Civil Engineering Reference
In-Depth Information
As indicated by Ingard (1953), the exact distribution of the velocity in the aperture located
at
x
3
=
0 is not known
apriori
. The simplest hypothesis involves considering a uniform
amplitude U for the
x
3
component of velocity and more elaborate models provide similar
results (Norris and Sheng 1989). The velocity distribution on the plane
x
3
=
0canbe
written
Y
(R
−
r)U
=
m,n
k
m,n
C
m,n
ωρ
0
cos
2
πmx
1
D
cos
2
πnx
2
D
(9.5)
In this equation, Y
(R
−
r)
is the unit step equal to 1 if
r<R
and0if
r>R
.
By multiplying both sides of this equation by cos
2
πmx
D
cos
2
πnx
D
and integrating over
the aperture, one obtains
v
m,n
R
2
π
D
2
4
U
cos
2
πmx
1
D
cos
2
πnx
2
D
k
m,n
C
m,n
ωρ
0
r
d
r
d
θ
=
(9.6)
0
0
where
v
m,n
=
0; and
v
0
,
0
=
1
/
4. The product cos
(
2
πmx
1
/D)
cos
(
2
πnx
2
/D)
can be rewritten (Ingard, 1953)
=
1if
m
=
0,
n
=
0;
ν
m,n
=
1
/
2if
m
=
0,
n
=
0or
m
=
0,
n
2
cos
2
πr
m
2
γ)
1
/
2
+
n
2
D
2
cos
2
π
mr
cos
θ
D
cos
2
πnr
sin
θ
D
1
=
sin
(θ
+
2
cos
2
πr
m
2
sin
(θ
−
γ)
(9.7)
1
/
2
+
n
2
D
2
1
+
where
γ
=
arctg
m/n
.
Using the relations:
cos
2
π
r
γ)
dθ
2
π
J
0
2
π
r
n
2
)
1
/
2
2
π
D
(m
2
n
2
)
1
/
2
sin
(θ
D
(m
2
+
±
=
+
(9.8)
0
=
J
1
2
π
R
n
2
)
1
/
2
R
J
0
2
π
r
n
2
)
1
/
2
rdr
RD
2
π(m
2
D
(m
2
D
(m
2
+
+
(9.9)
n
2
)
1
/
2
+
0
Equation (9.6) can be rewritten
n
2
)
1
/
2
J
1
2
π
R
n
2
)
1
/
2
D
2
4
URD
k
m,n
C
m,n
ωρ
0
D
(m
2
+
=
v
m,n
(9.10)
(m
2
+
and
C
m,n
is given by
4
v
m,n
URωρ
0
J
1
2
π
D
(m
2
+
n
2
)
1
/
2
C
m,n
=
(9.11)
D(m
2
+
n
2
)
1
/
2
k
m,n
For the case
m
=
n
=
0, Equation (9.6) yields
C
0
,
0
=
ωρ
0
Us/k
=
Z
c
sU
(9.12)
πR
2
/D
2
.
where
s
=
