source: anuga_work/publications/anuga_2007/anuga_validation.tex @ 5232

Last change on this file since 5232 was 5232, checked in by ole, 16 years ago

Added screenshots of circular island

File size: 26.1 KB
Line 
1% Use the standard \LaTeXe\ article style in 12pt Computer Modern font
2% on A4 paper by
3\documentclass[12pt,a4paper]{article}
4
5% Do \emph{not} change the width nor the height of the text from the
6% defaults set by this document class.
7%
8% The alternative which is closer to what we actually use is
9%\documentclass[11pt,a5paper]{article}
10%\usepackage[a5paper]{geometry}
11% Because it is a great size for on screen reading
12% and prints nicely on a4paper either 2up or booklet.
13
14% The preamble is to contain your own \LaTeX\ commands and to say
15% what packages to use.  Three useful packages are the following:
16\usepackage{graphicx} % avoid epsfig or earlier such packages
17
18
19\usepackage{url}      % for URLs and DOIs
20\usepackage{amsmath}  % many want amsmath extensions
21\usepackage{amsfonts}
22\usepackage{underscore}
23% Avoid loading unused packages (as done by some \LaTeX\ editors).
24
25% Create title and authors using \verb|\maketitle|.  Separate authors by
26% \verb|\and| and put addresses in \verb|\thanks| command with
27% \verb|\url| command \verb|\protect|ed.
28\title{On The Validation of A Hydrodynamic Model}
29
30\author{
31D.~S.~Gray\thanks{Natural Hazard Impacts Project, Geospatial and
32Earth Monitoring Division, Geoscience Australia, Symonston,
33\textsc{Australia}. \protect\url{mailto:Duncan.Gray@ga.gov.au}}\footnotemark[1]
34\and
35T.~Baldock\thanks{University of Queensland, Brisbane, \textsc{Australia}.
36\protect\url{mailto:tom.baldock@uq.edu.au}}\footnotemark[2]
37\and
38O.~M.~Nielsen\footnotemark[1]
39\and 
40M.~J.~Sexton\footnotemark[1]
41\and
42N.~Bartzis\footnotemark[1]
43\and
44S.~G.~Roberts\thanks{Department of Mathematics, Australian National University,
45Canberra, \textsc{Australia}. \protect\url{mailto:stephen.roberts@anu.edu.au}}}
46
47\date{30 May 2008}
48
49\newcommand{\ANUGA}{\textsc{ANUGA}}
50\newcommand{\Python}{\textsc{Python}}
51\newcommand{\VPython}{\textsc{VPython}}
52\newcommand{\pypar}{\textsc{mpi}}
53\newcommand{\Metis}{\textsc{Metis}}
54\newcommand{\mpi}{\textsc{mpi}}
55
56\newcommand{\UU}{\mathbf{U}}
57\newcommand{\VV}{\mathbf{V}}
58\newcommand{\EE}{\mathbf{E}}
59\newcommand{\GG}{\mathbf{G}}
60\newcommand{\FF}{\mathbf{F}}
61\newcommand{\HH}{\mathbf{H}}
62\newcommand{\SSS}{\mathbf{S}}
63\newcommand{\nn}{\mathbf{n}}
64
65\newcommand{\code}[1]{\texttt{#1}}
66
67
68
69
70\begin{document}
71
72% Use default \verb|\maketitle|.
73\maketitle
74
75% Use the \verb|abstract| environment.
76\begin{abstract}
77Modelling the effects on the built environment of natural hazards such
78as riverine flooding, storm surges and tsunami is critical for
79understanding their economic and social impact on our urban
80communities.  Geoscience Australia and the Australian National
81University have developed a hydrodynamic inundation modelling tool
82called \ANUGA{} to help simulate the impact of these hazards.
83The core of \ANUGA{} is a \Python{} implementation of a finite-volume method
84for solving the conservative form of the Shallow Water Wave equation.
85In this paper we describe the model, the architecture and a range of
86validations that have been carried out to establish confidence in the model.
87
88
89\ANUGA{} is available as Open Source to enable
90free access to the software and allow the scientific community to
91use, validate and contribute to the software in the future.
92
93%This method allows the study area to be represented by an unstructured
94%mesh with variable resolution to suit the particular problem.  The
95%conserved quantities are water level (stage) and horizontal momentum.
96%An important capability of ANUGA is that it can robustly model the
97%process of wetting and drying as water enters and leaves an area. This
98%means that it is suitable for simulating water flow onto a beach or
99%dry land and around structures such as buildings.
100
101
102\end{abstract}
103
104% By default we include a table of contents in each paper.
105%\tableofcontents
106
107% Use \verb|\section|, \verb|\subsection|, \verb|\subsubsection| and
108% possibly \verb|\paragraph| to structure your document.  Make sure
109% you \verb|\label| them for cross-referencing with \verb|\ref|\,.
110
111%\clearpage
112\section{Introduction}
113\label{sec:intro}
114
115The Indian Ocean tsunami on 26 December 2004 demonstrated the
116potentially catastrophic consequences of natural hazards.  While the
117scale of the impact from such events is not common, smaller-scale
118tsunami regularly threaten coastal communities
119around the world. Earthquakes which occur in the Java Trench near
120Indonesia (e.g.~\cite{TsuMIS1995} or \cite{Baldwin-2006}) and along
121the Puysegur Ridge to the south of New Zealand (e.g.~\cite{LebKC1998})
122have potential to generate tsunami that may threaten Australia's
123northwestern and southeastern coastlines. In addition, the
124preferential development of Australian coastal corridors means that
125inundation from hydrological disasters such as tsunami or storm-surge
126of even a few hundred metres beyond the shoreline has increased
127potential to cause significant disruption and loss. The extent of
128inundation is critically linked to the event, tidal conditions,
129bathymetry and topography and it not feasible to make impact
130predictions using heuristics alone.
131
132Hydrodynamic modelling allows impacts from flooding, storm-surge and
133tsunami to be better understood, their impacts to be anticipated and,
134with appropriate planning, their effects to be mitigated.  Geoscience
135Australia in collaboration with the Mathematical Sciences Institute,
136Australian National University, is developing a software application
137called \ANUGA{} to model the hydrodynamics of floods, storm surges and
138tsunami. These hazards are modelled using the conservative shallow
139water equations which are described in section~\ref{sec:model}. In
140\ANUGA{} these equations are solved using a finite volume method as
141described in section~\ref{sec:fvm}.  A more complete discussion of the
142method can be found in \cite{modsim2005} where the model and solution
143technique is validated on a standard tsunami benchmark data set
144or in \cite{Roberts2007} where parallelisation of ANUGA is discussed.
145This modelling capability is part of
146Geoscience Australia's ongoing research effort to model and
147understand the potential impact from natural hazards in order to
148reduce their impact on Australian communities (see \cite{Nielsen2006}).
149\ANUGA{} is currently being trialled for flood
150modelling (see \cite{Rigby2008}).
151
152Section~\ref{sec:software} describes the software implementation and
153the API while section~\ref{sec:validation} presents some
154validation results.
155
156
157\section{Model}
158\label{sec:model}
159
160The shallow water wave equations are a system of differential
161conservation equations which describe the flow of a thin layer of
162fluid over terrain. The form of the equations are:
163\[
164\frac{\partial \UU}{\partial t}+\frac{\partial \EE}{\partial
165x}+\frac{\partial \GG}{\partial y}=\SSS
166\]
167where $\UU=\left[ {{\begin{array}{*{20}c}
168 h & {uh} & {vh} \\
169\end{array} }} \right]^T$ is the vector of conserved quantities; water depth
170$h$, $x$-momentum $uh$ and $y$-momentum $vh$. Other quantities
171entering the system are bed elevation $z$ and stage (absolute water
172level) $w$, where the relation $w = z + h$ holds true at all times.
173The fluxes in the $x$ and $y$ directions, $\EE$ and $\GG$ are given
174by
175\[
176\EE=\left[ {{\begin{array}{*{20}c}
177 {uh} \hfill \\
178 {u^2h+gh^2/2} \hfill \\
179 {uvh} \hfill \\
180\end{array} }} \right]\mbox{ and }\GG=\left[ {{\begin{array}{*{20}c}
181 {vh} \hfill \\
182 {vuh} \hfill \\
183 {v^2h+gh^2/2} \hfill \\
184\end{array} }} \right]
185\]
186and the source term (which includes gravity and friction) is given
187by
188\[
189\SSS=\left[ {{\begin{array}{*{20}c}
190 0 \hfill \\
191 -{gh(z_{x} + S_{fx} )} \hfill \\
192 -{gh(z_{y} + S_{fy} )} \hfill \\
193\end{array} }} \right]
194\]
195where $S_f$ is the bed friction. The friction term is modelled using
196Manning's resistance law
197\[
198S_{fx} =\frac{u\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}\mbox{ and }S_{fy}
199=\frac{v\eta ^2\sqrt {u^2+v^2} }{h^{4/3}}
200\]
201in which $\eta$ is the Manning resistance coefficient.
202
203As demonstrated in our papers, \cite{modsim2005,Rob99l} these
204equations provide an excellent model of flows associated with
205inundation such as dam breaks and tsunamis.
206
207\section{Finite Volume Method}
208\label{sec:fvm}
209
210We use a finite-volume method for solving the shallow water wave
211equations \cite{Rob99l}. The study area is represented by a mesh of
212triangular cells as in Figure~\ref{fig:mesh} in which the conserved
213quantities of  water depth $h$, and horizontal momentum $(uh, vh)$,
214in each volume are to be determined. The size of the triangles may
215be varied within the mesh to allow greater resolution in regions of
216particular interest.
217
218\begin{figure}
219\begin{center}
220%\includegraphics[width=5.0cm,keepaspectratio=true]{step-five}
221\caption{Triangular mesh used in our finite volume method. Conserved
222quantities $h$, $uh$ and $vh$ are associated with the centroid of
223each triangular cell.} \label{fig:mesh}
224\end{center}
225\end{figure}
226
227The equations constituting the finite-volume method are obtained by
228integrating the differential conservation equations over each
229triangular cell of the mesh. Introducing some notation we use $i$ to
230refer to the $i$th triangular cell $T_i$, and ${\cal N}(i)$ to the
231set of indices referring to the cells neighbouring the $i$th cell.
232Then $A_i$ is the area of the $i$th triangular cell and $l_{ij}$ is
233the length of the edge between the $i$th and $j$th cells.
234
235By applying the divergence theorem we obtain for each volume an
236equation which describes the rate of change of the average of the
237conserved quantities within each cell, in terms of the fluxes across
238the edges of the cells and the effect of the source terms. In
239particular, rate equations associated with each cell have the form
240$$
241 \frac{d\UU_i }{dt}+ \frac1{A_i}\sum\limits_{j\in{\cal N}(i)} \HH_{ij} l_{ij} = \SSS_i
242$$
243where
244\begin{itemize}
245\item $\UU_i$ the vector of conserved quantities averaged over the $i$th cell,
246\item $\SSS_i$ is the source term associated with the $i$th cell,
247and
248\item $\HH_{ij}$ is the outward normal flux of
249material across the \textit{ij}th edge.
250\end{itemize}
251
252
253%\item $l_{ij}$ is the length of the edge between the $i$th and $j$th
254%cells
255%\item $m_{ij}$ is the midpoint of
256%the \textit{ij}th edge,
257%\item
258%$\mathbf{n}_{ij} = (n_{ij,1} , n_{ij,2})$is the outward pointing
259%normal along the \textit{ij}th edge, and The
260
261The flux $\HH_{ij}$ is evaluated using a numerical flux function
262$\HH(\cdot, \cdot ; \ \cdot)$ which is consistent with the shallow
263water flux in the sense that for all conservation vectors $\UU$ and normal vectors $\nn$
264$$
265H(\UU,\UU;\ \nn) = \EE(\UU) n_1 + \GG(\UU) n_2 .
266$$
267
268Then
269$$
270\HH_{ij}  = \HH(\UU_i(m_{ij}),
271\UU_j(m_{ij}); \mathbf{n}_{ij})
272$$
273where $m_{ij}$ is the midpoint of the \textit{ij}th edge and
274$\mathbf{n}_{ij}$ is the outward pointing normal, with respect to the $i$th cell, on the
275\textit{ij}th edge. The function $\UU_i(x)$ for $x \in
276T_i$ is obtained from the vector $\UU_k$ of conserved average values for the $i$th and
277neighbouring  cells.
278
279We use a second order reconstruction to produce a piece-wise linear
280function construction of the conserved quantities for  all $x \in
281T_i$ for each cell (see Figure~\ref{fig:mesh:reconstruct}. This
282function is allowed to be discontinuous across the edges of the
283cells, but the slope of this function is limited to avoid
284artificially introduced oscillations.
285
286Godunov's method (see \cite{Toro-92}) involves calculating the
287numerical flux function $\HH(\cdot, \cdot ; \ \cdot)$ by exactly
288solving the corresponding one dimensional Riemann problem normal to
289the edge. We use the central-upwind scheme of \cite{KurNP2001} to
290calculate an approximation of the flux across each edge.
291
292\begin{figure}
293\begin{center}
294%\includegraphics[width=5.0cm,keepaspectratio=true]{step-reconstruct}
295\caption{From the values of the conserved quantities at the centroid
296of the cell and its neighbouring cells, a discontinuous piecewise
297linear reconstruction of the conserved quantities is obtained.}
298\label{fig:mesh:reconstruct}
299\end{center}
300\end{figure}
301
302In the computations presented in this paper we use an explicit Euler
303time stepping method with variable timestepping adapted to the
304observed CFL condition.
305
306
307\section{Software Implementation}
308\label{sec:software}
309
310\ANUGA{} is mostly written in the object-oriented programming
311language \Python{} with computationally intensive parts implemented
312as highly optimised shared objects written in C.
313
314\Python{} is known for its clarity, elegance, efficiency and
315reliability. Complex software can be built in \Python{} without undue
316distractions arising from idiosyncrasies of the underlying software
317language syntax. In addition, \Python{}'s automatic memory management,
318dynamic typing, object model and vast number of libraries means that
319software can be produced quickly and can be readily adapted to
320changing requirements throughout its lifetime.
321
322The fundamental object in \ANUGA{} is the \code{Domain} which
323inherits functionality from a hierarchy of increasingly specialised
324classes starting with a basic structural Mesh to classes implementing
325the finite-volume scheme described in section \ref{sec:fvm}. Other classes
326are \code{Quantity} which represents values of one variable across the mesh
327along with their associated operations, \code{Geospatial_data} which
328represents georeferenced elevation data and a collection of \code{Boundary}
329classes which allows for a 'pluggable' way of driving the model.
330The conserved quantities updated automatically by the numerical scheme
331are stage (water level) $w$, $x$-momentum $uh$ and $y$-momentum
332$vh$. The quanitites elevation $z$ and friction $\eta$ are
333quantities that are not updated automatically but can be changed explicitly
334during run-time if the user wishes to do so.
335
336To set up a scenario the user specifies the study area along with any internal
337regions where increased mesh resolution is required. External edges may
338be labelled using symbolic tags which are subsequently used to bind
339boundary condition objects to tagged segments of the mesh boundary.
340The mesh is then generated using \ANUGA{}'s built-in mesh generator and
341converted into the \code{Domain} object which provides all methods used to
342setup and run the flow simulation. Figure \ref{fig:anuga mesh} shows an example of a mesh generated by \ANUGA{}.
343\begin{figure}
344\begin{center}
345\includegraphics[width=4in,keepaspectratio=true]{triangular-mesh}
346\caption{Triangular mesh used in our finite volume method. Conserved
347quantities $h$, $uh$ and $vh$ are associated with each triangular cell.}
348\label{fig:anuga mesh}
349\end{center}
350\end{figure}
351
352
353
354Next step is to setup initial conditions for each \code{Quantity} object. For
355the elevation $z$ this is typically obtained from bathymetric and
356topographic data sets. Setting initial values for quantities is done
357through the method \code{domain.set_quantity(name, X, location,
358region)} where name is the name of the quantity (e.g.\ 'stage',
359'xmomentum', 'ymomentum', 'elevation' or 'friction').  The variable X
360represents the source data for populating the quantity and may take
361one of the following forms:
362
363\begin{itemize}
364\item A constant value as in \code{domain.set_quantity('stage', 1)} which
365will set the initial water level to 1 m everywhere.
366\item Another quantity or a linear combination of quantities.  If \code{q1}
367and \code{q2} are two arbitrary quantities defined within the same domain,
368the expression \code{domain.set_quantity('stage', q1*(3*q2 + 5))} will set the stage
369quantity accordingly.  One common application of this would be to
370assign the stage as a constant depth above the bed elevation.
371\item An arbitrary function (or a callable object), \code{f(x, y)}, where \code{x}
372and \code{y} are assumed to be vectors. The quantity will be assigned values by
373evaluating \code{f} at each location within the mesh.
374\item An arbitrary set of points and associated values (wrapped into a
375Geospatial_data object). The points need not coincide with triangle
376vertices or centroids and a penalised least squares technique is
377employed to populate the quantity in a smooth and stable way.
378Since the least squares technique can be time consuming for large
379problems, \code{set_quantity} employs a caching technique which automatically
380decides whether to perform the computations or retrieve them from a
381cache.  This will typically speed up the build by several orders of
382magnitude after each computation has been performed once.
383\item A filename containing points and attributes.
384\item A Numerical Python array (or a list of numbers) ordered
385according to the internal data structure.
386\end{itemize}
387The parameter \code{location} determines whether the values should be
388assigned to triangle edge, midpoints or vertices and \code{region} allows the
389operation to be restricted to a region specified by a symbolic tag or
390a set of indices.
391
392Boundary conditions are bound to symbolic tags through the method
393\code{domain.set_boundary} which takes as input a lookup table (implemented
394as a Python dictionary) of the form \code{\{tag:~boundary_object\}}.
395The boundary objects are all assumed to be callable functions of vectors x
396and y.  Several predefined standard boundary objects are available and
397it is relatively straightforward to define problem-specific custom
398boundaries if needed.  The predefined boundary conditions include
399Dirichlet, Reflective, Transmissive, Temporal, and Spatio-Temporal
400boundaries.
401
402Forcing terms can be written according to a fixed protocol and added
403to the model using the idiom \code{domain.forcing_terms.append(F)} where \code{F} is
404assumed to be a user-defined callable object.
405
406When the simulation is running, the length of each time step is
407determined from the maximal speeds encountered and the sizes of
408triangles in order not to violate the CFL condition which specifies
409that no information should skip any triangles in one time step.  With
410large speeds and small triangles, time steps can become very small.
411In order to access the state of the simulation at regular time
412intervals, \ANUGA{} uses the method evolve:
413\begin{verbatim}
414For t in domain.evolve(yieldstep, duration):
415    <model interrogation and modification>
416\end{verbatim}
417The parameter \code{duration} specifies the time period over which
418evolve operates, and control is passed to the body of the for-loop at
419each fixed time step called \code{yieldstep}.  The internal workings
420of the numerical scheme and its variable time stepping are thus
421decoupled from the fixed time stepping of the evolve loop.  This means
422that the user of the API may access the model at fixed timesteps to
423e.g.\ store model outputs, interrogate quantities or change the model
424itself at runtime. The evolve method has been implemented using a
425Python generator hence the reference to 'yield' in the parameter name.
426
427%Figure \ref{fig:beach runup} shows a simulation of water flowing onto a
428%hypothetical beach with obstacles.
429%A number of complex patterns are captured in this example including a shock where water reflected off the wall far (at the right hand side) meets the main flow. Other physical features are the standing waves and interference patterns.
430%See the \ANUGA{} User Manual at \url{http://sourceforge.net/projects/anuga} for more details and examples.
431%\begin{figure}
432%\begin{center}
433%\includegraphics[width=4in,keepaspectratio=true]{runup}
434%\caption{A hypothetical runup scenario.}
435%\label{fig:beach runup}
436%\end{center}
437%\end{figure}
438
439
440
441\section{Validation}
442\label{sec:validation} The process of validating the \ANUGA{}
443application is in its early stages, however initial indications are
444encouraging.
445
446\subsection{Okushiri Wavetank Validation}
447
448As part of the Third International Workshop on Long-wave Runup
449Models in 2004 (\url{http://www.cee.cornell.edu/longwave}), four
450benchmark problems were specified to allow the comparison of
451numerical, analytical and physical models with laboratory and field
452data. One of these problems describes a wave tank simulation of the
4531993 Okushiri Island tsunami off Hokkaido, Japan \cite{MatH2001}. A
454significant feature of this tsunami was a maximum run-up of 32~m
455observed at the head of the Monai Valley. This run-up was not
456uniform along the coast and is thought to have resulted from a
457particular topographic effect. Among other features, simulations of
458the Hokkaido tsunami should capture this run-up phenomenon.
459
460\begin{figure}[htbp]
461%\centerline{\includegraphics[width=4in]{okushiri-gauge-5.eps}}
462\centerline{\includegraphics[width=4in]{ch5.png}}
463\centerline{\includegraphics[width=4in]{ch7.png}}
464\centerline{\includegraphics[width=4in]{ch9.png}}
465\caption{Comparison of wave tank and \ANUGA{} water stages at gauge
4665,7 and 9.}\label{fig:val}
467\end{figure}
468
469
470\begin{figure}[htbp]
471\centerline{\includegraphics[width=4in]{okushiri-model.jpg}}
472\caption{Complex reflection patterns and run-up into Monai Valley
473simulated by \ANUGA{} and visualised using our netcdf OSG
474viewer.}\label{fig:run}
475\end{figure}
476
477
478The wave tank simulation of the Hokkaido tsunami was used as the
479first scenario for validating \ANUGA{}. The dataset provided
480bathymetry and topography along with initial water depth and the
481wave specifications. The dataset also contained water depth time
482series from three wave gauges situated offshore from the simulated
483inundation area.
484
485Figure~\ref{fig:val} compares the observed wave tank and modelled
486\ANUGA{} water depth (stage height) at one of the gauges. The plots
487show good agreement between the two time series, with \ANUGA{}
488closely modelling the initial draw down, the wave shoulder and the
489subsequent reflections. The discrepancy between modelled and
490simulated data in the first 10 seconds is due to the initial
491condition in the physical tank not being uniformly zero. Similarly
492good comparisons are evident with data from the other two gauges.
493Additionally, \ANUGA{} replicates exceptionally well the 32~m Monai
494Valley run-up, and demonstrates its occurrence to be due to the
495interaction of the tsunami wave with two juxtaposed valleys above
496the coastline. The run-up is depicted in Figure~\ref{fig:run}.
497
498This successful replication of the tsunami wave tank simulation on a
499complex 3D beach is a positive first step in validating the \ANUGA{}
500modelling capability.
501
502\subsection{Manning's Friction Model Validation}
503 
504% Validation UQ friction
505% at X:\anuga_validation\uq_friction_2007
506% run run_dam.py to create sww file and .csv files
507% run plot.py to create graphs, and move them here
508\begin{figure}[htbp]
509\centerline{\includegraphics[width=4in]{uq-friction-depth}}
510\caption{Comparison of wave tank and \ANUGA{} water height at .4 m
511  from the gate, simulated using a Mannings friction of 0.0 and 0.1.}\label{fig:uq-friction-depth}
512\end{figure}
513
514 The bed friction is modelled in ANUGA using the Manning's
515 model. Validation of this model was carried out by comparing results
516 from ANUGA against experimental results from flume wave tanks. The
517experiments were carried out at the Gordon McKay Hydraulics Laboratory
518at St Lucia, University of Queensland.
519 
520%The Manning's friction model is
521
522%To validate the friction model
523
524\subsection{Stage and Velocity Validation in a Flume}
525% Validation UQ flume
526% at X:\anuga_validation\uq_sloped_flume_2008
527% run run_dam.py to create sww file and .csv files
528% run plot.py to create graphs heere automatically
529% The Coasts and Ports '2007 paper is in TRIM d2007-17186
530\begin{figure}[htbp]
531\centerline{\includegraphics[width=4in]{uq-flume-depth}}
532\caption{Comparison of wave tank and \ANUGA{} water height at .4 m
533  from the gate}\label{fig:uq-flume-depth}
534\end{figure}
535
536\begin{figure}[htbp]
537\centerline{\includegraphics[width=4in]{uq-flume-velocity}}
538\caption{Comparison of wave tank and \ANUGA{} water velocity at .45 m
539  from the gate}\label{fig:uq-flume-velocity}
540\end{figure}
541
542Flume experiments caried out at the University of Queensland has also
543been used for validating the water height and velocity predicted by
544\ANUGA{}.  The Flume was set up for Dam-break experiments, having a
545water reservior at one end.  The flume was glass-sided, 3m long, 0.4m
546in wide, and 0.4m deep, with a PVC bottom. The reservoir in the flume
547was 0.75m long.  For this experiment the reservoir water was 0.2m
548deep. At time zero the reservoir gate is opened and the water flows
549into the other side of the flume.  The water ran up a flume slope of
5500.03 m/m.  To accurately model the bed surface a Manning's friction
551value of 0.01, representing PVC was used.
552
553% Neale, L.C. and R.E. Price.  Flow characteristics of PVC sewer pipe.
554% Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE.
555% pp. 109-129.  1964.
556
557Acoustic displacement sensors that produced a voltage that changed
558with the water depth was positioned 0.4m from the reservoir gate. The
559water velocity was measured with an Acoustic Doppler Velocimeter 0.45m
560from the reservoir gate.  This sensor only produced reliable results 4
561seconds after the reservoir gate opened, due to limitations of the sensor.
562
563Figure~\ref{fig:uq-flume-depth} show that ANUGA predicts the actual
564water depth very well, with the exception of the fluid tip-region. The
565water velocity is also predicted accurately.
566
567\subsection{Runup of Solitary wave on circular island wavetank validation}
568
569\begin{figure}[htbp]
570\centerline{\includegraphics[width=4in]{circular1.png}}
571\centerline{\includegraphics[width=4in]{circular2.png}}
572\centerline{\includegraphics[width=4in]{circular3.png}}
573\centerline{\includegraphics[width=4in]{circular4.png}}
574\centerline{\includegraphics[width=4in]{circular5.png}}
575\centerline{\includegraphics[width=4in]{circular6.png}}
576\centerline{\includegraphics[width=4in]{circular7.png}}
577\centerline{\includegraphics[width=4in]{circular8.png}}
578\centerline{\includegraphics[width=4in]{circular9.png}}
579\centerline{\includegraphics[width=4in]{circular10.png}}
580\caption{Screenshots of the evolution of solitary wave around circular island.}
581\label{fig:circular screenshots}
582\end{figure}
583
584
585
586\subsection{MAYBE, 1D analytical validation}
587
588
589
590
591\section{Conclusions}
592\label{sec:6}
593\ANUGA{} is a flexible and robust modelling system
594that simulates hydrodynamics by solving the shallow water wave
595equation in a triangular mesh. It can model the process of wetting
596and drying as water enters and leaves an area and is capable of
597capturing hydraulic shocks due to the ability of the finite-volume
598method to accommodate discontinuities in the solution.
599\ANUGA{} can take as input bathymetric and topographic datasets and
600simulate the behaviour of riverine flooding, storm surge,
601tsunami or even dam breaks.
602Initial validation using wave tank data supports \ANUGA{}'s
603ability to model complex scenarios. Further validation will be
604pursued as additional datasets become available.
605The \ANUGA{} source code is available
606at \url{http://sourceforge.net/projects/anuga}.
607
608\bibliographystyle{plain}
609\bibliography{anuga-bibliography}
610
611
612
613
614\end{document}
Note: See TracBrowser for help on using the repository browser.