Environmental Engineering Reference
In-Depth Information
Here the momentum equation is expressed in its
conservative form. It is possible to substitute UA
for Q in Equations 12.1 and 12.2, and rearrange
Equation 12.2 to yield the mathematically correct
non-conservative form of the momentum equa-
tion. The use of the non-conservative form may,
however, lead to difficulties in its numerical so-
lution (see 'Introduction to numerical methods for
inundation modelling' below). Additional terms
can be added, such as inflow and loss terms (to the
mass conservation equation), and an inflow mo-
mentum (to
be many orders of magnitude smaller than those
obtained using the other approaches listed above).
These methods are based on much simpler repre-
sentations of the physical processes than 2Dmod-
els and the removal of the time discretization in
the computation. In addition, Pender (2006) also
refers to a so-called 0D class of inundation model-
ling methods, which are methods that do not
involve any modelling of the physical processes
of inundation. One may consider emulation tech-
niques making use of a limited number of training
runs of a hydraulic model (see, e.g., Beven et al.
2008) to belong to this category. Simple geometric
methods consisting in projecting river water
levels horizontally over a floodplain can also be
termed 0D as far as the modelling of floodplain
inundation is concerned. These may be applied to
both river and coastal inundation cases.
the momentum conservation
equation).
Frictional resistance acting on the flow is re-
presented by the friction slope S
f
. Several friction
slope models exist, namely:
f
8
gR
UU
S
f
¼
jj
ð
12
:
3
Þ
where f is the Darcy-Weisbach friction factor, and
R is the hydraulic radius (R
One-dimensional (1D) flow modelling
¼
A/P, where P is the
The St-Venant equations (Barr ´ de St-Venant 1871)
can be expressed as follows:
wetted perimeter);
1
C
2
R
U
jj
S
f
¼
ð
12
:
4
Þ
q
Q
x
þ
q
A
q
t
¼ 0
ð
12
:
1
Þ
q
where C is the Ch ´ zy coefficient; and
þ
Q
2
A
¼ 0
1
Q
q
1
h
A
q
A
q
g
q
n
2
R
4
t
þ
x
gS
0
S
f
S
f
¼
UU
jj
ð
12
:
5
Þ
q
x
q
=
3
ðiÞ
ðiiÞ
ðiiiÞ
ðivÞ
ðvÞ
ð
12
:
2
Þ
where n is the Manning-Gauckler or Manning's
coefficient ('Manning's n'). Manning's n is the
most commonly applied friction parameter in the
UK. The friction slope S
f
can be further expressed
as follows:
Equation 12.1 is the continuity or mass conserva-
tion equation, and Equation 12.2 is the momen-
tum conservation equation. In this, Q is the flow
discharge, A is the cross-section surface area, g is
the acceleration due to gravity, h is the cross-
sectional averaged water depth, S
0
is the bed slope
in the longitudinal direction and S
f
is the friction
slope (i.e. the slope of the energy line). The various
terms in the momentum conservation equation
are as follows:
(i) local acceleration term
(ii) advective acceleration term
(iii) pressure term
(iv) bed slope term
(v) friction slope term.
A
2
K
2
U
jj
S
f
¼
ð
12
:
6
Þ
where K, the channel conveyance, is expressed as:
AR
2=3
n
K
¼
ð
12
:
7
Þ
Theoretical assumptions for the St-Venant
equations to be valid include: (i) that the bed
slope is small; (ii) that the pressure is hydrostatic
(streamline curvature is
small and vertical
