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Rotating Thermal Flows in Natural and Industrial Processes

ISBN: 978-1-119-96079-9
540 pages
September 2012
Rotating Thermal Flows in Natural and Industrial Processes (1119960797) cover image

Rotating Thermal Flows in Natural and Industrial Processes provides the reader with a systematic description of the different types of thermal convection and flow instabilities in rotating systems, as present in materials, crystal growth, thermal engineering, meteorology, oceanography, geophysics and astrophysics. It expressly shows how the isomorphism between small and large scale phenomena becomes beneficial to the definition and ensuing development of an integrated comprehensive framework.  This allows the reader to understand and assimilate the underlying, quintessential mechanisms without requiring familiarity with specific literature on the subject. 

Topics treated in the first part of the book include: 

•           Thermogravitational convection in rotating fluids (from laminar to turbulent states);

•           Stably stratified and unstratified shear flows;

•           Barotropic and baroclinic instabilities;

•           Rossby waves and Centrifugally-driven convection;

•           Potential Vorticity, Quasi-Geostrophic Theory and related theorems;

•           The dynamics of interacting vortices, interacting waves and mixed (hybrid) vortex-wave states;

•           Geostrophic Turbulence and planetary patterns.

 

The second part is entirely devoted to phenomena of practical interest, i.e. subjects relevant to the realms of industry and technology,  among them:

 

•           Surface-tension-driven convection in rotating fluids;

•           Differential-rotation-driven (forced) flows;

•           Crystal Growth from the melt of oxide or semiconductor materials;

•           Directional solidification;

•           Rotating Machinery;

•           Flow control by Rotating magnetic fields;

•           Angular Vibrations and Rocking motions; 

Covering a truly prodigious range of scales, from atmospheric and oceanic processes and fluid motion in “other solar-system bodies”, to convection in its myriad manifestations in a variety of applications of technological relevance, this unifying text is an ideal reference for physicists and engineers, as well as an important resource for advanced students taking courses on the physics of fluids, fluid mechanics, thermal, mechanical and materials engineering, environmental phenomena, meteorology and geophysics.

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Preface xiii

Acknowledgements xvii

1 Equations, General Concepts and Nondimensional Numbers 1

1.1 The Navier-Stokes and Energy Equations 1

1.1.1 The Continuity Equation 2

1.1.2 The Momentum Equation 2

1.1.3 The Total Energy Equation 2

1.1.4 The Budget of Internal Energy 3

1.1.5 Closure Models 3

1.2 Some Considerations about the Dynamics of Vorticity 5

1.2.1 Vorticity and Circulation 5

1.2.2 Vorticity in Two Dimensions 7

1.2.3 Vorticity Over a Spherical Surface 8

1.2.4 The Curl of the Momentum Equation 10

1.3 Incompressible Formulation 10

1.4 Buoyancy Convection 13

1.4.1 The Boussinesq Model 13

1.4.2 The Grashof and Rayleigh Numbers 14

1.5 Surface-Tension-Driven Flows 14

1.5.1 Stress Balance 15

1.5.2 The Reynolds and Marangoni Numbers 16

1.5.3 The Microgravity Environment 18

1.6 Rotating Systems: The Coriolis and Centrifugal Forces 19

1.6.1 Generalized Gravity 20

1.6.2 The Coriolis, Taylor and Rossby Numbers 21

1.6.3 The Geostrophic Flow Approximation 22

1.6.4 The Taylor–Proudman Theorem 23

1.6.5 Centrifugal and Stratification Effects: The Froude Number 23

1.6.6 The Rossby Deformation Radius 24

1.7 Some Elementary Effects due to Rotation 25

1.7.1 The Origin of Cyclonic and Anticyclonic flows 25

1.7.2 The Ekman Layer 26

1.7.3 Ekman Spiral 28

1.7.4 Ekman Pumping 28

1.7.5 The Stewartson Layer 30

2 Rayleigh-Benard Convection with Rotation 33

2.1 Rayleigh-Benard Convection with Rotation in Infinite Layers 34

2.1.1 Linear Stability Analysis 35

2.1.2 Asymptotic Analysis 36

2.2 The Kuppers-Lortz Instability and Domain Chaos 38

2.3 Patterns with Squares 41

2.4 Typical Phenomena for Pr=

 

2.4.1 Spiral Defect Chaos and Chiral Symmetry 42

2.4.2 The Interplay between the Busse Balloon and the KL Instability 45

2.5 The Low-Pr Hopf Bifurcation and Mixed States 48

2.5.1 Standing and Travelling Rolls 50

2.5.2 Patterns with the Symmetry of Square and Hexagonal Lattices 52

2.5.3 Other Asymptotic Analyses 55

2.5.4 Nature and Topology of the Bifurcation Lines in the Space of Parameters (Pr) 56

2.6 Laterally Confined Convection 58

2.6.1 The First Bifurcation and Wall Modes 60

2.6.2 The Second Bifurcation and Bulk Convection 63

2.6.3 Square Patterns Driven by Nonlinear Interactions between Bulk and Wall Modes 64

2.6.4 Square Patterns as a Nonlinear Combination of Bulk Fourier Eigenmodes 67

2.6.5 Higher-Order Bifurcations 69

2.7 Centrifugal Effects 71

2.7.1 Stably Thermally Stratified Systems 71

2.7.2 Interacting Thermogravitational and Centrifugally Driven Flows 74

2.7.3 The Effect of the Centrifugal Force on Domain Chaos 84

2.8 Turbulent Rotating RB Convection 86

2.8.1 The Origin of the Large-scale Circulation 87

2.8.2 Rotating Vortical Plumes 89

2.8.3 Classification of Flow Regimes 91

2.8.4 Suppression of Large-scale Flow and Heat Transfer Enhancement 98

2.8.5 Prandtl Number Effects 102

 

3 Spherical Shells, Rossby Waves and Centrifugally Driven Thermal Convection 107

3.1 The Coriolis Effect in Atmosphere Dynamics 107

3.1.1 The Origin of the Zonal Winds 107

3.1.2 The Rossby Waves 110

3.2 Self-Gravitating Rotating Spherical Shells 114

3.2.1 Columnar Convective Patterns 115

3.2.2 A Mechanism for Generating Differential Rotation 119

3.2.3 Higher-Order Modes of Convection 121

3.2.4 Equatorially Attached Modes of Convection 126

3.2.5 Polar Convection 127

3.3 Centrifugally Driven Thermal Convection 128

4 The Baroclinic Problem 135

4.1 Energetics of Convection and Heuristic Arguments 136

4.2 Linear Stability Analysis: The Classical Eady’s Model 139

4.3 Extensions of the Eady’s model 148

4.4 Fully Developed Nonlinear Waveforms 154

4.5 The Influence of the Prandtl Number 162

4.6 The Route to Chaos 166

4.7 Hybrid Baroclinic Flows 172

4.8 Elementary Application to Atmospheric Dynamics 175

4.8.1 Spiralling Eddy Structures 176

4.8.2 The Baroclinic Life-Cycle and the ‘Barotropization’ Mechanism 177

4.8.3 The Predictability of Weather and Climate Systems 179

5 The Quasi-Geostrophic Theory 183

5.1 The Potential Vorticity Perspective 183

5.1.1 The Rossby-Ertel’s Potential Vorticity 183

5.1.2 The Quasi-Geostrophic (QG) Pseudo-Potential Vorticity 184

5.2 The Perturbation Energy Equation 189

5.3 Derivation of Necessary Conditions for Instability 191

5.3.1 The Rayleigh’s Criterion 192

5.3.2 The Charney–Stern Theorem 193

5.4 A Generalization of the Potential Vorticity Concept 195

5.4.1 The Origin of the Sheets of Potential Vorticity 196

5.4.2 Gradients of Potential Vorticity in the Interior 199

5.5 The Concept of Interlevel Interaction 201

5.6 The Counter-Propagating Rossby-Wave Perspective on Baroclinic Instability 205

5.6.1 The Heuristic Interpretation 206

5.6.2 A Mathematical Framework for the ‘Action-at-a-Distance’ Dynamics 208

5.6.3 Extension and Rederivation of Earlier Results 211

5.7 Barotropic Instability 215

5.8 Extensions of the CRW Perspective 218

5.9 The Over-reflection Theory and Its Connections to Other Theoretical Models 222

5.10 Nonmodal Growth, Optimal Perturbations and Resonance 225

5.11 Limits of the CRW Theory 229

6 Planetary Patterns 231

6.1 Jet Sets 232

6.2 A Rigorous Categorization of Hypotheses and Models 236

6.3 The Weather-Layer Approach 237

6.4 The Physical Mechanism of Vortex Merging 238

6.4.1 The Critical Core Size 240

6.4.2 Metastability and the Axisymmetrization Principle 241

6.4.3 Topology of the Streamline Pattern and Its Evolution 242

6.5 Freely Decaying Turbulence 246

6.5.1 Two-dimensional Turbulence 246

6.5.2 Invariants, Inertial Range and Phenomenological Theory 247

6.5.3 The Vortex-Dominated Evolution Stage 250

6.6 Geostrophic Turbulence 254

6.6.1 Relationship with 2D Turbulence 254

6.6.2 Vortex Stretching and 3D Instabilities 256

6.7 The Reorientation of the Inverse Cascade into Zonal Modes 258

6.7.1 A Subdivision of the Spectrum: Rossby Waves and Turbulent Eddies 258

6.7.2 Anisotropic Dispersion and Weak Nonlinear Interaction 259

6.7.3 The Stability of Zonal Mean Flow 262

6.8 Baroclinic Effects, Stochasting Forcing and Barotropization 262

6.9 Hierarchy of Models and Scales 264

6.9.1 The Role of Friction 264

6.9.2 The One-Layer Perspective and the Barotropic Equation 265

6.9.3 Classification of Models 266

6.9.4 Characteristic Wavenumbers 267

6.10 One-Layer Model 268

6.10.1 Historical Background 268

6.10.2 The Wavenumber Sub-space 276

6.11 Barotropicity, Baroclinicity and Multilayer Models 278

6.11.1 Eddy Variability and Zonally Averaged Properties 279

6.11.2 Polygonal Wave Structures 283

6.12 The Ocean–Jupiter Connection 286

6.13 Wave–Mean-Flow Dynamics 287

6.13.1 The Barotropic Instability of Rossby Waves 288

6.13.2 The Transition from Inflectional to Triad Resonance Instability 291

6.13.3 Destabilization of Mixed Rossby–Gravity Waves 296

6.13.4 Relaxation of the Triad Resonance Condition 299

6.13.5 Interaction with Critical Lines 300

6.14 Solitary Vortex Dynamics 302

6.14.1 The Zoo of Vortex Instabilities on the f-Plane 302

6.14.2 Free Vortices on the β Plane 309

6.14.3 Gyres and Rossby-Wave-Induced Gradual Vortex Decay 311

6.14.4 The Influence of Zonal Flow on Vortex Stability 317

6.15 Penetrative Convection Model 323

6.15.1 Limits of the Shallow Layer Approach 323

6.15.2 Differential Rotation and Deep Geostrophic Convection 324

6.16 Extension and Unification of Existing Theories and Approaches 330

6.16.1 The Classical Bowl-Based Experiment 330

6.16.2 Models with B Sign Reversal 333

6.16.3 Models with Scaling Discontinuities 337

6.16.4 Open Points and Future Directions of Research 342

7 Surface-Tension-Driven Flows in Rotating Fluids 345

7.1 Marangoni–Benard Convection 346

7.1.1 Classical Patterns and Theories 346

7.1.2 Stationary and Oscillatory Flows with Rotation 347

7.2 The Return Flow 352

7.3 The Hydrothermal Instability 354

7.3.1 A LSA Including the Effect of Rotation 356

7.4 The Annular Pool 360

7.4.1 Liquid Metals and Semiconductor Melts 363

7.4.2 Travelling and Stationary Waves 365

7.4.3 Transparent Organic Liquids 366

7.4.4 Modification of the Fundamental Hydrothermal Mechanism 368

8 Crystal Growth from the Melt and Rotating Machinery 371

8.1 The Bridgman Method 372

8.2 The Floating Zone 382

8.2.1 The Liquid Bridge 383

8.2.2 Rotating Liquid Bridge with Infinite Axial Extent 385

8.2.3 Rotation, Standing Waves and Travelling Waves 386

8.2.4 Self-Induced Rotation and PAS 390

8.3 The Czochralski Method 394

8.3.1 Spoke and Wave Patterns 396

8.3.2 Mixed Baroclinic-Hydrothermal States 399

8.3.3 Other Effects, Cold Plumes and Oscillating Jets 406

8.3.4 Geostrophic Turbulence 411

8.4 Rotating Machinery 413

8.4.1 The Taylor–Couette Flow 413

8.4.2 Cylinders with Rotating Endwalls 422

9 Rotating Magnetic Fields 431

9.1 Physical Principles and Characteristic Numbers 432

9.1.1 The Hartmann, Reynolds and Magnetic Taylor Numbers 432

9.1.2 The Swirling Flow 434

9.2 Stabilization of Thermo-gravitational Flows 438

9.3 Stabilization of Surface-Tension-Driven Flows 442

9.4 Combining Rotation and RMF 446

10 Angular Vibrations and Rocking Motions 449

10.1 Equations and Relevant Parameters 450

10.1.1 Characteristic Numbers 453

10.1.2 The Mechanical Equilibrium 454

10.2 The Infinite Layer 454

10.2.1 The Stability of the Equilibrium State 455

10.2.2 Combined Translational-Rotational Vibrations 460

10.3 The Vertical Coaxial Gap 462

10.4 Application to Vertical Bridgman Crystal Growth 467

References

 

473

Index

509

 

509

509

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“Given such a comprehensive review on this topic, this book can be highly recommended to crystal growth Researchers . . . It is also an ideal handbook for graduate students and researchers working in the field of fluid mechanics, geophysical and astrophysical fluid dynamics, thermal, mechanical and material science and engineering.”  (Journal of Experimental and Industrial Crystallography, 1 May 2013)

“It will be a useful resource for researchers in the appropriate fields of physics. An extensive, 36-page list of references supports the text. Summing Up: Recommended. Researchers and professionals.”  (Choice, 1 June 2013)

“As such, it should henceforth be considered as a reference work for any student, engineer and researcher in fluid mechanics, interested in either broadening their knowledge or in delving into one of the numerous subjects of interest developed here.”  (The International Journal of Geophysical & Astrophysical Fluid Dynamics, 1 April 2013)

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