Preferential concentration describes the tendency of heavy particles to accumulate in certain regions of a turbulent flow. This process has been hypothesized to play a role in particle growth which is of crucial importance in numerous physical and engineering applications. The efficiency of preferential concentration is known to depend on the ratio of the particle stopping time to the turbulent eddy turnover time, which is called the Stokes number. In this thesis, we investigate the role of turbulence on preferential concentration of heavy particles with Stokes number less than unity. We use Direct Numerical Simulations and adopt the two-fluid formalism, where the particulate phase is treated as a continuum. In the first work, we study a two-way coupled system in the particle-induced Rayleigh-Taylor instability, and observe the striking emergence of dense, filamentary particle structures. Most notably, we find that the particle concentration enhancement primarily depends on three properties of the system: the rms fluid velocity, the Stokes number, and the assumed particle diffusivity from the two-fluid equations. Additionally, we note that when preferential concentration is dominant, the probability distribution function of the particle concentration takes on a distinctive form, characterized by an exponential tail whose slope is related to the same three properties listed above. In the second part, we further extend our study to a regime in which turbulence is externally-driven, and verify that the results found in the first study also hold. In the final work, we use a box-counting algorithm to identify and extract key features of the dense particle structures. We find in particular that these structures have a large aspect ratio. We propose an advection-diffusion model to predict their thickness, and find preliminary evidence that suggests that their long dimension depends on the Taylor microscale.

Turbulence is ever-present: from flows in engineering, such as a wake behind a submarine, the boundary layer over an aircraft wing, and the swirl in an internal combustion engine, to flows in nature, such as convection in lakes, riffles on rivers and ocean currents. Turbulence can be found in flows at relatively small scales, such as blood flow in arteries and while mixing cream in a morning coffee, to flows at astrophysical scales, for instance, in accretion disks around stars or black holes. Because of their ubiquitous nature, progress in science and technology often hinges on progress in research on turbulent flows. In many situations described above, two lines of inquiry are of most interest. In the first direction, we are interested in quantities that are the ``net'' outcome of a fluid system, i.e., bulk quantities or global mean quantities such as drag force, rate of energy dissipation, mass, momentum and heat transport and mixing rate, which are usually long-time and volume averages and therefore depend only on the system's input parameters such as viscosity and diffusivity of the fluid, characteristics velocity scale, domain shape. The second direction, which is complementary to the first one, is the study of different structures in turbulent flows, for example, quantifying the range of scales and the energy distribution through this range in turbulent flows. In this thesis, we study a few problems that are related to and inspired by these two directions of questioning. While working on a problem, we always try to incorporate different perspectives: engineering, physics and mathematics. It is our intention to work at the interface of physical and mathematical fluid dynamics, as there appears to be great potential for an exchange of ideas that can eventually benefit both fields. On the one hand, having knowledge of various phenomenological theories from the physics literature gives one the advantage in tackling the various pressing problems considered in the mathematical community. On the other hand, putting various theoretical predictions on a rigorous mathematical footing can allow us to gain a deeper understanding of the physical mechanism/phenomenon. In accordance with this theme, below we describe the problems considered in this thesis, which is divided into two parts.

In the first part of the thesis, we are interested in quantifying bulk properties of turbulent flows, such as energy dissipation, drag force, heat and mass transfer. We obtain rigorous bounds on these quantities using a well-known technique known as the background method. We consider four problems in the first part: (1) uniform flow past a flat plate, (2) pressure-driven flow in a helical pipe, (3)  Taylor--Couette flow (flow between two independently rotating concentric cylinders), and (4) internally heated convection. In the flat plate study, we show that the energy dissipation rate for uniform flow past a flat plate remains bounded. This is the first and only example so far of an external flow problem (flow past a body) where such a bound has been established. In the second and third problems, we derive bound on mean quantities such as friction factor, volume flow rate, energy dissipation, torque on the cylinder and angular momentum transport not just as a function of the principal flow parameter, the Reynolds number, but more importantly, as a function of the geometry of the domain (i.e., curvature and torsion in the case of helical pipe flow and the ratio of the radii of two cylinders in Taylor--Couette flow). These studies are motivated by several engineering applications where the geometry of the domain plays an important role. In the fourth study, we consider the problem of convection between two solid boundaries driven by a source of internal heating and derive a bound on the mean vertical heat flux, an inquiry that is motivated, for example, by convection in the earth's mantle and the sun's radiative zone.

In the second part of the thesis, we are concerned with designing incompressible flows that possess some specific desired properties. The first problem in this direction is related to the optimal heat transport from a hot to a cold wall using a flow whose enstrophy is bounded by a given constant. The bound on the enstrophy can also be thought of as a bound on the power supply needed to generate this flow (using a body-force in the momentum equation) Navier--Stokes system. An upper bound on the heat transfer that scales as 1/3-power of the power supply had formally been derived previously, but whether a flow exists that transports heat at that rate remained an outstanding question. For this problem, we design three-dimensional branching flows to prove that the corresponding heat transfer saturates this known upper bound, which then establishes the exact asymptotic behavior of the optimal heat transport between two plates. Beyond the mathematical proof, our method also reveals why three-dimensional branching flows are so efficient in transferring heat. Finally, in the second part, we study a problem related to the nonuniqueness of flow maps in an ODE system for the class of velocity fields that are divergence-free and belong to Sobolev space $W^{1, p}$. We reprove and improve the known result that had been previously established using the method of convex integration. Our goal for this problem is simple: provide explicit constructions and use them to gain insights into the exact mechanism of the nonuniqueness of solutions of the ODE and the PDEs, transport and continuity equation with the same vector field. Beyond proving the nonuniqueness results, we anticipate that such explicit constructions will be helpful for designing velocity fields in the convection-diffusion equation or the body-forced Navier--Stokes equation to demonstrate the phenomenon of anomalous dissipation, an intrinsic characteristic of turbulent flows.

Oscillatory double diffusive convection (ODDC) is a double diffusive instability that occurs in fluids that are unstably stratified in temperature and stably stratified in chemical composition. Regions unstable to ODDC are common in the interiors of stars and giant planets, and knowing thermal and compositional transport through these regions is important for stellar and planetary evolution models. Using 3D direct numerical simulations, Rosenblum et al. 2011 first showed that ODDC can either lead to the spontaneous formation of convective layers, or remain in a state dominated by large scale gravity waves. Subsequent studies focused on identifying the conditions for layer formation (Mirouh et al. 2012), and quantifying transport through layered systems (Wood et al. 2013). This document includes 3 works that build on the results of these earlier studies. The subject of the first is transport through non-layered ODDC and shows that in the absence of layered convection, ODDC is dominated by large scale gravity waves that grow to the size of the domain. We find that while these gravity waves induce small amounts of turbulent mixing, turbulent transport through non-layered systems is not significant for the purposes of astrophysical modeling (unlike in layered convection). The second study pertains to ODDC in the presence of Coriolis forces, and shows that rotating systems can be categorized depending on the strength of the rotation. We find that in the slowly rotating regime, the presence of rotation does not significantly affect qualitative behavior, but leads to modest reductions in thermal and compositional transport, while in the fast rotation regime qualitative behaviors are radically different, and systems are dominated by vortices that affect thermal and compositional transport in complex ways. In the final work we study simulations of ODDC at non-layered parameters that are forced into a layered configuration by initial conditions. Our results show that measurements of thermal and compositional transport deviate from values predicted by oft-cited geophysical transport laws.

Double-diffusive convection at higher Prandtl numbers (Pr ~O(1) or larger) has been well studied in geophysical contexts, but detailed investigations of the low Prandtl number regimes (Pr << 1) which are relevant to most astrophysical scenarios have only recently become feasible. Since most low-Pr fluids in astrophysical scenarios are electrically conducting, it is possible that magnetic fields play a role in either enhancing or suppressing double-diffusive convection, but to date there have been no numerical investigations of such possibilities. Here we study the effects of both vertical (aligned with the gravitational axis) and horizontal background magnetic fields on the linear stability and nonlinear saturation of double-diffusive fingering, through a combination of theoretical work and direct numerical simulation (DNS). Both vertical and horizontal background magnetic fields are found to significantly enhance the fluid kinetic energy, vertical motion, and chemical flux relative to standard fingering convection, but the two cases differ considerably in their behavior. We focus mainly on the vertical case, finding that a vertical magnetic field suppresses the secondary shear instabilities between up- and down-flowing fingers such that saturation of the instability is delayed until significantly higher levels of vertical fluid motion are reached. This allows magnetized fingering convection to have significantly enhanced levels of turbulent mixing of chemical species with respect to the hydrodynamic case. Consequentially, magnetic effects offer a promising explanation of discrepancies between theoretical and observed mixing rates in low-mass red giant branch (RGB) stars.

Observational evidence points to the need for extra mixing in stars. Zahn (1992) proposed a turbulent mixing model due to shear instabilities and the model has been verified to be valid for non-rotating cases. It is not clear, however, whether Zahn’s model would still be valid in the presence of rotation. We use a triply-periodic Cartesian domain in the equator of a rotating star to examine this issue. We use the Boussinesq approximation, and assume the background temperature gradient to be constant, and the flow to experience a horizontal sinusoidal body force. A linear stability analysis reveals the existence of several regimes that are dominated by shear instabilities or GSF instabilities. Based on linear stability results, we run a set of numerical simulations for different control parameters including the rotation rate. At small rotation rates, we recover the previous results obtained in the non-rotating case. At higher rotation rates, we find regions governed by different dynamics that are not accounted for by Zahn’s model. In each case, we provide quantitative data on the heat and momentum transports induced by turbulence.

Low mass stars on the red giant branch (RGB) experience more mixing in their outer convection zone than what is predicted by stellar evolution theory. If there exists an inverse composition gradient on the external wing of the HBS shell after the first `dredge- up', then an unstable composition stratification along with the stable stratification played by entropy implies that double diffusive mixing processes should occur in stellar interiors and that some of the hidden mixing might be a result of double diffusive convection. We explore double diffusive modes in the case where there exists a lateral gradient in composition and entropy, in addition to a vertical gradient in order to understand their mixing rates. We find that under these circumstances, the mixing rates of laminar fingering modes and oscillatory modes may be able to address the missing mixing problem. In addition, we find that the `collective instability', oscillatory modes with no lateral gradients cannot explain the missing mixing on the RGB.

This thesis presents a new algorithm to mitigate cloud masking in the analysis of sea surface temperature (SST) data generated by remote sensing technologies such as infrared sensor satellites like the Level-2 Visible-Infrared Imager-Radiometer Suite (VIIRS). Cloud coverage interferes with the analysis of all remote sensing data using wavelengths shorter than ≈ 2 microns, significantly limiting the quantity of usable data and creating a biased geographical distribution towards equatorial and coastal regions. Prior studies have led to use of in-painting algorithms like Navier-Stokes but was typically only used up to 5% masking and had limited success. To address this issue, we propose an unsupervised machine learning algorithm called ENKI which uses a Vision Transformer with Masked Autoencoding to reconstruct pixels that are masked out by clouds. We train four different models of ENKI with training mask ratios (referred to as t) set to 10%, 35%, 50%, and 75% on a generated Ocean General Circulation Model (OGCM) dataset known as LLC4320. To evaluate performance we reconstruct LLC 4320 SST images at a patch masking ratio of 10%, 20%, 30%, 40%, 50% (referred to as p) and examine reconstruction qualitatively and statistically by calculating the root means squared error (RMSE) of reconstructed patches. Through our analysis we discover that edge patches contain a higher error rate and that a bias appears in some models when reconstructing images at p masking ratios away from their training mask ratio t. But we consistently find that at all levels of p masking ratios there is one or multiple models that create reconstructions with a mean RMSE of less than ≈ 0.03K which is lower than the estimated sensor error of VIIRS data which is ≈ 0.078 K for daytime, along scan, and ≈ 0.05 K for nighttime, along-scan. We also conclude the complexity of dynamics within an image and the p masking ratio affect RMSE with higher complexity and p masking seeing higher RMSE values. Critically, we also discover at a patch level that despite RMSE having some correlation to complexity, they are not directly proportional, and RMSE increases at a slower rate as complexity within a patch increases. Our analysis concludes that ENKI shows great promise in surpassing in-painting as a means of reconstructing cloud masking, and future research seeks to analyze ENKI’s capabilities in reconstructing real world data.

According to helioseismic inversions, the Sun exhibits two different rotational regimes. The inner radiative region rotates almost uniformly whereas the outer convection zone rotates differentially with the rotation rate decreasing with latitude. The transition region, which is located in the vicinity of the radiative-convective interface, is a very thin layer known as the solar tachocline. Both hydrodynamical and magnetohydrodynamical theories have been proposed to explain such a sharp rotational transition. This thesis presents and analyzes numerical simulations of the solar tachocline that explain the rotational regimes of the interior of the Sun as the result of the interaction between fluid in motion and magnetic fields.