3.1. Characterization of the Plunging Breaker
The generated plunging breaker occurred at
s, which were close to the designed focal location and time (
s). Figure 3
shows the front images of the breaking wave, which were taken by the GoPro camera. A large, steep wave crest was observed at t = 22.7 s (Figure 3
a) due to the focusing of the wave components. The wave crest advected faster than the wave, and an overturning jet occurred and fell on to the water surface, which led to the formation of a plunging breaker (Figure 3
b). The plunging breaker retained a large amount of energy and caused a water splash-up, which subsequently resulted in a secondary breaker (Figure 3
c). The wave kept propagating forward after breaking, and the water surface started to recover at t = 23.9 s (Figure 3
d). “Whitecaps” were observed during the breaking process, which are small bubbles formed by the entrained air due to the breaking wave.
shows side views of the breaking wave, which were taken by the GoPro camera. Each cell on the grid paper has a dimension of 3.85 (L) × 1.3 (W) cm. It is shown in Figure 4
a that the wave height was about 22.1 cm (17 cells) when the overturning jet hit the water surface at t = 23.2 s. The water splash-up caused by the major plunging breaker at t = 23.3 s had a height of about 15.6 cm (12 cells) (Figure 4
b). The wave propagated forward about 35 cm (9 cells) within 0.2 s, which corresponds to a phase speed of 1.75 m/s. The successive impingement between the breakers and water surface caused a large number of water ligaments and “white caps” (Figure 4
shows the temporal evolution of water levels at different locations measured by the sonic wave sensors. It can be observed that there were three large wave crests which passed through x = 2 m from t = 18 to 22 s. The number of wave crests which passed through x = 4 m decreased, but the wave crests became higher and steeper, which reflected the focusing process of the wave components. A very steep high wave crest was observed at x = 6 m around t = 22.7 s, which led to the occurrence of the plunging breaker at t = 23.2 s (indicated by the red dashed line; see also Figure 4
a,b). The wave height (maximum distance between trough and following crest) was around 0.40 m before breaking (x = 6 m). The water elevation at x = 8 m shows that a large wave crest propagated following the breaking wave, and the water surface recovered afterwards. The water elevations at different locations clearly show the propagation and focusing process of the waves.
The velocity components under the plunging breaker (at x = 7 m, and 23.5 cm below the mean water level 1.5 m) measured by Acoustic Doppler Velocimetry (ADV) are shown in Figure 6
. The velocities were relatively small in each direction before the occurrence of the plunging breaker. Large velocity fluctuations were observed in the horizontal and vertical direction after the occurrence of the plunging breaker (t > 23 s). The maximum forward velocity was about 0.3 m/s, and the backward velocity reached up to −0.6 m/s, which indicates a strong backflow caused by the breaker. A phase shift was observed between horizontal and vertical velocities. The fluctuation of the vertical velocities ranged from −0.3 to 0.3 m. After the passage of the wave train (t > 27 s), the velocities in each direction rapidly decayed. The velocities in the cross-tank direction were much smaller than the other directions during the whole test, which reflects the fact that the hydrodynamics of the breaker were nearly 2D. Other sets of measured velocity data can be found in the Supplementary Materials
3.2. Oil Dispersion under the Plunging Breaker
shows the snapshots of the oil dispersion experiments taken by the GoPro camera. The experiment setup before starting the wave paddle is shown in Figure 7
a. One hundred grams of Heidrun crude oil was released at x = 7.2 m, and the shadowgraph camera was deployed 1.2 m downstream of the crude oil (see also Figure 2
b). A plastic O-ring was used to prevent the oil from spreading, which was taken out of the tank right before the occurrence of the plunging breaker. The oil films dispersed into oil droplets after being hit by the plunging breaker (Figure 7
b within the yellow circle), and the dispersed oil droplets were transported downstream to the measurement window of the shadowgraph camera with the propagation of the wave (Figure 7
c). An underwater view of the dispersed oil droplets taken by a GoPro camera is shown in Figure 7
d, which demonstrates that droplets of various sizes were entrained in the water column to different depths. Figure 7
e shows a side view of the dispersed oil plume after the passage of the wave train. It appears in Figure 7
e that a large number of oil droplets were formed and entrained into the water column, and several oil films remained in the water surface (see the yellow rectangles).
shows an example of the photos taken by the shadowgraph camera. The dark disks indicate the oil droplets of various sizes, while the hollow circles indicate the air bubbles. The background was removed by setting a threshold of the gray value. The air bubbles were eliminated manually or taken out during the statistical process by checking the mean gray value (i.e., the mean gray value of a dark sphere is 255, and otherwise the value is less than 255) and circularity. The clustered droplets were broken up using the Watershed algorithm provided by ImageJ. Due to the resolution of the shadowgraph cameras, only droplets larger than 150 µm were considered during the analysis. The obtained droplet sizes were assigned to different size bins to form a droplet size distribution (DSD). The droplet size bins ranged from 150 to 2400 µm with a 150 µm interval. Thirty photos after the passage of the wave train (t = 35–38 s) were analyzed to obtain a temporally averaged DSD.
shows the volume and number of droplets in each size bin provided by ImageJ using the photos taken by the shadowgraph camera. The median value of a size bin was used to represent the bin in the results (i.e., we use 225 µm to represent a droplet bin of 150–300 µm). The captured volume-based DSD seemed to be bimodal (Figure 9
a). The standard deviation significantly increased for droplets larger than 1500 µm, which reflects the fact that the number of these droplets was generally small (but their volume was not). The volume-based DSD in Figure 9
a was found to fit a Gaussian distribution (
(µ = 1.2 mm, σ2
= 0.29 mm2
)) at the 5% significance level by the Kolmogorov–Smirnov test [34
]. The volume-based DSD is plotted on a normality sheet in Figure A2
, and the cumulative volume-based DSD was reported in Figure A3
. A Gaussian distribution was observed in numerous studies of DSD in reactors [33
]. Section 4
addresses this point in more detail. Figure 9
b shows the number of droplets as a function of the diameter in a log-log plot to allow comparison with the correlation reported in DS1988. The figure shows that droplets smaller than 1500 µm generally followed the DS1988 correlation given as N(d) ~ d−2.3
. However, for droplets larger than 1500 µm, the number of droplets decreased much faster with the diameter and may be approximated by N(d) ~ d−9.7
, as reported by Li et al. [11
] based on their measurements of oil droplets following a plunging breaker. Figure 9
also reports the simulated DSD using the VDROP model, which will be discussed in the following paragraphs. A similar DSD was found in the repeat experiment, which proves the repeatability of the experiment. The DSD in the repeat experiment could also be well approximated by the reported power–law correlations (see Figure S3
The energy dissipation rate ε in each direction was calculated based on Equations (3)–(5) using the measured velocities (Figure 6
). The results are shown in Figure 10
, and they display strong intermittency (sudden jumps), with high energy dissipation rates occurring mainly between 23 to 26 s, coinciding with the wave breaker. The energy dissipation rates in the horizontal (x) and vertical (z) directions were much higher than those in the cross-tank (y) direction, which reflects the two-dimensional nature of the wave motion (by design).
The breakup of oil droplets occurs at the µm-scale and within milliseconds (see the Discussion section of Zhao et al., 2014 Chemical Engineering Journal). In particular, it is well known that large ε values cause the breakup, but these occur over short durations (milliseconds and below) and cover only a small fraction of the volume. The latter has formed the basis for using a multifractal representation of ε, where the active or effective volume of mixing is only a small fraction (less than 5%) of the total volume [37
]. A discussion in the context of oil dispersion was also presented in more recent works [9
]. In our case, we needed to use a “design” or representative value of the energy dissipation rate ε to integrate the temporospatial interaction of the hydrodynamics and the oil plume. This is similar to the concept of the significant wave height in coastal engineering used as a design parameter for structures [41
], where the significant wave height is obtained as the average of the top one-third of wave heights. For this reason, we used the average of the top 30% of the ε values during the breaker (i.e., from 23 to 26 s). This was done in each direction to obtain the design ε, labeled
, reported also in Figure 10
. The values of
were 0.39 watts/kg along the tank (horizontal), 0.18 watts/kg vertical, and 0.07 watts/kg across the tank (horizontal).
As input to VDROP, the largest size of oil droplets observed in the experiment was 2400 µm (see Figure 9
a) and was adopted as the maximum droplet size in VDROP simulations. The size bin interval was taken as 150 µm, which was the same as that used in processing the experimental results. As reported in Figure 10
, the maximum
was equal to 0.39 watts/kg found in the horizontal direction along the tank. We elected to use the maximum value of the directions to characterize the breaker, which was due to the fact that breakage depends on the large values of ε in the field [43
]. Thus, neither adding ε in the three directions nor taking the average of the three directions was correct. We used the value of 0.4 watts/kg and varied the duration of the breaking from 0.1 to 0.5 s with an increment of 0.01 s to match the observed oil DSD. Figure 9
a shows that the agreement for the volume of oil droplets of each size is good for a duration of 0.15 s. The simulated number of oil droplets as a function of diameter also closely matched the observed results (see Figure 9
b). The VDROP simulation moderately underestimated the number of 225 and 1425 µm droplets and slightly overestimated the number of 2425 µm droplets. However, the difference was within the level of uncertainty of the data.
We also conducted a sensitivity analysis study using energy dissipations of 0.2, 0.4, and 0.6 watts/kg for time durations of 0.1, 0.15, and 0.2 s; thus, a total of nine simulations were run. The results are shown in Figure 11
and show that the volume of oil droplets transformed from large droplets to small droplets as the simulation time increased (moving down each column of Figure 11
). In addition, the DSD sometimes shifted from bimodal to unimodal (see results in the third column with an energy dissipation rate of 0.6 watts/kg). The breakup of large-size droplets was enhanced by increasing the energy dissipation rates (from right to left in each row of Figure 11
). Two simulations provided similar DSDs and showed reasonable agreements with the experimental observation, which were the cases which used a dissipation rate of 0.6 watts/kg for a duration of 0.1 s and a dissipation rate of 0.4 watts/kg for a duration of 0.15 s. This indicates that similar DSDs could be obtained by running VDROP using a larger energy dissipation rate for a shorter period or a smaller energy dissipation rate for a longer period.