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Convective Instability in Annular Systems

HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 List of abbreviations and...
 Abstract
 Introduction
 Literature review
 Modeling equations
 Linearized equations
 Spectral solution method
 Experimental design
 Results and discussion
 Conclusions and possible future...
 Appendices
 References
 Biographical sketch
 

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1 CONVECTIVE INSTABILITY IN ANNULAR SYSTEMS By DARREN MCDUFF A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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2 Copyright 2006 By Darren McDuff

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3 ACKNOWLEDGMENTS I begin by thanking my advisor, Dr. Ranga Narayanan, whose great ideas, support, and motivation have guided me through this work. I thank my mom, dad, and sister. They have always been the best of friends to me and been there for me during any difficult times in my life. On the subject of family, I should point out here that I first considered the idea of graduate sc hool as a result of my dad suggesting it. I am also grateful to those on my advisory board. These include Dr. Martin Volz, Dr. James Klausner, Dr. Dmitry Kopelevich, and Dr. Mark Orazem. Their different viewpoints and ideas about my research have helped me to impr ove my work and avoid potential mistakes. Dr. Volz is my mentor in the N.A.S.A. Graduate Student Researchers Program, and traveled from Marshall Space Flight Center in Huntsville, Alab ama, to be here for my presentations. Dr. Klausner comes from the Department of Mech anical and Aerospace Engineering at this university. Dr. Kopelevich and Dr. Orazem are from my own department. I must express my gratitude to Ken Reed. Ken is a talented machinist with T.M.R. Engineering. High-quality components made by T. M.R. were essential to the design of my experimental apparatus, and it was great to work with someone as helpful, knowledgeable, and professional as Ken throughout the project. Additionally, I thank Grard Labrosse, a prof essor from L.I.M.S.I. in Orsay, France. During visits with our research group here at the University of Florida, he has spent a great deal of time sharing his knowledge on co mputational fluid dynamics (in particular, spectral methods) and providing helpful answers to questions from my research group. I would also like to thank a couple of undergraduates who have worked with me on this project, particularly in the design of my experi mental apparatus and the process control program

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4 for it. Joe Cianfrone helped me to u nderstand the computer software, LabVIEWTM, and with that software, we designed a very effective process control program for my experiment. Joe also helped me to design a large elect rical circuit to send data from my experiment into a process control computer, and send process control signals from that computer to components of my experiment. Scott Suozzo helped me with th e design and re-design of several parts of my experimental apparatus, which helped the experi ment to run well enough to produce results that match theoretical predictions. Of course, I thank my research group and the other graduate students in this department whose advice and ideas have helped me get past difficult parts of my proj ect and helped me in my on-going education in this field. In partic ular, I thank Dr. Weidong Guo, of my research group, for spending a good deal of time discussing computational issues with me during the last couple of years. Lastly, I am deeply grateful to N.A.S.A. for supporting this work through a fellowship from the N.A.S.A. Graduate Student Re searchers Program (grant number NGT852941).

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5 TABLE OF CONTENTS ACKNOWLEDGMENTS………………………………………………………………………... 3 LIST OF TABLES……………………………………………………………………...………... 8 LIST OF FIGURES………………………………………………………………........................ 9 LIST OF ABBREVIATI ONS AND SYMBOLS……………………………………………….. 13 ABSTRACT………………………………………………………………………...................... 16 CHAPTER 1 INTRODUCTION………………………………………………………………………. 18 1.1 Physics of Convection…………………………………………………………... 18 1.1.1 Rayleigh Convection……………………………………………………. 19 1.1.2 Marangoni Convection……………………………………...................... 21 1.2 Rayleigh Number…………………………………………………...................... 24 1.3 Physical Explanation: Pattern Selection…………………………...................... 25 1.3.1 Laterally Unbounded System…………………………………………… 25 1.3.2 Rectangular System, Laterally Bounded, Periodic Lateral Boundary Conditions……………………................................................................. 28 1.3.3 Rectangular System, Laterall y Bounded, Non-Periodic Lateral Boundary Conditions…………………………………………………… 31 1.3.4 Cylindrical System……………………………………………………… 33 1.3.5 Annular System…………………………………………………………. 36 1.4 Annulus vs. Cylinder……………………………………………………………. 37 1.5 Application……………………………………………………………………… 40 2 LITERATURE REVIEW……………………………………………………………….. 50 2.1 Single Fluid Layers……………………………………………………………… 50 2.2 Multiple Fluid Layers…………………………………………………………… 56 3 MODELING EQUATIONS…………………………………………………………….. 61 3.1 Nonlinear Equations………………………………………………...................... 61 3.2 Scaling…………………………………………………………………………... 66 4 LINEARIZED EQUATIONS…………………………………………………………… 70 4.1 Linearization…………………………………………………………………….. 70 4.2 Expansion into Normal Modes………………………………………………….. 75

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6 5 SPECTRAL SOLUTION METHOD…………………………………………………… 80 5.1 Explanation of the Method……………………………………………………… 82 5.2 Application of the Method……………………………………………………… 89 6 EXPERIMENTAL DESIGN…………………………………………………………… 98 6.1 Goals in Experimental Design……………………………………...................... 98 6.2 Experimental Apparatus………………………………………………………… 98 6.2.1 Test Section…………………………………………………………….. 99 6.2.2 Bottom Temperature Bath…………………………………………….. 101 6.2.3 Top Temperature Bath……………………………................................ 102 6.2.4 Insulation………………………………………………………………. 103 6.2.5 Flow Visualization…………………………………………………….. 103 6.3 Process Control System……………………………………………………….. 104 6.4 Typical Experimental Procedure……………………………………………… 106 7 RESULTS AND DISCUSSION……………………………………………………….. 111 7.1 Cylindrical Systems………………………………………………..................... 111 7.1.1 Constant Viscosity Computations: Cylinder………………………….. 111 7.1.2 Non-Constant Viscosity Computations: Cylinder…………………….. 113 7.1.3 Experiments: Cylinder………………………………………………… 114 7.1.4 Comparison of Results: Cylinder……………………………………… 115 7.2 Annular Systems…………………………………………………...................... 116 7.2.1 Constant Viscosity Computations: Annulus…………………………... 117 7.2.2 Non-Constant Viscosity Computations: Annulus……………………... 121 7.2.3 Experiments: Annulus……………………………................................. 121 7.2.4 Comparison of Results: Annulus……………………………………… 122 7.3 Error Analysis………………………………………………………………….. 125 8 CONCLUSIONS AND POSSIBLE FUTURE STUDIES…………………………….. 159 8.1 Summary……………………………………………………………………….. 159 8.2 Future Studies………………………………………………………………….. 161 APPENDIX A THERMOPHYSICAL PROPERTIES………………………………………………… 163 B ADDITIONAL EXPLANATION OF EQUATIONS……………………..................... 165 B.1 Boussinesq Approximation…………………………………………………….. 165 B.2 Nonlinearities in the Governing Equations…………………………………….. 166 B.3 Characteristic Velocity and Characteristic Time…………………..................... 167

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7 C DEVELOPMENT OF MATHEMATICAL MODEL FOR THE CONSTANT VISCOSITY CASE……………………………………………………………………. 168 C.1 Nonlinear Equations……………………………………………….................... 168 C.2 Scaling…………………………………………………………………………. 170 C.3 Linearization…………………………………………………………………… 172 C.4 Expansion into Normal Modes………………………………………………… 174 D MATLAB PROGRAM GENERAL FLOW-DIAGRAM……………......................... 177 E EXAMPLE MATLAB PROGRAMS………………………………………………... 178 E.1 Physical Properties…………………………………………………………….. 178 E.2 Depths for Annular System……………………………………………………. 179 E.3 Main Program: Annular System with Temperature-Dependent Viscosity……. 179 F LABVIEWTM PROGRAM GENERAL FLOW-DIAGRAM………………………….. 191 REFERENCES………………………………………………………………………………… 192 BIOGRAPHICAL SKETCH…………………………………………………………………... 194

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8 LIST OF TABLES Table Page 5-1 Example of convergence of computed result with Nr and Nz…………………………… 94 5-2 Comparison of rectangular, 2-D, no-stress results with rectangular 1-D results……….. 94 5-3 Comparison of calculated cylindrical resu lts with results of Hardin et al……………… 94 7-1 Constant viscosity computational results: critical temperature difference and onset flow pattern for sets of cylindrical di mensions considered in experiments……………. 129 7-2 Non-constant viscosity co mputational results: critical temperature difference and onset flow pattern for sets of cylindrical dimensions considered in experiments……... 129 7-3 Experimental results: cri tical temperature difference and onset flow pattern for sets of cylindrical dimensions……………………………………………………………… 129 7-4 Summary of computational and experiment al results for cylindrical systems………… 130 7-5 Constant viscosity computational results: critical temperature difference and onset flow pattern for sets of annular dime nsions considered in experiments………………. 130 7-6 Non-constant viscosity co mputational results: critical temperature difference and onset flow pattern for sets of annular di mensions considered in experiments………… 130 7-7 Experimental results: cri tical temperature difference and onset flow pattern for sets of annular dimensions…………………………………………………………………. 131 7-8 Summary of computational and experi mental results for annular systems……………. 131 A-1 Thermophysical properties…………………………………………………………….. 164

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9 LIST OF FIGURES Figure Page 1-1 Simplified system diagram: heated from below……………………………………….. 43 1-2 Rayleigh convection……………………………………………………………………. 43 1-3 Marangoni convection………………………………………………………………….. 44 1-4 General diagram: critical Rayleigh number vs. disturbance wavelength……………… 44 1-5 Cross-sectional velocity pr ofiles of flow patterns with different numbers of convective rolls…………………………………………………………………………. 45 1-6 General stability diagram for buoyancy-driv en convection in a re ctangular, laterally bounded system, with periodic boundary conditions at lateral walls, for varying aspect ratio……………………………………………………………………………………… 45 1-7 Cross-sectional velocity pr ofiles of four-roll flow pattern s with non-uniform roll size… 46 1-8 General stability diagram for buoyancy-driv en convection in a re ctangular, laterally bounded system, with non-periodic boundary conditions at lateral walls, for varying aspect ratio……………………………………………………………………………… 46 1-9 Diagrams of m = 0, 1, 2, 3 flow patterns: “U”: upward flow, “D”: downward flow... 47 1-10 General stability diagram for buoyancy-driven convection in a cylindrical container, for varying aspect ratio…………………………………………………………………. 48 1-11 Convective flow pattern in an annulus (Stork & Mller 1974)………………………… 48 1-12 Example: computed flow profile fo r annular system, cross-sectional view…………… 49 1-13 Diagram of crystal growth system……………………………………………………… 49 3-1 System diagram: cylindrical and annular systems…………………………………….. 69 5-1 Discretization node s in a cylinder………………………………………………………. 95 5-2 Example: grid point spacing…………………………………………………………… 95 5-3 Diagram of matrix/vector arra ngement of discretized problem………………………… 96 5-4 Examples of flow patterns and parenthe tical notation for cylindrical systems…………. 97

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10 6-1 Simple experiment diagram……………………………………………………………. 108 6-2 Photograph of experi mental apparatus………………………………………………… 108 6-3 Photograph of flow-through top water bath of experimental apparatus………………. 109 6-4 Cross-sectional diag ram of test section………………………………………………... 109 6-5 Process control system layout…………………………………………………………. 110 7-1 Constant viscosity computational results: Racrit vs. A for cylindrical systems……….. 132 7-2 Constant viscosity computational results: Racrit vs. A for cylindrical systems, close-up view………………………………………………………………………….. 133 7-3 Computed velocity prof ile, cross-sectional view, Lz = 6.85 mm, Lr = 8.75 mm, m = 0:(0,2), [T]crit = 5.80 C…………………………………………………………. 134 7-4 Computed velocity prof ile, cross-sectional View, Lz = 7.18 mm, Lr = 11.51 mm, m = 1:(1,3), [T]crit = 5.04 C…………………………………………………………. 135 7-5 Computed velocity prof ile, cross-sectional view, Lz = 6.53 mm, Lr = 11.28 mm, m = 1:(1,3), [T]crit = 6.43 C…………………………………………………………. 136 7-6 Photo of onset flow pattern, Lz = 6.85 mm, Lr = 8.75 mm, m = 0:(0,2)……………….. 136 7-7 Photo of onset flow pattern, Lz = 7.18 mm, Lr = 11.51 mm, m = 1:(1,3)……………… 137 7-8 Photo of onset flow pattern, Lz = 6.53 mm, Lr = 11.28 mm, m = 1:(1,3)……………… 137 7-9 Constant viscosity computational results: Racrit vs. S for annular systems with A = .75…………………………………………………………………………………. 138 7-10 Constant viscosity computational results: Racrit vs. S for annular systems with A = 1.28………………………………………………………………………………... 139 7-11 Constant viscosity computational results: Racrit vs. S for annular systems with A = 1.28, close-up view…………………………………………………………………140 7-12 Constant viscosity computational results: Racrit vs. S for annular systems with A = 1.60………………………………………………………………………………… 141 7-13 Constant viscosity computational results: Racrit vs. S for annular systems with A = 1.60, close-up view…………………………………………………………………142

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11 7-14 Constant viscosity computational results: Racrit vs. S for annular systems with A = 1.73………………………………………………………………………………… 143 7-15 Constant viscosity computational results: Racrit vs. S for annular systems with A = 1.73, close-up view…………………………………………………………………144 7-16 Constant viscosity computational results: Racrit vs. S for annular systems with A = 2.90………………………………………………………………………………… 145 7-17 Constant viscosity computational results: Racrit vs. S for annular systems with A = 2.90, close-up view…………………………………………………………………146 7-18 Constant viscosity computational results: Racrit vs. S for annular systems with A = 3.40………………………………………………………………………………… 147 7-19 Constant viscosity computational results: Racrit vs. S for annular systems with A = 3.40, close-up view 1……………………………………………………………… 148 7-20 Constant viscosity computational results: Racrit vs. S for annular systems with A = 3.40, close-up view 2……………………………………………………………… 149 7-21 Computed velocity prof ile, cross-sectional view, Lz = 6.85 mm, Ro = 8.75 mm, S = .16, m = 0:(0,1), [T]crit = 7.08 C…………………………………………………150 7-22 Computed velocity prof ile, cross-sectional view, Lz = 6.85 mm, Ro = 8.75 mm, S = .30, m = 3:(3,0), [T]crit = 7.67 C…………………………………………………151 7-23 Computed velocity prof ile, cross-sectional view, Lz = 7.18 mm, Ro = 11.51 mm, S = .12, m = 0:(0,1), [T]crit = 5.57 C…………………………………………………152 7-24 Computed velocity prof ile, cross-sectional view, Lz = 7.18 mm, Ro = 11.51 mm, S = .50, m = 4:(4,0), [T]crit = 7.09 C…………………………………………………153 7-25 Computed velocity prof ile, cross-sectional view, Lz = 6.53 mm, Ro = 11.28 mm, S = .10, m = 2:(2,1), [T]crit = 7.21 C…………………………………………………154 7-26 Computed velocity prof ile, cross-sectional view, Lz = 6.53 mm, Ro = 11.28 mm, S = .40, m = 4:(4,0), [T]crit = 8.07 C…………………………………………………155 7-27 Photo of onset flow pattern, Lz = 6.85 mm, Ro = 8.75 mm, S = .16, m = 0:(0,1)……… 155 7-28 Photo of onset flow pattern, Lz = 6.85 mm, Ro = 8.75 mm, S = .30, m = 3:(3,0)……… 156 7-29 Photo of onset flow pattern, Lz = 7.18 mm, Ro = 11.51 mm, S = .12, m = 2:(2,1)…….. 156 7-30 Photo of onset flow pattern, Lz = 7.18 mm, Ro = 11.51 mm, S = .50, m = 4:(4,0)…….. 157

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12 7-31 Photo of onset flow pattern, Lz = 6.53 mm, Ro = 11.28 mm, S = .10, m = 2:(2,1)…….. 157 7-32 Photo of onset flow pattern, Lz = 6.53 mm, Ro = 11.28 mm, S = .40, m = 4:(4,0)…….. 158 D-1 MATLAB program general flow-diagram…………………………………………… 177 F-1 LabVIEWTM program general flow-diagram………………………………………….. 191

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13 LIST OF ABBREVIAT IONS AND SYMBOLS volumetric thermal expansion coefficient (C -1) z unit vector in z-direction amplitude of small perturbation thermal diffusivity (m2/s) eigenvalue of the eigenvalue form of the problem shown in Chapter 5 dynamic viscosity (kg/(m*s)) R reference dynamic viscosity (kg/(m*s)) kinematic viscosity (m2/s) density (kg/m3) R reference density (kg/m3) dimensionless inverse time constant transpose symbol for a matrix form of the x-direction dependence in a laterally bounded, rectangular system with periodic lateral boundary conditions dimensionless group which appears in the motion equation when considering the dependence of viscosity on temperature A aspect ratio for cylindr ical and annular systems Axz aspect ratio for rectangular systems A left-hand-side matrix of the eigenvalu e form of the problem shown in Chapter 5 B a constant related to the depende nce of viscosity on temperature (C -1) B right-hand-side matrix of the eigenvalue form of the problem shown in Chapter 5 Cp constant-pressure heat capacity (J/(kg*C))

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14 CV constant-volume heat capacity (J/(kg*C)) D differentiation matrix f function representing the temperature dependence of viscosity F function representing the temperature dependence of viscosity, in terms of dimensionless temperature g magnitude of gravity (m/s2) g gravity, in vector form (m/s2) k thermal conductivity (W/(m*C)) L.I.M.S.I. Laboratoire d'Informatique pour la Mcanique et les Sciences de l'Ingnieur Lr radius in a cylindrical system, or a nnular gap width in an annular system (m) Lx horizontal depth of fluid layer (m) Lz vertical depth of fluid layer (m) m azimuthal wave number NPr Prandtl number Nr parameter setting the number of di scretization nodes in the r-direction Nx parameter setting the number of di scretization nodes in the x-direction Nz parameter setting the number of di scretization nodes in the z-direction N.A.S.A. National Aeronautics and Space Administration p modified pressure (Pa) p characteristic pressure (Pa) P pressure (Pa) Ra Rayleigh number Racrit critical Rayleigh number Ro outer radius of annular system

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15 Ri inner radius of annular system S radius ratio for annular system S stress tensor t time (s) t characteristic time (s) tel time elapsed (s) (Appendix F) tseg,T time segment that each Tb set-point remains in effect for (s) (Appendix F) tseg,VCR time interval between repetitions of the video cassette recorder’s recording cycle (s) (Appendix F) T temperature (C) Tb temperature at botto m wall of system (C) Tb,sp set-point value for Tb (C) (Appendix F) Tb,sp,j for j = (1, 2, 3, …), series of set-point values input for Tb (C) (Appendix F) TR reference temperature (C) Tt temperature at top wall of system (C) T vertical temperature difference in fluid layer (C) [T]crit critical vertical temperatur e difference for convection (C) [T]guess guess-value for critical vertical temp erature difference for convection (C) vj component of velocity in the j-direction v velocity (m/s) v characteristic velocity (m/s) X eigenvector of the eigenvalue form of the problem shown in Chapter 5

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16 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CONVECTIVE INSTABILITY IN ANNULAR SYSTEMS By Darren McDuff December 2006 Chair: Ranga Narayanan Major Department: Chemical Engineering Since natural convection can occur in such a wide range of systems and circumstances, much remains to be learned about the convective behaviors that appear in some systems. For example, convection in fluid layers of shapes ot her than cylinders or rectangles can be quite interesting, and it is th is which is investigated in this research. This study focuses on Rayleigh convection, which is natural convection caused by buoyancy forces. The convective behavior of a system (referring to a fluid layer as the “system”) depends upon the shape and dimensions of the system, the vertical temperature difference across the system, the thermophysical properties of the fluid comprising the system, and the characteristics of the dist urbance given to the system. In particular, this research is concerned with the effects of th e size and shape of the system. One important application of this study is in semi-conducto r crystal growth processes, such as the vertical Bridgman growth method. This growth method is commonly used, for example, in growing the semi-conductor crystal, l ead-tin-telluride. Crys tal growth is usually thought of as a process in a cylindr ical container. It is of interest to le arn how the convective behavior of the system would differ if annular fluid layers were employed.

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17 The onset conditions for buoyancy-driven c onvection (critical ve rtical temperature difference and flow pattern) in bounded, vertic al, cylindrical systems and vertical, annular systems were determined both experimentally an d theoretically. In th e experiments, onset conditions were determined by flow visualization with tr acer particles. The theoretical analysis involves bifurcation theory and co mputations using spectral methods. Computations show that the physics of the a nnular system are similar to those of the cylindrical system. More rolls of convecting flui d are included as the co ntainer is widened, and rolls may exist with either an az imuthal or radial alignment. The relations between the annular container dimensions and the onset conditions for convection are determined and shown. The agreement between computations and experiments is generally good. Some discrepancy in this agreement arises in cases in which the annular ga p width is small, and a likely reason for this is explained.

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18 CHAPTER 1 INTRODUCTION This chapter provides an introduction to the physics of the convective phenomena investigated in this research. It then proceeds to a brief e xplanation of the dimensionless Rayleigh number which is very meaningful in re gard to buoyancy-driven convection. After this, an explanation is given regarding the comp eting physical phenomena which determine the number of convective cells that appear at the onset of convecti on and their sizes. Next, an explanation of why it is interesti ng to study the behavior of an a nnular system as compared to a cylindrical one is given. Lastl y, an example of an industrial a pplication to which this study has relevance is discussed. 1.1 Physics of Convection Many types of convective flows are possible, and many of them have been well studied. Convection can be classified as either forced or natural. In forced convection, flow is caused by external means. Flow of a liquid being pumped across a flat plate, for example, is forced convection. The flow that will be considered in this study, howe ver, is of the other variety. Natural convection is convection in which flow s are caused by forces that interact with thermophysical property variations For example, natural convection in a system can be caused by buoyancy, interfacial tension gradients, or a combination of the two. When buoyancy is the cause of flow, this is called Rayleigh convection; if interfacial tension gradients cause flow, this is called Marangoni convection. If a system (here, “system” refers to a fluid layer) is going to convect, then the temperature difference correspondin g to the onset of convection is determined by the direction from which the system is heated, the shape and dimensions of the system, the thermophysical properties of the fluid, and the ch aracteristics of the disturbance gi ven to the system. Regarding

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19 these factors, it is noteworthy th at the fluid’s thermophysical pr operties themselves may change significantly across the fluid la yer depending on the magnitude of the vertical temperature difference to which fluid layer is subjected. This research focuses on the effects of the shape and dimensions of the system on convective behavior Unless otherwise note d, the heating of the system referred to is uniform on the top and botto m surfaces. Also, for the physical explanations that follow, until otherwise noted, the systems discussed may be assumed unbounded in the lateral direction. 1.1.1 Rayleigh Convection A system capable of convection, in general, could be heated from below or above and include any number of liquid and vapor phases. For a physic al explanation of Rayleigh convection, though, all that need be considered is a single layer of fluid, unbounded in the horizontal direction, heated from below, with the top and bottom walls held at constant temperatures. Understanding the origin of convecti on in this system will be sufficient to predict what is likely to happen in more complex systems. More importantly, a single fluid layer, heated from below, is precisely the system examined in this research. Generally, the density of a flui d decreases as temperature is increased. This means that the density at the top of the flui d would be higher than that at th e bottom of the phase. Consider a z-axis along the vertical direction in th e fluid layer, with the top surface at z = Lz, and the bottom surface at z = 0. This simple system is shown as Figure 1-1. This top-heavy fluid layer is potentially uns table to gravity (shown in many figures as g ) if subjected to a perturbation. If a particular disturbance to the system causes an element of fluid near the top of the layer to be displaced downward, then it will have a tendency to continue moving downward toward its gravitationally appropria te resting place. As this element of fluid

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20 moves downward, it displaces the lighter fluid surrounding it upward due to mass conservation. The downward-moving fluid elements are heated on ce they reach the bottom of the system, and the upward-moving fluid elements are cooled once they reach the top of the system, and so the process repeats. This process results in rolls of circulating fluid, or conve ctive “cells,” that form cellular flow patterns. Since the lowest density in the phase is at the bottom, a fluid element displaced downward would continue to move down ward to z = 0 if its motion were not resisted in any way. Flow is resisted, however, by the dynamic viscosity of the fluid. Additionally, if the kinematic viscosity and thermal diffusivity of the fluid are sufficiently high, then the mechanical and thermal perturbations may quickly die out. It is when the vertical temperature difference across the system is increased beyond a certain critical value that this convection–Rayleigh convection–will occur; at that point, gravitationa l instability overcomes the damping effects of kinematic viscosity and thermal diffusivity. Rayleigh convection is stronger in a phase with lower kinematic viscosity and lower thermal diffus ivity; this is because mechanical and thermal disturbances are damped out less quickly in a ph ase having lower values for those properties. Now, it might appear that Rayleigh convecti on results simply from large enough vertical temperature gradients. In truth, however, even with a sufficiently large vertical temperature gradient present, Rayleigh convec tion can begin only if disturbances with transverse (horizontal) variation are imposed on the sy stem. A fluid element displace d by a disturbance must “feel” a relative difference in the density of the fluid hor izontally neighboring it, and this can occur only as a result of disturbances with transverse variation. Note that the disturbance which causes the onset of convection need not necessarily be a mechanical one. A thermal disturbance with transverse variation could just as easily cause convection to begin by making a certain fluid

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21 element warmer or cooler than the fluid hor izontally adjacent to it. Figure 1-2 shows a buoyancy-driven convective flow pattern occurring in a fluid phase. 1.1.2 Marangoni Convection Convection can occur in anothe r way, as well. The flow can be driven by interfacial tension gradients rather than buoyancy forces. This type of flow is call ed Marangoni convection. Since Marangoni convection is driven by gradients in interfacial tension, it is a phenomenon that can occur only in systems possessing multiple fluid phases which contact each other at interfaces. This research is conc erned with the simpler case of a si ngle fluid layer confined in all directions by solid walls. Thus, Marangoni c onvection could not occur in the system being researched. Still, a brief expl anation of Marangoni convection will now be given for the sake of giving a more complete explanation of the physics of convection. To this end, consider a system of two verti cally stacked fluids, in which one vertical boundary is heated so as to subject the system to a vertical temperature gradient. Again, assume the system is unbounded in the horizontal direction. In this hypothetical system, both the top and bottom walls are kept at consta nt temperatures and Marangoni convection can occur in this system regardless of which vertical boundary of the system is warmer. For the sake of making an explanation by example, assume that the botto m wall of the system is warmer than the top. Like Rayleigh convection, Marangoni convection re sults from perturbations with transverse variation. If the interfac e between the fluids is perturbed, then some interfacial regions could be pushed upward and some downward. This gives rise to a transverse temp erature gradient along the interface. Consider an inte rfacial region pushed upward. The f act that the system is heated from below means that this region now experiences a lower temperature than the regions of the interface adjacent to it. Typically, surface tension, like de nsity, increases with decreasing

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22 temperature. Thus, the interfacial region th at has been pushed upward has an increased interfacial tension and pulls fluid at the interface toward it. When this happens, fluid from the bulk phases must move in to replace fluid being pu lled away from regions of the interface that lowered in interfacial tension (when the system is heated from below, these regions are the troughs of the wavy interface). The flow patte rns in Marangoni convection, like those in Rayleigh convection, are cel lular. Since the in terfacial tension grad ients are generated by temperature differences along the interface, a high er vertical temperature difference across the system favors Marangoni convection. Figure 1-3 is a simple illustration of Marangoni convection. The replacement fluid arriving from the bulk ph ases is warm in the case of the lower fluid, and cool in the case of the upper fluid; this supply of replacement fluids can either strengthen or weaken the temperature gradient which fuels Marangoni convection. The effect that the fluid moving to the hot regions of the interface from the bulk phases has on this temperature gradient is dependent upon the amount of fluid from each phase that arrives at those locations and on the thermophysical properties of each phase. As with Rayleigh convection, dynamic viscosity resists flow and can damp the process. Marangoni convection, like Rayleigh convection, is stronger in a phase with lower kinematic viscosity and lower thermal diffusivity because a phase having lower values for those properties damps out disturbances less qui ckly. On the other hand, if these two properties have high enough values, a disturbance at the interface will di e out rather than amplif y to cause convection. Of course, the phase with a lower kinematic visc osity is not necessarily the phase that has a lower thermal diffusivity. Thus, the combination of these two properties determines which phase will convect more strongly. The phase which has a kinematic viscosity-thermal diffusivity

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23 combination more favorable to convection is the pha se from which it will be easiest for fluid to move in and replace that which is lost at hot interfacial regions during Marangoni convection; it is primarily from this phase that fluid is supplied to the hot regions of the interface. If this more strongly convecting phase is the warmer phase, th en the fluid it supplies to the hot interfacial regions is even warmer than what was orig inally present, and Marangoni convection is reinforced. In the opposite scenari o, cold fluid moves in to the hot regions of the interface; this cools those regions down and counteracts Marangoni convection. In the case where system of two stacked fluids is comprised of a vapor-liquid bilayer, rather than a liquid-liquid bilayer, it should be noted that a liquid can generally tran sport more heat by convection than can a vapor. This is quantitatively shown when a comparison is made between the product of density and heat capacity ( Cp) for liquids and vapors. Though the thermal conductivities of vapors and liquids may differ, this difference may be considered less significant. The reason for this is that the fluids are already conducting he at in the motionless initial stat e of the system (base state). While the effect of the difference in density multiplied by heat capacity becomes important once convection begins, the eff ect of the difference in thermal c onductivity is relatively unchanged. By this line of reasoning, repl acement fluid arriving to troughs on the interface from the liquid phase of a vapor-liquid bilayer system is likely to have more of an effect on the temperature gradient along the interface than replacement fluid arriving from the vapor phase of the bilayer. The same reasoning (involving the product of dens ity and heat capacity) also can be used in deciding which fluid will more strongly affect the interfacial temperature gradient in a liquidliquid bilayer system. It is also very interesting to note that Marangoni convection can occur in the system even if the interface remains flat. If a non-mechanical, thermal perturba tion is given to the interface,

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24 then some regions become hot and some cold. Fluid from the hot regions is pulled toward the cold regions by interfacial tension, and fluid at th e cold regions is displaced. As this happens, fluid from the bulk must move in to replace that lost at the hot in terfacial regions – this marks the start of convection. 1.2 Rayleigh Number The focus of this explanation will get b ack on its main course and address Rayleigh convection once again. Scaling of the equations that model a fluid layer heated from below leads to the following definition of the dimensionle ss Rayleigh number, which is shown as Equation 1.1. The Rayleigh (Ra) number is a quantity that relates th e factors determining whether or not buoyancy-driven convection occurs. * *3T L g Raz (1.1) The length scale associated with Rayleigh convection is th e vertical phase depth, shown as Lz. In this definition, is a volumetric thermal expansion coefficient, g is the magnitude of gravity, T is the vertical temperature difference across the layer, is the kinematic viscosity of the fluid, and is the thermal diffusivity of the fluid (which is equal to the thermal conductivity divided by the product of the density and heat capacity). A higher Rayleigh number corresponds to a sy stem that is more susceptible to buoyancydriven convection. Notice that the factors which favor buoyancy-driven convection are all grouped in the numerator, while those which oppos e the flow form the denominator. The thermal expansion coefficient represents the de gree to which the density of the fluid changes when it is heated, so of course a higher value for this works in the favor of convection. Gravity and the vertical temperature difference are the most obvious of the two components needed to drive Rayleigh convection. A larger vertical temperature difference makes the system more

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25 unstable to Rayleigh convection. Having a larger ve rtical phase depth means that the fluid feels a weaker effect from the no-slip conditions at the top and bottom walls and thus is less mechanically constricted, and can more easily flow in a convective pattern. This effect is quite strong, and accordingly, the Rayleigh number is proportional to the ve rtical phase depth raised to the third power. Kinematic viscosity and therma l diffusivity act to dissipate any disturbances given to the system and prevent convection. Thus, these two therm ophysical properties are located together in the denominator of the Rayleigh number. 1.3 Physical Explanation: Pattern Selection Some of the most interesting facets of the convective behavior studied in this research relate to the competition of certain physical phenomena to determine precisely which convective flow patterns arise at the onset of convection. 1.3.1 Laterally Unbounded System Imagine, once again, the system shown in Figur e 1-1. The system is constantly subjected to small mechanical and thermal disturbances. As mentioned earlier, on ly disturbances with transverse variation can cause the onset of convection. The disturba nces that the system receives are of an extremely large range of wavelengths especially considering the case in which the system is laterally unbounded. Still, only one of these many disturbances actually causes the onset of convection. It is this disturbance to which the system is most unstable. It is the wavelength of this disturbance which determines the wavelength of the onset convective flow pattern. It is interesting to ask what makes di sturbance of one wavelengt h more stabilizing or destabilizing to the system than another. For the moment, consider two possibilities in terms of the wavelength of a periodic, wave-shaped disturbance. One possibility is th at the disturbance is of very short wavelength,

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26 meaning that it is quite jagged and choppy in app earance. The second possibility is that the disturbance is of very long wavelength, and so it is very gently, gr adually sloping, with its minimum-points and maximum-points spread far ap art from one another. The disturbance of very short wavelength possesses a great deal of transverse variation, sinc e it repeats its lateral oscillations so frequently; this high degree of transverse varia tion favors convection. However, since the adjacent peaks and troughs of the wave-shaped disturbance are so close to one another, they easily diffuse their momentum and therma l energy into the horizon tally neighboring fluid, which dissipates the disturbance and stabilizes th e system to convection. For example, a fluid element which initially was heated and lowered in density relative to the fluid laterally adjacent to it, as a result of a disturbance of very s hort wavelength, would quickly match the density of the laterally adjacent fluid due to rapid diffusion of heat from the fluid element. Thus, insofar as diffusion of heat and momentum is concerned, the system is quite stable to disturbances which are of very short wavelength. The system reacts to a disturbance of very long wavelength a bit differently. A disturbance of very long wavelength is so laterally sp read out that the sort of rapid diffusive stabilization just desc ribed does not play a role. Co nsequently, the lateral density differences generated by a disturbance of very long wavelength remain present rather than dissipating into uniformity, whic h is a condition that favors buoyancy-driven convection. However, when the disturbance is of very long wavelength, the adjacent peaks and troughs of the wave-shaped disturbance are so far apart from one another that the lo cal density differences between adjacent fluid elements (the transverse variations which are needed to drive buoyancydriven convection) are too weak to cause and sustain buoyancy-driven convection. If a disturbance of very long wavele ngth were able to form a corr esponding cellular, convective flow pattern of a very long wavelength, the flow patter n would not be able to be sustained; this is

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27 because it would simply require too much energy to drive the fluid in the convective cells across the very long, nearly horizontal paths which would need to be traveled within each convective flow cell. So, in summary, short-wavelength di sturbances are not very destabilizing because they rapidly degrade by diffusi on, and long-wavelength disturban ces are not very destabilizing because they are spread out so wide ly in the lateral direction that the variations they induce in the thermophysical properties of th e fluid are not strong enough to dr ive flow. Thus, of the wide range of disturbances to whic h the system is subjected, the one disturbance which is most destabilizing to the system and causes the onset of convection will possess some intermediate wavelength, which is determined by the competi ng physical behaviors associated with shortwavelength and long-wavelengt h disturbances. The ve rtical phase depth, of course, plays a large role in governing which disturban ces are most unstable, as well. At this point, one could imagine the appearance of a simple graph relating the stability of the system, in terms of the critical Rayleigh number (Rayleigh number corre sponding to the critical vertical temperature difference at which convection begins), to dist urbances of different wavelengths. The graph (Figure 1-4) can be drawn using a general, dimensionless wavelength and the dimensionless critical Rayleigh number, which is Racrit; numerical values for either quantity are not necessary at this point. Once the system is at or above the critical ve rtical temperature difference and is subjected to the right disturbance, buoyanc y-driven convection begins. S till considering a one-dimensional system, unbounded in the lateral di rection, the flow pattern at th e onset of convection possesses the wavelength of the disturbance which caused convection.

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28 1.3.2 Rectangular System, Laterally Bounded, Periodic Lateral Boundary Conditions Now, consider a two-dimensional system. Th e two-dimensional system to be considered is the system that would be obtai ned if two lateral walls were simp ly added to the left and right sides of the system of Figure 1-1, with a nostress condition on veloci ty and an insulating condition on temperature imposed on each lateral wa ll. The horizontal width of this new system is called Lx and the vertical depth is called Lz. This system can be examined to understand the interesting physics governing convective pattern selection. Due to the stress-free conditions on veloci ty and the insulating conditions on temperature at the lateral walls, the x-direct ion dependence of the velocities and temperatures in the system, which shall be called could be expressed in terms of cosines, in the form xL x n cos n = 0, 1, 2, 3, …. (1.2) The number of convective cells which may form in the system, then, is related to Lx through this form of x-direction dependence, as is the size of th e cells which form. Notice that n is an integer, so not just any pattern may arise in a laterally b ounded system. Only those patterns which can be physically accommodated by the latera l width of the system may form, and so only disturbances with those shapes are physically admissible to the system. Of the set of disturbances that are physically admissible to th e system, only one will be the most destabilizing and, when the critical vertical temperature di fference has been exceeded, cause the onset of convection. In a laterally bounded system, for the same r easons as in a system without side walls, disturbances that are laterally very short or very long are not very destab ilizing. Thus, again the shape of the most destabilizing disturbance and onset flow pattern is determined by competition

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29 between physical behavi ors like those that were explai ned for short-wavelength and longwavelength disturbances. The size and shape of the onset flow patter n, of course, are heavily dependent on the dimensions of the system. In a system that is laterally very wide, the onset flow pattern would likely not be, for example, one very wide flow ce ll. A flow pattern such as that would not be sustainable for the physical reasons like thos e explained for distur bances of very long wavelength, which were already e xplained. The size of the system is typically described using the dimensionless aspect ratio. Fo r a rectangular system, the aspect ratio is the ratio of the width of the system to the vertical height of the system, and it may be called Axz. For a cylindrical system, the aspect ratio is the ratio of the radius of the system to the vertical height of the system, and it may simply be called A. For an annular system, the aspect ratio may also be called A, and it is the ratio of the outer annular radius (Ro) to the vertical he ight of the system. As the aspect ratio is varied, obviously th e stability of the system to any given disturbance is varied with it. For exampl e, a wave-shaped distur bance which induces a convective flow pattern consisting of two rolls of convecting fluid would be most destabilizing at some particular value of Axz, but if the aspect ratio were slig htly increased (making the container laterally wider or vertically shorter) or decrease d, then that same disturbance would not be as well accommodated by the lateral width of the syst em, and consequently would not be as well suited to destabilize the system. The reason for this is related to physic al behaviors like those described for disturbances of very short and very long wavelengths, as desc ribed for the laterally unbounded system. What is meant by a convective pa ttern consisting of tw o rolls of convecting fluid is clarified in the following cross-secti onal velocity profiles of a convecting, laterally bounded system (Figure 1-5).

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30 Continuing with the example, continuously increasing the aspect ratio would make the two-convective-roll-inducing distur bance less and less destabilizing to the system, and thus make the corresponding two-convective-ro ll onset flow pattern more and more difficult for the system to form and maintain; as an onset flow pattern becomes more and more difficult to form and maintain, the critical vertical temperature diffe rence corresponding to it increases, as does the corresponding critical Rayleigh number. At some sufficiently large aspe ct ratio, the system would find a disturbance that i nduces more than two convective rolls to be more destabilizing, and the corresponding onset flow pattern to be more energetically favorable. Supposing that increasing the Axz value caused the example system to b ecome more unstable to a disturbance with that induced four convective rolls rather than two, the correspondi ng onset flow pattern would consist of four rolls of convecting fluid. The stability of the system to the four-c onvective-roll-inducing di sturbance, and the degree of difficulty that the system would have in maintaining its co rresponding onset flow pattern, are again dependent on physical beha viors like the short-wa velength and longwavelength behaviors described for la terally unbounded systems. As the Axz value is further and further increased, this type of pa ttern-switching continues. If the Axz value of the example system were further increased, the system would eventually reach an Axz value at which it would be more favorable for the system to form and main tain an onset flow pattern with more than four rolls of convecting fluid. Supposing that the next flow pattern in th e example consists of six rolls of convecting fluid rather than four, then di fficulty that the system has in forming and maintaining this six-roll-patte rn would once again be determin ed by physical behaviors like the short-wavelength and long-wavelength behaviors th at have been explained earlier. Resulting from these physical behaviors, which corre spond to the size and shape of an imposed

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31 disturbance, the exact critic al temperature difference, and critical Rayleigh number, corresponding to each different flow pattern is dependent on the Axz value and has a minimum at some certain Axz value that is optimal in terms of how well a system with that aspect ratio physically accommodates the flow pattern. When a system is laterally unbounded, or a system has lateral walls that are stress-free and insula ted (which means the system behaves essentially like a laterally unbounded system), then the mini mum possible critical Rayleigh number for any possible flow pattern is the same. Based on all of this, a graph showing the relation between the critical Rayleigh number and the value of Axz is given below as Figure 1-6. Note that in a laterally unbounded system, or in a laterally bounded system with pe riodic lateral boundary conditions, the convective patterns that can form are comprised of convective rolls which are all of equal lateral width. As an example of what is meant by this, see the flow pattern diagrams in Figure 1-5. In this figure, the roman numerals represen t flow patterns with different numbers of convective rolls. Flow pattern “III” includes more convective rolls than “II” and flow pattern “II” includes more convective rolls than “I” does. Basically, aspect ratios at which the slope of the curve changes from positive to negative are asp ect ratios at which there is a transition to a higher number of convective rolls being present in the onset flow pattern. 1.3.3 Rectangular System, Laterally Bounded, Non-Periodic Lateral Boundary Conditions When running experiments or considering pr actical applications, one does not encounter systems with periodic boundary conditions at thei r lateral walls. The mo re realistic case of a laterally bounded, rectangular system with nonperiodic lateral boundary conditions will now be detailed.

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32 The behavior in this system is essentially the same as what was just explained for the rectangular system with peri odic lateral bounda ry conditions, but ther e are a couple of key differences: the convective rolls which form at the onset of convection are not necessarily of identical size to one another, and the minimum critical Raylei gh number for each possible flow pattern (considering di fferent values of Axz) decreases as Axz increases rather than maintaining a constant value. Since the convective rolls which form at onset are not necessarily of identical size to one another, the flow pattern with four convective rolls in Figure 15 could instead appear as shown in Figure 1-7. The second key difference of this case fr om the case in which the lateral boundary conditions are periodic is th at for higher and higher Axz values, the minimum critical temperature difference values become lower and lower. The r eason for this is that the no-slip effects of the side walls do less to stabilize the system as the aspect ratio is increased. As Axz approaches infinity, the critical temperature difference va lues for the flow patterns approach values corresponding to the well known crit ical Rayleigh number of 1708 th at is associated with the onset of buoyancy-driven convection in latera lly unbounded systems (Davis 1967). For the twodimensional, laterally bounded system, no-slip cond itions at the lateral wa lls, here is a general plot of the stability of the system, in terms of the critical Rayleigh number, to disturbances corresponding to different onset flow patterns; the plot is simply based on the reasoning explained above. Again, in this figure, the roman numerals re present flow patterns with different numbers of convective rolls, where pattern “III” includes mo re convective rolls than “II” and pattern “II” includes more convective rolls than “I.” Recall that aspect ratios at which the slope of the curve

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33 changes from positive to negative indicate transitions to onset flow patterns with higher numbers of convective rolls being present. 1.3.4 Cylindrical System So far, the example systems discussed have been assumed to be laterally bounded, rectangular systems or laterally unbounded systems. Radial walls in cy lindrical and annular systems significantly affect the cr itical conditions for the onset of convection. This research addresses primarily cylindrical and annular systems. To prev ent any possible confusion, note that throughout this dissertation, “cylindrical system” (or sometimes the term “open cylindrical system” may be used) refers to a fluid system bounded by a cylindrical container, in which the entire container is filled only with fluid, and does not include a center-piece as an annular container does. From this point onward, unless otherwise noted, the systems discussed will be within vertical, cylindrical containers (meaning that the direction of gravity is parallel to the axis cutting through the radial center of the cylinder) with finite oute r radii or within vertical, annular containers. To be completely clear, a vertical, annular contai ner is one which has an annular cross-section when viewed from above. Increasing the aspect ratio, A, in a cylindrical system has the effects on pattern selection which were just explained for the la terally bounded, rectangul ar system. As A is increased, the onset flow patterns include more a nd more convective rolls, so as to occupy the system in a more energetically favorable fashion. Also, as A approaches infinity, the valu e of the critical Rayleigh number approaches the value of 1708, which is well-known to be associ ated with laterally unbounded systems. There is, however, another as pect to pattern selection in cylindrical systems, and it will be explained now.

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34 In a rectangular system, the number of convec tive rolls which form can be related to the imposed disturbance’s size and shape in the xdirection. For recta ngular, laterally unbounded systems, the corresponding onset flow pattern can then be described by a wave number in the xdirection, inversely proportional to the wavelength of the flow patte rn in the x-direction. In all systems considered in this re search, whether rectangular, cyli ndrical, or annular, it is most energetically favorable for the system to form onset flow patterns with only one convective roll in the vertical direction; thus, there is no need to consider a z-direction wave number, z-direction wavelength, or the size and shape of a disturbance in the z-direction for this research. The flow pattern in a rectangular system, then, can be described simply by describing its size and shape in the x-direction. In a cylindrical system, flow patterns may ha ve different numbers of convective rolls in the radial direction, like the patt erns in rectangular systems c ould have different numbers of convective rolls in the x-direction. However, flow patterns in cylindrical systems may also have differing numbers of convective rolls in the azi muthal direction. The flow patterns in a cylindrical system must, of cour se, be periodic. The periodicity of the flow patterns in the azimuthal direction, and thus how many convectiv e rolls exist along the azimuthal direction, is characterized by an azimuthal wave number, called m. For example, the flow pattern described by the wave number m = 2 is one which repeats twice as one progresses once through the fluid layer in the azimuthal direction. For an m = 2 flow pattern, progressing 90 around the system in the azimuthal direction, from an arbitrary starting point, leads to a point that has the opposite vertical velocity of the star ting point. Likewise, for an m = 1 flow pattern, progressing 180 around the system in the azimuthal direction, from an arbitrary starting point, leads to a point that has the opposite vertical velocity of the starting point. An m = 0 flow pattern, though, is

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35 axisymmetric in the azimuthal direction. Figure 1-9 gives examples of flow patterns possessing some of the lowest m values. In these diagrams, the lette r “U” indicates upward velocities and the letter “D” indicates downward velocities. The intensity of the shading indicates the magnitude of the velocities, with darker shades representi ng higher velocities. At certain aspect ratios, the a cylindrical system favors certain azimuthal periodicities, and thus, certain m values. For each m value, the system may have a differing number of convective rolls in the r-dire ction, and how many there are depends once again upon physical behaviors like the short-wave length and long-wavelength behavi ors that have already been explained. The relation, for each m value, between the number of convective rolls, the critical Rayleigh number needed to obtain a particular number of rolls, a nd the aspect ratio appears much like what is shown in Figure 1-8. Ge nerally, in the cylindrical cases, only m = 0, 1, or 2 patterns were favored as onset patterns; th is means that in most cases, th e critical vertical temperature difference corresponding to the ons et of convection was lower for m = 0, 1, or 2 patterns than for patterns with other azimuthal wave numbers. As th e aspect ratio approaches infinity, the critical Rayleigh numbers for flow patterns of all m values approach the unbounded-system-value of 1708. A general graph showing the relation of flow patterns and th eir critical Rayleigh numbers to the aspect ratio in a cylindrical system is given as Figure 1-10. This graph was obtained by performing a series of computa tions, to determine the critical Rayleigh number corresponding to several values of the aspect ratio, using the computational program developed for obtaining computational results in this research. The comp utational solution method is detailed in Chapter 5, and the computational results ar e shown extensively in Chapter 7. On this diagram, starting from a given asp ect ratio and moving upward along the graph is analogous to increasing the vertic al temperature difference across the system. Once the first of

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36 the m = 0, 1, 2, or 3 curves has been reached, th en this is the critical Rayleigh number corresponding to the critical vertic al temperature difference. The m value of the bottom-most curve at a given aspect ratio is the m value for the onset flow pattern at that aspect ratio; this azimuthal wave number, since the critical temper ature difference and cri tical Rayleigh number corresponding to it are lowest, is the azimuthal wave number of di sturbances to which the system is most unstable. Notice that the curve for each particular m value looks much like the one in Figure 1-8, which reflects the effects of physics related to the sizes and shapes of different disturbances on the stability of the system. Graphs like Figure 1-10 are extremely helpful in research on buoyancy-driven convection. The reader is asked to notice that the graphs presenting co mputational results for buoyancy-driven instability in cylindrical systems in Chapter 7 are just like the one shown in Figure 1-10. Figure 1-10, intere stingly, is the correct graph fo r cylindrical systems of any dimensions, and it holds true re gardless of the set of thermophysical properties possessed by the fluid in the system. This will be further explained at the end of Chapter 4. Note that the stability diagrams in this research would, however, diffe r very slightly dependi ng on whether or not the variation of viscosity with te mperature is being considered. 1.3.5 Annular System Nearly everything that was ju st said for cylindrical system s in Section 1.3.4 applies for annular systems, as well. An exception is that as the outer radius of the system approaches infinity (and thus, the A value approaches infinity), one would expect the critical Rayleigh number to approach a value other than 1708, wh ich corresponds to a laterally unbounded system, since an inner radial wall is pr esent. Patterns in annular system s are described primarily by their azimuthal wave number, m. The patterns in annular systems may also include more than one

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37 convective roll in the radial direction, but this tends to happen only when the ratio of the inner annular radius (Ri) to the outer annular radius (Ro) is quite small. This ratio of radii is a new dimensionless parameter affecting pattern selecti on which was not present in the consideration of cylindrical systems. The ra dius ratio shall be called “S” and will be given by Equation 1.3. o iR R S (1.3) The effect of the radius ratio is the key difference between the a nnular system and the cylindrical system in terms of the formation and se lection of flow patterns. Stability diagrams of the relations between Racrit, the aspect ratio, and the radius ratio (S) for different wave numbers, and valid for systems with any set of thermophysi cal properties, could be made for the annular system, as well; they would appear somewhat lik e the one for the cylindrical system, shown as Figure 1-10. Such diagrams are shown in Chapter 7. As annular systems are the main focus of this research, they will be explained much more thoroughly in the following pages. 1.4 Annulus vs. Cylinder Certainly, there has been a good deal of research already done on natural convection in annular systems (Stork & Mller 1972, and Little field & Desai 1990). However, much of that work is considerably different from the curren t research. For example, many of those studies involve horizontal annuli rather than the vertical annuli addresse d in the current study. Secondly, of those past works which do address convection in vertical annuli, many consider the case in which heat is supplied to the system from the i nner rod of the annulus rather than the case considered in this study, which is that heat is supplied to the system from its bottom wall. The case in which heat is supplied from the bottom wall is highly relevant to industrial processes, and this will soon be discussed further.

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38 The radial walls of all cylindrical and annular systems considered in this research are assumed to be insulating. This assumption shall be maintained throughout this paper, as the radial walls in the experiments conducted for this research were made to be insulating. Whether the radial wall is treated as conducting or insula ting does have an effect on what the critical vertical temperature difference is for the onset of convection. Genera lly, the system becomes unstable at a lower vertical temperature difference if the radial walls are insulating. This makes sense because insulating radial walls do not allow a system to di ssipate thermal disturbances as easily as conducting radial walls would, and making the dissipation of ther mal disturbances more difficult makes the system more unstable to c onvection. As one might expect, the difference between the insulating and conducti ng cases is more pronounced in systems that span a smaller distance in the radial dir ection. In such systems, the radial wa lls are closer to th e interior of the system and so they have a relativ ely stronger effect on the behavior of the system. Also, whether the radial walls are insulating or conducting, it is obvious that if th e system is of smaller radial extent, then the system will be more stable to convection than a wider system of matching vertical depth; this is because of the relatively larger effect of the no-slip condition on velocity imposed at the radial walls. The addition of an inner bloc k of circular cross-section to the center of a cylindrical system transforms the cylindrical system into an annular system. The inner block of the annular system is insulating, just like the outer radial wall, and varying the radius of this circular inner block changes the gap width occupied by fluid with in the annular system. Changes in this gap width cause significant changes in the critical vertical temperatur e difference and flow pattern at the onset of convection in the annular system. Primary goals of this research were to better understand the phenomenon of buoyancy-driven conv ection in terms of how it differs in an

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39 annular system compared to a cylindrical system and how the onset conditions for convection in the annular system differ as the annular gap width varies. As one may expect, the addition of a circular inner block to the cy lindrical system does, because of the additional radial wall and corr esponding no-slip condition on velocity that it introduces, impart to the system a higher stability to c onvection. That is to say, a cylindrical system of a certain vertical dept h, which requires a certain vertical temperature difference for the onset of Rayleigh convection, woul d require a higher vertical te mperature difference to convect if some small, circular block were introduced to the center of the system, converting it to an annular system. The circular cen ter-block present in an annular system also affects the flow patterns which may appear at the onset of c onvection. Stork and Mll er (1974) have shown experimentally that in a one-flu id annular system, heated from below, convective cells can form such that they fill the annular gap in an azi muthally aligned arrangement that is rather reminiscent of the arrangement of spokes on a bicycle tire. An ex ample of this type of pattern presented in the work of Stork and Mller is s hown as Figure 1-11. In th is photograph of one of their experiments, the fluid convecting is a sili cone oil, and the flow is visible due to the presence of aluminum tracer powder in the oil. When the flow is horizontal, the aluminum particles align in such a way as to reflect more light, making t hose sections of the oil appear lighter in color; regions of the oil which are fl owing vertically appear darker in color. As the radius of the center-block in the a nnulus is increased, the system becomes more stable to buoyancy-driven convection, meaning th at a higher vertical temperature difference across the system is needed for c onvection to begin; this makes sens e because as the radius of the center-block is further and further increased, ther e is more and more wall-surface area (and more of its accompanying no-slip effect) compared to th e amount of fluid in the system. Also, as the

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40 radius of the center-block in the annulus is ch anged, the number of azimuthally aligned cells changes. Generally, a greater number of azimu thally aligned cells fo rm at the onset of convection as the radius of the center-block ap proaches the outer wall of the annulus, and a smaller number of azimuthally ali gned cells form at the onset of convection as the radius of the center-block is decreased. This research has shown that once the radius of the center-block decreases past a certain value, new rolls of convec ting fluid at the onset fo rm in radial alignment (meaning that they span the entire ty of the azimuthal direction and that they are concentric to the center-block of the annular container) in addition to the orig inal azimuthal arrangement. An example of this, generated by the computations done for this rese arch, is shown in Figure 1-12. In this flow profile, the velocity is scaled, negative velocities represent downward flow in the zdirection (which extends into and out of the plane of the diagram), and positive velocities represent upward flow in the z-direction. This cr oss-sectional velocity pr ofile represents a crosssectional region approximately half of the distance along the vertical depth of the system that it is computed for. If the radius of the center-block of the annular system is further and further decreased, the radial alignment of the convective cells become s more prominent compared to the azimuthal alignment of the cells, and the fl ow patterns become progressively more like those seen at the onset of convection in open, cylindrical systems. This transition in onset flow patterns was of great interest in this research. 1.5 Application Aside from its applications to the regular appearances of Rayleigh convection in nature, this study finds application in the drying of paint films, in small-scale fluidics, and notably, in semi-conductor crystal growth. The vertical Brid gman crystal growth method is commonly used

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41 for growing semi-conductor crystals such as lead -tin-telluride. Vertic al Bridgman growth involves a liquid semi-conductor melt layer, which lies atop the solid crysta l phase being formed by the solidification of the melt. The arrangement is heated from above so that the crystal at the bottom of the system grows upward (see Figure 1-13) Since solidification is occurring at the liquid-solid interface, the temperatur e at that location is constant. Of course, a system that is heated from above, such as this crystal growth system, will not be destabilized by the vertical temperature gradient in the system. Still, buoyancy-driven convection is possible in this system for anothe r reason. As solidificatio n occurs at the solidliquid interface, some portion of th e species that comprise the liquid melt is rejected into the liquid phase, near the interface. The species comp rising a liquid melt, such as lead-tin-telluride, are not all equal in weight. If the species rejected the most into the liquid near the interface is relatively light in weight, and makes the liquid region near the interface less dense than the remainder of the liquid melt phase above the interf acial region, then a top-heavy system results. Thus, due to solutal gradients, buoyancy forces can cause convection in the crystal growth system just like the systems previ ously described in this chapter. Convection in this system can cause impurities from the ampoule to be transmitted to the solid-liquid interface and this can cause flaws on the face of the forming crystal. Also, convection in the system can cau se the heat transfer at the solidification interface to be nonuniform. For these reasons, it is desired to know the conditions corres ponding to the onset of convection in this system so that it can be better understood, pred icted, controlled, and prevented. It should be mentioned that, in some cases, convection in a crysta l growth system may not be a problem, and may even be desirable. If a crystal is grow n at relatively low temperatures

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42 (unlike, for example, the high value of 1250 C at which gallium arseni de is grown), then convection helps transfer heat near the solidifica tion interface and relieves thermal stresses there, preventing the growing crystal from fracturing. Also, the transpor t of impurities from the walls of the crystal-growth container to the solidification interface does not tend to become a problem in these lower-temperature systems (in high-temper ature crystal growth systems, it can relate to constituents being released from the crysta l growth container by the degradation of the container). Thus, in some lowtemperature crystal growth system s, convection could clearly be beneficial. In vertical Bridgman crystal growth, the depth of the liquid ph ase is continuously changing as the melt solidifies in to the growing crystal, consuming a portion of the melt. As the phase depth changes during crysta l growth, so does the possible convective flow behavior. Thus, being able to predict the convective behavior of th e system at differing fluid depths is important. Some crystal growth processe s, such as the liquid-encap sulated growth of gallium arsenide, require very high temperatures. In this research, convective behavi or will be studied at more moderate temperatures, as the physics drivi ng the convection are the same regardless of the temperature of the system. Typically, crystal growth is c onducted using cylindrical fluid la yers. It is interesting to ask how the convective behavior of a crystal growth system would differ if annular fluid layers were employed rather than cylindrical layers. One might imagine that the presence of an extra solid wall at the center of the fluid layer woul d, by means of the no-sl ip condition it imposes on velocities at that location, impart to the crystal growth system a greater stability to convection. The additional center wall in an annular syst em would allow the system to suppress more disturbances than a cylindrical system. Furthermore, if the cr ystal were grown in an annular

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43 container with a sufficiently large annular center-p iece (relative to the outer annular radius), then any convective patterns which did arise would be very predictable and uniform sets of only azimuthally aligned cells, which could be easier to work with than the patterns that arise in cylindrical containers. Figure 1-1. Simplified system diagram: heated from below. Figure 1-2. Rayleigh convection. Cooler Fluid Warmer Fluid Disturbance z = 0 z = Lz Fluid Layer z Higher Density Thot Tcold x Lower Density g

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44 Figure 1-3. Marangoni convection. Figure 1-4. General diagram: critical Ra yleigh number vs. dist urbance wavelength. Racrit Wavelength Phase 2 Phase 1

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45 Figure 1-5. Cross-sectional ve locity profiles of flow pattern s with different numbers of convective rolls. Figure 1-6. General stability diagram for buoyancydriven convection in a rectangular, laterally bounded system, with periodic boundary condi tions at lateral walls, for varying aspect ratio. Racrit Axz I II III Two Convective Rolls Four Convective Rolls x z x z

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46 Figure 1-7. Cross-sectional velocity profiles of four-roll flow patterns with non-uniform roll size. Figure 1-8. General stability diagram for buoyancydriven convection in a rectangular, laterally bounded system, with non-periodic boundary conditions at lateral walls, for varying aspect ratio. Racrit Axz 1708 I II III Four Convective Rolls x z

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47 Figure 1-9. Diagrams of m = 0, 1, 2, 3 flow patterns: “U”: upward flow, “D”: downward flow. U D D U U U D U U U D

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48 Critical Ra vs. Aspect Ratio, Cylinder1700 1900 2100 2300 2500 2700 2900 0.71.21.72.22.73.2 Aspect RatioCritical Ra m = 0 m = 1 m = 2 m = 3 Figure 1-10. General stability diagram fo r buoyancy-driven convection in a cylindrical container, for varying aspect ratio. Figure 1-11. Convective flow pattern in an annulus (Sto rk & Mller 1974).

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49 Figure 1-12. Example: computed flow prof ile for annular system, cross-sectional view. Figure 1-13. Diagram of crystal growth system. Solid Crystal Liquid Melt H ot Cold g g

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50 CHAPTER 2 LITERATURE REVIEW Rayleigh convection in a single fluid layer has been a well -studied problem in fluid mechanics for many years, and it has branched out into many newer problems over the years. Systems with multiple fluid phases, in whic h Marangoni convection exists in addition to Rayleigh convection, have been stud ied, as well. Also, researchers have examined differences in convective behavior seen in systems of differing sh apes and sizes. It would be quite difficult to give a complete history of tremendous volume of research that has been done in this general field. In this section, a brief summary of some of the past research that may be considered most relevant to this study will be given. In Bnard’s classic experiments (1900), cel lular, hexagonal flow pa tterns in vertically thin fluid layers (about .5 mm to 1 mm deep), heated from below, and open to air at the top surface were found (Bnard 1900). Inspired by thes e experiments, about fifteen years later, Rayleigh investigated the instability that arises in a fluid layer heated from below (Rayleigh 1916). Originally, these researcher s attributed this flow instab ility to buoyancy forces – the phenomenon which would be termed Rayleigh convec tion. However, it turned out that the cells originally observed by Bnard were caused mo re by surface-tension gradients than buoyancy differences. Also, it was found that the veloc ity and heat transfer boundary conditions on the fluid layer, as well as the size and shape of th e system, had quite a significant impact on the critical conditions for the onset of convection. Th e early works of Bnard and Rayleigh inspired many more researchers to investigate th e convective behaviors of fluid layers. 2.1 Single Fluid Layers Single fluid layers can be st udied in the presence of th e Marangoni effect or in its absence. The former is more relevant to the co mputations and experiments in this thesis. Some

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51 of the earlier works in this area were that of Schmidt and Milverton in 1935 (Schmidt & Milverton 1935) and that of S ilveston (Silveston 1958) in 1958. Both of these works involved experiments in which a single laye r of liquid was heated from be low and the critical vertical temperature difference for the onset of convec tion was determined. Schmidt and Milverton determined the critical vertical temperature difference corresponding to the onset of convection by measuring the heat transfer th rough the liquid layer, while incr easing the vertical temperature difference, and then noting vertical temperature difference at which the heat transfer increased to a value higher than that of the initial state of pure conduction; th is increase indicated the added heat transfer which accompanied the onset of c onvection. Silveston determined the critical vertical temperature difference in experiments not only by heat transfer measurements, but also by optical observations. In 1961, in his book, Hydrodynamic and Hydromagnetic Stability, Chandrasekhar made a thorough mathematical analysis of the instability that arises solely due to buoyancy in a fluid layer heated from below (Chandrasekhar 1961). In 1967, Davis published a well-known comput ational study of the Rayleigh convection that can occur in a bounded, three-dimensional, rectangular box (Davis 1967). He used a Galerkin numerical scheme to model the behavior of the system, and determine the effects of lateral walls on the convective beha vior. He also observed the wi dely known effect of greatly increasing the horizontal dimensions of the contai ner, which is to cause the critical Rayleigh number to rapidly decrease to 1708. Lastly, as Chandrasekha r had done in his previously mentioned 1961 book, Davis explained that moderate ly-sized convective cells are preferred at onset to tall, narrow cells or wide, flat cells; this is because tall, narrow cells dissipate too large an amount of energy, and wide, flat cells require fluid particles to travel too great a horizontal distance before they can fall and release their po tential energy. This sort of reasoning was

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52 introduced in Chapter 1. In a 1970 paper (Cha rlson & Sani 1970) a nd an extension of it published in 1971 (Charlson & Sani 1971), Charls on and Sani did a mathematical study to determine the critical temperature difference for the onset of axisymmetric (1970) and nonaxisymmetric (1971) flow patterns in cylindrical fluid layers heated from below. Their study included quite a wide range of as pect ratios (ratio of container ra dius to container height) and addressed the cases of both insulating and conducting side walls. In 1972, Stork and Mller published an experimental study in which they studied a system quite similar to the one mathematically analyzed by Davis in 1967 (Stork & Mller 1972). They varied the lateral dimensions of their rectangular system and found their critical temperature difference results to be generally below those of Davis. They attr ibuted this discrepancy between experiment and theory to the effect of imperfect experimental control of the heat transfer boundary conditions at the lateral walls. In Koschmieder’s 1974 experi mental study, the heat flux through an oil layer before and after the onset of convection was m onitored; he was, however, unable to get an accurate measure of the heat tran sfer through the oil layer for cases with low critical, vertical temperature differences (Koschmieder 1974). Like this research, the 1974 experimental study of Stork and Mller addressed Rayleigh convection in annular systems. As in their 1972 work, they examined a fluid layer of 10 mm vertical depth, comprised of silicone oil mi xed with aluminum tracer powder, with the temperature at the top and bottom of the layer ca refully controlled. In some experiments, they considered simply a system of cylindrical cro ss-section, while in others, they added a centerpiece to the cylinder to create an an nular system. In all of their experiments, they detected the onset of convection by visual obser vation of the fluid layer and then obtained the critical vertical temperature difference by reading appropriately placed thermocouples. For the experiments on

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53 cylinders, onset conditions were determined for different aspect ratios; for the experiments on annuli, the onset conditions were determined fo r several different annu lar gap widths. As mentioned earlier, their experime nts on annuli resulted in flow pa tterns that looked rather like spokes of a wheel, with several rolls of convecting fluid (alway s an even number of them) aligned in the azimuthal direction. As the annular gap width was increased (meaning, for example, the center-piece of the annulus was ma de to be of smaller diameter), some quite interesting flow behavior was observed; for some large enough gap widths, there were more flow cells along the outer wall of the annulus than the inner wall. One important difference between the research in this thesis and the work of Stor k and Mller is that this research focuses on the transition in the types of onset flow patterns that are observed when the annular gap width becomes sufficiently large (see Section 1.4), wher eas this transition seemed to be more of a secondary observation in the work of Stork and Mller. In this resear ch, annular systems of certain dimensions were designe d specifically for investigating this transition in onset flow patterns. Sets of annular dimensions appropriate for this investigation were able to be chosen based on computations for the critical conditions run for annular systems. Another important difference between this research and that of Stor k and Mller is that this research includes a computation to accompany the experiments, and their work did not. In 1990, Hardin, Sani, Henry, and Roux published a computational study in which they determined the conditions at the onset of Raylei gh convection for cylindric al systems of several aspect ratios (Hardin, Sani, Henry, & Roux 1990). Also in 1990, Littlefield and Desai published a theoretical study in which they found the critical conditions for Rayleigh convection for an annular system with conducting side walls and with top and bottom walls that were assumed to be flat and free (Littlefield & Desai 1990). This

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54 research differs from the work of Littlefield and Desai in multiple ways. In this research, the computations address the more realistic case, with regard to a single-flui d-layer system, in which there is a no-slip condition on veloci ty at the walls; additionally, the radial walls in this research were considered as insulating rather than conducting, and the variati on of viscosity with temperature was accounted for. Another important difference of this research from the work of Littlefield and Desai is that it includes experiments which may be compared with the computations. The calculations of Littlefield and Desai matched qualitatively well with the experimental study of annular systems done by Stor k and Mller. Like the experimental study of Stork and Mller, their work showed that as the ratio of the inner radius to the outer radius becomes larger (meaning that the annular gap becomes narrower), more and more convective cells form in the azimuthal direction at onset (larger azimuthal wave num ber at onset). They reached the conclusion that as the ratio of the inner radius to the outer radius (the “radius ratio” which was earlier named S for this research) becomes larger the critical Rayleigh number (and critical vertical temperature diffe rence) approaches the value corr esponding to a vertical channel; they also determined that as the radius rati o becomes smaller, the critical Rayleigh number approaches the value corresponding to a vertical cylinder. Littlef ield and Desai noted that the convective patterns which develop in the annular system have a higher tendency to include new cells in the azimuthal direction than in the radi al direction. This can be explained, they say, by the physical argument that new cells which form in the azimuthal direction are more uniform in size (compared to one another and to cells alrea dy present) than new cells which form in the radial direction, and thus they form more eas ily because of the lowe r velocities and viscous stresses that they involve.

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55 At this point, to be clear, a list of the wa ys in which this research differs from the previous works on Rayleigh convection in si ngle-fluid-layer annular systems is given. This research includes corresponding co mputational and experimental work. In this research, dimensions of annular systems investigated computationally and experimentally were chosen particularly to focus on the transition in the types of onset flow patterns which occurs when the ratio of the inner annular radius to the outer annular radius is sufficiently small, and to determine what value for this ratio corresponds to the transition (when this ratio is small enough, the onset flow patterns can have one or more convective ro lls in the radial direction, while they generally have no convective roll s in the radial direction when this ratio is larger). Computations for this research include the variation of viscosity with temperature. Computations for this research address the realistic case in which there is a noslip condition on velocity at the walls. Computations for this research are for the case in which the radial walls are insulating. Another study of Rayleigh convection in an annular system was presented in 1990 by Ciliberto, Bagnoli, and Caponeri (Ciliberto, Bagnoli, & Caponeri 1990). Their study was experimental and it involved observing the convective behavior of an annular layer of silicone oil by shadowgraph. The flow patterns observed in their study consist of an azimuthally aligned series of cells, and are cons istent with the patterns seen by other researchers. In 1995, Zhao, Moates, and Narayanan pub lished an interesting study of Rayleigh convection in cylindrical systems (Zhao, Moat es, & Narayanan 1995). The study included both theoretical and experimental parts, which agr eed well with one another. The experimental apparatus used was similar to the one being used in this research. The bottom temperature of a layer of silicone oil was controlled by circulating hot water be low an aluminum plate, while the top temperature was controlled by circulating co oler water above a sapphire window. Flow was visualized by means of aluminum powder which was mixed in with the silicone oil. It is noteworthy that this study addr essed the changes in convective behavior which arise when one accounts for the variation of viscosity with temp erature rather than simply assuming it to be

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56 constant. It was seen that, depending on the aspect ratio of the system, th e variation of viscosity with temperature could have a si gnificant impact on not only the critical vertical temperature difference for the onset of convection, but also on the flow pattern at onset. This can be seen more later in this paper. The f act that the computations in this research account for the variation of viscosity with temperature when determining the critical conditions for convection is another noteworthy feature of this res earch, which may set it apart from previous works on annular systems. 2.2 Multiple Fluid Layers As mentioned, the studies on single fluid laye rs, for the fact that they exclude the Marangoni effect, are rather more relevant to this research. Still, some works on multiple fluid layers can be helpful in understanding what goes on in problems involving buoyancy-driven convection in single fluid layers, too. Buoyanc y-driven convection, afte r all, is generally occurring in problems with multiple fluid layers as well, even though the systems may be more complicated. Thus, a brief summary of it will no w be given, and so will be presented a more complete picture of what sorts of research have been done on convective phenomena over the years. Following experimental evidence obtained by Block in 1956 (Block 1956), Pearson, in 1958, produced a paper which proposed a mechanism to attribute convective flow of the type observed by Bnard to surface tension gradients rather than buoyancy forces (Pearson 1958). Pearson’s study focussed on a fluid layer, heated from below, laterally unbounded, with a rigid wall as its bottom surface, and a fr ee surface at the top. A passive gas was the upper phase in the study. In that study, the temperat ure of the bottom wall was held fixed, while the temperature at the upper surface was governed by a heat transf er boundary condition. At the top and bottom

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57 surfaces of the fluid layer, Pearson consid ered the effects of having both conducting and insulating behaviors with regard to heat transfer In Pearson’s paper, a dimensionless number called “B” showed up as a preliminary form of what would, in later years, be called the Marangoni number. The Marangoni number is surf ace tension-driven convection’s analog to the Rayleigh number of buoyancy-driven convection. Pearson’s analysis assumed that the upper free surface could not deflect. In 1959, a paper by Sternling and Scriven presented physical mechanisms and a corresponding simple mathematic al model to explain how several types of interfacial flows may develop in bilayer systems (Sternling & Scriven 1959). A 1964 paper by the same two authors extended Pearson’s work by considering the same system but allowing deflection of the free surface (Scriven & Stern ling 1964). Like Pearson’s work, the work of Sternling and Scriven involved solv ing for the behavior of the system in response to a small, imposed disturbance. None of the works, that were just described, by Pe arson, or Sternling and Scriven included an experimental component. The 1967 work of Koschmieder, however, was an experimental study of the convective behavior of a cylindrical layer of silicone oil, heated from below by a solid boundary, and cooled from above by a layer of air (Koschmied er 1967) In that study, the flow was visualized using aluminum powd er, and it was found that convective flow in the system initially took the form of concentric circular rolls, a nd subsequently transformed into a hexagonal pattern. Koschmieder was able to make an accurate determination of the wavelengths of the flows, as well as their de pendence on the dimensions of the fluid layer. Many studies involving multiple fluid layers were performed on systems comprised of one vapor layer atop one fluid layer. It is noteworthy th at in 1972, Zeren and Reynolds published a study in which they examined Rayl eigh and Marangoni convec tion in a liquid-liquid system (Zeren & Reynolds 1972). The liquids cons idered were benzene and water. Their study

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58 included both theoretical and experimental parts. They felt that the presence of contaminants in the interfacial region could have affected their experimental resu lts significantly. The work of Ferm and Wollkind, published in 1982, improved a nd extended the theoreti cal analysis given by Zeren and Reynolds (Ferm & Wollkind 1982). While it seems that most of the work done with annular systems has involved only single fluid layers, Bensimon, in 1988, published a study in which he experimentally examined the convective behavior in an annul ar layer of liquid with free bo undary conditions at the top and bottom surfaces (Bensimon 1988). He arranged this system by placing a layer of silicone oil on top of a layer of mercury and leav ing an air layer above the silicone oil. Flow visualization was accomplished in his study by the shadowgraph te chnique. In the shadowgraph images he presents in the paper, the flow patterns look quite similar to those that are seen in single-fluidlayer, annular systems. Returning to the works of Koschmieder, an entirely experimental work was published by Koschmieder and Prahl in 1990 (Koschmieder & Pr ahl 1990). It investigated the tendency of wide fluid layers, when heated from below and ope n to air at the top surface, to convect in a pattern of hexagonal cells. They examined silic one oil-air bilayers in small containers of differing shapes to find out what effects the differing container shapes had on the convective patterns that would form. They observed that as the width-to-height ratio for the containers increased, more and more convective cells would fo rm at the onset in order to fill the larger width. While determining the critical conditions by observation of the ve rtical heat transfer through the system is a nice met hod in terms of objectivity, Koschm ieder and Prahl point out that in their experiments with small fluid layers, the voltage created by the heat sensor was too small in comparison with outside electrical noise to be useful. Thus, the more desirable method (which

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59 is the method used in the current research) was to optically observe the fluid layers and note the first appearance of fluid motion as the critical point. A 1992 experimental work by Koschmieder and Switzer examined surface-tension driven convection using shadowgraphy (Koschmieder & Switzer 1992). In a 1995 paper by Zhao, Wagner, Narayana n, and Friedrich, a theoretical study is presented that addresses Rayleigh and Marangon i convection in fluid bilayer systems – both liquid-liquid, and liquid-vapor (Z hao, Wagner, Narayanan, & Friedr ich 1995). A wide range of cases are considered, including heating from belo w, heating from above, and the case in which solidification is occurring at the bottom surface of a liquid-liquid bi layer; the last case mentioned has strong similarity to crystal growth because of the solidification which it accounts for. The 1996 theoretical study of Dauby and Lebon addresse d the Rayleigh and Ma rangoni effects in a liquid-vapor bilayer system and included both linear and nonlinear analyses (Dauby & Lebon 1996). The linear analysis allows one to determine the critical vertical te mperature difference at which convection begins, as well as the flow pattern at the on set. The nonlinear analysis provides more specific information on the flow beha vior and can be used to predict supercritical behaviors. Most theoretical st udies, up to this point, had a ssumed laterally unbounded systems. That assumption allows separation of variables, an d thus an easier solution to the system of differential equations that model the problem. Of course, in actual experiments and other applications of these phenomena, lateral walls ar e present, and these walls can have important effects on the critical conditions an d onset flow patterns. Thus, a nother point of interest in the study by Dauby and Lebon is that they formed their mathematical model to include no-slip walls at the lateral boundaries. This is the most realistic boundary conditi on that could be enforced at those locations. Dauby and Lebon found their theo retical results to be in good qualitative

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60 agreement with the experimental results of Ko schmieder and Prahl. In 1997 by Kats-Demyanets, Oron, and Nepomnyashchy published a study, which ex amines convective behavior in tri-layer fluid systems, and thus is also quite applicable to the science of crystal growth (Kats-Demyanets, Oron, & Nepomnyashchy 1997). Johnson and Naraya nan published a tutorial in 1999 which explained five different mechanisms by which conv ection in two vertically stacked fluid layers could couple (Johnson & Narayanan 1999). Their tutori al also discussed the situation in which, at certain aspect ratios, due to the effects of side walls and the dimensions of the system, a convecting system may find two flow patterns equa lly favorable from an energy standpoint; as a result, the system would oscillate between the two patterns. A point such as this is called a codimension-two point.

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61 CHAPTER 3 MODELING EQUATIONS To predict the onset conditions of convecti on requires a model that respects the physics of the problem. Such a model would utilize mo mentum, mass, and energy equations. The inputs to the model would be the vertical phase dept h and thermophysical properties of the fluid, and the outputs would be the critic al temperature difference needed for convection, as well as the associated pattern of the flow. The details of su ch a model are given in this chapter, while later in this thesis, an explanation of the numerical scheme used to solve the system of modeling equations is given. In Appendix C, the modeli ng equations are presented and developed more fully for the simple case in which the fluid is assu med to have a viscosity constant with respect to temperature. The modeling equations presen ted and developed thr oughout this chapter and Chapter 4, however, are for a more physically accurate mathematical model, in which the variation of viscosity with temperature is included. To get started the nonlinear equa tions that govern flow in a c onvecting layer of fluid shall be introduced. These equations are nonlinear pr imarily on account of the dependence of velocity on temperature, coupled with the effects of the interactions between velocity and temperature fields on heat transport. 3.1 Nonlinear Equations The domain equations used to model the c onvective behavior of the system are the momentum equation, the energy eq uation, and the continuity e quation, which, respectively, are S g P v v t v (3.1) T k T v C t T CV V 2 (3.2)

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62 ) (v t (3.3) In these equations, is the density of the fluid, t is time, v is the velocity of the fluid, P is the pressure of the fluid, g is gravity, S is the stress tensor for the fluid, CV is the constantvolume heat capacity of the fluid, T is the temperature of the fluid, and k is the thermal conductivity of the fluid. The last term of the momentum equation has not been further simplified, at this point, because it will be needed in this form in or der to properly account fo r the dependence of the fluid’s viscosity on temperat ure. The stress tensor, S can be expanded as 2 ) ( 2v v S (3.4) The use of this expansion, however, will be postponed for now. In this expansion, is the dynamic viscosity of the fluid and represents a transposed matrix. Since the velocity gradients in this system will not be very large, the energy equation has been simplified by neglecting the viscous dissipation. The system of equations is analyzed in cylindrical coordinates, which are clearly th e natural choice for this problem; thus “r” denotes the radial direction, “ ” denotes the azimuthal direction, and “z” denotes the vertical direction. The system may be treated as periodic in the -direction (this is accomplished mathematically by the expansion of the variables into modes) with the oscillations of the form ime, in which m is the azimuthal wave number. Another approximation is to be made, and it a ffects the momentum equation. It is called the Boussinesq approximation and it addresses the va riation of density with temperature. This approximation says that variation of density with temperature is negligible insofar as the change

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63 in momentum or mass is concerned, but that it do es affect the acceleration in the system insofar as the body forces are concerned. The reasoning for introducing the Boussinesq approximation, and some notes on its applicability are e xplained in Appendix B as Section B.1. Once the Boussinesq approximation is applied to Equation 3.1, it becomes S g T T P v v t vR R R R )) ( 1 ( (3.5) In Equation 3.5, TR and R are a reference temperature and the fluid’s density at the reference temperature, respectively, and is the volumetric thermal expansion coefficient of the fluid phase. Note, also, that the Boussinesq ap proximation results in the disappearance of the time-derivative term of the continuity equation. At this point, multiple nonlinearities appear in the system of equations; a note about these nonlineari ties is given in the appendix as Section B.2. An additional modification to the momentum equation is to define a modified pressure, which is g P pR (3.6) Using this, the momentum equation may be written as S g T T p v v t vR R R R ) ( (3.7) Now, the expansion of S shall be introduced to the mome ntum equation. The result of doing so is ) ) ( ( ) ( ) (2 v v v g T T p v v t vR R R R (3.8) As mentioned, a goal of this calculation is to include the effect s of the variation of viscosity with temperature. Thus, the vi scosity at any locatio n can be written as

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64 )] ( [R RT T f (3.9) where f is some function of the difference betwee n the temperature at that location and the reference temperature. In this equation, R is the dynamic viscosity at the reference temperature. Note that, since has the same dimensions as R f must be dimensionless. Using this expansion, Equation 3.8 may be rewritten as ) ) ( ( ) ( ) (2 v v f v f g T T p v v t vR R R R R R (3.10) The assumed form for viscosity’s temperatur e dependence will be exponential, like the form assumed in the work of Zhao, Moates, an d Narayanan (Zhao, Moates, & Narayanan 1995). The form of this exponentia l temperature dependence is ) () (RT T B Re T T f (3.11) in which B is a constant that can be determined us ing measurements of the fluid’s viscosity obtained at a range of different temperatures. For temperatur es in degrees Celsius, the dimensions of B would be C1 In this system of three domain equations, it appears that five unknowns are present. These are the velocities in the r, and z directions, the pressure and the temperature. The continuity equation can eliminat e pressure by representing it in terms of velocities, and this reduces the number of unknowns to four. To solve for the convec tive behavior of a cylindrical system, constraints would be need ed on the velocities in the r, and z directions, as well as on the temperature, at the top wall (z = Lz), the bottom wall (z = 0), a nd the outer radial wall (r = Ro). Considering an annular system, however, four additional mathema tical constraints are

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65 required; these constraints on the three components of velocity and temperature are applied at the inner radial wall of the system (r = Ri). Figure 3-1 shows the geom etry being considered. The symbol Lr is used to denote the radius in a cylindr ical system, and also to denote the difference between Ro and Ri (Lr = Ro – Ri) for an annular system. This means that for an annular system, Lr is the width of the annular ga p that is filled with fluid. Thus, for the annular system, a total of sixt een constraints are needed, while only twelve are needed for the cylindrical system. These constraints are the boundary conditions. In their unscaled form, they are 0 z rv v v at z = 0, (3.12) bT T at z = 0, (3.13) 0 z rv v v at z = Lz, (3.14) tT T at z = Lz, (3.15) 0 z rv v v at r = Ro, (3.16) 0 r T at r = Ro, (3.17) and, when considering an annular system, the conditions at the inner annular wall, 0 z rv v v at r = Ri, (3.18) 0 r T at r = Ri (3.19) are needed. In these equations, vr, v, and vz are the components of velocity in the r, and z directions, respectively. The boundary conditions on velocity enforce no-slip behavior at the

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66 walls, and state that no flow may pass through the walls. The conditi ons on temperature at bottom and top walls keep the temperature fi xed at those locations; in these equations, Tb and Tt are the constant temperatures at the bottom wa ll and top wall, respectively. The conditions on temperature at the radial walls represent the insulating behavior at those boundaries. 3.2 Scaling Now that the complete set of domain equatio ns and boundary conditions has been listed, these equations will be made dimensionless. In scaling these equations, one has the option to choose several combinations of parameters to use in defining a char acteristic velocity ( v) and a characteristic time ( t). A brief discussion of how to choose these parameters is in Section B.3 of Appendix B. Lengths are scaled using the ver tical depth of the fluid. W ith that said, a list of the scaling relations used will now be presented. In these equations, the “hat” symbol indicates a dimensionless variable, and the “b ar” indicates a characteristic va lue for a variable. The scaling relations are v v v ˆ (3.20) T T T TR ˆ, (3.21) t bT T T (3.22) p p p ˆ (3.23) z RL v p, (3.24) t t t ˆ (3.25)

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67 zLˆ (3.26) In the above equations, Lz is the vertical depth of the flui d phase. As mentioned earlier, TR is the reference temperature. In simple cas es where the system of equations was solved without considering the dependen ce of viscosity on temperature, TR could be chosen to be the temperature at the top bo undary of the system (Tt), and the reference dynamic viscosity, R would simply be the dynamic viscos ity at that temperature. When the variation of viscosity with temperature is considered, however, TR is selected later, during the numerical solution of the system of equations, to correspond to the mean dynamic viscosity value of the fluid in its motionless base state, when the fl uid layer is at the cr itical vertical temperature difference, just before the onset of convection (the viscosity of the fluid varies from top to bottom along the motionless liquid layer corresponding to the vertical temperature gr adient). When the variation of viscosity with temperature is being considered, this mean dynamic viscosity value is the reference dynamic viscosity, R T is the overall temperature difference across the system. In addition to these scalings, f (Equation 3.11) should be re-expressed in terms of dimensionless temperature. This new version of f, which will be called F, is T T Be T Fˆ ) () ˆ (. (3.27) Applying these scalings to the domain equa tions, the following dimensionless (though the “hat” symbol will now be discarded) e quations are obtained (Equations 3.28-3.30): ) ) ( ( ) ( ) (2 2 2 v v F v F T v T g L p v v v L t v t Lz R z R R z R (3.28) T T v v L t T t Lz z2 2 (3.29)

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68 v 0, (3.30) in which the thermal diffusivity of the fluid, is equal to the fluid’s thermal conductivity divided by the product of its density and heat capacity. Unless otherwise indicated, all thermophys ical properties in these equations are considered at the reference temperature. The su bscript “R” has not been included on the symbols for all of these properties; however, it has been left on the symbols R and R because, in those cases, it arose from the expansions by which th e density and dynamic viscos ity of the fluid were defined. This is the reason, for example, that the subscript “R” is not included on the kinematic viscosity in Equation 3.28, even though that property is simply a ratio of the dynamic viscosity and the density taken at the refere nce temperature. Observe that if a substitution of the ratio of to Lz were made for v, then the coefficient of the temperature term in Equation 3.28 would be the Rayleigh number. Since many po ssible values could be assigned to v, depending on the relative values of system parameters, v has been left in the set of modeling equations. Again, a discussion regarding the definition of v is in Section B.3 of Appendix B. As for the scaling of the boundary conditi ons, the conditions on velocity remain unchanged in appearance, as do the conditions on te mperature at the radial walls. However, the conditions on temperature at the top and bottom walls appear slightly differently depending on whether or not the problem is being solved considering the variati on of viscosity with temperature. The reader is asked to refer to Appendix C for more information on the case in which this variation is not being considered. In cases in which the vari ation of viscosity with temperature is being considered, the dimensionl ess temperatures at th e vertical boundaries are dependent on TR. TR, though, is dependent on the vertical temperature gradient in the fluid, which means that it needs to be determined along with the critical vertical temperature difference

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69 for convection by an iterative approach (described in Chapter 5). Thus, for the non-constant viscosity case, the temperature boundary conditions at the top and bottom walls must be left in a more general form for now. In this general form, they are T T T TR b at z = 0, (3.31) T T T TR t at z = Lz. (3.32) This completes the presentation of the scal ed domain equations and boundary conditions. Next, the equations shall be simplified by linea rization and the removal of time dependence. Figure 3-1. System diagram: cylindrical and annular systems. z r r Fluid z = 0 z = Lz r = Ro r = 0 r = 0 r = Ri r = Ro Cylindrical System Annular System g

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70 CHAPTER 4 LINEARIZED EQUATIONS In this chapter, the process of simplifying the set of modeling equations by linearizing them around a motionless base state, and then elim inating their time dependence shall be shown. Since the goals of the calculations in this rese arch are to determine only the onset conditions for convection (critical vertical temp erature difference and flow pattern at onset) as opposed to the flow behavior at supercritical conditions, a linearized model is sufficient for the analysis. A nonlinear calculation would be nece ssary if the supercritical behavi or of the system were to be determined. Time dependence is being removed from the equations because a system is independent of time at critical conditions (also kn own as marginal stabilit y). Linearization and the removal of time dependence are accomplished by introducing a series perturbation expansion to the variables, then applying a further expans ion to the perturbed variables (expansion into modes), and then setting the inverse time constant in the expansion equal to zero. As mentioned earlier, the same process is presented for the le ss complex case in which the fluid is assumed to have a constant viscosity with respect to temperature is included in Appendix C. 4.1 Linearization The system of equations shall be lineari zed around the motionless base state of the system. In this motionless initial state, heat is transferred within the system only in the vertical direction and only by conduction. The form of the expansions wh ich shall be used in this li nearization, considering, for example, the expansion of velocity, is

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71 ... | 3 1 | 2 1 | |0 3 3 3 0 2 2 2 0 0 v v v v v (4.1) Replacing the expressions in brack ets, this may be rewritten as ... 3 1 2 13 3 2 2 1 0 v v v v v (4.2) The subscript “0” denotes values pertaining to the motionless base state, in which heat is transferred only by conduction, and the temperat ure and pressure gradients are only in the vertical direction. Clearly, then, 0v is equal to zero. This fact allows the cancellation of some terms in the linearized forms of the modeling equa tions. If the system begins to flow due to some perturbation, then the remaining terms in this series (1v ,2v etc.), which are the perturbed variables, represent the flow behavior of the sy stem. The amplitude of the perturbation that transforms the base state into the flowing, “perturbed” state is called ; this perturbation is taken to be extremely small. Considering a small pe rturbation allows a conclusion on the stability of the system to be reached using linearized equa tions. If a system is unstable to a small perturbation, it will, of course, be unstable to larger perturbations. If a system is stable to a small perturbation, then no such conclusion may be dr awn; in this case, the system could simply require a larger disturbance to become unstable. The linearized equations (comprised of terms that are first order in ) are an approximation of the behavior of the system and can be used to determine the conditions correspondi ng to the onset of convection a nd the pattern of flow at the onset. If it is desired to determine the magnitude s of the velocities in the convecting fluid, as

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72 opposed to simply determining their values relative to one another, then a nonlinear analysis is necessary. A disturbance of magnitude is mathematically applied to the system by expanding each variable in the modeling equations with the form shown in Equation 4.2. The set of linearized equations is then obtained by collecting only the te rms which are first order in The resulting linearized domain equations are ) ) ( ( ) ( ) (1 1 0 1 2 0 1 2 1 1 2 v v F v F T v T g L p t v t Lz R z R (4.3) 1 2 1 0 1 2T v z T v L t T t Lzz z (4.4) 10 v (4.5) In these equations, F0 is the same as F except that the exponentia l temperature value is now the base state temperature, T0. F0(T0), then, is 0) ( 0 0) (T T Be T F. (4.6) The linearized forms of the boundary conditions are 01 1 1 z rv v v at z = 0, (4.7) 01 T at z = 0, (4.8) 01 1 1 z rv v v at z = Lz, (4.9) 01 T at z = Lz, (4.10) 01 1 1 z rv v v at r = Ro, (4.11)

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73 01 r T at r = Ro, (4.12) and, when considering an annular system, 01 1 1 z rv v v at r = Ri, (4.13) 01 r T at r = Ri (4.14) hold. Linearization simplifies the boundary conditi ons on temperature at the top and bottom walls because it is the base state values of temperature, T0 at the top wall and bottom wall, which are equal to the fixed values of temperature at those boundaries. Substituting for F0 in the momentum equa tion, and making use of zz T T F T T F F 0 0 0 0 0 0 0, (4.15) in which z is the unit vector in the z-direction, br ings the momentum equation into the form ) ) ( ( ) ) ( ( ) (1 1 0 ) ( 1 2 ) ( 1 2 1 1 20 0 v v z T e T B v e T v T g L p t v t Lz T T B T T B z z R z R (4.16) Note that all factors on the right-hand-s ide of Equation 4.15 are dimensionless since 0F from which they arose, was already made dime nsionless in Chapter 3. For a more detailed and well given explanation of this pert urbation method, refer to the book titled Interfacial Instability by Johns and Narayanan (Johns & Nara yanan 2002). The momentum equation (Equation 4.16) is now rewritten as the three scal ar equations which are its components in the r, and z directions. The operators are expanded in cylindric al coordinates from this point

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74 onward. The scalar component equations compri sing the vector form of the momentum equation are the r-component of th e momentum equation, the -component of the momentum equation, and the z-component of the momentum equation, which, respectively, are ) ) ( ( 2 1 1 11 1 0 ) ( 1 2 1 2 2 2 2 2 2 2 2 ) ( 1 1 20 0 r v z v z T e T B v r v r z r r r r e r p t v t Lz r r rT T B T T B z (4.17) 1 ) ) ( ( 2 1 1 1 11 1 0 ) ( 1 2 1 2 2 2 2 2 2 2 2 ) ( 1 1 20 0 z rv r z v z T e T B v r v r z r r r r e p r t v t LT T B T T B z (4.18)

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75 ) ( 2 ) ) ( ( 1 11 2 1 0 ) ( 1 2 2 2 2 2 2 2 ) ( 1 1 20 0T v T g L z v z T e T B v z r r r r e z p t v t Lz T T B T T B zz z z (4.19) 4.2 Expansion into Normal Modes The set of domain equations now includes Equations 4.4, 4.5, 4.17, 4.18, and 4.19. A second expansion (expansion into modes) will now be applied to the variables, as well. This new expansion separates the r-direc tion and z-direction dependencie s of each variable from the direction and time dependencies. In the example expansion s hown as Equation 4.20, the new variable representing only the r-d irection and z-direction dependenc ies is marked with a “prime” symbol. This expansion assumes a periodic form for the -direction dependencies, which is sensible, since the cylindrical and annular system s being considered are indeed periodic in the -direction. The nature of the periodicity in the -direction is described by an azimuthal wave number, called m. Note that this wave number is dimensionless since it arises from dimensionless variables. The time dependence of each variable is assumed to have an exponential behavior. This exponent ial behavior with respect to time is governed by an inverse time constant, called is dimensionless since it arises from dimensionless equations; if unscaled, its units would be (1/time) Using pressure as an example, the form of the expansion is t ime z r p p ) (1 1. (4.20)

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76 The form of -direction dependency assumed in the expansion can be used because the periodic spatial dependen ce of the system in the -direction can be represented by a series of sines and cosines. When the system of mode ling equations is subjected to a perturbation ( ), the perturbation may either grow, resu lting in the onset of flow, or it may die out if the system is not at critical or supercritica l conditions. The value of is an indicator of the system’s response to a given perturbation. It should be noted that in certain cases, for which the system oscillates between convective flow patterns, the value of is imaginary. It can be mathematically shown, however, that is strictly real for the non-oscillatory cases considered in this research. If a disturbance applied to the system dies out, meaning that the system is stable, then has a negative value. If the system is unstable, a nd will flow when subjected to a perturbation, is positive. At marginal stability, however, when th e system initially becomes unstable, the system is independent of tim e and the value of is 0. The fact that = 0 at marginal stability allows the elimination of all time-derivative terms in the modeling equations on ce this expansion is applied. The final forms of the modeling equa tions (including all th ree components of the momentum equation, the energy eq uation, the continuity equati on, and the boundary conditions), with this expansion applied, and with the “prime” symbols dropped from the newly defined variables, are

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77 ) ) ( ( 2 ) 1 ( 1 01 1 0 ) ( 1 2 1 2 2 2 2 2 2 ) ( 10 0 r v z v z T e T B v r im v r m z r r r e r pz r rT T B T T B (4.21) ) ) ( ( 2 ) 1 ( 1 01 1 0 ) ( 1 2 1 2 2 2 2 2 2 ) ( 10 0 z rv r im z v z T e T B v r im v r m z r r r e p r imT T B T T B (4.22) ) ( 2 ) ) ( ( 1 01 2 1 0 ) ( 1 2 2 2 2 2 2 ) ( 10 0T v T g L z v z T e T B v z r m r r r e z pz T T B T T Bz z (4.23)

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78 1 2 2 2 2 2 2 1 01 0 T r m z r r r v z T v Lz (4.24) z rv z v r im v r r1 1 11 0 (4.25) 01 1 1 z rv v v at z = 0, (4.26) 01T at z = 0, (4.27) 01 1 1 z rv v v at z = Lz, (4.28) 01T at z = Lz, (4.29) 01 1 1 z rv v v at r = Ro, (4.30) 01 r T at r = Ro, (4.31) and, when considering an annular system, 01 1 1 z rv v v at r = Ri, (4.32) 01 r T at r = Ri (4.33) are needed. At this point, two important dimensionle ss groups which show up in the motion equation are F0 (see Equation 4.6) and z T e T BT T B 0 ) (0) (. (4.34) Now, the entire collection of modeling e quations (Equations 4.21 – 4.33) may be simultaneously, numerically solved on a computer as an eigenvalue problem, in which the

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79 eigenvalue is T. Since has been set equal to 0, the problem is being solved at the critical point for the onset of convection, and so the T value to be determined is the critical vertical temperature difference for convection, or [ T]crit. How exactly the solution for the [T]crit is carried out is the topic of the next chapter. Notice that the Prandtl number ( / ) does not appear in any of these equations. This may be surprising since one might e xpect that ratio of thermophysical properties to play a role in determining the critical conditions for convection. Now, recall that in Chapter 1 it was shown that the graph of critical Raylei gh number versus aspect ratio, for different azimuthal wave numbers, was identical for any system, regardless of the system’s thermophysical properties. This interesting fact can be clearly explained at this point. The kinema tic viscosity and thermal diffusivity of the system affect how quickly dist urbances may die out or grow. Thus, before the onset of convection, their values make more or less negative depending on how stable the system is, and after the onset of convection, their values make more or less positive depending on how rapidly a dest abilizing disturbance grew a nd how strongly the system is convecting in response to it. Precisely at th e onset of convection, however, which is the condition at which the modeling eq uations are being solved, the sy stem is independent of time ( = 0), and so the thermophysical properties play no role in determining the critical conditions for convection. If, rather than simply changi ng the thermophysical properties of the fluid, the insulating boundary conditions on temperature at the radial walls were changed to conducting boundary conditions, the graph of the critical Rayl eigh number versus aspect ratio would change (as explained in Chapter 1, the critical vertical temperature differences for convection are lower when insulating radial walls are consider ed rather than conduc ting radial walls).

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80 CHAPTER 5 SPECTRAL SOLUTION METHOD The goal of this chapter is to explain the spectral solution method used for computations in this research. The method is introduced and described in Section 5.1. Section 5.2 focuses on the application of the method. Rearranging the set of mode ling equations to be solved numerically as an eigenvalue problem for T is not too difficult. To do so, one mu st isolate the term of the z-directioncomponent of the momentum Equation 4.23 which includes T1. What is meant by isolating that term is arranging the equation so that that term is on the side of the equation opposite from all of the other terms. Supposing the side of the equation where the T1 term is placed is chosen to be the right-hand-side of the equation, then the rema ining terms of the z-dir ection-component of the momentum equation, and all nonz ero terms of all of the othe r domain equations and boundary conditions should be kept on the le ft-hand-sides of those equations. In general, what is being done here is that the problem is being recast in the form of a generalized eigenvalue problem for T. The form of this generalized eigenvalue problem is X B X A (5.1) Here, A is the matrix containi ng the coefficients of th e velocity components, temperature, and pressure from the left-h and-sides of the domain equations and boundary conditions. B as one would imagine, is the matrix of the coefficients of the velocity components, temperature, and pressure from the right-hand-sides of the set of modeling equations. X is the eigenvector; it is a column-vector including all three velocity components, temperature, and pressure. From this point onward, the subscript “1”’s used to indicate perturbed variables will be discarded, as nearly all vari ables referred to will be perturbed

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81 variables. The only exceptions to this would be the base state variables, which, for this reason, will still be denoted by a subscript “0 ” in all locations. The form of X is P T v v v Xz r. (5.2) of course, is the eigenvalue, which is T. Arranging the system of modeling equations in this form is straightforward when the variation of viscosity with temperature is not being considered. For the non-c onstant viscosity case, however, T is present in more locations than just the coefficient of T1 in the z-direction-component of the momentum equation; it is also present as an exponent in several terms. These terms, though, may be kept on the left-hand-side of the modeling equations rather than being placed on the right-hand-side of Equation 5.1. If the equations are arranged in this way, T can be determined by an iterative approach. In this iterative approach, a value must first be chosen for Tt, the temperature at the top of the fluid layer. In all experiments done for this research, this temperature was kept constant at 30.0 C, so that value was always substituted for Tt in the calculations. Next, a guess-value, which will be called [ T]guess, is substituted for T on the left-hand-side of Equation 5.1. Then, the eigenvalue ( T) on the right-hand-side of Equation 5.1 is solved for, and then its value is used to update the [ T]guess value on the left-hand-side of Equati on 5.1, and so on. The procedure is repeated until convergence, wh ich occurs quickly. Since TR and R are dependent on the vertical temperature gradient prior to the onset of convection, their values are updated as [ T]guess is updated. Note that updating the [ T]guess value on the left-hand-side of Equation 5.1 represents updating the value of viscosity, through its exponen tial temperature dependence.

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82 Also, note that in the case where the variation of viscosity with temperature is not accounted for, the viscosity value at 35.0 C, which is given in Appendix A, is used throughout the calculation. 5.1 Explanation of the Method Once the system of modeling e quations has been written in th e form of Equation 5.1, it can be used to numerically approximate the critical vertical temp erature difference for convection, and the onset flow pattern. As is typi cal of numerical solution methods, the first step is to consider discretized vers ions of the modeling equations, wh ich describe the behavior at a collection of individual points in the system as an approximation of the full behavior of the system. This means that each variable is written as a vector containing the values of that variable at a set of points within the system. For example, a cylindrical system co uld be discretized into sets of points (or “nodes”) in the r-direction and z-direction as s hown in Figure 5-1. Of course, the subscripted numbers here have nothing to do with the subscripted numbers that appeared during linearization in Chapter 4; they simply indicate the spatial ar rangement of the nodes. In this case, the cylinder was discretized into rows of four nodes in the r-direction, and columns of four nodes in the z-di rection. Clearly, these sets of nodes in the r-direction and zdirection form a grid of nodes in which the loca tion of any node can be specified with an index for the r-direction and an index fo r the z-direction. For this re ason, the discretization nodes are also referred to as “grid points.” Either directio n could have been discreti zed into any number of nodes. Of course, considering a larger number of nodes leads to a more accurate discretized model. The numbers of nodes created in the r-d irection and the z-direc tion are decided by the selection of parameters called Nr and Nz, respectively. Fo r the example above, Nr = 3 and Nz = 3. Note that discretizing the r-direc tion and z-direction means that r and z are written as vectors

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83 now rather than scalars. In general, the sets of nodes in the r-dire ction and z-direction, respectively, can be written as rNr r r r r 2 1 0, (5.3) zNz z z z z 2 1 0. (5.4) Due to the inclusion of r0 and z0, the numbers of nodes in th e r-direction a nd z-direction are actually (Nr + 1) and (Nz + 1), respectively. It should be me ntioned that, in this research, the treatment of the -direction was much simpler than the discretization applied to the r-direction and z-direction, due to the periodic form assumed for -direction dependencies; the treatment of the -direction will be discussed later. For now, the reader is free to imagine that the discussion here pertains to simply systems axisymmetric in the -direction, even though, in actuality, the treatment of the -direction is such that everything pres ented here applies to systems of any azimuthal periodicity. As indicated by Figure 5-1, when consid ering a system in two or more spatial dimensions, the indices of the nodes in the r-d irection and z-direction may be thought of as indices corresponding to different planes slicin g through the system, normal to the r-direction

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84 and the z-direction, respectively. Thus, discretizing the r-direction into (Nr + 1) nodes the and zdirection into (Nz + 1) nodes divides the system into a grid of nodes that includes a total of ((Nr + 1) (Nz + 1)) nodes. Again, note that th e computational treatment of the -direction has not yet been addressed; this means that the discretized fo rm of the system being discussed at the moment is only one plane in the -direction. Given this form of di scretization, every variable in the system must be written as a vector of its scalar values at every node of the discretized system. Temperature, for example, would be written in disc rete form as shown in Equation 5.5. To allow a shortened, more clear representation on paper, Equation 5.5 will show T in the transposed arrangement; note that it is ac tually a column-vector like r and z The transposed form of the discretized temperature is ] , , , , , , [, , , , ,1 0 1 1 1 1 0 0 0 1 0 0 z N r N z N z N r N r Nz r z r z r z r z r z r z r z r z rT T T T T T T T T T (5.5) The raised ellipsis in Equation 5.5 indicates the continuation of this row-vector on the next line. The discretization node s, although they may a ppear to be so in Figure 5-1, are not necessarily evenly distribut ed in the system. In fact, it can be quite beneficial to use nodes which are not evenly spaced. The sets of nodes used in this research are not evenly spaced; they are called the Gauss-Chebyshev-Lobatto grid points and the Gauss-Radau grid points (Trefethen 2000). Both sets of grid points are more highly clustered near the boundaries of the system than near the center of the system. It is highly benefici al to use these sets of clustered grid points as opposed to evenly spaced grid points because th e clustered grid points allow much quicker and more accurate numerical convergence. Using these se ts of grid points is also beneficial because, in the systems considered in this research, mu ch of the important c onvective flow behavior

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85 occurs near the edges of the system, and using se ts of grid points more clustered near the edges of the system thus allows the convective behavior of the system to be more easily captured by the discretized modeling equations. Gauss-Chebyshev-Lobatto grid points are a set of points spanning the range [-1,1] (including the endpoints at both -1 and 1), and so the actual distan ces within the system must be rescaled in order to fit this range. The set of Gauss-Radau grid points, however, spans the range (-1,1], meaning that it does include the endpoint at 1, but does not include the endpoint at -1. Simple diagrams showing the general appearan ce of the Gauss-Chebyshev-Lobatto and GaussRadau sets of grid points, considering the r-dir ection, and considering seven grid points, are shown below as Figure 5-2. To show how to determine the exact spacing of these clustered sets of grid points, the equations to generate the Gauss-Chebyshev-L obatto and Gauss-Radau points are shown below, considering the discretization of the r-direction. The equation to generate the Gauss-ChebyshevLobatto points is r r jN j N j r , 1 0 cos (5.6) and the equation to generate the Gauss-Radau points is r r jN j N j r , 1 0 1 2 2 cos* (5.7) A negative sign has been added to the GaussChebyshev-Lobatto equation so that that set of points corresponds with Figure 51. For the Gauss-Radau points, rj has been marked with a superscripted “*” to indicate that the order of the set of points generated by Equation 5.7 actually needs to be reversed so that the points correspond with Figure 5-1.

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86 In all calculations done for th is research, the Gauss-Chebys hev-Lobatto grid points were used in discretizing the z-direction. In calcu lations for annular systems, the Gauss-ChebyshevLobatto grid points were used for the discretization of the r-direction, as well. In calculations for cylindrical systems, however, Gau ss-Radau grid points were used in discretizing the r-direction. The reason for this will be explained now. The discretization in the r-d irection is done along the radius of the cylinder, and not the diameter. In the set of modeling equations for the cylindrical system, though, there are no boundary conditions at r = 0 (the interior endpoint of the discretization in the r-direction). Thus, includin g that location in the discretization of the rdirection would cause co mputational problems. This difficu lty can be dealt with by excluding the location r = 0 from the discretization. The use of Gauss-Radau grid points for the discretization in the r-direction accomplishes this because this set of grid points excludes one endpoint. Given that all variables must be written as vectors of their scalar values at every node of the discretized system, in the manner shown in Equation 5.5, it is clea r that the eigenvector, X will actually be a concatenated column-vector co mprised of the column-vectors for each system variable. This means that it is a vector in which the smaller column-vectors for each system variable are vertically arranged above and below one another, head -to-tail. In the discretized form of the problem, the matrices A and B of Equation 5.1 are ac tually large matrices comprised of many smaller matrices which operate on the discretized variables in X in accordance with the modeling equations. Each row of submatrices within A and B represents a different domain equation or boundary condition. Each submatrix in a row of submatrices representing a domain equation must operate on all grid points in the system, whereas each submatrix in a row of submatrices representi ng a boundary condition operates on only the grid

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87 points on the boundary where the condition is en forced. A diagram illustrating the general layout of the matrices and vectors of the disc retized, generalized eige nvalue problem is shown below as Figure 5-3. All differential operators in th e modeling equations must be expressed in matrix-form so that they may operate on the di scretized variables. The differe ntiation matrices can be derived from polynomial interpolation equations. When taking derivatives in a direction that was discretized with Gauss-ChebyshevLobatto grid points, th e differentiation matrix is different than the one that would be used for ta king derivatives in a direction that was discretized with GaussRadau grid points. An exampl e differentiation matrix, for r-dir ection differentiation on GaussChebyshev-Lobatto grid points (as given by Equation 5.6) with Nr = 3 is 1667 3 0000 4 3333 1 5000 0 0000 1 3333 0 0000 1 3333 0 3333 0 0000 1 3333 0 0000 1 5000 0 3333 1 0000 4 1667 33 ,rN rD, (5.8) where the subscript of Dr reflects the fact that this matrix is for Nr = 3. More details about how to obt ain differentiation matrices an d how to set up and apply the spectral solution method being used in this rese arch can be found in a helpful book by Trefethen, called Spectral Methods in MATLAB (2000). The discretization and discrete problem fo rmulation described so far are an accurate description of how to treat th e system of modeling equations on a given plane normal to the azimuthal direction (an r-z plane). Dealing with the -direction dependence of the physical system is mathematically quite simple because of the periodicity of the system in the -

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88 direction. Recall that the periodic -direction dependence of the sy stem was represented in the form shown in Equation 4.20. This results in the presence of the azimuthal wave number, m, in the final set of modeling equations. To account for the -direction dependen ce of the physical system, one must simply assume a value for m. To choose a value for m, of course, is to assume the exact periodic form (number of periods in the -direction) of the convective flow pattern at onset. If, for example, m is assumed to be equal to 2, then the onset flow pattern being sought is one which repeats twice as one progresses a single cycle through th e fluid layer in the direction. If m = 0 is assumed, then the onset flow pa ttern is one which does not vary in the direction at all; the m = 0 case is axisymmetric in the -direction. Once a value is set for m, and thus the particul ar periodic form of the onset flow pattern has been assumed, the discrete system of modeling equations in matrix-form (Figure 5-3) can be solved for the eigenvalue, T. This can be done numerically with an eigenvalue solver (for example, in MATLAB) that accommodates generalized eigenvalue problems. Note that, in this research, the left-hand-side matrix (the A matrix in Figure 5-3) was not of full rank. Another way of saying this is that the set of eigenvalues of the A matrix, itself, included some eigenvalues which were equal to zero. MATLAB was selected as the software to be used in performing the computa tions for this research. The MATLAB generalized eigenvalue solver used in this research, like most other generalized eigenv alue solvers that the author has encountered, required that the A matrix be of full rank. Consequently, if the matrices of discretized modeling equations were directly fed into a generalized eigenvalue solver, then the output would include several spurious values. The reasons for the appearance of the spurious values related to the redundancy of boundary conditions at some points. To read more about this type of complication, the reader is asked to see the 1993 work of Labrosse (Labrosse 1993). In

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89 order to eliminate spurious values from th e output of the eigenvalue calculation, the A matrix was put through a pre-conditioning process involvi ng singular-value decomposition. In this process, the A matrix was decomposed into three ma trices by singular-value decomposition, then the rows in the matrices corresponding to any eige nvalues equal to zero could be located, and finally those rows were filter ed out of further computations. The critical temperature diff erence and flow pattern at th e onset of convection can be determined by comparing the T values (eigenvalues) obtained for a range of different m values. When making this comparison, the lowest T value corresponds to the m value for which the fluid layer is most unstable to buoyancy forces. The lowest T value and the corresponding m value, thus, represent the cri tical temperature difference and flow pattern at the onset of convection, respectively. Note, also, that in or der to account for the vari ation of viscosity with temperature, it is necessary to use an iterative pr ocedure (as described earlier in this chapter) to obtain the T value corresponding to each m value. A flow-diagram displaying the gene ral procedure carri ed out in MATLAB to solve the system of modeling equations for onset conditions is given in Appendix D. A set of MATLAB programs were created to be used in the calculat ions for this research, and some examples of these programs are given in Appendix E. 5.2 Application of the Method As mentioned, considering a larger number of nodes make s the discretized model more accurate. If the number of disc retization nodes used to model the physical system were small enough, an inaccurate critical temperature diffe rence would be obtained from the eigenvalue calculation. If the number of nodes were, then, increased further and further, the critical temperature difference results given by the eige nvalue calculation woul d progressively converge

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90 toward the physically correct value. Eventually when considering a suff iciently high number of discretization nodes, the cr itical temperature difference obtain ed would be the physically correct value, and further increases to the number of nodes considered would have no impact on the critical temperature difference re sult obtained. A nice quality of the spectral method used here, though, is that not too large a number of nodes are actually needed for the computational solution to converge to an accurate solution. In all co mputations done for this research, 17 nodes were used in the r-direction and in the z-direction; this was always more than enough nodes to obtain a converged solution. Convergence of the comput ational results was, of course, verified by repeating computations with ev en higher numbers of discretization nodes. To exemplify this convergence behavior, a table is given below (Table 5-1) which shows the convergence of critical temperature difference values obtained using the author’s MATLAB program. The example system considered for this convergence tabl e is a fluid layer in a cylindrical container, with rigid, no-slip walls, insulated at the late ral boundary, and heated from below so as to facilitate Rayleigh convection. In particular, a case is being cons idered in which the size of the cylindrical system is such that the convectiv e flow pattern at onset is a one-cell pattern, axisymmetric in the -direction. The convergence of th e critical temperature difference ([T]crit) is shown with respect to both Nr and Nz. To confirm the validity of the set of computational programs written for this research, results obtained using the programs were checked against results for standard Rayleigh-Bnard problems (problems involving buoyancy-driven conv ection in systems heated from below) and some results obtained by other authors. As one check of the validity of the co mputational programs, a comparison was made between the critical temperature difference resu lts obtained for a two-di mensional, rectangular

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91 case (x-direction and z-directi on only) and a one-dimensional, rectangular case (z-direction only). In the two-dimensional cas e, the rectangular fluid layer was heated from below, with stress-free, insulated side walls and constant-temperature, no-slip walls as the top and bottom boundaries of the layer. The distance between the side walls for the two-dimensional case is called Lx. In the one-dimensi onal case, the fluid layer was heated from below, was bounded at the top and bottom edges by constant-tempera ture, no-slip walls, and was unbounded in the lateral direction. In neither cas e was the variation of viscosity w ith temperature considered. The two-dimensional case with stress-free side wa lls was expected to produce exactly the same critical temperature difference results as a corresponding one-dimensional case, in which the system had the same vertical depth as the two-dime nsional case. As Table 5-2 shows, it did. At first it may appear unusual that the one-dimens ional calculation could produce two different critical temperature difference resu lts for the same vertical depth (Lz = 7.2 mm). The reason this is possible has to do with the way the one-dimensi onal calculation is carried out. As explained in Chapter 1, a two-dimensional system with la teral walls can only physically accommodate a certain set of onset flow patterns, which is depend ent on the lateral width of the system. Thus, if the critical temperature difference result for a two-dimensional system with lateral walls is to be compared with the result fo r a corresponding one-dimensional calculation, it is only the disturbances of proper shape and size to indu ce these physically admissi ble onset flow patterns which may be considered in the one-dimensional calcula tion. It is because of this that the critical temperature difference results obtained from the one-dimensional calculation for the two cases in which Lz = 7.2 mm are different. Now that result s are being shown for systems of specified depths and boundary conditions, it is a good time to mention that the fluid used in all experiments and calculations for th is research is Dow Corning 200 1 Stoke silicone oil. Thus,

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92 its thermophysical properties are used in all calculati ons presented in this paper. A list of these thermophysical properties is gi ven in Appendix A. The depe ndence of temperature on the viscosity of the oil was determined using visc osity measurements taken with a Cole-Parmer 98936 series viscometer. The exponential equation fo r the viscosity of the oil as a function of temperature is included in Appendix A, as well as a description of the manner in which viscosity measurements were obtained. Except for the vi scosity, which was measured as described, the thermophysical properties of the silicone oil were based on information provided by Dow Corning. As an initial check of the ability of the computational programs to model cylindrical systems, critical temperature difference results were computed for buoyancy-driven convection in a cylindrical fluid layer, and compared to th e results given in a 1990 publication by Hardin et al. (Hardin, Sani, Henry, & Roux 1990). The part icular system considered for this comparison was a layer of silicone oil in a cylindrical containe r, heated from below, with no-slip walls at all boundaries, constant-temperature conditions at th e top an bottom boundaries, and insulation at the radial wall. Again, the variation of viscosity with temperature was not considered. Hardin et al. give several critical temperature difference results for this system, as well as the corresponding flow patterns. A co mparison of the results produced by the author’s computations for this system with those of Hardin et al. is given in Table 5-3. Th ese comparisons show the validity of the MATLAB programs written for the computations in this research. The results of Hardin et al. were given in terms of the dimens ionless Rayleigh number, so in order to compare with them, the author’s critical temperature differ ence results in this tabl e have been re-expressed in terms of the Rayleigh number, as well (as sh own in Equation 1.1). Since the variation of viscosity with temperature was not considered in this case, th e viscosity value used in the

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93 computation was simply the constant viscosity va lue at 35 C given in Appendix A. The aspect ratio is simply the ratio of the radius of the cy lindrical container to the vertical height of the container (Lr / Lz). The way in which convective flow patterns are named should be briefly discussed here, too. In general, they are named by their azimuthal wave number, m. For example, a ring-shaped patte rn axisymmetric in the -direction is called an m = 0 pattern. Optionally, when considering cylin drical systems, the flow pattern may be referred to with a parenthetical notation, in cluding two indices. A parenthetical system for naming the onset flow patterns in annular systems is given in Chapter 7. In the parenthetical notation for cylindrical systems, the first index is the azimuthal wave number and the second index represents the maximum number of convective rolls that can be counted across the diamet er of the cylindrical test section. Diagrams to exemplify the use of this parenthetical nota tion for flow patterns in cylindrical systems are shown as Figure 5-4. Th e diagrams are taken almost directly from the 1995 paper by Zhao, Moates, and Narayanan (Zhao, Moates, & Narayanan 1995). In Figure 5-4, a top view of each flow pattern is given, and belo w the top view of each patte rn is a side view of the same pattern. In the top views, the “X” i ndicates falling fluid and the “O” indicates rising fluid. Diagrams (a) and (c) in Figure 5-4 could simply be called m = 0 patterns if the parenthetical notation were not being used; li kewise, diagrams (b) and (d) could be called m = 1 patterns. Now, the method of obtaining computational re sults has been explained, and the validity of the computational method has been demonstrat ed. The computational results were compared with the results from experiments, and an expl anation of the experimental apparatus and the manner in which experimental results we re obtained is given in Chapter 6.

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94 Table 5-1. Example of converg ence of computed result with Nr and Nz. Nr Nz [T]crit (C) Nr Nz [ T]crit (C) 313 6.55 133Inf 413 5.55 1345.42 513 5.60 1355.55 613 5.59 1365.60 713 5.60 1375.60 813 5.60 1385.60 913 5.60 1395.60 1013 5.60 13105.60 1113 5.60 13115.60 1213 5.60 13125.60 1313 5.60 13135.60 Table 5-2. Comparison of rect angular, 2-D, no-stress results with rectangular 1-D results. [ T]crit (C), [ T]crit (C), Lx (mm) Lz (mm) 2-D, No-Stress 1-D 5 512.8712.87 7.2 7.24.314.31 23 7.24.334.33 9 92.212.21 Table 5-3. Comparison of calculated cylindrical results with results of Hardin et al. Predicted Flow Pattern, Author's Computation Predicted Flow Pattern, Hardin et al. Critical Rayleigh Number, Author's Computation Critical Rayleigh Number, Hardin et al. Aspect Ratio: .75, Lr = 4.5 mm, Lz = 6 mm (1,1) (1,1) 25902592 Aspect Ratio: 1, Lr = 6 mm, Lz = 6 mm (0,2) (0,2) 22602260 Aspect Ratio: 1.5, Lr = 9 mm, Lz = 6 mm (0,2) (0,2) 18951895 Aspect Ratio: 2.5, Lr = 15 mm, Lz = 6 mm (0,4) (0,4) 17801781

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95 Figure 5-1. Discretiza tion nodes in a cylinder. Figure 5-2. Example: grid point spacing. Gauss-ChebyshevLobatto Gauss-Radau r 0 = -1 r 1 = -.866 r 2 = -.5 r 3 =0 r 4 =.5 r 5 =.866 r 6 = 1 N r = 6 N r = 6 r 0 = -.9709 r 1 = -.7485 r 2 =-.3546 r 4 =.5681 r 5 =.8855 r 3 =.1205 r 6 = 1 r 0 z 0 r 1 z 0 r 2 z 0 r 3 z 0 r 0 z1 r 1 z1 r 2 z1 r 3 z1 r 0 z2 r 1 z2 r 2 z2 r 3 z2 r 0 z 3 r 1 z 3 r 2 z 3 r 3 z 3 r z

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96 Figure 5-3. Diagram of matrix/vector arrangement of discretized problem. = rv LHS Motion Equation, r-component, and velocity boundary conditions LHS Motion Equation, component, and velocity boundary LHS Motion Equation, z-component, and velocity boundary conditions LHS Energy Equation, and temperature boundary conditions LHS Continuity Equation RHS Motion Equation, r-component, and velocity boundary conditions RHS Motion Equation, component, and velocity boundary RHS Motion Equation, z-component, and velocity boundary conditions RHS Energy Equation, and temperature boundary conditions RHS Continuity Equation Left-Hand-Side Matrix (LHS), A Right-Hand-Side Matrix (RHS), B X X zv T P rv v zv T P v

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97 Figure 5-4. Examples of flow patterns and pa renthetical notation for cylindrical systems. (a) Lr / Lz = 1, Pattern: (0,2) (b) Lr / Lz = 1.8, Pattern: (1,3) (c) Lr / Lz = 2.5, Pattern: (0,4) (d) Lr / Lz = .75, Pattern: (1,1)

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98 CHAPTER 6 EXPERIMENTAL DESIGN After computational results were obtained in this research, they were compared with experimental results for systems identical to those considered in the computations. The apparatus which was designed to run experiments for this research is the subject of this chapter. Any possible errors associated with the apparatu s and its measurements are discussed in Chapter 7. 6.1 Goals in Experimental Design There were several requirements made of th e experimental system. One requirement was that the top and bottom temper atures of the fluid layer (Tt and Tb, respectively) were kept constant and uniform at desire d values. By monitoring and controlling the top and bottom temperatures of the fluid layer, the vertical temp erature difference across the fluid layer could be regulated. The radial walls in the experiments need ed to be insulating so that heat did not escape the experimental test fluid into the air surrounding the experiment. Lastly, some means of flow visualization needed to be employed in order to determine when the flui d layer was or was not convecting, so that the critical conditions for the onset of c onvection could be determined. Below, the means by which these goals were accomplished are described. 6.2 Experimental Apparatus As mentioned, the test fluid in the experiments was Dow Corning 200 1 Stoke silicone oil (thermophysical properties given in Appendix A). This fluid wa s chosen as the test fluid for this research because, for systems in the size range of those examined in this research, its thermophysical properties are su ch that the critical vertical temperature differences for convection were easily obtainable in experiments. The only th ermophysical properties of the silicone oil which vary signifi cantly with temperature are the density (enough to facilitate

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99 Rayleigh convection) and viscosity. In the experimental apparatu s, a lucite ring acted as the outer radial boundary of the silicon e oil test section. A lucite middle-piece was added when the test section was annular. The dimensions of th ese lucite pieces, thus set the sizes of the cylindrical and annular test sec tions. A copper plate at the top of a continuously stirred water bath was the bottom boundary of the test section, and a sapphire window at the bottom of a flowthrough water bath was the top plat e of the test section. To cr eate heated-from-below conditions, the temperature of the bottom water bath was adjusted to certain set-point values while the top water bath was always kept at a constant, cooler temperature. A process control comput er, running a LabVIEWTM process control program written by the author, sent signals to turn a heater in the experiment on and off as needed in order to keep the bottom water bath at desired set-point temper atures. The process control system built for the experiments in this research wi ll be further discussed in Sect ion 6.3. The test section was insulated to prevent heat loss in the radial direction. To allow flow visualization, a small amount of aluminum tracer powder was mixed into the sili cone oil. So that the flow behavior of the experimental system could be recorded and re viewed, a digital camcorder was mounted above the test section. More detailed descriptions of the components a nd features of the experimental apparatus, as well as some notes on the experi mental start-up proce dure, are given in the following subsections. A simple diagram of the apparatus is shown below as Figure 6-1, and a couple of photographs of the apparatu s follow as Figures 6-2 and 6-3. 6.2.1 Test Section As mentioned, a lucite ring acted as the out er radial boundary of th e test section which contained the silicone oil. The bottom boundary of the test section was a copper plate, which was the top surface of a stirred water bath that wi ll be described shortly. The top boundary of the

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100 test section was a sapphire window, which was th e bottom surface of a flow-through water bath, which also will be described. The lucite rings typically ranged from about 15 mm to 30 mm in diameter and about 6 mm to 8 mm in height. The sets of experiments for cylindrical systems were run before the sets for annular systems; th is procedure was used so that when the annular experiments were run, a nesting hole in the bottom c opper plate could be used to anchor the bottoms of the lucite center-pieces, which were added to transform the cylindrical systems to annular systems. A reason for selecting lucite as the material from whic h to construct the outerboundary rings and the center-pieces for the annular systems is it s thermal conductivity, which is very close to that of the silic one oil (see Appendix A). Since th e system of silicone oil being examined in this research was subjected to vertical temperature gradients, so, too, were the radial walls of the system. Thus, the f act that lucite has a thermal conduc tivity close to that of silicone oil made it an appealing choice as the material fr om which to construct the radial walls because it ensured that the vertical temperat ure gradients at the radial bounda ries of the test section would not differ from those in the interi or. This, and the fact that the lucite outer radial wall was surrounded by an insulating coating (which shall be further explained late r), ensured that heat would not flow into or out of the silicone oil in the test section in the radial direction. After placing the lucite ring onto the copper plate and before filling the test section with oil, a lucite clamp-piece was sc rewed down on top of the lucite te st section ring to tighten the ring down against the copper plate (the clamp-piece screws into the bottom water bath). This was an important measure taken to pr event any leakage of air/oil at th e bottom of the test section. Once the test section ring was tightened down to the copper plate, the test section was intentionally over-f illed with silicone oil. This was done so that the sapphire plate of the flowthrough top water bath could more easily be presse d down onto the top of th e test section without

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101 trapping air bubbles in the test sect ion. A simple cross-sectional diagram of the test section is given below as Figure 6-4. 6.2.2 Bottom Temperature Bath The temperature at the bottom surface of the test section was kept constant and uniform by a constantly stirred water bath. The body of th is bottom water bath was primarily lucite, but its top and bottom surfaces were copper plates. As mentioned, the top copper plate of this water bath acted as the bottom surface of the test se ction. The bottom copper plate rested on an electrical resistance heater, which was the mean s by which the temperature of the water in the bottom bath could be adjusted. This a rrangement can be seen in Figure 6-1. Three OMEGA thermistors, which were attached to the underside of the top copper plate of this bath, measured resistances in the wate r at that location and sent the resistance data to a process control computer. The resistance da ta, based on calibration equations relating the thermistors’ resistance readings to temperatures, was used to de termine the temperature at each thermistor. Since the thermist ors were on the underside of th e bath’s top copper plate, and copper is an extremely good heat conductor, they gave a very accurate reading of the temperature at the bottom of the test section. Depending on th e readings of these thermistors, the heater at the bottom surface of the water bath was turned on or off in order to maintain a set-point value for the temperature at the bottom of the test section. The temp erature of the bottom water bath typically needed to be in the ra nge of 30.0 C to 40.0 C. The bottom water bath also contained a small magnetic stir-bar so that the water within the bottom ba th could be mixed by a magnetic stirring plate, which was the surface upon which the electrical resistance heater rested. This mixing ensured that the temperature was uniform throughout this bath’s top copper plate, which

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102 was the bottom surface of the test section. Th e high thermal conductivity of copper also helped to ensure that the temperature was uniform at the bottom surface of the test section. 6.2.3 Top Temperature Bath The top surface of the test section was a sapphire window. This window was the bottom surface of a flow-through water bath. The body of the flow-through water bath was comprised of a thick aluminum bottom-piece and a transparen t top-piece made of pl exiglass; the aluminum bottom-piece had a circular section cut out of it, and that is where the sapphire window was located. The temperature of the flow-through wate r bath was set by water which flowed into it from a larger constant-temperature water bath. The larger constant-temperature bath was a NESLAB EX Series bath that was bought from Thermo Electron Corporation. The smaller, flow-through water bath had two water inlets and two water outlets, and this allowed a high volume of water to continuously pa ss over the top of the sapphire window. This, in turn, kept the sapphire window at a constant and uniform te mperature by removing any heat conducted to it from the test section. If the temperature of the smaller, flow-through water bath, and thus the value of Tt, were to be changed, it was the set-point temperature of the larger NESLAB bath which needed to be adjusted. During all experime nts for this research, the temperature at the top of the test section was maintained at 30.0 C The high thermal conductivity of sapphire (see Appendix A) ensured that the te mperature throughout the sapphire window remained uniform. Another important property of the sapphire was its transparency; the transp arency of the sapphire window and the transparency of the plexiglass top-piec e of the flow-through water bath were necessary to allow flow visualization by means of aluminum tracer powder. Two OMEGA thermistors placed in the flowing water’s inlet and outlet tubing were used to monitor the temperature within the flow-through water bath on the process control

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103 computer. In the experiments for this research, these thermistors were used to verify that the temperature in the top water bath was i ndeed constant at 30.0 C as desired. 6.2.4 Insulation In the experiments for this research, it was im portant that heat was not lost from the test section to the surrounding air through the outer radial wall of th e test section. To prevent this sort of heat exchange between the test section and the surroundings, a ring of foam insulation was placed snugly around the outer surface of each of the lucite rings used in the experimental runs. For the experiments on annular systems, at and near the onset of co nvection, no significant amount of heat would flow through the lucite center-pieces, and so it was not necessary to insulate them in any way. 6.2.5 Flow Visualization A small amount of aluminum tracer powder was mixed in with the test fluid so that convective flow patterns in the flui d layer could be seen. As explai ned earlier, when the oil flow was horizontal, the aluminum particles aligned so that they reflected more light, making the regions of horizontal flow app ear lighter in color; in regions where flow was vertical, the aluminum particles aligned such that they re flected less light, making those regions appear darker in color. As mentioned, the sapphire t op surface of the test section and the plexiglass top-piece of the upper flow-through water bath are both transparent, and this allo wed the flow patterns in the test section to be easily observed from above the test section. Th e experiments often ran continuously for a day or more, so it was desirabl e to have a way of recording video footage of the flow behavior (or lack of such behavior) in th e system for later viewing. To facilitate this, a digital camcorder, a television, a nd a video cassette recorder were included in the experimental

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104 design. The digital camcorder, mounted directly above the test section, continuously captured a top-view image of the test secti on as the experiment was running. This image was sent to the video cassette recorder. The circui try of the video cassette recorder was modified so that it could receive “Record” and “Stop” signals sent, throu gh a relay, from the process control computer. The LabVIEWTM process control program then, periodically throughout the course of each experiment, sent a series of si gnals to the video cassette recorder, causing it to intermittently record for periods of about five seconds. The re sult of this was a conveniently reviewable timelapsed video recording of the convective flow behavior observed over the entire experimental run. For real-time viewing of the flow behavior during the experiment, as well as viewing of the experimental video recordings a television was connected to the video cassette recorder. 6.3 Process Control System The process control system built for running this experiment includes a process control computer equipped with a LabVIEWTM process control program, a circuit board to interface between the computer and the components of th e experiment, and parts of the experimental apparatus, itself. The layout of the process cont rol system is shown as Figure 6-5. In this diagram, the dashed connecting lines represent signals which are part of the process control system. As Figure 6-5 shows, the process control com puter collected data from thermistors placed in and near the top and bottom water baths, between which was the test section. Note that three OMEGA thermistors were placed in the bottom, stirred water bath, and two OMEGA thermistors were placed in the tubing near the top, flow-through water bath. Before the thermistor readings reached the computer, they were routed through a process control circuit board built for this research. The most important components in the large system of circuitry

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105 were a set of National InstrumentsTM FieldPoint modules. The National InstrumentsTM FP-RTD122 module was used to collect data from th e thermistors, and the National InstrumentsTM FPRLY-422 module was used to send process contro l signals to components of the experiment. The readings from the thermistors are resi stances, not temperatures. Thus, calibration equations stored in the process control computer n eeded to be used to co nvert the resistance data to temperature data. The thermistors were placed such that their readings accurately represented the values of Tt and Tb. A LabVIEWTM process control program was writ ten by the author, and saved on the process control computer. The process control progr am allowed its user to input a series of setpoint values for the temperature of the lower ba th, and thus, the temperature of the lower surface of the test section. Recall that the top, flow-t hrough bath was kept at a constant temperature of 30.0 C for all experiments. Additionally, the program allowed its user to input a time-segment value, which determined how long each set-poin t value for the bottom water bath temperature was maintained before the next set-point was targ eted. Using “If” loops, this program sent “On” and “Off” signals to the heater beneath the bottom water bath, as needed, in order to keep that water bath at the current set-point temperature. To improve the ability of the process control program to keep the Tb value near the set-point, the process control could be set to behave in an advanced way (acting in anticip ation of upcoming changes in Tb) in a small range of temperatures around the Tb set-point. This small range of te mperatures is called the “deadband.” While the experiments ran, the program also kept track of how much time had elapsed; depending on the amount of elapsed time, the pr ogram sent signals causing the video cassette recorder to record segments of the video image of the test se ction which was being continuously captured by the digital camcorder.

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106 In addition to its powerful pro cess control capab ilities, LabVIEWTM also allows the creation of user-friendly front pages for proces s control programs. Thus, temperatures and temperature differences could easily be mon itored on continuously updated graphs during experimental runs. A flow-chart givi ng a very general layout of the LabVIEWTM process control program and how it worked is given as Appendix F. 6.4 Typical Experimental Procedure The purpose of this section is to briefly de scribe the typical pro cedure that was followed during the preparation of the experiment and the e xperimental runs for this research. First, a suitable mixture of silicone oil and the aluminum tracer powder needed to be made. When making this, it was important not to use too much tracer powder, so as not to significantly alter the properties of the silicone o il. Once this mixture was made, it was placed under the suction of a vacuum pump to remove any air bubbles in it. The next step in preparing the experiment was to fill the top and bottom water baths with water and make sure that no air bubbles were pres ent in them. Then, the bottom water bath (the copper plate on top of it, in particular) was made level, and the lucite ring (and middle-piece in annular cases) and lucite clamp-pi ece were assembled on top of it. Screws in the lucite clamppiece were used to tighten the lucite ring on to the top copper plate of the bottom water bath. Next, the test section was filled with silicone oi l. As mentioned earlier, the test section was purposely overfilled, which made it easier to avoi d trapping air bubbles in the test section when placing the sapphire window of the flow-through water bath on top of the test section. Once the flow-through water bath was put in place, comp leting the assembly of the test section, temperature control c ould be initiated.

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107 Usually, the bottom of the te st section was immediately he ated up enough to generate a strong convective flow under supercritical conditions Then, the vertical temperature difference was gradually, incrementally decreased by decreasing Tb. Each vertical temperature difference was maintained for a time segment often in the range of 1 to 4 hours, depending on the time scales associated with the systems being exam ined. The time scales, of course, depended strongly on the sizes of the systems. The comput ational predictions coul d be used to aid in deciding approximately what ranges of ver tical temperature differences to examine experimentally. Decreasing the vertical temper ature difference, over time, resulted in the weakening of convective flow. At some point near the critical temperature difference, the flow pattern would likely be the pattern predicte d computationally. Further decreasing the temperature difference in very small steps ma de it possible to determine the temperature difference at which convective flow stopped. Then, it was known that the last temperature difference value at which the fluid was convec tively flowing was the critical temperature difference. To further check this experiment al critical temperatur e difference result, the temperature difference could be increased slow ly from the no-flow state to determine the temperature difference at which fl ow restarted. As an additional check of the experimental result, experimental runs were repeated multiple times. The runs could easily last for longer than a day, depending on the time segment being used for each vertical temperature difference in the experiment. Thus, it was often helpful to make a video recording of the flow behavior at each vertical temperatur e difference and review the flow behavior using the video recording.

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108 Figure 6-1. Simple experiment diagram. Figure 6-2. Photograph of experimental apparatus. Heater Insulating Cement Lower Water Bath (Stirred) Test Section (Lucite) Lucite Clamp-Piece Sapphire Disk Digital Camcorder Upper Water Bath Water Flow Water Flow

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109 Figure 6-3. Photograph of flow-through top water bath of experimental apparatus. Figure 6-4. Cross-sectional diagram of te st section. Test Section Flow-Through Water Bath Sapphire Window Insulation Stirred Water Bath Lucite Ring Lucite ClampPiece Copper Plate Screw Threading

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110 Figure 6-5. Process control system layout. TV VCR Digital Camcorder Process Control Circuit Board Process Control Computer Flow-Through Water Bath Test Section Stirred Water Bath Heater (1) (2) (3) (4) (5) (6) (8) (7) (1): “Record” and “Stop” signals (2): Thermistor Readings (3): Thermistor Readings (4): “On” and “Off” signals (5): Thermistor Readings from (2) and (3) (6): “On” and “Off” signals for heater, depending on the values in (5), and “Record” and “Stop” signals for VCR, depending on amount of time elapsed (7): Video image into VCR (8): Video image into TV

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111 CHAPTER 7 RESULTS AND DISCUSSION In this research, the results that were sought through comput ations and experiments were the critical conditions for the onset of convectio n in the systems being st udied. These conditions included the critical vertical temperature difference correspondi ng to the onset of convection, and the corresponding convective flow patterns. Results were obtai ned both from calculations in which the variation of viscosity with temperatur e was not accounted for, and from calculations in which it was. The results from these computations were then compared with results obtained experimentally for the same systems that the com putations addressed. The test fluid considered in all computations and used in all experiments was Dow Corning 200 1 Stoke silicone oil; its properties are given in Appendix A. Possible erro rs in the results are noted on the tables of results, and the sources and quantification of the errors are explained in Section 7.3. 7.1 Cylindrical Systems This section presents the critical conditi ons for the onset of convection, determined computationally and experimentally, for cylindrical systems of three different sets of dimensions. The computational results are shown both for the case in which the variation of viscosity with temperature is included, and for the case in which it is not. Then, the experimental results are given. After that, the computa tional and experimental results ar e compared against each other. 7.1.1 Constant Viscosity Computations: Cylinder Since the top temperature of th e test section was always kept constant at 30.0 C, and the bottom temperature of the test section typically ranged from 30.0 C to 40.0 C, the constant viscosity value assumed for these calculations was the viscosity value for the oil at 35.0 C, which is the mean value in the 30.0 C – 40.0 C temperature range. The kinematic viscosity of the oil at 35.0 C is given in Appendix A. The re sults of the constant vi scosity computations for

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112 the three sets of cylindrical dimensions that we re considered in experi ments are given below as Table 7-1. In this table, the parenthetical notation for flow patte rns in cylindrical systems, which was introduced in Chapter 5, is used again to desc ribe the onset flow patterns. Remember that in this notation, the first index is the azimuthal wa ve number, and the second index is the maximum number of convective rolls that can be counted acro ss the diameter of the cylindrical test section. The experimental results corresponding to th ese calculations will be shown in Section 7.1.3. Carrying out the constant viscosity comput ation for the critical co nditions at a wide range of aspect ratios yields the st ability diagram shown as Figure 71. This diagram includes the stability curves for the onset flow patterns with m = 0, 1, 2, 3, 4, and 5. Figure 7-2 is from the same stability diagram, but is a close-up of a sm all portion of the diagram that better shows the behavior in that region. Notice that Figure 7-1 is much like the graph sh own in Chapter 1 as Figure 1-10. Figure 1-10, though, was created simply by logically considering the physics and pattern selection behavior of buoyancy-driven convection in a cyli ndrical system. Figure 7-1 shows that, even after working through a computati onal solution to obtain a stability diagram for this system, the result matches what could be predicted based on physical reasoning alone. This diagram is valid for any constant viscosity cylindrical syst em regardless of the exact dimensions or thermophysical properties of the system. In this diagram, as explained in Chapter 1, the points at which the slope of the curve for a particular m value changes from positive to negative correspond to transitions in the number of convectiv e rolls present in the onset flow pattern for that azimuthal wave number. For example, following the m = 0 curve, for the region of A (aspect ratio) between .7 and approximately 1.7, where an inflection occurs, the on set flow pattern would be (0,2) pattern. Advancing in A to the range between about 1.7 and 2.7, the onset flow pattern

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113 would be a (0,4) pattern. At even higher aspect ratios, a (0,6) pattern c ould be seen. Likewise, for increasing A, with the changes from positive to negative in the slope of the m = 1 curve, an onset m = 1 pattern would progress from a (1,1) pattern to a (1,3) pattern, and then to a (1,5) pattern, and so on. 7.1.2 Non-Constant Viscosity Computations: Cylinder Table 7-2 shows the results of the non-constant viscosity com putations for the three sets of cylindrical dimensions that we re considered in experiments. As stated, the temperature at the top of the test section for all computations and experiments was 30.0 C. In all computational cases for this research in which the variation of viscosity with temperature is taken into account, whether considering a cylindrical system or an annular system, the kinematic viscosity used in the Rayleigh number is the reference kinematic viscosity, which is a mean value based on the critical vertical temperature difference for each case. This is why, even though the critical vertical temperature difference fo r the non-constant viscosity case is noticeably different from that for the constant viscosity case, the critical Rayleigh numbers for the two cases can be nearly identical. The critical temperature diff erences calculated for the non-c onstant viscosity case are generally higher than those calcu lated for the constant viscosity case. This can be seen by comparing Table 7-1 with Table 7-2. Now, the co mputed velocity profiles for these three sets of dimensions will be shown. In these flow profiles, the velocities are scaled, negative velocities represent downward flow in the z-direction (whi ch extends into and out of the plane of the diagram), and positive velocities represent upward flow in the z-direction. All velocity profiles given in this chapter were computed using non-c onstant viscosity computations. Also, each of

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114 the cross-sectional velocity profiles shown here represents a cross-sectional region approximately half of the distance along the vertical depth of the system that it is computed for. When a non-constant viscosity computation for the critical conditions is carried out at a wide range of aspect ratios, a stability diagram relating the cri tical Rayleigh number to the dimensions of the system, much like Figure 7-1, can be obtained. Since the critical conditions determined from a non-constant viscosity comput ation are dependent on the vertical temperature difference in the system just before the onset of convection, which, for a given aspect ratio, is dependent on the vertical height of the system, the stability diagram differs very slightly from that for the constant viscosity case depending on the vertical height of the system being considered. Thus, in order to present the relati on between the system dime nsions and the critical Rayleigh number as simply and clearly as possi ble, for cylindrical systems and for annular systems, stability diagrams are presented only for the constant viscosity cases. Note that the vertical temperature difference in the system just before the onset of convection shows up in the dimensionless groups, F0 and given in Chapter 4. To read more about the effects of considering a non-constant viscosity in calculations, the reader is directed to the work of Zhao, Moates, and Narayanan (Zhao, Moates, & Narayanan 1995). 7.1.3 Experiments: Cylinder Experiments were run to determine the critic al vertical temperature difference and flow pattern for cylindrical systems of three sets of dimensions. The results of these experiments are given here and then compared to the computati onal results in Section 7. 1.4. First, a table containing these experimental results is given, a nd then photos of the onset flow patterns for each of the three cases are given. In this table, the value used for the kinematic viscosity in the Rayleigh number is, in each case, the value co rresponding to the mean temperature in the fluid

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115 just before the onset of convecti on (if the critical temperature di fference were 5.0 C, then since the top temperature was always ke pt at 30.0 C, the kinematic viscosity of the oil at 32.5 C would be used in calculating the Rayleigh number). Photos of the onset flow patterns follow (F igures 7-6 – 7-8). Re member that in these images, due to the use of aluminum powder as a tracer in the silicone oi l, the lighter-colored regions are regions of horizontal flow, and the da rker-colored regions ar e regions of vertical flow. 7.1.4 Comparison of Results: Cylinder To begin with, all results are consolidated in Table 7-4. In this table, the “% Error” column is an indication of the absolute value of the percentage of error between the experimental critical temperature difference and the critic al temperature differen ce obtained from the nonconstant viscosity computation. The error percentage is co mputed with respect to the nonconstant viscosity computational result. The error percentage has been adjusted by scaling out the portion of the error that resu lts from uncertainty in the ther mophysical properties assumed in computations (the thermophysical properties of the silicone oil used in ex periments are close to but do not exactly match the values assumed in co mputations). Exactly how the error percentage is computed, adjusted in this way, is explained in Section 7.3. Thus, the error percentages in Table 7-4 represent error arisi ng from sources other than ther mophysical propert y uncertainty; several possible sources of experimental error are explained in Section 7.3. Clearly, the non-constant viscos ity computational results are closer to the experimental results than are the constant vi scosity computational results. This is to be expected, though, because the non-constant viscosity form of the ma thematical model more realistically represents the thermophysical behavior of the experimental system than does the constant viscosity form.

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116 As Table 7-4 shows, the agreement between th e non-constant viscosity computations and the experimental results is quite good when consideri ng cylindrical systems. Now, the results for annular systems will be presented. 7.2 Annular Systems The critical conditions for th e onset of convection in annular systems, as determined computationally and experimentally, are given in this section. Critical conditions were determined for six different sets of dimensions. Included in these six ca ses are three subsets of two cases each. In each subset of two cases, the outer radius of the annular test section is the same, but the inner radius of the annular test section is relatively small in one case, and relatively large in the other. Again, critical conditions were determined computationally for the case in which the variation of viscosity with temperature is included, and for the case in which it is not. The sets of dimensions considered in the calcula tions are the same sets of dimensions considered experimentally. The dimensions for the three cases in which the ratio of the inner annular radius to the outer annular radius is small were chosen speci fically because the onset flow patterns in those systems would include at least one radially aligned convective roll Again, this approach is a point that distinguishes this res earch from the works of previous authors. It should be mentioned that the selection of the exact sizes of the inner annular pieces for the systems of small radius ratio were selected also, in part, based on the fact that inner pieces of too small a diameter would be too difficult to physically construct from lucite. After the presentation of the computational re sults, the experimental results are shown, and then the computational and experimental results are compared.

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117 7.2.1 Constant Viscosity Computations: Annulus As explained in Section 7.1.1, the viscos ity value assumed for the constant viscosity computations was the viscosity value for the oil at 35.0 C, which is given in Appendix A. Now, a new system for naming the onset flow patterns in annular systems shall be introduced. From this point onward, the onset flow patterns in annular systems may be referred to using a parenthetical notation similar to th e one used for naming the flow pa tterns in cylindrical systems. The parenthetical notation for fl ow patterns in annular systems, like the one for cylindrical systems, includes two indices, separated by a comma, within parentheses. The first index is the azimuthal wave number (which indicates the peri odicity of the flow pattern in the azimuthal direction), and the second index indicates th e maximum total number of radially aligned convective rolls which may be counted between th e inner annular radius and the outer annular radius of the test section. The type of arra ngement in which convective rolls are arranged like spokes on a wheel is called azimuthal alignment; if a convective roll spans the entire azimuthal extent of the test section and is concentric to th e inner annular piece, then this is called radial alignment. For example, Figure 1-11 in Chapter 1 is a (12,0) pattern, and Figure 1-12 is a (2,1) pattern. Many more examples of the use of this nomenclature arise when the computationally and experimentally obtained onset flow patterns for annular systems are shown. Note that, while traversing along the azimuthal direction, an up -flow section is always accompanied by a downflow section, and that the combination of an up-flow section and down-flow section adjacent in the azimuthal direction forms a convective cell. The azimuthal wave number relates more directly to convective cells than to convective rolls This is why, in Figure 1-11, for example, 24 convective rolls along the azimuthal direction co rrespond to an azimuthal wave number of only

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118 12. The results of the constant viscosity com putations for annular systems, using this new notation, are given below as Table 7-5. For any given aspect ratio, the constant vi scosity computation for the critical conditions can be carried out for a wide range of S values. Doing this for seve ral different aspect ratios elucidates the relations between the critical cond itions, the radius ratio, an d the aspect ratio for annular systems. These relations are illustrated be low in stability diagrams (Figures 7-9 – 7-20) for several fixed values of the aspect ratio, A. The values of A considered in the diagrams include .75, the three values co rresponding to the systems studied experimentally, 2.90, and 3.40. Also, as done in Section 7.1.1, close-up views of some of the more inte resting regions of the diagrams accompany the full diagrams. Just like the relation between Racrit and A for constant viscosity cylindrical system s, the relations between Racrit A, and S for constant viscosity annular systems are the same regardless of the thermophys ical properties or the specific dimensions of the system being considered. Again, this interes ting characteristic of the system stems from the fact that the system is time-inde pendent at the onset of convecti on. Also, as explained earlier, considering a viscosity non-cons tant with respect to temper ature would result in slight differences in the stability diagrams depending on the specific dimensions of the system. Some comments on the diagrams follow. Comparing the results for cyli ndrical systems in Table 7-4 w ith the results in Figures 711, 7-13, 7-15 in the regions of very small S values, it can be seen that as the center-piece of an annular system becomes extremely small in diam eter, the onset flow pa ttern in the annular system is the same as the onset pattern in the corresponding cylindr ical system. This seems quite reasonable since an annular system with a center-piece extremely small in diameter is nearly a cylindrical system. In most annul ar cases considered in this res earch, except for those at higher

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119 aspect ratios (such as 2.90 or 3. 40), the onset flow pattern at small S values is m = 0, 1, or 2. Onset flow patterns with azimuthal wave numbers of m = 0, 1, and 2 are the onset flow patterns most often seen in the cylindrical systems consider ed in this research, as well, as shown in Figure 7-1. Typically, the m = 0, 1, and 2 onset flow patterns include convective rolls in radial alignment in addition to azimuthally a ligned convective rolls (except for the m = 0 pattern which includes no azimuthally aligned convective rolls), and onset flow patterns with azimuthal wave numbers of m = 3 or higher, for the cases considered in this research, do not tend to include radially aligned convective rolls. At higher S values, the onset flow patterns in the annular systems considered are generally of high azimuthal wave number (such as m = 3, 4, 5, etc.) and include only azimuthally aligned convective rolls. At these high S values, a higher S value generally corresponds to a higher azimuthal wave number. This is plausible because increasing S narrows the area in which flow can occur, causing azimuthally aligned convective cells to stretch out in the azimuthal di rection to spatially fill the sy stem, and at certain high enough S values, it is simply more energetically favorable for new azimuthally aligned convective cells to form rather than to stretch the existing cells even further into extreme azimuthal lengths. All of this agrees with the observati ons of Littlefield and Desai th at an annulus with a small S value behaves more like a cylindrical system, and an annulus with a large S value behaves more like a long, rectangular system. In these stability diagrams, as is the case for cylindrical systems, the azimuthal wave number with the lowest curve at a given radi us ratio is the azimut hal wave number of the disturbance most destabilizing to the system at that radius ratio, and, thus, is the azimuthal wave number of the onset flow pattern at that radius ratio. Also, ch anges in the number of radially aligned convective rolls present in the onset flow pattern of a given azimuthal wave number are

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120 indicated by inflections in the stability curv es. However, by examining the computed onset velocity profiles while doing the computations for this research, it was seen that not all inflections in the stability curves correspond to th ese pattern transitions. Some of the less severe inflections simply correspond to slig ht variations in the instability of the system to a particular pattern with unchanging numbers of radially and azimuthally a ligned convective rolls. In general, the number of radially aligned convective rolls in the onset flow pattern for a given azimuthal wave number decreases as the radius ratio is increased. This is logical because in annular systems with higher radius ratios, it is energetically more and more difficult for a flow pattern with a greater number of radially ali gned convective rolls to ex ist due to the no-slip conditions on velocity imposed by the inne r and outer radial wa lls of the annulus. It has been explained, now, how the radius ratio in an annular syst em affects the numbers of azimuthally and radially aligned convective rolls present at the onset of convection. In the stability diagrams above, the effect of S on the number of azimuthally aligned rolls is obvious; the effect of S on the number of radially aligned rolls ha s been explained, t hough it is not quite so easy to see on the stability diagrams since some of the less severe inflections on the diagrams do not correspond to pattern transiti ons. Another key point is that the aspect ratio of an annular system determines the range of values over wh ich the numbers of azimuthally and radially aligned convective rolls vary in response to varia tions in the radius ratio. A system of large aspect ratio, since it is more spacious, will ge nerally have higher numbers of both azimuthally and radially aligned convective rolls present in its onset flow patterns than will a system of smaller aspect ratio. Stork and Mller’s experiments, for example, addressed large aspect ratios, such as A = 5, and the onset patterns they observed ha d quite large azimuthal wave numbers. As

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121 a side note, the MATLAB computational program wr itten for this research was able to predict the same onset flow patterns that St ork and Mller observed experimentally. 7.2.2 Non-Constant Viscosit y Computations: Annulus Table 7-6 presents the results of the non-cons tant viscosity computa tions for the sets of annular dimensions that we re considered in experiments. Agai n, the temperature at the top of the test section was 30.0 C. The computed velocity profiles for these a nnular cases are shown below (Figures 7-21 – 7-26). Again, the velocities in these profiles ar e scaled, negative veloci ties represent downward flow in the z-direction (which extends into a nd out of the plane of the diagram), and positive velocities represent upward fl ow in the z-direction. 7.2.3 Experiments: Annulus In this section, a ta ble containing the experimental results for annular systems is given. Then, photos of the onset flow patterns for each of the cases are given. In the table, the value used for the kinematic viscosity in the Raylei gh number is the value corresponding to the mean temperature in the fluid just be fore the onset of convection. Photos of the onset flow patterns observe d in the experiments on annular systems are given as Figures 7-27 – 7-32. Any noticeable asymmetries in these pattern s are, of course, simply due to small imperfections in the experimental system that cau se it to slightly differ from the theoretical system. If it is noticeable that some of these fl ow patterns appear to be more strongly flowing than others, the reason for that is simple. In several experimental cases, the computationally predicted onset flow pattern w ould show up at supercritical c onditions (in which the vertical temperature difference in the experiment ex ceeds the critical value determined by the

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122 experiment), and when this happened, the flow was even stronger than it would be at onset, allowing the pattern to be more easily viewed in photos. In other cases, however, the computationally predicted onset flow pattern woul d appear only near the critical temperature difference determined by the experiment, and woul d thus be flowing relatively weakly, and be less distinct in appearance. Figure 7-27 is the m = 0:(0,1) onset flow pattern seen in the annular container with Ro = 8.75 mm. The flow pattern was pa rticularly weak and sometimes difficult to discern in this particular case. A reason suspected for this is that the systems with Ro = 8.75 mm were the smallest in radial extent of all systems consider ed, and the combination of the radially confining dimensions of the system and the relatively larg e radial components of ve locity associated with the m = 0:(0,1) flow pattern resulted in much of the aluminum tracer powder in the test fluid quickly adhering to the radial wall s and bottom plate of the system. 7.2.4 Comparison of Results: Annulus Table 7-8 includes all of th e results obtained for annular systems. The “% Error” column, an indication of the absolute value of th e percentage of error between the experimental critical temperature difference and the critic al temperature difference obtained from the nonconstant viscosity computation, is calculate d with respect to the non-constant viscosity computational result. Again, the error has been adjusted to exclude the portion which arises from uncertainty in the thermophysical property values assumed in co mputations. Thus, the error percentages shown represent experimental error due to other sources; several possible error sources are described in Section 7. 3. Also, as expected, the nonconstant viscosity results more closely match the experimental results than do the constant viscosity results; this is because of

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123 the fact that the non-constant viscosity com putation more realisti cally represents the experimental system. As this table shows, the agreement between the computational and e xperimental critical temperature difference results is best in the cases in which S has a smaller value. Even though a full analysis of the errors in the results is gi ven in Section 7.3, a po ssible explanation for the poorer agreement between computations and experiments for some of the annular cases will be given here. The cases in which S is of higher value are the cases in which the center-block of the annulus is larger, and the annular gap width is smaller. It is suspected that the reason for the larger errors in the experimental critical temperature difference results for large S have to do with the small annular gap width. Slight imperfectio ns in the centering of the annular center-block within the outer annular wall of the experimental system could exist due to extremely small errors in machining, such as errors in the mach ining of the bottom plate of the test section (in terms of the placement of the notch that the annu lar center-block is nested in) or the lucite center-blocks of the annular syst ems. Any small imperfection in the centering of the annular center-block would slightly affect the critical co nditions for convection, but would have an even larger effect on the critical conditions for convec tion in cases in which the annular gap width is smaller, and thus could cause a more noticeable in crease in the experimental critical temperature difference in those cases. The reas oning behind this statement is as follows. In an annulus with an off-centered middle-piece, some region of th e annulus has a smaller gap width than the originally intended gap width, a nd the initiation of convective flow in this region, due to its higher degree of mechanical confinement, would be even more difficult than the initiation of flow elsewhere in the annular test section, and would consequently make the entire system more stable to convection. This effect would stre ngthen as the annular center-block increases in

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124 diameter, because annular center-blocks of larg er and larger diameter would introduce smaller and smaller minimum gap widths to a system in which the annular center-block is slightly offcentered. This effect, then, woul d not be expected to show up as strongly in cases with larger annular gap widths. The experiment al results, in fact, support this hypothesis. Of the three cases in which S is larger, the case in which th e annular gap width is largest (Lz = 6.53 mm, Ro = 11.28 mm, S = .40) has the smallest error of the three (s ee Table 7-8), the case with the second largest annular gap width (Lz = 6.85 mm, Ro = 8.75 mm, S = .30) has the second smallest error, and the case with the smallest annular gap width (Lz = 7.18 mm, Ro = 11.51 mm, S = .50) has the largest error. Since the imperfections present in each experimental run could slightly differ, the degree of off-centering in annular experiments was different in each run. Still, some test measurements of the annular gap width at differ ent locations within the test s ection showed that the degree of possible off-centering is very sm all (hundredths of a millimeter) co mpared to the width of the annular gap. Another obvious point of discussion in Table 7-8 is that, in one case, the experimentally observed onset flow pattern was not the computati onally predicted pattern. For that particular system, the computation shows the difference betw een the onset vertical temperature differences for the m = 0:(0,1) pattern, which was pr edicted computationally, and the m = 2:(2,1) pattern which was experimentally observed, to be about .2 C. This is outside of the range of experimental error. It is suspected that the diffe rence in patterns may be a result of imperfections related to the off-centeri ng of the annular center-block, which were just discussed, or perhaps some similarly small experimental imperfections Off-centering of the annular center-block would make it very difficult for the m = 0:(0,1) pattern, which is axisymmetric in the azimuthal direction, to form or be sustai ned (an annular system with an of f-centered center-block is, itself,

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125 not azimuthally axisymmetric). This would explain why, even for the case in which an m = 0:(0,1) was obtained experimentally the flow was very weak. A full error analysis follows in Section 7.3. All of these results are summarized in the first section of Chapter 8. 7.3 Error Analysis Unavoidably, some error is present in the expe rimental results. The main sources of error in the experimental results are error in the ther mophysical properties used in the computations, errors in the exact dimensions of the test sect ion (due to machining e rror), and error in the accuracy of temperature measurements. Imperfect leveling of the test section should also be mentioned as a source of potential error. One possible source of errors in the thermophys ical properties is error in the viscosity measurements taken to determine the dependence of the viscosity of the oil on temperature for the non-constant viscosity computations. Again, a Cole-Parmer 98936 series viscometer was used for viscosity measurements. The oil being te sted needed to be kept at several constant temperature values for set lengths of time (mor e details on the viscosity measurements are given in Appendix A). Its temperatur e was regulated by circulating water from the large NESLAB water bath mentioned in Chapter 6. Thus, the potential error in the viscosity measurements arises from error in the temperature control pr ovided by the NESLAB water bath. The NESLAB water bath provided great temperat ure control, keeping the temperat ure of the oil constant with an error of no more than approximately .02 C. This could cause a small error in the viscosity values used in the computations, but the maximum possible error in the viscosity is so extremely small (as shown by test computations) that it doe s not noticeably affect the computed critical temperature differences.

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126 The exact values of the thermophysical pr operties of silicone oil assumed for the computations (shown in Appendix A) are taken from data provided by Dow Corning and, thus, are reasonably close to the actual properties of the si licone oil used in experiments. Still, they are, of course, not perfect matc hes to the actual experimental thermophysical properties of the silicone oil. While there is no way to know the exact thermophysic al properties of the silicone oil used in experiments, the total experimental er ror can be adjusted to exclude the portion due to errors in the thermophysical prop erty values assumed in computat ions by comparing ratios of the results; this will now be explained. First, a ratio must be taken of the ([ T]crit Lz 3) value obtained from the constant viscos ity computation for a certain test case (call it Case 2) to the ([ T]crit Lz 3) value for a different test cas e (call it Case 1). Then, a ratio must be taken of the ([ T]crit Lz 3) value obtained from the experimental result from Case 2 to that from Case 1. Lastly, those two ratios must be compared. The “% Error” betw een the two ratios is calculated simply as 100 Ratio nal Computatio Ratio al Experiment Ratio nal Computatio The reasoning behind this process follows. Taking a ratio of ([T]crit Lz 3) values is the same as taking a ratio of the critical Rayleigh numbers, for the constant viscosity computational results and the experimental results, for a given pair of systems, with the thermophys ical properties divided out. The ratio of the critical Rayleigh numbers obtained from the cons tant viscosity computational results should be very close to the ratio of the critical Rayleigh numbers obtained from the experimental results. Since the thermophysical properties are divided out, any er ror between the ([T]crit Lz 3) ratio from the computational results and the ([ T]crit Lz 3) ratio from the experimental results must be due to errors in the experiment other than the error associated w ith uncertainty in the thermophysical properties. Thr oughout this explanation, the cons tant viscosity computations have been referred to rather than the non-consta nt viscosity computati ons. Still, the entire

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127 explanation applies for the non-constant viscos ity computations, as well, because the only difference in the thermophysical properties for the two cases is the viscosity value, and, as viscosity measurements show, the variation in visc osity over the vertical temperature differences considered here is reasonably small. This e rror-adjustment procedure was followed to obtain the error percentages shown in Tables 7-4 and 7-8 (which are based on the non-constant viscosity computational results). It is applied to every te st case except for one. The excluded test case is the reference case in this error-adjustment procedur e, and its results are used in the denominator of every ratio taken; in the above explanation, Case 1 is the reference case, and for the error percentages in Tables 7-4 and 7-8, the cylindrical case with an asp ect ratio of 1.28 is used as the reference case. Thus, the error percentage shown for the cylindr ical case of aspect ratio 1.28 does still include error due to uncertainty in assumed thermophysical properties, and it is calculated simply as 100 ] [ ] [ ] [crit crit critT nal Computatio T al Experiment T nal Computatio The significant error between the computational and experimental results for some of th e annular systems seen in Table 7-8, then, must be attributed to fact ors other than uncertaint y in the experimental thermophysical properties. In fact, it is likely that the relatively large error in those cases arises from the possible slight off-cente ring of the center-block of the a nnular system (this effect is described thoroughly in Section 7.2.4). The dimensions of the lucite rings and inne r annular pieces made for the experiment were measured as accurately as possibl e, with a depth micrometer, but still an error of approximately .025 mm (.0254 mm is a thousandth of an inch) aris es in the measured dimensions. Some test computations show that this error could typically l ead to an error of about .03 C. This will be included in the experimental results along with the next error to be discussed.

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128 Some error in temperature measurement was pres ent in the experimental results. Each of the OMEGA thermistors used in the experiment had an error of .15 C, but since multiple thermistors were used in both the bottom and top water baths, the overall error in the temperature reading at each of those locations was reduced. The error in the temp erature reading of the continuously stirred bottom water bath was reduced to .05 C si nce three thermistors were used to read the temperature in the bath. The consta nt temperature (30.0 C) of the top water bath, while it was monitored using two OMEGA thermi stors, was controlled by the large NESLAB water bath. The NESLAB bath was able to main tain a temperature within approximately .02 C of 30.0 C (and the thermistor s near the upper flow-t hrough water bath showed that there was not significant heat loss to the surroundings as the water flowed from the NESLAB bath to the upper flow-through water bath). Thus, a total error of .07 C is associated with experimental measurements of the vertical temperature differe nce. Adding this to the .03 C machiningrelated error gives a total error of .1 C for the experimental results. Note, then, that the “ .1 C” error indications on the tables of results in clude both the error in temperature measurement and the error due to uncertainties in the precision of machining. The critical temperature differences in the e xperiments were never any lower than 5.0 C, which means that the experimental error of .1 C was always an error of less than 2 %. This means that any portion of the experimental error gr eater than approximately 2 % (see Tables 7-4, 7-8) is due to factors other than those considered to reach the .1 C error estimate. One such factor, in the case of the annular systems, is the possible off-ce ntering of the annular centerblock, which has already been explained in Se ction 7.2.4. Another possible error source was imperfect leveling of the test section. Great ca re was taken in ensuring that the bottom plate of the test section was level before assembling the experimental apparatus on top of it. A bubble-

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129 level was placed on the bottom copper plate of the te st section, and its reading was used to guide adjustments to a leveling plate upon which the entire a pparatus sat. Still, th e level, itself could not be perfectly accurate. Also, it is likely that the tightening of the four screws which were used to tighten down the lucite clamp on the outer ring of the test section wa s not perfectly equal on all four screws, and thus, the top of the test se ction may not have been perfectly level either. Again, the results in this chapter are su mmarized at the beginning of Chapter 8. Table 7-1. Constant viscosity computational results: critical te mperature difference and onset flow pattern for sets of cylindrical di mensions considered in experiments. Onset Flow Pattern Lz (mm) Lr (mm) A [ T ]crit (C) Racrit m = 6.85 8.75 1.285.5819070:(0,2) 7.18 11.51 1.604.8218901:(1,3) 6.53 11.28 1.736.2318411:(1,3) Table 7-2. Non-constant viscos ity computational results: critical temperature difference and onset flow pattern for sets of cylindrical dimensions considered in experiments. Onset Flow Pattern Lz (mm) Lr (mm) A [ T ]crit (C) Racrit m = 6.85 8.75 1.285.8019060:(0,2) 7.18 11.51 1.605.0418891:(1,3) 6.53 11.28 1.736.4318401:(1,3) Table 7-3. Experimental results: critical temperature difference and onset flow pattern for sets of cylindrical dimensions. Onset Flow Pattern Lz (mm) Lr (mm) A [ T ]crit (C) Racrit m = 6.85 8.75 1.285.9 .119430:(0,2) 7.18 11.51 1.605.2 .119531:(1,3) 6.53 11.28 1.736.5 .118621:(1,3) In these results, Lz and Lr values have an error of .025 mm, and the error in [ T]crit includes a .03 C due to this, as well as a .07 C error from temperature measurement.

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130 Table 7-4. Summary of comput ational and experimental results for cylindrical systems. Constant Constant NonConstant NonConstant Visc., Visc., Visc., Visc., Expt. Expt. % T (C) Pattern T (C) Pattern T (C) Pattern Error Aspect Ratio: 1.28, Lz = 6.85 mm, Lr = 8.75 mm, Figures 7-3, 7-6, see note below table 5.58 0:(0,2) 5.80:(0,2) 5. 9 .10:(0,2) 0.03 3.48 Aspect Ratio: 1.60, Lz = 7.18 mm, Lr = 11.51 mm, Figures 7-4, 7-7 4.82 1:(1,3) 5.041:(1,3) 5. 2 .11:(1,3) 0.00 3.37 Aspect Ratio: 1.73, Lz = 6.53 mm, Lr = 11.28 mm, Figures 7-5, 7-8 6.23 1:(1,3) 6.431:(1,3) 6. 5 .11:(1,3) 0.00 2.24 In the experimental results, Lz and Lr values have an error of .025 mm, and the error in [ T]crit includes a .03 C due to this, as well as a .07 C error from temperature measurement. The range of erro r percentages shown corresponds to the indicated .1 C error in the experimental critical temp erature differences. Error percentages are shown with the contribution from uncertainty in the thermophysical properties scaled out. This scaling was done with the cylindrical case of aspect ratio 1.28 used as the reference case, so that case is the only case for which the error percentage still includes the effects of un certainty in the thermophysical properties. Table 7-5. Constant viscosity computational results: critical te mperature difference and onset flow pattern for sets of annular dime nsions considered in experiments. Onset Flow Pattern Lz (mm) Ro (mm) Ri (mm) A S [ T ]crit (C) Racrit m = 6.85 8.75 1.40 1.280.166.892355 0:(0,1) 6.85 8.75 2.62 1.280.307.512567 3:(3,0) 7.18 11.51 1.38 1.600.125.352099 0:(0,1) 7.18 11.51 5.76 1.600.506.902706 4:(4,0) 6.53 11.28 1.18 1.730.107.032079 2:(2,1) 6.53 11.28 4.51 1.730.407.932345 4:(4,0) Table 7-6. Non-constant viscos ity computational results: critical temperature difference and onset flow pattern for sets of annular dimensions considered in experiments. Onset Flow Pattern Lz (mm) Ro (mm) Ri (mm) A S [ T ]crit (C) Racrit m = 6.85 8.75 1.401.280.167.082353 0:(0,1) 6.85 8.75 2.621.280.307.672564 3:(3,0) 7.18 11.51 1.381.600.125.572098 0:(0,1) 7.18 11.51 5.761.600.507.092704 4:(4,0) 6.53 11.28 1.181.730.107.212077 2:(2,1) 6.53 11.28 4.511.730.408.072343 4:(4,0)

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131 Table 7-7. Experimental results: critical temperature difference and onset flow pattern for sets of annular dimensions. Onset Flow Pattern Lz (mm) Ro (mm) Ri (mm) A S [ T ]crit (C) Racrit m = 6.85 8.75 1.40 1.280.167.7 .12573 0:(0,1) 6.85 8.75 2.62 1.280.308.8 .12969 3:(3,0) 7.18 11.51 1.38 1.600.125.8 .12189 2:(2,1) 7.18 11.51 5.76 1.600.508.4 .13242 4:(4,0) 6.53 11.28 1.18 1.730.107.5 .12165 2:(2,1) 6.53 11.28 4.51 1.730.409.1 .12664 4:(4,0) In these results, Lz and Lr values have an error of .025 mm, and the error in [T]crit includes a .03 C due to this, as well as a .07 C error from temperature measurement. Table 7-8. Summary of com putational and experimental results for annular systems. Constant Constant NonConstant Non-Constant Visc., Visc., Visc., Visc., Expt. Expt. % T (C) Pattern T (C) Pattern T (C) Pattern Error A = 1.28, S = .16, Lz = 6.85 mm, Ro = 8.75 mm, Ri = 1.40, Figures 7-21, 7-27 6.89 0:(0,1) 7.080:(0,1) 7. 7 .10:(0,1) 5.51 – 8.29 A = 1.28, S = .30, Lz = 6.85 mm, Ro = 8.75 mm, Ri = 2.62, Figures 7-22, 7-28 7.51 3:(3,0) 7.673:(3,0) 8. 8 .13:(3,0) 11.41 13.97 A = 1.60, S = .12, Lz = 7.18 mm, Ro = 11.51 mm, Ri = 1.38, Figures 7-23, 7-29 5.35 0:(0,1) 5.570:(0,1) 5. 8 .12:(2,1) 0.56 4.09 A = 1.60, S = .50, Lz = 7.18 mm, Ro = 11.51 mm, Ri = 5.76, Figures 7-24, 7-30 6.90 4:(4,0) 7.094:(4,0) 8. 4 .14:(4,0) 15.13 17.90 A = 1.73, S = .10, Lz = 6.53 mm, Ro = 11.28 mm, Ri = 1.18, Figures 7-25, 7-31 7.03 2:(2,1) 7.212:(2,1) 7. 5 .12:(2,1) 0.81 3.54 A = 1.73, S = .40, Lz = 6.53 mm, Ro = 11.28 mm, Ri = 4.51, Figures 7-26, 7-32 7.93 4:(4,0) 8.074:(4,0) 9. 1 .14:(4,0) 9.55 11.98 In the experimental results, Lz, Ro, and Ri values have an error of .025 mm, and the error in [ T]crit includes a .03 C due to this, as well as a .07 C error from temperature measurement. The range of erro r percentages shown corresponds to the indicated .1 C error in the experimental critical temp erature differences. Error percentages are shown with the contribution from uncertainty in the thermophysical properties scaled out. This scaling was done with the cylindrical case of aspect ra tio 1.28 used as the reference case.

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132 Critical Ra vs. Aspect Ratio, Cylinder, Constant Viscosity1700 1900 2100 2300 2500 2700 2900 0.71.21.72.22.73.2 Aspect RatioCritical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 Figure 7-1. Constant viscosity computational results: Racrit vs. A for cylindrical systems.

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133 Critical Ra vs. Aspect Ratio, Cylinder, Constant Viscosity1740 1790 1840 1890 1940 2.52.72.93.13.33.5 Aspect RatioCritical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 Figure 7-2. Constant viscosity co mputational results: Racrit vs. A for cylindrical systems, closeup view.

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134 Figure 7-3. Computed velocity profile, cross-secti onal view, Lz = 6.85 mm, Lr = 8.75 mm, m = 0:(0,2), [ T]crit = 5.80 C.

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135 Figure 7-4. Computed velocity profile, cross-sectional View, Lz = 7.18 mm, Lr = 11.51 mm, m = 1:(1,3), [ T]crit = 5.04 C.

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136 Figure 7-5. Computed velocity profile, cross-sectional view, Lz = 6.53 mm, Lr = 11.28 mm, m = 1:(1,3), [ T]crit = 6.43 C. Figure 7-6. Photo of onset flow pattern, Lz = 6.85 mm, Lr = 8.75 mm, m = 0:(0,2).

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137 Figure 7-7. Photo of onset flow pattern, Lz = 7.18 mm, Lr = 11.51 mm, m = 1:(1,3). Figure 7-8. Photo of onset flow pattern, Lz = 6.53 mm, Lr = 11.28 mm, m = 1:(1,3).

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138 Critical Ra vs. S A = .75 1500 6500 11500 16500 21500 26500 00.10.20.30.40.50.60.70.8 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 Figure 7-9. Constant visc osity computational results: Racrit vs. S for annular systems with A = .75.

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139 Critical Ra vs. S A = 1.28 1500 2500 3500 4500 5500 6500 7500 8500 9500 00.10.20.30.40.50.60.70.8 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 Figure 7-10. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 1.28.

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140 Critical Ra vs. S, A = 1.281900 2100 2300 2500 2700 2900 3100 3300 3500 00.10.20.30.40.50.6 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 Figure 7-11. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 1.28, close-up view.

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141 Critical Ra vs. S A = 1.60 1500 2500 3500 4500 5500 6500 00.10.20.30.40.50.60.7 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 Figure 7-12. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 1.60.

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142 Critical Ra vs. S A = 1.60 1900 2100 2300 2500 2700 2900 00.10.20.30.40.50.6 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 Figure 7-13. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 1.60, close-up view.

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143 Critical Ra vs. S A = 1.73 1500 2000 2500 3000 3500 4000 4500 5000 00.10.20.30.40.50.60.7 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 Figure 7-14. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 1.73.

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144 Critical Ra vs. S A = 1.73 1900 2000 2100 2200 2300 2400 2500 2600 00.10.20.30.40.5 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 Figure 7-15. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 1.73, close-up view.

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145 Critical Ra vs. S A = 2.90 1700 1900 2100 2300 2500 00.10.20.30.40.50.60.7 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 m = 9 m = 10 Figure 7-16. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 2.90.

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146 Critical Ra vs. S A = 2.90 1750 1850 1950 2050 2150 2250 2350 00.10.20.30.40.50.6 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 m = 9 Figure 7-17. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 2.90, close-up view.

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147 Critical Ra vs. S A = 3.40 1700 1800 1900 2000 2100 2200 2300 240000.10.20.30.40.50.60.7 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 m = 9 m = 10 m = 11 m = 12 Figure 7-18. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 3.40.

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148 Critical Ra vs. S A = 3.40 1750 1800 1850 1900 1950 2000 00.10.20.30.40.5 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 m = 9 Figure 7-19. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 3.40, close-up view 1.

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149 Critical Ra vs. S A = 3.40 1850 1950 2050 2150 2250 23500.40.450.50.550.60.650.7 S (Radius Ratio)Critical Ra m = 0 m = 1 m = 2 m = 3 m = 4 m = 5 m = 6 m = 7 m = 8 m = 9 m = 10 m = 11 m = 12 Figure 7-20. Constant viscos ity computational results: Racrit vs. S for annular systems with A = 3.40, close-up view 2.

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150 Figure 7-21. Computed velocity profile, cross-sectional view, Lz = 6.85 mm, Ro = 8.75 mm, S = .16, m = 0:(0,1), [T]crit = 7.08 C.

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151 Figure 7-22. Computed velocity profile, cross-sectional view, Lz = 6.85 mm, Ro = 8.75 mm, S = .30, m = 3:(3,0), [ T]crit = 7.67 C.

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152 Figure 7-23. Computed velocity profile, cross-sectional view, Lz = 7.18 mm, Ro = 11.51 mm, S = .12, m = 0:(0,1), [ T]crit = 5.57 C.

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153 Figure 7-24. Computed velocity profile, cross-sectional view, Lz = 7.18 mm, Ro = 11.51 mm, S = .50, m = 4:(4,0), [ T]crit = 7.09 C.

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154 Figure 7-25. Computed velocity profile, cross-sectional view, Lz = 6.53 mm, Ro = 11.28 mm, S = .10, m = 2:(2,1), [T]crit = 7.21 C.

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155 Figure 7-26. Computed velocity profile, cross-sectional view, Lz = 6.53 mm, Ro = 11.28 mm, S = .40, m = 4:(4,0), [ T]crit = 8.07 C. Figure 7-27. Photo of onset flow pattern, Lz = 6.85 mm, Ro = 8.75 mm, S = .16, m = 0:(0,1).

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156 Figure 7-28. Photo of onset flow pattern, Lz = 6.85 mm, Ro = 8.75 mm, S = .30, m = 3:(3,0). Figure 7-29. Photo of onset flow pattern, Lz = 7.18 mm, Ro = 11.51 mm, S = .12, m = 2:(2,1).

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157 Figure 7-30. Photo of onset flow pattern, Lz = 7.18 mm, Ro = 11.51 mm, S = .50, m = 4:(4,0). Figure 7-31. Photo of onset flow pattern, Lz = 6.53 mm, Ro = 11.28 mm, S = .10, m = 2:(2,1).

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158 Figure 7-32. Photo of onset flow pattern, Lz = 6.53 mm, Ro = 11.28 mm, S = .40, m = 4:(4,0).

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159 CHAPTER 8 CONCLUSIONS AND POSSIBLE FUTURE STUDIES This chapter will first summarize some important findings of this research, and will then mention some possible future work that could be done in this area. 8.1 Summary The critical temperature differences determined by the non-constant viscosity computations were generally higher than t hose determined by the constant viscosity computations. As expected, since they more rea listically model the experimental system, in all cylindrical and annular cases, th e non-constant viscosity comput ations more closely match the experimental results than do the constant viscosity computations. Carrying out computations for the critical condi tions in cylindrical systems shows, also as expected, that the onset flow patterns in systems of larger aspect ratio include larger numbers of convective rolls so that they can spatially fill th e test section in a more energetically favorable manner. Even though it was known already, it is notable that th e relation between the aspect ratio and the critical Rayleigh number, for all azimuthal wave numbers, in constant viscosity cylindrical systems, is identical regardless of the exact dimensions of the system or the thermophysical properties of the system. The agreement between computations and experiments for cylindrical systems is very good in all cases considered. The progression of onset flow patterns with changing system dimensions in the annular case is a bit more complex. Stil l, similar to what was seen in cylindrical systems, the relations between the radius ratio, the as pect ratio, and the critical Rayl eigh number in constant viscosity annular systems are again identical, for all azimu thal wave numbers, regardless of the exact system dimensions or thermophysical properties. As explained, this is due to the timeindependence of the system at the onset of convection.

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160 In annular systems with small S values, as the center-block becomes very small in diameter, the onset flow pattern is the same as the onset pattern in the corresponding cylindrical system. In the cylindrical cases considered in th is research, onset flow patterns of the azimuthal wave numbers m = 0, 1, and 2 are most common, and it is generally these onset patterns which are seen in the annular systems with very small S values. Onset flow patterns with higher azimuthal wave numbers generally occurred in systems with larger S values. In the annular systems considered in this research, the onset pa tterns with azimuthal wave numbers 0, 1, and 2 typically included radially aligne d convective rolls as opposed to only azimuthally aligned rolls, and onset patterns with higher azimuthal wave numbers typically included only azimuthally aligned rolls. At high S values (greater than .30, for example), a higher S value generally corresponds to a higher azimuthal wave number. The number of radially ali gned convective rolls included in the onset flow pattern is generally lower at higher radius ratios (because at higher radius ratios, the annular gap width is smaller and there is not as much room for the ra dial motions of the radially aligned rolls). Changes in the number of radially aligned rolls present in the onset fl ow pattern for a given azimuthal wave number can be indicated by inflec tions in the curve for that azimuthal wave number on a stability diagram, though not all in flections indicate changes in the number of radially aligned rolls. The numbers azimuthally and radially aligned convective rolls which form at onset in an annular system are also dependent upon the aspect ratio of the system. An annular system with a larger aspect ratio will, of course, tend to ha ve onset flow patterns with larger numbers of azimuthally and radially aligned convective roll s because there is simply more space within which they may form and flow.

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161 Better agreement between the critical temperature differences determined by computations and those determined by experiment s is seen for the annular cases with smaller S values. The suspected reason for the poorer agreement when the annul ar center-block is larger is that slight imperfections in th e centering of the annular center-b lock within the outer annular wall are having a significant stabiliz ing effect on the system. It is also suspected, then, that the strength of this stabilizing effect is dependent on the originally intended gap width in each case, and that the effect should be more pronounced wh en the gap width is smaller. The reasoning behind this is explained in Chapter 7, and, esse ntially, it is that the st rong resistance to flow provided by the region of smallest annular gap widt h in an annular system with a slightly offcentered center-block stabili zes the entire system. The convectiv e behavior of an annular system with a slightly off-centered center-block is very interesting from the standpoint of physics, and it is certainly a topic open to furt her interpretation and study. The experimental results agree with the hypothesis that stabilizing eff ect introduced to annular system s by slight off-centering of the center-block is stronge r in cases for which the originally in tended annular gap width is smaller. Also, in one case (see Table 7-8), the onset flow pattern predicted by co mputation was not the pattern seen experimentally. This is likely to be a consequence of either the off-centering effect just mentioned, or some other small experimental imperfection. 8.2 Future Studies Many parameters in an annular problem could be varied in order to examine their effects on the stability of the system. One of the most obvious parameters to investigate, which has become of particular interest based on the result s of the experiments for this research, is the offcentering of the annular cen ter-block. It is suspec ted that off-centering th e annular center-block should increase the stability of the system to buoyancy-driven convection. It is not so easily

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162 predictable, though, how this woul d affect the onset flow patter ns, except that it would make axisymmetric onset patterns impossible. It would also be interesting to run experi ments on annular systems with extremely small center-blocks, to see ju st how close the critical conditions can get to those for corresponding open cylindrical systems. Another great study would be to investigat e Rayleigh convection in an annular system using a liquid metal as a test fluid. Such a study would better repres ent the thermophysical properties in a crystal growth sy stem. This would simply require choosing a metal that is liquid within a range of temperatures that would be ma nageable from an experimental standpoint, and that has thermophysical properties that, in some way, are simila r to those of an industrially important liquid metal. A study of Rayleigh an d Marangoni convection in an annular system with multiple vertically-stacked fluid layers coul d be applied quite directly to crystal growth applications, as well. Lastly, nonlinear calculati ons would provide valuable in formation about the convective behavior of an annular system that the linea rized computations shown here simply cannot provide. They would allow the determination of supercritical flow behavi ors, as wells as the exact velocities in the system. Hopefully some of these problems will be addr essed, because they have potential to be very interesting and could be very us eful in the crystal growth industry.

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163 APPENDIX A THERMOPHYSICAL PROPERTIES As mentioned, the dependence of viscosity on temperature was determined using a ColeParmer 98936 series viscometer. Essentially, what was involved in doing this was to measure the viscosity of the oil at a range of temperat ures between approximately 20.0 C and 50.0 C, with increments of 1.0 C near the middle of the range, and incr ements of 2.0 C near the low and high ends of the range. Each measured vi scosity value came from a measurement of the shearing effect created by rotati ng a metal spindle in a temperature-controlled chamber full of the silicone oil. The temperatur e of the oil chamber was regulated by circulating temperaturecontrolled water around it. The temperature of the circulating water was accurately controlled by the large NESLAB water bath. Except for the viscosity values, the silicone oil properties in Table A-1 are based on information provided by Dow Corning. In this research, the density and viscosity are the only properties being assumed to vary significantly with temperature (for Dow Corning 200 1 Stoke silicone oil, the variation with temperature in the other thermophys ical properties of interest is not significant). Thus, a temperature corres ponding to the thermophysical property value is shown only for those properties.

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164 Table A-1. Therm ophysical properties. Dow Corning 200 Material 1 Stoke Properties Silicone Oil Lucite Copper Sapphire Density (kg/m3) at 25 C 964---------------------------------------Thermal Conductivity (W/(m*(C))) 0.1548080.17 +++401 +++ 36.8192 ++++ Dynamic Viscosity (T) =.147105008 (kg/(m*s)), (T in C) + e-.017975325 T---------------------------------------Kinematic Viscosity (m2/s) at 35 C ++ 8.134296 10-5---------------------------------------Thermal Diffusivity (m2/s) 1.09039 10-7---------------------------------------Thermal Expansion Coefficient (C-1) 9.6 10-4---------------------------------------+ : The function for the dependence of dynamic viscosity on temperature is based on viscosity measurements taken by the author over a range of temper atures, using a ColeParmer 98936 series viscometer. This is th e varying viscosity which was used in calculations in which viscosity’s varia tion with temperature was considered. ++: The kinematic viscosity value given here is based on viscosity measurements taken by the author over a range of temperatures, using a Cole-Parmer 98936 series viscometer. This is the constant kinematic viscosity value which was used in calculations in which viscosity’s variation with te mperature was not considered. +++: As shown in Duane Johnson's dissertation (Johnson 1997). ++++: As shown in the 1974 paper by Koschmieder' s and Pallas (Koschmieder & Pallas 1974).

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165 APPENDIX B ADDITIONAL EXPLANATION OF EQUATIONS The following section provides further explan ation of the modeling equations and some of the modifications made to them. B.1 Boussinesq Approximation Obviously, for Rayleigh convection to be possibl e in a fluid, the fluid cannot be strictly incompressible (density must vary with temper ature). The Boussinesq approximation allows a fluid to be treated as incompressible in all te rms of the momentum equation except those relating to buoyancy forces. The approximation requires that the variation of density with temperature be accounted for only in the terms in which density is multiplied by gravitational acceleration; this is because, in any other terms, the effect of this density variation will be incomparably small in magnitude. The Boussinesq approximation is applied to Equation 3.1, which is S g P v v t v (3.1) so that it becomes S g T T T P v v t vR T R R R RR )) ( | 1 1 ( (B.1) in which TR and R are the reference temperatur e and density, respectively. The density variation term in this equa tion can be simplified by introducing the volumetric thermal expansion coefficient, which is RT RT | 1 (B.2) The volumetric thermal expansion coefficient is positive when the derivative of density with respect to temperature is negative. Th e momentum equation may then be written as

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166 S g T T P v v t vR R R R )) ( 1 ( (3.5) Once the Boussinesq approximation has been applied, a valid solution can be obtained only if the magnitude of the product of and T is much smaller than one, where T is the temperature difference from one vertical boundary of the fluid layer to the other. It can be seen by examining the thermophysical properties of the te st fluids used in this study, and considering the typical critical vertical te mperature differences seen in th is study, that the magnitude of is sufficiently small that *T will be much smaller than one. Also, to obtain a valid solution using the Boussinesq approximation, the magnitude of ( )/R must be much smaller than one, in which is the difference in density from one vertical boundary of the fluid layer to the other. The system of equations is being solved to determine the critical temperature difference at the onset of convection, and at th e onset of convection, the difference in density from one vertical boundary of the fluid layer to the other is quite small. An example of a situation in which the Boussinesq approximation is not valid is convection in stars. Stars convect inward due to their own gravitational forces and their values for the magnitude of ( ) /R are not much smaller than one. B.2 Nonlinearities in the Governing Equations For the system of Equations 3.2, 3.3, and 3.10 to have more solutions than the base state solution (motionless), a nonlineari ty must be present. At that point, an important nonlinearity can be seen. As shown by Equation 3.10, veloci ty is a function of te mperature. Thus, the T v term in Equation 3.2 is non linear. Of course, the v v term in the momentum equation is nonlinear, as well. This nonlinea rity is less important, though, b ecause the system of equations is being solved for the critical temperature difference at th e onset of conve ction and the

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167 velocities are of very small magnitude at that time, making v v to be of negligibly small magnitude. B.3 Characteristic Velocity and Characteristic Time In choosing which parameters are used to define the characteristic velocity ( v ) and characteristic time ( t), the goal is to make the characteristic velocity as small as possible. This is done to ensure that a ll of the flow behavior of the system (even the smallest velocities) are mathematically captured. Thus, the characteristic ve locity is chosen to be the ratio of the smaller of the kinematic viscosity and thermal diffusivity of the fluid phase to th e largest depth (vertical height, annular gap width, or diameter) of the flui d phase. The characteristic time is then defined using the same parameters. The forms of the char acteristic velocity and characteristic time are shown below, considering the example case that the fluid’s thermal diffusivity ( ) is smaller than its kinematic viscosity, and that the fluid is in a cylindric al container with a diameter ( D ) larger than its vertical height; this situation wa s quite typical for the cylindrical cases considered during this research. The typical value for the char acteristic velocity in the cylindrical cases in this research was D v (B.3) and the typical value for the characteristic time in the cylindrical cases in this research was 2D t (B.4) Since the characteristic time and velocity can be defined in seve ral ways depending on the thermophysical properties a nd dimensions of the fluid phase, the parameters used to define them will not be specified here.

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168 APPENDIX C DEVELOPMENT OF MATHEMATICAL MODEL FOR THE CONSTANT VISCOSITY CASE This section of the appendix presents the mathematical development of the modeling equations for the case in which the viscosity of the fluid is assumed to be constant with respect to temperature. As mentioned earlier, the viscos ity value used throughout the constant viscosity calculation is the value at 35.0 C, which is give n in Appendix A. The development shown here is parallel to what is given in Chapters 3 and 4, even though it is presente d more concisely. Note that the symbols used here are define d and explained in Chapters 3 and 4. C.1 Nonlinear Equations For this case, the original set of domain equati ons appears the same as it did in Chapter 3. This original set of domain equa tions includes the momentum e quation, the energy equation, and the continuity equation, which, respectively, are S g P v v t v (C.1) T k T v C t T CV V2 (C.2) ) ( v t (C.3) Again, the stress tensor, S can be expanded as 2 ) ( 2v v S (C.4) Applying the Boussinesq approximation bri ngs the momentum equation into the form S g T T P v v t vR R R R )) ( 1 ( (C.5)

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169 The Boussinesq approximation results in the disappearance of the time-derivative term of the continuity equation, as well. De fining a modified pressure, which is g P pR (C.6) further simplifies the momentum equation so that it becomes S g T T p v v t vR R R R ) ( (C.7) Now, substituting the expansion of the stress tensor into the momentum equation, and considering a constant viscosity, brings the momentum equation into the following familiar form: v g T T p v v t vR R R R 2) ( (C.8) Since this calculation does not include the variation of vi scosity with temperature, the dynamic viscosity at the reference temperature may be used as the dynamic viscosity in all equations, meaning that R (C.9) As explained in Chapter 3, a total of si xteen boundary conditions are needed for the annular system, while only twelve are needed for the cylindrical system. The unscaled forms of these boundary conditions are 0 z rv v v at z = 0, (C.10) bT T at z = 0, (C.11) 0 z rv v v at z = Lz, (C.12) tT T at z = Lz, (C.13)

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170 0 z rv v v at r = Ro, (C.14) 0 r T at r = Ro, (C.15) and, if considering an annular system, 0 z rv v v at r = Ri, (C.16) 0 r T at r = Ri. (C.17) C.2 Scaling A list of the scaling relations used is give n next. Dimensionless variables are marked using the “hat” symbol, while the “bar” indicate s a characteristic value for a variable. The scaling relations are v v v ˆ (C.18) T T T TR ˆ, (C.19) t bT T T (C.20) p p p ˆ (C.21) z RL v p, (C.22) t t t ˆ (C.23) zL ˆ (C.24) In this simple case, which does not consider the variation of viscos ity with temperature, TR will be chosen as the temperature at the top boundary of the system ( Tt). Applying these

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171 scalings to the domain equations yields the fo llowing dimensionless (though the “hat” symbol will now be discarded) equations: v T v T g L p v v v L t v t Lz R z R R z R 2 2 2) ( (C.25) T T v v L t T t Lz z 2 2 (C.26) v 0. (C.27) As in Chapter 3, all thermophysical propert ies in these equations are at the reference temperature unless otherwise noted. The subscript “R” is not included on the symbols for all of these properties. It is, however, left on the symbols R and R in order to better imitate the procedure shown in Chapter 3. Scaling leaves all boundary c onditions on velocity, as well as the boundary conditions on temperature at the radial walls, unchanged. S caling of the boundary cond itions also leads to a simplification of the conditions on temperature at the top and bottom walls Since the variation of viscosity with temperatur e is not being considered, TR has simply been chosen as Tt, and it is this which allows a simplification. The simp lified conditions on temperature at the top and bottom walls are 1 T at z = 0, (C.28) 0T at z = Lz (C.29) The reasoning for linearizing the system of modeling equations, and removing their time dependence, is explained thoroughly in Chapter 3. Here, this explanation is omitted and the focus is simply on the mathematical progression.

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172 C.3 Linearization The form of the linearization expansion to be used is the same whether the variation of viscosity with temperature is being considered or not. Again, the system is linearized around its motionless base state. The form of the expa nsion, considering the example case of expanding the velocity, is ... | 3 1 | 2 1 | |0 3 3 3 0 2 2 2 0 0 v v v v v (C.30) which may be rewritten as ... 3 1 2 13 3 2 2 1 0 v v v v v (C.31) The subscript “0” indicates values pertaining to the motionless base state, in which heat is transferred only in the verti cal direction by conduction, and so the modeling equations may be simplified by the fact that 0v is equal to zero. Applying this expansion to the modeling equations (which is actually the process of math ematically perturbing the equations with a small disturbance of magnitude ) results in the following set of linearized domain equations: 1 2 1 2 1 1 2) (v T v T g L p t v t Lz R z R (C.32) 1 2 1 0 1 2T v z T v L t T t Lzz z (C.33) 10 v (C.34)

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173 Linearizing the boundary conditions does not greatly change their appearance. The linearized boundary conditions are 01 1 1 z rv v v at z = 0, (C.35) 01 T at z = 0, (C.36) 01 1 1 z rv v v at z = Lz, (C.37) 01 T at z = Lz, (C.38) 01 1 1 z rv v v at r = Ro, (C.39) 01 r T at r = Ro, (C.40) and, if considering an annular system, 01 1 1 z rv v v at r = Ri, (C.41) 01 r T at r = Ri. (C.42) Now, the momentum equati on (Equation C.32) will be re written as its component equations in the r, and z directions. The operators will also be expanded in cylindrical coordinates. The components of the momentum equation are 2 1 1 11 2 1 2 2 2 2 2 2 2 2 1 1 2 v r v r z r r r r r p t v t Lr rz (C.43)

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174 2 1 1 1 11 2 1 2 2 2 2 2 2 2 2 1 1 2 rv r v r z r r r r p r t v t Lz (C.44) ) ( 1 11 2 1 2 2 2 2 2 2 2 1 1 2T v T g L v z r r r r z p t v t Lz zz z (C.45) C.4 Expansion into Normal Modes The variables will now be expanded into modes. In the example expansion below, the new variable representing only th e r-direction and z-direction de pendencies is marked with a “prime” symbol. The exponential time dependence wh ich each variable is assumed to possess is governed by an inverse time constant, called The form of the expansion, considering, for example, the expansion of pressure, is t ime z r p p ) (1 1. (C.46) At the onset of convection, the system is independent of time and the value of is 0. Thus, all time-derivative terms in the modeling equations may be eliminated once this expansion is applied. The final forms of the modeling equations for the case in which viscosity is considered constant with respect to temperature, with this expansion applied, and with the “prime” symbols dropped from the newly defined variables, are

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175 1 2 1 2 2 2 2 2 2 12 ) 1 ( 1 0 v r im v r m z r r r r pr, (C.47) rv r im v r m z r r r p r im1 2 1 2 2 2 2 2 2 12 ) 1 ( 1 0, (C.48) ) ( 1 01 2 1 2 2 2 2 2 2 1T v T g L v z r m r r r z pzz (C.49) 1 2 2 2 2 2 2 1 01 0 T r m z r r r v z T v Lz (C.50) z rv z v r im v r r1 1 11 0 (C.51) 01 1 1 z rv v v at z = 0, (C.52) 01 T at z = 0, (C.53) 01 1 1 z rv v v at z = Lz, (C.54) 01 T at z = Lz, (C.55)

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176 01 1 1 z rv v v at r = Ro, (C.56) 01 r T at r = Ro, (C.57) and, if considering an annular system, 01 1 1 z rv v v at r = Ri, (C.58) 01 r T at r = Ri. (C.59) This completes the presentation of the modeling equations for the case in which viscosity is considered constant with respect to temperat ure. These equations can be numerically solved on a computer as an eigenvalu e problem, for the eigenvalue T, as explained in Chapter 5.

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177 APPENDIX D MATLAB PROGRAM GENERAL FLOW-DIAGRAM Figure D-1. MATLAB program general flow-diagram. Input Thermophysical Properties, System Depths, N r N z, and T t Input [T]Guess Viscosity Value is Set According to [T]Guess For m = 0, 1, 2, 3, … T Value is Set as New [ T]Guess Value Based on Previous Iteration, is T Value Converged? YES NO Eigenvalue Solver Determines T and Ei g envecto r Output Critical T and Eigenvector for All m Values Considered Output Velocity Profile Graphs Based on Eigenvector

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178 APPENDIX E EXAMPLE MATLAB PROGRAMS This appendix presents, in their fu ll form, some examples of the MATLAB programs used for calculations in this research. Note that the programming done in MATLAB for this research was done with MATLAB 7 (Version 7.0.4.365, R14, Service Pack 2). The main program used when solving for the critical vertic al temperature difference and onset flow pattern for an annular system, including the variation of viscosity with temp erature, is presented third. The first and second programs presented are si mply files containing physical properties and depths, respectively, which ar e called upon by the main program. E.1 Physical Properties % PHYSICAL PROPERTIES % silicone oil disp('The physical properties of silicone oil (lower phase) will now be used.') disp('density (kg/m^3):') dens=964 disp('thermal conductivity (W/(m*K)):') k=0.154808 disp('kinematic viscosity (m^2/s):') %NOTE: This value has no relevance if the problem is being solved with the temperature-%dependence of viscosity being accounted for. kv=1e-4 disp('viscosity (kg/(m*s)):') visc=kv*dens disp('thermal diffusivity (m^2/s):') %(heat capacity is 1472.768000000 J/(kg*K)) td=1.09039e-7 disp('volumetric thermal expansion coefficient (K^(-1)):') tec=9.6e-4

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179 E.2 Depths for Annular System % DEPTHS disp('The dimensions being used.') %Vertical Phase Depth Lz=.00685292 %Inner Radius for Annulus Rin=.0175006/2*.17 %Outer Radius for Annulus Rout=.0175006/2 %Annular Gap Width Lr=Rout-Rin disp('Scaled depths:') %Set characteristic length. Unlike what is done in the written development of the equations, here Lchar is not simply replaced with Lz. The equations are left in a more general form, still containing “Lchar”. The particular value to be used for Lchar is set here. Lchar=Lr %Note, usually I have used Lr as Lchar even though Lz is usually used as Lchar in the written derivation %Scaled values for Rin, Rout, Lz, and Lr Rinscaled=Rin/Lchar Routscaled=Rout/Lchar Lzscaled=Lz/Lchar Lrscaled=Lr/Lchar %Actual span in non-spectral space of the r-direction a nd z-direction (spectral space spans from -1 to 1 in r-direction and z-direction when considering annulus) ro_leftboundary=Rin; ro_rightboundary=Rout; zo_lowerboundary=0; zo_upperboundary=Lz; E.3 Main Program: Annular System with Temperature-Dependent Viscosity clear all m=[]; output_ann_usv=[]; output_ann_usv_crit=[]; warningsmatrix=[]; Smin_dataset=[]; %Calling on other files for physical properties and depths propssiliconeoil35C_newoil

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180 dannsiliconeoil %In degrees Celsius: Tcold=30 %Singular value tolerances to later be used in removing spurious values. The tolerance choice is %kept at a fixed value (the value specified here) if desired. Also, a self-adjusting tolerance %can be set later. To choose a fixed tolerance, set tolchoice=1, and to choose an adjusting %tolerance, set tolchoice=2: tol1=1e-10; tolchoice=1; %What happens if an eigenvalue in the result has a very large imaginary part? %LIRT = 1 Stop Calculation, LIRT = 2 Warning Only LIRT=1; %The case for which it is desired to plot a velocity profile Nrforplot=13; Nzforplot=13; mforplot=3; %Tolerance for convergence of critic al temperature difference result with %respect to last iteration of visco sity's variation with temperature Tvcorrtol=1e-5 %Loops for N's for Nr=13 for Nz=13 %Loop for m's for m=0:15 %Guess Value for deltaT deltaTguess=7.3531; %Set initial value for the indicator of the converge nce of critical temperature difference result with %respect to last iteration of viscosity 's variation with temperature (Tvcorr) Tvcorr=1; %Loop for viscosity variation with respect to temperature while Tvcorr>=Tvcorrtol %Information for scaling original r coordinat es ("ro") to Chebyshev space ([-1:r:1]) %Lr=xo_upperboundary-xo_lowerboundary; %rchebyshev ("r") as a function of ro, in the form r=rm*ro+rb rmscaled=2/(Routscaled-Rinscaled); rm=2/(Rout-Rin); rb=-(Rin+Rout)/(Rout-Rin); %ro as a function of r=rchebyshev in the form ro=rom*r+rob rom=1/rm; rob=-rb/rm; %Chebyshev scale factors for di fferentiations of orders 1-2 rcsf1=rmscaled; rcsf2=rmscaled^2; %Generate grid points and differentiation matrix, r-direction

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181 %(Gauss-Lobatto points) if Nr==0, Dr=0; r=1; return, end r=cos(pi*(0:Nr)/Nr)'; r=flipud(r); c=[2; ones(Nr-1,1); 2].*(-1).^(0:Nr)'; R=repmat(r,1,Nr+1); dR=R-R'; Dr=(c*(1./c)')./(dR+(eye(Nr+1) )); %off-diagonal entries Dr=sparse(Dr-diag(sum(Dr'))); %diagonal entries Dr=rcsf1*Dr; %Dr matrix with Chebyshev scaling factor included Dr2=Dr^2; ro=rom*r+rob; %Information for scaling original z coordinates ("zo") to Chebyshev space ([-1:z:1]) %Lz=zo_upperboundary-zo_lowerboundary; %zchebyshev ("z") as a function of zo, in the form z=zm*zo+zb zmscaled=2/Lzscaled; zm=2/Lz; zb=1-zm*zo_upperboundary; %zo as a function of z=zchebyshev, in the form zo=zom*z+zob zom=1/zm; zob=-zb/zm; %Chebyshev scale factors for di fferentiations of orders 1-2 zcsf1=zmscaled; zcsf2=zmscaled^2; %Generate grid points and differentiation matrix, z-direction if Nz==0, Dz=0; z=1; return, end z=cos(pi*(0:Nz)/Nz)'; z=flipud(z); c=[2; ones(Nz-1,1); 2].*(-1).^(0:Nz)'; Z=repmat(z,1,Nz+1); dZ=Z-Z'; Dz=(c*(1./c)')./(dZ+(eye(Nz+1) )); %off-diagonal entries Dz=sparse(Dz-diag(sum(Dz'))); %diagonal entries Dz=zcsf1*Dz; %Dz matrix with Chebyshev scaling factor included Dz2=Dz^2; zo=zom*z+zob; zoscaled=zo./Lchar; %Setting Thot based on Tcold and deltaTguess Thot=Tcold+deltaTguess; %Determining the original (pre-onset) temperat ure profile in the system in the z-direction T0_zdir=(Thot-Tcold)/(Lz)*zo+Tcold; %Variation of viscosity with temperature for silicone oil vvfluid='silicone oil'; %Viscosity-Temperature curve determined by my measurements with a viscometer A0visc=.147105008294588; B0visc=-.0179753249042257; Bvisc=B0visc; %The viscosity at each location along the z-dire ction (prior to onset) in exponential form vv=A0visc*exp(B0visc*T0_zdir); %Finding the reference viscosity and reference temperature tempviscmat=[T0_zdir,vv];

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182 viscmean=mean(tempviscmat(:,2)); viscref=viscmean; %Now I will use viscref and any pair of vv(j), T0_zdir(j) to solve for Tref Tref=log(viscref/vv(1))/(B0visc)+T0_zdir(1); %Scaled vector of base state temperatur es (prior to onset) along the z-direction T0_zdir_scaled=(T0_zdir-Tref)/(deltaTguess); %To be consistent with the [-1:z:1] and [0:zo:Lz] (zo is z_original, which is z in original %coordinates) arrangement, I %should have the temperature and viscosity values corresponding %to z=Lz at the bottom of the viscosity vectors/matrices and those corresponding to z=0 at the %top. T0_zdir_scaled=flipud(T0_zdir_scaled); %Convert to matrix form for use later T0_zdir_scaled_full=kron(T0_zdir_scaled,ones(Nr+1,1)); %Top temperature, bottom temperature, and dT/dz in terms of scaled lengths: Tbottomscaled=(Thot-Tref)/(Thot-Tcold); Ttopscaled=(Tcold-Tref)/(Thot-Tcold); dT0dz=(Ttopscaled-Tbottomscaled)/Lzscaled; %Reference kinematic viscosity kvref=viscref/dens; %Characteristic velocity (m/s) and characteristic time (s): props=[kvref td]; depths=[Lr Lz]; vchar=min(props)/max(depths); tchar=(max(depths))^2/min(props) %Gravity in m/s^2: g=9.807; %Domain Equations %All variables shown in the following matrices are 1st order in epsilon %unless indicated as being base st ate variables by subscript "0" %Construct matrix operator, LHS (left-hand -side). Excluding boundary conditions. %Creating combination matrices including the reciprocal factors of r for use in the cylindrical %coordinate system equations roscaled=ro./Lchar; Dr2comb1=[]; eyercomb1=[]; eyercomb2=[]; eyematr=speye(Nr+1); for kk=1:(Nr+1), p=roscaled.^(-1); p2=roscaled.^(-2); Dr2comb1(kk,:)=Dr2(kk,:)+p(kk)*Dr(kk,:); eyercomb1(kk,:)=p2(kk)*eyem atr(kk,:); eyercomb2(kk,:) =p(kk)*eyematr(kk,:); end %NOTE ABOUT BC's: so, the way I have z set up, z=-1 is at the bottom (zo=0) %and z=1 is at the top (zo=Lz) %Motion r-dir %Rows will later be cut out of this matrix for the following BCs: %vr(r=Rin)=0,vr(r=Rout)=0,vr(z=0)=0,vr(z=Lz)=0 expterm=exp(Bvisc*deltaTguess*T0_zdir_scaled_full); %The coefficient of vr in the motion equation’s r-direction-component motr_vr=[(kron(speye(Nz+1),Dr2c omb1)+kron(Dz2,speye(Nr+1))((m^2+1)*kron(speye(Nz+1),eyercomb1)))+Bvisc *deltaTguess*dT0dz*k ron(Dz,speye(Nr+1))];

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183 for j=1:size(motr_vr,1), motr_vr(j,:)=motr_vr(j,:)*expterm(j);end %The coefficient of vtheta in the motion equation’s r-direction-component motr_vtheta=[(-2*i*m*kron( speye(Nz+1),eyercomb1))]; for j=1:size(motr_vtheta), motr_vtheta( j,:)=motr_vtheta(j,:)*expterm(j);end %The coefficient of vz in the motion equation’s r-direction-component motr_vz=[Bvisc*deltaTguess*dT0dz*kron(speye(Nz+1),Dr)]; for j=1:size(motr_vz), motr_vz(j ,:)=motr_vz(j,:)*expterm(j);end %The coefficient of T in the motion equation’s r-direction-component motr_T=[sparse((Nr+1)*(Nz+1),(Nr+1)*(Nz+1))]; %The coefficient of P in the motion equation’s r-direction-component motr_P=[-kron(speye(Nz+1),Dr)]; %Motion theta-dir %Rows will later be cut out of this matrix for the following BCs: %vtheta(r=Rin)=0,vtheta(r=Rout)= 0,vtheta(z=0)=0,vtheta(z=Lz)=0 mottheta_vr=[(2*i*m*kron(s peye(Nz+1),eyercomb1))]; for j=1:size(mottheta_vr), mottheta_v r(j,:)=mottheta_vr(j,:)*expterm(j);end mottheta_vtheta=[( kron(speye(Nz+1), Dr2comb1)+kron(D z2,speye(Nr+1))((m^2+1)*kron(speye(Nz+1),eyercomb1)))+Bvisc *deltaTguess*dT0dz*k ron(Dz,speye(Nr+1))]; for j=1:size(mottheta_vtheta) mottheta_vtheta(j,:)=motthet a_vtheta(j,:)*expterm(j);end mottheta_vz=[Bvisc*deltaTguess*dT0dz*i*m*kron(speye(Nz+1),eyercomb2)]; for j=1:size(mottheta_vz), mottheta_vz( j,:)=mottheta_vz(j ,:)*expterm(j);end mottheta_T=[sparse((Nr+1)* (Nz+1),(Nr+1)*(Nz+1))]; mottheta_P=[-i*m*kron(speye(Nz+1),eyercomb2)]; %Motion z-dir %Rows will later be cut out of this matrix for the following BCs: %vz(r=Rin)=0,vz(r=Rout)= 0,vz(z=0)=0,vz(z=Lz)=0 motz_vr=[sparse((Nr+1)*(Nz+1),(Nr+1)*(Nz+1))]; motz_vtheta=[sparse((Nr+1)* (Nz+1),(Nr+1)*(Nz+1))]; motz_vz=[(kron(speye(Nz+1),Dr 2comb1)+kron(Dz2,speye(Nr+1))(m^2*kron(speye(Nz+1),eyerco mb1)))+Bvisc*deltaTguess*dT0dz*2*kron(Dz,speye(Nr+1))]; for j=1:size(motz_vz), motz_vz( j,:)=motz_vz(j,:)*expterm(j);end motz_T=[sparse((Nr+1)*(Nz+1),(Nr+1)*(Nz+1))]; motz_P=[-kron(Dz,speye(Nr+1))]; motz=[motz_vr,motz_vtheta,motz_vz,motz_T,motz_P]; %Energy %Rows will later be cut out of this matrix for the following BCs: %dT/dr(r=Rin)=0,dT/dr(r=Rout )=0,T(z=0)=0,T(z=Lz)=0 ener_vr=[sparse((Nr+1)*(N z+1),(Nr+1)*(Nz+1))]; ener_vtheta=[sparse((Nr+1)* (Nz+1),(Nr+1)*(Nz+1))]; ener_vz=[-Lchar*vchar/td*dT0dz*kro n(speye(Nz+1), speye(Nr+1))]; ener_T=[kron(speye(Nz+1),D r2comb1)+kron(Dz2,speye(Nr+1))-(m^ 2*kron(speye(Nz+1 ),eyercomb1))]; ener_P=[sparse((Nr+1)*(Nz+1),(Nr+1)*(Nz+1))]; %Continuity cont_vr=[kron(speye(Nz+1),Dr)+ kron(speye(Nz+1),eyercomb2)]; cont_vtheta=[i*m*kron(speye(Nz+1),eyercomb2)]; cont_vz=[kron(Dz,speye(Nr+1))]; cont_T=[sparse((Nr+1)*(Nz+1),(Nr+1)*(Nz+1))]; cont_P=[sparse((Nr+1)*(Nz+1),(Nr+1)*(Nz+1))]; %Boundary Conditions

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184 %BC1: vr(r=Rin)=0 preBC1=[sparse(1,Nr),1]; BC1=[sparse((Nz-1),Nr+1),kron(speye(Nz-1), preBC1),sparse((Nz-1),(Nr+1)),sparse((Nz1),4*(Nr+1)*(Nz+1))]; %BC2: vr(r=Rout)=0 preBC2=[1,sparse(1,Nr)]; BC2=[sparse((Nz-1),Nr+1),kron(speye(Nz-1), preBC2),sparse((Nz-1),(Nr+1)),sparse((Nz1),4*(Nr+1)*(Nz+1))]; %BC3: vr(z=0)=0 BC3=[speye(Nr+1),sparse(Nr+1,(Nz)*(Nr+1)),sparse(Nr+1,4*(Nr+1)*(Nz+1))]; %BC4: vr(z=Lz)=0 BC4=[sparse(Nr+1,(Nz)*(Nr+1)),speye( Nr+1),sparse(Nr+1,4*(Nr+1)*(Nz+1))]; %BC5: vtheta(r=Rin)=0 BC5=[sparse((Nz-1),(Nr+1)*(Nz+1)),sparse((Nz-1),Nr+1),kron(speye(Nz-1),preBC1),sparse((Nz1),(Nr+1)),sparse((Nz-1),3*(Nr+1)*(Nz+1))]; %BC6: vtheta(r=Rout)=0 BC6=[sparse((Nz-1),(Nr+1)*(Nz+1)),sparse((Nz-1),Nr+1),kron(speye(Nz-1),preBC2),sparse((Nz1),(Nr+1)),sparse((Nz-1),3*(Nr+1)*(Nz+1))]; %BC7: vtheta(z=0)=0 BC7=[sparse(Nr+1,(Nr+1)*(Nz+1)),sp eye(Nr+1),sparse(Nr+1,(Nz)*(Nr+1 )),sparse(Nr+1,3*(Nr+1)*(Nz+1))]; %BC8: vtheta(z=Lz)=0 BC8=[sparse(Nr+1,(Nr+1)*(Nz+1)),sp arse(Nr+1,(Nz)*(Nr+1)), speye(Nr+1),sparse(Nr+ 1,3*(Nr+1)*(Nz+1))]; %BC9: vz(r=Rin)=0 BC9=[sparse((Nz-1),2*(Nr+1)*(Nz+1)),sparse((Nz-1),Nr+1),kron(speye(Nz-1),preBC1),sparse((Nz1),(Nr+1)),sparse((Nz-1),2*(Nr+1)*(Nz+1))]; %BC10: vz(r=Rout)=0 BC10=[sparse((Nz-1),2*(Nr+1)*(Nz+1)),sparse((Nz1),Nr+1),kron(s peye(Nz-1),preBC2 ),sparse((Nz1),(Nr+1)),sparse((Nz-1),2*(Nr+1)*(Nz+1))]; %BC11: vz(z=0)=0 BC11=[sparse(Nr+1,2*(Nr+ 1)*(Nz+1)),speye( Nr+1),sparse(Nr+1,(Nz)*(Nr+1)),sparse(Nr+1,2*(Nr+1)*(Nz+ 1))]; %BC12: vz(z=Lz)=0 BC12=[sparse(Nr+1,2*(Nr+ 1)*(Nz+1)),sparse(Nr+1,(Nz)*(Nr+1)),s peye(Nr+1),s parse(Nr+1,2*(Nr+1)*(Nz+ 1))]; %BC13: dT/dr(r=Rin)=0 preBC13=[Dr(Nr+1,:)]; BC13=[sparse((Nz-1),3*(Nr+1)*(Nz+1)),sparse((Nz-1),(Nr+1)),kron(speye(Nz-1),preBC13),sparse((Nz1),(Nr+1)),sparse((Nz-1),(Nr+1)*(Nz+1))]; %BC14: dT/dr(r=Rout)=0

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185 preBC14=[Dr(1,:)]; BC14=[sparse((Nz-1),3*(Nr+1)*(Nz+1)),sparse((Nz-1),(Nr+1)),kron(speye(Nz-1),preBC14),sparse((Nz1),(Nr+1)),sparse((Nz-1),(Nr+1)*(Nz+1))]; %BC15: T(z=0)=0 BC15=[sparse(Nr+1,3*(Nr+ 1)*(Nz+1)),speye( Nr+1),sparse(Nr+1,(Nr+1)*(Nz)),sparse(Nr+1,(Nr+1)*(Nz+1)) ]; %BC16: T(z=Lz)=0 BC16=[sparse(Nr+1,3*(Nr+ 1)*(Nz+1)),sparse(Nr+1,(Nr+1)*(Nz)),s peye(Nr+1),s parse(Nr+1,(Nr+1)*(Nz+1)) ]; %Now I will enforce the boundary conditions in each equation by replacing rows in the domain %equation matrices: %motr: vr(r=Rin)=0,vr(r=Rout)=0,vr(z=0)=0,vr(z=Lz)=0 motr(1:(Nr+1),:)=BC3; motr((Nr+1)*(Nz+1)-(Nr+1)+1:(Nr+1)*(Nz+1),:)=BC4; jj=0; iii=[]; for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1; motr(ii,:)=BC1(jj,:); motr(iii,:)=BC2(jj,:); end %mottheta: vtheta(r=Rin)=0,vtheta(r= Rout)=0,vtheta(z=0)=0,vtheta(z=Lz)=0 mottheta(1:(Nr+1),:)=BC7; mottheta((Nr+1)*(Nz+1)-(Nr+1)+1:(Nr+1)*(Nz+1),:)=BC8; jj=0; iii=[]; for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1; mottheta(ii,:)=BC5(jj,:); mottheta(iii,:)=BC6(jj,:); end %motz: vz(r=Rin)=0,vz(r=R out)=0,vz(z=0)=0,vz(z=Lz)=0 motz(1:(Nr+1),:)=BC11; motz((Nr+1)*(Nz+1)-(Nr+1)+1:(Nr+1)*(Nz+1),:)=BC12; jj=0; iii=[]; for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1; motz(ii,:)=BC9(jj,:); motz(iii,:)=BC10(jj,:); end %ener: dT/dr(r=Rin)=0,dT/dr(r =Rout)=0,T(z=0)=0,T(z=Lz)=0 ener(1:(Nr+1),:)=BC15; ener((Nr+1)*(Nz+1)-(Nr+1)+ 1:(Nr+1)*(Nz+1),:)=BC16; jj=0; iii=[]; for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1;

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186 ener(ii,:)=BC13(jj,:); ener(iii,:)=BC14(jj,:); end %Final form of LHS matrix LHS=[motr;mottheta;motz;ener;cont]; %Form RHS (right-hand-side) matrix motrRHS=[sparse((Nr+1)*(Nz+1),5*(Nr+1)*(Nz+1))]; motthetaRHS=[sparse((Nr+1)*(Nz+1),5*(Nr+1)*(Nz+1))]; motzRHS=[sparse((Nr+1)*(Nz+1),3*(Nr+1)*(Nz+1)),((tec*g*Lchar^2)/(kvref*vchar))*kron(speye(Nz+1),speye(Nr+1)),sparse((Nr+1)*(Nz+1),(Nr+1)*(Nz+1))]; enerRHS=[sparse((Nr+1)*(Nz +1),5*(Nr+1)*(Nz+1))]; contRHS=[sparse((Nr+1)*(Nz+1),5*(Nr+1)*(Nz+1))]; %Now I will enforce the boundary conditions in each equation in the RHS: %motr: vr(r=Rin)=0,vr(r=Rout)=0,vr(z=0)=0,vr(z=Lz)=0 motrRHS(1:(Nr+1),:)=spars e(Nr+1,5*(Nr+1)*(Nz+1)); motrRHS((Nr+1)*(Nz+1)-(Nr+1)+1:(Nr+1)*(Nz+1),:)=sparse(Nr+1,5*(Nr+1)*(Nz+1)); jj=0; iii=[]; for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1; motrRHS(ii,:)=sparse(1,5*(Nr+1)*(Nz+1)); motrRHS(iii,:)=spar se(1,5*(Nr+1)*(Nz+1)); end %mottheta: vtheta(r=Rin)=0,vtheta(r= Rout)=0,vtheta(z=0)=0,vtheta(z=Lz)=0 motthetaRHS(1:(Nr+1),:)=sparse(Nr+1,5*(Nr+1)*(Nz+1)); motthetaRHS((Nr+1)*(Nz+1)-(Nr+1)+1:(Nr+1) *(Nz+1),:)=sparse(Nr+ 1,5*(Nr+1)*(Nz+1)); jj=0; iii=[]; for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1; motthetaRHS(ii,:)=spa rse(1,5*(Nr+1)*(Nz+1)); motthetaRHS(iii,:)=spa rse(1,5*(Nr+1)*(Nz+1)); end %motz: vz(r=Rin)=0,vz(r=R out)=0,vz(z=0)=0,vz(z=Lz)=0 motzRHS(1:(Nr+1),:)=spars e(Nr+1,5*(Nr+1)*(Nz+1)); motzRHS((Nr+1)*(Nz+1)-(Nr+1)+1:(Nr+1)*(Nz+1),:)=sparse(Nr+1,5*(Nr+1)*(Nz+1)); jj=0; iii=[]; for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1; motzRHS(ii,:)=spar se(1,5*(Nr+1)*(Nz+1)); motzRHS(iii,:)=sparse(1,5*(Nr+1)*(Nz+1)); end %ener: dT/dr(r=Rin)=0,dT/dr(r =Rout)=0,T(z=0)=0,T(z=Lz)=0 enerRHS(1:(Nr+1),:)=sparse( Nr+1,5*(Nr+1)*(Nz+1)); enerRHS((Nr+1)*(Nz+1)-(Nr+1) +1:(Nr+1)*(Nz+1),:)=spars e(Nr+1,5*(Nr +1)*(Nz+1)); jj=0; iii=[];

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187 for ii=((Nr+1)+1):(Nr+1):((Nr+1)*(Nz+1)-2*(Nr+1)+1) iii=ii+Nr; jj=jj+1; enerRHS(ii,:)=sparse(1,5*(Nr+1)*(Nz+1)); enerRHS(iii,:)=sparse (1,5*(Nr+1)*(Nz+1)); end %Final form of RHS matrix RHS=[motrRHS;motthetaRHS;m otzRHS;enerRHS;contRHS]; %Use singular value decomposition on LHS matrix to make it easier to work with/deal with %spurious values U=[]; S=[]; V=[]; [U,S,V]=svd(full(LHS)); Ssort=sort(diag(S)); %Examining and sorting singular values for ii=1 : ((Nr+1)*(Nz+1)*5) if abs(imag(Ssort(ii))) >= 1e-9, disp('Error: Large imaginary part in sing ular value'), return, else, end end Smin=Ssort(1:20); Smin_dataset=[Smin_dataset;Nr;Nz;m;Smin]; save S.mat S; save U.mat U; save V.mat V; save Smin.mat Smin %Eigenvalue calculation, taking into account t he singular value tolerance specified earlier %Inverting the singular values, but f iltering out those singular values which are smaller than the %singular value tolerance spec ified earlier. Sinv1=[]; if tolchoice==1; for ii=1 : ((Nr+1)*(Nz+1)*5) if abs(real(S(ii,ii))) <= tol1 ;Sinv1(ii,ii) = 0; else, Sinv1(ii,ii) = 1/S(ii,ii);end,end,else,end if tolchoice==2; for ii=1 : ((Nr+1)*(Nz+1)*5) if abs(real(S(ii,ii))) <= tolmod ;Sinv1(ii,ii) = 0; else, Sinv1(ii,ii) = 1/S(ii,ii);end,end,else,end %Creating modified RHS matrix by multiplying by the inverse of matrices created in the singular value decomposition of the LHS matrix RHSmod1=Sinv1*inv(U)*full(RHS); %Solving the original generalized eigenvalue problem using matrices created and modified according to the singular value decomposition of the original LHS matrix [result1evec,result1]=eig(V',RHSmod1); result1=diag(result1); %”result1” are the eigenvalues. The following section examines them. result1real=real(result1); result1posrealmod=[]; jj=0; %% %Setting non-positive eigenvalues equal to infinity so that they are not considered when sorting %the eigenvalues for kk=1:length(result1); %% if result1real(kk)<=0; result1posrealmod(kk)= inf; else result1posrealmod(kk)=result1real(kk); end, end

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188 %Sorting the eigenvalues and findi ng the critical eigenvalue [sortresult1mod,sortresult1modi ndex]=sort(result1posrealmod); loccrit=sortresult1modindex(1); %Obtaining the eigenvector which corr esponds to the critical eigenvalue result1eveccrit=result1evec(:,loccrit); %Obtaining the vr, vtheta, vz, and T por tions of the critical eigenvector vr=result1eveccrit(1 :((Nr+1)*(Nz+1))); vtheta=result1eveccrit((Nr+1)* (Nz+1)+1:(2*(Nr+1)*(Nz+1))); vz=result1eveccrit(2*(Nr+1)* (Nz+1)+1:(3*(Nr+1)*(Nz+1))); T=result1eveccrit(3*(Nr+1)*(Nz+1)+1:(4*(Nr+1)*(Nz+1))); %Renaming the critical value resultmin1=sortresult1mod(1); deltaTcrit=resultmin1; %Creating a compact and clear display of the output of the calculation with its corresponding %input parameters output_ann_usv=[output_ann_usv; [m Nr Nz deltaTguess deltaTcrit]]; [m Nr Nz deltaTguess deltaTcrit] %Indicator of the convergence of critic al temperature difference result with %respect to last iteration of visco sity's variation with temperature Tvcorr=abs((deltaTcrit-deltaTguess)/deltaTguess) %Setting the new value of deltaTguess for iteration deltaTguess=deltaTcrit; %For use in possible text displays Nrstring=num2str(Nr); Nzstring=num2str(Nz); mstring=num2str(m); deltaTstring=num2str(deltaTcrit); %If the critical temperature difference value is conv erged with respect to changes in viscosity due to its temperature-dependence: if Tvcorr=1e-7, disp('Error: Large imag inary part in critical value.'), return, else, end,%% else, end if LIRT==2, if abs(imag(resultmin1))>=1e-7, warningstr=['Large imag inary part in critical value: Nr=',Nrstring,', Nz=',Nzstring,', m=',mstring,', del taT=',deltaTstring,' filler: ']; strs izedeficiency=90-si ze(warningstr,2); filler=zeros(1,strsize deficiency); warningstr =[warningstr,filler], warningsmatrix=[warningsmatr ix;warningstr]; else, end,%% else, end %Output display including all outputs calculated for each input m, Nr, Nz output_ann_usv_crit=[out put_ann_usv_crit; [m Nr Nz deltaTcrit]]; output_ann_usv_crit %If this is the case for which it is desired to plot a velocity profile, then the following procedure begins

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189 if Nr==Nrforplot & Nz==Nzforplot & m==mforplot, theta=[0:2*pi/64:2*pi]'; halfNz=round((Nz+1)/2); rgrid=[kron(ones(Nz+1,1),kron( ro',ones(1,length(theta))))]; thetagrid=[kron(ones(Nz+1,1),kro n(ones(1,length(r)),theta'))]; zgrid=[kron(zo,ones(1,(N r+1)*length(theta)))]; xcartgrid=[]; ycartgrid=[]; zcartgrid=[]; for j=1:(Nz+1) for k=1:((Nr+1)*length(theta)) xcartgrid(j,k)=rgrid(j,k )*cos(thetagrid(j,k)); ycartgrid(j,k)=rgrid(j,k)*sin(thetagrid(j,k)); zcartgrid(j,k)=zgrid(j,k); end end vrfull=2*real(kron(vr,exp(i*m*theta))); vthetafull=2*real(kron(vt heta,exp(i*m*theta))); vzfull=2*real(kron(vz,exp(i*m*theta))); Tfull=2*real(kron(T,exp(i*m*theta))); vrmatrix=reshape(vrfull,(Nr+ 1)*length(theta),Nz+1)'; vthetamatrix=reshape(vthetafull, (Nr+1)*length(theta),Nz+1)'; vzmatrix=reshape(vzfull,(Nr+ 1)*length(theta),Nz+1)'; Tmatrix=reshape(Tfull,(Nr+ 1)*length(theta),Nz+1)'; vxcartmatrix=[]; vycartmatrix=[]; vzcartmatrix=[]; Tcartmatrix=[]; for j=1:(Nz+1) for k=1:((Nr+1)*length(theta)) vxcartmatrix(j,k)=vrmatrix(j,k)*cos(thetagrid (j,k))-vthetamatrix(j, k)*sin(thetagrid(j,k)); vycartmatrix(j,k)=vrmatrix(j,k) *sin(thetagrid(j,k))+vthetamat rix(j,k)*cos(thetagrid(j,k)); vzcartmatrix(j,k)=vzmatrix(j,k); Tcartmatrix(j,k)=Tmatrix(j,k); end end %Three-dimensional plot of velocity vectors quiver3(xcartgrid,ycartgrid,zcartgrid, vxcartmatrix,vycartmatrix,vzcartmatrix) title(['Velocity Profile, at deltaTc rit= ',deltaTstring,' degrees C']) xlabel('x') ylabel('y') zlabel('z') rotate3d on %Two-dimensional plot of cross-sect ion of velocity profile (this gra ph is generally the most %helpful) vzcrosssection=vzmatrix(halfNz,:)'; Tcrosssection=Tma trix(halfNz,:)'; vzcrosssectionmatrix=r eshape(vzcrosssection, length(theta),Nr+1)'; Tcrosssectionmatrix=reshape(Tcro sssection,length(theta),Nr+1)'; xcartgridcrosssectionmatrix=[]; ycartgridcrosssetionmatrix=[]; xcartgridcrosssection=xcartgrid(halfNz,:)'; ycartgridcrosssection=ycartgrid(halfNz,:)'; xcartgridcrosssectionmatrix=reshape(xca rtgridcrosssection,le ngth(theta),Nr+1)'; ycartgridcrosssectionmatrix=reshape(yca rtgridcrosssection,le ngth(theta),Nr+1)'; figure

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190 contourf(xcartgridcrosssectionmatrix,ycartg ridcrosssectionmatrix,vzcrosssectionmatrix) colorbar('southoutside') %NOTE: The velocities determined and plotted up to this point are dimensionless. %To recover the dimensional, unscaled velocity values, must simply %multiply scaled velocity results by the characteristic velocity, vchar. %This will simply change the magnitudes of the velocity values, but the %flow pattern will, of course, be the same. The unscaled velocities could %be plotted in exactly the same way that the scaled values are. vrmatrix_unscaled=vrmatrix.*vchar; vthetamatrix_unscaled=vthetamatrix.*vchar; vzmatrix_unscaled=vzmatrix.*vchar; vxcartmatrix_unscaled=vxcartmatrix.*vchar; vycartmatrix_unscaled=vycartmatrix.*vchar; vzcartmatrix_unscaled=vzcartmatrix.*vchar; vzcrosssectionmatrix_unscale d=vzcrosssectionmatrix.*vchar; %Ending of if loops else, end, else, end %Ending of while loop for viscosity variation end %Ending of m loop end %Ending of Nz loop end %Ending of Nr loop end %Final output display output_ann_usv_crit warningsmatrix

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191 APPENDIX F LabVIEWTM PROGRAM GENERAL FLOW-DIAGRAM Figure F-1. LabVIEWTM program general flow-diagram. Input tseg,T, tseg,VCR, Tb,sp,j (for j = 1, 2, 3, …) Select First Value (j = 1) for Tb,sp,j Thermistor Data Continuously in from Experiment Calibration Equations Temperature Data from Experiment Average the Tb Temperature Data into One Tb Value Temperature Control Loop Is tel > tseg,T? YES Advance to Next Tb,sp,j Value (Add 1 to j Value) and Reset tel to 0 Start Adding to tel NO Is Tb < Tb,sp,j? Heater “On” Signal Sent to Experiment YES NO Heater “Off” Signal Sent to Experiment Experiment VCR Control Loop Is j tel = j tseg,VCR? YES NO Nothing Signal Causing VCR to Record for Five Seconds VCR Initial Value of tel is 0

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192 REFERENCES BNARD, H. 1900 Rev. Gen. Sci. Pure Appl. 11, 1261, 1309. BENSIMON, D. 1988 Physical Review A 37 (1), 200. BLOCK, M. J. 1956 Nature (London) 178, 650. CHANDRASEKHAR, S. 1961 Hydrodynamic and Hydromagnetic Stability Dover Publications, Inc., New York. CHARLSON, G. S., & SANI, R. L. 1970 Int. J. Heat and Mass Transfer 13, 1479. CHARLSON, G. S., & SANI, R. L. 1971 Int. J. Heat and Mass Transfer 14, 2157. CILIBERTO, S., BAGNOLI, F., & CAPONERI, M. 1990 Il Nuovo Cimento 12 D (N. 6), 781. DAUBY, P. C., & LEBON, G. 1996 J. Fluid Mech. 329, 25. DAVIS, S. H. 1967 J. Fluid Mech. 30 (part 3), 465. FERM, E. N., & WOLLKIND, D. J. 1982 J. Non-Equilib. Thermodyn. 7, 169. HARDIN, G. R., SANI, R. L., HENRY, D., & ROUX, B. 1990 International J. Num. Methods in Fluids 10, 79. JOHNS, L. E., & NARAYANAN, R. 2002 Interfacial Instability Springer, New York. JOHNSON, D. 1997 Geometric Effects on Bilaye r Convection in Cylindrical Containers. Ph.D. Dissertation, University of Florida, Gainesville, Florida. JOHNSON, D., & NARAYA NAN, R. 1999 CHAOS 9, 124. KATS-DEMYANETS, V., ORON, A., & NEPOMNYASHCHY, A. A. 1997 European Journal of Mechanics, B/Fluids 16, No. 1, 49. KOSCHMIEDER, E. L. 1967 J. Fluid Mech. 30 (part 1), 9. KOSCHMIEDER, E. L., & PALLAS, S. G. 1974 Int. J. Heat and Mass Transfer 17, 991. KOSCHMIEDER, E. L., & PRAHL, S. A. 1990 J. Fluid Mech. 215, 571. KOSCHMIEDER, E. L., & SWITZER, D. W. 1992 J. Fluid Mech. 240, 533. LABROSSE, G. 1993 Computer Methods in Applie d Mechanics and Engineering 106, 353.

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193 LITTLEFIELD, D. L., & DESAI, P. V. 1990 J. Heat Transfer 112, 959. PEARSON, J. R. A. 1958 J. Fluid Mech. 4, 489. RAYLEIGH, LORD 1916 Phil. Mag. 52, 529. SCHMIDT, R. J., & MILV ERTON, S. W. 1935 Proc. Roy. Soc. (London) A 152, 586. SCRIVEN, L. E., & STER NLING, C. V. 1964 J. Fluid Mech. 19, 321. SILVESTON, P. L. 1958 Forsch. Ing. Wes. 24, 29, 59. STERNLING, C. V., & SCRIVEN, L. E. 1959 A.I.Ch.E. Journal 5 (4), 514. STORK, K., & MLLER, U. 1972 J. Fluid Mech. 54 (part 4), 599. STORK, K., & MLLER, U. 1974 J. Fluid Mech., 71, 231. TREFETHEN, L. N. 2000 Spectral Methods in MATLAB S.I.A.M., Philadelphia. ZEREN, R. W., & REYNOLDS, W. C. 1972 J. Fluid Mech. 53, 305. ZHAO, A. X., MOATES, F. C ., & NARAYANAN, R. 1995 Phys. Fluids 7 (7), 1576. ZHAO, A. X., WAGNER, C., NARAYAN AN, R., & FRIEDRICH, R. 1995 Proceedings: Mathematical and Physical Sciences 451, No. 1942, 487.

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194 BIOGRAPHICAL SKETCH The author was born on August 28, 1980 at La ngley Air Force Base in Virginia. He received a B.S. in chemical engineering at Loui siana Tech University in 2002. After this, he attended the University of Flor ida from 2002 to 2006 to earn a Ph.D. in chemical engineering. During his time in graduate school, he was awarded a fellowship by N.A.S.A. through the Graduate Student Researchers Program.


Permanent Link: http://ufdc.ufl.edu/UFE0017102/00001

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Title: Convective Instability in Annular Systems
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0017102:00001

Permanent Link: http://ufdc.ufl.edu/UFE0017102/00001

Material Information

Title: Convective Instability in Annular Systems
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0017102:00001


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Table of Contents
    Title Page
        Page 1
        Page 2
    Acknowledgement
        Page 3
        Page 4
    Table of Contents
        Page 5
        Page 6
        Page 7
    List of Tables
        Page 8
    List of Figures
        Page 9
        Page 10
        Page 11
        Page 12
    List of abbreviations and symbols
        Page 13
        Page 14
        Page 15
    Abstract
        Page 16
        Page 17
    Introduction
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    Literature review
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    Modeling equations
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    Linearized equations
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    Spectral solution method
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    Biographical sketch
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Full Text





CONVECTIVE INSTABILITY IN ANNULAR SYSTEMS


By

DARREN MCDUFF



















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2006































Copyright 2006

By

Darren McDuff









ACKNOWLEDGMENTS

I begin by thanking my advisor, Dr. Ranga Narayanan, whose great ideas, support, and

motivation have guided me through this work.

I thank my mom, dad, and sister. They have always been the best of friends to me and

been there for me during any difficult times in my life. On the subject of family, I should point

out here that I first considered the idea of graduate school as a result of my dad suggesting it.

I am also grateful to those on my advisory board. These include Dr. Martin Volz, Dr.

James Klausner, Dr. Dmitry Kopelevich, and Dr. Mark Orazem. Their different viewpoints and

ideas about my research have helped me to improve my work and avoid potential mistakes. Dr.

Volz is my mentor in the N.A.S.A. Graduate Student Researchers Program, and traveled from

Marshall Space Flight Center in Huntsville, Alabama, to be here for my presentations. Dr.

Klausner comes from the Department of Mechanical and Aerospace Engineering at this

university. Dr. Kopelevich and Dr. Orazem are from my own department.

I must express my gratitude to Ken Reed. Ken is a talented machinist with T.M.R.

Engineering. High-quality components made by T.M.R. were essential to the design of my

experimental apparatus, and it was great to work with someone as helpful, knowledgeable, and

professional as Ken throughout the project.

Additionally, I thank Gerard Labrosse, a professor from L.I.M.S.I. in Orsay, France.

During visits with our research group here at the University of Florida, he has spent a great deal

of time sharing his knowledge on computational fluid dynamics (in particular, spectral methods)

and providing helpful answers to questions from my research group.

I would also like to thank a couple of undergraduates who have worked with me on this

project, particularly in the design of my experimental apparatus and the process control program









for it. Joe Cianfrone helped me to understand the computer software, Lab VIEWTM, and with that

software, we designed a very effective process control program for my experiment. Joe also

helped me to design a large electrical circuit to send data from my experiment into a process

control computer, and send process control signals from that computer to components of my

experiment. Scott Suozzo helped me with the design and re-design of several parts of my

experimental apparatus, which helped the experiment to run well enough to produce results that

match theoretical predictions.

Of course, I thank my research group and the other graduate students in this department

whose advice and ideas have helped me get past difficult parts of my project and helped me in

my on-going education in this field. In particular, I thank Dr. Weidong Guo, of my research

group, for spending a good deal of time discussing computational issues with me during the last

couple of years.

Lastly, I am deeply grateful to N.A.S.A. for supporting this work through a fellowship

from the N.A.S.A. Graduate Student Researchers Program (grant number NGT852941).









TABLE OF CONTENTS

A C K N O W L E D G M EN T S .................................................................................... 3

L IST O F T A B L E S .................................. ...................................................... 8

L IST O F F IG U R E S ......................................................................................... 9

LIST OF ABBREVIATIONS AND SYMBOLS.................................. ............... 13

A B S T R A C T ....................................................................................................... 1 6

CHAPTER

1 IN TR O D U C TIO N ................................................................... .......... 18

1.1 Physics of Convection ...................................... ................ ........... 18
1.1.1 Rayleigh Convection ................................. .............. .......... 19
1.1.2 M arangoni C onvection............................................ ... .............. 21
1.2 R ayleigh N um b er...................................................... ... .............. 24
1.3 Physical Explanation: Pattern Selection......................... 25
1.3.1 Laterally Unbounded System................................................ 25
1.3.2 Rectangular System, Laterally Bounded, Periodic Lateral Boundary
Conditions ........... .. .. ........... .. ..... ...... .............. 28
1.3.3 Rectangular System, Laterally Bounded, Non-Periodic Lateral
Boundary Conditions .............................................. .......... 31
1.3.4 Cylindrical System ................................................. ......... 33
1.3.5 A nnular System .................................................... .......... 36
1.4 Annulus vs. Cylinder ........................................ ............... ........... 37
1.5 Application ................................................................... .......... 40

2 LITERATURE REVIEW ........................................................... ........... 50

2.1 Single Fluid Layers ......................................................... .......... 50
2.2 Multiple Fluid Layers ........................................................ .......... 56

3 MODELING EQUATIONS................................................................. 61

3.1 N onlinear E quations.................. .......................................................... 61
3 .2 S calling ........................................................................ .. . .... 66

4 LINEARIZED EQUATIONS............................................................ 70

4.1 Linearization ..................................... ..................... .................... 70
4.2 Expansion into Normal Modes................................... ............ ...... 75









5 SPECTRAL SOLUTION METHOD.................................................. 80

5.1 Explanation of the M ethod ................................................. ......... 82
5.2 Application of the M ethod ................................................. ......... 89

6 EXPERIMENTAL DESIGN.............................................................. 98

6.1 G oals in Experim ental D esign............................................. ................ 98
6.2 Experimental Apparatus ...................................... ......................... 98
6.2.1 Test Section ..................................... ............................... 99
6.2.2 Bottom Temperature Bath............................. .................. 101
6.2.3 Top Temperature Bath.................................. 102
6.2 .4 Insulation ........................................................... . .... 103
6.2.5 Flow Visualization .................................. ............................ 103
6.3 Process C control System ................................................... .......... 104
6.4 Typical Experimental Procedure.................................................. 106

7 RESULTS AND DISCUSSION........................................................... 111

7.1 Cylindrical System s ............................ .. ...... .... ............................ 111
7.1.1 Constant Viscosity Computations: Cylinder.............................. 111
7.1.2 Non-Constant Viscosity Computations: Cylinder....................... 113
7.1.3 Experiments: Cylinder ............................. .............. .......... 114
7.1.4 Comparison of Results: Cylinder....................................... 115
7.2 A nnular System s .................................................... .... ................... 116
7.2.1 Constant Viscosity Computations: Annulus.............................. 117
7.2.2 Non-Constant Viscosity Computations: Annulus....................... 121
7.2.3 Experiments: Annulus......... ........... .............. 121
7.2.4 Comparison of Results: Annulus.......................................... 122
7.3 Error A analysis .............................................................. .......... 125

8 CONCLUSIONS AND POSSIBLE FUTURE STUDIES............................... 159

8 .1 S u m m ary ................................................................................... 1 5 9
8.2 Future Studies .............................................................. .......... 161

APPENDIX

A THERMOPHYSICAL PROPERTIES......................................................... 163

B ADDITIONAL EXPLANATION OF EQUATIONS......................... 165

B. 1 Boussinesq Approximation ........................... ........... 165
B.2 Nonlinearities in the Governing Equations......................................... 166
B.3 Characteristic Velocity and Characteristic Time........... .............. 167









C DEVELOPMENT OF MATHEMATICAL MODEL FOR THE CONSTANT
V ISC O SIT Y C A SE ............................................................................... 168

C 1 N onlinear E equations .......................................................... .............. 168
C.2 Scaling ................................................................................ .. 170
C .3 L inearization ......................................................... ........ ........... 172
C.4 Expansion into Normal M odes................................................... 174

D MATLAB PROGRAM GENERAL FLOW-DIAGRAM ..................... 177

E EXAM PLE M ATLAB PROGRAM S......................................................... 178

E.1 Physical Properties ........................................................ ........... 178
E.2 D epths for Annular System ................................ .. .......... ......... 179
E.3 Main Program: Annular System with Temperature-Dependent Viscosity....... 179

F LABVIEWTM PROGRAM GENERAL FLOW-DIAGRAM.............................. 191

R E F E R E N C E S ............................................................................................. 19 2

BIOGRAPHICAL SKETCH ............................................................... ........... 194































7









LIST OF TABLES


Table Page

5-1 Example of convergence of computed result with N, and Nz................ ................ 94

5-2 Comparison of rectangular, 2-D, no-stress results with rectangular 1-D results......... 94

5-3 Comparison of calculated cylindrical results with results of Hardin et al................ 94

7-1 Constant viscosity computational results: critical temperature difference and onset
flow pattern for sets of cylindrical dimensions considered in experiments.............. 129

7-2 Non-constant viscosity computational results: critical temperature difference and
onset flow pattern for sets of cylindrical dimensions considered in experiments....... 129

7-3 Experimental results: critical temperature difference and onset flow pattern for sets
of cylindrical dimensions ......................................................... .......... 129

7-4 Summary of computational and experimental results for cylindrical systems.......... 130

7-5 Constant viscosity computational results: critical temperature difference and onset
flow pattern for sets of annular dimensions considered in experiments................. 130

7-6 Non-constant viscosity computational results: critical temperature difference and
onset flow pattern for sets of annular dimensions considered in experiments.......... 130

7-7 Experimental results: critical temperature difference and onset flow pattern for sets
of annular dimensions ............................................... .............. .......... 131

7-8 Summary of computational and experimental results for annular systems.............. 131

A-i Thermophysical properties ....................................................... ........... 164









LIST OF FIGURES


Figure Page

1-1 Simplified system diagram: heated from below........................................... 43

1-2 Rayleigh convection ................................................................ .......... 43

1-3 M arangoni convection .............................................................. .......... 44

1-4 General diagram: critical Rayleigh number vs. disturbance wavelength................ 44

1-5 Cross-sectional velocity profiles of flow patterns with different numbers of
convective rolls ...................................................................... ......... 45

1-6 General stability diagram for buoyancy-driven convection in a rectangular, laterally
bounded system, with periodic boundary conditions at lateral walls, for varying aspect
r a tio ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ... ... ... ... ... ... 4 5

1-7 Cross-sectional velocity profiles of four-roll flow patterns with non-uniform roll size... 46

1-8 General stability diagram for buoyancy-driven convection in a rectangular, laterally
bounded system, with non-periodic boundary conditions at lateral walls, for varying
aspect ratio ............................................................. .............. ......... 46

1-9 Diagrams of m = 0, 1, 2, 3 flow patterns: "U": upward flow, "D": downward flow... 47

1-10 General stability diagram for buoyancy-driven convection in a cylindrical container,
for varying aspect ratio ............................................................. .......... 48

1-11 Convective flow pattern in an annulus (Stork & Muller 1974)........................... 48

1-12 Example: computed flow profile for annular system, cross-sectional view............. 49

1-13 Diagram of crystal growth system ................... ... ................ ....... ..... ..... 49

3-1 System diagram: cylindrical and annular systems........................................ 69

5-1 Discretization nodes in a cylinder ................................................. .......... 95

5-2 Exam ple: grid point spacing .......................................................... .......... 95

5-3 Diagram of matrix/vector arrangement of discretized problem........................... 96

5-4 Examples of flow patterns and parenthetical notation for cylindrical systems........... 97









6-1 Simple experiment diagram ....................................................... ........... 108

6-2 Photograph of experimental apparatus....................................... ...... .... 108

6-3 Photograph of flow-through top water bath of experimental apparatus................. 109

6-4 Cross-sectional diagram of test section................................................ 109

6-5 Process control system layout ..................................................... .......... 110

7-1 Constant viscosity computational results: Racrt vs. A for cylindrical systems......... 132

7-2 Constant viscosity computational results: Racrt vs. A for cylindrical systems,
close-up view ........................................................................ .......... 133

7-3 Computed velocity profile, cross-sectional view, Lz = 6.85 mm, Lr = 8.75 mm,
m = 0:(0,2), [ A T]crit = 5.80 C .................................................... .......... 134

7-4 Computed velocity profile, cross-sectional View, Lz = 7.18 mm, Lr = 11.51 mm,
m = 1:(1,3), [ AT]rit = 5.04 C .................................... ................ .......... 135

7-5 Computed velocity profile, cross-sectional view, Lz = 6.53 mm, Lr = 11.28 mm,
m = 1:(1,3), [ A T]crit = 6.43 C .................................................... .......... 136

7-6 Photo of onset flow pattern, Lz = 6.85 mm, Lr = 8.75 mm, m = 0:(0,2).................. 136

7-7 Photo of onset flow pattern, Lz = 7.18 mm, Lr= 11.51 mm, m = 1:(1,3)................ 137

7-8 Photo of onset flow pattern, Lz = 6.53 mm, Lr = 11.28 mm, m = 1:(1,3)................ 137

7-9 Constant viscosity computational results: Racrt vs. S for annular systems with
A = .7 5 .............................................................................................. 1 3 8

7-10 Constant viscosity computational results: Racrt vs. S for annular systems with
A = 1 .2 8 ............................................................................................. 1 3 9

7-11 Constant viscosity computational results: Racrt vs. S for annular systems with
A = 1.28, close-up view ............................................................ .......... 140

7-12 Constant viscosity computational results: Racrt vs. S for annular systems with
A = 1 .6 0 ............................................................................................. 1 4 1

7-13 Constant viscosity computational results: Racrt vs. S for annular systems with
A = 1.60, close-up view ............................................ ................ .......... 142









7-14 Constant viscosity computational results: Racrct vs. S for annular systems with
A = 1 .7 3 ............................................................................................. 1 4 3

7-15 Constant viscosity computational results: Racrt vs. S for annular systems with
A = 1.73, close-up view ........................................................... ........... 144

7-16 Constant viscosity computational results: Racrct vs. S for annular systems with
A = 2 .9 0 ............................................................................................. 1 4 5

7-17 Constant viscosity computational results: Racrct vs. S for annular systems with
A = 2.90, close-up view ........................................................... .......... 146

7-18 Constant viscosity computational results: Racrct vs. S for annular systems with
A = 3 .4 0 ............................................................................................. 1 4 7

7-19 Constant viscosity computational results: Racrct vs. S for annular systems with
A = 3.40, close-up view 1 ......................................................... .......... 148

7-20 Constant viscosity computational results: Racrct vs. S for annular systems with
A = 3.40, close-up view 2 ......................................................... .......... 149

7-21 Computed velocity profile, cross-sectional view, Lz = 6.85 mm, Ro = 8.75 mm,
S= .16, m = 0:(0,1), [AT]crit= 7.08 C ......................................................... 150

7-22 Computed velocity profile, cross-sectional view, Lz = 6.85 mm, Ro = 8.75 mm,
S= .30, m = 3:(3,0), [AT]crit= 7.67 C ......................................................... 151

7-23 Computed velocity profile, cross-sectional view, Lz = 7.18 mm, Ro = 11.51 mm,
S= .12, m = 0:(0,1), [AT]crit= 5.57 C ......................... ................................ 152

7-24 Computed velocity profile, cross-sectional view, Lz = 7.18 mm, Ro = 11.51 mm,
S= .50, m = 4:(4,0), [AT]crit = 7.09 C..................................... 153

7-25 Computed velocity profile, cross-sectional view, Lz = 6.53 mm, Ro = 11.28 mm,
S= .10, m = 2:(2,1), [AT]crit= 7.21 C..................................... 154

7-26 Computed velocity profile, cross-sectional view, Lz = 6.53 mm, Ro = 11.28 mm,
S = .40, m = 4:(4,0), [ A T]crit = 8.07 C..................................... 155

7-27 Photo of onset flow pattern, Lz= 6.85 mm, Ro= 8.75 mm, S= .16, m = 0:(0,1) ....... 155

7-28 Photo of onset flow pattern, Lz = 6.85 mm, Ro= 8.75 mm, S= .30, m = 3:(3,0) ....... 156

7-29 Photo of onset flow pattern, Lz = 7.18 mm, Ro = 11.51 mm, S= .12, m = 2:(2,1)........ 156

7-30 Photo of onset flow pattern, Lz = 7.18 mm, Ro = 11.51 mm, S= .50, m = 4:(4,0)........ 157









7-31 Photo of onset flow pattern, Lz= 6.53 mm, Ro = 11.28 mm, S= .10, m = 2:(2,1)........ 157

7-32 Photo of onset flow pattern, Lz= 6.53 mm, Ro = 11.28 mm, S= .40, m = 4:(4,0)........ 158

D-1 MATLAB program general flow-diagram..................................... .......... 177

F-i LabVIEW TM program general flow-diagram.................................... .......... 191









LIST OF ABBREVIATIONS AND SYMBOLS

a volumetric thermal expansion coefficient (C -1)

93 unit vector in z-direction

E amplitude of small perturbation

ic thermal diffusivity (m2/s)

A eigenvalue of the eigenvalue form of the problem shown in Chapter 5

p/ dynamic viscosity (kg/(m*s))

P/R reference dynamic viscosity (kg/(m*s))

v kinematic viscosity (m2/s)

p density (kg/m3)

PR reference density (kg/m3)

U dimensionless inverse time constant

r transpose symbol for a matrix

form of the x-direction dependence in a laterally bounded, rectangular system
with periodic lateral boundary conditions

0 dimensionless group which appears in the motion equation when considering the
dependence of viscosity on temperature

A aspect ratio for cylindrical and annular systems

As aspect ratio for rectangular systems

A left-hand-side matrix of the eigenvalue form of the problem shown in Chapter 5

B a constant related to the dependence of viscosity on temperature (C -1)

B right-hand-side matrix of the eigenvalue form of the problem shown in Chapter 5

CP constant-pressure heat capacity (J/(kg*OC))









Cv constant-volume heat capacity (J/(kg*OC))

D differentiation matrix

f function representing the temperature dependence of viscosity

F function representing the temperature dependence of viscosity, in terms of
dimensionless temperature

g magnitude of gravity (m/s2)

g gravity, in vector form (m/s2)

k thermal conductivity (W/(m*OC))

L.I.M.S.I. Laboratoire d'Informatique pour la Mecanique et les Sciences de l'Ingenieur

Lr radius in a cylindrical system, or annular gap width in an annular system (m)

Lx horizontal depth of fluid layer (m)

Lz vertical depth of fluid layer (m)

m azimuthal wave number

Npr Prandtl number

Nr parameter setting the number of discretization nodes in the r-direction

Nx parameter setting the number of discretization nodes in the x-direction

Nz parameter setting the number of discretization nodes in the z-direction

N.A.S.A. National Aeronautics and Space Administration

p modified pressure (Pa)

p characteristic pressure (Pa)

P pressure (Pa)

Ra Rayleigh number

Racrit critical Rayleigh number

Ro outer radius of annular system









R, inner radius of annular system

S radius ratio for annular system

S stress tensor

t time (s)

t characteristic time (s)

tel time elapsed (s) (Appendix F)

tseg,T time segment that each Tb set-point remains in effect for (s) (Appendix F)

tseg, VCR time interval between repetitions of the video cassette recorder's recording cycle
(s) (Appendix F)

T temperature (C)

Tb temperature at bottom wall of system (C)

Tb,sp set-point value for Tb (C) (Appendix F)

Tb,spj forj = (1, 2, 3, ...), series of set-point values input for Tb (C) (Appendix F)

TR reference temperature (oC)

Tt temperature at top wall of system (oC)

A T vertical temperature difference in fluid layer (oC)

[ A T]crit critical vertical temperature difference for convection (oC)

[ A T]guess guess-value for critical vertical temperature difference for convection (oC)

vj component of velocity in the j-direction

Velocity (m/s)

V characteristic velocity (m/s)

X eigenvector of the eigenvalue form of the problem shown in Chapter 5









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CONVECTIVE INSTABILITY IN ANNULAR SYSTEMS

By

Darren McDuff

December 2006

Chair: Ranga Narayanan
Major Department: Chemical Engineering

Since natural convection can occur in such a wide range of systems and circumstances,

much remains to be learned about the convective behaviors that appear in some systems. For

example, convection in fluid layers of shapes other than cylinders or rectangles can be quite

interesting, and it is this which is investigated in this research.

This study focuses on Rayleigh convection, which is natural convection caused by

buoyancy forces. The convective behavior of a system (referring to a fluid layer as the

"system") depends upon the shape and dimensions of the system, the vertical temperature

difference across the system, the thermophysical properties of the fluid comprising the system,

and the characteristics of the disturbance given to the system. In particular, this research is

concerned with the effects of the size and shape of the system.

One important application of this study is in semi-conductor crystal growth processes,

such as the vertical Bridgman growth method. This growth method is commonly used, for

example, in growing the semi-conductor crystal, lead-tin-telluride. Crystal growth is usually

thought of as a process in a cylindrical container. It is of interest to learn how the convective

behavior of the system would differ if annular fluid layers were employed.









The onset conditions for buoyancy-driven convection (critical vertical temperature

difference and flow pattern) in bounded, vertical, cylindrical systems and vertical, annular

systems were determined both experimentally and theoretically. In the experiments, onset

conditions were determined by flow visualization with tracer particles. The theoretical analysis

involves bifurcation theory and computations using spectral methods.

Computations show that the physics of the annular system are similar to those of the

cylindrical system. More rolls of convecting fluid are included as the container is widened, and

rolls may exist with either an azimuthal or radial alignment. The relations between the annular

container dimensions and the onset conditions for convection are determined and shown. The

agreement between computations and experiments is generally good. Some discrepancy in this

agreement arises in cases in which the annular gap width is small, and a likely reason for this is

explained.









CHAPTER 1
INTRODUCTION

This chapter provides an introduction to the physics of the convective phenomena

investigated in this research. It then proceeds to a brief explanation of the dimensionless

Rayleigh number which is very meaningful in regard to buoyancy-driven convection. After this,

an explanation is given regarding the competing physical phenomena which determine the

number of convective cells that appear at the onset of convection and their sizes. Next, an

explanation of why it is interesting to study the behavior of an annular system as compared to a

cylindrical one is given. Lastly, an example of an industrial application to which this study has

relevance is discussed.

1.1 Physics of Convection

Many types of convective flows are possible, and many of them have been well studied.

Convection can be classified as either forced or natural. In forced convection, flow is caused by

external means. Flow of a liquid being pumped across a flat plate, for example, is forced

convection. The flow that will be considered in this study, however, is of the other variety.

Natural convection is convection in which flows are caused by forces that interact with

thermophysical property variations. For example, natural convection in a system can be caused

by buoyancy, interfacial tension gradients, or a combination of the two. When buoyancy is the

cause of flow, this is called Rayleigh convection; if interfacial tension gradients cause flow, this

is called Marangoni convection.

If a system (here, "system" refers to a fluid layer) is going to convect, then the

temperature difference corresponding to the onset of convection is determined by the direction

from which the system is heated, the shape and dimensions of the system, the thermophysical

properties of the fluid, and the characteristics of the disturbance given to the system. Regarding









these factors, it is noteworthy that the fluid's thermophysical properties themselves may change

significantly across the fluid layer depending on the magnitude of the vertical temperature

difference to which fluid layer is subjected. This research focuses on the effects of the shape and

dimensions of the system on convective behavior. Unless otherwise noted, the heating of the

system referred to is uniform on the top and bottom surfaces. Also, for the physical explanations

that follow, until otherwise noted, the systems discussed may be assumed unbounded in the

lateral direction.

1.1.1 Rayleigh Convection

A system capable of convection, in general, could be heated from below or above and

include any number of liquid and vapor phases. For a physical explanation of Rayleigh

convection, though, all that need be considered is a single layer of fluid, unbounded in the

horizontal direction, heated from below, with the top and bottom walls held at constant

temperatures. Understanding the origin of convection in this system will be sufficient to predict

what is likely to happen in more complex systems. More importantly, a single fluid layer, heated

from below, is precisely the system examined in this research.

Generally, the density of a fluid decreases as temperature is increased. This means that

the density at the top of the fluid would be higher than that at the bottom of the phase. Consider

a z-axis along the vertical direction in the fluid layer, with the top surface at z = Lz, and the

bottom surface at z = 0. This simple system is shown as Figure 1-1.

This top-heavy fluid layer is potentially unstable to gravity (shown in many figures as g)

if subjected to a perturbation. If a particular disturbance to the system causes an element of fluid

near the top of the layer to be displaced downward, then it will have a tendency to continue

moving downward toward its gravitationally appropriate resting place. As this element of fluid









moves downward, it displaces the lighter fluid surrounding it upward due to mass conservation.

The downward-moving fluid elements are heated once they reach the bottom of the system, and

the upward-moving fluid elements are cooled once they reach the top of the system, and so the

process repeats. This process results in rolls of circulating fluid, or convective "cells," that form

cellular flow patterns. Since the lowest density in the phase is at the bottom, a fluid element

displaced downward would continue to move downward to z = 0 if its motion were not resisted

in any way. Flow is resisted, however, by the dynamic viscosity of the fluid. Additionally, if the

kinematic viscosity and thermal diffusivity of the fluid are sufficiently high, then the mechanical

and thermal perturbations may quickly die out. It is when the vertical temperature difference

across the system is increased beyond a certain critical value that this convection-Rayleigh

convection-will occur; at that point, gravitational instability overcomes the damping effects of

kinematic viscosity and thermal diffusivity. Rayleigh convection is stronger in a phase with

lower kinematic viscosity and lower thermal diffusivity; this is because mechanical and thermal

disturbances are damped out less quickly in a phase having lower values for those properties.

Now, it might appear that Rayleigh convection results simply from large enough vertical

temperature gradients. In truth, however, even with a sufficiently large vertical temperature

gradient present, Rayleigh convection can begin only if disturbances with transverse (horizontal)

variation are imposed on the system. A fluid element displaced by a disturbance must "feel" a

relative difference in the density of the fluid horizontally neighboring it, and this can occur only

as a result of disturbances with transverse variation. Note that the disturbance which causes the

onset of convection need not necessarily be a mechanical one. A thermal disturbance with

transverse variation could just as easily cause convection to begin by making a certain fluid









element warmer or cooler than the fluid horizontally adjacent to it. Figure 1-2 shows a

buoyancy-driven convective flow pattern occurring in a fluid phase.

1.1.2 Marangoni Convection

Convection can occur in another way, as well. The flow can be driven by interfacial

tension gradients rather than buoyancy forces. This type of flow is called Marangoni convection.

Since Marangoni convection is driven by gradients in interfacial tension, it is a phenomenon that

can occur only in systems possessing multiple fluid phases which contact each other at

interfaces. This research is concerned with the simpler case of a single fluid layer confined in all

directions by solid walls. Thus, Marangoni convection could not occur in the system being

researched. Still, a brief explanation of Marangoni convection will now be given for the sake of

giving a more complete explanation of the physics of convection.

To this end, consider a system of two vertically stacked fluids, in which one vertical

boundary is heated so as to subject the system to a vertical temperature gradient. Again, assume

the system is unbounded in the horizontal direction. In this hypothetical system, both the top and

bottom walls are kept at constant temperatures and Marangoni convection can occur in this

system regardless of which vertical boundary of the system is warmer. For the sake of making

an explanation by example, assume that the bottom wall of the system is warmer than the top.

Like Rayleigh convection, Marangoni convection results from perturbations with transverse

variation. If the interface between the fluids is perturbed, then some interfacial regions could be

pushed upward and some downward. This gives rise to a transverse temperature gradient along

the interface. Consider an interfacial region pushed upward. The fact that the system is heated

from below means that this region now experiences a lower temperature than the regions of the

interface adjacent to it. Typically, surface tension, like density, increases with decreasing









temperature. Thus, the interfacial region that has been pushed upward has an increased

interfacial tension and pulls fluid at the interface toward it. When this happens, fluid from the

bulk phases must move in to replace fluid being pulled away from regions of the interface that

lowered in interfacial tension (when the system is heated from below, these regions are the

troughs of the wavy interface). The flow patterns in Marangoni convection, like those in

Rayleigh convection, are cellular. Since the interfacial tension gradients are generated by

temperature differences along the interface, a higher vertical temperature difference across the

system favors Marangoni convection. Figure 1-3 is a simple illustration of Marangoni

convection.

The replacement fluid arriving from the bulk phases is warm in the case of the lower

fluid, and cool in the case of the upper fluid; this supply of replacement fluids can either

strengthen or weaken the temperature gradient which fuels Marangoni convection. The effect

that the fluid moving to the hot regions of the interface from the bulk phases has on this

temperature gradient is dependent upon the amount of fluid from each phase that arrives at those

locations and on the thermophysical properties of each phase.

As with Rayleigh convection, dynamic viscosity resists flow and can damp the process.

Marangoni convection, like Rayleigh convection, is stronger in a phase with lower kinematic

viscosity and lower thermal diffusivity because a phase having lower values for those properties

damps out disturbances less quickly. On the other hand, if these two properties have high

enough values, a disturbance at the interface will die out rather than amplify to cause convection.

Of course, the phase with a lower kinematic viscosity is not necessarily the phase that has a

lower thermal diffusivity. Thus, the combination of these two properties determines which phase

will convect more strongly. The phase which has a kinematic viscosity-thermal diffusivity









combination more favorable to convection is the phase from which it will be easiest for fluid to

move in and replace that which is lost at hot interfacial regions during Marangoni convection; it

is primarily from this phase that fluid is supplied to the hot regions of the interface. If this more

strongly convecting phase is the warmer phase, then the fluid it supplies to the hot interfacial

regions is even warmer than what was originally present, and Marangoni convection is

reinforced. In the opposite scenario, cold fluid moves in to the hot regions of the interface; this

cools those regions down and counteracts Marangoni convection. In the case where system of

two stacked fluids is comprised of a vapor-liquid bilayer, rather than a liquid-liquid bilayer, it

should be noted that a liquid can generally transport more heat by convection than can a vapor.

This is quantitatively shown when a comparison is made between the product of density and heat

capacity (p Cp) for liquids and vapors. Though the thermal conductivities of vapors and

liquids may differ, this difference may be considered less significant. The reason for this is that

the fluids are already conducting heat in the motionless initial state of the system (base state).

While the effect of the difference in density multiplied by heat capacity becomes important once

convection begins, the effect of the difference in thermal conductivity is relatively unchanged.

By this line of reasoning, replacement fluid arriving to troughs on the interface from the liquid

phase of a vapor-liquid bilayer system is likely to have more of an effect on the temperature

gradient along the interface than replacement fluid arriving from the vapor phase of the bilayer.

The same reasoning (involving the product of density and heat capacity) also can be used in

deciding which fluid will more strongly affect the interfacial temperature gradient in a liquid-

liquid bilayer system.

It is also very interesting to note that Marangoni convection can occur in the system even

if the interface remains flat. If a non-mechanical, thermal perturbation is given to the interface,









then some regions become hot and some cold. Fluid from the hot regions is pulled toward the

cold regions by interfacial tension, and fluid at the cold regions is displaced. As this happens,

fluid from the bulk must move in to replace that lost at the hot interfacial regions this marks the

start of convection.

1.2 Rayleigh Number

The focus of this explanation will get back on its main course and address Rayleigh

convection once again. Scaling of the equations that model a fluid layer heated from below leads

to the following definition of the dimensionless Rayleigh number, which is shown as Equation

1.1. The Rayleigh (Ra) number is a quantity that relates the factors determining whether or not

buoyancy-driven convection occurs.

a*g*L3 *AT
Ra = (11)
v*K

The length scale associated with Rayleigh convection is the vertical phase depth, shown

as Lz. In this definition, a is a volumetric thermal expansion coefficient, g is the magnitude of

gravity, AT is the vertical temperature difference across the layer, v is the kinematic viscosity

of the fluid, and K is the thermal diffusivity of the fluid (which is equal to the thermal

conductivity divided by the product of the density and heat capacity).

A higher Rayleigh number corresponds to a system that is more susceptible to buoyancy-

driven convection. Notice that the factors which favor buoyancy-driven convection are all

grouped in the numerator, while those which oppose the flow form the denominator. The

thermal expansion coefficient represents the degree to which the density of the fluid changes

when it is heated, so of course a higher value for this works in the favor of convection. Gravity

and the vertical temperature difference are the most obvious of the two components needed to

drive Rayleigh convection. A larger vertical temperature difference makes the system more









unstable to Rayleigh convection. Having a larger vertical phase depth means that the fluid feels

a weaker effect from the no-slip conditions at the top and bottom walls, and thus is less

mechanically constricted, and can more easily flow in a convective pattern. This effect is quite

strong, and accordingly, the Rayleigh number is proportional to the vertical phase depth raised to

the third power. Kinematic viscosity and thermal diffusivity act to dissipate any disturbances

given to the system and prevent convection. Thus, these two thermophysical properties are

located together in the denominator of the Rayleigh number.

1.3 Physical Explanation: Pattern Selection

Some of the most interesting facets of the convective behavior studied in this research

relate to the competition of certain physical phenomena to determine precisely which convective

flow patterns arise at the onset of convection.

1.3.1 Laterally Unbounded System

Imagine, once again, the system shown in Figure 1-1. The system is constantly subjected

to small mechanical and thermal disturbances. As mentioned earlier, only disturbances with

transverse variation can cause the onset of convection. The disturbances that the system receives

are of an extremely large range of wavelengths, especially considering the case in which the

system is laterally unbounded. Still, only one of these many disturbances actually causes the

onset of convection. It is this disturbance to which the system is most unstable. It is the

wavelength of this disturbance which determines the wavelength of the onset convective flow

pattern. It is interesting to ask what makes disturbance of one wavelength more stabilizing or

destabilizing to the system than another.

For the moment, consider two possibilities in terms of the wavelength of a periodic,

wave-shaped disturbance. One possibility is that the disturbance is of very short wavelength,









meaning that it is quite jagged and choppy in appearance. The second possibility is that the

disturbance is of very long wavelength, and so it is very gently, gradually sloping, with its

minimum-points and maximum-points spread far apart from one another. The disturbance of

very short wavelength possesses a great deal of transverse variation, since it repeats its lateral

oscillations so frequently; this high degree of transverse variation favors convection. However,

since the adjacent peaks and troughs of the wave-shaped disturbance are so close to one another,

they easily diffuse their momentum and thermal energy into the horizontally neighboring fluid,

which dissipates the disturbance and stabilizes the system to convection. For example, a fluid

element which initially was heated and lowered in density relative to the fluid laterally adjacent

to it, as a result of a disturbance of very short wavelength, would quickly match the density of

the laterally adjacent fluid due to rapid diffusion of heat from the fluid element. Thus, insofar as

diffusion of heat and momentum is concerned, the system is quite stable to disturbances which

are of very short wavelength. The system reacts to a disturbance of very long wavelength a bit

differently. A disturbance of very long wavelength is so laterally spread out that the sort of rapid

diffusive stabilization just described does not play a role. Consequently, the lateral density

differences generated by a disturbance of very long wavelength remain present rather than

dissipating into uniformity, which is a condition that favors buoyancy-driven convection.

However, when the disturbance is of very long wavelength, the adjacent peaks and troughs of the

wave-shaped disturbance are so far apart from one another that the local density differences

between adjacent fluid elements (the transverse variations which are needed to drive buoyancy-

driven convection) are too weak to cause and sustain buoyancy-driven convection. If a

disturbance of very long wavelength were able to form a corresponding cellular, convective flow

pattern of a very long wavelength, the flow pattern would not be able to be sustained; this is









because it would simply require too much energy to drive the fluid in the convective cells across

the very long, nearly horizontal paths which would need to be traveled within each convective

flow cell. So, in summary, short-wavelength disturbances are not very destabilizing because

they rapidly degrade by diffusion, and long-wavelength disturbances are not very destabilizing

because they are spread out so widely in the lateral direction that the variations they induce in the

thermophysical properties of the fluid are not strong enough to drive flow. Thus, of the wide

range of disturbances to which the system is subjected, the one disturbance which is most

destabilizing to the system and causes the onset of convection will possess some intermediate

wavelength, which is determined by the competing physical behaviors associated with short-

wavelength and long-wavelength disturbances. The vertical phase depth, of course, plays a large

role in governing which disturbances are most unstable, as well. At this point, one could

imagine the appearance of a simple graph relating the stability of the system, in terms of the

critical Rayleigh number (Rayleigh number corresponding to the critical vertical temperature

difference at which convection begins), to disturbances of different wavelengths. The graph

(Figure 1-4) can be drawn using a general, dimensionless wavelength and the dimensionless

critical Rayleigh number, which is Raccrt; numerical values for either quantity are not necessary at

this point.

Once the system is at or above the critical vertical temperature difference and is subjected

to the right disturbance, buoyancy-driven convection begins. Still considering a one-dimensional

system, unbounded in the lateral direction, the flow pattern at the onset of convection possesses

the wavelength of the disturbance which caused convection.









1.3.2 Rectangular System, Laterally Bounded, Periodic Lateral Boundary Conditions

Now, consider a two-dimensional system. The two-dimensional system to be considered

is the system that would be obtained if two lateral walls were simply added to the left and right

sides of the system of Figure 1-1, with a no-stress condition on velocity and an insulating

condition on temperature imposed on each lateral wall. The horizontal width of this new system

is called Lx and the vertical depth is called L This system can be examined to understand the

interesting physics governing convective pattern selection.

Due to the stress-free conditions on velocity and the insulating conditions on temperature

at the lateral walls, the x-direction dependence of the velocities and temperatures in the system,

which shall be called Vi, could be expressed in terms of cosines, in the form

n Kx
SCOS n = 0, 1,2, 3, .... (1.2)


The number of convective cells which may form in the system, then, is related to Lx

through this form of x-direction dependence, as is the size of the cells which form. Notice that n

is an integer, so not just any pattern may arise in a laterally bounded system. Only those patterns

which can be physically accommodated by the lateral width of the system may form, and so only

disturbances with those shapes are physically admissible to the system. Of the set of

disturbances that are physically admissible to the system, only one will be the most destabilizing

and, when the critical vertical temperature difference has been exceeded, cause the onset of

convection.

In a laterally bounded system, for the same reasons as in a system without side walls,

disturbances that are laterally very short or very long are not very destabilizing. Thus, again the

shape of the most destabilizing disturbance and onset flow pattern is determined by competition









between physical behaviors like those that were explained for short-wavelength and long-

wavelength disturbances.

The size and shape of the onset flow pattern, of course, are heavily dependent on the

dimensions of the system. In a system that is laterally very wide, the onset flow pattern would

likely not be, for example, one very wide flow cell. A flow pattern such as that would not be

sustainable for the physical reasons like those explained for disturbances of very long

wavelength, which were already explained. The size of the system is typically described using

the dimensionless aspect ratio. For a rectangular system, the aspect ratio is the ratio of the width

of the system to the vertical height of the system, and it may be called Az. For a cylindrical

system, the aspect ratio is the ratio of the radius of the system to the vertical height of the system,

and it may simply be called A. For an annular system, the aspect ratio may also be called A, and

it is the ratio of the outer annular radius (Ro) to the vertical height of the system.

As the aspect ratio is varied, obviously the stability of the system to any given

disturbance is varied with it. For example, a wave-shaped disturbance which induces a

convective flow pattern consisting of two rolls of convecting fluid would be most destabilizing at

some particular value of A but if the aspect ratio were slightly increased (making the container

laterally wider or vertically shorter) or decreased, then that same disturbance would not be as

well accommodated by the lateral width of the system, and consequently would not be as well

suited to destabilize the system. The reason for this is related to physical behaviors like those

described for disturbances of very short and very long wavelengths, as described for the laterally

unbounded system. What is meant by a convective pattern consisting of two rolls of convecting

fluid is clarified in the following cross-sectional velocity profiles of a convecting, laterally

bounded system (Figure 1-5).









Continuing with the example, continuously increasing the aspect ratio would make the

two-convective-roll-inducing disturbance less and less destabilizing to the system, and thus make

the corresponding two-convective-roll onset flow pattern more and more difficult for the system

to form and maintain; as an onset flow pattern becomes more and more difficult to form and

maintain, the critical vertical temperature difference corresponding to it increases, as does the

corresponding critical Rayleigh number. At some sufficiently large aspect ratio, the system

would find a disturbance that induces more than two convective rolls to be more destabilizing,

and the corresponding onset flow pattern to be more energetically favorable. Supposing that

increasing the Axz value caused the example system to become more unstable to a disturbance

with that induced four convective rolls rather than two, the corresponding onset flow pattern

would consist of four rolls of convecting fluid.

The stability of the system to the four-convective-roll-inducing disturbance, and the

degree of difficulty that the system would have in maintaining its corresponding onset flow

pattern, are again dependent on physical behaviors like the short-wavelength and long-

wavelength behaviors described for laterally unbounded systems. As the Axz value is further and

further increased, this type of pattern-switching continues. If the A, value of the example

system were further increased, the system would eventually reach an Az value at which it would

be more favorable for the system to form and maintain an onset flow pattern with more than four

rolls of convecting fluid. Supposing that the next flow pattern in the example consists of six rolls

of convecting fluid rather than four, then difficulty that the system has in forming and

maintaining this six-roll-pattern would once again be determined by physical behaviors like the

short-wavelength and long-wavelength behaviors that have been explained earlier. Resulting

from these physical behaviors, which correspond to the size and shape of an imposed









disturbance, the exact critical temperature difference, and critical Rayleigh number,

corresponding to each different flow pattern is dependent on the Ax value and has a minimum at

some certain Axz value that is optimal in terms of how well a system with that aspect ratio

physically accommodates the flow pattern. When a system is laterally unbounded, or a system

has lateral walls that are stress-free and insulated (which means the system behaves essentially

like a laterally unbounded system), then the minimum possible critical Rayleigh number for any

possible flow pattern is the same. Based on all of this, a graph showing the relation between the

critical Rayleigh number and the value of Az is given below as Figure 1-6. Note that in a

laterally unbounded system, or in a laterally bounded system with periodic lateral boundary

conditions, the convective patterns that can form are comprised of convective rolls which are all

of equal lateral width. As an example of what is meant by this, see the flow pattern diagrams in

Figure 1-5.

In this figure, the roman numerals represent flow patterns with different numbers of

convective rolls. Flow pattern "III" includes more convective rolls than "II" and flow pattern

"II" includes more convective rolls than "I" does. Basically, aspect ratios at which the slope of

the curve changes from positive to negative are aspect ratios at which there is a transition to a

higher number of convective rolls being present in the onset flow pattern.

1.3.3 Rectangular System, Laterally Bounded, Non-Periodic Lateral Boundary Conditions

When running experiments or considering practical applications, one does not encounter

systems with periodic boundary conditions at their lateral walls. The more realistic case of a

laterally bounded, rectangular system with non-periodic lateral boundary conditions will now be

detailed.









The behavior in this system is essentially the same as what was just explained for the

rectangular system with periodic lateral boundary conditions, but there are a couple of key

differences: the convective rolls which form at the onset of convection are not necessarily of

identical size to one another, and the minimum critical Rayleigh number for each possible flow

pattern (considering different values of Axz) decreases as Axz increases rather than maintaining a

constant value.

Since the convective rolls which form at onset are not necessarily of identical size to one

another, the flow pattern with four convective rolls in Figure 1-5 could instead appear as shown

in Figure 1-7.

The second key difference of this case from the case in which the lateral boundary

conditions are periodic is that for higher and higher A, values, the minimum critical temperature

difference values become lower and lower. The reason for this is that the no-slip effects of the

side walls do less to stabilize the system as the aspect ratio is increased. As A, approaches

infinity, the critical temperature difference values for the flow patterns approach values

corresponding to the well known critical Rayleigh number of 1708 that is associated with the

onset of buoyancy-driven convection in laterally unbounded systems (Davis 1967). For the two-

dimensional, laterally bounded system, no-slip conditions at the lateral walls, here is a general

plot of the stability of the system, in terms of the critical Rayleigh number, to disturbances

corresponding to different onset flow patterns; the plot is simply based on the reasoning

explained above.

Again, in this figure, the roman numerals represent flow patterns with different numbers

of convective rolls, where pattern "III" includes more convective rolls than "II" and pattern "II"

includes more convective rolls than "I." Recall that aspect ratios at which the slope of the curve









changes from positive to negative indicate transitions to onset flow patterns with higher numbers

of convective rolls being present.

1.3.4 Cylindrical System

So far, the example systems discussed have been assumed to be laterally bounded,

rectangular systems or laterally unbounded systems. Radial walls in cylindrical and annular

systems significantly affect the critical conditions for the onset of convection. This research

addresses primarily cylindrical and annular systems. To prevent any possible confusion, note

that throughout this dissertation, "cylindrical system" (or sometimes the term "open cylindrical

system" may be used) refers to a fluid system bounded by a cylindrical container, in which the

entire container is filled only with fluid, and does not include a center-piece as an annular

container does. From this point onward, unless otherwise noted, the systems discussed will be

within vertical, cylindrical containers (meaning that the direction of gravity is parallel to the axis

cutting through the radial center of the cylinder) with finite outer radii or within vertical, annular

containers. To be completely clear, a vertical, annular container is one which has an annular

cross-section when viewed from above.

Increasing the aspect ratio, A, in a cylindrical system has the effects on pattern selection

which were just explained for the laterally bounded, rectangular system. As A is increased, the

onset flow patterns include more and more convective rolls, so as to occupy the system in a more

energetically favorable fashion. Also, as A approaches infinity, the value of the critical Rayleigh

number approaches the value of 1708, which is well-known to be associated with laterally

unbounded systems. There is, however, another aspect to pattern selection in cylindrical

systems, and it will be explained now.









In a rectangular system, the number of convective rolls which form can be related to the

imposed disturbance's size and shape in the x-direction. For rectangular, laterally unbounded

systems, the corresponding onset flow pattern can then be described by a wave number in the x-

direction, inversely proportional to the wavelength of the flow pattern in the x-direction. In all

systems considered in this research, whether rectangular, cylindrical, or annular, it is most

energetically favorable for the system to form onset flow patterns with only one convective roll

in the vertical direction; thus, there is no need to consider a z-direction wave number, z-direction

wavelength, or the size and shape of a disturbance in the z-direction for this research. The flow

pattern in a rectangular system, then, can be described simply by describing its size and shape in

the x-direction.

In a cylindrical system, flow patterns may have different numbers of convective rolls in

the radial direction, like the patterns in rectangular systems could have different numbers of

convective rolls in the x-direction. However, flow patterns in cylindrical systems may also have

differing numbers of convective rolls in the azimuthal direction. The flow patterns in a

cylindrical system must, of course, be periodic. The periodicity of the flow patterns in the

azimuthal direction, and thus how many convective rolls exist along the azimuthal direction, is

characterized by an azimuthal wave number, called m. For example, the flow pattern described

by the wave number m = 2 is one which repeats twice as one progresses once through the fluid

layer in the azimuthal direction. For an m = 2 flow pattern, progressing 900 around the system in

the azimuthal direction, from an arbitrary starting point, leads to a point that has the opposite

vertical velocity of the starting point. Likewise, for an m = 1 flow pattern, progressing 1800

around the system in the azimuthal direction, from an arbitrary starting point, leads to a point that

has the opposite vertical velocity of the starting point. An m = 0 flow pattern, though, is









axisymmetric in the azimuthal direction. Figure 1-9 gives examples of flow patterns possessing

some of the lowest m values. In these diagrams, the letter "U" indicates upward velocities and

the letter "D" indicates downward velocities. The intensity of the shading indicates the

magnitude of the velocities, with darker shades representing higher velocities.

At certain aspect ratios, the a cylindrical system favors certain azimuthal periodicities,

and thus, certain m values. For each m value, the system may have a differing number of

convective rolls in the r-direction, and how many there are depends once again upon physical

behaviors like the short-wavelength and long-wavelength behaviors that have already been

explained. The relation, for each m value, between the number of convective rolls, the critical

Rayleigh number needed to obtain a particular number of rolls, and the aspect ratio appears much

like what is shown in Figure 1-8. Generally, in the cylindrical cases, only m = 0, 1, or 2 patterns

were favored as onset patterns; this means that in most cases, the critical vertical temperature

difference corresponding to the onset of convection was lower for m = 0, 1, or 2 patterns than for

patterns with other azimuthal wave numbers. As the aspect ratio approaches infinity, the critical

Rayleigh numbers for flow patterns of all m values approach the unbounded-system-value of

1708. A general graph showing the relation of flow patterns and their critical Rayleigh numbers

to the aspect ratio in a cylindrical system is given as Figure 1-10. This graph was obtained by

performing a series of computations, to determine the critical Rayleigh number corresponding to

several values of the aspect ratio, using the computational program developed for obtaining

computational results in this research. The computational solution method is detailed in Chapter

5, and the computational results are shown extensively in Chapter 7.

On this diagram, starting from a given aspect ratio and moving upward along the graph is

analogous to increasing the vertical temperature difference across the system. Once the first of









the m = 0, 1, 2, or 3 curves has been reached, then this is the critical Rayleigh number

corresponding to the critical vertical temperature difference. The m value of the bottom-most

curve at a given aspect ratio is the m value for the onset flow pattern at that aspect ratio; this

azimuthal wave number, since the critical temperature difference and critical Rayleigh number

corresponding to it are lowest, is the azimuthal wave number of disturbances to which the system

is most unstable. Notice that the curve for each particular m value looks much like the one in

Figure 1-8, which reflects the effects of physics related to the sizes and shapes of different

disturbances on the stability of the system.

Graphs like Figure 1-10 are extremely helpful in research on buoyancy-driven

convection. The reader is asked to notice that the graphs presenting computational results for

buoyancy-driven instability in cylindrical systems in Chapter 7 are just like the one shown in

Figure 1-10. Figure 1-10, interestingly, is the correct graph for cylindrical systems of any

dimensions, and it holds true regardless of the set of thermophysical properties possessed by the

fluid in the system. This will be further explained at the end of Chapter 4. Note that the stability

diagrams in this research would, however, differ very slightly depending on whether or not the

variation of viscosity with temperature is being considered.

1.3.5 Annular System

Nearly everything that was just said for cylindrical systems in Section 1.3.4 applies for

annular systems, as well. An exception is that as the outer radius of the system approaches

infinity (and thus, the A value approaches infinity), one would expect the critical Rayleigh

number to approach a value other than 1708, which corresponds to a laterally unbounded system,

since an inner radial wall is present. Patterns in annular systems are described primarily by their

azimuthal wave number, m. The patterns in annular systems may also include more than one









convective roll in the radial direction, but this tends to happen only when the ratio of the inner

annular radius (R1) to the outer annular radius (Ro) is quite small. This ratio of radii is a new

dimensionless parameter affecting pattern selection which was not present in the consideration of

cylindrical systems. The radius ratio shall be called "S" and will be given by Equation 1.3.

S = R, (1.3)
R

The effect of the radius ratio is the key difference between the annular system and the

cylindrical system in terms of the formation and selection of flow patterns. Stability diagrams of

the relations between Racrt, the aspect ratio, and the radius ratio (S) for different wave numbers,

and valid for systems with any set of thermophysical properties, could be made for the annular

system, as well; they would appear somewhat like the one for the cylindrical system, shown as

Figure 1-10. Such diagrams are shown in Chapter 7. As annular systems are the main focus of

this research, they will be explained much more thoroughly in the following pages.

1.4 Annulus vs. Cylinder

Certainly, there has been a good deal of research already done on natural convection in

annular systems (Stork & Muller 1972, and Littlefield & Desai 1990). However, much of that

work is considerably different from the current research. For example, many of those studies

involve horizontal annuli rather than the vertical annuli addressed in the current study. Secondly,

of those past works which do address convection in vertical annuli, many consider the case in

which heat is supplied to the system from the inner rod of the annulus rather than the case

considered in this study, which is that heat is supplied to the system from its bottom wall. The

case in which heat is supplied from the bottom wall is highly relevant to industrial processes, and

this will soon be discussed further.









The radial walls of all cylindrical and annular systems considered in this research are

assumed to be insulating. This assumption shall be maintained throughout this paper, as the

radial walls in the experiments conducted for this research were made to be insulating. Whether

the radial wall is treated as conducting or insulating does have an effect on what the critical

vertical temperature difference is for the onset of convection. Generally, the system becomes

unstable at a lower vertical temperature difference if the radial walls are insulating. This makes

sense because insulating radial walls do not allow a system to dissipate thermal disturbances as

easily as conducting radial walls would, and making the dissipation of thermal disturbances more

difficult makes the system more unstable to convection. As one might expect, the difference

between the insulating and conducting cases is more pronounced in systems that span a smaller

distance in the radial direction. In such systems, the radial walls are closer to the interior of the

system and so they have a relatively stronger effect on the behavior of the system. Also, whether

the radial walls are insulating or conducting, it is obvious that if the system is of smaller radial

extent, then the system will be more stable to convection than a wider system of matching

vertical depth; this is because of the relatively larger effect of the no-slip condition on velocity

imposed at the radial walls.

The addition of an inner block of circular cross-section to the center of a cylindrical

system transforms the cylindrical system into an annular system. The inner block of the annular

system is insulating, just like the outer radial wall, and varying the radius of this circular inner

block changes the gap width occupied by fluid within the annular system. Changes in this gap

width cause significant changes in the critical vertical temperature difference and flow pattern at

the onset of convection in the annular system. Primary goals of this research were to better

understand the phenomenon of buoyancy-driven convection in terms of how it differs in an









annular system compared to a cylindrical system, and how the onset conditions for convection in

the annular system differ as the annular gap width varies.

As one may expect, the addition of a circular inner block to the cylindrical system does,

because of the additional radial wall and corresponding no-slip condition on velocity that it

introduces, impart to the system a higher stability to convection. That is to say, a cylindrical

system of a certain vertical depth, which requires a certain vertical temperature difference for the

onset of Rayleigh convection, would require a higher vertical temperature difference to convect

if some small, circular block were introduced to the center of the system, converting it to an

annular system. The circular center-block present in an annular system also affects the flow

patterns which may appear at the onset of convection. Stork and Muller (1974) have shown

experimentally that in a one-fluid annular system, heated from below, convective cells can form

such that they fill the annular gap in an azimuthally aligned arrangement that is rather

reminiscent of the arrangement of spokes on a bicycle tire. An example of this type of pattern

presented in the work of Stork and Miller is shown as Figure 1-11. In this photograph of one of

their experiments, the fluid convecting is a silicone oil, and the flow is visible due to the

presence of aluminum tracer powder in the oil. When the flow is horizontal, the aluminum

particles align in such a way as to reflect more light, making those sections of the oil appear

lighter in color; regions of the oil which are flowing vertically appear darker in color.

As the radius of the center-block in the annulus is increased, the system becomes more

stable to buoyancy-driven convection, meaning that a higher vertical temperature difference

across the system is needed for convection to begin; this makes sense because as the radius of the

center-block is further and further increased, there is more and more wall-surface area (and more

of its accompanying no-slip effect) compared to the amount of fluid in the system. Also, as the









radius of the center-block in the annulus is changed, the number of azimuthally aligned cells

changes. Generally, a greater number of azimuthally aligned cells form at the onset of

convection as the radius of the center-block approaches the outer wall of the annulus, and a

smaller number of azimuthally aligned cells form at the onset of convection as the radius of the

center-block is decreased. This research has shown that once the radius of the center-block

decreases past a certain value, new rolls of convecting fluid at the onset form in radial alignment

(meaning that they span the entirety of the azimuthal direction and that they are concentric to the

center-block of the annular container) in addition to the original azimuthal arrangement. An

example of this, generated by the computations done for this research, is shown in Figure 1-12.

In this flow profile, the velocity is scaled, negative velocities represent downward flow in the z-

direction (which extends into and out of the plane of the diagram), and positive velocities

represent upward flow in the z-direction. This cross-sectional velocity profile represents a cross-

sectional region approximately half of the distance along the vertical depth of the system that it is

computed for.

If the radius of the center-block of the annular system is further and further decreased, the

radial alignment of the convective cells becomes more prominent compared to the azimuthal

alignment of the cells, and the flow patterns become progressively more like those seen at the

onset of convection in open, cylindrical systems. This transition in onset flow patterns was of

great interest in this research.

1.5 Application

Aside from its applications to the regular appearances of Rayleigh convection in nature,

this study finds application in the drying of paint films, in small-scale fluidics, and notably, in

semi-conductor crystal growth. The vertical Bridgman crystal growth method is commonly used









for growing semi-conductor crystals such as lead-tin-telluride. Vertical Bridgman growth

involves a liquid semi-conductor melt layer, which lies atop the solid crystal phase being formed

by the solidification of the melt. The arrangement is heated from above so that the crystal at the

bottom of the system grows upward (see Figure 1-13). Since solidification is occurring at the

liquid-solid interface, the temperature at that location is constant.

Of course, a system that is heated from above, such as this crystal growth system, will not

be destabilized by the vertical temperature gradient in the system. Still, buoyancy-driven

convection is possible in this system for another reason. As solidification occurs at the solid-

liquid interface, some portion of the species that comprise the liquid melt is rejected into the

liquid phase, near the interface. The species comprising a liquid melt, such as lead-tin-telluride,

are not all equal in weight. If the species rejected the most into the liquid near the interface is

relatively light in weight, and makes the liquid region near the interface less dense than the

remainder of the liquid melt phase above the interfacial region, then a top-heavy system results.

Thus, due to solutal gradients, buoyancy forces can cause convection in the crystal growth

system just like the systems previously described in this chapter.

Convection in this system can cause impurities from the ampoule to be transmitted to the

solid-liquid interface and this can cause flaws on the face of the forming crystal. Also,

convection in the system can cause the heat transfer at the solidification interface to be non-

uniform. For these reasons, it is desired to know the conditions corresponding to the onset of

convection in this system so that it can be better understood, predicted, controlled, and

prevented.

It should be mentioned that, in some cases, convection in a crystal growth system may

not be a problem, and may even be desirable. If a crystal is grown at relatively low temperatures









(unlike, for example, the high value of 1250 C at which gallium arsenide is grown), then

convection helps transfer heat near the solidification interface and relieves thermal stresses there,

preventing the growing crystal from fracturing. Also, the transport of impurities from the walls

of the crystal-growth container to the solidification interface does not tend to become a problem

in these lower-temperature systems (in high-temperature crystal growth systems, it can relate to

constituents being released from the crystal growth container by the degradation of the

container). Thus, in some low-temperature crystal growth systems, convection could clearly be

beneficial.

In vertical Bridgman crystal growth, the depth of the liquid phase is continuously

changing as the melt solidifies into the growing crystal, consuming a portion of the melt. As the

phase depth changes during crystal growth, so does the possible convective flow behavior. Thus,

being able to predict the convective behavior of the system at differing fluid depths is important.

Some crystal growth processes, such as the liquid-encapsulated growth of gallium

arsenide, require very high temperatures. In this research, convective behavior will be studied at

more moderate temperatures, as the physics driving the convection are the same regardless of the

temperature of the system.

Typically, crystal growth is conducted using cylindrical fluid layers. It is interesting to

ask how the convective behavior of a crystal growth system would differ if annular fluid layers

were employed rather than cylindrical layers. One might imagine that the presence of an extra

solid wall at the center of the fluid layer would, by means of the no-slip condition it imposes on

velocities at that location, impart to the crystal growth system a greater stability to convection.

The additional center wall in an annular system would allow the system to suppress more

disturbances than a cylindrical system. Furthermore, if the crystal were grown in an annular








container with a sufficiently large annular center-piece (relative to the outer annular radius), then

any convective patterns which did arise would be very predictable and uniform sets of only

azimuthally aligned cells, which could be easier to work with than the patterns that arise in

cylindrical containers.


Tcold


zLX
X


Higher Density Fluid Layer



Lower Density


I hot

Figure 1-1. Simplified system diagram: heated from below.


z Lz






Z 0


Figure 1-2. Rayleigh convection.









Upliase
.......... ...............
"H O" T- 6* 11 IIIIIIIIIIIIIIIIIIIIIlllllllllliiI .
I",
Phase I I


Figure 1-3. Marangoni convection.


Wavelength

Figure 1-4. General diagram: critical Rayleigh number vs. disturbance wavelength.









z t-_O


Two Convective Rolls


zt


Four Convective Rolls


Figure 1-5. Cross-sectional velocity profiles of flow patterns with different numbers of
convective rolls.


Racrit


t I
' I I
' ~I
II III


Axz


Figure 1-6. General stability diagram for buoyancy-driven convection in a rectangular, laterally
bounded system, with periodic boundary conditions at lateral walls, for varying
aspect ratio.


t% 4-


x








x










zt x


Four Convective Rolls


Figure 1-7. Cross-sectional velocity profiles of four-roll flow patterns with non-uniform roll
size.


1708


Axz


Figure 1-8.


General stability diagram for buoyancy-driven convection in a rectangular, laterally
bounded system, with non-periodic boundary conditions at lateral walls, for varying
aspect ratio.










m=0 @
D






m=2 1





m=3 a,


Figure 1-9. Diagrams of m 0, 1, 2, 3 flow patterns: "U": upward flow, "D": downward flow.










Critical Ra vs. Aspect Ratio, Cylinder


.......------- m = 2


::::::::::::::::::::::::::::::: m = 3


Figure 1-10.


General stability diagram for buoyancy-driven convection in a cylindrical
container, for varying aspect ratio.


Figure 1-11. Convective flow pattern in an annulus (Stork & Miuller 1974).


2900

2700

2500

2300

2100

1900

1700


0.7


--m = 0


1.2 1.7 2.2 2.7 3.2
Aspect Ratio


---- m= 1











Example -Flow Pattern in Annular System


0.01

0.008

0.006






0.004






-0.01

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 0.008 0.01
x-direction



-0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Velocity (scaled) Il.i i Velocities are Downward in z-direction and Positive Velocities are Upward in z-direction


Figure 1-12. Example: computed flow profile for annular system, cross-sectional view.


Hot


Liquid Melt






Solid Crystal


Cold

Figure 1-13. Diagram of crystal growth system.









CHAPTER 2
LITERATURE REVIEW

Rayleigh convection in a single fluid layer has been a well-studied problem in fluid

mechanics for many years, and it has branched out into many newer problems over the years.

Systems with multiple fluid phases, in which Marangoni convection exists in addition to

Rayleigh convection, have been studied, as well. Also, researchers have examined differences in

convective behavior seen in systems of differing shapes and sizes. It would be quite difficult to

give a complete history of tremendous volume of research that has been done in this general

field. In this section, a brief summary of some of the past research that may be considered most

relevant to this study will be given.

In Benard's classic experiments (1900), cellular, hexagonal flow patterns in vertically

thin fluid layers (about .5 mm to 1 mm deep), heated from below, and open to air at the top

surface were found (Benard 1900). Inspired by these experiments, about fifteen years later,

Rayleigh investigated the instability that arises in a fluid layer heated from below (Rayleigh

1916). Originally, these researchers attributed this flow instability to buoyancy forces the

phenomenon which would be termed Rayleigh convection. However, it turned out that the cells

originally observed by Benard were caused more by surface-tension gradients than buoyancy

differences. Also, it was found that the velocity and heat transfer boundary conditions on the

fluid layer, as well as the size and shape of the system, had quite a significant impact on the

critical conditions for the onset of convection. The early works of Benard and Rayleigh inspired

many more researchers to investigate the convective behaviors of fluid layers.

2.1 Single Fluid Layers

Single fluid layers can be studied in the presence of the Marangoni effect or in its

absence. The former is more relevant to the computations and experiments in this thesis. Some









of the earlier works in this area were that of Schmidt and Milverton in 1935 (Schmidt &

Milverton 1935) and that of Silveston (Silveston 1958) in 1958. Both of these works involved

experiments in which a single layer of liquid was heated from below and the critical vertical

temperature difference for the onset of convection was determined. Schmidt and Milverton

determined the critical vertical temperature difference corresponding to the onset of convection

by measuring the heat transfer through the liquid layer, while increasing the vertical temperature

difference, and then noting vertical temperature difference at which the heat transfer increased to

a value higher than that of the initial state of pure conduction; this increase indicated the added

heat transfer which accompanied the onset of convection. Silveston determined the critical

vertical temperature difference in experiments not only by heat transfer measurements, but also

by optical observations. In 1961, in his book, Hydrodynamic andHydromagnetic Stability,

Chandrasekhar made a thorough mathematical analysis of the instability that arises solely due to

buoyancy in a fluid layer heated from below (Chandrasekhar 1961).

In 1967, Davis published a well-known computational study of the Rayleigh convection

that can occur in a bounded, three-dimensional, rectangular box (Davis 1967). He used a

Galerkin numerical scheme to model the behavior of the system, and determine the effects of

lateral walls on the convective behavior. He also observed the widely known effect of greatly

increasing the horizontal dimensions of the container, which is to cause the critical Rayleigh

number to rapidly decrease to 1708. Lastly, as Chandrasekhar had done in his previously

mentioned 1961 book, Davis explained that moderately-sized convective cells are preferred at

onset to tall, narrow cells or wide, flat cells; this is because tall, narrow cells dissipate too large

an amount of energy, and wide, flat cells require fluid particles to travel too great a horizontal

distance before they can fall and release their potential energy. This sort of reasoning was









introduced in Chapter 1. In a 1970 paper (Charlson & Sani 1970) and an extension of it

published in 1971 (Charlson & Sani 1971), Charlson and Sani did a mathematical study to

determine the critical temperature difference for the onset of axisymmetric (1970) and non-

axisymmetric (1971) flow patterns in cylindrical fluid layers heated from below. Their study

included quite a wide range of aspect ratios (ratio of container radius to container height) and

addressed the cases of both insulating and conducting side walls. In 1972, Stork and Muller

published an experimental study in which they studied a system quite similar to the one

mathematically analyzed by Davis in 1967 (Stork & Muller 1972). They varied the lateral

dimensions of their rectangular system and found their critical temperature difference results to

be generally below those of Davis. They attributed this discrepancy between experiment and

theory to the effect of imperfect experimental control of the heat transfer boundary conditions at

the lateral walls. In Koschmieder's 1974 experimental study, the heat flux through an oil layer

before and after the onset of convection was monitored; he was, however, unable to get an

accurate measure of the heat transfer through the oil layer for cases with low critical, vertical

temperature differences (Koschmieder 1974).

Like this research, the 1974 experimental study of Stork and Muller addressed Rayleigh

convection in annular systems. As in their 1972 work, they examined a fluid layer of 10 mm

vertical depth, comprised of silicone oil mixed with aluminum tracer powder, with the

temperature at the top and bottom of the layer carefully controlled. In some experiments, they

considered simply a system of cylindrical cross-section, while in others, they added a center-

piece to the cylinder to create an annular system. In all of their experiments, they detected the

onset of convection by visual observation of the fluid layer and then obtained the critical vertical

temperature difference by reading appropriately placed thermocouples. For the experiments on









cylinders, onset conditions were determined for different aspect ratios; for the experiments on

annuli, the onset conditions were determined for several different annular gap widths. As

mentioned earlier, their experiments on annuli resulted in flow patterns that looked rather like

spokes of a wheel, with several rolls of convecting fluid (always an even number of them)

aligned in the azimuthal direction. As the annular gap width was increased (meaning, for

example, the center-piece of the annulus was made to be of smaller diameter), some quite

interesting flow behavior was observed; for some large enough gap widths, there were more flow

cells along the outer wall of the annulus than the inner wall. One important difference between

the research in this thesis and the work of Stork and Muller is that this research focuses on the

transition in the types of onset flow patterns that are observed when the annular gap width

becomes sufficiently large (see Section 1.4), whereas this transition seemed to be more of a

secondary observation in the work of Stork and Muller. In this research, annular systems of

certain dimensions were designed specifically for investigating this transition in onset flow

patterns. Sets of annular dimensions appropriate for this investigation were able to be chosen

based on computations for the critical conditions, run for annular systems. Another important

difference between this research and that of Stork and Muller is that this research includes a

computation to accompany the experiments, and their work did not.

In 1990, Hardin, Sani, Henry, and Roux published a computational study in which they

determined the conditions at the onset of Rayleigh convection for cylindrical systems of several

aspect ratios (Hardin, Sani, Henry, & Roux 1990).

Also in 1990, Littlefield and Desai published a theoretical study in which they found the

critical conditions for Rayleigh convection for an annular system with conducting side walls and

with top and bottom walls that were assumed to be flat and free (Littlefield & Desai 1990). This









research differs from the work of Littlefield and Desai in multiple ways. In this research, the

computations address the more realistic case, with regard to a single-fluid-layer system, in which

there is a no-slip condition on velocity at the walls; additionally, the radial walls in this research

were considered as insulating rather than conducting, and the variation of viscosity with

temperature was accounted for. Another important difference of this research from the work of

Littlefield and Desai is that it includes experiments which may be compared with the

computations. The calculations of Littlefield and Desai matched qualitatively well with the

experimental study of annular systems done by Stork and Muller. Like the experimental study of

Stork and Miller, their work showed that as the ratio of the inner radius to the outer radius

becomes larger (meaning that the annular gap becomes narrower), more and more convective

cells form in the azimuthal direction at onset (larger azimuthal wave number at onset). They

reached the conclusion that as the ratio of the inner radius to the outer radius (the "radius ratio"

which was earlier named S for this research) becomes larger, the critical Rayleigh number (and

critical vertical temperature difference) approaches the value corresponding to a vertical channel;

they also determined that as the radius ratio becomes smaller, the critical Rayleigh number

approaches the value corresponding to a vertical cylinder. Littlefield and Desai noted that the

convective patterns which develop in the annular system have a higher tendency to include new

cells in the azimuthal direction than in the radial direction. This can be explained, they say, by

the physical argument that new cells which form in the azimuthal direction are more uniform in

size (compared to one another and to cells already present) than new cells which form in the

radial direction, and thus they form more easily because of the lower velocities and viscous

stresses that they involve.









At this point, to be clear, a list of the ways in which this research differs from the

previous works on Rayleigh convection in single-fluid-layer annular systems is given.

> This research includes corresponding computational and experimental work.
> In this research, dimensions of annular systems investigated computationally and
experimentally were chosen particularly to focus on the transition in the types of
onset flow patterns which occurs when the ratio of the inner annular radius to the
outer annular radius is sufficiently small, and to determine what value for this
ratio corresponds to the transition (when this ratio is small enough, the onset flow
patterns can have one or more convective rolls in the radial direction, while they
generally have no convective rolls in the radial direction when this ratio is larger).
> Computations for this research include the variation of viscosity with temperature.
> Computations for this research address the realistic case in which there is a no-
slip condition on velocity at the walls.
> Computations for this research are for the case in which the radial walls are
insulating.

Another study of Rayleigh convection in an annular system was presented in 1990 by

Ciliberto, Bagnoli, and Caponeri (Ciliberto, Bagnoli, & Caponeri 1990). Their study was

experimental and it involved observing the convective behavior of an annular layer of silicone oil

by shadowgraph. The flow patterns observed in their study consist of an azimuthally aligned

series of cells, and are consistent with the patterns seen by other researchers.

In 1995, Zhao, Moates, and Narayanan published an interesting study of Rayleigh

convection in cylindrical systems (Zhao, Moates, & Narayanan 1995). The study included both

theoretical and experimental parts, which agreed well with one another. The experimental

apparatus used was similar to the one being used in this research. The bottom temperature of a

layer of silicone oil was controlled by circulating hot water below an aluminum plate, while the

top temperature was controlled by circulating cooler water above a sapphire window. Flow was

visualized by means of aluminum powder which was mixed in with the silicone oil. It is

noteworthy that this study addressed the changes in convective behavior which arise when one

accounts for the variation of viscosity with temperature rather than simply assuming it to be









constant. It was seen that, depending on the aspect ratio of the system, the variation of viscosity

with temperature could have a significant impact on not only the critical vertical temperature

difference for the onset of convection, but also on the flow pattern at onset. This can be seen

more later in this paper. The fact that the computations in this research account for the variation

of viscosity with temperature when determining the critical conditions for convection is another

noteworthy feature of this research, which may set it apart from previous works on annular

systems.

2.2 Multiple Fluid Layers

As mentioned, the studies on single fluid layers, for the fact that they exclude the

Marangoni effect, are rather more relevant to this research. Still, some works on multiple fluid

layers can be helpful in understanding what goes on in problems involving buoyancy-driven

convection in single fluid layers, too. Buoyancy-driven convection, after all, is generally

occurring in problems with multiple fluid layers, as well, even though the systems may be more

complicated. Thus, a brief summary of it will now be given, and so will be presented a more

complete picture of what sorts of research have been done on convective phenomena over the

years.

Following experimental evidence obtained by Block in 1956 (Block 1956), Pearson, in

1958, produced a paper which proposed a mechanism to attribute convective flow of the type

observed by Benard to surface tension gradients rather than buoyancy forces (Pearson 1958).

Pearson's study focused on a fluid layer, heated from below, laterally unbounded, with a rigid

wall as its bottom surface, and a free surface at the top. A passive gas was the upper phase in the

study. In that study, the temperature of the bottom wall was held fixed, while the temperature at

the upper surface was governed by a heat transfer boundary condition. At the top and bottom









surfaces of the fluid layer, Pearson considered the effects of having both conducting and

insulating behaviors with regard to heat transfer. In Pearson's paper, a dimensionless number

called "B" showed up as a preliminary form of what would, in later years, be called the

Marangoni number. The Marangoni number is surface tension-driven convection's analog to the

Rayleigh number of buoyancy-driven convection. Pearson's analysis assumed that the upper

free surface could not deflect. In 1959, a paper by Sternling and Scriven presented physical

mechanisms and a corresponding simple mathematical model to explain how several types of

interfacial flows may develop in bilayer systems (Sternling & Scriven 1959). A 1964 paper by

the same two authors extended Pearson's work by considering the same system but allowing

deflection of the free surface (Scriven & Sternling 1964). Like Pearson's work, the work of

Sternling and Scriven involved solving for the behavior of the system in response to a small,

imposed disturbance. None of the works, that were just described, by Pearson, or Sternling and

Scriven included an experimental component. The 1967 work of Koschmieder, however, was an

experimental study of the convective behavior of a cylindrical layer of silicone oil, heated from

below by a solid boundary, and cooled from above by a layer of air (Koschmieder 1967). In that

study, the flow was visualized using aluminum powder, and it was found that convective flow in

the system initially took the form of concentric circular rolls, and subsequently transformed into

a hexagonal pattern. Koschmieder was able to make an accurate determination of the

wavelengths of the flows, as well as their dependence on the dimensions of the fluid layer.

Many studies involving multiple fluid layers were performed on systems comprised of

one vapor layer atop one fluid layer. It is noteworthy that in 1972, Zeren and Reynolds

published a study in which they examined Rayleigh and Marangoni convection in a liquid-liquid

system (Zeren & Reynolds 1972). The liquids considered were benzene and water. Their study









included both theoretical and experimental parts. They felt that the presence of contaminants in

the interfacial region could have affected their experimental results significantly. The work of

Ferm and Wollkind, published in 1982, improved and extended the theoretical analysis given by

Zeren and Reynolds (Ferm & Wollkind 1982).

While it seems that most of the work done with annular systems has involved only single

fluid layers, Bensimon, in 1988, published a study in which he experimentally examined the

convective behavior in an annular layer of liquid with free boundary conditions at the top and

bottom surfaces (Bensimon 1988). He arranged this system by placing a layer of silicone oil on

top of a layer of mercury and leaving an air layer above the silicone oil. Flow visualization was

accomplished in his study by the shadowgraph technique. In the shadowgraph images he

presents in the paper, the flow patterns look quite similar to those that are seen in single-fluid-

layer, annular systems.

Returning to the works of Koschmieder, an entirely experimental work was published by

Koschmieder and Prahl in 1990 (Koschmieder & Prahl 1990). It investigated the tendency of

wide fluid layers, when heated from below and open to air at the top surface, to convect in a

pattern of hexagonal cells. They examined silicone oil-air bilayers in small containers of

differing shapes to find out what effects the differing container shapes had on the convective

patterns that would form. They observed that as the width-to-height ratio for the containers

increased, more and more convective cells would form at the onset in order to fill the larger

width. While determining the critical conditions by observation of the vertical heat transfer

through the system is a nice method in terms of objectivity, Koschmieder and Prahl point out that

in their experiments with small fluid layers, the voltage created by the heat sensor was too small

in comparison with outside electrical noise to be useful. Thus, the more desirable method (which









is the method used in the current research) was to optically observe the fluid layers and note the

first appearance of fluid motion as the critical point. A 1992 experimental work by Koschmieder

and Switzer examined surface-tension driven convection using shadowgraphy (Koschmieder &

Switzer 1992).

In a 1995 paper by Zhao, Wagner, Narayanan, and Friedrich, a theoretical study is

presented that addresses Rayleigh and Marangoni convection in fluid bilayer systems both

liquid-liquid, and liquid-vapor (Zhao, Wagner, Narayanan, & Friedrich 1995). A wide range of

cases are considered, including heating from below, heating from above, and the case in which

solidification is occurring at the bottom surface of a liquid-liquid bilayer; the last case mentioned

has strong similarity to crystal growth because of the solidification which it accounts for. The

1996 theoretical study of Dauby and Lebon addressed the Rayleigh and Marangoni effects in a

liquid-vapor bilayer system and included both linear and nonlinear analyses (Dauby & Lebon

1996). The linear analysis allows one to determine the critical vertical temperature difference at

which convection begins, as well as the flow pattern at the onset. The nonlinear analysis

provides more specific information on the flow behavior and can be used to predict supercritical

behaviors. Most theoretical studies, up to this point, had assumed laterally unbounded systems.

That assumption allows separation of variables, and thus an easier solution to the system of

differential equations that model the problem. Of course, in actual experiments and other

applications of these phenomena, lateral walls are present, and these walls can have important

effects on the critical conditions and onset flow patterns. Thus, another point of interest in the

study by Dauby and Lebon is that they formed their mathematical model to include no-slip walls

at the lateral boundaries. This is the most realistic boundary condition that could be enforced at

those locations. Dauby and Lebon found their theoretical results to be in good qualitative









agreement with the experimental results of Koschmieder and Prahl. In 1997 by Kats-Demyanets,

Oron, and Nepomnyashchy published a study, which examines convective behavior in tri-layer

fluid systems, and thus is also quite applicable to the science of crystal growth (Kats-Demyanets,

Oron, & Nepomnyashchy 1997). Johnson and Narayanan published a tutorial in 1999 which

explained five different mechanisms by which convection in two vertically stacked fluid layers

could couple (Johnson & Narayanan 1999). Their tutorial also discussed the situation in which,

at certain aspect ratios, due to the effects of side walls and the dimensions of the system, a

convecting system may find two flow patterns equally favorable from an energy standpoint; as a

result, the system would oscillate between the two patterns. A point such as this is called a

codimension-two point.









CHAPTER 3
MODELING EQUATIONS

To predict the onset conditions of convection requires a model that respects the physics

of the problem. Such a model would utilize momentum, mass, and energy equations. The inputs

to the model would be the vertical phase depth and thermophysical properties of the fluid, and

the outputs would be the critical temperature difference needed for convection, as well as the

associated pattern of the flow. The details of such a model are given in this chapter, while later

in this thesis, an explanation of the numerical scheme used to solve the system of modeling

equations is given. In Appendix C, the modeling equations are presented and developed more

fully for the simple case in which the fluid is assumed to have a viscosity constant with respect to

temperature. The modeling equations presented and developed throughout this chapter and

Chapter 4, however, are for a more physically accurate mathematical model, in which the

variation of viscosity with temperature is included.

To get started the nonlinear equations that govern flow in a convecting layer of fluid shall

be introduced. These equations are nonlinear primarily on account of the dependence of velocity

on temperature, coupled with the effects of the interactions between velocity and temperature

fields on heat transport.

3.1 Nonlinear Equations

The domain equations used to model the convective behavior of the system are the

momentum equation, the energy equation, and the continuity equation, which, respectively, are


p- +pW =-VP+pg + V.-S, (3.1)
at


pC -+pCi- VT= kV2T, (3.2)
at









tp V
at


(3.3)


In these equations, p is the density of the fluid, t is time, i is the velocity of the fluid, P

is the pressure of the fluid, g is gravity, S is the stress tensor for the fluid, Cv is the constant-

volume heat capacity of the fluid, Tis the temperature of the fluid, and k is the thermal

conductivity of the fluid.

The last term of the momentum equation has not been further simplified, at this point,

because it will be needed in this form in order to properly account for the dependence of the

fluid's viscosity on temperature. The stress tensor, S, can be expanded as


s + (Vw
^ Vv~f~v


(3.4)


The use of this expansion, however, will be postponed for now. In this expansion, p is

the dynamic viscosity of the fluid and r represents a transposed matrix. Since the velocity

gradients in this system will not be very large, the energy equation has been simplified by

neglecting the viscous dissipation. The system of equations is analyzed in cylindrical

coordinates, which are clearly the natural choice for this problem; thus "r" denotes the radial

direction, "0 denotes the azimuthal direction, and "z" denotes the vertical direction. The

system may be treated as periodic in the 0-direction (this is accomplished mathematically by the

imo
expansion of the variables into modes), with the oscillations of the form e in which m is the

azimuthal wave number.

Another approximation is to be made, and it affects the momentum equation. It is called

the Boussinesq approximation and it addresses the variation of density with temperature. This

approximation says that variation of density with temperature is negligible insofar as the change









in momentum or mass is concerned, but that it does affect the acceleration in the system insofar

as the body forces are concerned. The reasoning for introducing the Boussinesq approximation,

and some notes on its applicability are explained in Appendix B as Section B. 1.

Once the Boussinesq approximation is applied to Equation 3.1, it becomes


R + R9 VV = -VP+ pR(1 a(T TR)) + V. S. (3.5)

In Equation 3.5, TR and PR are a reference temperature and the fluid's density at the

reference temperature, respectively, and a is the volumetric thermal expansion coefficient of the

fluid phase. Note, also, that the Boussinesq approximation results in the disappearance of the

time-derivative term of the continuity equation. At this point, multiple nonlinearities appear in

the system of equations; a note about these nonlinearities is given in the appendix as Section B.2.

An additional modification to the momentum equation is to define a modified pressure, which is

- Vp = -VP + pRg. (3.6)

Using this, the momentum equation may be written as

PR -+PR V Vp- pR(T TR )g + V.S. (3.7)
at

Now, the expansion of S shall be introduced to the momentum equation. The result of

doing so is


PR + PRV Vv = VP PRa(T R) + /V2V +
0t (3.8)



As mentioned, a goal of this calculation is to include the effects of the variation of

viscosity with temperature. Thus, the viscosity at any location can be written as









S=/[f (T TR )], (3.9)

wherefis some function of the difference between the temperature at that location and the

reference temperature. In this equation, /R is the dynamic viscosity at the reference

temperature. Note that, since p has the same dimensions as /R mustt be dimensionless.

Using this expansion, Equation 3.8 may be rewritten as


PR +p PRV*V =-Vp-pRa(T-TR)g+Puf V32 +
at (3.10)

PR(Vf)-(VV + (VVY) .


The assumed form for viscosity's temperature dependence will be exponential, like the

form assumed in the work of Zhao, Moates, and Narayanan (Zhao, Moates, & Narayanan 1995).

The form of this exponential temperature dependence is

f(T TR) eB(T-TR), (3.11)

in which B is a constant that can be determined using measurements of the fluid's viscosity

obtained at a range of different temperatures. For temperatures in degrees Celsius, the

dimensions of B would be -
oC

In this system of three domain equations, it appears that five unknowns are present.

These are the velocities in the r, 0, and z directions, the pressure, and the temperature. The

continuity equation can eliminate pressure by representing it in terms of velocities, and this

reduces the number of unknowns to four. To solve for the convective behavior of a cylindrical

system, constraints would be needed on the velocities in the r, 0, and z directions, as well as on

the temperature, at the top wall (z = Lz), the bottom wall (z = 0), and the outer radial wall (r =

Ro). Considering an annular system, however, four additional mathematical constraints are









required; these constraints on the three components of velocity and temperature are applied at the

inner radial wall of the system (r = R,). Figure 3-1 shows the geometry being considered. The

symbol Lr is used to denote the radius in a cylindrical system, and also to denote the difference

between Ro and R, (Lr = Ro R,) for an annular system. This means that for an annular system, Lr

is the width of the annular gap that is filled with fluid.

Thus, for the annular system, a total of sixteen constraints are needed, while only twelve

are needed for the cylindrical system. These constraints are the boundary conditions. In their

unsealed form, they are

Vr =Vo =Vz =0 atz =0, (3.12)

T = Tb at z= 0, (3.13)


Vr = Vo = Vz = 0 atz=z (3.14)

T=T, atz =L, (3.15)


Vr V = Vz = 0 atr =R, (3.16)

OT
-= 0 atr =Ro, (3.17)
Ar

and, when considering an annular system, the conditions at the inner annular wall,

Vr =v =vz =0 atr =R,, (3.18)

OT
-=0 atr =R, (3.19)
9r

are needed.

In these equations, vr, V0, and vz are the components of velocity in the r, 0, and z

directions, respectively. The boundary conditions on velocity enforce no-slip behavior at the









walls, and state that no flow may pass through the walls. The conditions on temperature at

bottom and top walls keep the temperature fixed at those locations; in these equations, Tb and T,

are the constant temperatures at the bottom wall and top wall, respectively. The conditions on

temperature at the radial walls represent the insulating behavior at those boundaries.

3.2 Scaling

Now that the complete set of domain equations and boundary conditions has been listed,

these equations will be made dimensionless. In scaling these equations, one has the option to

choose several combinations of parameters to use in defining a characteristic velocity (V) and a

characteristic time (T). A brief discussion of how to choose these parameters is in Section B.3

of Appendix B.

Lengths are scaled using the vertical depth of the fluid. With that said, a list of the

scaling relations used will now be presented. In these equations, the "hat" symbol indicates a

dimensionless variable, and the "bar" indicates a characteristic value for a variable. The scaling

relations are

v=-, (3.20)
v

T=T- T, (3.21)
AT

AT =Tb Tt, (3.22)


P, (3.23)



p =R, (3.24)
L


t (3.25)
t









V = LV. (3.26)

In the above equations, Lz is the vertical depth of the fluid phase. As mentioned earlier,

TR is the reference temperature. In simple cases where the system of equations was solved

without considering the dependence of viscosity on temperature, TR could be chosen to be the

temperature at the top boundary of the system (Tt), and the reference dynamic viscosity, / R,

would simply be the dynamic viscosity at that temperature. When the variation of viscosity with

temperature is considered, however, TR is selected later, during the numerical solution of the

system of equations, to correspond to the mean dynamic viscosity value of the fluid in its

motionless base state, when the fluid layer is at the critical vertical temperature difference, just

before the onset of convection (the viscosity of the fluid varies from top to bottom along the

motionless liquid layer corresponding to the vertical temperature gradient). When the variation

of viscosity with temperature is being considered, this mean dynamic viscosity value is the

reference dynamic viscosity, /IR AT is the overall temperature difference across the system.

In addition to these scalings, f(Equation 3.11) should be re-expressed in terms of

dimensionless temperature. This new version off which will be called F, is

F(T) = eB(AT)T (3.27)

Applying these scalings to the domain equations, the following dimensionless (though the

"hat" symbol will now be discarded) equations are obtained (Equations 3.28-3.30):

pRL2 PRLv aL2g(AT)T 2
PR LZ-+ RLz v -Vv = -Vp z -(AT) T + FV2v +
R//R t /R U V (3.28)
(VF). (Vf + (Vf)T)


LOT+ Ly f VT =V2T,
K/CTat /C


(3.29)









0=V-v, (3.30)

in which K-, the thermal diffusivity of the fluid, is equal to the fluid's thermal conductivity

divided by the product of its density and heat capacity.

Unless otherwise indicated, all thermophysical properties in these equations are

considered at the reference temperature. The subscript "R" has not been included on the symbols

for all of these properties; however, it has been left on the symbols PR and /R because, in those

cases, it arose from the expansions by which the density and dynamic viscosity of the fluid were

defined. This is the reason, for example, that the subscript "R" is not included on the kinematic

viscosity in Equation 3.28, even though that property is simply a ratio of the dynamic viscosity

and the density taken at the reference temperature. Observe that if a substitution of the ratio of

K to Lz were made for V, then the coefficient of the temperature term in Equation 3.28 would be

the Rayleigh number. Since many possible values could be assigned to 7, depending on the

relative values of system parameters, iV has been left in the set of modeling equations. Again, a

discussion regarding the definition of T is in Section B.3 of Appendix B.

As for the scaling of the boundary conditions, the conditions on velocity remain

unchanged in appearance, as do the conditions on temperature at the radial walls. However, the

conditions on temperature at the top and bottom walls appear slightly differently depending on

whether or not the problem is being solved considering the variation of viscosity with

temperature. The reader is asked to refer to Appendix C for more information on the case in

which this variation is not being considered. In cases in which the variation of viscosity with

temperature is being considered, the dimensionless temperatures at the vertical boundaries are

dependent on TR. TR, though, is dependent on the vertical temperature gradient in the fluid,

which means that it needs to be determined along with the critical vertical temperature difference









for convection by an iterative approach (described in Chapter 5). Thus, for the non-constant

viscosity case, the temperature boundary conditions at the top and bottom walls must be left in a

more general form for now. In this general form, they are


T =b R at z = 0, (3.31)
AT

T -T
S=at z = (3.32)
AT

This completes the presentation of the scaled domain equations and boundary conditions.

Next, the equations shall be simplified by linearization and the removal of time dependence.



Cylindrical System Annular System

r = Ro r = 0 Fluidr = Ri r = 0 r = Ro











Figure 3-1. System diagram: cylindrical and annular systems.




Figure 3-1. System diagram: cylindrical and annular systems.









CHAPTER 4
LINEARIZED EQUATIONS

In this chapter, the process of simplifying the set of modeling equations by linearizing

them around a motionless base state, and then eliminating their time dependence shall be shown.

Since the goals of the calculations in this research are to determine only the onset conditions for

convection (critical vertical temperature difference and flow pattern at onset) as opposed to the

flow behavior at supercritical conditions, a linearized model is sufficient for the analysis. A

nonlinear calculation would be necessary if the supercritical behavior of the system were to be

determined. Time dependence is being removed from the equations because a system is

independent of time at critical conditions (also known as marginal stability). Linearization and

the removal of time dependence are accomplished by introducing a series perturbation expansion

to the variables, then applying a further expansion to the perturbed variables (expansion into

modes), and then setting the inverse time constant in the expansion equal to zero. As mentioned

earlier, the same process is presented for the less complex case in which the fluid is assumed to

have a constant viscosity with respect to temperature is included in Appendix C.

4.1 Linearization

The system of equations shall be linearized around the motionless base state of the

system. In this motionless initial state, heat is transferred within the system only in the vertical

direction and only by conduction.

The form of the expansions which shall be used in this linearization, considering, for

example, the expansion of velocity, is









0 + |+ 2 +
ac =s=0 2! a 2 =0


1 3 3V (4.1)

3 a.3 6 = 0


Replacing the expressions in brackets, this may be rewritten as

v = v0 + +-2v2 +- 3v3 +.... (4.2)
2! 3!

The subscript "0" denotes values pertaining to the motionless base state, in which heat is

transferred only by conduction, and the temperature and pressure gradients are only in the

vertical direction. Clearly, then, f0 is equal to zero. This fact allows the cancellation of some

terms in the linearized forms of the modeling equations. If the system begins to flow due to

some perturbation, then the remaining terms in this series (v3, v2, etc.), which are the perturbed

variables, represent the flow behavior of the system. The amplitude of the perturbation that

transforms the base state into the flowing, "perturbed" state is called E; this perturbation is taken

to be extremely small. Considering a small perturbation allows a conclusion on the stability of

the system to be reached using linearized equations. If a system is unstable to a small

perturbation, it will, of course, be unstable to larger perturbations. If a system is stable to a small

perturbation, then no such conclusion may be drawn; in this case, the system could simply

require a larger disturbance to become unstable. The linearized equations (comprised of terms

that are first order in E) are an approximation of the behavior of the system and can be used to

determine the conditions corresponding to the onset of convection and the pattern of flow at the

onset. If it is desired to determine the magnitudes of the velocities in the convecting fluid, as









opposed to simply determining their values relative to one another, then a nonlinear analysis is

necessary.

A disturbance of magnitude E is mathematically applied to the system by expanding

each variable in the modeling equations with the form shown in Equation 4.2. The set of

linearized equations is then obtained by collecting only the terms which are first order in E. The

resulting linearized domain equations are

PRL2 -Vp aLg(AT) FV



L 2 Ot L L aT V2T
(4.3)



L\0 __ V 2T (4 .4)
ht7 t K Oz

0 = V.-. (4.5)

In these equations, Fo is the same as F except that the exponential temperature value is

now the base state temperature, To. Fo(To), then, is

Fo (T) = e B (AT) (4.6)

The linearized forms of the boundary conditions are

S= 0 = Vlz = 0 atz = 0, (4.7)

T, =0 atz=0, (4.8)

v1, = V =lz = 0 atz =L, (4.9)

T, = 0 atz =Lz, (4.10)

V1, = o 0 VIz = 0 at r = Ro, (4.11)









S= 0 at r = Ro, (4.12)
ar

and, when considering an annular system,

V1r =V0 = Vz = 0 atr =R, (4.13)

aT
0 atr =R, (4.14)
9r

hold.

Linearization simplifies the boundary conditions on temperature at the top and bottom

walls because it is the base state values of temperature, To at the top wall and bottom wall, which

are equal to the fixed values of temperature at those boundaries.

Substituting for Fo in the momentum equation, and making use of

VFo = VT = g (4.15)
aTo aTo az

in which 3 is the unit vector in the z-direction, brings the momentum equation into the form


PRL Z _v a L + zg(AT) B(AT) +
PtRt 8t U V
T (4.16)
(B(AT)eB(AT)T O z ), *(Vf1 + (Vf)
8z

Note that all factors on the right-hand-side of Equation 4.15 are dimensionless since

VFo, from which they arose, was already made dimensionless in Chapter 3. For a more detailed

and well given explanation of this perturbation method, refer to the book titled Interfacial

Instability by Johns and Narayanan (Johns & Narayanan 2002). The momentum equation

(Equation 4.16) is now rewritten as the three scalar equations which are its components in the r,

0, and z directions. The V operators are expanded in cylindrical coordinates from this point








onward. The scalar component equations comprising the vector form of the momentum equation

are the r-component of the momentum equation, the 06-component of the momentum equation,

and the z-component of the momentum equation, which, respectively, are



vt t Or


1 02 a2
r2 92 z2


(4.17)


(B(AT)eB(AT)To


LU Vt


eB(AT)To


-) -+ ,
)[r +


r --
r D0


1 a2 a2
r2 92 z2


T& &
az az


17-
I
2 Yio


1 &V
r 00


(4.18)


eB(AT)To









I I, p ,
--- -+
vU t at 0z

B(AT)TO 02 1 a 1 02 02
e -(-+---+ +_ Vl +
Or 2 r Or r2 O 2 16 1
S ] 1 (4.19)


(B(AT)e )(A) T) 2 + a g T, .


4.2 Expansion into Normal Modes

The set of domain equations now includes Equations 4.4, 4.5, 4.17, 4.18, and 4.19. A

second expansion (expansion into modes) will now be applied to the variables, as well. This new

expansion separates the r-direction and z-direction dependencies of each variable from the 0 -

direction and time dependencies. In the example expansion shown as Equation 4.20, the new

variable representing only the r-direction and z-direction dependencies is marked with a "prime"

symbol. This expansion assumes a periodic form for the 06 -direction dependencies, which is

sensible, since the cylindrical and annular systems being considered are indeed periodic in the

06 -direction. The nature of the periodicity in the 06 -direction is described by an azimuthal wave

number, called m. Note that this wave number is dimensionless since it arises from

dimensionless variables. The time dependence of each variable is assumed to have an

exponential behavior. This exponential behavior with respect to time is governed by an inverse

time constant, called c c is dimensionless since it arises from dimensionless equations; if

unsealed, its units would be (1/time). Using pressure as an example, the form of the expansion is

= im + (420)
/01 P 1 Z ) e (4.20)









The form of 06 -direction dependency assumed in the expansion can be used because the

periodic spatial dependence of the system in the 0-direction can be represented by a series of

sines and cosines. When the system of modeling equations is subjected to a perturbation (E), the

perturbation may either grow, resulting in the onset of flow, or it may die out if the system is not

at critical or supercritical conditions. The value of a is an indicator of the system's response to

a given perturbation. It should be noted that in certain cases, for which the system oscillates

between convective flow patterns, the value of a is imaginary. It can be mathematically shown,

however, that a is strictly real for the non-oscillatory cases considered in this research. If a

disturbance applied to the system dies out, meaning that the system is stable, then a has a

negative value. If the system is unstable, and will flow when subjected to a perturbation, a is

positive. At marginal stability, however, when the system initially becomes unstable, the system

is independent of time and the value of a is 0. The fact that a = 0 at marginal stability allows

the elimination of all time-derivative terms in the modeling equations once this expansion is

applied. The final forms of the modeling equations (including all three components of the

momentum equation, the energy equation, the continuity equation, and the boundary conditions),

with this expansion applied, and with the "prime" symbols dropped from the newly defined

variables, are












(m2 + 1)


(B(AT)eB(AT)To


&0 &


im
0 =-- p+
r


eB(A)T)o


a2
or2


2im
r2 I


1 a
+-r
9


(B(AT)e(AT)To




Dz


B(AT)TO 2
e
-1 r 2


r 1r
r&r


(B(AT)e B(AT)
8z


a2
(2


(m2 +1)
2+ v +
r2 le


im
r


m2 2
--2+-- v- +
2 2 1,


82 a-g(AT)
Dz V


0=- +
Or


a2
or2


a2
+
&2


r r


2im
_ r2 1 1


(4.21)


(4.22)


(4.23)


B(AT)To


Sr









O=L 9 To [ 92 1 9 92 m2
-TTO V 0-+-- 2 2 T, (4.24)
cK 9z 9r r 9r z2 r


8 ^ 1 im 8
0= + V1, + 1+ 1V, (4.25)


Vlr, = Vl = Viz = 0 atz = 0, (4.26)

T, = 0 at z = 0, (4.27)

Vi. = V10 = Vz = 0 atz =Lz, (4.28)


T, = 0 at z = L, (4.29)

vl,. = V = ~ = 0 atr =Ro, (4.30)


- = 0 at r = Ro, (4.31)
Or

and, when considering an annular system,

v. = V0 = vz = 0 atr = R, (4.32)


= 0 at r = R, (4.33)
Or

are needed.

At this point, two important dimensionless groups which show up in the motion equation

are Fo (see Equation 4.6) and


Q = B(AT)eBATTO (4.34)


Now, the entire collection of modeling equations (Equations 4.21 4.33) may be

simultaneously, numerically solved on a computer as an eigenvalue problem, in which the









eigenvalue is A T. Since c has been set equal to 0, the problem is being solved at the critical

point for the onset of convection, and so the A Tvalue to be determined is the critical vertical

temperature difference for convection, or [ A T]crit. How exactly the solution for the [ A T]crit is

carried out is the topic of the next chapter.

Notice that the Prandtl number (v/Ki) does not appear in any of these equations. This

may be surprising since one might expect that ratio of thermophysical properties to play a role in

determining the critical conditions for convection. Now, recall that in Chapter 1 it was shown

that the graph of critical Rayleigh number versus aspect ratio, for different azimuthal wave

numbers, was identical for any system, regardless of the system's thermophysical properties.

This interesting fact can be clearly explained at this point. The kinematic viscosity and thermal

diffusivity of the system affect how quickly disturbances may die out or grow. Thus, before the

onset of convection, their values make c more or less negative depending on how stable the

system is, and after the onset of convection, their values make U more or less positive

depending on how rapidly a destabilizing disturbance grew and how strongly the system is

convecting in response to it. Precisely at the onset of convection, however, which is the

condition at which the modeling equations are being solved, the system is independent of time

(o = 0), and so the thermophysical properties play no role in determining the critical conditions

for convection. If, rather than simply changing the thermophysical properties of the fluid, the

insulating boundary conditions on temperature at the radial walls were changed to conducting

boundary conditions, the graph of the critical Rayleigh number versus aspect ratio would change

(as explained in Chapter 1, the critical vertical temperature differences for convection are lower

when insulating radial walls are considered rather than conducting radial walls).









CHAPTER 5
SPECTRAL SOLUTION METHOD

The goal of this chapter is to explain the spectral solution method used for computations

in this research. The method is introduced and described in Section 5.1. Section 5.2 focuses on

the application of the method.

Rearranging the set of modeling equations to be solved numerically as an eigenvalue

problem for A Tis not too difficult. To do so, one must isolate the term of the z-direction-

component of the momentum Equation 4.23 which includes T1. What is meant by isolating that

term is arranging the equation so that that term is on the side of the equation opposite from all of

the other terms. Supposing the side of the equation where the T, term is placed is chosen to be

the right-hand-side of the equation, then the remaining terms of the z-direction-component of the

momentum equation, and all nonzero terms of all of the other domain equations and boundary

conditions should be kept on the left-hand-sides of those equations. In general, what is being

done here is that the problem is being recast in the form of a generalized eigenvalue problem for

A T. The form of this generalized eigenvalue problem is

AX =ABX. (5.1)

Here, A is the matrix containing the coefficients of the velocity components,

temperature, and pressure from the left-hand-sides of the domain equations and boundary

conditions. B, as one would imagine, is the matrix of the coefficients of the velocity

components, temperature, and pressure from the right-hand-sides of the set of modeling

equations. X is the eigenvector; it is a column-vector including all three velocity components,

temperature, and pressure. From this point onward, the subscript "l"'s used to indicate

perturbed variables will be discarded, as nearly all variables referred to will be perturbed









variables. The only exceptions to this would be the base state variables, which, for this reason,

will still be denoted by a subscript "0" in all locations. The form of X is

Vr
Vo
X= v. (5.2)
T
P

A, of course, is the eigenvalue, which is A T. Arranging the system of modeling

equations in this form is straightforward when the variation of viscosity with temperature is not

being considered. For the non-constant viscosity case, however, A Tis present in more locations

than just the coefficient of T, in the z-direction-component of the momentum equation; it is also

present as an exponent in several terms. These terms, though, may be kept on the left-hand-side

of the modeling equations rather than being placed on the right-hand-side of Equation 5.1. If the

equations are arranged in this way, A T can be determined by an iterative approach. In this

iterative approach, a value must first be chosen for Tt, the temperature at the top of the fluid

layer. In all experiments done for this research, this temperature was kept constant at 30.0 C, so

that value was always substituted for Tt in the calculations. Next, a guess-value, which will be

called [A T]guess, is substituted for A Ton the left-hand-side of Equation 5.1. Then, the

eigenvalue (A T) on the right-hand-side of Equation 5.1 is solved for, and then its value is used to

update the [A T]guess value on the left-hand-side of Equation 5.1, and so on. The procedure is

repeated until convergence, which occurs quickly. Since TR and '/R are dependent on the

vertical temperature gradient prior to the onset of convection, their values are updated as

[ A T]guess is updated. Note that updating the [ A T]guess value on the left-hand-side of Equation 5.1

represents updating the value of viscosity, through its exponential temperature dependence.









Also, note that in the case where the variation of viscosity with temperature is not accounted for,

the viscosity value at 35.0 C, which is given in Appendix A, is used throughout the calculation.

5.1 Explanation of the Method

Once the system of modeling equations has been written in the form of Equation 5.1, it

can be used to numerically approximate the critical vertical temperature difference for

convection, and the onset flow pattern. As is typical of numerical solution methods, the first step

is to consider discretized versions of the modeling equations, which describe the behavior at a

collection of individual points in the system as an approximation of the full behavior of the

system. This means that each variable is written as a vector containing the values of that variable

at a set of points within the system. For example, a cylindrical system could be discretized into

sets of points (or "nodes") in the r-direction and z-direction as shown in Figure 5-1. Of course,

the subscripted numbers here have nothing to do with the subscripted numbers that appeared

during linearization in Chapter 4; they simply indicate the spatial arrangement of the nodes.

In this case, the cylinder was discretized into rows of four nodes in the r-direction, and

columns of four nodes in the z-direction. Clearly, these sets of nodes in the r-direction and z-

direction form a grid of nodes in which the location of any node can be specified with an index

for the r-direction and an index for the z-direction. For this reason, the discretization nodes are

also referred to as "grid points." Either direction could have been discretized into any number of

nodes. Of course, considering a larger number of nodes leads to a more accurate discretized

model. The numbers of nodes created in the r-direction and the z-direction are decided by the

selection of parameters called Nr and N, respectively. For the example above, Nr = 3 and Nz = 3.

Note that discretizing the r-direction and z-direction means that r and z are written as vectors









now rather than scalars. In general, the sets of nodes in the r-direction and z-direction,

respectively, can be written as



ro


F = r2
(5.3)

rv




z1


(5.4)

ZNz


Due to the inclusion of ro and zo, the numbers of nodes in the r-direction and z-direction

are actually (Nr + 1) and (Nz + 1), respectively. It should be mentioned that, in this research, the

treatment of the 0 -direction was much simpler than the discretization applied to the r-direction

and z-direction, due to the periodic form assumed for 06 -direction dependencies; the treatment of

the 0-direction will be discussed later. For now, the reader is free to imagine that the discussion

here pertains to simply systems axisymmetric in the 0 -direction, even though, in actuality, the

treatment of the 06 -direction is such that everything presented here applies to systems of any

azimuthal periodicity.

As indicated by Figure 5-1, when considering a system in two or more spatial

dimensions, the indices of the nodes in the r-direction and z-direction may be thought of as

indices corresponding to different planes slicing through the system, normal to the r-direction









and the z-direction, respectively. Thus, discretizing the r-direction into (Nr + 1) nodes the and z-

direction into (Nz + 1) nodes divides the system into a grid of nodes that includes a total of ((Nr +

1) (Nz + 1)) nodes. Again, note that the computational treatment of the 0 -direction has not yet

been addressed; this means that the discretized form of the system being discussed at the moment

is only one plane in the 0 -direction. Given this form of discretization, every variable in the

system must be written as a vector of its scalar values at every node of the discretized system.

Temperature, for example, would be written in discrete form as shown in Equation 5.5. To allow

a shortened, more clear representation on paper, Equation 5.5 will show T in the transposed

arrangement; note that it is actually a column-vector like F and f The transposed form of the

discretized temperature is

r =[T T IT I ... IT IT I .
ro,zo r8,zo *' r,Zo ro,zz r,z >* *>

T T I T 1 (5.5)
rNr,z Z i ro,zz Trl,ZNz ,ZNrz (5.5)

The raised ellipsis in Equation 5.5 indicates the continuation of this row-vector on the

next line. The discretization nodes, although they may appear to be so in Figure 5-1, are not

necessarily evenly distributed in the system. In fact, it can be quite beneficial to use nodes which

are not evenly spaced. The sets of nodes used in this research are not evenly spaced; they are

called the Gauss-Chebyshev-Lobatto grid points and the Gauss-Radau grid points (Trefethen

2000). Both sets of grid points are more highly clustered near the boundaries of the system than

near the center of the system. It is highly beneficial to use these sets of clustered grid points as

opposed to evenly spaced grid points because the clustered grid points allow much quicker and

more accurate numerical convergence. Using these sets of grid points is also beneficial because,

in the systems considered in this research, much of the important convective flow behavior








occurs near the edges of the system, and using sets of grid points more clustered near the edges

of the system thus allows the convective behavior of the system to be more easily captured by

the discretized modeling equations.

Gauss-Chebyshev-Lobatto grid points are a set of points spanning the range [-1,1]

(including the endpoints at both -1 and 1), and so the actual distances within the system must be

rescaled in order to fit this range. The set of Gauss-Radau grid points, however, spans the range

(-1,1], meaning that it does include the endpoint at 1, but does not include the endpoint at -1.

Simple diagrams showing the general appearance of the Gauss-Chebyshev-Lobatto and Gauss-

Radau sets of grid points, considering the r-direction, and considering seven grid points, are

shown below as Figure 5-2.

To show how to determine the exact spacing of these clustered sets of grid points, the

equations to generate the Gauss-Chebyshev-Lobatto and Gauss-Radau points are shown below,

considering the discretization of the r-direction. The equation to generate the Gauss-Chebyshev-

Lobatto points is


r =-cos j = 0,1,...,N,, (5.6)
TN)

and the equation to generate the Gauss-Radau points is


r1 = cos 2 j = 0,1,... Nr (5.7)
2Nr +1 I

A negative sign has been added to the Gauss-Chebyshev-Lobatto equation so that that set

of points corresponds with Figure 5-1. For the Gauss-Radau points, r, has been marked with a

superscripted "*" to indicate that the order of the set of points generated by Equation 5.7 actually

needs to be reversed so that the points correspond with Figure 5-1.









In all calculations done for this research, the Gauss-Chebyshev-Lobatto grid points were

used in discretizing the z-direction. In calculations for annular systems, the Gauss-Chebyshev-

Lobatto grid points were used for the discretization of the r-direction, as well. In calculations for

cylindrical systems, however, Gauss-Radau grid points were used in discretizing the r-direction.

The reason for this will be explained now. The discretization in the r-direction is done along the

radius of the cylinder, and not the diameter. In the set of modeling equations for the cylindrical

system, though, there are no boundary conditions at r = 0 (the interior endpoint of the

discretization in the r-direction). Thus, including that location in the discretization of the r-

direction would cause computational problems. This difficulty can be dealt with by excluding

the location r = 0 from the discretization. The use of Gauss-Radau grid points for the

discretization in the r-direction accomplishes this because this set of grid points excludes one

endpoint.

Given that all variables must be written as vectors of their scalar values at every node of

the discretized system, in the manner shown in Equation 5.5, it is clear that the eigenvector, X,

will actually be a concatenated column-vector comprised of the column-vectors for each system

variable. This means that it is a vector in which the smaller column-vectors for each system

variable are vertically arranged above and below one another, head-to-tail. In the discretized

form of the problem, the matrices A and B of Equation 5.1 are actually large matrices

comprised of many smaller matrices which operate on the discretized variables in X in

accordance with the modeling equations. Each row of submatrices within A and B represents a

different domain equation or boundary condition. Each submatrix in a row of submatrices

representing a domain equation must operate on all grid points in the system, whereas each

submatrix in a row of submatrices representing a boundary condition operates on only the grid









points on the boundary where the condition is enforced. A diagram illustrating the general

layout of the matrices and vectors of the discretized, generalized eigenvalue problem is shown

below as Figure 5-3.

All differential operators in the modeling equations must be expressed in matrix-form so

that they may operate on the discretized variables. The differentiation matrices can be derived

from polynomial interpolation equations. When taking derivatives in a direction that was

discretized with Gauss-Chebyshev-Lobatto grid points, the differentiation matrix is different than

the one that would be used for taking derivatives in a direction that was discretized with Gauss-

Radau grid points. An example differentiation matrix, for r-direction differentiation on Gauss-

Chebyshev-Lobatto grid points (as given by Equation 5.6) with Nr = 3 is

Dr,N,=3 =

-3.1667 4.0000 -1.3333 0.5000
-1.0000 0.3333 1.0000 -0.3333
(5.8)
0.3333 -1.0000 -0.3333 1.0000
0.5000 1.3333 -4.0000 3.1667

where the subscript of Dr reflects the fact that this matrix is for N = 3.

More details about how to obtain differentiation matrices and how to set up and apply the

spectral solution method being used in this research can be found in a helpful book by Trefethen,

called Spectral Methods in MATLAB (2000).

The discretization and discrete problem formulation described so far are an accurate

description of how to treat the system of modeling equations on a given plane normal to the

azimuthal direction (an r-z plane). Dealing with the 0 -direction dependence of the physical

system is mathematically quite simple because of the periodicity of the system in the 0-









direction. Recall that the periodic 0 -direction dependence of the system was represented in the

form shown in Equation 4.20. This results in the presence of the azimuthal wave number, m, in

the final set of modeling equations. To account for the 06 -direction dependence of the physical

system, one must simply assume a value for m. To choose a value for m, of course, is to assume

the exact periodic form (number of periods in the 0-direction) of the convective flow pattern at

onset. If, for example, m is assumed to be equal to 2, then the onset flow pattern being sought is

one which repeats twice as one progresses a single cycle through the fluid layer in the 0 -

direction. Ifm = 0 is assumed, then the onset flow pattern is one which does not vary in the 0 -

direction at all; the m = 0 case is axisymmetric in the 06-direction.

Once a value is set for m, and thus the particular periodic form of the onset flow pattern

has been assumed, the discrete system of modeling equations in matrix-form (Figure 5-3) can be

solved for the eigenvalue, A T. This can be done numerically with an eigenvalue solver (for

example, in MATLAB) that accommodates generalized eigenvalue problems.

Note that, in this research, the left-hand-side matrix (the A matrix in Figure 5-3) was not

of full rank. Another way of saying this is that the set of eigenvalues of the A matrix, itself,

included some eigenvalues which were equal to zero. MATLAB was selected as the software

to be used in performing the computations for this research. The MATLAB generalized

eigenvalue solver used in this research, like most other generalized eigenvalue solvers that the

author has encountered, required that the A matrix be of full rank. Consequently, if the matrices

of discretized modeling equations were directly fed into a generalized eigenvalue solver, then the

output would include several spurious values. The reasons for the appearance of the spurious

values related to the redundancy of boundary conditions at some points. To read more about this

type of complication, the reader is asked to see the 1993 work of Labrosse (Labrosse 1993). In









order to eliminate spurious values from the output of the eigenvalue calculation, the A matrix

was put through a pre-conditioning process involving singular-value decomposition. In this

process, the A matrix was decomposed into three matrices by singular-value decomposition,

then the rows in the matrices corresponding to any eigenvalues equal to zero could be located,

and finally those rows were filtered out of further computations.

The critical temperature difference and flow pattern at the onset of convection can be

determined by comparing the A Tvalues (eigenvalues) obtained for a range of different m values.

When making this comparison, the lowest A Tvalue corresponds to the m value for which the

fluid layer is most unstable to buoyancy forces. The lowest A Tvalue and the corresponding m

value, thus, represent the critical temperature difference and flow pattern at the onset of

convection, respectively. Note, also, that in order to account for the variation of viscosity with

temperature, it is necessary to use an iterative procedure (as described earlier in this chapter) to

obtain the A Tvalue corresponding to each m value.

A flow-diagram displaying the general procedure carried out in MATLAB to solve the

system of modeling equations for onset conditions is given in Appendix D. A set of MATLAB

programs were created to be used in the calculations for this research, and some examples of

these programs are given in Appendix E.

5.2 Application of the Method

As mentioned, considering a larger number of nodes makes the discretized model more

accurate. If the number of discretization nodes used to model the physical system were small

enough, an inaccurate critical temperature difference would be obtained from the eigenvalue

calculation. If the number of nodes were, then, increased further and further, the critical

temperature difference results given by the eigenvalue calculation would progressively converge









toward the physically correct value. Eventually, when considering a sufficiently high number of

discretization nodes, the critical temperature difference obtained would be the physically correct

value, and further increases to the number of nodes considered would have no impact on the

critical temperature difference result obtained. A nice quality of the spectral method used here,

though, is that not too large a number of nodes are actually needed for the computational solution

to converge to an accurate solution. In all computations done for this research, 17 nodes were

used in the r-direction and in the z-direction; this was always more than enough nodes to obtain a

converged solution. Convergence of the computational results was, of course, verified by

repeating computations with even higher numbers of discretization nodes. To exemplify this

convergence behavior, a table is given below (Table 5-1) which shows the convergence of

critical temperature difference values obtained using the author's MATLAB program. The

example system considered for this convergence table is a fluid layer in a cylindrical container,

with rigid, no-slip walls, insulated at the lateral boundary, and heated from below so as to

facilitate Rayleigh convection. In particular, a case is being considered in which the size of the

cylindrical system is such that the convective flow pattern at onset is a one-cell pattern,

axisymmetric in the 06-direction. The convergence of the critical temperature difference

([ A T]crit) is shown with respect to both Nr and Nz.

To confirm the validity of the set of computational programs written for this research,

results obtained using the programs were checked against results for standard Rayleigh-Benard

problems (problems involving buoyancy-driven convection in systems heated from below) and

some results obtained by other authors.

As one check of the validity of the computational programs, a comparison was made

between the critical temperature difference results obtained for a two-dimensional, rectangular









case (x-direction and z-direction only) and a one-dimensional, rectangular case (z-direction

only). In the two-dimensional case, the rectangular fluid layer was heated from below, with

stress-free, insulated side walls, and constant-temperature, no-slip walls as the top and bottom

boundaries of the layer. The distance between the side walls for the two-dimensional case is

called Lx. In the one-dimensional case, the fluid layer was heated from below, was bounded at

the top and bottom edges by constant-temperature, no-slip walls, and was unbounded in the

lateral direction. In neither case was the variation of viscosity with temperature considered. The

two-dimensional case with stress-free side walls was expected to produce exactly the same

critical temperature difference results as a corresponding one-dimensional case, in which the

system had the same vertical depth as the two-dimensional case. As Table 5-2 shows, it did. At

first it may appear unusual that the one-dimensional calculation could produce two different

critical temperature difference results for the same vertical depth (Lz = 7.2 mm). The reason this

is possible has to do with the way the one-dimensional calculation is carried out. As explained in

Chapter 1, a two-dimensional system with lateral walls can only physically accommodate a

certain set of onset flow patterns, which is dependent on the lateral width of the system. Thus, if

the critical temperature difference result for a two-dimensional system with lateral walls is to be

compared with the result for a corresponding one-dimensional calculation, it is only the

disturbances of proper shape and size to induce these physically admissible onset flow patterns

which may be considered in the one-dimensional calculation. It is because of this that the critical

temperature difference results obtained from the one-dimensional calculation for the two cases in

which Lz = 7.2 mm are different. Now that results are being shown for systems of specified

depths and boundary conditions, it is a good time to mention that the fluid used in all

experiments and calculations for this research is Dow Coming 200 1 Stoke silicone oil. Thus,









its thermophysical properties are used in all calculations presented in this paper. A list of these

thermophysical properties is given in Appendix A. The dependence of temperature on the

viscosity of the oil was determined using viscosity measurements taken with a Cole-Parmer

98936 series viscometer. The exponential equation for the viscosity of the oil as a function of

temperature is included in Appendix A, as well as a description of the manner in which viscosity

measurements were obtained. Except for the viscosity, which was measured as described, the

thermophysical properties of the silicone oil were based on information provided by Dow

Corning.

As an initial check of the ability of the computational programs to model cylindrical

systems, critical temperature difference results were computed for buoyancy-driven convection

in a cylindrical fluid layer, and compared to the results given in a 1990 publication by Hardin et

al. (Hardin, Sani, Henry, & Roux 1990). The particular system considered for this comparison

was a layer of silicone oil in a cylindrical container, heated from below, with no-slip walls at all

boundaries, constant-temperature conditions at the top an bottom boundaries, and insulation at

the radial wall. Again, the variation of viscosity with temperature was not considered. Hardin et

al. give several critical temperature difference results for this system, as well as the

corresponding flow patterns. A comparison of the results produced by the author's computations

for this system with those of Hardin et al. is given in Table 5-3. These comparisons show the

validity of the MATLAB programs written for the computations in this research. The results of

Hardin et al. were given in terms of the dimensionless Rayleigh number, so in order to compare

with them, the author's critical temperature difference results in this table have been re-expressed

in terms of the Rayleigh number, as well (as shown in Equation 1.1). Since the variation of

viscosity with temperature was not considered in this case, the viscosity value used in the









computation was simply the constant viscosity value at 35 C given in Appendix A. The aspect

ratio is simply the ratio of the radius of the cylindrical container to the vertical height of the

container (L/ Lz). The way in which convective flow patterns are named should be briefly

discussed here, too. In general, they are named by their azimuthal wave number, m. For

example, a ring-shaped pattern axisymmetric in the 0-direction is called an m = 0 pattern.

Optionally, when considering cylindrical systems, the flow pattern may be referred to with a

parenthetical notation, including two indices. A parenthetical system for naming the onset flow

patterns in annular systems is given in Chapter 7. In the parenthetical notation for cylindrical

systems, the first index is the azimuthal wave number and the second index represents the

maximum number of convective rolls that can be counted across the diameter of the cylindrical

test section. Diagrams to exemplify the use of this parenthetical notation for flow patterns in

cylindrical systems are shown as Figure 5-4. The diagrams are taken almost directly from the

1995 paper by Zhao, Moates, and Narayanan (Zhao, Moates, & Narayanan 1995). In Figure 5-4,

a top view of each flow pattern is given, and below the top view of each pattern is a side view of

the same pattern. In the top views, the "X" indicates falling fluid and the "0" indicates rising

fluid. Diagrams (a) and (c) in Figure 5-4 could simply be called m = 0 patterns if the

parenthetical notation were not being used; likewise, diagrams (b) and (d) could be called m = 1

patterns.

Now, the method of obtaining computational results has been explained, and the validity

of the computational method has been demonstrated. The computational results were compared

with the results from experiments, and an explanation of the experimental apparatus and the

manner in which experimental results were obtained is given in Chapter 6.










Table 5-1. Example of convergence of computed result with Nr and Nz.


Nr Nz [AT]crit(C)
3 13 6.55
4 13 5.55
5 13 5.60
6 13 5.59
7 13 5.60
8 13 5.60
9 13 5.60
10 13 5.60
11 13 5.60
12 13 5.60
13 13 5.60


Nr Nz [A Ticrit (oC)
13 3 Inf
13 4 5.42
13 5 5.55
13 6 5.60
13 7 5.60
13 8 5.60
13 9 5.60
13 10 5.60
13 11 5.60
13 12 5.60
13 13 5.60


Table 5-2. Comparison of rectangular, 2-D, no-stress results with rectangular 1-D results.


[ A T]crit (OC),
L. (mm) L, (mm) 2-D, No-Stress


[ A T]crit (OC),
1-D


5 5 12.87 12.87
7.2 7.2 4.31 4.31
23 7.2 4.33 4.33
9 9 2.21 2.21


Table 5-3. Comparison of calculated cylindrical results with results of Hardin et al.


Predicted Flow
Pattern,
Author's
Computation


Predicted
Flow Pattern,
Hardin et al.


Critical Rayleigh
Number,
Author's
Computation


Critical Rayleigh
Number, Hardin
et al.


Aspect Ratio: .75, Lr = 4.5 mm, Lz = 6 mm
(1,1) (1,1) 2590 2592
Aspect Ratio: 1, Lr = 6 mm, Lz = 6 mm
(0,2) (0,2) 2260 2260
Aspect Ratio: 1.5, Lr = 9 mm, Lz = 6 mm
(0,2) (0,2) 1895 1895
Aspect Ratio: 2.5, Lr= 15 mm, Lz = 6 mm
(0,4) (0,4) 1780 1781











ro, Z3

t ri, r2,Z3 2 3,Z3



r z- >- -
SroZ 1,Z2 r2,Z2 3, Z2




ro,zi r,z r2,ZI r3,Z
^^__ ___ --- -i- --- ------ -


o ro,Z-o rZo0 r2,Zo rZO



Figure 5-1. Discretization nodes in a cylinder.


Gauss-Chebyshev- =6
Lobatto -

r -.866 r5 .866


ro -1 r2 -.5 r3 0 r4 .5 r6- 1



Gauss-Radau Nr=6

r, -.7485 r5 .8855


ro -.9709 r2 -.3546 r3 .1205 r4 .5681 r6 1


Figure 5-2. Example: grid point spacing.










Left-Hand-Side Matrix (LHS), A
LHS Motion Equation, r-component,
and velocity boundary conditions
LHS Motion Equation, 0 -
component, and velocity boundary
LHS Motion Equation, z-component,
and velocity boundary conditions
LHS Energy Equation, and
temperature boundary conditions
LHS Continuity Equation


rV






T

P


Right-Hand-Side Matrix (RHS), B

Figure 5-3. Diagram of matrix/vector arrangement of discretized problem.


RHS Motion Equation, r-component,
and velocity boundary conditions
RHS Motion Equation, 0 -
component, and velocity boundary
RHS Motion Equation, z-component,
and velocity boundary conditions
RHS Energy Equation, and
temperature boundary conditions
RHS Continuity Equation


Vr






T

P











[OO
(a) L, / L, = 1, Pattern: (0,2)


o0D
(b) L,/L = 1.8, Pattern: (1,3)


(c) L,r / L = 2.5, Pattern: (0,4) (d) L, / L, = .75, Pattern: (1,1)
Figure 5-4. Examples of flow patterns and parenthetical notation for cylindrical systems.









CHAPTER 6
EXPERIMENTAL DESIGN

After computational results were obtained in this research, they were compared with

experimental results for systems identical to those considered in the computations. The

apparatus which was designed to run experiments for this research is the subject of this chapter.

Any possible errors associated with the apparatus and its measurements are discussed in Chapter

7.

6.1 Goals in Experimental Design

There were several requirements made of the experimental system. One requirement was

that the top and bottom temperatures of the fluid layer (T, and Tb, respectively) were kept

constant and uniform at desired values. By monitoring and controlling the top and bottom

temperatures of the fluid layer, the vertical temperature difference across the fluid layer could be

regulated. The radial walls in the experiments needed to be insulating so that heat did not escape

the experimental test fluid into the air surrounding the experiment. Lastly, some means of flow

visualization needed to be employed in order to determine when the fluid layer was or was not

convecting, so that the critical conditions for the onset of convection could be determined.

Below, the means by which these goals were accomplished are described.

6.2 Experimental Apparatus

As mentioned, the test fluid in the experiments was Dow Coming 200 1 Stoke silicone

oil (thermophysical properties given in Appendix A). This fluid was chosen as the test fluid for

this research because, for systems in the size range of those examined in this research, its

thermophysical properties are such that the critical vertical temperature differences for

convection were easily obtainable in experiments. The only thermophysical properties of the

silicone oil which vary significantly with temperature are the density (enough to facilitate









Rayleigh convection) and viscosity. In the experimental apparatus, a lucite ring acted as the

outer radial boundary of the silicone oil test section. A lucite middle-piece was added when the

test section was annular. The dimensions of these lucite pieces, thus, set the sizes of the

cylindrical and annular test sections. A copper plate at the top of a continuously stirred water

bath was the bottom boundary of the test section, and a sapphire window at the bottom of a flow-

through water bath was the top plate of the test section. To create heated-from-below conditions,

the temperature of the bottom water bath was adjusted to certain set-point values while the top

water bath was always kept at a constant, cooler temperature.

A process control computer, running a Lab VIEWTM process control program written by

the author, sent signals to turn a heater in the experiment on and off as needed in order to keep

the bottom water bath at desired set-point temperatures. The process control system built for the

experiments in this research will be further discussed in Section 6.3. The test section was

insulated to prevent heat loss in the radial direction. To allow flow visualization, a small amount

of aluminum tracer powder was mixed into the silicone oil. So that the flow behavior of the

experimental system could be recorded and reviewed, a digital camcorder was mounted above

the test section. More detailed descriptions of the components and features of the experimental

apparatus, as well as some notes on the experimental start-up procedure, are given in the

following subsections. A simple diagram of the apparatus is shown below as Figure 6-1, and a

couple of photographs of the apparatus follow as Figures 6-2 and 6-3.

6.2.1 Test Section

As mentioned, a lucite ring acted as the outer radial boundary of the test section which

contained the silicone oil. The bottom boundary of the test section was a copper plate, which

was the top surface of a stirred water bath that will be described shortly. The top boundary of the









test section was a sapphire window, which was the bottom surface of a flow-through water bath,

which also will be described. The lucite rings typically ranged from about 15 mm to 30 mm in

diameter and about 6 mm to 8 mm in height. The sets of experiments for cylindrical systems

were run before the sets for annular systems; this procedure was used so that when the annular

experiments were run, a nesting hole in the bottom copper plate could be used to anchor the

bottoms of the lucite center-pieces, which were added to transform the cylindrical systems to

annular systems. A reason for selecting lucite as the material from which to construct the outer-

boundary rings and the center-pieces for the annular systems is its thermal conductivity, which is

very close to that of the silicone oil (see Appendix A). Since the system of silicone oil being

examined in this research was subjected to vertical temperature gradients, so, too, were the radial

walls of the system. Thus, the fact that lucite has a thermal conductivity close to that of silicone

oil made it an appealing choice as the material from which to construct the radial walls because it

ensured that the vertical temperature gradients at the radial boundaries of the test section would

not differ from those in the interior. This, and the fact that the lucite outer radial wall was

surrounded by an insulating coating (which shall be further explained later), ensured that heat

would not flow into or out of the silicone oil in the test section in the radial direction.

After placing the lucite ring onto the copper plate and before filling the test section with

oil, a lucite clamp-piece was screwed down on top of the lucite test section ring to tighten the

ring down against the copper plate (the clamp-piece screws into the bottom water bath). This

was an important measure taken to prevent any leakage of air/oil at the bottom of the test section.

Once the test section ring was tightened down to the copper plate, the test section was

intentionally over-filled with silicone oil. This was done so that the sapphire plate of the flow-

through top water bath could more easily be pressed down onto the top of the test section without