Geoscience Reference
In-Depth Information
Integration over
z
produces
h
h
∂
u
]
0
−
x
dz
+
[
w
φ
l
dz
=
0
(5.10)
∂
0
0
and insertion of condition (5.4) for a solid bottom surface
h
h
∂
u
|
z
=
h
∂
h
x
+
∂
h
x
dz
+
u
t
−
φ
l
dz
=
0
(5.11)
∂
∂
∂
0
0
By virtue of Leibniz's rule (see Appendix) for the differentiation of an integral, the first
term can be rewritten as
h
∂
∂
|
z
=
h
∂
h
udz
−
u
(5.12)
x
∂
x
0
The equation of continuity becomes finally
∂
h
∂
∂
t
+
(
Vh
)
−
i
=
0
(5.13)
∂
x
in which
i
is the net lateral inflow per unit width of flow, which results from the integration
of
φ
l
in (5.11). Equation (5.13) was probably first derived by Dupuit (1863; p. 149) for
i
=
0.
5.2.2
Conservation of momentum
The conservation of momentum at a point in a moving Newtonian fluid is given by the
Navier-Stokes equation. When the flow is turbulent, this is conveniently transformed
into the Reynolds equation for the mean quantities. The Reynolds equation can be readily
obtained from Equation (1.12), by replacing each of the dependent variables by the sum
of its mean and fluctuation, both in the turbulence sense, and by subsequently applying
the time-averaging operation over a suitable time period. For the two-dimensional case
of incompressible flow under consideration, and with a source inflow
φ
l
, the component
of Equation (1.12) parallel to the bottom surface can be written as follows
∂
u
∂
x
+
φ
l
+
w
∂
∂
u
u
u
1
ρ
p
2
u
t
+
z
=−
θ
−
x
+
ν
∇
−
∇·
v
)
u
g
sin
(
(5.14)
∂
∂
∂
∂
in which
v
=
(
u
i
v
j
+
w
k
) is the turbulent fluctuation in the velocity vector
v
+
=
v
). Observe that (5.14), without the last two terms on the right-hand side, is in
the form of Euler's equation (1.11); these two terms represent respectively the stresses
due to viscosity and the Reynolds stresses due to the turbulence. To obtain a momentum
equation in terms of the average velocity
V
defined in (5.8), it is necessary to integrate
(5.14) over
z
, as follows. For convenience, first, the zero quantity, consisting of (5.9)
multiplied by
u
, is added to (5.14) to obtain
∂
(
v
+
x
u
2
+
u
∂
∂
∂
∂
1
ρ
∂
p
2
u
t
+
(
w
u
)
=−
g
sin
θ
−
x
+
ν
∇
−
(
∇·
v
)
u
(5.15)
∂
z
∂
