**4. Results and discussion**

One of the great advantages of the numerical models is their ability to show the evolutions of vorticity and turbulence quantities in the spatial and temporal domains, which are too expensive to be investigated by experiments. Using the SPH computational results, the turbulent kinetic energy distributions are shown in **Figures 7a**–**e** and **8a**–**e**, respectively, for the spilling (T1) and plunging (T2) waves. For both breakers, the turbulence quantity has the largest values near the free surface and decreases into the water column. However, the results highlight that there exist fundamental differences in the dynamics of turbulence between the spilling and plunging breakers, which can be related to the processes of wave breaking production.

For the spilling wave (T1), higher turbulence levels are mainly concentrated in the breaking wave front and the highest turbulence level appears in the roller region (**Figure 7d**). After the breaking, as the wave propagates forward, the turbulence kinetic energy decreases (**Figure 7e**). Instead, the turbulence levels increase rapidly after the wave breaking for the plunging case (T2) as shown in **Figure 8c**–**e**.

**Figure 7.**

*Instantaneous turbulence intensity distributions in the SPH simulation of spilling wave (T1): (a) before; (b)–(c) during and (d)–(e) after breaking.*

**Figure 8.**

*Instantaneous turbulence intensity distributions in the SPH simulation of plunging wave (T2): (a) before; (b)–(c) during and (d)–(e) after breaking.*

The maximum turbulence level is generated as the plunging jet touches down on the wave trough (**Figure 8d**) in sections 46–45 (**Figure 2**); After the breaking, the roller continues to spread downwards and therefore high turbulence levels are generated beneath the free surface after breaking (**Figure 8e**).

Using the SPH computational results, the vorticity maps are shown in **Figures 9a**–**e** and **10a**–**e**, respectively, for the spilling and plunging waves. Vorticity is defined as

$$
\rho = \left(\frac{\partial u}{\partial \mathbf{z}}\right) - \left(\frac{\partial v}{\partial \mathbf{x}}\right) \tag{8}
$$

and is computed using instantaneous values of the horizontal and vertical velocity. As noted by several authors [77, 78], for both breakers (T1 and T2), when the breaking begins, positive vorticity occupies the whole region of the surface roller

### **Figure 9.**

*Instantaneous values of ω distributions in the SPH simulation of spilling wave (T1): (a) before; (b)–(c) during and (d)–(e) after breaking.*

and spreads out over the whole water column. However, the vorticity levels increase rapidly after the wave breaking for the plunging case (T2) due to the strong impingement of the jet on the forward trough, inducing a propagation of the positive vorticity towards the bottom (**Figure 10c**–**e**).

Moreover, the results highlight that there exist differences in the dynamics of vorticity between the spilling and plunging breakers. In fact, only during spilling formation (T1), small structures of negative vorticity are generated, instead when the plunging breaker (T2) occurs the fluid is relatively free of negative vorticity regions.

**Figures 11** and **12** show the comparison between the instantaneous map of vorticity and of the surface parallel convective acceleration for the spilling and plunging waves (T1 and T2) when the breaking begins at time step of **Figures 9b** and **10b**, respectively. The surface parallel convective acceleration here has been computed following [79]. As noted by Dabiry and Gharib [80], for both breakers (T1 and T2), a flow deceleration (**Figures 11b** and **12b**) occurs in the same location where peaks of positive vorticity appear (**Figures 11a** and **12a**). Therefore, the present results confirm the findings by Dabiri and Gharib [80] that the vorticity is convected due to the sharp velocity gradient of the fluid near the free surface with respect to the fluid below.

*Hydrodynamics of Regular Breaking Wave DOI: http://dx.doi.org/10.5772/intechopen.94449*

### **Figure 10.**

*Instantaneous values of ω distributions in the SPH simulation of spilling wave (T2): (a) before; (b)–(c) during and (d)–(e) after breaking.*

**Figure 11.** *SPH simulation of spilling wave (T1): (a) Vorticity map and (b) surface-parallel convective acceleration map.*

**Figure 12.**

*SPH simulation of plunging wave (T2): (a) Vorticity map and (b) surface-parallel convective acceleration map.*
