Geoscience Reference
In-Depth Information
1
Fig. 5.3
Velocity profile for
steady laminar free
surface flow on a
sloping plane. The
velocity
u
h
is the
maximal value at the
water surface where
z
=
h
.
0.8
z/h
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
u/u
h
Re
500 can be taken as a typical value. The case of rectangular channels
with finite width has been studied by Woerner
et al.
(1968).
The effect of rainfall impact at the free surface on the friction slope was considered
by Yoon and Wenzel (1971). Their experimental results over smooth surfaces can be
summarized by a simple formula proposed in Brutsaert (1972),
≡
(
Vh
/ν
)
=
gS
f
h
2
=
ν
3
P
e
<
,
S
0
≤
.
V
for Re
800
0
03
(5.33)
cS
f
+
where
P
is the rainfall intensity in cm h
−
1
and
c
,
d
,
e
are constants, whose values were
estimated to be
c
=
5
.
36
,
d
=
0
.
16 and
e
=
0
.
36. With additional data Shen and Li
(1973) subsequently concluded that
c
=
2
.
32
,
d
=
0 and
e
=
0
.
40.
5.3.2
Turbulent flow
For Reynolds numbers larger than 1000, free surface flow in nature may generally be
considered fully turbulent and rough, with little or no effect from the impact of lateral
inflow. Unlike laminar flow, however, even when it is steady and uniform, there is no
exact solution available for turbulent flow. As was the case for the analogous turbulence
closure problem in the atmosphere discussed in Chapter 2, the main difficulty here is
the presence of the second moments of the velocity fluctuations, that is the Reynolds
stress, in Equations (5.14) and (5.29). At present, the only practical way to eliminate (or
determine) this unknown shear stress
τ
zx
is either to invoke similarity theory or to rely
entirely on empirical results.
Similarity for two-dimensional turbulent flow
Uniform two-dimensional turbulent flow can be considered as a fully developed boundary
layer. For rough flow conditions, which are usually the case for flow over a natural land-
surface, the velocity profile in the inner region of the neutral boundary layer is given by
Equations (2.40) or (2.41). Strictly speaking, one can apply this logarithmic profile only in
