Geoscience Reference
In-Depth Information
Equation (16.58)
for the measured difference signal, generalized to a sensor with a
nonzero time constant, becomes
e
i
κ
·
x
e
i
κ
·
r
1
−
θ
m
(
x
,
r
,τ
w
,U
1
)
=
iκ
1
τ
w
U
1
)
dZ(
κ
)
(
1
−
e
i
κ
·
x
dZ
d
(
κ
,
r
,τ
w
,U
1
),
=
(16.68)
with
dZ
d
the Fourier-Stieltjes components of the measured difference signal.
Likewise,
Eq. (16.59)
becomes
2 [1
−
cos
(
κ
·
r
)
]
φ
d
(
κ
)
=
φ(
κ
),
(16.69)
(κ
1
τ
w
U
1
)
2
1
+
and
Eq. (16.60)
, rewritten for the measured difference variance, becomes
φ
d
(
κ
)d
κ
=
2 [1
−
cos
(
κ
·
r
)
]
(θ
m
)
2
=
φ(
κ
)d
κ
.
(16.70)
(κ
1
τ
w
U
1
)
2
1
+
Experimentalists can be reluctant to separate two sensors only in the streamwise
direction because this can put one sensor in thewake of the other. Thus, in ge
neral the
separation vector
r
in
Eq. (16.70)
has a lateral component, whichmeans that
(θ
m
)
2
cannot be reduced to an integral of a transfer function times the one-dimensional
spectrum.
One can simulate a sensor separation
r
in the streamwise direction by using a
single sensor sampled at two times separated by
r/U
1
:
c(
x
+
r, t )
−
c(
x
,t)
c(
x
,t
−
r/U
1
)
−
c(
x
,t).
(16.71)
In this case
Eq. (16.70)
becomes
∞
∞
2 [1
−
cos
(κ
1
r)
]
2 [1
−
cos
(κ
1
r)
]
(κ
1
τ
w
U
1
)
2
F
1
(κ
1
)dκ
1
.
(16.72)
If the separation
r
is in the inertial range of scales, the one-dimensional spectrum
F
1
falls as
κ
−
5
/
3
(c
m
)
2
=
φ(
κ
)d
κ
=
(κ
1
τ
w
U
1
)
2
1
+
1
+
0
−∞
in the region that contributes to the integral
(16.72)
. The transfer
1
function 2 [1
−
peak is at
κ
1
=
π/r
, the second at 3
π/r
. Aminimum requirement is that the sensor
time constant
τ
w
be small enough that the transfer function 1
(κ
1
τ
w
U
1
)
2
is unity
+
in the first pass band of the difference filter. Thus, we require
(κ
1
τ
w
U
1
)
2
1when
κ
1
r
=
2
π,
(16.73)
