<p>We study Ginzburg–Landau equations for a complex vector order parameter to a two-component system. We discuss the existence, uniqueness, asymptotics, monotonicity and stability of solutions by extending Alama-Bronsard-Mironescu's results in a more general case.</p> / Doctor of Philosophy (PhD)

A wind tunnel based investigation into the effects of unsteady yaw angles on the aerodynamics of a simple automotive body has been carried out to increase the understanding of the effects of unsteady onset conditions similar to those experienced in normal driving conditions. Detailed flow field measurements have been made using surface pressure tappings and PIV around a simple automotive model in steady state conditions and these have been compared to measurements made whilst the model was oscillating in the yaw plane. The oscillating motion was created by a motored crank which was used to produce consistent and repeated motion which produced a reduced frequency that indicated that a quasi-static response should be expected. The PIV data are used to compare the wake flow structures and the surface pressures are used to infer aerodynamic loads and investigate the development of the flow structures across the surfaces of the model. This includes a comprehensive comparison of the surface pressures on the sides of the model during a transient and quasi-static yaw angel oscillation. These results show differences between the two test conditions with the oscillating model results containing hysteresis and the greatest differences in the flow field occurring on the leeside of the model. Two configurations of the same model with different rear pillar geometries were used to isolate model specific effects. Square rear pillars create strong and stable trailing vortices which are less affected by the model motion whereas radiused rear pillars created weaker and less steady vortices that mixed with the quasi-2D wake behind the model base and were affected to a greater extent by the model motion. The unsteadiness in the trailing vortex separation feeds upstream into the A-pillar vortex demonstrating that small geometry changes at the rear can affect the entire flow field around the model.

We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.

Final accepted manuscripts of parts 4 and 5 from Roadmap on Structured Light, authored by Masud Mansuripur, College of Optical Sciences, The University of Arizona.

In this thesis we solve the Gross-Pitaevskii equation numerically in order to model the response of trapped Bose-Einstein condensed gases to perturbations by electromagnetic fields. First, we simulate output coupling of pulses from the condensate and compare our results to experiments. The excitation and separation of eigen-modes on flow through a constriction is also studied. We then move on to the main theme of this thesis: the important subject of quantised vortices in Bose condensates, and the relation between Bose-Einstein condensation and superfluidity. We propose methods of producing vortex pairs and rings by controlled motion of objects. Full three-dimensional simulations under realistic experimental conditions are performed in order to test the validity of these ideas. We link vortex formation to drag forces on the object, which in turn is connected with energy transfer to the condensate. We therefore argue that vortex formation by moving objects is intimately related to the onset of dissipation in superfluids. We discuss this idea in the context of a recent experiment, using simulations to provide evidence of vortex formation in the experimental scenario. Superfluidity is also manifest in the property of persistent currents, which is linked to vortex stability and dynamics. We simulate vortex line and ring motion, and find in both cases precessional motion and thermodynamic instability to dissipation. Strictly speaking, the Gross-Pitaevskii equation is valid only for temperatures far below the BEG transition. We end the thesis by describing a simple finite- temperature model to describe mean-field coupling between condensed and non- condensed components of the gas. We show that our hybrid Monte-Carlo/FFT technique can describe damping of the lowest energy excitations of the system. Extensions to this model and future research directions are discussed in the conclusion.

Parameters governing separation control with sweeping jet actuators over a deflected flap are investigated experimentally on a generic "Multiple Flap Airfoil" (MFA). The model enables an extensive variation of geometric and aerodynamic parameters to aid the scaling of this novel flow control method to full-size applications.Sweeping jets exit from discrete, millimeter-scale nozzles distributed along the span and oscillate from side-to-side. The sweeping frequency is almost linearly dependent on the supplied flowrate per actuator. The measured thrust exerted by a row of actuators agrees well with vectored momentum calculations. Frequency and thrust measurements suggest that the jet velocity is limited to subsonic speeds and that any additional increase in flowrate causes internal choking of the flow.Neither the flowrate nor the momentum input is found to be a sole parameter governing the lift for varying distance between adjacent actuators. However, the product of the mass flow coefficient and the square root of the momentum coefficient collapses the lift onto a single curve regardless of the actuator spacing. Contrary to other actuation methods, separation control with sweeping jets does not exhibit any hysteresis with either momentum input or flap deflection. A comparison between sweeping and non-sweeping jets illustrates the superior control authority provided by sweeping jets. Surface flow visualization on the flap suggests the formation of counter-rotating pairs of streamwise vortices caused by the interaction of neighboring jets.The actuation intensity required to attach the flow increases with increasing downstream distance from the main element's trailing edge and increasing flap deflection. No obvious dependence of the ideal actuation location on actuator spacing, flap deflection, angle of attack, or actuation intensity is found within the tested range. Comparisons between experimental and numerical results reveal that the inviscid flow solution appears to be a suitable predictor for the effectively and efficiently obtainable lift of a given airfoil configuration. The flap size affects the achievable lift, the accompanying drag, and the required flap deflection and actuation intensity. By controlling separation, the range of achievable lift coefficients is doubled without significant penalty in drag even when considering a safety margin for the maximum applicable incidence.

L'objectif de cette thèse est d'étudier la formation de tourbillons dans la zone morte des disques protoplanétaires. Un code numérique 3D compressible a été mis au point et utilisé pour cette étude. Deux instabilités hydrodynamiques sont envisagées pour former les tourbillons: l'instabilité de Rossby et l'instabilité barocline.La première entraine la fragmentation d'une sur-densité annulaire en une chaîne de tourbillons qui se rattrapent les uns les autres et finissent par fusionner en un seul tourbillon qui reste stable sur de très longues durées lorsque son rapport d’aspect est suffisamment grand, et possède une structure quasi bidimensionnelle. En revanche, les tourbillons tridimensionnels de petits rapport d'aspect sont affectés par l’instabilité elliptique qui les détruits en quelques rotations. Seuls persistent ceux de grand rapport d'aspect.L'instabilité barocline, fondamentalement non linéaire, produit des tourbillons à partir de perturbations d'amplitude finies ; ces tourbillons sont ensuite amplifiés et fusionnent en tourbillons plus gros si le disque est stratifié de façon instable et s’il permet aussi le transfert de chaleur. Deux types de transfert thermique ont été envisagés pour étudier cette instabilité qui conduit alors à des différences significatives dans la structure des tourbillons formés. Le rapport d'aspect étant lié à la vorticité, l'amplification des tourbillons se traduit par une diminution de leur rapport d'aspect, et les rend donc sujet à l'instabilité elliptique. Cependant, ils ne sont pas détruit et gardent une structure tourbillonnaire grâce à l'amplification barocline. / The objective of this thesis is to study the formation of vortices in the dead-zone of protoplanetary disks. A 3D compressible numerical code has been performed and used for this study. Two hydrodynamical instabilities are considered for vortex formation: the Rossby wave instability and the baroclinic instability.The first one leads tp the fragmentation of an annular bump into a chain of vortices that catch one another and merge in a single vortex; this vortex remains stable on very long durations when its aspect ratio is large enough and has a quasi two-dimensional structure. In contrast, tridimensional small aspect ratios vortices are affected by the elliptical instability and are destroyed in a few rotation periods. Only vortices with large aspect ratios can survive.The baroclinic instability, a basically non-linear one, can produce vortices from small amplitude perturbations; these vortices are then amplified and merge in bigger vortices if the disk is unstably stratified and also permits heat transfer. Two types of heat transfer have been considered leading to significant differences in the structures of the resulting vortices. As aspect ratio and vorticity are strongly related, the baroclinic amplification reduces the aspect ratio and, so, make the vortex sensitive to the elliptical instability. However, such vortices are not destroyed and keep a vertical structure thanks to the baroclinic amplification.

Este trabalho lida com o escoamento incompressível ao redor de pares de cilindros rígidos e imóveis, posicionados de forma alinhada em relação a uma corrente uniforme. São dois os objetivos desta pesquisa: o primeiro é estabelecer relações de causalidade entre características físicas do escoamento e as mudanças observadas nas forças e no campo fluido com as variações do número de Reynolds (Re) e do espaçamento entre os corpos; e o segundo consiste em compreender a influência mútua entre tridimensionalidades e interferência no escoamento. Utilizou-se o método de elementos espectrais para realizar simulações bi e tridimensionais do escoamento. Os espaçamentos entre centros (lcc) analisados vão de 1,5 a 8 diâmetros, e eles são comparados com o caso de um cilindro isolado. A faixa de Re vai de 160 a 320, compreendendo a transição na esteira. O foco foi dado nas instabilidades de pequena escala (modos A e B). Dados referentes ao número de Strouhal, coeficiente de arrasto médio, flutuação do coeficiente de sustentação e correlação axial são apresentados. Com auxílio de visualizações do escoamento, são propostos mecanismos que explicam o fenômeno de interferência, refletido no comportamento das curvas. Os resultados mostraram que simulações bidimensionais são insuficientes para se prever a combinação (Re, lcc) de inversão do arrasto. Verificou-se também que, quando o espaçamento é menor do que o crítico, o processo de transição na esteira se dá de forma diferente do observado para um cilindro isolado. / This work deals with the incompressible flow around pairs of rigid and immovable circular cylinders in tandem arrangements. There are two goals in this research: the first one is to find causality relationships between physical characteristics of the flow and the changes that are observed in the forces and in the flow field with the variation of the Reynolds number (Re) and the distance between the bodies; and the second one is to comprehend the mutual influence between three-dimensional structures and interference. The spectral element method was employed to carry out two- and three-dimensional simulations of the flow. The centre-to-centre distance (lcc) of the investigated configurations varies between 1.5 and 8 diameters, and they are compared to the isolated cylinder case. The Re range goes from 160 to 320, covering the transition in the wake. We focused in the small scale instabilities (modes A and B). Data of Strouhal number, mean drag coefficient, RMS of the lift coefficient and axial correlation are presented. With aid of flow visualizations, we propose mechanisms to explain the interference phenomenon, which is reflected in the behaviour of the graphics. The results show that two-dimensional simulations are not sufficient to predict the (Re, lcc) pair correspondent to the drag inversion point. We also verified that, in the cases where lcc is lower than the critical spacing, the transition in the wake happens in a way different from the one observed in the flow around a single cylinder.

O objetivo desta pesquisa é o estudo do uso potencial de multi-winglets adaptativas para obter reduções no arrasto induzido através da variação do diedro das winglets. Os vórtices gerados nas pontas das asas são um produto inevitável da presença da sustentação, ou seja, é um custo a pagar pela força que mantém as aeronaves no ar. Diferentes estudos têm demonstrado que o escoamento presente nas pontas das asas pode ser redirecionado usando pequenas superfícies aerodinâmicas reduzindo assim o arrasto induzido. O modelo testado era constituído por uma asa retangular construída a partir de um perfil NACA 653 - 018 dotado de três winglets tipo \"tip-sails\", que são pequenas asas sem enflechamento ao 25% da corda. Os testes realizados foram para um regime de Número de Reynolds de 357.000. Os resultados foram analisados através da análise da sustentação, arrasto e mapeamento da esteira através da técnica de anemometria de fio quente. / The aim of this research is the study of the potential use of adaptive multi-winglets to reduce induced drag through variations of winglet cant angles. The vortices generated at the wing tips are an inevitable product of the presence of lift, that is, they represent the price paid for the presence for the force that keeps the aircraft in the air. Different studies have shown that the flow over the wing-tip can be redirected using small aerodynamics surfaces, thereby reducing the induced drag. The model tested is composed of a rectangular wing using a NACA 653 - 018 profile with three winglets called \"tip-sails\", which are small wings without sweep at 25% chord. The tests were made at a Reynolds number of 350,000. The results are analyzed in terms of lift and drag and mapping of the wake using hot wire anemometry techniques.

A maior parte do sinal propagante para oeste nos oceanos é explicada pelas ondas de Rossby baroclínicas. Porém, vórtices de mesoescala podem interferir na identificação dessas ondas. A maior adversidade em se distinguir o sinal dessas feições é que os vórtices parecem propagar-se com uma velocidade aproximada à velocidade de fase das ondas. Um dos objetivos do presente trabalho é caracterizar os sinais propagantes para oeste em termos da sua velocidade de fase no Atlântico. A análise se baseia em de dados das anomalias da altura da superfície do mar (ASM) e da temperatura de superfície do mar (TSM) derivados dos altímetros TOPEX/Poseidon e Jason-1 e radiômetro TRMM/TMI. As anomalias de ASM e TSM foram filtradas por um conjunto de filtros de resposta impulsiva finita (FIR) para eliminar o sinais sazonais, interanuais e sinais de alta frequência. A análise de correlação cruzada entre as matrizes zonais-temporais de ASM e TSM foi feita para limitar as conclusões aos sinais presentes simultaneamente em ambas as bases de dados. A velocidade de fase das ondas de Rossby foi estimada via transformada de Radon aplicada às matrizes de correlação cruzada. Um máximo local solitário sobre a origem foi observado nos diagramas de correlação e associado à presença de vórtices de mesoescala. Porém este máximo se alonga com a mesma inclinação correspondente à velocidade de fase das ondas de Rossby. Isto sugere que estes vórtices podem propagar-se sobrepostos às ondas. As velocidades de propagação dos vórtices são estimadas através do ajuste de uma função de decaimento exponencial no tempo e na distância zonal. Análises preliminares da transformada de Fourier mostraram que os sinais propagantes para oeste são predominantemente anuais, embora picos de frequências semi-anuais são observadas nestes espectros. Em decorrência da evidência de que os sinais propagantes para oeste são compostos simultaneamente por vórtices e ondas de Rossby, um filtro baseado na transformada de Radon e sua inversa foi desenvolvido para separar o sinal de feições com simetria circular. O filtro de vórtice circular foi aplicado em três áreas onde se localizam a Corrente das Agulhas, Confluências Brasil-Malvinas e Corrente do Golfo. Com base na análise visual pode-se afirmar que o filtro identificou com sucesso vórtices circulares nas três áreas estudadas, tanto nos dados de ASM como nos de TSM. As velocidades de fase das ondas de Rossby foram ligeiramente mais rápidas, em média, que as velocidades dos vórtices em todas as três áreas, cerca de 10% na ASM e 13% na TSM. As velocidades calculadas após a aplicação do filtro de vórtice circular apresentaram um viés positivo em relação as obtidas via correlações cruzadas. A maior diferença na estimativa das velocidades dos vórtices foi de 21% e nas ondas 25%, ambas na região da Corrente das Agulhas. Baseado nas evidências apresentadas é possível afirmar que vórtices podem se propagar com velocidades semelhantes à velocidade das ondas de Rossby do primeiro modo baroclínico. O lag das correlações cruzadas indicaram que o processo físico que relaciona a variabilidade nos dados de ASM à dos TSM é a advecção causada pela passagem de uma onda planet´aria do primeiro modo baroclínico. Esta advecção pode ser horizontal ou vertical dependendo do processo dominante que ocorre numa dada região. / In the oceans, most of the westward propagating signal is explained by baroclinic Rossby waves. However, mesoscale vortices can interfere in the identification of these waves. The main observational issue is to distinguish eddies from the wave-like propagating signals, since the former propagates with a speed that approximately matches the phase speed of baroclinic Rossby waves. The objective of the present study is to characterize the westward propagating signals in terms of their propagation speeds in the Atlantic. The analysis is based on satellite derived sea surface height (SSH) and sea surface temperature (SST) anomalies from the TOPEX/Poseidon and Jason-1 altimeters and TRMM/TMI radiometer records. The SSH and SST anomaly maps were filtered with a set of finite impulse response filters to eliminate the seasonal and interannual cycles and high frequency signals. The cross-correlation analysis between SSH and SST longitude-time matrices was performed to limit the conclusions to the features that appear simultaneously in both datasets. The of Rossby wave phase speed was estimated via Radon transform applied to the longitude-time cross-correlation matrices. A single local maximum was was observed at the origin of the cross-correlation diagrams and associated to mesoscale vortices. However, this maximum spreads along the same slope that characterizes the the westward Rossby wave phase speed. This suggests that vortices propagate superimposed to Rossby waves. The propagating speed of the vortices is estimated from the linear fit of an exponential decay function. A preliminary Fourier analysis show that the westward propagating signals are predominantly annual, yet peaks in the semiannual frequencies are observed. The evidence that the westward propagating signals are composed simultaneously of vortices and Rossby waves motivated the development of a filter based in the Radon transform and its inverse, to isolate the signal associated to circularly symmetric features. This circular vortex filter was applied in three areas that portray the Agulhas Current, the Brazil-Malvinas Confluence, and the Gulf Stream. Based on visual analysis one can affirm that the circular filter sucessfully identified vortices in three areas, both in the SSH and in the SST data. The phase speeds of Rossby waves were, on average, slightly faster than vortices speeds in the three areas, approximately 10% in SSH and 13% in SST. The speeds calculated after the circular vortex filter was applied presented a positive bias in relation to those obtain from cross correlations. The largest difference in the vortices speeds was 21% and in the wave speeds 25%, both in Agulhas Current region. Based on the present evidences it is possible to state that vortices can propagate with speeds similar to those of first-mode baroclinic Rossby waves. The cross-correlation lag suggests that the physical process that links the variability of the SSH to that of the SST is the advection generated by the passage of a first-mode baroclinic planetary wave. This advection can be horizontal or vertical depending of the dominant process in a given region.