Geoscience Reference
In-Depth Information
If the sensor path vector
L
is oriented in the
x
1
(streamwise) direction the path-
averaging transfer function can be taken out of this integral. In order to avoid flow
distortion this orientation is not generally used, however, so typically the integral
in
(16.54
) must be carried out over
κ
2
and
κ
3
. This is usually done numerically.
If the sensor averages over a rectangular area
A
whose sides are
x
1
and
x
2
,
the spectral relation is
φ(
κ
)
sin
2
(κ
1
x
1
/
2
)
(κ
1
x
1
/
2
)
2
sin
2
(κ
2
x
2
/
2
)
(κ
2
x
2
/
2
)
2
φ
m
(
κ
,A)
=
.
(16.55)
If the sensor averages over a volume
V
whose sides are
x
1
,
x
2
,
x
3
,the
spectral relation is
φ(
κ
)
sin
2
(κ
1
x
1
/
2
)
(κ
1
x
1
/
2
)
2
sin
2
(κ
2
x
2
/
2
)
(κ
2
x
2
/
2
)
2
sin
2
(κ
3
x
3
/
2
)
(κ
3
x
3
/
2
)
2
φ
m
(
κ
,V)
=
.
(16.56)
In general the effect of spatial averaging on other statistics is more difficult (or
perhaps not possible) to determine analytically.
16.2.2 Response of scalar structure-function sensors
In the same way we can analyze the response of a “structure-function” sensor.
Its output is the difference of turbulent quantity at two points separated in space.
When the separation distance
r
falls in the inertial range of scales such two-point
difference variances can be used to infer dissipation rates
(Chapter 7)
.
Again suppressing the dependence on time, we write
e
i
κ
·
(
x
+
r
)
dZ(
κ
),
e
i
κ
·
x
dZ(
κ
).
c(
x
˜
+
r
)
=
c(
x
)
˜
=
(16.57)
The difference signal is
e
i
κ
·
x
e
i
κ
·
r
1
dZ(
κ
)
c(
x
,
r
)
˜
=˜
c(
x
+
r
)
−˜
c(
x
)
=
−
e
i
κ
·
x
dZ
d
(
κ
,
r
).
=
(16.58)
From
(16.58
) the spectra of the scalar difference and the scalar are related by
(e
i
κ
·
r
1
)(e
−
i
κ
·
r
dZ
d
(
κ
)dZ
d
(
κ
)
1
) dZ(
κ
)dZ
∗
(
κ
)
φ
d
(
κ
)d
κ
=
=
−
−
=
2 [1
−
cos
(
κ
·
r
)
]
φ(
κ
)d
κ
.
(16.59)