Geoscience Reference
In-Depth Information
the lower 10% to 20% of the boundary layer thickness; in the outer reaches of the boundary
layer, some type of velocity defect law is more appropriate. However, in two-dimensional
free surface flow this defect law (see, for example, Keulegan, 1938) is often assumed to be
ln
z
−
d
0
h
−
d
0
u
k
u
h
−
u
=−
(5.34)
which implies that Equation (2.40) can be used over the whole depth of the flow. This
is apparently only an approximation, but in laboratory type turbulent boundary layers the
difference between Equations (2.40) or (5.34) and observed velocity profiles in the outer
region is quite small (see, for example, Hinze, 1959, p. 473; Monin and Yaglom, 1971,
pp. 300-301, 315-317; Kisisel
et al.
, 1973). Thus, in light of (2.41) and (5.30), the mean
velocity can be expressed roughly (for
h
z
0
)as
ln
h
−
d
0
z
0
1
(
gS
f
)
1
/
2
(
h
−
d
0
)
kh
1
/
2
V
=
−
(5.35)
or (when also
h
d
0
),
(
gS
f
h
)
1
/
2
k
V
=
[ln (
h
/
z
0
)
−
1]
(5.36)
As an aside, it should be mentioned that one possible problem with the application of
(2.40) to open channel flow is that
z
0
may be a function of Froude number Fr
=
V
/
(
gh
)
1
/
2
(Iwagaki, 1954; Chow, 1959); this means that the structure of the turbulent boundary layer
of water flowing down a slope may also be affected by gravity,
g
, beside the variables that
were considered in the derivation of Equation (2.40). A second potential difficulty is the
effect of the raindrop impact. For instance, Kisisel
et al.
(1973) showed that for shallow
flows with
h
values around 15 mm and a high rainfall intensity of
P
=
125 mm h
−
1
, the
measured velocity profiles (
u
h
−
u
) were logarithmic, but the resulting
V
values would be
only about half the magnitude predicted by (5.36). For less shallow flows, however, the
effect of rainfall impact is likely to be small.
For certain applications it has on occasion been found convenient to express the turbulent
velocity profile by a simple power function of height, instead of by (2.40). Among the more
recent forms, for
z
d
0
, the following has been used
z
z
0
m
u
=
C
p
u
∗
(5.37)
in which
C
p
and
m
are constants. The use of power functions to describe wind speed profiles
in the lower atmosphere goes back at least to the work of Stevenson in the 1870s (see
Brutsaert, 1982, for a review). An equation similar to (5.37) was implicit in the work of
Prandtl and Tollmien (1924; see also Brutsaert, 1993), and it has subsequently been applied
by many in the solution of various turbulent transport problems. The parameters
C
p
and
m
may be determined by fitting Equation (5.37) to the more accurate (2.41) over the range of
heights
z
that are of interest; values of
m
tend to lie in the range from 1
/
6to1
/
8, with a
typical value of 1
/
7, and
C
p
is of the order of
m
−
1
. Integration of (5.37) according to (5.8)
and substitution of the friction velocity
u
∗
(see (2.32)) by (5.30) yields the average velocity
C
p
g
1
/
2
(
m
S
1
/
2
f
h
m
+
1
/
2
V
=
(5.38)
+
1)
z
0
A derivation of Equation (5.38), somewhat different from the present one, was first published
by Keulegan (1938). The main point of interest is that (5.38) can serve as a theoretical basis
for some of the empirical equations for
S
f
that will be reviewed next.
