Environmental Engineering Reference
In-Depth Information
Two-dimensional (2D) flow modelling
accelerations are negligible); and (iii) that the ef-
fects of boundary friction and turbulence can be
accounted for by representations of channel con-
veyance derived for steady-state flow.
An alternative form of the momentum conser-
vation equation can be obtained by rearranging
Equation 12.2:
The two-dimensional form of the shallow water
equations can be expressed as:
U
q
F
G
q
q
t
þ
q
x
þ
q
y
¼
H
ð
12
:
10
Þ
q
2
where x and y are the two spatial dimensions, and
the vectors U, F, G, H are defined as follows:
1
1
S
0
q
h
gA
q
Q
A
gA
q
Q
q
S
f
¼
x
ð
12
:
7
Þ
q
q
x
t
0
@
1
A
0
@
1
A
;
F
¼
hu
h
It is justified in many applications to neglect the
last term in Equation 12.7, yielding the quasi-
steady form of the momentum equation. In most
rivers, the flow is subcritical and all acceleration
terms can also be neglected, to yield the so-called
diffusive wave equation (Julien 2002):
g
h
2
hu
2
U
¼
hu
;
2
þ
hv
huv
0
@
1
A
;
H
¼
0
@
1
A
hv
huv
g
h
2
0
G
¼
gh
ð
S
0
x
S
fx
Þ
h
q
x
S
f
¼
S
0
q
hv
2
gh
ð
S
0
y
S
fy
Þ
2
þ
ð
12
:
8
Þ
ð
12
:
11
Þ
A further simplification can be applied by neglect-
ing the pressure term, retaining only:
In this u and v are the depth-averaged velocities in
the x and y directions, and S
ox
and S
oy
are the bed
slopes in the x and y directions. Assuming the use
of Manning's n, the friction slopes in the x and y
directions can be expressed as:
S
f
¼
S
0
ð
12
:
9
Þ
which is referred to as the kinematic wave
equation.
One of the principal strengths of 1D river
models is their capability to simulate flows over
and through a large range of hydraulic structures
such as weirs, gates, sluices, etc. (Evans et al.
2007). Recent and ongoing research advances in
1D modelling include enhanced conveyance es-
timation techniques and afflux estimation tech-
niques. The Conveyance Estimation System
(Samuels et al. 2002; HR Wallingford 2003) fo-
cused particularly on the effects of riverine veg-
etation, the momentum exchange between the
river channel and floodplain flows, and the
behaviour of natural shaped channels. It is im-
plemented in a number of commercial packages
such as ISIS and InfoWorks-RS. The Afflux Esti-
mation System (Lamb et al. 2006) is an improved
methodology for the prediction of the increase in
water level upstream of a structure (caused by
energy losses at high flows through bridges and
culverts).
n
2
u
p
n
2
v
p
u
2
þ
v
2
u
2
þ
v
2
S
fx
¼
and
S
fy
¼
h
4=3
h
4=3
ð
12
:
12
Þ
Equation 12.10 reverts the 1D St-Venant equa-
tions by ignoring the velocity component and
gradients in the y direction and multiplying
by the depth-averaged channel width. Equa-
tion 12.10 is expressed in conservative form, and
similarly with the 1D St-Venant equations, a
non-conservative
formulation can also
be
derived.
A number of terms can be added to Equa-
tion 12.10 to represent a wider range of physical
processes that may contribute to floodplain flow.
These include viscosity, the Coriolis effect, wind
shear stresses, wall friction stresses, inflow vol-
ume and inflow momentum. Viscosity terms F
d
and G
d
to be added to F and G respectively can be
expressed as follows:
