Geoscience Reference
In-Depth Information
Moreover, it results from both the geometric relationships and earlier analysis of
velocity changes in the horizontal plane (see Fig.
3
) that:
t
1
t
2
¼
B
1
B
2
(21)
It is convenient to assume for further calculation (to simplify notation) that
the
y
axis is oriented from the point at the river bank toward the meridian.
Then the velocity description for areas (1) and (2) is obtained in the following
form:
For area (1)
2
4
3
5
2
4
3
5
y
1
ln
y
B
1
3
2
B
1
þ
z
ln
H
3
2
H
þ
e
3
=
2
e
3
=
2
u
x
ð
y
;
z
Þ¼u
01
1
þ
1
þ
(22)
U
3
U
1
ln
ln
For area (2)
2
4
3
5
2
4
3
5
y
2
ln
y
B
2
3
2
B
2
þ
3
z
ln
H
2
H
þ
u
x
ð
y
;
z
Þ¼u
02
1
þ
e
3
=
2
1
þ
e
3
=
2
(23)
U
2
U
3
ln
ln
and where
u
01
and
u
02
denote the averaged velocities in areas (1) and (2),
respectively.
At the meridian, there must be
u
x
y
1
;
ðÞ¼u
x
y
2
;
z
ð
z
Þ¼u
max
ð
z
Þ
for
y
1
¼
B
1
or
y
2
¼
B
2
. Hence, there is an additional equation for calculating the relationship
between
u
01
and
u
02
. The obtained relation reads:
2
4
3
5
¼ u
02
2
4
3
5
1
1
e
3
=
2
e
3
=
2
u
01
1
þ
1
þ
(24)
U
1
U
2
2
ln
2
ln
Furthermore,
Q
¼ u
0
B
H
¼ u
01
B
1
H
þ u
02
B
2
H
(25)
In the further analysis, it is convenient to put:
1
e
i
¼
;
i
¼
;
;
1
2
3
U
i
e
3
=
2
(26)
2
ln
which gives:
1
þ e
1
u
02
¼ u
01
(27)
1
þ e
2
