Chin H. Wu and Aifeng Yao
Dept. of Civil and Environmental Engineering, University of Wisconsin, Madison,
Kuang-An Chang
Department of Civil Engineering, Texas A&M University
The co-existence of wave breaking and current is very common in marine environments. It is known that wave breaking provides an important conduit for the exchange of gas, water vapor, energy, and momentum between air-water interface.
The existence of current may further enhance these exchanges. Extensive reviews of breaker classification, the mechanisms of breaking, and the evolution of wave geometry are given by Peregrine (1991) for shallow-water waves and by Melville
(1996) for deep-water waves. For non-breaking surface waves with co-existent current, the wave-current interactions (Peregrine, 1976, Jonsson, 1990, and Thomas and Klopman 1997) is usually considered as an extension of the fundamental wave theories. A handful of detailed experiments (Lee 1990, Skyner and Easson 1998, and
Swan et al. 2001) have been conducted to show the significant effects of current profiles and directions on surface waves. However, it is unclear if currents can affect the breaking mechanisms or processes, in particular, the onset of unsteady wave breaking. Therefore, further study is needed to understand kinematics of wave breaking on flowing currents.
A Digital Particle Image Velocimetry (DPIV) and wave gages were used to measure velocity field and surface displacement of unsteady deep-water wave breaking on different strengths of following currents. The experiments were conducted in a newly renovated two-dimensional wave-current flume in the
Environmental Fluid Mechanics Laboratory at the University of Wisconsin-Madison.
The flume has a dimension of 46 m long, 0.91 m wide and 1.22 m deep, and was filled with 0.6 m water depth. The flume was carefully leveled and smooth bed conditions were maintained. The waves were generated by two servo-controlled flat, bottom-hinged paddles located at the upstream of the flume. At the downstream end of the flume, a beach, made up of high absorption horsehair material, with a slope of
1:10 was constructed. The beach was tested with less than 5% reflection coefficient.

Three current speeds (5 cm/s, 9cm/s and 15cm/s) were achieved by a controllable speed pump and pipe system. The water was pumped from at the end of the flume then was directed to an upstream settling well sitting behind the wavemaker. The following current was supplied by a bed opening in front of the wavemaker. The current profiles of the three stations ( x = 6 m, 10 m, and 14 m, where x = 0 is defined at the neutral position of the wavemaker) was measured to be nearly uniform using a
Sontek 16Mhz Micro-Acoustic Doppler Velocimeter.
The unsteady, single, and different types (incipient, spiller, and plunger) of deep-water wave breaking on the following currents was generated by a wave focusing method (Rapp and Melville, 1990) with further consideration of Doppler shift relation in the presence of currents. The modified wave focusing method used a specific range of wave components and predetermined phases from the Doppler shift dispersion relationship,
=
+ k ⋅ U c
(where
is the apparent wave frequency relative to a stationary observer, k is the wavenumber, and
is the intrinsic wave frequency observed in a moving frame at velocity U c
), to allow constructive interference so that all wave components traveling on the currents were still able to focus at one spatial location. To examine the effect of currents on wave breaking, the same apparent frequency band, of 32 components evenly spaced with the frequency range, 0.686 Hz ~ 1.274 Hz, was used for experimental cases listed in Table1. Using the modified method, a very repeatable single wave breaker on the different strength of the following currents was able to be generated at the same location.
Our newly developed real-time DPIV system was used to measure the velocity field of wave breaking on the following currents. The DPIV system consists of an illumination source (a dual New Wave Research Gemini Nd:YAG lasers), a Berkeley
Pulse Generator, a SMD 1M30P CCD camera with a 60 mm Nikon lens, and a realtime computer recording system that can directly record acquired images on a
StreamStor 180GB hard-drive storage system. The lasers, with 120 ml/pulse and a wavelength of 532 nm, can be triggered by four sequences of TTL signals provided by the Pulse Generator that in turn is triggered by one TTL signal. A thin laser sheet was generated by directing the laser beam through a 25 mm focal length cylindrical lens and a 1000 mm focal length spherical lens. The flow was Seeded with round polyamide particles (Vestosint 2157, Crea Nova) with a mean diameter of 57 µ m and a density of 1.01 g/cm 3 . To capture the successive wave crest motion immediately

before the focusing/breaking point, the field-of-view was set at 24 ~ 25 cm. The time interval between two consecutive laser pulses was 2.5 ms and lens aperture was set at f/4.0. The two lasers were allowed to fire for 20 sec before the image acquisition started. The images were acquired an Imaging Technology PC-DIG frame grabber board installed in a dual-processor Pentium III 710 computer. When two lasers fire alternatively at a frequency of 15 Hz, the camera can capture 30 images with 12 bits and a spatial resolution of 1024 × 1024 pixels per second.
To synchronize the DPIV system with the wavemaker, we used a master
Pentium 100 computer with a AT-AO-6 data acquisition board (National Instrument).
The AT-AO-6 board has 6 analog output channels and 8 digital I/O channels, which was controlled by a LabVIEW program. Specified TTL signals were sent from the master computer to trigger wave generation system and the DPIV system. The paddle motion of the wavemaker is controlled by a linear servo amplifier system, which in turn was driven by our modified wave focusing output signals from the AT-AO-6
DAQ board.
In addition to velocity measurements, an array of surface piercing resistance wave gauges was used to measure the water surface displacement at/near the wave breaking location with an interval of 15 cm. The total measurement points span 90 cm. A resolution of 0.2 mm by the wave gauges was calibrated with an amplifier and a DAS1602 data acquisition board (Keithley Metrabyte) installed on the third Pentium
200 computer. This computer for surface displacement measurement was also synchronized with the master computer with the AT-AO-6 board by a triggering TTL circuitry so that both surface displacement and velocity measurements started at the same time when the wavemaker motion was initiated. In all the experiments, the water surface displacement was sampled at 200 Hz that was checked to be necessary to capture the very rapid wave profile evolution of wave breaking. For the wave breaking in the presence of current, the desired current was generated first and at lease
30 minutes elapsed before the wave generation.
To reveal the onset of wave breaking, both geometric criteria (based on the geometry of wave shape, Kjeldsen and Myrhaug 1979) and kinematic criteria (based on the characteristics of wave motions, particle velocity and wave phase speed) were used. It was observed that the geometric criteria of wave breaking on different strength of following currents could vary a wide range of values, suggesting that

existence of following currents in the field can change surface steepness at breaking.
These effects may explain the range of local wave shape criteria reported from field observations (Kjeldsen and Myrhaug 1979) and explain why geometric local wave shape parameters have not proved to be stable limits for ocean wave breaking.
Using a Hilbert transform technique and the linear wave theory for surface displacement data, Wu and Nepf (2001) successfully showed that wave breaking appears when the particle velocity, u , is larger than the wave phase speed, C . In our experiments of wave breaking on the following current, we found that wave breaking occurs when the total particle velocity, u+ U c
, is larger than the apparent wave phase speed, Ca, using a modified Hilbert transform with consideration of a Doppler shift.
Therefore, the single-point kinematic criterion evaluated using a modified Hilbert transform can be used as a robust indicator for ocean breaking waves with or without currents. However, one shortcoming of the Hilbert transform technique is the assumption of the linear wave theory, which is well known to be invalid at steep/breaking waves. It is also unclear that why the modified Hilbert transform with a simple wave theory can provide a good breaking indicator.
To further examine the kinematic criteria, we obtained the velocity at the crest of wave breaking on the following currents using the DPIV images. The images were analyzed by an interrogated 64 x 64 pixels window with 50% overlapping using a hybrid digital particle tracking velocimetry technique (Cowen and Monismith 1997).
Figure 2a and 2b show the velocity field for both spiller breakers with and without a 5 cm/s following current reached 1.8 m/s, respectively. A consecutive images were used to estimate the phase speed with 1.78 m/s at the onset of wave breaking. However, the results for the geometric steepness were slightly different. These results by the DPIV technique indicate that currents can affect the geometric steepness, but not the kinematic breaking at the onset of unsteady wave breaking. Furthermore, the velocity field and the phase speed estimated by the DPIV method reveal that the modified
Hilbert transform underestimated both the particle velocity and phase speed up to
30%. Since both are underestimated, their ratio, | v
| / C , can still be close to the actual ratio, which explains why a simple linear wave theory assumption with the
Hilbert transform can still provide a good breaking indicator. Future study for opposing currents on the unsteady wave breaking is under way.

References
Cowen, E.A., Monismith, S.G., "A hybrid digital particle tracking velocimetry technique," Experiments in Fluids , 22, 199 - 211, 1997.
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Table 1 Experimental conditions for different currents cases
(m/s)
Apparent frequency range (Hz)
Intrinsic frequency range (Hz)
Wavenumber k c
(m -1 )
A 0 0.686~1.474 0.686~1.474 4.7264
B 5 0.686~1.474 0.663~1.426 4.4360
C 9 0.686~1.474 0.647~1.391 4.2311
D 15 0.686~1.474 0.626~1.345 3.9715
Figure 2a Spiller with a 5cm/s following current Figure 2b Spiller without a following current