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Elastic wave scattering from randomly rough surfaces and a smooth surface are essentially different. For ultrasonic nondestructive evaluation (NDE) the scattering from defects with smooth surfaces has been extensively studied, providing fundamental building blocks for the current inspection techniques. However, all realistic surfaces are rough and the roughness exists in two dimensions. It is thus very important to understand the rough surface scattering mechanism, which would give insight for practical inspections. Knowledge of the stochastics of scattering for different rough surfaces would also allow the detectability of candidate rough defects to be anticipated. Hence the main motivation of this thesis is to model and study the effect of surface roughness on the scattering field, with focus on elastic waves. The main content of this thesis can be divided into three contributions. First of all, an accurate numerical method with high efficiency is developed in the time domain, for computing the scattered waves from obstacles with arbitrary shapes. It offers an exact solution which covers scenarios where approximation-based algorithms fail. The method is based on the hybrid idea to combine the finite element (FE) and boundary integral (BI) methods. The new method efficiently couples the FE equations and the boundary integral formulae for solving the transient scattering problems in both near and far fields, which is implemented completely in the time domain. Several numerical examples are demonstrated and sufficiently high accuracy is achieved with different defects. It enables the possibility for Monte Carlo simulations of the elastic wave scattering from randomly rough surfaces in both 2D and 3D. The second contribution relates to applying the developed numerical method to evaluate the widely used Kirchhoff approximation (KA) for rough surface scattering. KA is a high-frequency approximation which limits the use of the theory for certain ranges of roughness and incidence/scattering angles. The region of validity for elastic KA is carefully examined for both 1D and 2D random surfaces with Gaussian spectra. Monte Carlo simulations are run and the expected scattering intensity is compared with that calculated by the accurate numerical method. An empirical rule regarding surface parameters and angles is summarized to establish the valid region of both 2D and 3D KA. In addition, it is found that for 3D scattering problems, the rule of validity becomes stricter than that in 2D. After knowing the region of validity, KA is applied to investigate how the surface roughness affects the statistical properties of scattered waves. An elastodynamic Kirchhoff theory particularly for the statistics of the diffused field is developed with slope approximations for the first time. It provides an analytical expression to rapidly predict the expected angular distribution of the scattering intensity, or the scattering pattern, for different combinations of the incidence/scattering wave modes. The developed theory is verified by comparison with numerical Monte Carlo simulations, and further validated by the experiment with phased arrays. In particular the derived formulae are utilized to study the effects of the surface roughness on the mode conversion and the 2D roughness caused depolarization, which lead to unique scattering patterns for different wave modes.

The next generation of cosmological surveys will measure the large-scale structure (LSS) of the universe with unprecedented statistical power, covering large cosmological volumes and yielding constraints on cosmological parameters competitive with the cosmic microwave background (CMB), letting us test the standard model of cosmology at percent levels or below across cosmic history, from the recombination era to the present. Unlike CMB experiments, which probe the early universe in the linear regime, these new surveys will map the distribution of matter and galaxies at late times where the effects of nonlinearities are significant. To reliably extract fundamental physics information from this data will require theoretical models that can make accurate predictions on large, cosmological scales while being robust against the effects of nonlinear physics of small scales, due not just to gravitational collapse but also the astrophysics of galaxy formation whose precision modeling currently eludes us.In this dissertation we develop perturbation theory (PT) models for two key observables in upcoming surveys: the redshift-space clustering of galaxies in spectroscopic surveys and the cross correlation of galaxy clustering with weak gravitational lensing. Perturbation theory at the linear level has an indispensable role in CMB analyses, and the past decade has seen rapid developments in \textit{nonlinear} perturbation theories of large-scale structure, in part due to a reinterpretation of PT through the lens of effective field theories that has allowed for a robust and theoretically consistent treatment of the effects of non-perturbative small-scale physics on large-scale clustering. We review these developments in Chapter 1 before diving into the study of galaxy velocities and redshift-space distortions in Eulerian and Lagrangian perturbation theory (LPT) in Chapter 2. We use this knowledge to construct a model of the redshift-space galaxy 2-point function at 1-loop in perturbation theory featuring a full infrared (IR) resummation of large-scale displacements and velocities in Chapter 3. In Chapter 4, we use the same mathematical techniques to model density-field reconstruction, a technique used to sharpen the baryon acoustic oscillations (BAO) signal in spectroscopic surveys.The PT techniques described above are quite versatile and, in Chapter 5 and 6, we use them ``out-of-the-box'' to study non-standard features in galaxy clustering due to relative perturbations between dark matter and baryons after recombination as well as exotic early-universe scenarios such as non-standard inflation and early dark energy, with a particular focus on IR resummation in LPT. In Chapter 7 we exploit this same resummation of displacements for a different purpose: by combining the perturbative Lagrangian bias expansion with exact displacements solved-for in N-body simulations we construct a model for galaxy-matter cross correlations in weak lensing surveys that significantly extend the reach of PT models without requiring any additional assumptions about the galaxy-halo connection.To complete this dissertation, we apply the LPT models constructed in the preceding chapters to existing data in Chapters 8 and 9. In the former, we use the predictions of LPT to jointly model the ``full shape'' of pre- and post-reconstruction galaxy 2-point functions in the Baryon Oscillation Spectroscopic Survey (BOSS), showing that this combined analysis can be performed directly at the data level as opposed to the ex post facto approach of earlier analyses. In the latter, we further add in cross correlations with the weak lensing of the CMB from the Planck satellite, and, by modeling the cross correlation with the same LPT formalism as the galaxy power spectrum, do so within a consistent dynamical framework and with minimal additional parameters. Together, these pilot analyses demonstrate that the LPT formalism developed in this dissertation, and more generally perturbation theory, can offer a robust and pragmatic choice for future analyses of cosmological data, modeling the large-scale structure of the universe on cosmologically interesting scales to well within the statistical requirements set by future surveys.

Wastewater discharged by industrial plants, including paper mills, petroleum refineries, chemical processing and other manufacturing facilities pose serious environmental concerns. In order to effectively remove mixed types of pollutants, including fine solids, emulsified oil and dissolved chemicals, a multi-stage loop-flow flotation column (MSTLFLO) has been developed. The loop flow action in the MSTLFLO column provides favorable hydrodynamic conditions, which promotes local in-stage mixing and enhances the bubble-particle contacts. It overcomes the slow collection problems normally encountered in conventional column flotation. Furthermore, with the addition of suspended adsorbent particles, the MSTLFLO column can function as an adsorptive flotation device to remove dissolved chemicals from wastewater. In this study, an experimental investigation on a simulated wastewater system containing emulsified mineral oil, suspended particles (powdered activated carbon or glass beads), and dissolved phenol (as a representative chemical) has been carried out using the MSTLFLO flotation column. Test results show that the separation efficiencies of emulsified oil and fine particles are greater than 90% while the phenol removal approaches the limiting value of equilibrium adsorption. Thus, the potential application of MSTLFLO process for simultaneous removal of mixed pollutants from industrial wastewater has been demonstrated. The rate of oil/fine particle separations in MSTLFLO column obeys the non-linear kinetics. The kinetic constants are correlated in terms of a two-parameter (gas holdup and bubble size) expression. A process simulation program based on the classic tank-in-series model has been established. Experimental results for the removal of both individual and mixed components in MSTLFLO process are shown to be in excellent agreement with values predicted by the numerical simulation. In addition, a comparative study of oil removal in a 4-in and a 12-in MSTLFLO column has yielded a simple geometric scale-up scheme in term of the ratio of column diameters. The findings of this study are intended to provide an engineering design basis in exploring future applications of the MSTLFLO flotation process for industrial wastewater treatment.