UC Davis

Air entrainment is the incorporation of pockets of air into the flow due to the strong agitation of the free surface. It occurs in natural flows such as hydraulic jumps, steep mountain rivers, breaking waves, and waterfalls, as well as in man-made structures such as smooth and stepped chutes, and has significant implications for both natural systems and industrial applications. Experimental techniques encounter inherent limitations due to scale effects, as well as challenges arising from the inability or high degree of inaccuracy of traditional measuring devices in highly aerated flows. Numerical methods are also limited, either due to the significant computational costs associated with scale-resolving techniques (SRS) or the elevated sensitivity to calibration parameters in time-averaged techniques. This thesis aims to deepen our understanding of the process of air entrainment in free-surface flows, and to advance the current capabilities of simulatingself-aerated flows using numerical methods.

The first part of this thesis is devoted to the development of a novel theoretical/numerical model for the simulation of self-aerated flows under a Reynolds-Averaged Navier-Stokes (RANS) framework. The new formulation is based on a three-phase mixture approach composed of a continuous air phase, a bubble phase, and a continuous water phase. A mass transfer mechanism that does not depend on an entrainment function and does not require calibration accounts for the incorporation of air into the flow. A modification in the formulation of the Volume-of-Fluid algorithm allows one to capture the increase in water depth due to the presence of bubbles. The proposed formulation recovers the traditional Volume-of-Fluid formulation for free surface flows in the absence of bubbles, allowing the model to represent simultaneously the aerated and not aerated regions of a flow. Governing equations for the mixture are derived from mass and momentum conservation equations for each phase, and a numerical algorithm that ensures the boundedness of the numerical solution is proposed. The model is tested and validated using four experimental cases: a degassing tank, a bubble plume, a plunging jet, and a stepped spillway, showing very satisfactory results.

In a second stage of the development of the model, the impact of the turbulence closure on the onset of air entrainment and the distribution of bubble concentration is investigated. To achieve this, a criterion to define the occurrence of air entrainment based on the balance between disturbing and stabilizing energies is implemented. It is shown that incompressible formulations RANS closures lead to severe overprediction of air entrainment due to unphysical transport of turbulent kinetic energy generated in the air phase to the water phase. In contrast, combining compressible turbulence closures with the energy balance criterion enables accurate prediction of the regions where air is entrained for stepped spillways and plunging jets. The choice of turbulence closure significantly influences the proposed criterion, emphasizing the importance of selecting an appropriate closure for an accurate description of the air entrainment process. Based on the conducted tests, the standard $k-\epsilon$ and buoyancy modified $k-\epsilon$ models better predict the onset of air entrainment and bubble distribution in stepped spillways, while the $k-\omega$ SST model proves to be more effective in capturing air entrainment at the impingement point in plunging jets. The new methodology provides a significant advance in the current capabilities for simulating self-aerated flows, expands the capabilities of numerical models in predicting air entrainment, and provides valuable insights into the effects and interrelations of the turbulence modeling, the air entrainment occurrence criteria, and the bubble transport equations.

In the second part of this thesis, an analysis of a Detached-eddy simulation of the flow past stepped spillways in three dimensions is presented, in order to investigate the coherent structures conducive to the phenomenon of air entrainment. The analyses focus on the spatial distributions of vorticity and velocity, as well as time series of the vorticity component in the transverse direction. A new index, $V_n$, is proposed in order to represent the spatial location of patches of vorticity magnitude, showing by definition that time-averaged values of such index $(\overline{V}_n)$ constitute the fraction of time in which the vorticity exceeds $n$ s$^{-1}$. When such average values are plotted for the central plane of the spillway, they strongly agree with plots of turbulent kinetic energy, conclusively connecting the vorticity patches with the turbulence intensities. The spatial evolution of velocity and vorticity components in a surface located at the outer edge of the boundary layer indicates an important development of turbulence, manifested by large instantaneous values of the main flow variables. Three-dimensional plots of iso-surfaces of constant $V_n$ and of turbulent kinetic energy show a similar growth rate, providing further evidence of the interconnection of variables. These results suggest that steps "compensate" the decay of turbulence by generating vorticity patches in between the steps, which then become released to the flow and reach positions close to the free surface. Additional results obtained coupling a Detached-eddy simulation closure with a Volume of Fluid method show big deformations on the free surface resulting from the interaction of the interface with large-scale turbulent structures. These deformations evolve until they become unstable and trap pockets of air, with associated time scales of the order of $10^{-2}$ s.

Finally, the two techniques utilized in this thesis (RANS and SRS) are compared, both showing excellent agreement with the experimental data. The RANS model requires user input, and is more sensitive to calibration parameters, while the SRS model provides good results without any calibration. The RANS model, at its present stage of development, provides information about bubble concentration and level of bulking. The SRS model not only gives this information but also bubble size distribution (limited by the grid resolution) and rate of entrapped to entrained air. The advantages of the SRS model come with a computational cost that is $\mathcal{O}(10^3)$ times of the RANS model and a cell count $\mathcal{O}(10^2)$ times higher. The SRS model still presents several limitations regarding the sub-grid scales. Both techniques have advantages and disadvantages that may suit different applications in the Hydraulic Engineering field. Future lines of research for both types of models are highlighted at the end of this thesis.