Detuning and wave breaking during nonlinear surface wave focusing

Ocean Engineering 113 (2016) 215–223

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Detuning and wave breaking during nonlinear surface wave focusing Dianyong Liu a, Yuxiang Ma a,n, Guohai Dong a, Marc Perlin b a b

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, 116023 Dalian, China Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 6 February 2015 Accepted 26 December 2015 Available online 14 January 2016

Using an energy focusing technique, several transient wave trains were generated in a two-dimensional wave flume to investigate their nonlinear evolution. By increasing the initial wave steepness (S0), while all other parameters remained the same, non-breaking through breaking conditions were achieved. The experimental results demonstrated that the waves’ initial steepness affected the focusing process significantly. As expected, the nonlinear effects on wave packets with a low initial steepness were very small. However, for cases with large initial steepness, nonlinearity increased causing the wave packet to focus and break prematurely, a “detuning” process. As a result of the detuning process more than one breaker may occur, and more importantly when this ensues, energy loss was altered. It is found that for S0 o0.274, only one breaker occurred, and the energy loss increased with increased initial steepness. For larger steepness, two breakers occurred and the total energy loss remained nearly constant (
17%) as the initial steepness was increased. As the initial steepness was increased still further, a large plunging breaking was obtained with additional energy loss (4%). These results suggest that the “detuning” has a pronounced influence not only on the resulting wave field, but on the resulting energy loss. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Nonlinearity Detuning Wave breaking Breaking position Spectra Energy dissipation

1. Introduction Real sea waves often exhibit breaking on the ocean surface. Breaking represents a strongly nonlinear phenomenon encompassing different stages: incipient breaking, developing breaking, subsiding breaking and residual breaking (Babanin, 2011). The process limits the height of waves and dissipates wave energy, converting it into currents and turbulent kinetic energy (Drazen and Melville, 2009). In addition, theses currents and large impact forces before and at breaking have been long known to cause hazards to navigation vessels and marine structures (Smith and Swan, 2002). However, due to the complexity of the breaking process, a fundamental understanding of wave breaking has been elusive even though water waves have been studied scientifically for a very long time. For several decades now, a significant effort has been devoted to the study of breaking waves (Babanin, 2009; Baldock et al., 1996; Benjamin and Feir, 1967; Clauss, 2002; Rapp and Melville, 1990; Thorpe, 1993; Tian et al., 2012; Tulin and Waseda, 1999; Wu and Nepf, 2002; Wu and Yao, 2004; Zhang and Melville, 1990). Laboratory experiments, which can be conducted in a well-controlled environment, are a popular method wherein high-precision measurements can be obtained to further our understanding of the dynamics and n

Corresponding author. E-mail address: [email protected] (Y. Ma).

http://dx.doi.org/10.1016/j.oceaneng.2015.12.048 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

kinematics of wave breaking. On the other hand, the complexities associated with breaking waves make detailed numerical simulations or theoretical analysis very difficult. Likewise, field measurements of breaking waves are problematic due to the difficulty of measuring accurately the appropriate parameters in intermediate and deep water. As regards lab experiments, there are three primary methods used for generating breaking waves. The first method, which is appropriate for shallow water breaking, is via shoaling, where waves propagate toward a beach and wave height increases until breaking occurs (Mukaro et al., 2013). The second method takes advantage of frequency dispersion, and generates waves that near a prescribed spatial position (Kway et al., 1997; Rapp and Melville, 1990; Yao and Wu, 2005), while the third technique uses the modulation instability (Babanin et al., 2007; Benjamin and Hasselmann, 1967; Longuet-Higgins, 1978). The first method is used to study near-shore breaking while the other two techniques are employed in the study of intermediate and deep water wave breaking. In this study, in the presence of intermediate and deep water, the method of dispersive focusing in a laboratory flume was adopted. In view of the many important investigations of wave breaking (see Perlin et al. (2013) for a recent review), the prediction of the breaking onset of surface waves is crucial. Various threshold criteria for predicting the onset of breaking have been proposed, for example based on wave geometry (i.e. wave steepness, vertical/horizontal asymmetry, etc.), kinematics (i.e. crest fluid velocity) and dynamics (i.e. downward acceleration at the wave crest, local energy growth

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rate, etc.). Intuitively, the criterion based on crest fluid velocity (i.e. a kinematic breaking criterion) is perhaps the simplest notion, i.e. if the horizontal crest particle velocity U exceeds the wave phase speed C, wave breaking occurs. However, the application of this criterion is non-trivial because of the difficulties associated with measuring U and C during the rapidly evolving breaking process. Regardless, kinematic criteria have drawn much attention. Particle image velocimetry (PIV) was used by Perlin et al. (1996) to measure the crest particle velocity U of plunging waves demonstrating that wave breaking had occurred when U/C reached 0.74. Stansell and MacFarlane (2002) investigated two different types of breaking waves and found that when plunging occurred U/C was equal to 0.81, while it was 0.95 for spilling waves. These investigations illustrated that the kinematic criterion seems inappropriate as a robust predictive standard. In addition, Oh et al. (2005) conducted experiments to detect breaking in deep-water in the presence of strong wind. They concluded that the kinematic criterion is also inadequate for predicting breaking under strong wind. Compared to the study of the geometric criteria predicting breaking onset, criteria based on the evolution of the local energy have achieved better results. Song and Banner (2002) proposed a dimensionless local energy growth rate δ(t) to predict the onset of breaking waves through numerical study. This predictive parameter reflects the steepness of the maximal carrier waveform associated with the increased local carrier wave number. A threshold range for δ(t) of (1.47 0.1)*10  3 was suggested to distinguish wave breaking from non-breaking. Thereafter, a detailed laboratory experiment with the same initial conditions as Song and Banner’s numerical study was conducted by Banner and Peirson (2007). This study substantiated the proposed breaking threshold. Then, Tian et al. (2008) evaluated the breaking criterion experimentally by different wave groups in a two-dimensional wave tank. It was found that the breaking criterion of Song and Banner (2002) is sensitive to the choice of the local wave number; they suggested that the local wave number should be determined from the breaking crest. However, this wave breaking criterion is difficult to apply, especially for the estimation of the local wave energy density. Recently, a further simplification of the diagnostic parameter proposed by Song and Banner (2002) was investigated and confirmed to provide a good approximation by Tian et al. (2010). Hence, this simplified approximation is used herein. As has been clearly established by researchers, wave breaking plays a dominant role in dissipating the energy of ocean surface waves. Although significant effort has been expended to estimate the energy dissipation due to breaking via field and laboratory experiments, our present understanding of the energy dissipation resulting from wave breaking is far from adequate. Rapp and Melville (1990) systematically studied breaking waves in deep water by the dispersive focusing technique and estimated the energy dissipation by measuring the surface elevation upstream and downstream of breaking. They reported that during breaking the spilling process could dissipate 10% of the initial energy, while this value was 25% for the plunging case. Specifically, they found that the energy dissipation was from mostly the high end of the first harmonic band (f/fp ¼1 2; fp is the spectral peak frequency). Later, Kway et al. (1997) investigated deep-water breaking generated by focusing wave groups with three different spectral shapes and found that the energy losses varied from

14% to 22% for different spectral distributions. The so-called Hybrid Wave Model (HWM) was used to eliminate the bound wave components from measured surface elevations by Meza et al. (2000) in their investigation of energy dissipation; this illustrated that the energy dissipation is almost exclusively from wave components at frequencies higher than the spectral peak frequency (wave components below the peak frequency gained a small portion
12% of energy). Wu and Yao (2004) investigated the energy dissipation of breakers on currents and showed that, in the presence of a strong opposing current, most of the dissipated energy in the higher frequency wave components transferred to the lower frequency band. As mentioned above, a majority of studies on wave breaking that are undertaken in the laboratory use a computer-controlled wave-maker to focus an isolated breaking wave at a prescribed location by moving that position in space or in time according to superposition using linear theory. As one adjusts the steepness of the wave train, nonlinearity causes the temporal/spatial location to change. However, the evolution of this process and the subsequent dissipation that occurs is not clear. Recently, a study of weakly three-dimensional breaking by Liu et al. (2015) found that the breaking occurred prematurely compared with the linear focusing position. (This is not surprising as nonlinearity in gravity waves causes the wave speed to increase, and has been noted by others for two-dimensional experiments.) They attributed it to the focusing wave components "detuning", which is a nonlinear phenomenon that inhibits wave components from focusing at one location/time simultaneously, thus forming a large breaking wave. When relatively low-frequency wave components overtake relatively high-frequency wave components upstream of the prescribed focusing location, premature breaking may occur. More importantly, the remaining energy in the various wave components will continue to interact, and may cause a second breaker to occur during the "detuning" process. (In fact, for shallow water, this detuning phenomenon has been studied numerically by Pelinovsky et al. (2000) using the Korteweg–de Vries equation.) Therefore, it is necessary to investigate “detuning” during wave focusing in laboratory experiments. In experimental programs, wave groups are forced often from non-breaking through breaking by increasing the initial wave steepness. In this manuscript, we investigate the detuning process in two dimensions. The paper is organized as follows: subsequent to the introduction, the experimental facilities and the wave generation technique are described carefully; then, the experimental results and observations are shown in Section 3; and lastly, the conclusions are presented.

2. Experiments 2.1. Experimental facility The experiments were conducted at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, China. The experimental setup is shown in Fig. 1. This flume is 50 m long, 2 m wide and was used with a water depth of 0.6 m. A piston-type wave-maker is installed at one end of the flume, and in the following x ¼0 m is defined as the mean position of the

Fig. 1. Experimental setup.

D. Liu et al. / Ocean Engineering 113 (2016) 215–223

wave-maker. At the far end of the flume, a wave absorbing beach is present to help mitigate reflections. To obtain two-dimensional experiments, a splitter wall was mounted along the flume to divide it into two longitudinal sections; the side with 0.8 m width was used for experiments. Data were recorded as time series of wave surface elevation by the probes used in the basin at 17 locations along the flume. The locations of these probes are marked in Fig. 1 by vertical bars. Moreover, prior to and during the experiment, every wave probe was examined and calibrated for accuracy, which is of the order 71 mm. Additionally, a video camera (Sony HDR-XR500E) was placed outside the glass walls of the facility with its view perpendicular to the sidewall to record the spatial variation of the surface elevations in the vicinity of the breaking locations. This imager was able to record time series of spatial information in a swath 0.5 m in longitudinal extent. 2.2. Generation of focusing waves In this study, the dispersive focusing method was employed to generate the breaking waves. According to the linear wave theory (LWT), the surface elevation of different frequencies at an arbitrary point (x, y) can be expressed as

ηðx; tÞ ¼

N X

ai cosðki x  ωi t þ εi Þi ;

ð1Þ

i¼1

where N is the total number of wave components, which was 64 in this study; and ai, ki, εi and ωi are the corresponding wave amplitude, wave number, phase shift and radian frequency of the ith wave component, respectively. The wave number of each component ki is given by the linear, gravity wave dispersion relation ðωi Þ2 ¼ ki g tanhðki hÞ;

ð2Þ

where g is the gravitational acceleration and h is water depth. It was assumed that the waves were focused at a spatial position, xf ¼ 9 m, and at a specified time tf ¼27 s. Hence, at the focused point, the phase shift for each wave component is

εi ¼ ki xf  ωi t f þ 2mπ m ¼ 0;

7 1;

7 2; …

ð3Þ

Substituting Eq. (3) into Eq. (1) and letting m ¼0, the wave elevation at an arbitrary point can be written as

ηðx; tÞ ¼

N X

ai cos ½ki ðx  xf Þ  ωi ðt  t f Þ:

ð4Þ

i¼1

obtain the transfer function: first, the cyclic frequency band [f1,fn] was uniformly spaced using 64 discrete frequencies; then, three different amplitudes were selected for a given frequency, fi, as input amplitudes and the transfer function |T(fi)| was calculated by dividing output amplitudes by input amplitudes. It was shown that the results were almost the same for three different experiments at a given frequency, and therefore the average value was regarded as the final transfer function. It is well known, however, that nonlinearity can significantly affect the accuracy of the transfer function calculated from such linear assumptions. As a result, the amplitude spectra calculated from the experiments do not match identically the theoretical case. Hence, an iterative procedure was employed (i.e. the amplitude calculated by the transfer function was used as the input signal in the next iteration, and after roughly three iterations, the transfer function wellapproximated the relations between input and output signals). Then, a cubic spline interpolation through the measured transfer function was employed to calculate the amplitude of every frequency component, as can be seen in Fig. 2. To assess the transfer function acquired by iteration, a comparison of the desired and measured amplitude spectra of the surface elevation at x ¼3.5 m is carried out using an FFT, and is shown in Fig. 3. The solid lines show the measured spectra, while the dash lines show the computed/desired spectra. Two cases are shown in the figure: one case, F1, is for the small initial steepness while the second case is for the large initial steepness, F19, where breaking occurred downstream. For the case with low steepness (S0 ¼ 0.173), the measured spectrum is very similar to the theory. However, for the case with relatively large steepness (S0 ¼0.341), although not that obvious from the figure, the nonlinear wave interactions among various harmonics contribute to significant deviations downstream between the actual wave field and the desired wave field, with energy shifted from the high frequency end of the first harmonic band (f/fp ¼ 1  1.75) towards the higher frequencies. This energy transfer to high frequency components has been observed also in previous studies upstream of the focused/breaking position (Baldock et al., 1996; Kway et al., 1997), suggesting that the energy transfer has already occurred near the wave-maker – that is, it occurs very rapidly. Also, as expected, it is clear that the initial steepness plays a significant role in the process of nonlinear interaction downstream. Regardless, the deviation between the spectra near the wave maker in both cases was fairly small. The same procedure was applied for the other experimental cases. It is noted that through the application of an iterative method an acceptable agreement between the measured

That is, the theoretical displacement of the paddle for generating focused waves can be described as

ηðtÞ ¼

N X

ai cos ½  ki xf  ωi ðt t f Þ:

ð5Þ

i¼1

To obtain a steady focusing wave group and inhibit breaking prematurely, the constant wave steepness spectrum first used by Perlin et al. (1996) but implemented as in Nepf et al. (1997) was adopted 1 ,0 64 X @ ai ¼ A ð6Þ ki 1=kj A; j¼1

in which A is the amplitude at the focused point. As is well known, the wave-maker transfer function, which reflects the relation of amplitudes and phases of the wave-maker displacement at a given frequency to the corresponding parameters of the propagating wave (Shemer et al., 2007), needs to be employed to precisely determine the wave-maker drive signal. In our experiments, a method of frequency response was applied to

217

Fig. 2. The transfer function as determined from the iterative procedure.

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D. Liu et al. / Ocean Engineering 113 (2016) 215–223

and the desired wave forms was obtained. Hence the evolution of the surface downstream can be studied further (see below).

2.3. Wave conditions To investigate the “detuning” due to nonlinear interactions during wave focusing, it is necessary to ensure that the initial phase speeds between the physical experiment and the linear prediction agree. Hence, prior to the experiment, a comparison of the phase speeds was conducted for regular waves and it is presented in Fig. 4. The experimental phase speeds were estimated from the distance between the first two probes and the propagation time. It is found that for low frequency conditions (f o0.8 Hz) the actual wave speed varied as compared with the theoretical wave speed, which may result in some wave components not reaching the focusing location at the prescribed time. Because of this, only the frequency band f¼ 0.8–1.4 Hz is considered in the present investigation. A priori, kh ranges from 1.66 through 4.74 corresponding to intermediate and deep water conditions. Detailed information of the wave trains used herein is listed in Table 1. For each experimental case, three repeated measurements

Table 1 Wave parameters. In this table, fp is the peak wave frequency; fs ¼Σ(fnan2)/Σ(an2), is the spectrally weighted wave frequency as defined by Drazen et al. (2008) and Tian P et al. (2010); ks is the wave number corresponding to fs; and S0 ¼ ks an. (S0 is the initial wave steepness used, and is calculated from the probe data immediately adjacent to the wavemaker.) (This sentence is just beneath the Table 1.) Case

fp

N

f

fs

ks

S0

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21

0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83 0.83

64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64

0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4 0.8–1.4

1.034 1.035 1.029 1.035 1.031 1.035 1.031 1.030 1.031 1.031 1.031 1.031 1.031 1.030 1.032 1.030 1.030 1.029 1.029 1.029 1.029

4.357 4.361 4.316 4.364 4.330 4.362 4.333 4.324 4.332 4.329 4.331 4.326 4.331 4.324 4.333 4.322 4.321 4.316 4.315 4.312 4.312

0.173 0.190 0.200 0.224 0.233 0.248 0.249 0.258 0.265 0.274 0.280 0.288 0.297 0.303 0.312 0.318 0.326 0.332 0.341 0.346 0.354

were collected with a record length of 80 s and a sample interval of 0.005 s.

3. Experimental results 3.1. Definitions 3.1.1. Criteria for breaking onset Song and Banner (2002) proposed a dimensionless local energy growth rate, δ(t), to predict the onset of wave breaking. A threshold, δc ¼(1.4 70.1)*10  3, was suggested to differentiate waves that eventually break from those that do not break. The parameter δ(t) was defined as Fig. 3. Amplitude spectra at x ¼3.5 m for two different cases. The solid curves show the measured spectra, while the dash curves show the computed spectra.

δðtÞ ¼

1 D〈μðtÞ〉 ; Dt

ωc

ð7Þ

where ωc is the initial mean carrier wave frequency; μ(t)¼ (E/ρg)k2 is a variable that reflects the relation between local wave energy and wave number. (The syntax used in Eq. (7) follows that used by and explained in Song and Banner, 2002.) Here, E is the depth integrated local wave energy density at the maximum surface elevation, and k is the local wave number. 〈μ(t)〉 describes the local mean value of the upper and lower envelopes of μ(t). Hence the growth rate, D〈μ(t)〉/Dt, mirrors a mean energy convergence rate to or from the wave group energy maximum, associated with nonlinear interaction between different wave components (Banner and Peirson, 2007). However, the calculation of the local wave energy density from experiments is quite difficult as the measurement of particle velocities (e.g. via PIV) required to calculate kinetic energy is hampered by the appearance of two-phase flow during active breaking. Therefore, Tian et al. (2008) further investigated the Song and Banner (2002) parameter and proposed the following (adopted herein): Fig. 4. Wave phase speeds at x¼ 3.5 m for the frequencies in the wave packet. The squares represent theoretical linear phase speeds; the circles represent the speeds as measured at the crest; the triangles represent the speeds measured in the troughs.

δbr ¼ BS2b

ωb : ωs

ð8Þ

Based on Tian et al. (2008) the local wave number, kb ¼ π/| xzu  xzd|, which is calculated from consecutive zero-up (xzu) and

D. Liu et al. / Ocean Engineering 113 (2016) 215–223

219

Fig. 5. Evolution of the wave packet along the flume. (a) x ¼633 cm; (b) x¼ 714 cm; and (c) x ¼ 845 cm. The dash curves represent the measured surface elevations; the solid curves represent the linear surface elevation predictions.

3.1.2. Breaking intensity Breaking intensity is another important characteristic of wave breaking, which indicates the amount of the energy dissipation in the breaking process. Here, a spectral definition of breaking intensity (Babanin, 2011) is used as Z f2 Z f2 Z f2 Eloss ¼ 16 Sup ðf Þdf  16 Sdown ðf Þdf ¼ 16s Sup ðf Þdf ; ð9Þ f1

Fig. 6. Initial steepness versus breaking position. The circles represent spilling breakers while the triangles are indicative of plunging breakers. With an increase of the initial steepness (i.e. Region II), the blue filled circles represent the breakers that occurred first, the black markers at the same steepness represent the second occurrence of breaking. In Region III, the red triangles are the first occurrence of breaking while the blue symbols represent the second breakers that occurred at those steepnesses. The two largest steepness waves in Region II, 0.312 and 0.318, that have blue symbols only broke once. The horizontal bars, blue and red, represent positions located behind I-beams that retain the glass sidewalls and are part of the tank wall structure, and hence prevent more exact downstream distance measurements. The red vertical bars on the abscissa represent the positions x ¼633 cm, x¼ 714 cm, x¼ 845 cm respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

zero-down (xzd) crossings spanning the maximum crest, is used in this study. Then the corresponding angular wave frequency, ωb, is determined using the linear dispersion relation, while ωs ¼ 2πfs is determined from a spectrally weighted wave frequency (fs). Sb ¼kbΣan is the local wave steepness, and B is a proportionality constant equal to 8.77*10  3 (see Fig. 6 in Tian et al. (2010) for details).

f1

f1

where Eloss is the energy dissipation due to breaking, and Sup(f) and Sdown(f) correspond to the wave frequency spectra up- and downstream of the breaking event (i.e. the influx and efflux of wave energy to and from a control volume), which are obtained from the fast Fourier transform (FFT) of the surface elevations measured along the flume. In the equation, the unknown coefficient s reflects the breaking intensity. In using Eq. (9), the lower and upper cutoff frequencies, f1 and f2, are set to 0.5fp and 2.5fp respectively, beyond which the energy was confirmed negligible. The multiplier, 16, is introduced corresponding to the definition of the significant wave height Hs sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z Hs ¼ 4

1

0

Sðf Þdf :

ð10Þ

3.2. Evolution of the surface elevation Fig. 5 presents the temporal water surface elevations measured at three downstream locations each for cases with initial steepness of 0.248, 0.274, 0.303 and 0.326, respectively. In each inset, the solution derived from the linear sum of the individual wave components is embedded for comparison. In this study, the surface elevation measured at the first station (x¼3.5 m) was Fourier decomposed to determine the amplitude and phase of the individual wave components. Thus, the linear surface elevation at any other position is available. The evolution of the surface elevations at different locations, i.e. x¼ 633 cm, 745 cm and 845 cm, respectively are also shown in Fig. 5 where dash curves represent the measured surface and solid

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D. Liu et al. / Ocean Engineering 113 (2016) 215–223

curves represent the linear solution. As presented in Fig. 5, the condition noted as “breaking” represents the breaker just occurred there. Also shown in Fig. 5 insets is “pre-breaking”, which is used to represent that a breaking wave occurred upstream. During the evolution to breaking of case S0 ¼0.248, the measured wave crests became more peaked while the adjacent troughs flattened, resulting in an increased mismatch with the linear solution. Prior to wave breaking (see Fig. 5(c), S0 ¼ 0.248) the measured wave front was nearly vertical. As expected all of these figures indicate that nonlinear evolution contributes to wave detuning. In addition as also could be expected, with the increase of initial steepness, large deviations between the measured profiles and the linear superposition occurred sooner demonstrating how the initial steepness affects wave detuning. This detuning implies that the phase velocity of the wave components has changed as is also known from Stokes wave solutions. 3.3. Breaking position In this study different wave packets were generated with a linear dispersive focusing technique with the same theoretical focusing point (xf ¼ 9 m) and time (tf ¼27 s) for varying initial steepness. As wave–wave interaction predicts, it is found that the focusing process is far from simple linear superposition. Here, we determine the wave breaking position as a point where the wave has reached its stability limit, but has not begun the irreversible breaking process. The experiments were conducted with increasingly larger initial steepness; the following sections present the main results. Fig. 6 illustrates the breaking positions of different initial steepness packets along the flume. The abscissa represents the distance from the mean position of the wave-maker; the ordinate represents the initial wave steepness. As is shown by the dashed lines in the figure, the wave steepness can be divided into three regions: Region I with S0 o0.274 exhibited only one breaker; Region II, where S0 spanned 0.274–0.326 had two breakers occur with increasing wave steepness except for the two largest steepnesses; and Region III, S0 40.326, in which two breakers also occurred but closer to the wave-maker. During the experiments when S0 was in the range of 0.224–0.248, while it is equivalent to 0.339 in Tian et al. (2010) as a critical value for breaking onset, the breaking occurred immediately before the linear focusing point (x ¼900 cm). Note that it is difficult to precisely determine the exact location of incipient breaking. As the steepness was increased further, the break point also moved upstream compared to the linear focusing point with what appeared by eye to be a more severe spilling. However, the linear and actual focusing points were nearly identical (not shown in Fig. 6) when the initial steepness was small and the breaking did not occur. This result is similar to that reported by Baldock et al. (1996) and Ma et al. (2009), which both showed that the divergence between the measured surface elevations and the linear superposition was small with low wave steepness. As S0 is increased to 0.274, two breakers occurred (within Region II as can be seen in Fig. 6) with the first break point shifted significantly toward the wave maker, more than 1.2Ls (Ls ¼2π/ks) compared to the linear focusing point. While the second break point, which is the only breaker in the Region I, is shifted downstream. (That is, all the black dots and all the blue symbols are assumed to be the same breaking wave as it moves upstream/downstream in the experiment. For example, as the steepness was increased to 0.274, a premature breaker shown by the blue dot for that steepness occurred upstream, but the primary breaker at that steepness is still shown by the black dot.) Within Region II of the graph, an interesting phenomenon occurred – the first break point moved upstream (i.e. the blue symbols that appear) while the second break point moved downstream (i.e. the black dots), until the second breaker disappears, which

Fig. 7. Breaking parameter δbr versus initial wave steepness S0. The horizontal line denotes the threshold, (1.4 70.1) * 10  3, for wave breaking onset as proposed by Song and Banner (2002). The symbols correspond to those in Fig. 6.

occurred near the linear focusing point with a further increase of the input wave steepness. This is due to the nonlinear interaction. Clearly, not all of the wave components participate in the first breaker (accept for the two largest steepness in Region II) because of detuning, and therefore there is sufficient energy remaining in the wave packet that another breaker occurred downstream. In a similar way, when S0 reaches 0.326, a new breaker occurred closer to the wave-maker, and the first break point in Region II becomes the second break point in Region III. These results demonstrate the nonlinear influence and detuning during focusing. 3.4. Examination of the breaking parameter As the wave packets propagated downstream and the energy focused, breaking occurred, and hence the breaking parameter proposed by Song and Banner (2002) that can distinguish wave breaking from non-breaking is examined here. Results for different initial wave steepness are shown in Fig. 7. In this study the wave breaking parameter, δbr, is approximated by Eq. (8); in addition, the parameters are obtained from the recorded video where incipient breaking has occurred, although no multi-phase flow was evident yet. As can be seen in Fig. 7, the breaking parameter δbr values are all greater than the threshold when incipient breaking occurred, which suggests that the non-dimensional parametric mean growth rate remained sufficiently large when multiple breaking happened. Moreover, it can be seen that a strong correlation is present between the breaking parameter, δbr, prior to violent breaking, and the breaking severity (see Figs. 10 and 11). At initial wave steepness S0 less than 0.274 only one breaker occurs; also δbr is found to increase with increasing initial wave steepness. However, beyond this S0, two breakers occurred, and the breaking parameter of the first breaker (blue dots) increased while the breaking parameter corresponding to the second breaker (black dots) decreased. (It is evident from Figs. 10 and 11 that the trend in the variation of the breaking parameter was consistent with that of breaking severity.) 3.5. Evolution of the frequency spectrum To assess the energy dissipation during the breaking event, the frequency spectra of surface elevations along the wave flume were computed and are presented in Fig. 8. Fig. 8(a) shows the spectral evolution near the peak frequencies while Fig. 8(b) exhibits the variation of the higher- and lower-frequency components in addition to the peak frequencies. In the figure, the spectral density, S(f), is non-dimensionalized by kp2fp and the frequency, f, is nondimensionalized by the peak frequency (fp) of the input spectrum.

D. Liu et al. / Ocean Engineering 113 (2016) 215–223

221

Fig. 8. Evolution of the wave frequency spectra for Case F10: (a) linear scale and (b) logarithmic scale.

Fig. 9. A comparison of frequency spectra at two breaking positions: (a) linear scale and (b) logarithmic scale.

The frequency spectrum at x¼ 3.5 m was used as the reference spectrum (dash curves). The data that comprise Fig. 8 exhibit a typical set of breaking wave spectra. As the wave packet propagates downstream, it is found that the focusing process is far from simple linear superposition even though prior to initial breaking (x ¼7.14 m), the energy gain in the higher frequency components (f/fp 41.7) is evident at the expense of the energy in the frequency range f/ fp ¼ 1.0–1.7. This energy transfer to the high frequency components has been observed in previous studies (Kway et al., 1997; Rapp and Melville, 1990). Near x ¼7.14 m and x ¼8.45 m, two wave breakers occurred and the energy transfer reached a maximum, i.e. the slope of the high frequency components (f/fp 41.7) is the least as is clearly shown in Fig. 9, which is consistent with Kway et al. (1997). The spectral energy loss during both breaking events is from the wave components of frequencies higher than the peak frequency,

while the energy in the lower and higher frequency band increases. However, in the downstream defocusing process, the increased energy in the high frequency band returns almost to its initial level, perhaps due to the fact that the energy transfers to high frequency band is reversible. In addition, it is found that the energy in the peak frequency (f/fp ¼1.0) also increases close to the actual breaking positions (x ¼7.14 m and x ¼8.45 m), but the gained energy also disappears downstream. As reported by Meza et al. (2000), who investigated the free wave energy dissipation by decoupling the bound wave components, the energy loss or gain near the peak frequency is insignificant. This result indicates that the contributions to the resultant amplitude from bound-wave components are not negligible even near the peak frequency when breaking occurs. Nevertheless, it is strange that between the two breakers at x ¼8.18 m, the energy in the peak frequency does not increase.

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Fig. 10. Total energy dissipation as a function of the initial wave steepness S0.

Fig. 11. A comparison of breaking dissipation as a function of steepness when two breaking waves occurred. The filled circles denote the upstream breaking wave; the hollow circles denote the second/downstream breake.

According to our measurements, across the low frequency wave components (f/fp o 0.9) energy variation was observed also, with small energy gains. In previous studies (Meza et al., 2000; Rapp and Melville, 1990), this result also was reported. However, no definitive explanation has been made regarding the energy gain in this frequency band. Hence, this may be attributed to the nonlinear energy transfer from the high frequency end of the first harmonic band (f/fp ¼ 1.0–1.7) during the focusing process. 3.6. Energy dissipation Energy dissipation due to breaking is another obvious important quantity and as such has attracted much interest. However, assessing the actual energy dissipation in nonlinear wave groups has been elusive. With the assumption of linear wave theory, the energy dissipation can be inferred from surface elevation measurements as in Rapp and Melville (1990). To estimate the energy loss, a control volume was chosen (5.84 m ox o9.35 m), within which all of the breaking occurred. Based on the assumption that all of the breaking dissipation is contained in the control volume, the energy flux upstream and downstream of the control volume can be approximated by F¼ ρgCgη2, where Cg is the group velocity. The time integration of the energy flux provides the total energy, E (x)¼ ρgCg〈η2〉. Here, E(x) is the total energy and 〈η2〉 denotes the integration of the surface elevation squared over the time required

for the wave group to pass. Thus, the energy loss can be expressed by ΔE/E0 ¼〈Δη2〉/〈η(x0)2〉, wherein, Cg is assumed to be constant along the flume as in Rapp and Melville (1990). Here, ΔE is the energy loss; E0 is the pre-breaking energy density at the reference location x0 ¼3.5 m. It is noted that the wave packet energy obtained by means of integration of energy flux has been known to exhibit some variability; therefore, energy assessment using this technique may contain errors, which contribute to small oscillations of energy in regions along the wave propagation direction. This would be true in wave focusing/breaking regions where nonlinearity increases, and it may be attributed to failure of the linear assumption. However, it is still a robust way to estimate the wave energy with an accuracy of O[(ka)2] (Rapp and Melville, 1990), especially far downstream and far upstream of the breaking region as was done here. A graph of energy dissipation with varying initial wave steepness, S0, is shown in Fig. 10, where the loss is non-dimensionalized by the measured variance at the upstream position (x ¼3.5 m). At wave steepness S0 less than 0.2, no visible breaking occurred, and the energy dissipation was due essentially to sidewall boundarylayers, bottom friction and highly dissipative capillary waves generated on the sidewalls (Perlin and Schultz, 2000). At S0 E0.2, the plot exhibits an obvious rise, which indicates that either nonlinear frictional dissipation is occurring, or that incipient breaking may have occurred although not visible. Once this threshold breaking is exceeded, a relatively small increase in initial wave steepness will lead to the appearance of energetic breaking. With an increase in initial steepness, a spilling breaker first appears with energy loss varying from 5% to 17% referenced to the initial case, i.e. E0 at x ¼3.5 m. It is observed, however, that beyond this point, S0 E0.274 (where two breakers occurred for a given steepness), the energy loss seems independent of the initial steepness and is constant with an increases of initial steepness. On the other hand, to obtain a plunging breaker, a large increase in S0 is required, and at this steepness, there occurred an additional energy loss over the spilling breaker. Additionally, when S0 exceeds 0.318, two breakers occurred for the same steepness, with at least one a plunger. As the initial wave steepness is increased further, the total energy loss of the two breakers remains essentially constant (to within 4%). It is interesting that the energy loss is nearly constant although two breakers occurred. As such, an additional probe was installed between the two breakers and a further comparison in the variation of energy loss was determined as illustrated in Fig. 11. In the case of S0 E 0.274, two breakers occurred in the experiment, with the breaking dissipation for each almost the same (about 9%). However, as is evident in Fig. 11, with an increase of initial steepness the breaking energy loss of the first breaker increased, while the breaking dissipation of the second breaker decreased rapidly until its disappearance, which is in agreement with the visual observations. Due to “detuning” the first wave broke “prematurely” (i.e. some components of the wave packet had not yet arrived when breaking occurred), and these absent components then focused downstream causing another breaker. This result shows that the phenomenon of “detuning” may have significant influence on the energy loss.

4. Conclusions A series of experiments on the evolution of two-dimensional dispersive-focusing wave groups is presented. During the experiments, both non-breaking and breaking wave groups were generated wherein the individual wave components were focused to a point in space and time according to linear theory. Temporal variation of the surface elevation was measured, which is used to

D. Liu et al. / Ocean Engineering 113 (2016) 215–223

examine the evolution of wave groups and the wave frequency spectra. The energy dissipation was determined for the breaking wave groups using wave probes at fixed locations along the wave flume. As expected, it is found that the focusing process is far more complicated than simple linear superposition, and “detuning” occurs due to the nonlinear interactions; this resulted in groups that were focused at more than one location and, more importantly, more than one breaker occurred. The role of initial steepness in the evolution of the wave groups was investigated. While small initial steepness does not affect the focusing point in space, large initial steepness can significantly change the kinematics and dynamics of the wave groups. With an increase of initial steepness, the nonlinear interactions are not negligible and result in a significant increase of wave height upstream of the prescribed focusing point. Beyond the value of S0 E 0.248, obvious breaking occurred and the breaking positions seem to be appropriate: when one breaker occurred, the breaking position moved upstream with an increase of initial steepness; however, with a further increase in initial steepness, two breaking events occurred – the first breaking position has a tendency to move upstream while the second breaking position trended to move downstream. This suggests that detuning appears to dominate the laboratory evolution of wave groups with large initial steepness, i.e. the individual wave components no longer focused at one point in space and more importantly, the nonlinear interactions contributed further to the detuning. An FFT was used to study the spatial evolution of frequency spectra for breaking wave packets during the focusing process. During the focusing process, the transfer of the energy was from the high end of the first harmonic band (f/fp ¼1.0–1.7) to the high frequency band (f/fp 4 1.7). And the frequency spectra in the high frequency band (f/fp 42) almost returned to their initial level further downstream. When the frequency spectra for two breaking waves were investigated, it was found that the energy dissipation from breaking arises from the high end of the first harmonic band (f/fp ¼ 1.0–1.7) as in previous studies. In addition, an energy increase in the low frequency band (f/fp o0.9) was observed. The results of the total energy loss estimated from surface elevations are reported also. As is well known, it is found that the energy loss from breaking depends strongly on the initial wave steepness S0. More interestingly, when two breakers occurred, the energy loss is almost constant. This loss is as much as 17% of the initial energy for spilling, while more than 21% can be dissipated during a plunging breaking event. In addition, a comparison of breaking energy loss due to two breakers shows that with an increase of initial steepness, the breaking dissipation of the first breaker increased, while that of the second breaker decreased rapidly until its disappearance, indicating that detuning has significant influence on the energy loss.

Acknowledgment This research is supported financially by the National Basic Research Program (Grant no. 2011CB013701), the National Natural Science Foundation of China (Grant nos. 51422901, 11172058, 51221961) and a Foundation for the Author of National Excellent Doctoral Dissertation of PR China (Grant no. 201347).

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