Experimental Setup

A set of experiments was examined at two different scales in two narrow wave flumes of different length but similar aspect ratio. Mid-scale experiments were conducted using a two dimensional wave flume (1.0 m wide, 1.5 m high and 50.0 m long) at Osaka City University. A fixed, impermeable 1/30 slope was installed with the toe 20 from the wavemaker. The water depth was 1.0 m, and regular wave trains were generated by a PC controlled wavemaker with an active wave absorber. A detail of the mid-scale experimental setup is shown in Figure 5 where $x_s$ is the horizontal coordinate from the shoreline ($x_b=0$), $x_b$ is the horizontal coordinate in the direction of wave propagation with $x_b=0$ at breaking point (B.P.), and $z$ is the vertical coordinate positive upward with $z=0$ at the still water level. To verify the scale effects, the small-scale experiments were also conducted with the same hydrodynamic conditions following Froude similarity law.

$$\frac{T_{small}}{T_{mid}} = \sqrt{ \frac{h_{small}}{h_{mid}} }$$ (5)

where $T_{small}$ and $T_{mid}$ are wave periods for the small and mid-scale experiments and $h_{small}$ and $h_{mid}$ are water depths for the small and mid-scale experiments, respectively. Small-scale experiments were conducted using a narrow wave flume (0.5 m wide, 0.5 m high and 20.0 m long) with a 1/30 slope also at Osaka City University. The water depth of the small-scale experiments was 0.2 m, and regular wave trains were generated by mechanically controlled wavemaker.

Figure 5: Illustration of experimental setup

Figure 6: Coordinate of ADV probe

Mid-Scale Experiments

For mid-scale experiments, the water surface elevation water velocities, void fractions and bubble chord length distributions were measured by an array mounted with two wave gages, an ADV and DVP (see Figure 5). The offshore side wage gauge was used as a trigger for starting data acquisition of the ADV and DVP, and the onshore side wave gauge measured surface elevation at the point where the ADV and DVP were located. The data acquisition of the wave gauge, ADV and DVP was started when the water surface became higher than the still water level at offshore side wave gauge. The DVP measures not only bubble chord length but also horizontal bubble velocity. Due to the instrument limitations, the ADV was mounted at a 30 degree incline to the array, horizontally (Figure 6). Therefore, the measured velocity vectors were transformed to the normal coordinate using rotation matrix. The mean and turbulence components of the velocity are extracted from despiked data using phase ensemble method. Figure 7 shows an example of the instantaneous time series of the surface elevation, water velocities and void fraction at 1.0 m from B.P. and $z=0$ (still water level) for Case 1 (experimental conditions will be described later). The panel (a) in Figure 7 indicates the surface elevation, the panels (b)-(d) are horizontal, vertical and cross sectional velocity components, respectively. The panel (e) is instantaneous time series of void fraction and solid and dashed lines in the panel are the void fractions of front and rear senor of the DVP (see Figure 1). As mentioned in the previous section, the measured ADV data were filtered by the 3D phase space method (Wahl, 2003; Mori et al., 2007). The spike noise in the ADV recorded data were excluded clearly as shown as in Figure 7. The DVP and ADV were synchronized by the surface displacement of the wave gage at the measurement location. The air-liquid interface phase velocity is calculated by the temporal difference between the front and rear sensor. All velocity components are relatively smooth and therefore void fraction is also remains constant in this case.

Figure 7: Example of instantaneous time series for Case 1 at 1.0 m from B.P. at $z=0$ cm. (a) free surface elevation, (b) horizontal velocity: $u$, (c) vertical velocity: $w$, (d) cross sectional velocity: $v$, (e) void fraction: $\alpha$ (solid line: front sensor, dashed line: rear sensor)

The DVP was mounted with the measurement array and offset by 1 cm to the onshore side of ADV measurement volume. The three cases of regular waves were run, characterized by their breaking type to give spilling (Case 1), spilling/plunging (Case 2), and plunging (Case 3) breakers. Table 1 summaries the wave statistics for Case 1 to 3 where $T$ is the wave period, $H_0$ is the deep-water wave height, $H_b$ is the breaking wave height at the breaking point (B.P.), $h_b$ is the water depth at the breaking point, and $X_s$ is the length of the surf zone. Figure 8 shows a vertical sectional view of the measurement points for the three cases. The measurements were traversed horizontally 8-10 points and vertically 2-5 points depends on the wave height and local water depth. The measurements locations were mainly selected near the water surface because where is highly aerated region due to wave breaking. For each trial, measurements were conducted for several minutes to establish equilibrium conditions of breaker position. Data were recorded at each locations for 50 waves. The wave gauge and ADV were sampled at 25 Hz, and the DVP were sampled at 5 kHz.

Table 1: Characteristics of wave statistics for mid-scale experiments ($T$: incident wave period, $H_0$: incident wave height, $L_0$: incident wave length, $H_b$: wave height at breaking point, $h_b$: water depth at breaking point, $X_s$: width of surf zone, $\xi _0$: surf similarity parameter)

Case	Type	$T$[s]	$H_0$[cm]	$L_0$[m]	$H_0/L_0$	$H_b$[cm]	$h_b$[cm]	$X_s$[m]	$\xi _0$
Case 1	Spilling	1.6	16.3	3.55	0.046	16.5	16.8	5.04	0.16
Case 2	Spilling/Plunging	2.0	11.5	4.85	0.024	12.0	12.5	3.74	0.22
Case 3	Plunging	3.8	12.2	10.25	0.012	12.6	13.0	3.90	0.30

Table 2: Characteristics of wave statistics for small-scale experiments ($T$: incident wave period, $H_0$: incident wave height, $L_0$: incident wave length, $H_b$: wave height at breaking point, $h_b$: water depth at breaking point, $X_s$: width of surf zone, $\xi _0$: surf similarity parameter)

Case	Type	$T$[s]	$H_0$[cm]	$L_0$[m]	$H_0/L_0$	$H_b$[cm]	$h_b$[cm]	$X_s$[m]	$\xi _0$
Case 1	Spilling	1.0	6.1	1.37	0.045	6.5	8.5	2.66	0.16
Case 2	Spilling/Plunging	1.2	4.3	1.77	0.024	5.0	8.4	2.60	0.22

Figure 8: Measurement locations of mid-scale experiments ($+$: crest and trough location, $\bullet$: location of measurement points)

Small-Scale Experiments

The small-scale experiments were scaled using the Froude similarity law as already shown in previously. The water depth of the small-scale experiments was 0.3 m. Therefore, the spatial and temporal scales were reduced 3/8 and $\sqrt{3/8}$ of the mid-scale experiments, respectively. For small-scale experiments, only the water surface elevation, void fractions and bubble chord distributions were measured to avoid the influence of the ADV probe on turbulence characteristics. Moreover, due to the limitation of wavemaker, the long period incident wave case (Case 3) was excluded from the experiments. Therefore, only Case 1 and 2 were conducted to measure the void fractions and bubbles for the small-scale experiments.