We investigate the potential of inverse methods for retrieving adequate information about the rate kernel functions of cluster-cluster aggregation processes from mass density distribution data. Since many of the classical physical kernels have fractional order exponents the ability of an inverse method to appropriately represent such functions is a key concern. In early chapters, the properties of the Smoluchowski Coagulation Equation and its simulation using Monte Carlo techniques are introduced. Two key discoveries made using the Monte Carlo simulations are briefly reported. First, that for a range of nonlocal solutions of finite mass spectrum aggregation systems with a source of mass injection, collective oscillations of the solution can persist indefinitely despite the presence of significant noise. Second, that for similar finite mass spectrum systems with (deterministic) stable, but sensitive, nonlocal stationary solutions, the presence of noise in the system can give rise to behaviour indicative of phase-remembering, noise-driven quasicycles. The main research material on inverse methods is then presented in two subsequent chapters. The first of these chapters investigates the capacity of an existing inverse method in respect of the concerns about fractional order exponents in homogeneous kernels. The second chapter then introduces a new more powerful nonlinear inverse method, based upon a novel factorisation of homogeneous kernels, whose properties are assessed in respect of both stationary and scaling mass distribution data inputs.

We study the dynamics, kinematics and statistics of resonant and quasiresonant three-wave interactions appearing in models of geophysical flow. In these dispersive wave systems, the phenomenon of nonlinear resonance broadening plays a significant role across all three different branches of wave turbulence theory: from the statistical, to the discrete, and even the mesoscopic, formed as an intermediate regime between the two. The principal aim of this thesis is to understand the processes by which resonance broadening can induce a transition between each of these three different regimes. Beginning with the discrete case, we study two variants of the isolated triad: one with a constant additive forcing term; and the other in the presence of detuning. We provide a detailed analysis of both of these systems, covering their integrability and boundedness properties, showing that for almost all initial conditions the motion remains quasi-periodic and periodic respectively. Interestingly, we show that moderate amounts of detuning can actually promote energy exchange, increase the period and in rare instances cease to be periodic at all; each of these statements are contrary to what was previously thought. This motivates a more detailed study into the kinematics of resonance broadening. By analysing how the set of quasi-resonant modes develops under increased broadening, we show that a percolation-like transition exists, independent of the dispersion relationship used. At critical levels of broadening, we see the emergence of a single quasi-resonant cluster that begins to dominate the entire system. We argue that the formation of this cluster provides a way of characterising the turbulent state of the system, distinguishing between the discrete and statistical regimes. Through direct numerical simulation of the Charney-Hasegawa-Mima equation, we then assess whether this view is truly representative of the underlying dynamics. Here we find that the generation of quasi-resonantly excited modes can be detected through the statistical measures of total correlation and mutual information. We conclude by suggesting that these techniques have an incredible potential to infer the signature of both resonant and quasi-resonant clusters in fully realised turbulent systems, and yet are also subtle enough to detect qualitative changes in the underlying dynamics between different interacting modes.

<p> In previous papers about searches on star graphs several patterns have been made apparent; the speed up only occurs when graphs are ''tuned'' so that their time step operators have degenerate eigenvalues, and only certain initial states are effective. More than that, the searches are never faster than order square root of N time. In this thesis the problem is defined rigorously, the causes for all of these patterns are identified, sufficient and necessary conditions for quadratic-speed searches for any connected subgraph are demonstrated, the tolerance of these conditions is investigated, and it is shown that (unfortunately) we can do no better than order square root of N time. Along the way, a useful formalism is established that may be useful in future work involving highly symmetric graphs.</p><p> The tools and techniques so derived are then used to demonstrate that tree graphs can be used for the computation of Boolean functions. The philosophy of Farhi's work on the continuous-time NAND tree is applied to a discrete-time walk with any (AND, OR, NAND, or NOR) gate at each vertex. Tentative results show that the vast majority of possible Boolean functions on <i>N</i> bits can be calculated in order square root of N time.</p>

This thesis reports research carried out towards numerical simulation of hypersonic flows around space-plane shapes. For high speed flows around conical geometries, a locally conical approximation was introduced, which resulted in locally conical Navier-Stokes equations. In order to achieve accuracy and efficiency for steady state solutions, various methods were investigated. Based on the MacCormack implicit scheme and the Beam-Warming implicit scheme, two implicit procedures were developed to solve the locally conical Navier-Stokes equations (LCNSE). A new implicit boundary treatment was introduced in the MacCormack implicit scheme. The source term in the governing equations was treated explicitly. A simplified Beam-Warming implicit scheme was developed for its application to the LCNSE. Accuracy of the two schemes was investigated. The time step dependence of steady state solution with MacCormack-type schemes was analyzed and a procedure to reduce the error was proposed. To further accelerate the convergence to the steady state, two multigrid methods were applied to the two implicit schemes respectively. An extention of Ni-type multigrid method was developed to accelerate the MacCormack implicit scheme, and the FAS multigrid method was employed to accelerate the simplified Beam-Warming implicit scheme. In parallel, a new approach for fast steady state solution - sparse quasi-Newton method - was proposed to avoid difficulties in linearization associated with implicit schemes for general CFD problems. Formulation was given for three-point and five-point spatial discretization schemes. Preliminary results of a nozzle problem with van Leer's flux splitting and Harten's TVD high shock-resolution schemes illustrated significantly faster convergence to steady state with the sparse quasi-Newton approach than those with corresponding implicit operators of van Leer and Harten. Numerical simulations by solving LCNSE with the two implicit schemes developed in this study were carried out on hypersonic flows around a cone, on the leeside of a delta wing and beneath/over a cone-delta-wing combination. Detailed structures of the complex flow interaction were well predicted including the existence of embedded shock waves and secondary vortices. Comparison with available experimental data was made. Euler solutions were also carried out to compare with the N-S solutions. In the present hypersonic delta wing flow simulation, different phenomena were found than would have been expected from the Miller and Wood classification in the lower speed range. The numerical simulation of hypersonic viscous flows around a cone-delta-wing combination was the first flow field simulation around such a shape representing wing-body interference. It was found that the complexity of the flow field results from the shock-shock, shock-boundary layer and shock-vortex interactions in the flow field. High local heating and its cause were revealed near the corner on both the windward side and the leeward side surfaces of the geometry.

No description available.

Transient growth is quantitatively examined in two prototype separated flows using Direct Numerical Simulation (DNS). Separated flows typically exhibit regions of convective instability due to the inflectional velocity profiles inherent in the shear flow. This can lead to the transient growth of small disturbances by many orders of magnitude. After reviewing the mathematical tools and numerical techniques required, we present an analysis of transient growth in an axisymmetric pipe with a 1:2 diametral expansion. A direct method is used to calculate the optimal transient energy growth for specified time horizons and Reynolds numbers up to Re=1200, and low-order azimuthal wavenumber m. At each Re the maximum growth is in azimuthal mode m=1 and this maximum is found to increase exponentially with Re. The time evolution of optimal perturbations is presented and shown to correspond to sinuous oscillations of the shear layer. Finally, full three-dimensional DNS with the in flow perturbed with Gaussian white-noise conforms the presence of the structures determined by the transient growth analysis. The second prototype flow considered is the cylinder wake in the subcritical regime. Large energy growth is observed at Reynolds numbers close to the onset of global instability and the optimal perturbations which lead to this growth are determined. Three-dimensional spanwise perturbations are also examined and it is found that, except for short time horizons, the zero wavenumber is dominant. Furthermore, performing accurate linear and transient growth analysis is found to be highly dependent on the size of the computational domain. Adjoint eigenmodes extend far upstream of the cylinder necessitating a long in flow. More importantly, constrictions in the cross-stream direction are found to distort the basic flow, which has a substantial effect on the accuracy of the analysis. Transition in pipe flow is a topic for which there is still relatively little understanding. Pus are small regions of turbulence observed close to the transitional Reynolds number. A gradually expanding pipe is proposed as a means to effectively produce turbulent puffs and study their creation and decay.

This work presents a large scale multifractal analysis of the electronic state in the vicinity of the localisation-delocalisation transition in the three-dimensional Anderson model of localisation using high-precision data and very large system sizes of up to L3 = 2403. The multifractal analysis is implemented using box- and system- size scaling of the generalized inverse participation ratios employing typical and ensemble averaging techniques. The statistical analysis in this study has shown that in the thermodynamic limit a proposed symmetry relation in the multifractal exponents is true for the 3D Anderson model in the orthogonal universality class. Better agreement with the symmetry is found when using system-size scaling with ensemble averaging in which a more complete picture of the multifractal spectrum f(α) is also obtained. A complete profile of f(α) has negative fractal dimensions and shows the contributions coming from the tails of the distribution. Various boxpartitioning approaches have been carefully studied such as the use of cubic and non-cubic boxes, periodic boundary conditions to enlarge the system, and single and multiple origins in the partitioning grid. The most reliable method is equal partitioning of a system into cubic boxes which has also been shown to be the least numerically expensive. Furthermore, this work gives an expression relating f(α) and the probability density function (PDF) of wavefunction intensities. The relation which contains a finite-size correction provides an alternative and simpler method to obtain f(α) directly from the PDF in which f(α) is interpreted as the scaleinvariant distribution at criticality. Finally, a generalization of standard multifractal analysis which is applicable to the critical regime and not just at the critical point is presented here. Using this generalization together with finite-size scaling analysis, estimates of critical disorder and critical exponent based on exact diagonalization have been obtained that are in excellent agreement, supporting for the first time previous results of transfer matrix calculations.

This study is concerned with the prediction of the fluid-flow, chemical reactions and heat transfer processes in an industrial off-gas ducting system. A mathematical model is developed and then applied to predict the processes occurring in the off-gas ducting system. Particular attention is focussed on the two-phase thermal behaviour and the chemical reactions. A three-dimensional, two-phase numerical solution technique is used to solve the governing time-averaged partial differential equations. The model includes equations for turbulence, chemical reactions and two-phase thermal radiation. The calculations are performed for a particulate phase comprising non-reacting particles and a gaseous phase comprising chemically reacting gases. Both exothermic and endothermic reactions are considered. The effects of thermal radiation, particle solidification, chemical reactions and heat transfer on the two-phase flow are introduced and examined in detail. Predictions are made for an extensive range of parameters. The effects of these parameters on the off-gas ducting system are quantified. Comparisons are made between predicted results and experimental data when available and agreement is reasonable. The models developed can be easily incorporated into general-purpose fluid-flow packages. The procedure is general, and allows two-phase, two- or three-dimensional computations. Industrial plant can be modelled realistically on minicomputers at moderate costs. Convergence can normally be obtained with ease. It is concluded that for the cases studied, thermal radiation is a dominant factor in the calculation of the heat losses and that the particle contribution to these losses is small compared with that of the gases. The model indicates that the strongly temperature dependent reaction rates have a dominant influence in determining optimal operating conditions.

A mathematical model has been developed, simulating various aspects of an iron blast furnace, for the purpose of analysing its behaviour. This involved the simulation of a counter current compressible gas flow, through a packed bed, dealing with the momentum and thermal energy of both phases. Directional resistances were added to the gas momentum, so as to account for the interphase friction caused by the packed bed. This enabled the prediction of the cohesive zone geometry, together with the active coke and stack, thus providing an important step for a successful analysis. The availability of multi-phase codes to solve such a system was limited and those existing being inadequate to represent these kinds of problems. What resulted was, the development of an algorithm to solve for two phases (gas and solids) with interspersed counter current flow, where the solids behaved as a packed bed. The algorithm developed is an enhanced version of existing algorithms. As well as the numerical model, a physical model of the raceway was developed, using dry ice particles to simulate the packed bed. The sublimation properties of the ice give a more realistic simulation to coke combustion, compared to the use of inert particles. The results of the experiment brought to light the effects of particle-particle interaction as being most significant in enabling the solids bed to move freely, around and into the raceway. From numerical modelling results, it is concluded that the ore:coke charging profile plays a dominant role in furnace behaviour. More interestingly, the gas distribution was not affected by raceway geometries when the cohesive zone was not in the immediate vicinity. It was therefore concluded that, the size and shape of the raceway zone has little influence on the gas distribution in the iron blast furnace.

The existence of discrete properties is shown in the landscape of the Free-Fermionic Heterotic-String vacua. These were discovered via the classification of the SO(10) GUT gauge group and its subgroups, such as, the Pati-Salam, the Flipped SU(5) and the SU(4)×SU(2)×U(1) models. The classification is carried out by fixing a set of basis vectors and then varying the GGSO projection coefficients entering the one-loop partition function. The analysis of the models is facilitated by deriving algebraic expressions for the GGSO projections to enable a computerised analysis of the entire String spectrum and the scanning of large spaces of vacua. The analysis reveals an abundance of 3 generation models with exophobic String vacua. This is observed with the SO(10) and the Pati-Salam models. Contrary to this, the Flipped SU(5) models contained no exophobic vacua with an odd number of generations. Moreover, it is also observed that the SU(4)×SU(2)×U(1) models are substantially more constrained and that no generations exist. The analysis of the SU(3)×U(1)×SU(2)×U(1) and the SU(3)×U(1)×SU(2)×SU(2) models are being examined, which is work in progress, that are expected to generate further interesting phenomenology.