Environmental Engineering Reference
In-Depth Information
where
ψ = ψ ¯
η ,
x
ξ 2
, ¯
η
ch c
A
¯
2
2
D At .
ξ ,
¯
ch
=
(
At
)
(
y
η )
¯
ξ ,
¯
Note that, by a polar coordinate transformation,
1
2 d
ξ
d
η =
2
π
At
.
2
2
(
At
)
(
x
ξ )
(
y
η )
D At
Therefore
1
t
t
e
e
u
(
x
,
y
,
t
)=
2
τ 0
A ψ ·
ch2
π
At
=
2
τ 0 t
ψ ·
ch
.
2
π
Finally,
u t (
x
,
y
,
0
)=
t = 0
1
1
2
t
0
t
0
τ 0 e
e
·
K
( ψ )
d
ξ
d
η +
K
( ψ )
d
ξ
d
η
2
π
A
t
D At
D At
τ 0
τ 0
t ψ ·
ch ·
1
1
2
t
t
τ 0 e
e
=
2
K
( ψ )
d
ξ
d
η +
2
2
π
At
2
π
A
D At
t = 0
A
+ ψ ·
ch
·
2
π
= ψ (
x
,
y
) ,
where we have used lim
t
0 ψ = ψ (
x
,
y
)
and lim
t
0 ch
=
1.
2. The u in Eq. (5.69)
Rewrite Eq. (5.69) into
0
1
1
A
t
t
0
τ 0 e
e
u
=
K
( ϕ )
d
ξ
d
η +
K
( ϕ )
d
ξ
d
η ,
(5.71)
4
π
A
2
π
t
D At
D At
where
ch A
2
2
2
(
At
)
(
x
ξ )
(
y
η )
K
( ϕ )=
(
ϕ ( ξ , η ) .
At
)
2
(
x
ξ )
2
(
y
η )
2
Note that
η =
K
( ϕ )
d
ξ
d
t ( ψ ·
ch
) ·
2
π
At
+ ψ ·
ch
·
2
π
A
.
(5.72)
t
C At
 
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