In the present doctoral work we have developed and tested a model for dissolved oxygen (DO) transfer from water to underlying flat and cohesive sediment beds populated with DO-absorbing bacteria. The model couples Large-Eddy Simulation (LES) of turbulent transport in the water-column, a biogeochemical model for DO transport and consumption in the sediment, and Darcy’s Law for the pore water-driven solute dispersion and advection. The model’s predictions compare well against experimental data for low friction-Reynolds numbers (Re). The disagreement for higher Re is investigated by progressively increasing the complexity of the model. A sensitivity analysis shows that the sediment-oxygen uptake (or demand, SOD) is approximately proportional to the bacterial content of the sediment layer, and varies with respect to fluid dynamics conditions, in accordance to classic high-Schmidt-number mass-transfer laws. The non- linear transport dynamics responsible for sustaining a statistically steady SOD are investigated by temporal- and-spatial correlations and with the aid of instantaneous visualizations: the near-wall coherent structures modulate the diffusive sublayer, which exhibits complex spatial and temporal filtering behaviours; its slow and quasi-periodic regeneration cycle determines the streaky structure of the DO field at the sediment-water interface (SWI), retained in the deeper layers of the porous medium. Oxygen depletion dynamics are then simulated by preventing surface re-areation with turbulent mixing driven by an oscillating low-speed current — an idealization of hypolimnetic DO depletion in the presence of a non-equilibrium periodic forcing. The oxygen distribution exhibits a self-similar pattern of decay with, during the deceleration phase, oscillations modulated by the periodic ejection of peaks of high turbulent mass flux (pumping oxygen towards the SWI), generated at the edge of the diffusive sublayer at the end of the acceleration phase. These fronts of highly turbulent mixing propagate away from the SWI, at approximately constant speed, in layers of below-average oxygen concentration. Finally, the model has been tested in a real geophysical framework, reproducing published in-situ DO measurements of a transitional flow in the bottom boundary layer of lake Alpnach. A simple model for the SOD is then derived for eventual inclusion in RANSE biogeochemical management-type models for similar applications. / Thesis (Ph.D, Mechanical and Materials Engineering) -- Queen's University, 2012-07-04 11:13:24.936

In this thesis, we study homogenization of optimal control problems in various oscillatory domains. Specifically, we consider four types of domains given in Figure 1 below. Figure 1: Oscillating Domains The thesis is organized into six chapters. Chapter 1 provides an introduction to our work and the rest of the thesis. The main contributions of the thesis are contained in Chapters 2-5. Chapter 6 presents the conclusions of the thesis and possible further directions. A brief description of our work (Chapters 2-5) follows: Chapter 2: Asymptotic behaviour of a fourth order boundary optimal control problem with Dirichlet boundary data posed on an oscillating domain as in Figure 1(A) is analyzed. We use the unfolding operator to study the asymptotic behavior of this problem. Chapter 3: Homogenization of a time dependent interior optimal control problem on a branched structure domain as in Figure 1(B) is studied. Here we pose control on the oscillating interior part of the domain. The analysis is carried out by appropriately defined unfolding operators suitable for this domain. The optimal control is characterized using various unfolding operators defined at each branch of every level. Chapter 4: A new unfolding operator is developed for a general oscillating domain as in Figure 1(C). Homogenization of a non-linear elliptic problem is studied using this new un-folding operator. Using this idea, homogenization of an optimal control problem on a circular oscillating domain as in Figure 1(D) is analyzed. Chapter 5: Homogenization of a non-linear optimal control problem posed on a smooth oscillating domain as in Figure 1(C) is studied using the unfolding operator.

Mathematical theory of partial differential equations (PDEs) is a pretty old classical area with wide range of applications to almost every branch of science and engineering. With the advanced development of functional analysis and operator theory in the last century, it became a topic of analysis. The theory of homogenization of partial differential equations is a relatively new area of research which helps to understand the multi-scale phenomena which has tremendous applications in a variety of physical and engineering models, like in composite materials, porous media, thin structures, rapidly oscillating boundaries and so on. Hence, it has emerged as one of the most interesting and useful subject to study for the last few decades both as a theoretical and applied topic. In this thesis, we study asymptotic analysis (homogenization) of second-order partial differential equations posed on an oscillating domain. We consider a two dimensional oscillating domain (comb shape type) consisting of a fixed bottom region and an oscillatory (rugose) upper region. We introduce optimal control problems for the Laplace equation. There are mainly two types of optimal control problems; namely distributed control and boundary control. For distributed control problems in the oscillating domain, one can apply control on the oscillating part or on the fixed part and similarly for boundary control problem (control on the oscillating boundary or on the fixed part the boundary). We consider all the four cases, namely distributed and boundary controls both on the oscillating part and away from the oscillating part. The present thesis consists of 8 chapters. In Chapter 1, a brief introduction to homogenization and optimal control is given with relevant references. In Chapter 2, we introduce the oscillatory domain and define the basic unfolding operators which will be used throughout the thesis. Summary of the thesis is given in Chapter 3 and future plan in Chapter 8. Our main contribution is contained in Chapters 4-7. In chapters 4 and 5, we study the asymptotic analysis of optimal control problems namely distributed and boundary controls, respectively, where the controls act away from the oscillating part of the domain. We consider both L2 cost functional as well as Dirichlet (gradient type) cost functional. We derive homogenized problem and introduce the limit optimal control problems with appropriate cost functional. Finally, we show convergence of the optimal solution, optimal state and associate adjoint solution. Also convergence of cost-functional. In Chapter 6, we consider the periodic controls on the oscillatory part together with Neumann condition on the oscillating boundary. One of the main contributions is the characterization of the optimal control using unfolding operator. This characterization is new and also will be used to study the limiting analysis of the optimality system. Chapter 7 deals with the boundary optimal control problem, where the control is applied through Neumann boundary condition on the oscillating boundary with a suitable scaling parameter. To characterize the optimal control, we introduce boundary unfolding operators which we consider as a novel approach. This characterization is used in the limiting analysis. In the limit, we obtain two limit problems according to the scaling parameters. In one of the limit optimal control problem, we observe that it contains three controls namely; a distributed control, a boundary control and an interface control.

Mathematical theory of homogenization of partial differential equations is relatively a new area of research (30-40 years or so) though the physical and engineering applications were well known. It has tremendous applications in various branches of engineering and science like : material science ,porous media, study of vibrations of thin structures, composite materials to name a few. There are at present various methods to study homogenization problems (basically asymptotic analysis) and there is a vast amount of literature in various directions. Homogenization arise in problems with oscillatory coefficients, domain with large number of perforations, domain with rough boundary and so on. The latter one has applications in fluid flow which is categorized as oscillating boundaries. In fact ,in this thesis, we consider domains with oscillating boundaries. We plan to study to homogenization of certain optimal control problems with oscillating boundaries. This thesis contains 6 chapters including an introductory Chapter 1 and future proposal Chapter 6. Our main contribution contained in chapters 2-5. The oscillatory domain under consideration is a 3-dimensional cuboid (for simplicity) with a large number of pillars of length O(1) attached on one side, but with a small cross sectional area of order ε2 .As ε0, this gives a geometrical domain with oscillating boundary. We also consider 2-dimensional oscillatory domain which is a cross section of the above 3-dimensional domain. In chapters 2 and 3, we consider the optimal control problem described by the Δ operator with two types of cost functionals, namely L2-cost functional and Dirichlet cost functional. We consider both distributed and boundary controls. The limit analysis was carried by considering the associated optimality system in which the adjoint states are introduced. But the main contribution in all the different cases(L2 and Dirichlet cost functionals, distributed and boundary controls) is the derivation of error estimates what is known as correctors in homogenization literature. Though there is a basic test function, one need to introduce different test functions to obtain correctors. Introducing correctors in homogenization is an important aspect of study which is indeed useful in the analysis, but important in numerical study as well. The setup is the same in Chapter 4 as well. But here we consider Stokes’ Problem and study asymptotic analysis as well as corrector results. We obtain corrector results for velocity and pressure terms and also for its adjoint velocity and adjoint pressure. In Chapter 5, we consider a time dependent Kirchhoff-Love equation with the same domain with oscillating boundaries with a distributed control. The state equation is a fourth order hyperbolic type equation with associated L2-cost functional. We do not have corrector results in this chapter, but the limit cost functional is different and new. In the earlier chapters the limit cost functional were of the same type.