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Memoria para optar al título de Ingeniero Civil Mecánico / Desde hace más de un siglo se ha estudiado el flujo en el intersticio entre dos cilindros que giran a velocidades angulares independientes. Las condiciones naturales de simetría en el caso en que los dos cilindros son coaxiales permitieron obtener campos analíticos de presión y velocidad (que se utilizaron para determinar la viscosidad de los fluidos), modelar la formación de estructuras vorticiales para ciertas configuraciones y describir en general el sistema. Hasta la fecha no se ha encontrado bibliografía con referencia a haber liberado esta condición de coaxialidad. En este trabajo se estudia un sistema tipo Taylor-Couette pero con el cilindro interior libre, es decir, acoplado al cilindro externo sólo por el fluido. Para esto, se construyó un montaje experimental que controla la velocidad de giro del cilindro externo, y se midió el flujo interno y el comportamiento del cilindro interno. Además, se hacen modelaciones numéricas del sistema. Se encontró que el sistema establece 4 regímenes para diferentes velocidades de excitación, de los cuales 3 ya habían sido reportados para esferas pequeñas en flujos rotatorios. El cuarto régimen, corresponde a la levitación estable del cilindro interno en el flujo, este trabajo se centra principalmente en la descripción de este régimen. Se hicieron PIV y simulaciones numéricas del flujo tridimensional del sistema en el régimen de levitación. En las secciones donde no hay intersección con el cilindro interno se encuentra un flujo tipo rotación sólida y en el resto un flujo compuesto por 3 circulaciones principales. La posición de levitación queda caracterizada por un radio de equilibrio, que depende exclusivamente de la geometría del sistema, y una posición angular que varía con la masa y la velocidad de excitación del sistema. Para velocidades de excitación relativamente bajas, la velocidad angular del cilindro interno es opuesta a la velocidad del cilindro externo y para velocidades mayores de excitación, el cilindro interno deja de rotar.

Magíster en Ciencias de la Ingeniería, Mención Recursos y Medio Ambiente Hídrico / Los ensayos realizados en un reómetro rotacional tienen como límite superior de medición la velocidad angular crítica para asegurar un flujo axisimétrico. Para velocidades mayores a ésta, el flujo deja de ser concéntrico y comienza la transición hacia un flujo helicoidal. Este fenómeno fue estudiado para fluidos newtonianos en 1923 por Geoffrey I. Taylor, quien pudo determinar analíticamente la velocidad crítica de estabilidad para el flujo rotacional. Los supuestos realizados en este estudio no permiten extender sus resultados a fluidos no newtonianos, en partículas fluidos con partículas. Las mezclas con partículas tienen ciertas características que los diferencian de los fluidos newtonianos homogéneos; como interacciones entre las fases sólidas y liquidas, migración de partículas, sus efectos sobre los perfiles de velocidad y viscosidad. En el presente trabajo, se estudió el comienzo de inestabilidades de Taylor-Couette para una mezcla no coloidal, con concentración hasta 50\% y relaciones de radio interior y exterior r_i/r_e entre 0.71 y 0.88. Se desarrolló una instalación experimental de Taylor-Couette, con rotación del cilindro interior, capaz de controlar la velocidad de giro y medir el torque en el cilindro interior, identificando de esta manera la velocidad crítica de transición. Este estudio es complementado con un análisis de estabilidad lineal que considera las particularidades de este tipo de mezclas. Finalmente, se compararon los resultados de ambas etapas del estudio con el caso base de un fluido newtoniano homogéneo, determinando así el efecto de las partículas sobre la estabilidad de flujo.

Dans ce travail, nous représentons des résultats expérimentaux sur la transition vers les rouleaux turbulents dans l’écoulement de Couette-Taylor, l’écoulement produit entre deux cylindres coaxiaux tournant indépendamment l’un par rapport à l’autre. Une fois la géométrie et la nature du fluide sont fixes, l’écoulement est gouverné par deux paramètres de contrôle, les nombres de Reynolds intérieur et extérieur 〖Re〗_o et 〖Re〗_i associés à la rotation des cylindres extérieur et intérieur respectivement. La variation de ces paramètres confère à l’écoulement une variété de régimes, décrits par Coles [7] et par Andereck et al. [13]. Dans le cas de la contra-rotation, nous avons identifié trois régimes principaux sur le diagramme d'Andereck et al. [13] En considérant la variation de 〖Re〗_i pour une valeur fixe de〖 Re〗_o, l’écoulement est laminaire pour des faibles〖 Re〗_i. Il devient instable pour des valeurs plus élevées de〖 Re〗_i. Avant d'atteindre la turbulence, l'écoulement passe par un régime de coexistence laminaire-turbulent [7 13 14 16]. Pour notre série de mesures, nous avons fixé le nombre de Reynolds extérieur à 〖Re〗_o=-4368 et nous avons varié 〖Re〗_i du régime laminaire vers le régime turbulent. Pour 3000 < 〖Re〗_i < 4000, les diagrammes spatio-temporels indiquent la présence de structures cohérentes désordonnées. Ces structures sont bien organisées dans le temps et dans l’espace pour 4000 <〖 Re〗_i < 10000, et deviennent stationnaires pour des 〖Re〗_i plus grands [35]. Cette étude a été réalisée à l’aide d’une technique de visualisations à l'aide du kalliroscope, ainsi que par des mesures de vitesse par PIV stéréoscopique et LDV. Ceci nous permet ensuite de calculer les différentes quantités moyennes (énergie cinétique, contrainte de Reynolds, temps et longueur de corrélation, ...). / In this work we report experimental results on the transition to the turbulent vortices in the Couette-Taylor flow, the flow produced between independently rotating coaxial cylinders. Once the geometry and the nature of the fluid are fixed, the flow is gouverned by two control parameters, the outer and the inner Reynolds numbers 〖Re〗_o and 〖Re〗_i associated with the rotation of the outer and inner cylinders respectively. The variation of these parameters produces a large variety of regimes, which have been described by Coles [7], and Andereck et al. [13]. In the counter-rotating case, we have identified three main regimes on the diagram of Andereck et al. [13] When considering the variation of 〖Re〗_ifor a fixed value of〖 Re〗_o, the flow is laminar for low〖 Re〗_i. It becomes unstable for higher values of 〖Re〗_i. Before reaching turbulence, the flow passes by a regime of laminar-turbulent coexistence [7 13 14 16]. For our series of measurements, we fixed the outer Reynolds number at 〖Re〗_o=-4368, and varied 〖Re〗_ifrom the laminar to the turbulent regime. For 3000 < 〖Re〗_i< 4000, the space-time diagrams indicate the occurrence of disordered coherent structures. These structures are then well organized in time and space for 4000 < 〖Re〗_i< 10000, and become stationnary for the highest 〖Re〗_i [35]. These regimes are studied by visualizations using kalliroscope, as well as measurements of the velocity by stereoscopic PIV and LDV. This later allows us to calculate the various mean quantities (kinetic energy, Reynolds stress, time and length of correlation, etc…).

Magnetorotational instability (MRI) is one of the most important and most common instabilities in astrophysics. Today it is widely accepted that it serves as a major source of turbulent viscosity in accretion disks, the most energy efficient objects in the universe. The importance of the MRI for astrophysics has been realized only in recent fifteen years. However, originally it was discovered much earlier, in 1959, in a very different context. Theoretical flow of a conducting liquid confined between differentially rotating cylinders in the presence of an external magnetic field was analyzed. The central conclusion is that the additional magnetic field parallel to the axis of rotation can destabilize otherwise stable flow. Theory of non-magnetized fluid motion between rotating cylinders has much longer history, though. It has been studied already in 1888 and today such setup is usually referred as a Taylor-Couette flow. To prove experimentally the existence of MRI in a magnetized Taylor-Couette flow is a demanding task and different MHD groups around the world try to achieve it. The main problem lies in the fact that laboratory liquid metals which are used in such experiments are characterized by small magnetic Prandtl number. Consequently rotation rates of the cylinders must be extremely large and vast amount of technical problems emerge. One of the most important difficulties is an influence of plates enclosing the cylinders in any experiment. For fast rotation the plates tend to dominate the whole flow and the MRI can not be observed. In this thesis we discuss a special helical configuration of the applied magnetic field which allows the critical rotation rates to be much smaller. If only the axial magnetic field is present, the cylinders must rotate with angular velocities corresponding to Reynolds numbers of order Re ≈ 10^6. With the helical field this number is dramatically reduced to Re ≈ 10^3. The azimuthal component of the magnetic field can be easily generated by letting an electric current through the axis of rotation, In a Taylor-Couette flow the (primary) instability manifests itself as Taylor vortices. The specific geometry of the helical magnetic field leads to a traveling wave solution and the vortices are drifting in a direction determined by rotation and the magnetic field. In an idealized study for infinitely long cylinders this is not a problem. However, if the cylinders have finite length and are bounded vertically by the plates the situation is different. In this dissertation it is shown, with use of numerical methods, that the traveling wave solution also exists for MHD Taylor-Couette flow at finite aspect ratio H/D, H being height of the cylinders, D width of the gap between them. The nonlinear simulations provide amplitudes of fluid velocity which are helpful in designing an experiment. Although the plates disturb the flow, parameters like the drift velocity indicate that the helical MRI operates in this case. The idea of the helical MRI was implemented in a very recent experiment PROMISE. The results provided, for the first time, an evidence that the (helical) MRI indeed exists. Nevertheless, the influence of the vertical endplates was evident and the experiment can be, in principle, improved. Exemplary methods of reduction of the end-effect are here proposed. Near the vertical boundaries develops an Ekman-Hartmann layer. Study of this layer for the MHD Taylor-Couette system as well as its impact on the global flow properties is presented. It is shown that the plates, especially if they are conducting, can disturb the flow far more then previously thought also for relatively slow rotation rates. / Die magnetische Scherinstabilitaet (engl. MRI) ist eine sehr häufig in der Astrophysik anzutreffende Instabilität. Es wird heute weithin angenommen, dass sie die Ursache für die turbulente Viskosität in Akkretionsscheiben ist, den Objekten mit der höchsten Energieeffizienz im Kosmos. Die Bedeutung der MRI ist erst in den letzten fünfzehn Jahren klargeworden. Entdeckt wurde sie jedoch schon viel früher, im Jahre 1959 in einem völlig anderen physikalischen Kontext. Die Strömung in einer leitfähigen Flüssigkeit zwischen differentiell rotierenden Zylindern unter dem Einfluss eines externen Magnetfeldes wurde theoretisch untersucht. Die Schlussfolgerung war, dass das zugesetzte Magnetfeld eine sonst stabile Strömung destabilisieren kann. Die Geschichte der Theorie von Strömungen zwischen Zylindern reicht bis ins Jahr 1888 zurück. Heute wird ein solcher Aufbau üblicherweise als Taylor-Couette-Strömung bezeichnet. Ein System rotierender Zylinder, zwischen denen sich flüssiges Metall befindet, war Gegenstand des kürzlich durchgeführten Experiments PROMISE. Die Ergebnisse belegen zum ersten Mal experimentell die Existenz der MRI. Um die notwendigen Drehzahlen gering zu halten, wurde ein spezielles, helikales Magnetfeld angelegt. Gegenstand dieser Dissertation ist die theoretische Behandlung der magnetohydrodynamischen Taylor-Couette-Strömung, ähnlich der des Experiments PROMISE. Insbesondere der Einfluss der vertikalen Ränder (Deckel) wird untersucht. Es wird gezeigt, dass die MRI auch in Zylindern mit endlicher Höhe und mit begrenzenden Deckeln einsetzt. In der Nähe der vertikalen Ränder bildet sich eine Ekman-Hartmann-Schicht. Die Untersuchung dieser Schicht im Zusammenhang mit dem MHD-Taylor-Couette-System sowie ihr Einfluss auf die globalen Strömungseigenschaften werden vorgestellt. Es wird gezeigt, dass die Deckel - insbesondere wenn sie elektrisch leitend sind - die Strömung stärker beeinflussen können als bisher angenommen, selbst bei den geringen Drehzahlen. Es werden Methoden zur Verringerung dieser unerwünschten Effekte vorgeschlagen.

Cette thèse porte sur les effets combinés de l’excentricité et du débit axial sur les propriétés de stabilité linéaire de l’écoulement de Couette circulaire avec cylindre extérieur fixe. Cet écoulement intervient, entre autres, lors du forage de puits de pétrole. Une méthode pseudospectrale est mise en oeuvre pour calculer l’écoulement de base, stationnaire et invariant suivant la direction axiale, ainsi que les modes normaux d’instabilité. L’écoulement est régi par quatre paramètres adimensionnels : rapport de rayons _ et excentricité e pour la géométrie, nombres de Reynolds azimuthal et axial, Re et Rez, pour la dynamique. La première partie de l’étude est consacrée aux propriétés de stabilité temporelle. Il apparaît que l’excentricité repousse le seuil d’instabilité convective vers de plus fortes valeurs de Re. L’effet de l’advection axiale sur le seuil est principalement stabilisant également. L’excentricité a pour conséquence de déformer la structure des modes par rapport au cas concentrique. Le mode au plus fort taux de croissance temporelle est ainsi constitué de tourbillons de Taylor « pseudo-toroïdaux » lorsque le débit axial est nul, et de structures « pseudo-hélicoïdales » d’ordre azimuthal croissant lorsque Rez augmente. Les résultats sont qualitativement similaires lorsque l’on change le rapport de rayons. Les prédictions théoriques sont en bon accord avec les quelques résultats expérimentaux disponibles. Dans une seconde partie, l’instabilité absolue est étudiée par application d’un critère de point selle à la relation de dispersion. Le débit axial a pour effet d’inhiber fortement l’instabilité absolue, d’origine centrifuge, et la valeur de Re au seuil est typiquement supérieure à celle de Rez d’un ordre de grandeur. L’effet de l’excentricité est plus complexe : légère stabilisation aux faibles valeurs de e, puis déstabilisation marquée aux excentricités modérées lorsque Rez est suffisament grand, et enfin stabilisation lorsque e croît davantage. Contrairement au cas de l’instabilité convective, le mode dominant l’instabilité absolue correspond à l’écoulement tourbillonnaire « pseudo-toroïdal » pour toute la gamme de paramètres considérée. / This work is concerned with the combined effects of eccentricity and pressure-driven axial flow on the linear stability properties of circular Couette flow with a fixed outer cylinder. An example of this flow can be found in oil-well drilling operations. A pseudospectral method is implemented to compute the basic flow, steady and homogeneous in the axial direction, as well as the normal modes of instability. There are four non-dimensional parameters: the radius ratio _ and the eccentricity e for the geometry, the azimuthal and axial Reynolds numbers, Re and Rez, for the dynamics. The first part of the study is devoted to the temporal stability properties. It is found that eccentricity pushes the convective instability threshold towards higher values of Re. The effect of axial advection on the threshold also tends to be stabilising. Eccentricity deforms the modes structure compared to the concentric case. As a result, the mode with the largest temporal growth rate takes the form of ‘pseudo-toroidal’ Taylor vortices in the absence of axial flow, and ‘pseudo-helical’ structures with increasing azimuthal order as Rez becomes larger. Results are qualitatively similar for different radius ratios. Agreement with the few available experimental data is good. In a second part, absolute instability is studied by applying the pinch-point criterion to the dispersion relation. Axial flow is found to strongly inhibit absolute instability, the mechanism of which being centrifugal, and the value of Re at the threshold is typically one order of magnitude larger than that of Rez. The effect of eccentricity is more complex: weak stabilisation for low values of e, marked destabilisation for moderate eccentricities and high enough Rez, and finally stabilisation as e is further increased. Unlike temporal instability, the dominant absolutely unstable mode is the ‘pseudo-toroidal’ Taylor vortex flow over the whole range of parameter space considered.

Magnetohydrodynamic (MHD) flows in annular channels are of great current interest due to experimental search for the so-called magnetorotational instability (MRI) which is important for astrophysical applications (accretion disk physics, magnetic dynamo effect). <p>The main point of MRI experiments is to study the stability of liquid metal rotating in an external magnetic field. Two different types of fluid rotation are proposed: Taylor-Couette flow between rotating coaxial cylinders and electrically driven flow in transverse magnetic field. The implementation of MRI experiments and explanation of experimental results requires a theoretical study of the equilibrium and the stability of MHD flow in an annular channel. This is one of the main tasks of present thesis.<p>For study of equilibrium Taylor-Couette and electrically driven flows, a numerical code is developed which is based on the finite difference scheme with Jacobi iterations. The structure of flows is calculated for different parameters of the experiment. Effect of the inertia on the rotation profiles is investigated in detail. The approximate analytical expressions are obtained for radial profiles of rotation that can be used for optimization of the experimental device for MRI investigation. Equilibrium Taylor-Couette and electrically driven flows are compared from the perspective of experimental studies of MRI.<p>The spectral stability of electrically driven flow is studied by solving the eigen-value problem. This study is performed in the frames of both ideal and dissipative MHD models. It is shown that electrically driven flow can be destabilized through the mechanism of MRI if fluid velocity exceeds some instability threshold, which is determined by non-axisymmetric modes. The obtained results are compared with available experimental data.<p>A general variational method is developed for the stability study of MHD flows of ideal compressible fluids. It is shown that the linearized dynamics of such fluids has an infinite set of invariants. A necessary and sufficient stability criterion can be obtained after inclusion of one or several such invariants in analysis. An analytical example is presented to confirm the fruitfulness of the developed method.

Magnetohydrodynamic (MHD) flows in annular channels are of great current interest due to experimental search for the so-called magnetorotational instability (MRI) which is important for astrophysical applications (accretion disk physics, magnetic dynamo effect). <p>The main point of MRI experiments is to study the stability of liquid metal rotating in an external magnetic field. Two different types of fluid rotation are proposed: Taylor-Couette flow between rotating coaxial cylinders and electrically driven flow in transverse magnetic field. The implementation of MRI experiments and explanation of experimental results requires a theoretical study of the equilibrium and the stability of MHD flow in an annular channel. This is one of the main tasks of present thesis.<p>For study of equilibrium Taylor-Couette and electrically driven flows, a numerical code is developed which is based on the finite difference scheme with Jacobi iterations. The structure of flows is calculated for different parameters of the experiment. Effect of the inertia on the rotation profiles is investigated in detail. The approximate analytical expressions are obtained for radial profiles of rotation that can be used for optimization of the experimental device for MRI investigation. Equilibrium Taylor-Couette and electrically driven flows are compared from the perspective of experimental studies of MRI.<p>The spectral stability of electrically driven flow is studied by solving the eigen-value problem. This study is performed in the frames of both ideal and dissipative MHD models. It is shown that electrically driven flow can be destabilized through the mechanism of MRI if fluid velocity exceeds some instability threshold, which is determined by non-axisymmetric modes. The obtained results are compared with available experimental data.<p>A general variational method is developed for the stability study of MHD flows of ideal compressible fluids. It is shown that the linearized dynamics of such fluids has an infinite set of invariants. A necessary and sufficient stability criterion can be obtained after inclusion of one or several such invariants in analysis. An analytical example is presented to confirm the fruitfulness of the developed method.

Zugl.: Cottbus, Techn. University, Diss.

Techn. Universiẗat, Diss., 2003--Berlin.