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the transit time of flow of mean speed
U
1
past a body of length
a
. Using turbu-
lence scaling he argued that if the “error parameter”
p
=
(u/U
1
)(a/)
1
/
3
1,
with
u
and
the turbulent velocity and length scales of the approach flow, is small
Eq. (16.97)
holds also for the low-pass-filtered scalar signal:
∂ c
d
∂t
(
x
m
,t
∂ c
∂t
(
x
0
,t),
(u/U
1
)(a/)
1
/
3
+
t)
=
p
=
1
.
(16.99)
If so, then for
p
1 the low-pass-filtered time series in the free stream and in the
distortion region differ only by a fixed time delay, so their frequency spectra are
identical below the probe frequency
f
p
.
In an aircraft application with
u
1ms
−
1
,
U
1
∼
100 m s
−
1
,sothat
u/U
1
∼
∼
10
−
2
,and
a
300 m, so that
(a/)
1
/
3
10
−
1
, for example, then the
∼
1m,
∼
∼
10
−
3
, which one expects is small enough for
Eq. (16.99)
to
hold. Thus the frequency spectra of the scalar in the region of flow distortion and
in the free stream should be the same. In tower applications the situation is less
clear because both
u/U
1
and
a/
are apt to be larger, making the error parameter
p
larger. If, for example, we measured at the flight height but from a very tall tower,
then in a convective ABL
u/U
1
could increase to 3
error parameter,
p
∼
10
−
1
, say. If the probe were
×
10
−
2
;the
error parameter is 30 times the aircraft value. If we measured from a tower but in
the surface layer, say,
(a/)
1
/
3
could be larger yet. Thus
p
could be still larger for
in-situ
measurements in the unstable surface layer.
0
.
2 m, say) then
(a/)
1
/
3
10
−
1
,making
p
smaller (
a
∼
∼
0
.
9
×
∼
3
×
16.3.2.3 Quasi-steady turbulent flow, high frequencies
Wyngaard
(
1988
) also extended his analysis to high frequencies (relative to the
probe frequency
U
1
/a
). He found that if the parameter
ua/(U
1
λ)
, with
λ
the Taylor
microscale, is small, the classical rapid-distortion analysis (
Batchelor
,
1960
;
Hunt
,
1973
) can be extended to scalars. As the flowapproaches the body along a streamline
on its centerline, the decreasing fluid velocity causes fluid particles to decrease
their streamwise separation, as in duct flow with a sharp increase in cross-sectional
area - a
rapid expansion
. This is found to amplify the streamwise wavenumber
spectrum of the scalar in the inertial range. The off-axis flow experiences a
rapid
contraction
, which attenuates the inertial-range wavenumber spectrum. However,
Taylor's hypothesis in the form
κ
1
=
2
πf/U
1
, plus the constraint that both spectra
integrate to the variance, show that the frequency spectrum is unchanged.
Questions on key concepts
16.1 Explain the essence of the process by which a conservation equation for a
turbulent field is converted to an evolution equation for its power spectral
