Dans ce travail de thèse, une méthode de pénalisation en volume pour la simulation d'écoulements magnétohydrodynamiques (MHD) en domaines confinés est présentée. Les équations incompressibles de la MHD résistives sont résolues par le truchement d'un solveur pseudo-spectral parallèlisé. La pénalisation en volume est une méthode de frontières immergées, caractérisée par une grande flexibilité dans le choix de la géométrie de l'écoulement. Dans le cas présent, elle permet d'utiliser des conditions aux limites non-périodiques dans un schéma pseudo-spectral Fourier. La méthode numérique est validée et sa convergence est quantifiée pour des écoulements hydrodynamiques et MHD, en deux et trois dimensions, en comparant les résultats numériques à ceux de la littérature et à des solutions analytiques. Dans un second temps, la génération spontanée de moment cinétique et magnétique est étudiée pour des écoulements MHD confinés 2D et 3D. L'influence du nombre de Reynolds et du rapport des énergies cinétique/magnétique est explorée, ainsi que les différences induites par les conditions aux limites. Le fait que l'axisymétrie des frontières résulte en un terme de pression non-nul est primordial pour engendrer de grandes valeurs du moment cinétique. L'exclusivité de cette auto-organisation aux écoulements 2D est étudiée en considérant la MHD 3D en présence d'un fort champ magnétique axial. La suite est consacrée à la simulation d'un fluide conducteur dans un cylindre avec un forçage magnétique axial et poloidal. En faisant varier l'amplitude du forçage poloidal, différents états dynamiques sont atteints. Enfin, l'effet du nombre de Prandtl sur le seuil des instabilitées est étudié. / A volume penalization method for the simulation of magnetohydrodynamic (MHD) flows in confined domains is presented. Incompressible resistive MHD equations are solved in 3D by means of a parallelized pseudo-spectral solver. The volume penalization technique is an immersed boundary method, characterized by a high flexibility in the choice of the geometry of the considered flow. In the present case, it allows the use of conditions different from periodic boundaries in a Fourier pseudo-spectral scheme. The numerical method is validated and its convergence is assessed for two- and three-dimensional hydrodynamical and MHD flows by comparing the numerical results with those of the literature or analytical solutions. Then, the spontaneous generation of kinetic and magnetic angular momentum is studied for confined 2D and 3D MHD flows. The influence of the Reynolds number and of the ratio of kinetic/magnetic energies is explored, as well as the differences induced by the boundary conditions. The fact that axisymmetric borders introduce a non-zero pressure term in the evolution equation of the angular momentum is essential to generate large values of the angular momentum. It is investigated whether this self-organization is exclusively observed in 2D flows by considering 3D MHD in the presence of a strong axial magnetic field. The last part is devoted to the simulation of a conducting fluid in a periodic cylinder with imposed axial and poloidal magnetic forcing, implying a resulting magnetic field. By varying the amplitude of the poloidal forcing, different dynamical states can be achieved. The effect of the Prandtl number on the threshold of the instabilities is then studied.

Pseudo-spectral optimal solvers are used to optimize numerically a performance index of a dynamical system with differential constraints. Although these solvers are commonly used for space vehicles and space launchers for trajectory optimization, few experimental papers exist on optimal control of small airships. The objective of this thesis is to evaluate the use of a pseudo-spectral optimal control solver for generating dynamically constrained, minimal time trajectories. A dynamical model of a small airship is presented, with its experimental virtual mass, drag and motor experimentally modeled. The problems are solved in PSOPT, a pseudo-spectral optimal control code. Experimental tests with a small scale model are performed to evaluate the generated paths. Although drift occurs, as a consequence of an open loop control, the vehicle is capable of following the path. This results of this thesis may find uses in verifying how close to optimal discreet path planners are, to plan complex trajectories on short distances, or to generate dynamic maneuverer such as take-off or landing. Ultimately, improving path planning of small airships will improve their safety, maneuverability and flight-time, which makes them fit for scientific monitoring, for search and rescue, or as mobile telecommunications platforms.

This document presents the results of the master thesis conducted from April 2019 to October 2019 under the direction of CNES engineer Eric Bourgeois, as part of the KTH Master of Science in Aerospace Engineering curriculum. Within the framework of development studies for the Callisto demonstrator, this master thesis aims at studying and developing possible guidance strategies for the boosted landing. Two main approaches are described in this document : • Adaptive pseudo-spectral interpolation • Convex optimization The satisfying results yielded give strong arguments for choosing the latter as part of the Callisto GNC systems and describe possible implementation strategies as well as complementary analyses that could be conducted. / Denna rapport presenterar resultaten av ett examensarbete som genomfördes från april till oktober 2019 under ledning av CNES-ingenjören Eric Bourgeois, som en del av en masterexamen i flyg- och rymdteknik från KTH, Kungliga tekniska högskolan. Inom ramen för utvecklingsstudier för bärraketen Callisto syftar detta arbete att studera och utveckla möjliga reglerstrategier för Callistos landing som kontrolleras med raketer. Två huvudsakliga metoder beskrivs: • Adaptiv pseudospektral interpolering • Konvex optimering Resultaten ger starka argument för att välja den senare av dessa två metoder för Callistos reglersystem och beskriver möjliga implementeringsstrategier samt vilka kompletterande analyser som bör genomföras

Turbulence in the ionosphere is important to understand because it can negatively affect communication signals. This work examines different scenarios in the ionosphere in which turbulence may develop. The two main causes of turbulence considered in this work are the gradient drift instability (GDI) and the Kelvin-Helmholtz instability (KHI). The likelihood of the development of the GDI during the August 17, 2017 total solar eclipse is studied numerically. This analysis uses the ``Sami3 is Also a Model of the Ionosphere" (SAMI3) model to study the effect of the eclipse on the plasma density. The calculated GDI growth rates are small compared to how quickly the eclipse moves over the Earth. Therefore, the GDI is not expected to occur during the solar eclipse. A novel 2D electrostatic pseudo-spectral fluid model is developed to study the growth of these two instabilities and the problem of ionospheric turbulence in general. To focus on the ionospheric turbulence, a set of perturbed governing equations are derived. The model accurately captures the GDI growth rate in different limits; it is also benchmarked to the evolution of instability development in different collisional regimes of a plasma cloud. The newly developed model is used to study if the GDI is the cause of density irregularities observed in subauroral polarization streams (SAPS). Data from Global Positioning System (GPS) scintillations and the Super Dual Auroral Radar Network (SuperDARN) are used to examine the latitudinal density and velocity profiles of SAPS. It is found that the GDI is stabilized by velocity shear and therefore will only generate density irregularities in regions of low velocity shear. Furthermore, the density irregularities cannot extend through regions of large velocity shear. In certain cases, the turbulence cascade power laws match observation and theory. The transition between the KHI and the GDI is studied by understanding the effect of collisions. In low collisionality regimes, the KHI is the dominant instability. In high collisionality regimes, the GDI is the dominant instability. Using nominal ionospheric parameters, a prediction is provided that suggests that there exists an altitude in the upper textit{F} region ionosphere above which the turbulence is dominated by the KHI. / Doctor of Philosophy / In the modern day, all wireless communication signals use electromagnetic waves that propagate through the atmosphere. In the upper atmosphere, there exists a region called the ionosphere, which consists of plasma (a mixture of ions, electrons, and neutral particles). Because ions and electrons are charged particles, they interact with the electromagnetic communication signals. A better understanding of ionospheric turbulence will allow for aid in forecasting space weather as well as improve future communication equipment. Communication signals become distorted as they pass through turbulent regions of the ionosphere, which negatively affects the signal quality at the receiving end. For a tangible example, when Global Positioning System (GPS) signals pass through turbulent regions of the ionosphere, the resulting position estimate becomes worse. This work looks at two specific causes of ionospheric turbulence: the gradient drift instability (GDI) and the Kelvin-Helmholtz instability (KHI). Under the correct background conditions, these instabilities have the ability to generate ionospheric turbulence. To learn more about the GDI and the KHI, a novel simulation model is developed. The model uses a method of splitting the equations such that the focus is on just the development of the turbulence while considering spatially constant realistic background conditions. The model is shown to accurately represent results from previously studied problems in the ionosphere. This model is applied to an ionospheric phenomenon known as subauroral polarization streams (SAPS) to study the development of the GDI and the KHI. SAPS are regions of the ionosphere with large westward velocity that changes with latitude. The shape of the latitudinal velocity profile depends on many other factors in the ionosphere such as the geomagnetic conditions. It is found that for certain profiles, the GDI will form in SAPS with some of these examples matching observational data. At higher altitudes, the model predicts that the KHI will form instead. While the model is applied to just the development of the GDI and the KHI in this work, it is written in a general manner such that other causes of ionospheric turbulence can be easily studied in the future.

L'étude théorique de la transition vers la turbulence d'écoulements en tuyau de fluides non newtoniens rhéofluidifiants (fluides de Carreau) est menée, avec l'approche consistant à calculer des «~structures très cohérentes~» sous la forme d'«~ondes non linéaires~». Pour cela un code pseudo-spectral de type Petrov-Galerkin, permettant de suivre des solutions ondes non linéaires tridimensionnelles dans l'espace des paramètres par continuation, est développé. Ce code est validé par comparaison à des résultats existants en fluide newtonien, et grâce à un test de consistance en fluide non newtonien. Une convergence spectrale exponentielle est obtenue dans tous les cas. Ce code est utilisé pour chercher (guidé par des résultats expérimentaux récents) de nouvelles solutions de nombre d'onde azimutal fondamental égal à 1, sans succès pour l'instant. Par contre des solutions de nombre d'onde azimutal fondamental égal à 2 ou 3 sont obtenues par continuation à partir du cas newtonien. La rhéofluidification induit, en termes de nombres de Reynolds critiques, un retard à l'apparition de ces ondes par rapport au cas newtonien. Ce retard est caractérisé, et le parallèle est fait avec divers résultats expérimentaux qui montrent un retard à l'apparition de bouffées turbulentes en fluides non newtoniens / The transition to turbulence in pipe flows of shear-thinning fluids is studied theoretically. The method used is the computation of `exact coherent structures' that are tridimensional nonlinear waves. For this purpose a pseudo-spectral Petrov-Galerkin code is developped, which also allows to follow solution branches in the parameter space with continuation methods. This code is validated by recovering already published results in the Newtonian case, and by a consistency test in the non-Newtonian case. A spectral exponential convergence is obtained in all cases. This code is used to seek (guided by recent experimental results) new solutions of fundamental azimuthal wavenumber equal to 1,without success at the time being. On the contrary solutions with a fundamental azimuthal wavenumber equal to 2 and 3 are obtained by continuation from the Newtonian case. The shear-thinning effects induce, in terms of critical Reynolds numbers, a delay for the onset of these waves, as compared with the Newtonian case. This delay is characterized. An analogy is made with various experimental results that show a delay in the transition to turbulence, more precisely, in the onset of `puffs', in non-Newtonian fluids

The complexity of numerical ocean circulation models requires careful checking with a variety of test problems. The purpose of this paper is to develop a test problem involving Rossby and gravity waves in a two-layer fluid in a channel. The goal is to compute very accurate solutions to this test problem. These solutions can then be used as a part of the checking process for numerical ocean circulation models. Here, Chebychev pseudo-spectral methods are used to solve the governing equations with a high degree of accuracy. Chebychev pseudo-spectral methods can be described in the following way: For a given function, find the polynomial interpolant at a particular non-uniform grid. The derivative of this polynomial serves as an approximation to the derivative of the original function. This approximation can then be inserted to differential equations to solve for approximate solutions. Here, the governing equations reduce to an eigenvalue problem with eigenvectors and eigenvalues corresponding to the spatial dependences of modal solutions and the frequencies of those solutions, respectively. The results of this method are checked in two ways. First, the solutions using the Chebychev pseudo-spectral methods are analyzed and are found to exhibit the properties known to belong to physical Rossby and gravity waves. Second, in the special case where the two-layer model degenerates to a one-layer system, some analytic solutions are known. When the numerical solutions are compared to the analytic solutions, they show an exponential rate of convergence. The conclusion is that the solutions computed using the Chebychev pseudo-spectral methods are highly accurate and could be used as a test problem to partially check numerical ocean circulation models. / Graduation date: 2012

Dans le cadre du projet européen AERONEWS pour le développement de méthodes de Contrôle Non Destructif, des structures aéronautiques complexes fissurées sont analysées par spectroscopie ultrasonore d'ondes élastiques des propriétés non linéaires classique et non classique. Le coefficient non linéaire est mesuré dans des échantillons homogènes bi-couches par modulation de phase calibrée en contact. Une approche phénoménologique du comportement hystérétique d'un matériau fissuré est réalisée avec l'espace de Preizach-Mayergoysz. Un algorithme numérique pseudo-spectral 3D, utilisant les notations de Kelvin, valide les méthodes de localisation de défaut, NEWS-TR et TR-NEWS, combinant le retournement temporel acoustique avec le traitement non linéaire des signaux. La signature non linéaire est extraite avec l'inversion d'impulsion ultrasonore. Une instrumentation est créée pour adapter expérimentalement ces méthodes de détection de défaut.

The Poiseuille Flow is the flow of a viscous incompressible fluid in a channel between two infinite parallel plates. The behaviour of flow is properly described by the well-known Navier-Stokes Equations. The fact that Navier-Stokes equations are partial differential equations makes their solution difficult. They can rarely be solved in closed form. On the other hand, numerical techniques can be applied successfully to the well-posed partial differential equations. In the present study pseudo-spectral method is implemented to analyze the Poiseuille Flow. The pseudo-spectral method is a high-accuracy numerical modelling technique. It is an optimum choice for the Poiseuille flow analysis due to the flows simple geometry. The method makes use of Fourier Transform and by handling operations in the Fourier space reduces the difficulty in the solution. Fewer terms are required in a pseudo-spectral orthogonal expansion to achieve the same accuracy as a lower order method. Karhunen-Lo&egrave / ve (KL) decomposition is widely used in computational fluid dynamics to achieve reduced storage requirements or construction of relatively low-dimensional models. In this study the KL basis is extracted from the flow field obtained from the direct numerical simulation of the Poiseuille flow.

A prestudy indicated superior accuracy and convergence properties of apseudo-spectral method compared to a spline-based method implemented byCòrdoba et al. in 2005 when solving the α-patches problem. In this thesis wefurther investigate the numerical properties of the pseudo-spectral method and makeit more robust by implementing the Nonequispaced Fast Fourier Transform. Wepresent a more detailed overview and analysis of the pseudo-spectral method and theα-patches problem in general and conclude that the pseudo-spectral method issuperior in regards to accuracy in periodic settings.

Large amplitude internal solitary waves in the coastal ocean are commonly modelled with the Korteweg-de Vries (KdV) equation or a closely related evolution equation. The characteristic feature of these models is the solitary wave solution, and it is well documented that these provide the basic paradigm for the interpretation of oceanic observations. However, often internal waves in the ocean survive for several inertial periods, and in that case, the KdV equation is supplemented with a linear non-local term representing the effects of background rotation, commonly called the Ostrovsky equation. This equation does not support solitary wave solutions, and instead a solitary-like initial condition collapses due to radiation of inertia-gravity waves, with instead the long-time outcome typically being an unsteady nonlinear wave packet. The KdV equation and the Ostrovsky equation are formulated on the assumption that only a single vertical mode is used. In this thesis we consider the situation when two vertical modes are used, due to a near-resonance between their respective linear long wave phase speeds. This phenomenon can be described by a pair of coupled Ostrovsky equations, which is derived asymptotically from the full set of Euler equations and solved numerically using a pseudo-spectral method. The derivation of a system of coupled Ostrovsky equations is an important extension of coupled KdV equations on the one hand, and a single Ostrovsky equation on the other hand. The analytic structure and dynamical behaviour of the system have been elucidated in two main cases. The first case is when there is no background shear flow, while the second case is when the background state contains current shear, and both cases lead to new solution types with rich dynamical behaviour. We demonstrate that solitary-like initial conditions typically collapse into two unsteady nonlinear wave packets, propagating with distinct speeds corresponding to the extremum value in the group velocities. However, a background shear flow allows for several types of dynamical behaviour, supporting both unsteady and steady nonlinear wave packets, propagating with the speeds which can be predicted from the linear dispersion relation. In addition, in some cases secondary wave packets are formed associated with certain resonances which also can be identified from the linear dispersion relation. Finally, as a by-product of this study it was shown that a background shear flow can lead to the anomalous version of the single Ostrovsky equation, which supports a steady wave packet.