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In order to meet new stringent emissions and noise reduction targets, the turbomachinery community is increasingly concerned with addressing inefficiencies in aeroengines, and of particular interest is the issue of sound generated through aeroacoustic processes within the components of the turbomachine. At the same time, precipitous increases in computing power over the last thirty years are driving larger and more detailed simulations. Increasing data output from these requires the development of new techniques that are able to identify the underlying processes and facilitate future efforts for optimisation. This thesis seeks to address this by creating a direct-adjoint framework that can be applied in two sub-categories of flow problems commonly found in turbomachinery aeroacoustics, and lays the groundwork for future efforts in the efficient analysis and optimisation of these flows. For this, the in-house, high-order, time-domain pseudo-wave-based aeroacoustics code, previously used for isolated airfoil self-noise studies, has been extensively modified and adapted for application in turbomachinery geometries. Using this code as a starting point, self-excited instabilities and the resulting acoustic fields are simulated within a representative compressor blade row, at off-design conditions. Prior experiments and numerical calculations of the flow at these conditions have confirmed the presence of an unsteady reattaching laminar separation bubble on the suction side of the blade, shedding vortices that are subsequently scattered by the blunt trailing edge. Nonlinear flow analyses are carried out first within a single blade passage, followed by a linear impulse response analysis. A detailed linear direct-adjoint mean flow global stability analysis is performed, and the direct spectrum obtained and correlated with the nonlinear pressure signals, showing excellent agreement. The direct global modes are categorised and corresponding adjoint modes leveraged to compute structural sensitivities with wavemaker regions. The sensitivity analysis confirms the presence of aeroacoustic feedback loops between the pressure and suction sides of the blade, and is consistent with earlier observations in similar studies of isolated airfoils. The theory of block-circulant matrices is then detailed and exploited to achieve computationally efficient decompositions of the governing system matrices, and the previous global stability analysis reproduced with ten passages per period window. An increase in linear growth rates is observed, new low-frequency structures identified and analysed in the context of new feedback loop pathways which propagate disturbances throughout the domain. The effects of aerodynamic mistuning are then briefly addressed. Rotor-stator interaction is then considered in the context of an adjoint sensitivity problem. For this, a new high-order, time-domain adjoint sliding-plane interface treatment, which is capable of modelling and propagating self-excited instabilities and tonal blade row interaction noise, is developed and extensively validated. This work concludes with the application of this theory to a simplified rotor-stator sensitivity analysis within the context of a time-domain nonlinear-adjoint looping algorithm. Sensitivities identified here are briefly discussed. Applicability of these methods to different codes and numerical treatments is highlighted throughout this thesis.

Wrinkling is a universal instability occurring in a wide variety of engineering and biological materials. Despite extensive study across different systems, a full description is still lacking, particularly in the case of growing materials. Most studies consist of some combination of a substantially simplified elastic model, an analysis that does not go beyond first order, and numerical simulations in commercial software packages that do not necessarily correspond to the mathematical system studied. This thesis addresses all three of these shortcomings by providing a systematic analysis of a fully hyperelastic bilayer past the linear stability threshold into the weakly nonlinear regime, along with a carefully discretised numerical bifurcation analysis of the system. For comparison, we assume that wrinkling is generated either by the isotropic growth of the film or by the lateral compression of the entire film-substrate system (both in plane strain). We adopt a stream-function-based formulation and perform an exhaustive linear analysis of the wrinkling problem for all stiffness ratios and under a variety of additional boundary and material effects. Namely, we consider the effect of added pressure, surface tension, an upper substrate, and fibres. We obtain analytical estimates of the instability in the two asymptotic regimes of long and short wavelengths. We then carry out a weakly-nonlinear analysis to derive an amplitude equation that describes the evolution of the wrinkling amplitude beyond the bifurcation point, followed by a comprehensive numerical bifurcation analysis of the problem using the finite element method. We demonstrate excellent agreement between the weakly-nonlinear analysis and the numerical experiments and are also able to directly solve for the bifurcation point in our discretised system and characterise the effect of implementation details such as the aspect ratio of the computational domain on the observed bifurcation point. We then explore solutions of the amplitude equation in the case that the wrinkling amplitude is allowed to vary over long spatial and/or temporal scales. Finally, we demonstrate that our numerical experiments are able to identify secondary bifurcations in the system similar to those observed in experiments for which analytical methods have thus far been unable to provide a complete explanation.

This thesis consists of two parts. In the first, we develop two new strategies for spatial white noise and Gaussian-Mat\'ern field sampling that work within a non-nested multilevel (quasi) Monte Carlo (ML(Q)MC) hierarchy. In the second, we apply the techniques developed to quantify the level of uncertainty in a new stochastic model for tracer transport in the brain. The new sampling techniques are based on the stochastic partial differential equation (SPDE) approach, which recasts the sampling problem as the solution of an elliptic equation driven by spatial white noise. We present a new proof of an a priori error estimate for the finite element (FEM) solution of the white noise SPDE. The proof does not require the approximation of white noise in practice, and includes higher order elliptic operators and p-refinement. Within the SPDE approach, the efficient sampling of white noise realisations can be computationally expensive. In this thesis, we present two new sampling techniques that can be used to efficiently compute white noise samples in a FEM-MLMC and FEM-MLQMC setting. The key idea is to exploit the finite element matrix assembly procedure and factorise each local mass matrix independently, hence avoiding the factorisation of a large matrix. In a multilevel framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. In the MLQMC case, the QMC integrand variables must also be ordered in order of decaying importance to achieve fast convergence with respect to the number of samples. We express white noise as a Haar wavelet series whose hierarchical structure naturally exposes the leading order dimensions. We split this series in two terms which we sample via a hybrid standard Monte Carlo/QMC approach. We demonstrate the efficacy of our sampling methods with numerical experiments. In a multilevel setting, a good coupling is enforced and the telescoping sum is respected. In the MLQMC case, the asymptotic convergence rate is the same as standard Monte Carlo, but significant computational gains are obtained in practice thanks to a pre-asymptotic QMC-like regime. In the final part of the thesis, we employ a combination of the methods presented to solve a PDE with random coefficients describing tracer transport within the interstitial fluid of the brain. Numerical simulations support the claim that diffusion alone cannot explain the penetration of tracers within deep brain regions as observed in clinical experiments, even when uncertainties in the diffusivity have been accounted for. A convective velocity field may however increase tracer transport, provided that a directional structure is present in the interstitial fluid circulation.

Multiphase flow through porous media can be modelled as a complex system of partial differential equations. Such models can be used to optimize the recovery of oil and gas from subsurface reservoirs. In the case of highly viscous oils, thermal recovery techniques are typically used to enhance their extraction. To simulate this, models describing the flow of fluids (typically oil, water, and gas) are coupled with a model for heat flow. Thermal reservoir simulation entails solving these highly coupled systems. Their complexity and the computational effort needed to solve them motivate the need for highly efficient solvers. In reservoir simulation, most of the computational time is spent on solving linearized systems with a preconditioned Krylov subspace iterative method. Industry-standard preconditioning techniques are based on the approach introduced by Wallis in 1983, the Constrained Pressure Residual method (CPR). This preconditioner is a two-stage process involving the solution of a restricted pressure system. While initially designed for isothermal reservoir simulation, CPR is also the standard for thermal cases. However, its treatment of the conservation of energy equation does not incorporate heat diffusion, which is often dominant in thermal cases. We are interested in preconditioners specifically designed for thermal reservoir simulation. In this thesis, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted pressure-temperature system is solved in the first stage. To study the effects of both pressure and temperature on fluid and heat flow, we first consider a model of non-isothermal single-phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Then, we extend this method for multiphase flow as a solver for the first stage of CPTR. We present a comparison of the algorithmic performance of the different preconditioning approaches under mesh refinement and parallelization.

Two-phase flows arise in many areas of application such as in the study of coastal and hydraulic processes. Often the fluids involved can be modelled by incompressible phases which have disparate physical properties such as density and viscosity. We utilise a two-phase flow model of immiscible Newtonian fluids. A key part of this model is a set of variable coefficient Navier-Stokes equations. This thesis focuses on the numerical solution of the linear systems which arise after linearisation of these equations. Solving these systems often dominates the computation time when running simulations. One of the challenges in solving the model is that the density and viscosity coefficients are discontinuous and can have large jumps between the two phases. In this work we consider preconditioned iterative Krylov methods to solve the large and sparse linear systems and pay particular attention to incorporating the highly varying coefficients into the block preconditioners that we propose. We will see that such considerations can be essential in order to obtain good performance. An important issue is the scalability of the solution methodology. Here, we will study how the convergence of the iterative solver depends on a grid parameter which controls the refinement of the computational mesh. We will see that the novel preconditioners we propose can lead to convergence which is effectively independent of the grid parameter. We also investigate dependence on other model parameters such as the Reynolds number as well as the density and viscosity ratios between the two fluids. Another topic we examine is the use of a multipreconditioned iterative method allowing more than one preconditioner to be used simultaneously. Our results using this approach show some promising features. Finally, we consider an implementation within a more realistic model used in practice for simulating complex air-water flows. In particular, we will provide results for a problem modelling the breaking of a dam.

We use a continuum phase field model to study pattern formation in Diblock Copolymers (DBCs), with an emphasis on terracing in thin films. Phase separation in DBCs is an important process in nanotechnology that yields an array of highly regular structures on very fine length scales that are of enormous utility. Terraces are ubiquitous in thin film DBC melts. It was long believed that modelling these structures was an intractable problem, but recent attempts to do so using the Self Consistent (Mean) Field Theory (SCFT) were successful. In this thesis, we use the simpler and more flexible Ohta-Kawasaki (non-local Cahn-Hilliard) Density Functional Theory (DFT) model, derived from the SCFT, with obstacle homogeneous, and long-range energy components. Our main result is that such models can be used to capture terraces in the simplified 2D scenarios we examine, and that these structures form as an inevitable consequence of a simple energy minimisation process. We consider a 2-phase model for DBC micro-structure evolution, which we analyse rigorously and comprehensively. We study stationary and dynamic (gradient flow) variants of the problem, using the latter as the basis for an efficient but provably convergent numerical method for simulating DBC evolution. Our theoretical analysis and numerical method are based on a common Moreau-Yosida regularisation of the homogeneous energy component. We use the Semi-Smooth Newton (SSN) method to resolve the non-linearities in our numerical method. Comprehensive numerical experiments confirm our theoretical predictions, including a Method of Manufactured Solutions (MMS) test for the order of convergence. We propose a complete specification of the equally spaced global energy minimisers of the symmetric, 2-phase problem in 1D, and offer a practical algorithm for finding them. We explicitly characterise the Order-Disorder Transition (ODT) in parameter space. We extend our 2-phase model and methods to a 3-phase situation involving a copolymer/homopolymer blend, in which the free surface of the copolymer can adjust to form terraces. We develop energy estimates for energy-minimising 1D morphologies in the sharp interface limit, which we then combine via a double-tangent construction to analyse low energy, 2D structures. We argue that thin films with terraces have a lower free energy than their flat counterparts. Time-dependent numerical simulations in 2D confirm these predictions and provide insight into the evolution of these structures. We replicate and extend recent SCFT results, by performing a numerical experiment in which a terrace forms from a flat initial state. The model, tools and methods developed in this thesis should serve as a solid foundation for researching more complex, multi-step 3D terraces.