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 1 x 1 / 2 )
1 x 1 / 2 ) 2
sin 2 2 x 2 / 2 )
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 1 x 1 / 2 )
1 x 1 / 2 ) 2
sin 2 2 x 2 / 2 )
2 x 2 / 2 ) 2
sin 2 3 x 3 / 2 )
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)
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