Environmental Engineering Reference
In-Depth Information
condition (u.
D
<
1). Implicit schemes are not
subject to such stringent limitations, but time
steps are nevertheless limited by considerations
of accuracy.
t/
D
wetting and drying (the prediction of the bound-
aries of inundation) is a specific challenge in in-
undation modelling because flood depths are
usually very small along most floodplain inunda-
tion boundaries. Model instabilities occur very
easily, often due to the fact that friction slope
formulae (see, e.g., Equation 12.12) diverge for very
small flow depths. A number of approaches have
been proposed (see, e.g., Begnudelli and Sanders
2006, or Bates and Horritt 2005 for a comprehen-
sive review of the issue), all of which are a com-
promise between stability, accuracy and mass
conservation.
x
Challenges in numerical modelling
The shallow water equations are non-linear, i.e.
they do not satisfy the principle of superposition.
One of the implications is that shallow water
flows are subject to shock waves, which are to be
understood as discontinuous solutions of the shal-
lowwater equations (Toro 2001). Shocks on flood-
plains are mainly encountered in the form of
hydraulic jumps, i.e. transitions for supercritical
to subcritical flows. These may be caused by local
changes in terrain topography (diminution of bot-
tom slope, lateral expansion), or by the effect of
bottom friction. An important challenge in the
numerical resolution of the shallow water equa-
tions is the prediction of the location (celerity) of
flood wave fronts and discontinuities (shocks).
This is an area where considerable progress has
been achieved over the last two decades, mainly
through the use of the so-called shock-capturing
methods. Some of the well-known shock-captur-
ing methods (see Toro 2001) used in inundation
modeling include the MacCormack method
(Liang et al. 2007a), the Lax-Wendroff method,
Total-Variation Diminishing (TVD) schemes,
Monotonic upstream-centred Schemes for Con-
servation Laws (MUSCL) based on the Godunov
approach, and Essentially Non-Oscillatory (ENO)
schemes. Schemes belonging to the class of ap-
proximate Riemann solvers (Roe 1981; Toro 1999)
are also increasing in popularity.
Two other major areas of ongoing research are
related to (i) the treatment of source terms, and
(ii) the modelling of wetting and drying. Source
terms (H in Equation 12.11) arising from the bed
slope dominate in applications to real floodplains,
so that the discretization approaches for the
flux term and the source terms must ensure an
appropriate balance (see, e.g., Garcia-Navarro and
Vazquez-Cendon 2000), otherwise spuriousmove-
ment can be generated by the numerical model
even in a body of water at rest. The modelling of
Computational grids
Amesh or grid is a collection of points (or vertices)
where the variables defining the flow condition
(velocity, depth or water level) are computed as
outlined above under 'Classes of numerical meth-
ods'. Closely positioned vertices give a fine or
high-resolution grid, and widely spaced vertices
give a coarse or low-resolution grid. The resolution
may also vary in space. The computational effi-
ciency of a numerical model is directly related to
the number of equations that need to be solved and
therefore to the resolution of the grid.
A structured grid is a grid that can be concep-
tually represented on a rectangular matrix (i.e. the
numerical program can effectively make use of
rectangular matrices to store the flow variables
involved in the computation). Any point in the
matrix is physically connected to the four points
on either side. A structured gridwhere the vertices
are physically at regular intervals apart is called a
structured square grid (Fig. 12.2a). A boundary-
fitted grid is a structured grid that makes use of
irregular intervals between vertices (Fig. 12.2b).
An unstructured grid is a grid that cannot be
represented on a rectangularmatrix (Fig. 12.3). The
points that constitute such a grid are kept as lists
of (x,y,z) coordinates and details on how the points
are connected to each other are recorded in a
database. The flow variables computed by the
model are also stored in the form of lists. The
attraction of unstructured grid models lies in
the possibility to follow irregular floodplain
