{\frac12 \left(\sqrt{1+  8 \left(\frac{\sigma}{c_1}\right)^2}-1  \right)},
+\end{equation}
+and 
+\begin{equation}
+-2c_0 = u_2 -2c_2.
+\end{equation} 
+The value of $h_2$ is calculated using relation $c_2=\sqrt{gh_2}$\,.
+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 $h_0=20$ and $h_1=10$ 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. 
 
 \begin{figure}[h]