The concept of Magneto-Hydrodynamic (MHD) propulsion can be used to implement a propeller-less propulsion system for marine vehicles. The basic principle behind MHD is to use the (Lorentz) force produced by the interaction of electric and magnetic fields to generate thrust on a conducting fluid in motion. Electrodes are lined up along the walls of the duct which act as the source of the electric field. Seawater acts as the conducting medium for the current when it passes through the duct. This medium is then subjected to a strong magnetic field within the duct, thereby producing an axial force, i.e., an axial thrust. Propulsion systems based on MHD require virtually no mechanical components, therefore a good application would be to design a propulsor which produces very little noise for small underwater vehicles. Results of a preliminary feasibility study on this application are presented in this thesis. An approximate, consistent MHD propulsion theoretical model to assess the performance of a MHD propulsor for small underwater vehicles is introduced and analyzed. The model is generalized from the hydrodynamic point of view to consider inlet and outlet diffusers. The general model is applied systematically varying the main design parameters with respect to a given autonomous underwater vehicle (AUV) size. The results show that larger magnetic fields, longer propulsor lengths and smaller inlet flow speeds are preferred to get the highest propulsion efficiency and thrust. To check the consistency of the theoretical model, experiments are conducted. The results of these experiments show an approximate relation between the theoretical equations and the actual phenomenon. / Master of Science / In recent years, there has been an increase in the usage of small autonomous (unmanned) underwater vehicles (AUV) for various purposes such as exploration, mining and military applications. Most of these AUVs use the conventional system of a motor and propeller to drive the vehicle. This thesis proposes a different method of propulsion, one without any mechanical moving parts such as a rotor or a motor, for certain applications of these AUVs. The proposed system uses the concept of Magneto-Hydrodynamics (MHD) to propel the vehicle using an interaction between the applied magnetic and electric fields inside the propulsion channel. These applied fields produce a force (Lorentz) on the fluid that is present in the channel, thereby creating thrust to propel the vehicle. In the present case, the fluid is the electrically conducting seawater. Since, propulsion systems based on MHD require no mechanical components, they produce very little noise and are ideal for applications that require stealth. A feasibility study on this application is introduced, analyzed and presented in this thesis. Parameters such as applied fields, propeller configurations, and propeller shape and size are varied with respect to a given AUV size, to understand how each variable effects the system. The results show that larger magnetic fields, longer propulsor lengths and smaller inlet flow speeds are preferred to get the highest propulsion efficiency and thrust. To check the consistency of the theoretical model, experiments are conducted. The results of these experiments show an approximate relation between the theoretical equations and the actual phenomenon.

The main idea behind the work presented in this thesis is to investigate if it is possible to find a mechanism that leads to surface magnetic field concentrations and could operate under solar conditions without postulating the presence of magnetic flux tubes rising from the bottom of the convection zone, a commonly used yet physically problematic approach. In this context we study the ‘negative effective magnetic pressure effect’: it was pointed out in earlier work (Kleeorin et al., 1989) that the presence of a weak magnetic field can lead to a reduction of the mean turbulent pressure on large length scales. This reduction is now indeed clearly observed in simulations. As magnetic fluctuations experience an unstable feedback through this effect, it leads, in a stratified medium, to the formation of magnetic structures, first observed numerically in the fifth paper of this thesis. While our setup is relatively simple, one wonders if this instability, as a mechanism able to concentrate magnetic fields in the near surface layers, may play a role in the formation of sunspots, starting from a weak dynamo-generated field throughout the convection zone rather than from strong flux tubes stored at the bottom. A generalization of the studied case is ongoing. / <p>At the time of the the doctoral defence the following paper was unpublished and had a status as follows: Paper nr 7: Submitted</p>

Magnetohydrodynamics (MHD) studies the dynamics of an electrically conducting fluid under the influence of a magnetic field. Many astrophysical phenomena are related to MHD, and computer simulations are used to model these dynamics. In this thesis, we conduct MHD simulations of non-radiative black hole accretion as well as fast magnetic reconnection. By performing large scale three dimensional parallel MHD simulations on supercomputers and using a deformed-mesh algorithm, we were able to conduct very high dynamical range simulations of black hole accretion of Sgr A* at the Galactic Center. We find a generic set of solutions, and make specific predictions for currently feasible observations of rotation measure (RM). The magnetized accretion flow is subsonic and lacks outward convection flux, making the accretion rate very small and having a density slope of around $-1$. There is no tendency for the flows to become rotationally supported, and the slow time variability of the RM is a key quantitative signature of this accretion flow. We also provide a constructive numerical example of fast magnetic reconnection in a three-dimensional periodic box. Reconnection is initiated by a strong, localized perturbation to the field lines and the solution is intrinsically three-dimensional. Approximately $30\%$ of the magnetic energy is released in an event which lasts about one Alfv\'en time, but only after a delay during which the field lines evolve into a critical configuration. In the co-moving frame of the reconnection regions, reconnection occurs through an X-like point, analogous to the Petschek reconnection. The dynamics appear to be driven by global flows rather than local processes. In addition to issues pertaining to physics, we present results on the acceleration of MHD simulations using heterogeneous computing systems \cite. We have implemented the MHD code on a variety of heterogeneous and multi-core architectures (multi-core x86, Cell, Nvidia and ATI GPU) using different languages (FORTRAN, C, Cell, CUDA and OpenCL). Initial performance results for these systems are presented, and we conclude that substantial gains in performance over traditional systems are possible. In particular, it is possible to extract a greater percentage of peak theoretical performance from some heterogeneous systems when compared to x86 architectures.

Magnetohydrodynamics (MHD) studies the dynamics of an electrically conducting fluid under the influence of a magnetic field. Many astrophysical phenomena are related to MHD, and computer simulations are used to model these dynamics. In this thesis, we conduct MHD simulations of non-radiative black hole accretion as well as fast magnetic reconnection. By performing large scale three dimensional parallel MHD simulations on supercomputers and using a deformed-mesh algorithm, we were able to conduct very high dynamical range simulations of black hole accretion of Sgr A* at the Galactic Center. We find a generic set of solutions, and make specific predictions for currently feasible observations of rotation measure (RM). The magnetized accretion flow is subsonic and lacks outward convection flux, making the accretion rate very small and having a density slope of around $-1$. There is no tendency for the flows to become rotationally supported, and the slow time variability of the RM is a key quantitative signature of this accretion flow. We also provide a constructive numerical example of fast magnetic reconnection in a three-dimensional periodic box. Reconnection is initiated by a strong, localized perturbation to the field lines and the solution is intrinsically three-dimensional. Approximately $30\%$ of the magnetic energy is released in an event which lasts about one Alfv\'en time, but only after a delay during which the field lines evolve into a critical configuration. In the co-moving frame of the reconnection regions, reconnection occurs through an X-like point, analogous to the Petschek reconnection. The dynamics appear to be driven by global flows rather than local processes. In addition to issues pertaining to physics, we present results on the acceleration of MHD simulations using heterogeneous computing systems \cite. We have implemented the MHD code on a variety of heterogeneous and multi-core architectures (multi-core x86, Cell, Nvidia and ATI GPU) using different languages (FORTRAN, C, Cell, CUDA and OpenCL). Initial performance results for these systems are presented, and we conclude that substantial gains in performance over traditional systems are possible. In particular, it is possible to extract a greater percentage of peak theoretical performance from some heterogeneous systems when compared to x86 architectures.

Sunspots and active regions are two of the many manifestations of the solar magnetic field. This field plays an important role in causing phenomena such as coronal mass ejections, flares, and coronal heating. Therefore, it is important to study the origin of sunspots and active regions and determine the underlying mechanism which creates them. It is believed that flux tubes rising from the bottom of the convection zone can create sunspots. However, there are still unanswered questions about this model. In particular, flux tubes are expected to expand as they rise, hence their strength weakens and some sort of reamplification mechanism must complement this model to match the observational properties of sunspots. To compensate for the absence of such an amplification mechanism, the field strength of the flux tubes, when at the bot- tom of the convection zone, must be far stronger than present dynamo models can explain. In the last few years, there has been significant progress toward a new model of magnetic field concentrations based on the negative effective mag- netic pressure instability (NEMPI) in a highly stratified turbulent plasma. NEMPI is a large-scale instability caused by a negative contribution to the total mean-field pressure due to the suppression of the total turbulent pressure by a large-scale magnetic field. In this thesis, I study for the first time NEMPI in the presence of a dynamo-generated magnetic field in both spherical and Carte- sian geometries. The results of mean-field simulations in spherical geometry show that NEMPI and the dynamo instability can act together at the same time such that we deal with a coupled system involving both NEMPI and dynamo effects simultaneously. I also consider a particular two-layer model which was previously found to lead to the formation of bipolar magnetic structures with super-equipartition strength in the presence of a dynamo-generated field. In this model, the turbulence is forced in the entire domain, but the forcing is made helical in the lower part of the domain, and non-helical in the upper part. The study of such a system in spherical geometry showed that, when the stratification is strong enough, intense bipolar regions form and, as time passes, they expand, merge and create giant structures. To understand the underlying mechanism of the formation of such intense, long-lived bipolar structures with a sharp boundary, we performed a systematic numerical study of this model in plane parallel geometry by varying the magnetic Reynolds number, the scale separation ratio, and Coriolis number. Finally, I investigate the formation of the current sheet between bipolar regions and reconnection of oppositely orientated magnetic field lines and demonstrate that for large Lundquist numbers, S, the reconnection rate is nearly independent of S – in agreement with recent studies in identical settings.

Liquid metals (LMs) are compelling candidates for use as plasma facing components (PFCs) in fusion devices to mitigate heat loading, limit damage due to erosion, and possibly breed tritium. When used as electrodes, such as in z-pinch devices, PFCs are subject to large current and magnetic flux densities resulting in large Lorentz forces. Furthermore, if the PFCs are LM, the forces excite wave behavior that has not previously been investigated. The work presented here first characterizes the response of LMs to current pulses which peak between 50 and 200 kA and generate magnetic pressures between 0.5 and 5 MPa. High-speed videography records the liquid metal free surface during and after the current pulse and captures a fast moving, annular jet of LM emerging from the main body. The vertical velocities of the jet range from 0.6 to 5.3 m/s which is consistent with hydrodynamic predictions. Ejection of small droplets is observed from the LM immediately after the current pulse, preceding the LM jet, with velocities ranging from −3.1 to 18.9 m/s in the vertical direction and −14.3 to 6.3 m/s in the radial. A statistical model is developed to predict the likelihood of certain LM PFC material contaminating a core plasma and the severity in such an event. Lastly, effectiveness of bulk wave movement mitigation is investigated with two solid barrier designs, a cylindrical and conical baffle. These designs were fabricated after an iterative design process with assistance from hydrodynamic simulations. A cylindrical baffle design is shown to be preferable for integration into future fusion devices for the reduced likelihood of interference with plasma column formation. / Doctor of Philosophy / Liquid metals are considered for use as a coating on the interior surfaces in nuclear fusion reactors because they can remove heat, reduce damage, and generate additional fuel for the reactor. There has been very little research on what happens to the liquid metal when large amounts of electric current pass through it, as would be necessary in some designs. The work presented here first shows the liquid responds to large amounts of electric current with a fast moving, ring-shaped jet that correlates to the specific amount of current used. A theoretical relationship is used to relate the jet to hydrodynamic scenarios with solid bodies entering liquids. Small droplets are also observed sprayed from the LM earlier in time and the likelihood and severity of liquid metal contaminating the fusion core is analyzed. Finally, solid barriers are used to slow down the jet and minimize the mass it contains. To reduce the likelihood that the jet interferes with the fusion core, certain characteristics of barriers are identified as being preferable for use in plasma devices.

The dissertation explores a framework for numerically simulating the deformation of the liquid metal wall's free surface in Z-pinch fusion devices. This research is conducted in the context of utilizing liquid metals as plasma-facing components in fusion reactors. In the Z-pinch fusion process, electric current travels through a plasma column and enters into a pool of liquid metal. The current flowing through the liquid metal generates Lorentz force, which deforms the free surface of the liquid metal. Modeling this phenomenon is essential as it offers insights into the feasibility of using liquid metal as an electrode wall in such fusion devices. The conventional magneto-hydrodynamic (MHD) formulation aims at modeling the situation where an external magnetic field is applied to flows involving electrically conducting liquids, with the initial magnetic field is known and then evolved over time through magnetic induction equation. However, in Z-pinch fusion devices, the electric current is directly injected into a conducting liquid. In these situations, an analytical expression for the magnetic field generated by the applied current is not readily available, necessitating numerical calculations. Moreover, the deformation of the liquid metal surface changes the geometry of the current path over time and the resulting magnetic field. By directly solving the Maxwell equations in combination with Navier-Stokes equations, it becomes possible to predict the magnetic field even when the fluid is in motion. In this dissertation, a numerical framework utilizing the Maxwell-Navier-Stokes system is explored to successfully capture the deformation of the liquid metal's free surface due to applied electric current. / Doctor of Philosophy / In this dissertation, a method is described that uses a computer to simulate how the initially stable, flat surface of liquid metal deforms when subjected to electrical currents in Z-pinch fusion devices, a specific type of nuclear fusion technology. Z-pinch fusion devices generate plasma, a hot fluid-like substance, through the nuclear fusion process, triggered and maintained by strong pulsated current. There's a growing interest in using liquid metal as the first layer of material to isolate the hot plasma from the rest of the nuclear fusion reactor body, rather than solid materials, due to its unique benefits. However, the Z-pinch fusion process, by introducing electric currents through the liquid metal layer, induces a Lorentz force that consequently deforms the surface of the liquid metal. Developing a tool to predict this deformation is vital as it aids in evaluating the potential of using liquid metal as a plasma-facing layer over solid materials in these fusion devices. The simulation tools presented in this dissertation are able to successfully captures the dynamics of how the liquid metal surface deforms under the impact of electrical currents.

The finite element method (FEM) is a widely-used numerical tool in the fields of structural dynamics, acoustics and electromagnetics. In this work, our goal is to develop robust FEM strategies for solving problems in the areas of acoustics, structures and electromagnetics, and then extend these strategies to solve multi-physics problems such as magnetohydrodynamics and structural acoustics. We now briefly describe the finite element strategies developed in each of the above domains. In the structural domain, we show that the trapezoidal rule, which is a special case of the Newmark family of algorithms, conserves linear and angular momenta and energy in the case of undamped linear elastodynamics problems, and an ‘energy-like measure’ in the case of undamped acoustic problems. These conservation properties, thus, provide a rational basis for using this algorithm. In linear elastodynamics variants of the trapezoidal rule that incorporate ‘high-frequency’ dissipation are often used, since the higher frequencies, which are not approximated properly by the standard displacement-based approach, often result in unphysical behavior. Instead of modifying the trapezoidal algorithm, we propose using a hybrid FEM framework for constructing the stiffness matrix. Hybrid finite elements, which are based on a two-field variational formulation involving displacement and stresses, are known to approximate the eigenvalues much more accurately than the standard displacement-based approach, thereby either bypassing or reducing the need for high-frequency dissipation. We show this by means of several examples, where we compare the numerical solutions obtained using the displacementbased and hybrid approaches against analytical solutions. We also present a monolithic formulation for the solution of structural acoustic problems based on the hybrid finite element approach. In the area of electromagnetics, since our goal is to ultimately couple the electromagnetic analysis with structural or fluid variables in a ‘monolithic’ framework, we focus on developing nodal finite elements rather than using ‘edge elements’. It is well-known that conventional nodal finite elements can give rise to spurious solutions, and that they cannot capture singularities when the domains are nonconvex and have sharp corners. The commonly used remedies of either adding a penalty term or using a potential formulation are unable to address these problems satisfactorily. In order to overcome this problem, we first develop several mixed finite elements in two and three dimensions which predict the eigenfrequencies (including their multiplicities) accurately, even for non-convex domains. In this proposed formulation, no ad-hoc terms are added as in the penalty formulation, and the improvement is achieved purely by an appropriate choice of the finite element spaces for the different variables. For inhomogeneous domains, ‘double noding’ is used to enforce the appropriate continuity conditions at an interface. Although the developed mixed FEM works very accurately for all 2D geometries and regular Cartesian 3D geometries, it has so far not yielded success for curved 3D geometries. Therefore, for 3D harmonic and transient analysis problems, we propose and use a modified form of the potential formulation that overcomes the disadvantages of the standard potential method, especially on non-convex domains. Electromagnetic radiation and scattering in an exterior domain traditionally involved imposing a suitable absorbing boundary condition (ABC) on the truncation boundary of the numerical domain to inhibit reflection from it. In this work, based on the Wilcox asymptotic expansion of the electric far-field, we propose an amplitude formulation within the framework of the nodal FEM, whereby the highly oscillatory radial part of the field is separated out a-priori so that the standard Lagrange interpolation functions have to capture a relatively gently varying function. Since these elements can be used in the immediate vicinity of the radiator or scatterer (with few exceptions which we enumerate), it is more effective compared to methods of imposing ABCs, especially for high-frequency problems. We show the effectiveness of the proposed formulation on a wide variety of radiation and scattering problems involving both conducting and dielectric bodies, and involving both convex and non-convex domains with sharp corners. The Time Domain Finite Element Method (TDFEM) has been used extensively to solve transient electromagnetic radiation and scattering problems. Although conservation of energy in electromagnetics is well-known, we show in this work that there are additional quantities that are also conserved in the absence of loading. We then show that the developed time-stepping strategy (which is closely related to the trapezoidal rule) mimics these continuum conservation properties either exactly or to a very good approximation. Thus, the developed numerical strategy can be said to be ‘unconditionally stable’ (from an energy perspective) allowing the use of arbitrarily large time-steps. We demonstrate the high accuracy and robustness of the developed method for solving both interior and exterior domain radiation problems, and for finding the scattered field from conducting and dielectric bodies. In the field of magneto-hydrodynamics, we develop a monolithic strategy based on a continuous velocity-pressure formulation that is known to satisfy the Babuska-Brezzi (BB) conditions. The magnetic field is interpolated in the same way as the velocity field, and the entire formulation is within a nodal finite element framework. Both transient and steady-state formulations are developed for two- and three-dimensional geometries. An exact linearization of the monolithic strategy ensures that rapid (quadratic) convergence is achieved within each time (or load) step, while the stable nature of the interpolations used ensure that no instabilities arise in the solution. Good agreement with analytical solutions, even with the use of very coarse meshes, shows the efficacy of the developed formulation.

Beaucoup de procédés utilisés dans l'industrie sidérurgique (coulée de lingots,coulée continue, …) peuvent générer des défauts : macro-ségrégation, mauvaises propriétés de la microstructure, défauts surfaciques. Ces problèmes peuvent être résolus par un contrôle de la température et de l’écoulement d'acier liquide. Le brassage électromagnétique (EMS) est une technique largement utilisée pour contrôler l’écoulement d'acier liquide par l’imposition d'un champ électromagnétique. Cette technique est complexe car elle couple plusieurs types de problèmes physiques:écoulement multiphasique, solidification,transfert de chaleur et induction électromagnétique à basse fréquence.En outre, l’approche expérimentale est difficile de par la dimension,l'environnement et le coût des procédés considérés. Pour ces raisons, des simulations numériques efficaces sont nécessaires pour comprendre les applications EMS et améliorer les procédés évoqués. L'objectif de cette thèse est de développer une méthodologie numérique robuste,efficace et précise pour la simulation multi-physique de l'EMS, en particulier pour le brassage dans le moule dans le cadre de la coulée continue d'acier. Cette méthodologie a été mise en oeuvre dans le code commercial THERCAST® pour être utilisé dans le cadre d’applications industrielles / Many of the processes used in thesteelmaking industry (e.g. ingot casting,continuous casting, …) can lead todefects: macro-segregation, poormicrostructure properties, surfacedefects. These issues can be solved bycontrolling the temperature and the flowof molten steel. Electromagnetic stirring(EMS) is a widely used technique to steerthe flow of liquid steel by thesuperimposition of an electro-magneticfield. This application is complex becauseit couples several physical problems:multi-phase flow, solidification, heattransfer and low frequency electromagneticinduction. In addition,experimental work is difficult because ofthe size, environment and cost of theconsidered processes. For thesereasons, efficient and effective numericalsimulations are needed to understandEMS applications and improve theaforementioned processes.The objective of this thesis is to developa robust, efficient and accurate numericalprocedure for the multi-physicssimulation of EMS, especially for in-moldstirring in the framework of continuouscasting of steel. This procedure has beenimplemented in the commercial codeTHERCAST® in order to be used forindustrial applications.

The transport of cosmic rays in the heliosphere is studied by making use of a newly developed modulation model. This model employes stochastic differential equations to numerically solve the relevant transport equation, making use of this approach’s numerical advantages as well as the opportunity to extract additional information regarding cosmic ray transport and the processes responsible for it. The propagation times and energy losses of galactic electrons and protons are calculated for different drift cycles. It is confirmed that protons and electrons lose the same amount of rigidity when they experience the same transport processes. These particles spend more time in the heliosphere, and also lose more energy, in the drift cycle where they drift towards Earth mainly along the heliospheric current sheet. The propagation times of galactic protons from the heliopause to Earth are calculated for increasing heliospheric tilt angles and it is found that current sheet drift becomes less effective with increasing solar activity. Comparing calculated propagation times of Jovian electrons with observations, the transport parameters are constrained to find that 50% of 6 MeV electrons measured at Earth are of Jovian origin. Charge-sign dependent modulation is modelled by simulating the proton to anti-proton ratio at Earth and comparing the results to recent PAMELA observations. A hybrid cosmic ray modulation model is constructed by coupling the numerical modulation model to the heliospheric environment as simulated by a magneto-hydrodynamic model. Using this model, it is shown that cosmic ray modulation persists beyond the heliopause. The level of modulation in this region is found to exhibit solar cycle related changes and, more importantly, is independent of the magnitude of the individual diffusion coefficients, but is rather determined by the ratio of parallel to perpendicular diffusion. / PhD (Space Physics), North-West University, Potchefstroom Campus, 2013