Environmental Engineering Reference
In-Depth Information
and minimizing the expression
WðZþluÞ
, where
vector
u
is the gradient of function
W
from the
previous
other links
other links
D
*
iteration,
u ¼rWðZ
Þ
,
and
scalar
U
node
2
.
Once the system (Equation 13.15) is solved, the
discharge at any link end can be determined from
Equations 13.12 or 13.13. For pipe/channel links,
where flowdirection is from the node to a link, the
discharge thus obtained becomes the boundary
condition at that channel end. Otherwise, the
water level calculated from the energy conserva-
tion equation or from the free-outflow condition
becomes the boundary condition.
Finally, these boundary conditions are added
to the system of Equations 13.8 and 13.9, which
is rearranged to a tridiagonal form and solved
using a double-sweep technique and the Newton-
Raphson method. This system of equations forms
one global iterationwithin one time step, and since
linearization is applied at several stages within the
process, typically two to three global iterations
are sufficient to meet the convergence criteria.
Pressurized flow is simulated by the open-slot
method and supercritical flow is treated using
Havno's approximations. These techniques are
also used in standard commercial packages such
as InfoWorks CS, MOUSE and others.
l ¼u rW= p u
j
j
node
Fig. 13.14
Network elements. U, upstream network
node; D, downstream network node; i, N, index and
number of computational cross-section, respectively.
conservation equation into Equations 13.10
and 13.11 transforms those into:
Q
j
þ1
1
f
0
1
Z
j
þ1
g
0
1
Z
j
þ1
D
¼
þ
þh
1
ð
13
:
12
Þ
v
Q
j
þ1
N
f
0
N
Z
j
þ1
g
0
N
Z
j
þ1
þh
0
N
¼
þ
ð
13
:
13
Þ
v
D
where indices U and D denote network nodes at
upstream and downstream channel ends, respec-
tively (Fig. 13.14). After possibly some lineariza-
tion, relationships for other link types (weirs,
pumps) can be expressed in the same form as well.
For each node, if F
Þ > 0
, Equation 13.1 is
solved by the Euler modified method:
ð
Z
j
F
j
þ
j
þ1
F
j
þ1
j
þ
D
t
2
j
Z
j
þ1
Z
j
¼
ð
13
:
14
Þ
where
j¼
righthand side of Equation 13.1. For
point-type junctions (where F
¼
0), Equation 13.1
The urban inundation model (UIM)
reduces
¼ 0
. Substitution of Equa-
tions 13.12 and 13.13 for all joining links into
Equation 13.14, and doing so for all the nodes,
leads to a system that can be written in matrix
form:
to
j
j
þ1
The UIM has been developed as a 2D non-inertia
model derived from the St-Venant equations, with
the inertial terms neglected by assuming the ac-
celeration terms of the water flow over the land
surface are relatively small compared to gravita-
tion and friction terms. The continuity and mo-
mentum equations are written as:
p z ¼ q
ð
13
:
15
Þ
where
Z¼
a vector containing unknown water
levels at network nodes and
p,q¼
coefficient ma-
trices. As node matrix
p
can be rather large but
with a large number of zero terms, it may be
banded into a row-indexed sparse storage form.
Then the system (Equation 13.15) is solved by the
conjugate gradient method considering the func-
tion:
d
ud
q
vd
q
q
t
þ
q
x
þ
q
y
¼
q
ð
13
:
17
Þ
q
n
2
u
p
q
ð
d
þ
z
Þ
u
2
þ
v
2
þ
¼ 0
ð
13
:
18
Þ
x
d
q
n
2
v
p
q
ð
d
þ
z
Þ
q
u
2
þ
v
2
1
2
p Zq
2
þ
¼ 0
ð
13
:
19
Þ
WðZÞ
ð
13
:
16
Þ
j
j
y
d
4
=
3
