Void Fraction and Bubble Spectra

The characteristics of entrained air in the surf zone will be examined in this section. First, the horizontal (onshore-offshore) distributions of time-averaged void fraction $\overline {\alpha }$ for Case 1 and 2 are shown in Figure 9. The solid line indicates the void fraction for the mid-scale experiments and dotted line indicates the void fraction for the large scale, respectively. The maximum values of $\overline {\alpha }$ are 0.19 for Case 1 and 0.24 for Case 2. The maximum value decreases rapidly as the measurement location moves down to the still water level. The difference of the maximum $\overline {\alpha }$ between $z/H_b=0.066$ and 0.0 are four times for Case 1 and ten times for Case 2. Cox and Shin (2003) reported exponential decay of the void fraction in time. Therefore, it should be noted that the instantaneous void fraction is several times larger than time-averaged value. The locations of $\overline {\alpha }$'s peak for each vertical layer are about $6 \times H_b$ and $4 \times H_b$ from the B.P. for Case 1 and 2, respectively. For the spilling breaker (Case 1), the difference of $\overline {\alpha }$ between the mid- and small-scale experiments is about two times at most. Moreover, for spilling/plunging breaker (Case 2), the difference of $\overline {\alpha }$ between the mid- and small-scale experiments is larger than spilling case and becomes four times more at most. This is because the strong wave breaking entraps a larger volume of air and this large-size bulk of air can rise faster than a small one. In addition, there is the second peak of void fraction in Case 2 of the mid-scale experiments. This is due to the second splash of breaking wave in the mid-scale experiments but was not clearly observed in the small-scale experiments. The relative size of the surf zone between the small and mid-scale experiments were 0.53 in Case 1 and were 0.70 were Case 2. These were not followed Froude similarity law. Therefore, the scale effects of the void fraction on Case 2 is stronger than that of Case 1.

Figure 9: Horizontal variations of time-averaged void fraction $\overline {\alpha }$ (solid line: mid-scale experiments, dotted line: small-scale experiments). (a) Case 1. (b) Case 2

The scale effects on void fraction are significant in comparison with two different scale experiments. Entrapped air by breaking waves is split into small bubbles due to the shear and turbulence of surrounding flows. The early work of this problem for the general condition had been discussed by Kolmogorov (1949) and Hinze (1955) in the middle of the last century and a large bibliography has been generated (i.e. Martínez-Bazán et al., 1999). Therefore, it is of interest to verify the influence of the void fraction on the bubble size spectra. The spatial changes of bubble size spectra for large scale bubbles (chord length $d>0.1$mm) above the still water level are shown in Figure 10. The horizontal axes are bubble size $d$ (chord length) and normalized distance from the breaking point $x_b/H_b$, respectively, and the vertical axis $N_d(d,x)$ is the number of bubbles per unit square per one wave period. The left hand side and right hand side in Figure 10 are the spatial variation of the measured bubble size $d$ for the mid- and small-scale experiments, respectively. The bubble size spectra increase rapidly from the B.P. in Case 1 of the mid-scale experiments. After that, the number of small size bubbles decreases monotonically towards the shoreline, although the time-averaged void fraction remains constant as shown in Figure 9 (a). The difference between Case 1 and 2 of the mid-scale experiment is clear. The bubble size spectra in Case 2 of the mid-scale experiment show two clear peaks spatially, and the locations of peaks appear at the same location of the time-averaged void fraction as shown in Figure 9 (b). The difference of the spatial distribution of bubble size spectra between Case 1 and Case 2 of the mid-scale experiment is hard to explain but advection effect is one possibility. There is no clear peak and the continuous shape of the spectra can be observed in the small-scale experiments due to statistical fluctuation. Thus, the power-law scaling of bubble size spectra is discussed next paragraph.

Figure 10: Horizontal variations of bubble chord length spectra at $z/H_b=0.033$ (horizontal axes: distance from B.P. and bubble size, vertical axis: number of bubble per one wave period). (a) Case 1 (left panel: mid-scale experiments, right panel: small-scale experiments). (b) Case 2 (ditto).

To check the power-law scaling of the bubble size spectra, Figure 11 shows the PDF of time-space averaged bubble size from top to bottom and offshore to onshore side.

$$P(d){\rm d}d = \frac{1}{N} \sum_{x} N_d(d,x) \Delta d$$ (6)

where $P(d)$ is the PDF, $N$ is total number of measured bubble per one wave period and $\Delta d$ is the increment of the bubble chord length of bubble size spectra. The spectra are normalized by the total number of bubbles for each case. Therefore the number of bubbles are ignored in the figure. The notations of $d_m$ in the figure indicate the mean chord length, and the power-law scaling of bubble size $d$ and linear lines are given power-law curves. The power-law of the bubble size spectrum tail was estimated by the least square method in a log-scale domain. The mean chord lengths for mid- and small-scale experiments are 0.56 cm and 0.51 cm in Case 1 and 0.56 cm and 0.61 cm in Case 2, respectively. Thus, there is no influence of scale and breaker type on the mean chord length. On the other hand, the power-law scalings are also similar to each case and the different experimental scales. The power-law scalings are in between $d^{-1.2\sim-1.9}$. The Weber number $We$ can be defined as

$$We = \frac{\rho_w}{\gamma}u^2 D$$ (7)

where $\rho_w$ is the water density, $\gamma$ is the surface tension coefficient, $D$ is the bubble diameter, and $u$ is turbulence velocity of the bubble scale $D$. The turbulence velocity $u$ of the bubble scale in the inertial range can be given by $u=2\varepsilon^{2/3}D^{2/3}$ where $\varepsilon$ is the energy dissipation rate. Then, the Hinze scale $D_H$ (Deane and Stokes, 2002) of bubble splitting can be given by

$$D_H = 2^{-\frac{8}{5}} c \left( \frac{\rho_w}{\gamma} \right) \varepsilon^{-\frac{2}{5}}$$ (8)

where $c=(We^c)^{3/5}$ and $We^c$ the critical Weber number. Garrett et al. (2000) proposed semi-empirical $-10/3$ power-law scaling for the bubble which is larger than the Hinze scale based on the discussion of bubble fragmentation in the strong turbulence flow below the trough level. The measured power-law scaling of bubble size spectra is wider than Garrett et al. (2000). On the other hand, Deane and Stokes (2002) measured a $-3/2$ power-law scaling, smaller than the Hinze scale (1mm) of the acoustically active phase near the crest. The power-law scaling of the time-averaged chord length shown in Figure 11 is similar to the power-law scaling of the small-scale bubble of Deane and Stokes (2002). Although we measured the bubble chord length, the theory is formulated for the bubble diameter. Therefore, the power-law of the spectra of DVP measured chord length can be changed into bubble diameter spectra as discussed in a previous section. Therefore, the estimation of the Hinze scale is important to discuss in detail. Following Hinze (1995), $c=0.363$ if $D_H$ is defined such that 95% of the air is contained in bubbles with a radius less than $D_H$. Battjes and Janssen (1978) formulated the depth-averaged energy dissipation rate $\overline{\varepsilon}$ for surf zone breaking waves as

$$\overline{\varepsilon} = \frac{g}{4}\frac{H^2}{Th}$$ (9)

where $h$ is the water depth. Similar formulations have also been made by the other researcher with different coefficient and parameters (e.g. Feddersen and Trowbridge, 2005). For the mid-scale and small-scale experiments of Case 1, $\overline{\varepsilon}$ are 0.254 and 0.0414 W/kg by Battjes and Janssen theory and it gives $D_h=2.01$ and 4.28 mm, respectively. However, the energy dissipation rate shows a dependence on water depth (e.g. Ting and Kirby, 1996). Svendsen (1987) modeled the dissipation rate through the relation

$$\varepsilon = - \frac{C_D}{l}k^{\frac{3}{2}}$$ (10)

where $C_D$ is the empirical constant and $l$ is the length scale, using $l=h/5$ for the spilling breaker. The measured $k$ was $0.05 \sim 0.15 {m^2/s^2}$ which will be shown later, and it gives a dissipation rate of $0.05 \sim 0.26$ W/kg. Similar experiments on 1/35 plain slope was conducted by Ting and Kirby (1996). Their breaking wave height $H_b$ was 0.163 m and is almost equivalent to our Case 1, although their water depth was different. The maximum energy dissipation measured by Ting and Kirby (1996) was 0.01 W/kg below the trough level. Therefore, the measured or estimated energy dissipation above still water level is ten times larger than below the trough level. If the bubble chord length can be approximated by bubble diameter, the experimental data of the mid-scale and small-scale experiments shows similar chord length spectrum profiles with different peak chord length. The scale differences in this study only play peak chord length difference with the same profile of size distribution. The detail mechanisms and relationship between the Hinze scale and energy dissipation should keep for future study.

Figure 11: PDF of time-space averaged bubble chord length and their statistics (dashed line and $h=0.8$: mid-scale experiments, dotted line and $h=0.3$: small-scale experiments). (a) Case 1 (left: linear scale, right: log scale). (b) Case 2 (ditto).