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Thermal Convection Experiments in Liquid Metal Flows with and Without Thermal convection experiments in liquid metal flows with and without magnetic field Dissertation zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) vorgelegt der Fakultät für Maschinenbau der Technischen Universität Ilmenau von Herrn M.Sc. Till Zürner geboren am 1. Oktober 1990 in Dresden, Deutschland 1. Gutachter: Univ.-Prof. Dr. rer. nat. habil. Jörg Schumacher 2. Gutachter: Dr. rer. nat. Sven Eckert 3. Gutachter: Prof. Dr. Alban Pothérat Tag der Einreichung: 2. Juli 2019 Tag der wissenschaftlichen Aussprache: 12. November 2019 urn:nbn:de:gbv:ilm1-2019000433 Abstract The interaction of electrically conducting fluid flows with magnetic fields appears in numerous natural phenomena and technical applications. Since the relevant fluids – such as liquid metals and plasmas – are generally very hot, the flows are often accompanied or even driven by thermal convection. The study of this so-called magnetoconvection is thus of interest for a number of physical systems. Two aspects are investigated in this thesis. The first concerns the case when an imposed magnetic field does not alter the fluid flow. The second case explores the changes of the flow structure and global transport properties in the presence of strong magnetic fields. The first point is relevant for inductive measurement techniques, which are required to probe the flow without disturbing it. Here, the size of the fluid volume affected by a localised magnetic field is of major importance. This topic is investigated theoretically by deriving an algorithm to calculate the penetration depth of the magnetic field into the medium. This allows the prediction of a magnetic field strength, above which a flow is significantly disturbed. The theoretical results are verified for the measurement method of local Lorentz force velocimetry which is applied to a vertical convection flow. The second point is investigated experimentally for a Rayleigh-Bénard convection system that is subject to a homogeneous vertical magnetic field. The set-up consists of a cylindrical cell of aspect ratio one. The large-scale flow structure is monitored using temperature measurements and ultrasound Doppler velocimetry. The evolution of the flow with increasing magnetic field strength is classified into different regimes and compared with theoretical predictions, and numerical simulations. Global transport properties of the flow concerning its momentum, and the heat passing through the fluid are analysed and their behaviour is interpreted in light of the aforementioned flow regimes. Additionally, a new theoretical model is developed to predict the turbulent heat and momentum transfer in the fluid by extending the Grossmann-Lohse theory for the classical Rayleigh-Bénard convection setting by the effects of a vertical magnetic field. Experimental data of the present study and from literature are used to verify and enhance the model, and to identify relevant physical mechanisms responsible for the observed results. i Zusammenfassung Die Wechselwirkung zwischen elektrisch leitfähigen Fluiden und Magnetfeldern tritt in zahl- reichen natürlichen Phänomenen und technischen Anwendungen auf. Weil die dabei relevanten Medien – meist Flüssigmetalle oder Plasmen – im Allgemeinen sehr heiß sind, werden die Strö- mungen meist von thermischer Konvektion begleitet oder werden sogar von dieser getrieben. Das Phänomen der sogenannten Magnetokonvektion ist damit von Interesse für eine große Anzahl physikalischer Systeme. Die vorliegende Arbeit untersucht hierbei zwei Aspekte. Zum einen wird der Fall betrachtet, wenn ein aufgeprägtes Magnetfeld das Strömungsfeld nicht verändert. Zum anderen werden die Modifizierungen von Strömungsstruktur und globalen Transporteigenschaf- ten durch starke Magnetfelder untersucht. Der erste Fall ist wichtig für induktive Messtechniken, welche die Bewegung eines Medi- ums untersuchen müssen, ohne dieses dabei zu stören. Die Größe des Fluidvolumens, welches von einem örtlich begrenzten Magnetfeld beeinflusst wird, ist hier ein äußerst wichtiger Faktor. Dieses Thema wird untersucht, indem die Eindringtiefe des Magnetfeldes in das Medium theo- retisch hergeleitet wird. Das erlaubt die Vorhersage einer Magnetfeldstärke, oberhalb derer eine Strömung maßgeblich gestört wird. Die theoretischen Ergebnisse werden mittels experimenteller Messungen überprüft. Dazu wird die Messmethode der lokalen Lorentzkraft-Anemometrie auf eine vertikale Konvektionsströmung angewandt. Für den zweiten Fall wird das System der Rayleigh-Bénard Konvektion unter einem homoge- nen, vertikalen Magnetfeld experimentell untersucht. Der Aufbau besteht aus einer zylindrischen Zelle mit einem Aspektverhältnis von eins. Die großskalige Struktur der Strömung wird mittels Temperaturmessungen und Ultraschall Doppler Anemometrie überwacht. Die Entwicklung der Strömung mit ansteigender Magnetfeldstärke kann in verschiedene Regime kategorisiert und mit theoretischen Vorhersagen sowie numerischen Simulationen verglichen werden. Globale Trans- porteigenschaften des Systems bezüglich Impuls und übertragener Wärme werden analysiert und ihr Verhalten anhand der zuvor gefundenen Strömungsregime interpretiert. Zusätzlich wird ein theoretisches Modell entwickelt um den turbulenten Wärme- und Im- pulstransport vorherzusagen. Dazu wird die Großmann-Lohse Theorie für klassische Rayleigh- Bénard Konvektion durch den Effekt eines vertikalen Magnetfeldes erweitert. Die experimentel- len Daten aus der vorliegenden Arbeit und aus der Literatur werden genutzt, um dieses Modell zu verifizieren und zu optimieren. Dabei werden physikalische Prozesse identifiziert, welche maßgeblich zu den beobachteten Ergebnissen beitragen. iii Contents Abstract i Nomenclature vii 1. Introduction 1 1.1. Motivation . .1 1.2. Thermal convection . .2 1.2.1. Predicting heat & momentum transfer . .6 1.2.2. The Grossmann-Lohse theory . .9 1.3. Magnetoconvection . 11 1.4. Flow measurement techniques in liquid metals . 14 1.4.1. Lorentz force velocimetry . 16 1.4.2. Ultrasound Doppler velocimetry . 19 1.5. Scientific objectives of the thesis . 19 2. Local Lorentz force velocimetry in vertical convection experiments 21 2.1. Vertical convection . 21 2.2. Experimental set-up . 22 2.3. The measurement volume of LLFV . 23 2.4. The influence of a magnet on the convective flow . 25 2.5. Force scaling . 27 2.6. Concluding remarks . 29 3. Theory of heat and momentum transport in magnetoconvection 31 3.1. Extension of the Grossmann-Lohse theory by magnetic dissipation . 31 3.1.1. Regime transitions . 33 3.1.2. Fitting of the model parameters . 35 3.2. Initial theoretical results . 36 3.3. Concluding remarks . 39 4. Classical Rayleigh-Bénard convection experiments 41 4.1. Experimental set-up . 41 4.2. Large-scale flow . 44 4.2.1. The flow oscillation frequency . 48 4.2.2. Interplay of the torsion and sloshing mode . 51 4.3. Global transport properties . 55 4.3.1. Heat transfer . 56 4.3.2. Momentum transfer . 57 v Contents 4.4. Concluding remarks . 59 5. Rayleigh-Bénard magnetoconvection experiments 61 5.1. Experimental set-up . 62 5.2. Evolution of the large-scale flow . 62 5.2.1. Weak magnetic fields: Modification of the LSC . 64 5.2.2. Intermediate magnetic fields: Cellular flow structure . 68 5.2.3. Strong magnetic fields: Magnetic wall modes . 72 5.2.4. Flow regime map and transitions . 75 5.3. Global transport properties . 77 5.4. Grossmann-Lohse theory for magnetoconvection . 83 5.4.1. Revision of the theoretical model . 83 5.4.2. Revised theoretical results . 85 5.4.3. Outlook . 87 6. Summary and outlook 89 A. The onset of magnetoconvection 93 B. Calculation of the Lorentz force 95 B.1. General case . 95 B.2. The penetration depth of LLFV . 98 C. Power law fitting and error estimation 101 Bibliography 105 List of Publications . 105 References . 105 Acknowledgements 113 vi Nomenclature Latin characters Symbol Unit Description A m2 Horizontal cell cross-section a − Blasius boundary layer parameter B T Magnetic flux density b T Induced magnetic flux density c m s−1 Speed of sound −1 −1 cp J kg K Isobaric heat capacity D m Cell diameter d m Penetration depth of a local Lorentz force velocimeter E V m−1 Electric field ex, ey, ez − Cartesian unit basis vectors F N Force f N m−3 Force density f s−1 Frequency G − Green’s function g m s−2 Gravitational acceleration H m Cell height Ha − Hartmann number Hac − Onset of convection in an infinite fluid layer HaCh − Chandrasekhar limit HaH − Onset of convection in a cylindrical cell of aspect ratio 1 h m Vertical position of a permanent magnet above a fluid surface j A m−2 Electric current density k m−1 Wavenumber L m Width of a fluid layer l m Half side length of a cubic permanent magnet lm m Characteristic scale of a magnetic field M A m−1 Magnetisation m kg Mass N − Interaction parameter Nu − Nusselt number n − Unit normal vector on a surface P~ − Relative contribution of a velocity field to the total Lorentz force Pm − Magnetic Prandtl number vii Nomenclature Symbol Unit Description Pr − Prandtl number p N m−2 Pressure Q − Chandrasekhar number Q_ W Heat flux R m Cell radius Ra − Rayleigh number Rac − Onset of convection in an infinite fluid layer RaCh − Chandrasekhar limit RaH − Onset of convection in a cylindrical cell of aspect ratio 1 Re − Reynolds number Rm − Magnetic Reynolds number r m Spatial vector r m Radial coordinate S m2 Surface s m Spatial vector on a surface T ◦C Temperature T¯ ◦C Average fluid temperature T~ − Normalised temperature − Pseudo-temperature profile (normalised) T t s Time U m s−1 Characteristic velocity V m3 Volume v m s−1 Velocity w N s m−4 Sensitivity function
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