PIV measurements of accelerations in water waves
Abstract submitted to PIV and Modeling Water Wave Phenomena, Cambridge, UK,
17-19 April 2002
Jensen, A., Huseby, M., Clamond, D., Pedersen, G. and Grue, J.
Mechanics Division, Department of Mathematics,
University of Oslo
The PIV technique has proven to be eÆcient for velocity measurements for a wide
spectrum of applications. Apart from cases involving turbulence, aeration etc., velocities are often reported to be measured within 1-2%, say. Accelerations, on the other
hand, are much more diÆcult to obtain from PIV data and are reported less frequently
in the literature (Chang & Liu, 1998; Chang et al., 1999; Jakobsen et al., 1997; Jensen
et al., 2001; Christensen & Adrian, 2001; Dong et al., 2001). The accelerations can be
extracted from two, or more, velocity elds at slightly dierent times by computing
temporal as well as spatial gradients. Clearly, such procedures will yield results that
are very sensitive to the noise that is inherent in the velocity measurements.
We will present a series of cases where acceleration measurements have been attempted, with a variable degree of success:
1. Periodic waves of small amplitude, Jensen et al. (2001)
2. Leading part of wave trains, Huseby et al. (2000)
3. Run-up on beaches, Jensen et al. (2002)
4. Solitary waves
5. Extreme events in irregular seas
6. Focusing waves
Accelerations will be computed by virtual tracing techniques as well as direct methods
based on explicit calculation of gradients.
Jensen et al. (2001) investigated waves that were close to pure Stokes waves. For
comparison of the experiments with third-order Stokes theory, we may exploit that
the magnitude of v and @ v=@t of the Stokes waves are independent of the horizontal
coordinate (x). We thus derive
_ j@ v=@tj =
agk
1
agk
h
ut2 + vt2
i1
2
= exp(ky );
(1)
where (u; v ) = v, (ut; vt ) = @ v=@t, a is the rst harmonic amplitude and k is calculated
from the dispersion relation ! 2 = gk (1 + a2k 2). ! is found from the time history of the
1
y
h
0.1
0.1
0.05
0.05
0
0
−0.05
−0.05
−0.1
−0.1
−0.15
−0.15
−0.2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
0
_
0.2
0.4
0.6
0.8
1
_
Figure 1: Acceleration measurements, ! = 8:95s 1 , ak = 0:16 and kh = 4:9. Left:
'+' are 20 columns of acceleration measurements from the crest, ak = 0:16. Right: 40
columns, ak = 0:09. and _ (y ) dened in the text. Solid line exp(ky ).
elevation, using the zero up-crossing period. In gure 1 we have plotted _ due to the
measurements for two dierent ak 's and compared with the analytic exp(ky ).
Irregular wave trains are investigated and generated from a JONSWAP spectrum
with signicant wave height Hs = 0:065 m, mean wave period T1 = 0:78 s and a peak
enhancement factor = 3:3. A time history of the wave elevation, measured by wave
gauges, is depicted in gure 2. The position of the gauge corresponds to x = 0 in the
velocity eld shown in gure 3.
The wave train is repeated three times and the acceleration proles are shown in the
right panel of gure 3. The inner accelerations (solid are local acceleration and dashed
lines are particle accelerations, right panel) are computed using the Cauchy integral
formula (see Clamond & Grue, 2001). The red (@v=@t) and black (Dv=dt) symbols
are vertical accelerations and green (Du=dt) and blue (@u=@t) are the horizontal ones
(experimental). The dierent symbols are due to each repetition and the local wave
steepness is kmax = 0:31 (max is the maximum elevation). The local acceleration
eld @ v=@t may be estimated from two velocity elds by evaluating
v(x; t2)
v(x; t1)
t2
t1
(2)
and the particle acceleration eld Dv=dt is estimated using a tracing technique (Jensen
et al., 2002). Even though there is scatter, signicant acceleration pattern do emerge
from the gure. At the workshop, corresponding results will be presented also for the
experiments listed on page 1.
2
1.2
8
6
4
[cm] 2
0
−2
−4
275
280
285
290
295
8
6
4
[cm] 2
0
−2
−4
282.8
283
283.2
283.4
283.6
283.8
t[s]
284
284.2
Figure 2: Upper panel: JONSWAP spectrum with Hs = 0:065, T1 = 0:78 and = 3:3.
Lower panel: Enlarged subsection from upper panel. The position corresponds to x = 0
in gure 3 and is 814; 5cm from the reference position of the paddle.
15
0.4
−56cm/s
10
0.2
y [cm]5
ky
0
0
−0.2
−5
−10
−5
0
5
10
x[cm]
−2.5
−2
−1.5
−1
a=(! max e
2
−0.5
kmax
0
0.5
)
Figure 3: Measured velocities/accelerations in a irregular wave train, kmax = 0:31.
Left panel: Velocity vector plot on a PIV-image at t = 283; 4s. Right panel: Acceleration proles. Red (@v=@t) and black (Dv=dt) symbols are vertical and green (Du=dt)
and blue (@u=@t) are horizontal accelerations. Solid are local and dashed lines are
particle acceleration (analytical, colored black). Solid magenta line is exp(ky ).
3
References
Chang, K.-A., Cowen, E. A. & Liu, P. L.-F. 1999 A multi-pulsed PTV technique
for acceleration measurement. In 3th International Symposium on Particle Image
Velocimetry, Santa Barbara, USA, 16-18 Sept..
Chang, K.-A. & Liu, P. L.-F. 1998 Velocity, acceleration and vorticity under a
breaking wave. Physics of uids 10, 327{329.
Christensen, K. T. & Adrian, R. J. 2001 Measurements of instantaneous acceleration elds using particle image velocimtery. In 4th International Symposium on
Particle Image Velocimetry, Gottingen, Germany, 17-19 Sept..
Clamond, D. & Grue, J. 2001 A fast method for fully nonlinear water-wave computations. J. Fluid Mech. 447, 337{355.
Dong, P., Hsu, T.-Y., Atsavapranee, P. & Wei, T. 2001 Digital particle image
accelerometry. Experiments in Fluids 30, 626{632.
Huseby, M., Jensen, A. & Grue, J. 2000 An experimental investigation of ringing
loads on a vertical cylinder in transient waves. In Proceedings 15th International
Workshop on Water Waves and Floating Bodies (ed. T. Miloh & G. Zilman), pp. 75
{ 78. Faculty of Engineering, Tel{Aviv University, Israel.
Jakobsen, M. L., Dewhirst, T. P. & Greated, C. A. 1997
Particle image
velocimetry for predictions of acceleration elds and force within uid ows. Meas.
Sci. Technol. 8, 1502{1516.
Jensen, A., Pedersen, G. K. & Wood, D. J. 2002 An experimental study of wave
run-up at a steep beach. Submitted to J. Fluid Mech.
Jensen, A., Sveen, J. K., Grue, J., Richon, J.-B. & Gray, C. 2001 Accelerations in water waves by extended particle image velocimetry. Experiments in Fluids
30, 500{510.
4