Thesis (Ph.D. (Applied Mathematics)) -- University of Limpopo, 2020 / The flow of electrically conducting fluids in the presence of a magnetic field has wide applications in science, engineering and technology. Examples of the applications include industrial processes such as the cooling of reactors, extrusion of plastics, purification of crude oil, medical applications, aerodynamics and many more. The induced magnetic field usually act as a flow control mechanism, especially under intense heat. In this study a couple stress fluid in a channel will be used as the working fluid. Channel flow and heat transfer characteristics of couple stress fluids find applications in processes such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plates in a bath, tribology of thrust bearings and lubrication of engine rod bearings. One major characteristic that distinguishes the couple stress fluid from other non-Newtonian fluids is the inclusion of size-dependent microstructure that is of mechanical significance. As such, the couple stress constitutive model is capable of describing the couple stresses, the effect of body couples and the nonsymmetric tensors manifested in several real fluids of technological importance. A fully developed laminar magnetohydrodynamic (MHD) flow of an incompressible couple stress fluid through a vertical channel due to a steady-periodic temperature on the channel plates is investigated. Specifically, the effects of couple stresses and internal heat generation on MHD natural convection flow with steady-periodic heat input, the impact of magnetic field induction on the buoyancy-induced oscillatory flow of couple stress fluid with varying heating and a mixed convective two dimensional flow of unsteady MHD couple stress fluid through a channel field with porous medium are studied. Analytical methods and the semi-analytic Adomian decomposition method will be used to solve the resulting non-linear differential equations governing the flow systems. Useful results for velocity, temperature, skin friction and Nusselt number are obtained and discussed quantitatively. The effects of the various flow governing parameters on the flow field are investigated.

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In this Thesis the theoretical understanding of oscillations in coronal structures is developed. In particular, coronal loops are modelled as magnetic slabs of plasma. The effect of introducing inhomogeneities on the frequency of oscillation is studied. Current observations indicate the existence of magnetohydrodynamic (MHD) modes in the corona, so there is room for improved modelling of these modes to understand the physical processes more completely. One application of the oscillations, on which this Thesis concentrates, is coronal seismology. Here, the improved theoretical models are applied to observed instances of coronal MHD waves with the aim of determining information regarding the medium in which these waves propagate. In Chapter two, the effect of gravity on the frequency of the longitudinal slow MHD mode is considered. A thin, vertical coronal slab of magnetised plasma, with gravity acting along the longitudinal axis of the slab is studied, and the effect on the frequency of oscillation for the uniform, stratified and structured cases is addressed. In particular, an isothermal plasma, a two-layer plasma and a plasma with a linear temperature profile are studied. Here, a thin coronal loop, with its footpoints embedded in the chromosphere-photosphere is modelled, and the effects introduced by both gravity and the structuring of density at the footpoint layers are studied. In this case, gravity increases the frequency of oscillation and causes amplification of the eigenfunctions by stratification. Furthermore, density enhancements at the footpoints cause a decrease in the oscillating frequency, and can inhibit wave propagation, depending on the parameter regime. In Chapter three, the effects introduced to the transverse fast MHD mode when gravity acts across a thin coronal slab of magnetised plasma are considered. This study concentrates on the modification of the frequency due to the dynamical effect of gravity in the equation of motion, neglecting the effect of stratification. Here, gravity causes a reduction of the oscillating frequency of the fundamental fast mode, and increases the lower cutoff frequency. In effect, for this configuration, gravity allows the transition between body and surface modes, in a slab geometry. It is found, in these two studies, that each harmonic is affected in a unique manner due to structuring or stratification of density. With this knowledge, in Chapter four, a new parameter is derived; P1/2P2, the ratio of the period of the fundamental harmonic of oscillation to twice the period of its first harmonic. This parameter is shown to be a measure of the longitudinal structuring of density along a coronal loop, and the departure of this ratio from unity can yield information regarding the lengthscales of the structure. This process is highlighted using the known observations, indicating that P1/2P2 may prove to be a useful diagnostic tool for coronal seismology. Finally, in Chapter five, outwardly propagating coronal slow MHD modes are observed and are used to infer coronal parameters. The possibility of using these oscillations to infer near-resolution lengthscales in coronal loops -- fine-scale strands -- is also discussed. TRACE observations are used to determine the average period, phase speed, detection length, amplitude and energy flux for the propagating slow MHD mode. The indication is that the source of these oscillations appears very localised in space, and the driver only acts for a few periods, suggesting the perturbations are driven by leaky p-modes (solar surface modes).

While magnetohydrodynamic (MHD) instabilities are considered one of the intriguing topics in tokamak physics, a feasibility study was conducted in the Saskatchewan Torus-Modified (STOR-M) tokamak to investigate the global MHD activities during the normal (L-mode) and improved (H-mode) confinement regimes. The experimental setup consists of 32 discrete Mirnov coils arranged into four poloidal arrays and mounted on STOR-M at even toroidal distances. The perturbed magnetic field fluctuations during STOR-M discharges were acquired and processed by the Fourier transform (FT), the wavelet analysis and the singular value decomposition (SVD) techniques. In L-mode discharges, the poloidal MHD mode numbers varied from 2 to 4 with peak frequencies in the range 20-40 kHz. The dominant toroidal modes were reported between 1 and 2 oscillating at frequencies 15-35 kHz. In another experiment, a noticeable MHD suppression was observed during the H-mode-like phase induced by the compact torus (CT) injection into STOR-M. However, a burst-like mode called the gong mode was triggered prior to the H-L transition, followed by coherent Mirnov oscillations. Mirnov oscillations with strong amplitude modulations were observed in the STOR-M tokamak. Correlations between Mirnov signals and soft x-ray (SXR) signals were found.

While magnetohydrodynamic (MHD) instabilities are considered one of the intriguing topics in tokamak physics, a feasibility study was conducted in the Saskatchewan Torus-Modified (STOR-M) tokamak to investigate the global MHD activities during the normal (L-mode) and improved (H-mode) confinement regimes. The experimental setup consists of 32 discrete Mirnov coils arranged into four poloidal arrays and mounted on STOR-M at even toroidal distances. The perturbed magnetic field fluctuations during STOR-M discharges were acquired and processed by the Fourier transform (FT), the wavelet analysis and the singular value decomposition (SVD) techniques. In L-mode discharges, the poloidal MHD mode numbers varied from 2 to 4 with peak frequencies in the range 20-40 kHz. The dominant toroidal modes were reported between 1 and 2 oscillating at frequencies 15-35 kHz. In another experiment, a noticeable MHD suppression was observed during the H-mode-like phase induced by the compact torus (CT) injection into STOR-M. However, a burst-like mode called the gong mode was triggered prior to the H-L transition, followed by coherent Mirnov oscillations. Mirnov oscillations with strong amplitude modulations were observed in the STOR-M tokamak. Correlations between Mirnov signals and soft x-ray (SXR) signals were found.

We consider the stabilized finite element method for solving the incompressible Navier-Stokes equations and the magnetohydrodynamic (MHD) equations in two dimensions. The well-known instabilities arising from the application of standard Galerkin finite element method are eliminated by using the stabilizing subgrid method (SSM), the streamline upwind Petrov-Galerkin (SUPG) method, and the two-level finite element method (TLFEM). The domain is discretized into a set of regular triangular elements. In SSM, the finite-dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the solution, a two-level finite element method with a stabilizing subgrid of a single node is described and its applications to the Navier-Stokes equations and MHD equations are displayed. This constitutes the main original contribution of this thesis. Numerical approximations employing the proposed algorithms are presented for some benchmark problems. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. The stabilized finite element method of SUPG type is applied to the unsteady Navier-Stokes equations together with a finite element discretization in the time domain. Thus, oscillations in the solution and the need of very small time increment are avoided in obtaining stable solutions.

La turbulence homogène et isotrope fut formalisée par Kolmogorov (1941), à l'aide d'une analyse dimensionnelle. Il parvint à démontrer que la densité spectrale de l'énergie cinétique, $E(k)$ suivait une loi en $k^{-5/3}$. Ce comportement est connu sous le nom de cascade de Kolmogorov. Dans de nombreux contexte géophysique ou astrophysiques, l'hélicité cinétique joue un rôle important. Parker (1955) a notamment démontré que l'hélicité cinétique pouvait contribuer à l'amplification d'un champ magnétique pour des écoulements conducteurs. Brissaud {it et al} (1973) ont alors tenté de déterminer l'influence que l'hélicité cinétique pouvait avoir sur les spectres d'énergie cinétique. Brissaud {it et al} (1973) suggèrent l'existence d'une cascade pour laquelle les spectres d'énergie cinétique suivent une loi en $k^{-7/3}$. Dans la première partie de ce manuscrit nous allons confirmer à l'aide de simulations numériques directes (DNS) l'existence d'une loi asymptotique en $k^{-7/3}$. Nous aurons également recourt à la décomposition en modes hélicitaires afin d'analyser de manière approfondie la physique qui régit ces écoulements. Dans de nombreux écou-le-ments géophysique ou astrophysiques, la turbulence est très forte, et une très large gamme d'échelles est impliquée. Bien que la puissance des calculateurs ait considérablement augmentée ces dernières années, il n'est toujours pas possible de simuler l'ensemble de cette gamme d'échelles pour des configurations réalistes. Une solution connue sous le nom de Large Eddy Simulations (LES) permet de réaliser des simulations de ce type d'écoulement. Concrètement, lors de la réalisation d'une LES, les grandes échelles de l'écoulement sont résolues, et les interactions entre les grandes et les petites échelles de l'écoulement sont modélisées. Divers modèles de turbulence existent déjà pour la réalisation de LES en turbulence. Néanmoins leurs limites ne sont pas toujours bien connues dans le cadre de la turbulence magnétohydrodynamique (MHD), c'est-à-dire pour les fluides conducteurs de l'électricité que l'on rencontre en géophysique ou astrophysique. Dans la seconde partie de ce manuscrit nous allons donc évaluer les performances fonctionnelles (voir Sagaut (2002)) de ces différents modèles dans des configurations correspondant à des dynamos turbulentes, c'est-à-dire à des régimes où un champ magnétique est généré par un fluide conducteur animé d'un mouvement turbulent. Nous étudierons notamment la capacité des modèles LES à reproduire les échanges énergétiques entre grandes et petites échelles. Pour ce faire, nous réaliserons plusieurs DNS, pour des écoulements non-hélicitaires (menant à des dynamos de petites échelles) et des écoulements hélicitaires (menant à des dynamos de grandes échelles). `A l'aide d'une opération de filtrage, nous calculerons les transferts sous-mailles exacts, puis les comparerons aux prédictions fournies par les modèles. Finalement nous réaliserons des LES à l'aide des différents modèles et nous les comparerons aux DNS filtrées. / Homogeneous and isotropic turbulence was first formalized by Kolmogorov (1941), through dimensional analysis. He managed to show that the spectral density of kinetic energy, $E(k)$, was following a $k^{-5/3}$ law. This behaviour is known as Kolmogorov's cascade. For many geophysical and astrophysical flow, kinetic helicity plays an important role. For instance, Parker (1955) showed that for conductive fluids such as Sun, kinetic helicity could contribute to amplify the magnetic field. Brissaud {it et al} (1973) tried to show that kinetic helicity could have an influence on the spectral density of kinetic energy. Through dimensional analysis they suggested the existence of a cascade for which the kinetic energy spectra would follow a $k^{-7/3}$ law. In the first part of this thesis we will confirm thanks to Direct Numerical Simulations (DNS) the existence of such an asymptotic limit in $k^{-7/3}$. We will also use helical decomposition to perform a deep analysis of the physics encountered within such flows. In several geophysical and astrophysical fluids, turbulence is very strong, and involves a large range of scales. Despite the strong development of computational resources the last few decades, it remains impossible to simulate this range of scales for realistic configurations. One solution is known as Large Eddy Simulations (LES). While a LES is performed, only the large scales of the flow are resolved, and the interactions between large and small scales are modeled. Several turbulence models have been developed for LES of turbulence. Nevertheless, the limitations of these models are not always well known for magnetohydrodynamic (MHD) turbulence, i.e for conductive fluids that can be encoutered in geophysics and astrophysics. In the second part of this thesis we will evaluate the functional performances (see Sagaut (2002)) of these models for several flow configurations involving turbulent dynamo action, i.e when a magnetic field is amplified though the action of a turbulent conductive fluid. In particular we will study the capabilities of LES models to reproduce energy exchanges between large and small scales. In order to do so, we will perform several DNS, for both non-helical flows (i.e leading to small scale dynamo) and helical flows (i.e leading to large scale dynamo). Thanks to a filtering operation we will compute the exact subgrid-scale transfers and compare them to the predictions given by several models. Finally we will achieve LES using subgrid-scale models and we will compare them to filtered DNS.

Cette thèse s'inscrit dans le cadre d'un projet R&T CNES. Elle concerne l'étude d'un actionneur appelé roue d'inertie, qui fait partie intégrante de l'ensemble SCAO (Système de Contrôle d'Attitude et d'Orbite). Les nouvelles roues proposées, dites Magnétohydrodynamique (MHD) à Conduction, présentent un volant d'inertie fluidique sous forme d'un canal torique, dans lequel un métal liquide conducteur à fort potentiel inertiel est mis en mouvement sous l'effet d'un champ électromagnétique. Contrairement aux roues actuelles, elles n'ont pas de roulements ni d'arbre mécanique ce qui permet un gain en espace, un éloignement idéal de la masse inertielle, et une durée de vie théoriquement illimitée. Aussi, de par la viscosité naturelle du fluide, elles ne présentent pas de non-linéarité autour de la vitesse nulle ce qui évite une perte de précision sur le contrôle du couple de réaction, et donc du pointage du satellite. Le travail réalisé pendant la thèse porte sur l'appréhension des phénomènes MHD consistant en un couplage entre les lois de la Mécanique des Fluides et celles de l’Electromagnétisme, au travers de la loi d’Ohm généralisée. A partir d'hypothèses axisymétriques, et dans le cadre des milieux incompressibles et d’un écoulement laminaire, un modèle générale 3D a pu être établie. Puis une formulation 1D cylindrique a permis une résolution analytique, et une autre en 2D axisymétrique, par résolution numérique en différences fines, a permis l'amélioration de la précision des résultats. Ce modèle a permis de comprendre que deux approches étaient possibles pour la conception et plus particulièrement la commande de l'actionneur. Cette résolution, faisant l'objet du développement d'un code numérique, a d'abord porté sur les équations en régime permanent, puis en temporel, afin de caractériser l'actionneur du point de vue de ses deux modes de fonctionnement. La réalisation d’un prototype a permis de quantifier la validité de la modélisation d’un point de vue dynamique. / This thesis is part of a CNES R&T project. It's related to the study of an actuator called flywheel, which is part of AOCS (Attitude an Orbit Control System). The proposed new wheels, said Magnetohydrodynamic (MHD), are constituted by a fluid flywheel in the form of a ring channel, in which a conductive and high inertial potential liquid metal is driven through an electromagnetic field (Lorentz's force). Unlike current wheels, among others types DC brushless motor, MHD wheels, whose rotor is the fluid, have neither bearings nor mechanical shaft. This allows space saving, an ideal distance of the inertial mass, and a theoretically unlimited lifespan. Moreover, thanks to the natural viscosity of the fluid, they do not present a non-linearity around the zero speed which avoids a loss of precision in the reaction torque's control, and therefore the satellite pointing. The work for the thesis focuses on the apprehension of MHD phenomena. Indeed, MHD is a coupling between fluid mechanics' laws (Navier-Stokes, etc.) and Maxwell's equations, through the Lorentz force. From a number of assumptions, and as part of incompressible environment, a genral 3D model has been established. Then a 1D cylindrical formulation allowed an analytical resolution and another 2D axisymmetric one, by finite differences resolution, helped to improve results. This model allow us to understand that both approaches were possible for the design and especially the actuator control. The resolution, which is subjected to the development of a numerical code, first focused on the equations in steady state, then in dynamic, to characterize the actuator in terms of its two operating modes. The realization of a prototype has quantified the validity of the model from a dynamic point of view.

In this thesis I study a variety of two-dimensional turbulent systems using a mixed analytical, phenomenological and numerical approach. The systems under consideration are governed by the two-dimensional Navier-Stokes (2DNS), surface quasigeostrophic (SQG), alpha-turbulence and magnetohydrodynamic (MHD) equations. The main analytical focus is on the number of degrees of freedom of a given system, defined as the least value $N$ such that all $n$-dimensional ($n$ ≥ $N$) volume elements along a given trajectory contract during the course of evolution. By equating $N$ with the number of active Fourier-space modes, that is the number of modes in the inertial range, and assuming power-law spectra in the inertial range, the scaling of $N$ with the Reynolds number $Re$ allows bounds to be put on the exponent of the spectrum. This allows the recovery of analytic results that have until now only been derived phenomenologically, such as the $k$[superscript(-5/3)] energy spectrum in the energy inertial range in SQG turbulence. Phenomenologically I study the modal interactions that control the transfer of various conserved quantities. Among other results I show that in MHD dynamo triads (those converting kinetic into magnetic energy) are associated with a direct magnetic energy flux while anti-dynamo triads (those converting magnetic into kinetic energy) are associated with an inverse magnetic energy flux. As both dynamo and anti-dynamo interacting triads are integral parts of the direct energy transfer, the anti-dynamo inverse flux partially neutralises the dynamo direct flux, arguably resulting in relatively weak direct energy transfer and giving rise to dynamo saturation. These theoretical results are backed up by high resolution numerical simulations, out of which have emerged some new results such as the suggestion that for alpha turbulence the generalised enstrophy spectra are not closely approximated by those that have been derived phenomenologically, and new theories may be needed in order to explain them.

Les instabilités fluides peuvent dégrader le confinement du plasma au sein des tokamaks. Étant données les échelles spatio-temporelles, on choisit les modèles fluides obtenus à partir de la dérivation des modèles cinétiques. On dérive plusieurs modèles hiérarchiques de la MagnétoHydroDynamique (MHD) et en particulier les modèles de la MHD réduite du Current Hole et de l'équilibre de Grad-Shafranov. Une des difficulté de l'ensemble de ces modèles est de respecter l'équation modélisant l'absence de monopôles magnétiques. Pour assurer cette condition en tout point du domaine, le champ magnétique est réécrit avec un potentiel vecteur. L'utilisation de potentiels fait apparaître des équations faisant intervenir des dérivées d'ordre supérieurs. La stratégie numérique développée est l'utilisation de la méthode des éléments finis avec des éléments C1 de Bell. Sur un maillage non structuré, ces éléments ont l'intérêt de présenter une base réduite définir exclusivement avec des variables aux noeuds du maillage. Les modèles de MHD réduite du Current Hole et de Grad-Shafranov ont été résolus avec ces éléments. La résolution du cas test de Grad-Shafranov avec les conditions de bords exactes a permis d'obtenir l'ordre optimale de 5. La résolution du système du Current Hole avec ces éléments, validée par l'obtention du paramètre η1/3, a permis l'observation de développement d'instabilités en dents de scies. / Fluid instabilities can degrade plasma confinement in tokamaks. Given the spatial and temporal scales, we choose the fluid models obtained from the derivation of kinetic models. We derived several hierarchical models of MagnetoHydroDynamic (MHD) and in particular models of reduced MHD like the Current Hole and the Grad-Shafranov equilibrium. One of the difficulty of all these models is to respect the absence of magnetic monopoles equation. To ensure this condition at any point, the magnetic field is rewritten with a vector potential. The use of vector portential implies that higher order derivatives appear in the equation. The numerical strategy is developed using the finite element method with C1 Bell's elements. On a unstructured mesh, these have the advantage to present a reduced basis with degrees of freedom defined exclusively on the nodes of the mesh. The reduced MHD models of the Current Hole and Grad-Shafranov have thus been resolved with these elements. The resolution of a Grad-Shafranov test case with exact boundary conditions yields the optimal order of 5. The resolution of the Current Hole system with thesse elements has been validated by obtaining physical parameter η1/3 and allowed the observation of the development of sawtooth instabilities.