Void Fraction and Turbulence

The entrapped air induced by wave breaking generates strong turbulence near the free surface. Cox and Shin (2003) reported the dependence of void fraction on turbulent intensity in the bore region of surf zone waves. Therefore, the relationship between void fraction and turbulence is very significant. Moreover, it is also important for turbulence modeling. The simultaneous observations of void fraction and turbulence for the breaking waves of the mid-scale experiments were conducted in this study. Therefore, the relationship between the void fraction and turbulence will be examined in this section.

Using the phase-averaged method the velocity data were divided by the mean and turbulence components of the velocity. Figure 12 shows an example of phase-averaged time series of surface elevation and turbulence components at $z/H_b=0.0333$ (above still water level) for Case 1. The notation of $\sigma_{u}^2$, $\sigma_{v}^2$ and $\sigma_{w}^2$ are turbulent stress for velocity components $(u,v,w)$. Similar to the laboratory experiments by Ting and Kirby (1994), Cox et al. (1994) and Cox and Shin (2003). They showed that the temporal variation of $\sigma_u^2$ in the surf zone fluctuated in the range $400<\sigma_u^2<500$ $({\rm cm}^2/{\rm s}^2)$. Figure 12 shows that the magnitude of $\sigma_{u}^2$ is similar at the still water level $z/H_b=0.0$ but is large above the still water level. The peak of turbulent stress is located at the front of crest and decreases linearly as time passes. This is a typical time series of turbulence characteristics of this and other laboratory experiments on depth-limited breaking waves.

Figure: Example of phase-averaged surface elevation $\langle \eta \rangle$ and turbulence components 1.0m from B.P. in Case 1. (a) Free surface $\eta$ (solid) and standard deviation $\eta \pm \sigma _\eta$, (b)-(d) temporal variation of $\sigma _u$ (thick solid), $\sigma _v$ (thin solid) and $\sigma _w$ (dotted) and (e) the phase-averaged void fraction $\langle \alpha \rangle$

Figure 13 shows the horizontal relationship between time-averaged void fraction: $\overline {\alpha }$ and time-averaged total kinetic energy (TKE): $\overline {k}$ for Case 1-3. The total kinetic energy and time-averaged TKE are defined by

$$k = \frac{1}{2} (\sigma_u^2 + \sigma_v^2 + \sigma_w^2)$$ (11)

$$\overline{k} = \overline{k} = \frac{1}{2} (\overline{\sigma_u^2 + \sigma_v^2 + \sigma_w^2})$$ (12)

The spatial variations of time-averaged void fraction $\overline {\alpha }$ and time-averaged TKE: $\overline {k}$ show similar tendency, although the peak of $\overline {\alpha }$ appears $0.2 \times H_b \sim 2 \times H_b$ offshore than the peak of $\overline {k}$. The short spatial lag between $\overline {\alpha }$ and $\overline {k}$ might be associated with the entrapped air generated turbulence. The spatial variations of the turbulence and void fraction are highly correlated below $z/H_b=0.013$. The entrained bulk of air and bubbles generate turbulence and turbulence splits the coarse bubbles into fine bubbles in this phase. Moreover, TKE and the void fraction increased ten times from $z/H_b=-0.013$ to $z/H_b=0.033$, thus two-phase flow characteristics is remarkable near the wave crest. In addition, there is significant difference between the turbulence and void fraction $x_b/H_b>0.4$ at $z/H_b=0.033$ in Case 1. The void fraction at $z/H_b=0.033$ keeps high level after reached at its peak but $\overline {k}$ is decreased in this region. This vertical level is re-aerated region of breaking wave induced bubble and is corresponds to silent-phase by Deane and Stokes classification. Neglecting the short spatial lag between void fraction and TKE, the direct comparison between time-averaged void fraction: $\overline {\alpha }$ and normalized time-averaged TKE: $\overline {k}$ of the all measuring locations is shown in Figure 14. For small value $\overline {\alpha }$ or $\overline {k}$, the relationship between them shows high correlation and it becomes scattered at high void fraction or TKE region. Overall, the relationship between time-averaged void fraction and time-averaged TKE is clear. In addition, the incident wave dependence on the relationship between $\overline {\alpha }$ and $\overline {k}$. The similar analysis has been examined by Cox and Shin (2003) but their results were more scattered than us. This is based on the limitation of accurate turbulence measurements near the wave crest. The relation between the void fraction and TKE, the scale dependence of void fraction and the further theoretical consideration will be required finding the universal relationship between the turbulence characteristics of breaking wave and void fraction.

Figure 13: Horizontal relationship between time-averaged void fraction $\overline {\alpha }$ (solid line) and time-averaged TKE: $\overline {k}$ (dotted line) in Case 1

Figure 14: Relationship between time-averaged TKE:$\overline {k}$ and time-averaged void fraction $\overline {\alpha }$. ($\bullet$: Case 1, $+$: Case 2, $\square$: Case 3)