# Numerical Simulation of Micro-Bubbles Dispersion by Surface Waves
## 4 Results and Discussion
### Bubbles Dispersion
The profiles of the mean bubble concentration (or void fraction), [\(C\)], and \(x\) and \(z\) components of the concentration fluxes, [\(CV_{x}\)] and [\(CV_{z}\)], were evaluated in DNS according to Equations (28) and (29) for different bubble size, and both with and without wind stress at the water surface (Figure 7). Figure 7 also compares numerical solution for the fluxes with their parameterizations in the form:  
\[[CV_{x}]=V_{sd}[C]\exp(-2 kz), \tag{31}\]  
\[[CV_{z}]=V_{t}[C], \tag{32}\]  
where \(V_{sd}\) is the bubble drift velocity at the water surface (\(V_{sd}\approx 0.02\) and \(V_{sd}\approx 0.03\) for the zero (Figure 7b) and non-zero (Figure 7e) wind-induced-drift cases, respectively); \(V_{t}\) is a terminal velocity determined by numerical solution of the following equation:  
\[0=-\frac{f(\text{Re}_{b})}{\tau}V_{t}+2g\delta_{iz}, \tag{33}\]  
which is obtained from the bubble equation of motion, Equation (5), rewritten for a bubble riking with constant (terminal) velocity, \(V_{t}\), in a quiescent water. Equation is solved by the [PERSON]'s method [26]. Figure 7: Profiles of mean bubble concentration, [\(C\)], (**a**,**d**); and horizontal (**b**,**e**) and vertical (**c**,**f**) components of the concentration (void fraction) fluxes, [\(CV_{x}\)] and [\(CV_{z}\)], normalized with [\(C\)], obtained in DNS for different bubble size, \(d\), (200 \(\upmu\)m, blue color; 300 \(\upmu\)m, red; and 400 \(\upmu\)m, magenta). Top panels (**a**–**c**) are for the cases without the wind-induced surface drift (\(U_{d}=0\)), and bottom panels (**d**–**f**) are for finite \(U_{d}\). Parameterizations for the fluxes in panels (**b**,**c**) and (**e**,**f**), Equations (31) and (32), are shown in dotted and dashed line, respectively. Wave slope \(ka=0.2\). Figure 7 shows that in our numerical experiment, the bubble concentration decreases exponentially with depth, as observed in natural oceanic conditions [12]. Thus, the injection mechanism employed in our DNS adequately reproduces the required void-fraction vertical distribution. The parameterization for the horizontal concentration flux, \([CV_{x}]\), in the absence of the wind-induced surface drift (\(U_{d}=0\), (b), Equation (31)) is obtained using the Stokes-drift solution [1] adapted here for a somewhat smaller surface drift velocity (\(V_{sd}\approx 0.02\)) as compared to the classical surface Stokes-drift velocity (\(c(ka)^{2}\approx 0.04\) for the considered surface-wave slope \(ka=0.2\)). Note that a similar effect of a reduced Stokes drift velocity was observed in DNS of a free-propagating, non-breaking surface waves [31]. In the presence of the wind-induced surface drift (\(U_{d}\approx 0.04\)), the same parameterization is used but with \(V_{sd}\approx 0.03\). Both parameterizations agree well with DNS results. The vertical component of the void-fraction flux, \([CV_{z}]\), is well predicted by Equation (32) in both (zero and non-zero wind induced drift) cases. That means that neither wave-induced irrotational motions nor turbulence affect on average the bubble rising rate, so that the void fraction vertical flux is analogous to that in a quiescent water.