Abstract Turbulent Shear Flow in a Rapidly Rotating

ABSTRACT Title of dissertation: TURBULENT SHEAR FLOW IN A RAPIDLY ROTATING SPHERICAL ANNULUS Daniel S. Zimmerman, Doctor of Philosophy, 2010 Dissertation directed by: Professor Daniel P. Lathrop Department of Physics This dissertation presents experimental measurements of torque, wall shear stress, pressure, and velocity in the boundary-driven turbulent flow of water between concentric, independently rotating spheres, commonly known as spherical Couette flow. The spheres' radius ratio is 0.35, geometrically similar to that of Earth's core. The measurements are performed at unprecedented Reynolds number for this geometry, as high as fifty-six million. The role of rapid overall rotation on the turbulence is investigated. A number of different turbulent flow states are possible, selected by the Rossby number, a dimensionless measure of the differential rotation. In certain ranges of the Rossby number near state borders, bistable co-existence of states is possible. In these ranges the flow undergoes intermittent transitions between neighboring states. At fixed Rossby number, the flow properties vary with Reynolds number in a way similar to that of other turbulent flows. At most parameters investigated, the large scales of the turbulent flow are characterized by system-wide spatial and temporal correlations that co-exist with intense broadband velocity fluctuations. Some of these wave-like motions are iden- tifiable as inertial modes. All waves are consistent with slowly drifting large scale patterns of vorticity, which include Rossby waves and inertial modes as a subset. The observed waves are generally very energetic, and imply significant inhomogene- ity in the turbulent flow. Increasing rapidity of rotation as the Ekman number is lowered intensifies those waves identified as inertial modes with respect to other velocity fluctuations. The turbulent scaling of the torque on inner sphere is a focus of this dissertation. The Rossby-number dependence of the torque is complicated. We normalize the torque at a given Reynolds number in the rotating states by that when the outer sphere is stationary. We find that this normalized quantity can be considered a Rossby-dependent friction factor that expresses the effect of the self-organized flow geometry on the turbulent drag. We predict that this Rossby-dependence will change considerably in different physical geometries, but should be an important quantity in expressing the parameter dependence of other rapidly rotating shear flows. TURBULENT SHEAR FLOW IN A RAPIDLY ROTATING SPHERICAL ANNULUS by Daniel S. Zimmerman Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2010 Advisory Committee: Professor Daniel P. Lathrop, Chair/Advisor Professor Peter Olson Professor William Dorland Professor Saswata Hier-Majumder Professor James Duncan c Copyright by Daniel S. Zimmerman 2010 Dedication Dedicated to my wife, Jess, who has accompanied me through the latter half of my extended graduate career with patience, encouragement, and interest in my work. ii Acknowledgments First and foremost, I must acknowledge our core experimental team, consisting of my advisor, Dan Lathrop, my fellow graduate student Santiago AndrésTriana, and technical supervisor Don Martin, who has most certainly earned his unofficial lab title of \Ubërtechnician." A well known turbulence experimentalist, upon visiting our lab and seeing our experiment for the first time, asked, before anything else, how large our technical staff was. At this time the permanent technical staff for the three meter experiment then consisted solely of the four of us. We have spent years together, starting when the experimental space was full of other experiments. We have brainstormed, sketched, engineered, and constructed together, and we are all rightfully proud of the machine we've built. However, while each of us has assumed many roles, we could not do it ourselves. I'd like to acknowledge Jim Weldon, a long time acquaintance of Dan Lathrop's and indispensable engineering consult on the design of rapidly rotating equipment, and Allen Selz of Pressure Sciences Incorporated in Richardson, TX, whose stress analysis of the outer sphere gives us confidence that we may safely reach our design speed goals. Furthermore, I would like to acknowledge the fine fabrication job done on the outer sphere by Central Fabricators in Cincinnati, OH, and the efforts of the other machine and fabrication shops, especially Motion Systems in Warren, MI who built our large, custom pulley and Chesapeake Machine in Baltimore, MD who were responsible for the fabrication of many large parts. I'd like to acknowledge the great crew of young scientists that Dan Lathrop iii has assembled here at the University of Maryland over the years. I have enjoyed working with and learning from Woodrow Shew, Dan Sisan, Barbara Brawn, Nicolas Mujica and others on various smaller experiments. I'd like to especially thank Doug Kelley and Matthew Adams, who are respectively the previous and current researchers on the 60 cm experiment , without whom the insights from magnetic data would not be possible, and Matt Paoletti who was undertaking a similar program of torque measurement in Taylor-Couette flow, and with whom I had several helpful discussions. Thanks to IREAP technician Jay Pyle for additional technical support and machining advice, and also to Dr. John Rodgers, who provides much insight into the issues of taking good data in a noisy world. I'd like to acknowledge all of those who have tirelessly supported my scientific leanings throughout my life, especially my mother Nancy and her late father, John D. Sipple. When I was young, he gave me a hefty science encyclopedia. This single volume was nearly more than I could lift at the time, probably nine inches thick, and bound with steel bolts. Perhaps that influenced my eventual choice of Ph.D. project. And last, I cannot thank my wife Jess enough. Two years ago, when we were married after several years together, we requested that the guests take it easy with physical gifts since \Dan was going to get his Ph.D. soon," and we expected to relocating. Jess has been patient, understanding, encouraging, and constantly interested in what I do, helping me through the inevitable frustrations of this long project. iv CONTENTS Table of Contents :::::::::::::::::::::::::::::::::: vii List of Tables :::::::::::::::::::::::::::::::::::: viii List of Figures ::::::::::::::::::::::::::::::::::: xiii List of Abbreviations :::::::::::::::::::::::::::::::: xiv 1. Introduction and Theoretical Background :::::::::::::::::: 1 1.1 Turbulence.................................1 1.1.1 The Problem With Turbulence.................1 1.1.2 Homogeneity, Isotropy, and Universality of Statistics.....3 1.2 Outline of the Dissertation........................6 1.3 Rotating Fluid Flow...........................7 1.3.1 Equations of Motion.......................7 1.3.2 Two-Dimensionality....................... 10 1.3.3 Inertial Waves........................... 11 1.3.4 Inertial Modes........................... 14 1.3.5 Rossby Waves........................... 18 1.3.6 Ekman Layers: Pumping and Stability............. 23 1.4 Rotating Turbulence........................... 28 1.4.1 Rotating vs. 2D Turbulence................... 28 1.4.2 Inertial Waves in Turbulence: Linear Propagation....... 30 1.4.3 Inertial Waves in Turbulence: Nonlinear Interaction...... 36 1.4.4 Inertial Modes and Rossby Waves in Turbulence........ 38 1.5 Summary................................. 42 2. Motivation and Prior Work :::::::::::::::::::::::::: 44 2.1 Earth's Dynamo.............................. 44 2.2 Atmospheric Turbulence......................... 52 2.3 Prior work on Spherical Couette..................... 57 2.3.1 Differential Rotation vs. Convection.............. 57 2.3.2 Non-Rotating Laminar States.................. 59 2.3.3 Turbulent Equatorial Jet..................... 63 2.3.4 Stewartson Layer and Instabilities................ 65 2.3.5 Turbulent Sodium Spherical Couette.............. 66 3. Experimental Apparatus :::::::::::::::::::::::::::: 69 3.1 Introduction................................ 69 3.2 Design Motivation............................ 70 3.2.1 Dimensionless Parameter Goals................. 70 3.2.2 Experiment Limitations..................... 77 3.3 Machinery Overview........................... 79 3.3.1 Rotating Frame.......................... 79 3.3.2 Lab Frame............................. 84 3.3.3 Motor Drive System....................... 89 3.4 Mechanical Engineering......................... 90 3.4.1 Support Frame.......................... 90 3.4.2 Static Stress Analysis....................... 101 3.4.3 Vibrational Mode Analysis.................... 102 3.4.4 Additional Contributions..................... 108 3.5 Instrumentation and Control....................... 114 3.5.1 Data and Control Overview................... 116 3.5.2 Rotating Instrumentation.................... 119 3.5.3 Speed Encoders.......................... 126 3.5.4 Wall Shear Stress Array..................... 128 3.5.5 Torque Sensor........................... 136 3.6 Sixty Centimeter Apparatus....................... 142 4. Results I: Turbulent Flow States ::::::::::::::::::::::: 146 4.1 Introduction................................ 146 4.2 Ro = 1, Outer Sphere Stationary.................... 149 4.2.1 Flow Properties.......................... 149 4.2.2 Torque............................... 165 4.2.3 Summary............................

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