Geoscience Reference
In-Depth Information
After dividing side by side, the basic relationship will be obtained in the
following form:
2
4
3
5
H
þ
ln
H
3
2
u
x
ð
z
Þ¼u
0
1
þ
e
3
=
2
(10)
U
ln
The following symbols were used in the above equations:
e
¼
the base of
Naperian logarithm,
U
u
0
¼
depth averaged water velocity. It is to be stressed that the von Karmann constant is
still a matter of research (Nowh
1989
).
Equation (
10
) gives good results until
z
/
H
¼
parameter of the model,
u
x
ð
z
Þ¼
water velocity,
>
0.02. Moreover, it can be noticed
that for
z
/
H
¼
0.30701
,
there is
u
x
¼ u
0
. An example of the curve according to (
10
)
is presented in Fig.
1
.
Formula (
10
) is especially suitable for experimental data analyses, because if
there is a set of field measurement data
f
u
i
;
z
i
H
=
g
, it is possible to calculate constant
values of
U
and
u
0
in this formula, using the least square method.
P
Y
i
P
u
i
P
u
i
YðÞ
2
P
u
ðÞ
P
Y
i
P
YðÞ
P
u
i
YðÞ
½
u
0
¼
(11)
P
Y
i
P
u
i
P
u
i
YðÞ
¼
2
½
U
e
3
=
2
P
YðÞ
P
u
ðÞ
P
uðÞ
P
u
ln
(12)
2
YðÞ
where
3
2
z
H
þ
ln
z
Y
ð
z
Þ¼
H
;
Y
i
¼
YzðÞ
(13)
Fig. 1 Plot of curves
u
x
z
ð
;
U
Þ
for different values of
U
