Geoscience Reference
In-Depth Information
With (5.16), (5.18) and (5.20), the integral of (5.15) becomes finally
hg
∂
S
0
∂
∂
∂
∂
h
x
(
V
2
h
)
(
Vh
)
+
+
x
+
S
f
−
=
0
(5.21)
t
∂
which is the momentum equation of the hydraulic theory of free surface flow. This result
is often written in an alternative form, which is obtained by subtraction of continuity
(5.13) multiplied by
V
, and by subsequent division by
h
; thus the momentum equation
is often written as follows
∂
g
∂
S
0
V
∂
V
∂
V
h
iV
h
=
t
+
x
+
x
+
S
f
−
+
0
(5.22)
∂
∂
Equations (5.13 and (5.22) are known as the shallow water equations; a simpler
version was first presented by Saint Venant in the nineteenth century, so that they are
often named after him. To recapitulate briefly, the shallow water equations are based
on the following assumptions. (i) The pressure distribution in the water is hydrostatic
leading to Equation (5.5); (ii) the bed slope
S
0
is constant and small, which leads from
Equation (5.6) to (5.7), and allows replacement of sin
S
0
; (iii) the effects
of viscous and turbulent stresses can be parameterized and combined in a friction slope
S
f
, defined in Equation (5.20); (iv) the velocity is not very dependent on
z
, so that
θ
by tan
θ
=−
β
c
in
Equation (5.19) can be taken as unity.
Example 5.1. Steady flow
The meaning of the different terms in Equation (5.22) can be illustrated by considering
steady flow conditions in the absence of lateral inflow. Thus, after putting both
∂
V
/∂
t
and
i
equal to zero, (5.22) can be readily integrated over a flow distance
δ
x
to yield
V
1
V
2
2
+
gh
1
+
gS
0
δ
x
=
2
+
gh
2
+
gS
f
δ
x
(5.23)
in which the subscripts 1 and 2 refer to the entrance and exit of the flow section
x
.
Figure 5.2 shows the balance of the left- and right-hand sides of this equation. Recall
that the bed slope is sufficiently small, so that
x
, the coordinate along the bed, can be
represented as horizontal in the figure. Because the integration of a force (or rate of
momentum change) over distance yields work, the terms of Equation (5.23) may be
considered as different forms of energy. In open channel hydraulics the energy per unit
weight with respect to the channel bottom is called the
specific energy
; in the present
notation this is [
h
δ
V
2
(2
g
)]. As shown in Figure 5.2, its elevation defines the
energy
grade line
(EGL); the friction slope
S
f
is the slope of the energy grade line. The quantity
[
z
+
/
g
)] in any cross section defines the
hydraulic grade line
(HGL); because this
is equal to the water depth
h
, the HGL coincides with the water surface.
+
p
/
(
ρ
Equations (5.13) and (5.22) were derived for two-dimensional flow, i.e. an infinitely
wide channel. It can, however, readily be shown that for a channel with finite cross
section of arbitrary shape, but wide enough so that the flow is approximately still
