<p>Cavitation is ubiquitous in nature and scientific application where it might hinder through noise, vibration or erosion which eventually leads to reduced performance. Similarly, rising bubbles are employed in several industrial applications to homogenize the fluid. This thesis sheds light on special applications of these two phenomena. </p> <p>Once a bubble has been captured by a vortex core, the low (sometimes negative) pressure in the core causes the cavitation bubble to elongate axially while the radius of the bubble oscillates with time. Three dimensional compressible Navier-Stokes equations with surface tension are numerically solved using an all-mach solver on Basilisk software. The bubble dynamics can be categorised into separate stages: spherical growth, pinching, elongation and fragmentation. As the cylindrical bubble grows, it increases the vortex core radius. The flow and the bubble dynamics are strongly coupled. The effect of changing cavitation number and bubble to vortex size ratio has been explored. The bubble sizes and dynamics at different time steps have also been recorded. When the pressure in the core is negative, the bubble continues to grow axially forming a long tube, which is also observed in experiments. In oceans, density varies with depth due to varying salinity and temperature gradient, which prevents the vertical exchange of heat, carbon, dissolved oxygen, and nutrients as well as blooms the population of harmful bacteria such as cyanobacteria. The rising motion of a single or cluster of bubbles creates an upflow that can cause homogenization or destratification. Confined bubble columns are used for microelectronic cooling as well as in chemical reactors for mixing stratified fluids without any mechanical agitation or power. To begin realizing this complex multi-phase flow system to better understand mixing, we start with a simplified problem of a single air bubble rising in a confined Hele-Shaw channel. We performed a time-resolved stereoscopic Particle Image Velocimetry (PIV) measurement to characterize the bubble wake. Pure water and varying salt concentration were used to achieve a linear density stratification corresponding to Froude numbers (Fr) ranging from 22.1 to 40.7. Due to the large velocity dynamic range for PIV, we enhanced the signal to noise ratio of our correlation planes with pyramid correlation. We found a significant out of plane velocity component in both homogenous and stratified fluid in the vicinity of the bubble, which was assumed to be negligible in previous studies with confined fluid. The wake of the bubble carries the higher density fluid to the top, which later releases from the wake to form the reverse jet. This buoyant jet has been characterized for different Fr. Eulerian coherent structures are also considered to describe the flow. The rising bubble generates vortices that shed downstream and decay with varying timescales for different Fr. The difference in the coherent structures and decay coefficient leads to a different level of mixing with Fr. The scope of this research is in applications homogenizing the stratified flow using rising bubbles. </p>

Continental slope channels, which serve as the primary conduits for sediment transport into the deep marine, occasionally become sites of sediment deposition with excellent reservoir potential. Increasingly reported in the literature are subsurface channel fills exhibiting shingled seismic reflectors that are interpreted to have formed by lateral channel migration. In lower Isaac Formation channels inclined strata are observed but at a lateral scale that is far below industry-seismic detection. Distinctively these flat-based channels are filled with coarse-grained sandstone that transitions abruptly and obliquely upwards into thin, fine grained turbidites. Like rivers, lateral accretion in Isaac channels is interpreted to be the result of the interaction of inertial and pressure forces, but in highly turbulent, highly density-stratified turbidity currents. This resulted in the formation of two superimposed secondary circulation cells that caused enhanced erosion on the outer bank and preferential deposition of coarse-grained sediment along the inner bank.

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This thesis considers the problem of hypothesis testing for umbrella alternatives when there are two groups, or strata, of observations. The proposed methods extend a previously established general framework of hypothesis testing based on rankings to stratified datasets by first aligning the strata. The tests based on the Spearman and Kendall distances between ranking vectors lead to the traditional aligned-rank tests and new methods which account for “misalignment” under the alternative hypothesis. Asymptotic null distributions and simulation studies are given for the Spearman distance. Diagnostic tools for the misalignment issue are illustrated alongside the proposed tests on a dataset of IQ scores of coma patients. Extensions to three or more strata and ”adaptive” tests are provided as future research directions.

The effects of propagation of a short internal gravity wave through an inertia wave on internal wave stability is analyzed and parameterized. The interactions are specifically between a short wave packet and a large inertia wave packet. The short wave packet is a wave bounded with a Gaussian envelope with high frequencies and scales in the hundreds of meters horizontally and tens of meters vertically. The inertia wave packet is also an enveloped wave but with frequencies close to the rotation of the earth and scales in the thousands of meters in the horizontal and hundreds of meters in the vertical. The wave-wave interactions are modeled using ray theory and 2d non-linear numerical models. Ray tracing is used because it is less computationally expensive, however it fails at regions of strong refraction also known as caustics. To measure stability the steepness is calculated from the 2d non-linear methods and it is compared with estimates found in the linear theory. It is determined that the estimates of the short wave steepness from linear theory are qualitatively comparable. A quantifiable comparison, although more difficult, resulted in adjustment factors to the ray tracing results. It is also found that for the particular cases modeled, convective instabilities are predominant and the influence of the shear exerted by the large inertia wave is insignificant. Instability time scales are included in the stability analysis and estimates of overturning and wave-breaking are developed for different wave-wave interactions. From the stability analysis it is found that in general the faster the short wave propagates the more likely it is to conform to both of the conditions required for wave breaking (i.e presence of instabilities and instability time scales longer than the timescale of the short wave).

Internal waves exist everywhere in stratified fluids - fluids whose density changes with depth. The two largest bodies of stratified fluid are the atmosphere and ocean. Internal waves are generated from a variety of mechanisms. One common mechanism is wind forcing over repeated sinusoidal topography, like a series of hills. When modeling these waves, linear theory has been employed due to its ease and low computational cost. However, recent research has shown that non-linear effects, such as boundary layer separation, may have a dramatic impact on wave generation. This research has consisted of experimentation on sloped, sinusoidal hills. As of yet, no experimental research has been done to characterize internal wave generation when repeated sinusoidal hills lie on a sloped surface such as a continental slope or a foothill. In order to perform this experiment, a laboratory was built which employed the synthetic schlieren method of wave visualization. Measurements were taken to find wind speed, boundary layer thickness, and density perturbation. From these data, an analysis was performed on wave propagation angle, wave amplitude, and pressure drag. The result of the analysis shows that when wind blows across a series of sloped sinusoidal hills, fluid becomes trapped in the troughs of the hills resulting in a lower apparent forcing amplitude. The generated waves contain less energy than linear predictions. Additionally, the sloped hills produce waves which propagate at an angle away from the viewer. A necessary correction, which shifts from the reference frame of the observer to the reference plane of the waves is described. When this correction is applied, it is shown that linear theory may only be applied for low Froude numbers. At high Froude numbers, the effect of the boundary layer is great enough that the wave characteristics deviate significantly from linear theory predictions. The analyzed data agrees well with previous studies which show a similar deviation from linear theory.

It is well-known that the basic properties of a bivariate spline space such as dimension and approximation order depend on the geometric structure of the partition. The dependence of geometric structure results in the fact that the dimension of a C1 cubic spline space over an arbitrary triangulation becomes a well-known open problem. In this paper, by employing a new group of smoothness conditions and conformality conditions, we determine the dimension of bivariate C1 cubic spline spaces over a so-called even stratified triangulation.

This work can be roughly divided into two parts. Initially, it may be considered a continuation of the very interesting research on the topic of Lattice-Valued Convergence Spaces given by Jager [2001, 2005]. The alternate axioms presented here seem to lead to theorems having proofs more closely related to standard arguments used in Convergence Space theory when the Lattice is L = f0; 1g:Various Subcategories are investigated. One such subconstruct is shown to be isomorphic to the category of Lattice Valued Fuzzy Convergence Spaces defined and studied by Jager [2001]. Our principal category is shown to be a topological universe and contains a subconstruct isomorphic to the category of probabilistic convergence spaces discussed in Kent and Richardson [1996] when L = [0; 1]: Fundamental work in lattice-valued convergence from the more general perspective of monads can be found in Gahler [1995]. Secondly, diagonal axioms are defned in the category whose objects consist of all the lattice valued convergence spaces. When the latter lattice is linearly ordered, a diagonal condition is given which characterizes those objects in the category that are determined by probabilistic convergence spaces which are topological. Certain background information regarding filters, convergence spaces, and diagonal axioms with its dual are given in Chapter 1. Chapter 2 describes Probabilistic Convergence and associated Diagonal axioms. Chapter 3 defines Jager convergence and proves that Jager's construct is isomorphic to a bireáective subconstruct of SL-CS. Furthermore, connections between the diagonal axioms discussed and those given by Gahler are explored. In Chapter 4, further categorical properties of SL-CS are discussed and in particular, it is shown that SL-CS is topological, cartesian closed, and extensional. Chapter 5 explores connections between diagonal axioms for objects in the sub construct δ(PCS) and SL-CS. Finally, recommendations for further research are provided.

We develop a viscoelastic version of the volume of fluid algorithm for tracking deformable interfaces. The code uses the piecewise linear interface calculation method to reconstruct the interface, the continuous surface force formulation to model interfacial tension forces and utilizes the semi-implicit Stokes solver (enabling computations at low Reynolds numbers). The algorithm is primarily designed to simulate the flow of superposed fluids and the drop in a flow problem in 2D shear flows of viscoelastic and/or Newtonian fluids. The code is validated against linear stability theory for the two-layer flow case and against experimental and other documented numerical investigations for the droplet-matrix case. / Ph. D.

This thesis studies the implementation of an Explicit Algebraic Reynolds-Stress Model(EARSM) for Atmospheric Boundary Layer (ABL) in an open source ComputationalFluid Dynamics (CFD) software, OpenFOAM, following the guidance provided by thewind company ENERCON that aims to make use of this novel model to improvesites’ wind-field predictions. After carefully implementing the model in OpenFOAM,the EARSM implementation is verified and validated by testing it with a stratifiedCouette flow case. The former was done by feeding mean flow properties, takenfrom OpenFOAM, in a python tool containing the full EARSM system of equationsand constants, and comparing the resulting flux profiles with the ones extracted bythe OpenFOAM simulations. Subsequently, the latter was done by comparing theprofiles of the two universal functions used by Monin-Obukhov Similarity Theory(MOST) for mean velocity and temperature to the results obtained by Želi et al. intheir study of the EARSM applied to a single column ABL, in “Modelling of stably-stratified, convective and transitional atmospheric boundary layers using the explicitalgebraic Reynolds-stress model” (2021). The verification of the model showed minordifferences between the flux profiles from the python tool and OpenFOAM thus, themodel’s implementation was deemed verified, while the validation step showed nodifference in the unstable and neutral stratification cases, but a significant discrepancyfor stably stratified flow. Nonetheless, the reason behind the inconsistency is believedto be related to the choice of boundary conditions thus, the model’s implementationitself is considered validated. Finally, the comparison between the EARSM and the k − ε model showed thatthe former is able to capture the physics of the flow properties where the latter failsto. In particular, the diagonal momentum fluxes resulting from the EARSM reflectthe observed behaviour of being different from each other, becoming isotropic withaltitude in the case of unstable stratification, and having magnitude u′u′ &gt; v′v′ &gt; w′w′ for stably stratified flows. On the other hand, the eddy viscosity assumption used bythe k − ε model computes the diagonal momentum fluxes as being equal to each other.Moreover, the EARSM captures more than one non-zero heat flux component in theCouette flow case, which has been observed to be the case in literature, while the eddydiffusivity assumption used by the k − ε model only accounts for one non-zero heat fluxcomponent.