Geoscience Reference
In-Depth Information
be established:
Q
=
f
(
z
s
,
up
)
(5.106)
The first-order Taylor series expansion of Eq. (5.106) reads
−
∂
f
f
∗
−
Q
∗
δ
Q
z
s
,
up
δ
h
up
=
(5.107)
∂
For the downstream control flow, the following relation of orifice-like flow is
usually used:
A
2
g
(
z
s
,
up
−
z
s
,
down
)
Q
=
(5.108)
K
L
where
K
L
is the coefficient of energy loss at the hydraulic structure.
Because it cannot handle the situation of
z
s
,
up
≤
z
s
,
down
, Eq. (5.108) is reformu-
lated as
K
L
2
g
Q
|
Q
|
z
s
,
up
−
z
s
,
down
=
(5.109)
A
2
which is then expanded as
Q
∗
|
Q
∗
|
A
∗
2
K
L
2
g
z
s
,
up
+
z
s
,
down
+
δ
h
up
−
δ
h
down
=−
(5.110)
Note that the stage and discharge increments originated from the term on the right-
hand side of Eq. (5.109) are ignored in Eq. (5.110). They may be included for the sake
of completion.
A dam structure may have various flow passage facilities, such as spillways, sluice
gates, and power generators. The flows through these facilities may be free overflow
and/or under control. Thus, the stage-discharge rating relation for a dam structure
may be Eqs. (5.106), (5.108), or a combination of them.
In addition, the water stage or flowdischarge measured at a damand other structures
can be used as the internal condition. If a time series of the water stage is known:
z
s
,
up
=
z
s
(
t
)
(5.111)
the stage increment at the upstream point is determined by
z
n
+
1
s
,
up
z
s
,
up
δ
h
up
=
−
(5.112)
If a time series of the flow discharge is known:
Q
up
=
Q
(
t
)
(5.113)
