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size was 17 cm (1600 pixels) alongtank by 12 cm (1186 pixels) in the vertical for the first
series of measurements. In the second set of measurements, the camera was set farther back
and the image size/pixel resolution was correspondingly larger. Velocities were computed
using an adaptive correlation technique in which the initial interrogation area was 256
×
256
pixels and in three successive refinement steps the final horizontal resolution was 1
2mm.
Based on the image pixel size and elapsed time between images, the uncertainty of each
velocity observation was
.
s(
Hyun
et al.
,
2003
).
Fifteen wave periods were recorded for each of the wave trains whose wave amplitudes
a
0
ranged from 6mm through 32
±
0
.
03m
/
5mm (at this amplitude waves started to break at and
before the measurement point, and therefore this measurement was not taken into consid-
eration here). Wave amplitudes were recorded at the measurement location using a digital
laser elevation gauge.
The PIV system provided spatial distributions of the
u
(horizontal, parallel to the tank's
axis) and
.
(vertical) components of the two-dimensional velocity, and thus avoided the
usual problem of converting the frequency velocity spectra into wavenumber spectra
P
w
)
(see
Section 5.3.4
and
eq. 5.66
), in the oscillatory wave flow where mean velocity is close
to zero and therefore there is no steady advective velocity to scale turbulence frequency
(
k
ω
into turbulence wavenumber
k
(
Lumley & Terray
,
1983
;
Agrawal
et al.
,
1992
). If isotropic
turbulence is generated, the velocity spectra, at the scales where its intensity dominates
over the orbital oscillations, are expected to exhibit a
k
−
5
/
3
Kolmogorov interval at small
scales (high wavenumbers)
k
k
s
where
k
s
characterises the source of energy in the
dynamic system (e.g.
Monin & Yaglom
,
1971
). In our case of the monochromatic waves,
k
s
≈
k
1
.
5
hz
=
9
.
82 rad
/
m, and therefore the wavenumber scales of
k
=
20-2600 rad
/
m,
resolved by the PIV frame, provided a sufficiently broad band.
WRN
(7.70)
for 1
5mm and water of 20
◦
C
ranged from 330 to 5900 and therefore the transition to turbulence could be expected.
This transition depends strongly on the vertical distance
z
according to
(7.72)
. Directly
at the surface, however, PIV measurements were not possible as, in order to obtain the
wavenumber spectrum, the entire horizontal layer of measured velocities has to be in the
water. For consistency of comparisons, the layer at 30mm from the still surface was chosen
as a compromise which was close enough to the surface, but below the wave troughs for
all recorded non-breaking waves.
It was expected that the turbulence, subjected to periodic forcing, will exhibit complex
and modulated behaviour (e.g.
Bos
et al.
,
2007
). Close to the critical Reynolds number,
the turbulence observed was also highly intermittent in both the space and time domains.
At the surface wave amplitude of
a
0
=
.
5 Hz waves of surface amplitudes
a
0
=
6-25
.
5mm at each of the ten recorded phases of the
wave period, the Kolmogorov interval in the velocity spectra appeared from 0 to 3 times
over the duration of 15 periods. Most frequently, it was observed at the rear face of the
wave profile, close to the instant of zero down-crossing, and therefore this phase of the
wave was chosen for further analysis.
Examples of the wavenumber spectra recorded during the down-crossing, with the turbu-
lence, are demonstrated in
Figure 7.23
. Here, a few
P
u
(
22
.
k
)
spectra exhibiting the
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