Geoscience Reference
In-Depth Information
5.2
HYDRAULIC THEORY: SHALLOW WATER EQUATIONS
In most situations of free surface flow encountered in hydrology, it is possible to make
certain simplifications. The main assumption is that the path- or streamlines are only
slightly curved so that the accelerations normal to the direction of mean flow are neg-
ligible. This means that the pressure distribution may be taken as hydrostatic along the
z
-direction, i.e. normally to the bottom, or
∂
p
z
+
γ
cos
θ
=
0
(5.5)
∂
where
θ
is the slope angle of the bottom and
γ
(
≡
ρ
g
) is the specific weight of the water.
With
z
=
0 at the bottom surface, the integral of (5.5) is
p
=
γ
cos
θ
(
h
−
z
)
(5.6)
where as before
h
=
h
(
x
,
t
)
=
(
z
s
−
z
b
) is the water depth measured normally to the
bottom. If the bed slope angle
is constant in the main direction of flow, Equation (5.6)
yields immediately upon differentiation
∂
θ
p
θ
∂
h
x
=
γ
cos
(5.7)
∂
∂
x
As this pressure gradient is not a function of
z
, the corresponding acceleration of the
w
ater particles is independent of
z
as well. Therefore the velocity parallel to the bottom
u
preserve
s
its
de
pendence on
z
, independently of
x
and
t
. Accordingly, it is permissible
to replace
u
=
u
(
x
,
z
,
t
) by its average over
z
, namely
V
=
V
(
x
,
t
) defined by
h
1
h
V
=
udz
(5.8)
0
These two simplifications, namely the hydrostatic pressure distribution and the
assumption of an average velocity
V
, constitute the basis of the so-called hydraulic
theory of free surface flow; as will become clear below, it reduces the two-dimensional
problem to a one-dimensional problem. The theory is usually referred to as shallow
water theory or the theory of long waves. It consists of reducing the continuity and the
momentum or Reynolds equations to the shallow water equations. This will be shown
in what follows.
5.2.1
Equation of continuity
The equation of continuity of an incompressible fluid is given by Equation (1.9). For
turbulent flow this can be equally applied to the mean and to the turbulent velocity
components; if there is also a source inflow
φ
l
at the point under consideration, the
equation of continuity for the mean velocity components becomes in the case of two-
dimensional motion
∂
u
x
+
∂w
z
−
φ
l
=
0
(5.9)
∂
∂
