# Numerical Simulation of Micro-Bubbles Dispersion by Surface Waves
## 4 Results and Discussion
### Bubbles Dispersion  
The trajectories of individual bubbles of different sizes (with diameters \(d\) = 200 \(\upmu\)m and 400 \(\upmu\)m) obtained using DNS without and with wind-induced surface drift, are presented in Figures 5 and 6, respectively. The figures also show the depth-dependence of the forces imposed on the bubbles by the surrounding water (drag, fluid acceleration, and lift) along the trajectories. Figure 2: Instantaneous distribution of the vorticity modus, \(\omega\), obtained using DNS in central (\(x\),\(z\)) (**a**,**d**) and (\(y\),\(z\)) (**c**,**f**) planes and in a near-surface (\(x\), \(y\)) plane at \(\eta=-0.05\) (**b**,**e**). Panels (**a**–**c**) are for the no-bubbles case, and panels (**d**–**f**) are for DNS with bubbles with diameter \(d=400\) μm. In both cases, surface drift velocity, Equation (21), is added at the waved boundary accounting for the wind stress. Wave slope \(ka=0.2\). Figure 4: Mean profiles of water velocity fluctuations obtained using DNS with wave slope \(ka=0.2\) and different bubbles diameter, \(d\). Panels (**a–c**) are for the case \(U_{d}=0\) and panels (**d–f**) are for the non-zero surface drift case. Figure 3: Phase-averaged water velocity components, \(U_{x}\) (**a,b**) and \(U_{z}\) (**c,d**), obtained using DNS with wave slope \(ka=0.1\) (**a,c**) and \(ka=0.2\) (**b,d**) with and without bubbles (black and blue lines, respectively). Dashed line shows analytical solution for the deep-water surface wave [1]; \(U_{d}=0\). Figures 5 and 6 show that the influence of the surface-wave-induced motions on the bubbles dynamics increases as they rise toward the water surface: the trajectories, being almost straight-vertical at \(z/\lambda<0.5\), oscillate with an increasing amplitude as the depth decreases and start spiraling sufficiently close to the surface (Figures 5a,e and 6a,e). The bubbles experience a mean drift in the direction of the surface wave propagation. This drift is similar to the classical Stokes drift of the Lagrangian fluid particles in a surface wave and caused by the decreasing dependence of the wave motion amplitude on depth [29]. As  
Figure 5: Trajectories of individual bubbles (**a**,**e**) and \(x\) and \(z\) components of forces imposed on the bubble by the surrounding water [drag \(F_{d}\) (**b**,**f**); fluid acceleration \(F_{a}\) (**c**,**g**); lift \(F_{L}\) (**d**,**h**)] obtained using DNS without surface-induced wind drift, \(U_{d}=0\). Top and bottom panels are for the bubble diameter \(d=400\) and \(200\) μm, respectively. Wave slope \(ka=0.2\). For convenience, in panels (**b**,**f**) the mean value of the \(z\)-component of \(F_{d}\) is subtracted. expected, the drift becomes more pronounced in the presence of the wind-induced surface drift (Figure 6). In the considered case, however, this drift is also modified by the bubbles' ascent due to buoyancy, so that larger bubbles (\(d=400\)\(\upmu\)m) are less subject to this mean Lagrangian drift since their terminal rise velocity is almost 4 times larger as compared to 200 \(\upmu\)m- bubbles (cf. Figure 7 below). As a result, their dynamics are less affected by the oscillatory wave-field motion. In this aspect, there is some analogy between the observed reduction of the drift due to bubble rising and the \"crossing trajectories\" effect governing the dispersion of inertial particles by turbulence, where particles' settling reduces their dispersion by ambient turbulence [30]. Figure 6: The same as in Figure 5 but obtained using DNS with finite \(U_{d}\). The analysis of forces governing bubbles dynamics reveals that the drag force, \(F_{d}\), and the fluid-acceleration force, \(F_{a}\), are both of the same order, and much larger as compared to the lift force, \(F_{L}\). The forces, \(F_{a}\) and \(F_{d}\), increase monotonically as the bubbles rise toward the water surface, whereas the lift force becomes non-zero only in close vicinity of the surface, where the fluid motion is not strictly irrotational due to the presence of wave-induced turbulence [17].