Environmental Engineering Reference
In-Depth Information
The first two terms at the left-hand side, namely ( v
±
c )( ( v
±
2 c ) /∂x )
+
±
=
g ( S o
±
( v
2 c ) /∂t
S f ), represents the rate of change of ( v
2 c ) from
the view point of two observers, one moving in the x
t plane with velocity
( v
+
c ).
The path of the two imaginary observers can be plotted on the x
c ) and the other moving with a velocity ( v
t plane
and a complete solution will be obtained for any prescribed unsteady flow
situation. Only in simple cases does the process lead to explicit solutions,
but in more complex cases numerical methods may be used without any
great difficulty (see Figure 2.16).
A
t
B
C
Figure 2.16. Method of
characteristics in the x and t
plane, with an explicit solution.
x
Along the line with the direction d x/ d t
=
( v
±
c ), the expression
d x/ d t
=
( v
±
c ) holds and along the line with the direction d x/ d t
=
v
c
the
expression
d x/ d t
=
( v
±
c )
holds.
The
line
with
the
direction
d x/ d t
c ) is called the positive characteristic ( c + ) and the other
is the negative characteristic ( c ).
Integration of d x/ d t
=
( v
±
=
( v
±
c ) gives that v
+
2 c
=
constant. The equa-
tion d x/ d t
=
( v
±
c ) means that v
±
c can be treated as constant for t .If
d x/ d t
=
( v
±
c ) then u
=
v
±
2 c
+
g ( S f
S o ) t . The reciprocal value gives
d t/ d x
t space.
Thus in this specific case that two observers move with two velocities,
namely ( v
=
1 /v
±
c which defines two lines in x
2 c ) appear to remain constant.
The results are two families of curves in the x
±
2 c ), the two velocities ( v
±
t plane, which have inverse
+
slopes, namely ( v
c ) and ( v
c ). These lines are the characteristic lines.
Summary of the four characteristic equations of unsteady flow
in a rectangular channel
The method of characteristics gives two lines:
+
one line has a slope 1 / ( v
c )
the other line has a slope 1 / ( v
c )
 
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