Wave Acceleration Measurement Using PTV Technique
Kuang-An Chang
Department of Civil Engineering
Texas A&M University, College Station, TX 77843-3136, U.S.A.
Edwin A. Cowen and Philip L.-F. Liu
School of Civil and Environmental Engineering
Cornell University, Ithaca, NY 14853, U.S.A.
A particle tracking velocimetry (PTV) based measurement technique is
developed for determination of wave acceleration. Based on the illumination of seeding
particles three times over two frames and continuously tracking an individual particle
over the three distinct illuminations, two velocity elds and a Lagrangian acceleration
are obtained. The technique not only provides a method for acceleration measurement
but also reduces the constraint on the time separation between adjacent illumination
pulses imposed by the camera framing rate in a single pulsed triple-frame technique to
a much shorter camera speci c frame transfer time. Monte Carlo simulations are used
to test the technique for a deep water wave.
Abstract
1. Introduction
The simultaneous measurements of velocity and acceleration are crucial in understanding wave forces. For example, acceleration data is important in designing an o shore
structure (Gudmestad, 1998). A simple example is seen from the Morison's equation
(Morison , 1950) in which the inertia force contains the uid acceleration, Du=Dt.
Given simultaneous information on velocity and acceleration, the inertia and drag force
distributions can be determined, provided the inertia and drag coeÆcients are known.
Although the acceleration eld can be calculated directly from two successive velocity
measurements using PIV, reports of quantitative acceleration measurements are relatively rare. This may be the result of a comparatively small number of researchers
working on the direct measurement of force with respect to the number interested in
the determination of velocity. In addition, hardware constraints have also played a role
in the relative inactivity of acceleration measurement. Typical digital camera framing
rates coupled with current pulsed laser repetition rates are too low for single-pulsed
image determination of the velocity eld at suÆciently short temporal spacing to allow
even moderate accelerations to be measured.
In this study we present a multi-pulsed acceleration measurement technique based
on PTV. The velocity and Lagrangian acceleration are obtained. Monte Carlo simuet al.
1
(a)
(b)
Camera
Camera
Laser
Laser
(c)
1
0.5
0
20
60
40
40
20
60
Figure 1: Trigger timing diagrams, (a) the single-pulsed image-triple trigger signal,
(b) the double-pulsed/single-pulsed trigger signal; and (c) a typical cross-correlation
result of (b).
lation results are used to conclude the technique on the determination of the velocity
and acceleration elds.
2. The acceleration measurement technique
The most simple and direct method for determining acceleration is to analyzing three
successive images. This gives two velocity elds so the acceleration can be calculated.
The time separation, t, is basically one half (or greater) of the frame time. However,
most cameras with reasonable high resolution (e.g., 1K 1K pixels) have framing rate
of about 30 or less. For example, a typical 30 frame per second (fps) camera gives t 17 ms. This renders t being too large to determine acceleration of interest in water
wave studies. If moving to a single-frame triple-pulsed technique , the limitation on
t can be removed but new problems develop: traditional autocorrelation directional
ambiguity needs to be handled.
In addition to the time separation issue, the selection between PIV and its modi ed
method, particle tracking velocimetry (PTV), is also need to be considered. Traditional
PIV gives Eulerian acceleration if two velocity elds are used. Finite di erences are
needed to compute the convective acceleration, u ru. Errors will inevitably be magni ed, especially when the spatial grid, x and y, between velocity vectors is large.
On the other hand, PTV tracks an individual particle so Lagrangian acceleration is
obtained { no need to calculate the convection term in acceleration determination.
Therefore, PTV is used in this study.
2
Figure 2: Particle tracking of the image triple. Particle images 1 and 2 are on the rst
frame while particle image 3 is on the second frame. The dotted line is the particle
trajectory.
To overcome the time separation problem mentioned above, a two-frame doublepulsed/single-pulsed technique is introduced for acceleration measurement. The hybrid
technique is based on three illumination pulses in two successive frames. Unlike the
three-frame single-pulsed method (trigger signal is shown in gure 1(a)), the rst frame
is double-pulsed and second frame is single-pulsed, as shown in gure 1(b). The doublepulsed rst frame is correlated with the single-pulsed second frame resulting in two
correlation peaks. The two peaks are of the same order of magnitude, which allows
their identi cation above the background noise level set by random particle image
correlations as seen in gure 1(c). Assuming t is small and the particle displacement is
(a reasonable assumption given the physics and the enhanced statistical
probability of randomly getting unidirectional displacement), the individual particles
can be tracked based on the displacements estimated from the two correlation peaks.
The de nition of \unidirection" is that the distance between the second and third
particle images, jd23 j, is shorter than that between the rst and third particle images,
jd13 j, i.e.,
jd23j < jd13 j
(1)
This means that the correlation peak closer to the origin is from the correlation of
the second and third particle images, while the other correlation peak is from the
correlation of the rst and third particle images. This allows the ow direction to be
identi ed unambiguously from the correlation such as in gure 1(c).
The resulting particle image triple from the double/single-pulsed frame pair is
illustrated in gure 2. Since the rst frame is double-pulsed (particle images 1 and
2 in gure 2), the two velocity displacements are relative to the second single-pulsed
frame (particle image 3), i.e., d13 and d23. The acceleration can then be calculated
from these two displacements as
Du u23 u12 2d23 d13
a=
' t = t2
(2)
Dt
unidirectional
3
in which u23 = d23=t, and u12 = d12 =t = (d13 d23 )=t.
The particle tracking technique used in this study is based on Cowen and Monismith
(1997). The mean displacement is estimated with, for example, a 6464 subwindow
using a cross-correlation based PIV algorithm (note two displacements are identi ed at
this point). The correlation is determined iteratively with the second-frame subwindow
dynamically located based on the previous iteration displacement estimate. By assuming the image-triple is unidirectional, the closer peak (to the origin) is the correlation
peak between the second and the third particle images while the farther peak is the
correlation peak between the rst and the third particle images. A smaller subwindow
can then be used to obtain higher resolution in velocity. Subsequently, the PIV results
are used to track individual particles (called PTV) using both displacement estimates.
A local median lter technique (Westerweel, 1994) is applied to both the PIV and PTV
results to remove stray vectors. The nal step is the determination of the acceleration
by di erencing the velocity vectors when two vectors are found to terminate at the
same particle in the second image. The accuracy of the acceleration is determined
by the sum of the errors in the two displacement vectors as mentioned previously. In
general the displacements have a subpixel accuracy of 0.1 pixel. Therefore, the error
is approximately 0.2 pixels in magnitude in acceleration measurements.
3. Applications to water waves
To date only very crude acceleration measurements (Chang and Liu, 1998; Kjeldsen
, 1998) or less accurate PIV based algorithms with the use of multiple cameras
have been reported (Jakobsen
, 1997). In this section we intend to verify the
proposed method through Monte Carlo simulations. In the future we will present
further validation of the method with laboratory data.
The acceleration measurement technique is tested here by using the simulation
under linear water waves. Linear wave theory is used to calculate the particle displacement in synthetic images. A deep water sinusoidal wave is tested. The wave has wave
parameters as follows: water depth h = 20 cm, wave period T = 0:5 s, wavelength
= 39 cm, and wave height H = 4 cm. The eld of view (FOV), with a 1024 1024
pixel resolution, is 66 cm2, from trough level at y = 2 cm downward to y = 8
cm, and t is 5 ms. The simulated velocity elds are shown in gures 3(a) and 3(b)
with the resulting acceleration shown in gure 3(c). The r.m.s. error in acceleration is
about 0.15 pixel in each direction (0.0018 g in physical unit) . Note that the error has
strong dependence on velocity gradient. The wave parameters were chosen to represent
a case that has relatively large velocity gradients. If a shallow water wave is used, the
error is typically smaller and about 0.1 pixel due to smaller velocity gradients.
et al.
et al.
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15 cm/s
−2
−3
−3
−4
−4
−5
−5
−6
−6
−7
−7
−8
0
2
4
15 cm/s
(b)
y (cm)
y (cm)
(a)
−8
0
6
2
4
x (cm)
x (cm)
−2
0.12 g
(c)
−3
y (cm)
−4
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x (cm)
Figure 3: Vector elds of the deep water wave, (a) velocity from the 2nd and 3rd
pulses, (b) velocity from the 1st and 3rd pulses, (c) acceleration.
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4. Conclusions
A double-pulsed/single-pulsedparticle image velocimetry technique with particle tracking capability is developed for simultaneous velocity and acceleration measurements.
The technique uses only one camera which, in comparison with the existing multicamera techniques, reduces not only the error due to image registration between images taken from di erent cameras but also reduces the cost due to the need of only
one camera. In addition, this technique can signi cantly reduce t, the time separation between two adjacent particle illumination pulses, to an order of magnitude or
more smaller than a three-frame single-pulsed method. The technique was successfully
tested through the Monte Carlo simulations of small amplitude water waves.
References
Chang, K. A. & Liu, P. L.-F. (1998) \Velocity, acceleration and vorticity under a
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Cowen, E. A. & Monismith, S. G. (1997) \A hybrid digital particle tracking velocimetry technique."
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Gudmestad, O. T. (1998) \On the importance of understanding ocean wave kinematics for calculation of dynamics and loads on o shore structures."
, ed. J. Zhang, p. 1-8
Jakobsen, M. L., Dewhirst, T. P. & Greated, C. A. (1997) \Particle image velocimetry
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Kjeldsen, S. K., Bonmarin, P., Skafel, M. G. & Drennan, W. M. (1998) \Lagrangian
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Morison, J. R., O'Brien, M. P., Johnson, J. W. & Schaaf, S. A. (1950) \The forces
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Westerweel, J. (1994) \EÆcient detection of spurious vectors in particle image velocimetry data."
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