Environmental Engineering Reference
In-Depth Information
0
@
1
A
0
@
1
A
wave equations (Bradbrook et al. 2004). This is
appropriate where the flow is predominantly driv-
en by local water surface slope and momentum
effects are less important, as is often the case in the
context of UK fluvial floodplains. Such modelling
approaches and recent practical applications (e.g.
Bradbrook et al. 2005) are discussed in Hunter
et al. (2007).
0
0
e
h
q
h
q
u
q
u
q
x
e
F
d
¼
and
G
d
¼
y
v
v
h
q
h
q
e
e
q
x
q
y
ð
12
:
13
Þ
where the subscript d stands for 'diffusion' as these
terms are analogous to a diffusion process. In this,
e
is the so-called viscosity coefficient, which ac-
counts for the combined effect of (i) kinematic
viscosity, (ii) the turbulent eddy viscosity and
(iii) the apparent viscosity due to the velocity
fluctuations about the vertical average. The con-
tribution of the kinematic viscosity to the value
of
e
is typically at least an order of magnitude
smaller than the turbulent eddy viscosity and for
this reason is neglected.
The apparent viscosity due to the velocity fluc-
tuations about the vertical average is recognized as
a much more significant contributor to the value
of
e
(Alcrudo 2004). However, this effect is poorly
understood and is therefore also neglected in
most applications. The turbulent eddy diffusivity
has been the object of significant research (see
Rodi 1980), but in the context of flood modelling
it is generally not considered an important param-
eter (Alcrudo 2004). For overland flow conditions
it is unlikely that the eddy viscosity will have a
major effect on model predictions as friction will
dominate. It may, however, have a significant
effect upon local high-resolution predictions
(Danish Hydraulic Institute 2007a) for flow in and
around structures.
Coriolis effects (which account for the effects of
the Earth's rotation) are considered negligible in
the context of flood inundation studies. Wind
shear stresses may result in non-negligible effects
on water depths in very large floodplains but their
prediction is intimately dependent on the ability
to predict wind strength and direction. Wall
friction terms are only relevant in very-high-
resolution modelling studies and are therefore
rarely included.
It is possible to neglect the acceleration terms in
the 2Dshallow-water equations (the terms involv-
ing u and v in U, F and G) to yield the 2D diffusion
Introduction to numerical methods
for inundation modelling
Classes of numerical methods
Numerical modelling consists of replacing the
differential equations such as the shallow water
equations by a set of algebraic equations. The
process of representing space and time using a
finite set of points in the space-time domain and
converting the differential equations into algebra-
ic equations is called discretization. The many
numerical methods in existence can be split into
classes depending on the discretization strategy,
i.e. the specific approach applied to do this. The
great majority of methods used to solve the shal-
low water equations fall into one of three discre-
tization strategies: Finite Difference, Finite
Element, and Finite Volume methods.
Finite difference methods Finite Difference
methods rely on Taylor series expansions to ex-
press the value taken by a variable (h, u, v, etc.) at a
given point, as a function of the values at neigh-
bouring points and of local derivatives of increas-
ing orders. These Taylor series are then combined
to yield approximate expressions for the deriva-
tives involved in the shallowwater equations, as a
function of a finite number of neighbouring point
values. The accuracy of the approximations made
in doing this can be controlled by the order to
which the Taylor series expansions are developed
(the order of the so-called truncation), which is
also linked to the number of neighbouring points
involved. In 2D, the implementation of Finite
Difference methods is significantly more straight-
forward on a structured grid (see 'Computational
grids' below). This explains to some extent why
