Geoscience Reference
In-Depth Information
are required to solve for the two variables. These equations were first formulated by the Barre
de St. Venant in terms of a mass balance and a balance of momentum. Using the cross-sectional
area of the flow to represent storage per unit length of channel, then as shown in Box 2.3, the
mass balance is given by
∂A
∂t
=−
∂Q
∂x
+
q
∂vA
∂x
+
=−
q
A
∂v
v
∂A
=−
∂x
−
∂x
+
q
where
x
is distance downslope or downstream and
q
[L
2
T
−1
]is the net lateral inflow rate per
unit length of channel.
A second equation linking
v
and
A
can be developed from the momentum balance of the
flow. The control volume approach of Box 2.3 can also be used to derive the momentum
balance which may be expressed in words as:
spatial change
loss in
temporal change
spatial change
in hydrostatic
+ potential
−
friction =
in local
+ in momentum
pressure
energy
loss
momentum
flux
or
∂Ap
∂x
+
∂Av
∂t
∂Av
2
∂x
−
gAS
o
−
P
=
+
(B5.6.1)
where
g
[LT
−2
] is the acceleration due to gravity,
S
o
[
] is the channel bed slope,
P
is the
wetted perimeter of the channel [L],
is the boundary shear stress [ML
−1
T
−2
], and
p
is the
local hydrostatic pressure at the bed [ML
−1
T
−2
]. We can substitute for
p
and
as
p
−
=
gh
(B5.6.2)
and
=
gR
h
S
f
(B5.6.3)
where
h
[L] is an average depth of flow,
S
f
[
] is the friction slope which is a function of the
roughness of the surface or channel, and
R
h
[L] is the hydraulic radius of the flow (
−
A/P
).
With these substitutions, and dividing through by
under the assumption that the fluid is
incompressible, the momentum equation may be rearranged in the form
∂Av
∂t
+
=
∂Av
2
∂x
∂Agh
∂x
+
=
gA
(
S
o
−
S
f
)
(B5.6.4)
The friction slope is usually calculated by assuming that the rate of loss of energy is ap-
proximately the same as it would be under uniform flow conditions at the same water surface
slope so that one of the
uniform flow
equations holds locally in space and time. The Manning
equation (5.7) is often used, but an alternative is the Darcy-Weisbach uniform flow equation
2
g
f
S
f
R
h
0
.
5
v
=
(B5.6.5)
where
f
is the Darcy-Weisbach resistance coefficient, so that an alternative form of the mo-
mentum equation can then be written as:
∂Av
2
∂x
∂Av
∂t
+
∂Agh
∂x
gP
f
2
g
v
2
+
=
gAS
o
−
(B5.6.6)
