12 Computational Modelling in Hydraulic and Coastal Engineering
water elevation z N = N·Δz (Figure 2.1). The plot of V(z i ) versus z i is known
as the rating curve.
V z
S z
S z z
N
i
i
i
i N
( )
[ ( ) ( )]
=
+
−
=
=
∑
1
2
1
1
∆
(2.2)
The formulation of the mathematical model describing the volumetric
changes of reservoir water as a result of inflow and outflow discharges can
be derived by using the continuity equation and a stage–discharge relationship for the design outflow. The continuity equation implies that for an
infinitesimal time (Δt) the difference between the inflowing (Q in Δt) and the
outflowing (Q out Δt) volume of water will account for any accumulation or
depletion of the reservoir water storage. Thus,
S(z)Δz = Q in Δt − Q out Δt
(2.3)
By assuming that outflow is occurring through an orifice of surface area A,
then according to Torricelli’s formula, the outflow depends on the hydraulic
head (z) measured from the center of the orifice to the water surface,
Q
C A gz
out
orifice
o
=
2
(2.4)
where C o is the discharge coefficient with an average value of 0.7.
If outflow is occurring over a weir, then
Q
C B z z
out
weir
w
w
=
−
(
)
.
1 5
for z > z w , and Q out
weir
= 0 for z < z w
(2.5)
z
i + 1
N
i
Δz
S
S i+1
S i
Maximum water surface elevation z N
Reference datum
Reservoir
Dam
z N = H o
0
Inflow Q in
Outflow Q out
Bed
z i
Figure 2.1 Schematic representation of reservoir operation.
water elevation z N = N·Δz (Figure 2.1). The plot of V(z i ) versus z i is known
as the rating curve.
V z
S z
S z z
N
i
i
i
i N
( )
[ ( ) ( )]
=
+
−
=
=
∑
1
2
1
1
∆
(2.2)
The formulation of the mathematical model describing the volumetric
changes of reservoir water as a result of inflow and outflow discharges can
be derived by using the continuity equation and a stage–discharge relationship for the design outflow. The continuity equation implies that for an
infinitesimal time (Δt) the difference between the inflowing (Q in Δt) and the
outflowing (Q out Δt) volume of water will account for any accumulation or
depletion of the reservoir water storage. Thus,
S(z)Δz = Q in Δt − Q out Δt
(2.3)
By assuming that outflow is occurring through an orifice of surface area A,
then according to Torricelli’s formula, the outflow depends on the hydraulic
head (z) measured from the center of the orifice to the water surface,
Q
C A gz
out
orifice
o
=
2
(2.4)
where C o is the discharge coefficient with an average value of 0.7.
If outflow is occurring over a weir, then
Q
C B z z
out
weir
w
w
=
−
(
)
.
1 5
for z > z w , and Q out
weir
= 0 for z < z w
(2.5)
z
i + 1
N
i
Δz
S
S i+1
S i
Maximum water surface elevation z N
Reference datum
Reservoir
Dam
z N = H o
0
Inflow Q in
Outflow Q out
Bed
z i
Figure 2.1 Schematic representation of reservoir operation.