Civil Engineering Reference
In-Depth Information
the channel. Channel cross section can be of any shape and change along its length
with changes in liquid level, cross-sectional area of the liquid in the channel denoted
by
(
)
h
=
. The depth of the channel and basin are assumed to be small compared
with the wavelength. We write the Euler equation in the form of
x
t
¶
¶
=
¶ r¶
u
1
p
(2)
t
x
1
¶
=-
r¶
p
g
(3)
z
where
ρ
--density,
p
--pressure
g
--acceleration of free fall. Quadratic in velocity
members omitted, since the amplitude of the waves is still considered low [2].
From the second equation we have that at the free surface
(
)
z
=
h
x
,
t
(where)
p
=
p
should be satisfied:
0
(
)
p
=
p
+
ρ
g
h
−
z
(4)
0
Substituting this expression in equation (2), we obtain to determine
u
and
h
we use
the continuity equation for the case under consideration.
¶
u
¶
h
=-
g
¶
(5)
¶
t
x
Consider the volume of fluid contained between two planes of the cross-section of the
canal at a distance
dx
from each other. Per unit time through a cross-section
x
enter the
amount of fluid, equal to
( )
x
hu
. At the same time through the section
x + dx
there is
forth coming
( )
x
hu
+
. Therefore, the volume of fluid between the planes is changed to
dx
( )
¶
-=
¶
By virtue of incompressibility of the liquid is a change could occur only due to changes
in its level. Changing the volume of fluid between these planes in a unit time is equal
hu
( ) ( )
hu
hu
dx
x
+
dx
x
x
¶
¶
h
dx
t
Consequently, we can write:
( )
( )
¶
hu
¶
hu
¶
h
¶
h
dx
=-
dx
and
+ = > -¥< <¥
0,
t
0,
x
or,
(6)
¶
x
¶
t
¶
x
¶
t
Since
=
hh
where a
h
--denotes the ordinate of the free liquid surface (Figure
8.1), in a state of relative equilibrium and evolving the influence of gravity is
+
ξ
¶x
+=
¶
u
h
0
(7)
0
t
¶
x
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