Turbulent flows occur in various fields and are a central, yet an extremely complex, topic in fluid dynamics. Understanding the behaviour of fluids is vital for multiple research areas including, but not limited to: biological models, weather prediction, and engineering design models for automobiles and aircraft. In this thesis, we study a number of fundamental problems that arise in 2D turbulent flows, using the 2D Navier-Stokes system. Introducing optimization techniques for systems described by partial differential equations (PDE), we frame these problems such that they can be solved using computational methods. We utilize adjoint calculus to build the computational framework to be implemented in an iterative gradient flow procedure, using the "optimize-then-discretize" approach. Pseudospectral methods are employed for solving PDEs in a numerically efficient manner. The use of optimization methods together with computational mathematics in this work provides an illuminating perspective on fluid mechanics. We first apply these techniques to better understand enstrophy dissipation in 2D Navier-Stokes flows, in the limit of vanishing viscosity. By defining an optimization problem to determine optimal initial conditions, multiple branches of local maximizers were obtained each corresponding to a different mechanism producing maximum enstrophy dissipation. Viewing this quantity as a function of viscosity revealed quantitative agreement with an analytic bound, demonstrating the sharpness of this bound. We also introduce an extension of this problem, where enstrophy dissipation is maximized in the context of kinetic theory using the Boltzmann equation. Secondly, these PDE-constrained optimization techniques were used to probe the fundamental limitations on the performance of the Leith eddy-viscosity closure model for 2D Large-Eddy Simulations of the Navier-Stokes system. Obtained by solving an optimization problem with a non-standard structure, the results demonstrate the optimal eddy viscosities do not converge to a well-defined limit as regularization and discretization parameters are refined, hence the problem of determining an optimal eddy viscosity is ill-posed. Further extending the problem of finding optimal eddy-viscosity closures, we consider imposing an additional nonlinear constraint on the control variable in the problem, in the form of requiring the time-averaged enstrophy be preserved. To address this problem in a novel way, we employ adjoint calculus to characterize a subspace tangent to the constraint manifold, which allows one to approximately enforce the constraint. Not only do we demonstrate that this produces better results when compared to the case without constraints, but this also provides a flexible computational framework for approximate enforcement of general nonlinear constraints. Lastly in this thesis, we introduce an optimization problem to study the Kolmogorov-Richardson energy cascade, where a pathway towards solutions is outlined. / Thesis / Doctor of Philosophy (PhD)

This study has two parts. In the first part we develop a computational approach to the solution of an inverse modelling problem concerning the material properties of electrolytes used in Lithium-ion batteries. The dependence of the diffusion coefficient and the transference number on the concentration of Lithium ions is reconstructed based on the concentration data obtained from an in-situ NMR imaging experiment. This experiment is modelled by a system of 1D time-dependent Partial Differential Equations (PDE) describing the evolution of the concentration of Lithium ions with prescribed initial concentration and fluxes at the boundary. The material properties that appear in this model are reconstructed by solving a variational optimization problem in which the least-square error between the experimental and simulated concentration values is minimized. The uncertainty of the reconstruction is characterized by assuming that the material properties are random variables and their probability distribution estimated using a novel combination of Monte-Carlo approach and Bayesian statistics. In the second part of this study, we carefully analyze a number of secondary effects such as ion pairing and dendrite growth that may influence the estimation of the material properties and develop mathematical models to include these effects. We then use reconstructions of material properties based on inverse modelling along with their uncertainty estimates as a framework to validate or invalidate the models. The significance of certain secondary effects is assessed based on the influence they have on the reconstructed material properties. / Thesis / Doctor of Philosophy (PhD)

<p>In this work we propose and validate a computational method for reconstructing constitutive relations (material properties) in complex multiphysics phenomena based on incomplete and noisy measurements which is applicable to different problems arising in nonequilibrium thermodynamics and continuum mechanics. The parameter estimation problem is solved as PDE–constrained optimization using a gradient–based technique in the optimize–then–discretize framework. The reconstructed material properties taken as an example here are the transport coefficients characterizing diffusion processes such as the viscosity and the thermal conductivity, and we focus on problems in which these coefficients depend on the state variables in the system. The proposed method allows one to reconstruct a smooth constitutive relation defined over a broad range of the dependent variable. This research is motivated by questions arising in the computational analysis and optimization of advanced welding processes which involves modelling complex alloys in the liquid phase at high temperatures.</p> / Doctor of Philosophy (PhD)