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.

Rivers and their floodplains are a complex system that presents numerous challenges when coupled with human adaptation. The many adaptations humans have made to the landscape have come at the cost of aquatic ecology. Efforts have gone underway to mitigate those costs and begin to repair our relationship with the aquatic landscape via restoration and re-engineering systems to benefit habitat. Of the myriad of tools at our disposal to investigate different restoration strategies, this thesis explores the application of hydrodynamic models to ecosystems and how the linkages between numerical models can evaluate ecological outcomes.

The application of this thesis has centered around the Sacramento-San Joaquin Delta, a focal point of California’s Central Valley and a critical ecosystem in the state. The Sacramento-San Joaquin Delta has many “islands” which are essentially parcels of land that have levees surrounding them to protect against flooding from the Detla’s channels. One island in the Delta is the McCormack-Williamson Tract, which lies just below the confluence of the Cosumnes and Mokelumne Rivers. The McCormack-Williamson Tract is the study area for Chapters 2 and 3, where a hydrodynamic model was develope and calibrated to investigate natural phenomena. The models were calibrated against observed flood events and then used to compare against field data collected in the study systems within the model calibration period. By pairing observed field data with replicated modeling data, linkages between source water distribution, age, and general hydrodynamic features were drawn. These linkages provide useful tools in the field of floodplain restoration engineering.

Chapter 2 demonstrates a hydrodynamic model’s ability to resolve the spatio-temporal distribution of the water from the convergence of the Mokelumne and Cosumnes Rivers as they advance through the McCormack-Williamson Tract and its surrounding channels and sloughs. This modeling effort was coupled with an isotopic mixing model using water sampling in the study area, and shows that the two methods agree on source water distribution.

The subsequent chapter (Chapter 3) that focuses on the McCormack-Williamson Tract evaluates a modeled water age against field-collected zooplankton abundance. Although this study did not find a robust relationship between the two variables, the chapter lays out the methodology used to investigate the relationship.

The second study area centered around a key feature in the Delta’s hydrography - the Yolo Bypass. The Yolo Bypass is a major floodplain to convey flood waters from the Feather, American, and Sacramento Rivers around major cities and dwellings and into the Delta. Chapter 4 outlines the development and calibration of a model of the Yolo Bypass and discusses the applications the model can have to answer ongoing questions about the study system.

Long-term hydrodynamic and salinity transport modeling of the Sacramento–San Joaquin Delta is needed to evaluate the future Delta in terms of the California co-equal goals of ecosystem health and reliable water supply. While 2-D and 3-D hydrodynamic and water quality models are by definition better suited to modeling a complex network of tidally influenced flows under future conditions, a 1-D model is more computationally efficient in narrowing the large variety of multiple-year simulations required into a more manageable task. Still, a 1-D model of sea level rise in an estuary must account for the three-dimensional effects where increased depths will affect density driven (baroclinic) circulation and tidal dispersion of salt. In this paper, we use a simplified Delta network model with a tidally averaged computational approach to quickly perform multi-year simulations for sea level rise. The 1-D model uses tidal dispersion coefficients developed from 3-D hydrodynamic models. The resulting model is capable of performing very fast simulations over a wide range of conditions, providing guidance on what should be explored in depth with more detailed, but slower models.

Comparisons of unimpaired Delta inflow with the historical case show that the south Delta and San Joaquin River would be much fresher without exports, while the Sacramento River would be fresher in spring and more saline in the fall. Sea level rise will increase salinity throughout the Delta over time. With peripheral conveyance of export, water salinity will intrude upstream in the Sacramento River, be slightly lower up the San Joaquin River and increase in the south Delta. With sea level rise, peripheral conveyance will have similar trends to changes to the historical case, but export salinity will be improved by the peripheral conveyance component. A larger peripheral conveyance can benefit both the ecosystem and exports if managed properly.

Lakes are globally relevant for socio-economic needs as they provide fresh water sources for water supply and recreational activities, can be sites of critical sustenance fisheries, are often places of great natural beauty, and have cultural importance for many groups. In addition, lakes support diverse biota. The ecosystem balance of these natural waters is linked to the lake's physical processes that drive the transport and mixing of carbon, nutrients, sediments, and oxygen. For much of the year, thermal stratification acts as a barrier to vertical transport. At the same time, the existence of a thermal stratification allows for the occurrence of the upwelling process, which can potentially promote vertical transport and mixing. The upwelling phenomenon can alter the water quality in the littoral zone, and nutrient fluxes have been reported during the active upwelling phase. Conceptual models describe the flow structures to follow either a non-rotational or a fully rotational conceptual model, with the former applied to small lakes and systems of moderate size. Despite recognizing lake hydrodynamics influenced by the Coriolis force in lakes of moderate size, the intrinsic details of the rotational influence during lake upwelling are poorly understood, and the thresholds for the validity of the conceptual models have not been defined to date.

This dissertation uses fine-resolution numerical simulations with a lake model to quantify the Coriolis force effects during the active and relaxation phases of lake upwelling and to detail the physical processes observed in lakes that exhibit partial rotational influence in the flow structures. The numerical simulations conclude that lakes that show Coriolis force effects do not follow the binary perspective of upwelling, and rotational flow structures describe lake response to wind forcing during weak stratification. Furthermore, the analysis of the flow and temperature patterns of over 300 numerical simulations of lakes resulted in the definition of a third conceptual model for lake upwelling and allowed to define the thresholds for the validity of the conceptual models for upwelling following a basin-wide parameterization. Finally, dissolved oxygen dynamics driven by the physical processes during the upwelling relaxation stage were addressed using field observations taken between 2015 and 2018 in a lake of moderate size. The analyses of the available data for a total of 18 upwellings conclude that flow structures during the upwelling relaxation stage play a relevant role in increasing dissolved oxygen levels in deep waters of lakes. Flow paths from these flow structures also indicate that the increase in oxygen may occur across the whole extent of the upwelled region instead of the effects limited to the littoral zone, as is the case for the water quality dynamics during the active phase of upwelling.

The chapters of this dissertation serve as an integral step toward a better understanding of the upwelling phenomenon and concomitant water quality dynamics in lakes, especially in those exhibiting partial rotational influence. In addition, the results from Chapter 3 served as the foundation for creating a unique warning system of physical lake conditions that has been prototyped for the Lake Tahoe community.

This work shows how differential forms and boundary conditions can be conveniently expressedin an appropriately constructed orthogonal or near-orthogonal coordinate system. As a special case, two coordinate systems are constructed that map a R3 rectangle to two irregularly shaped open channels and perturbation methods are used to handle a Laplacian equation in different magnitudes. Additionally, other attempts at constructing such coordinate systems are presented along with their challenges.

Fecal contamination from non-point sources in coastal and estuarine water bodies is a problem of increasing concern. Water monitoring alone is sometimes insufficient in providing a clear picture of the fecal contamination of a water body. Well-formulated and developed mathematical and numerical transport models, on the contrary, predict continuous concentrations of microbial indicators under diverse scenarios of interest, and they can quantify fecal source contributions on a land use basis (human versus livestock or wildlife).

The present project has demonstrated the utility of quantitative analyses of fecal contamination in water bodies via new experimental measurement techniques as well as mathematical/numerical modeling of the fate and transport of biological contaminants. San Pablo Bay was selected because there is abundant knowledge about the estuarine system and its tributaries and because of the availability of an existing 3-dimensional model. Environmental monitoring of San Pablo Bay was performed with a Microbial Source Tracking (MST) approach that utilized a validated large-volume (100 L) hollowfiber ultrafiltration method to concentrate water, which was then tested with quantitative real-time PCR assays to identify and quantify fecal contamination contributed from general and human-, cow- and dog-specific sources. Monitoring of human viral pathogens (adeno- and enteroviruses) was also performed. Additional samples were also collected to enumerate microbial indicators (E. coli and enterococci). This was the first time that DNA-based molecular assays, which had been previously validated with fecal samples and wastewater effluents in freshwater studies, were applied to an estuarine environment.

Monitoring results indicated low-level general and human-derived fecal contamination in the bay, while cow- and dog-derived contamination was not detected, except for one sample which contained dog-specific genetic marker. Human viruses were also below the sample detection limit. The pollution was more likely to come from surrounding urban areas or wastewater treatment facilities than from agricultural farm land or wildlife areas. Another application of the study included the validation of quantitative sea bird-specific molecular DNA assays for the purpose of microbial source tracking to enumerate the contribution of sea birds to fecal contamination. The assay was found to possess adequate sensitivity and specificity and could be applied in the future to archived environmental samples to estimate the contribution from sea birds to bacterial fecal loads in San Pablo Bay and its tributaries.

A suite of theoretical and computational tools, ranging from one-dimensional (1-D) to three-dimensional (3-D) models, was developed during this project for the analysis of the fate and transport of Bacteroidales gene markers. These models were paired with existing visualization tools. Flow conditions for the sampling dates were simulated and results compared with velocity observations. Exploratory assessments of water quality in the San Pablo Bay allowed for an incipient characterization of the transport conditions in San Pablo Bay. Future work includes the optimization of the water-quality sub-module of the code in order to a) include sediment transport in the bay; b) obtain more reliable results 3 under a wide set of boundary conditions; and c) be able to simulate a wider set of waterquality parameters. Also, the authors would like to add a user-friendly interface to the code to facilitate decision making by coastal managers.