Geoscience Reference
In-Depth Information
If we ensemble average both sides of the linear expansion
(16.85)
we find
u
i
(
x
)
0
u
i
(
x
,t)
˜
˜
.
(16.88)
This says that to first order in the fluctuating velocity, the mean flow at the mea-
surement point is simply that which exists there when the flow is nonturbulent.
Differences do appear at second order.
If we subtract
Eq. (16.88)
from
Eq. (16.85)
we find an equation for the distorted
turbulent velocity:
u
i
(
x
,t)
a
ij
(
x
)u
j
(t)
=
a
i
1
(
x
)u
1
(t)
+
a
i
2
(
x
)u
2
(t)
+
a
i
3
(
x
)u
3
(t).
(16.89)
This says that probe-induced flow distortion can cause attenuation or amplification
in a measured turbulence signal as well as crosstalk (signal mixing).
Symmetries of the turbulence probe can simplify the problem, however. For
0 on that centerplane
(Problem 16.12)
. Similarly, symmetry
about a vertical (
x
1
,x
3
) centerplane makes
a
12
and
a
32
vanish there.
The effects of probe-induced flow distortion ca
n be
particularly serious for tur-
bulent fluxes. Say, for example, we are measuring
u
1
u
3
on a tower in a horizontally
homogeneous surface layer. We'll assume a typical situation: lateral symmetry
of the probe about a vertical centerplane, but no corresponding vertical symmetry
about a horizontal centerplane. Thus when the mean wind vector lies in this vertical
centerplane
a
12
and
a
32
are zero. But the asymmetry about the horizontal midplane
induces non-zero
a
13
and
a
33
so that
u
1
(
x
,t)
=
a
11
(
x
)u
1
(t)
+
a
13
(
x
)u
3
(t),
u
3
(
x
,t)
=
a
13
(
x
)u
1
(t)
+
a
33
(
x
)u
3
(t).
(16.90)
Thus the measured stress is
u
1
u
3
(
x
)
a
11
a
13
u
1
+
a
13
a
33
u
3
.
=
(a
11
a
33
+
a
13
a
31
) u
1
u
3
+
(16.91)
Figure 16.6
shows the measured stress upstream of a right circular cylinder as
calculated with
Eq. (16.91)
. Here the symmetry about the horizontal midplane
causes
a
13
to vanish at
x
3
=
0 and minimizes the errors there. Elsewhere the errors
can be quite serious, even several diameters upstream. Hogstrom's (
1982
) results
confirm this.
Wyngaard
(
1987
) showed that the errors in vertical scalar fluxes produced by this
distortion of the turbulent velocity field can also be minimized by designing the
