Geoscience Reference
In-Depth Information
By using Leibniz's rule (see Appendix), one can write the integral of the first term on
the left of Equation (5.15) as
h
∂
u
∂
∂
∂
|
z
=
h
∂
h
dz
=
(
Vh
)
−
u
(5.16)
t
t
∂
t
0
In the same way, by also using surface condition (5.4) and the assumption that
h
u
2
dz
V
2
h
=
(5.17)
0
one obtains the integral of the second term on the left of (5.15) as
h
(
u
2
)
∂
∂
∂
∂
|
z
=
h
∂
h
x
(
V
2
h
)
|
z
=
h
+
dz
=
−
w
u
u
(5.18)
x
∂
t
0
O
n account of the definition of
V
in (5.8), the assumption of (5.17) can be valid only if
u
is uniform, that is, constant along
z
. With a no-slip condition at
z
0, this is never the
case; nevertheless, in turbulent open channel flow, which is well mixed in the vertical,
it is usually an acceptable approximation. However, for laminar and transitional flows,
a correction coefficient (often associated with the name of Boussinesq; see Bakhmeteff,
1941), namely,
=
h
V
)
2
dz
β
c
=
(
u
/
/
h
(5.19)
0
may have to be applied to the first term on the right-hand side of (5.18), i.e. the advective
acceleration term.
The integration of the other terms in Equation (5.15) is straightforward. The third
term on the left yields upon integration simply the product of
w
and
u
at
z
=
h
.If
it can be assumed that
θ
is small, the first term on the right can be approximated by
replacing
S
0
, which is the slope of the bottom surface, so that its
integral becomes (
gS
0
h
). Similarly, the pressure gradient in the second term on the right
can be replaced by the water depth gradient on account of (5.7), provided the slope angle
θ
−
sin
θ
by
−
tan
θ
=
is small enough.
In the hydraulic approach to free surface flow, the integral of the last two terms
of Equation (5.15) is usually expressed in terms of the friction slope
S
f
, as a closure
parameterization to account for the effects of the viscosity and the turbulence. For the
present case of two-dimensional flow, this is
dz
∂
z
2
dz
h
h
2
u
2
u
(
u
u
)
∂
w
u
)
∂
x
2
+
∂
∂
+
∂
(
ν
−
=−
ghS
f
(5.20)
∂
∂
x
z
0
0
