Rotating Thermal Flows in Natural and Industrial Processes

Dr Marcello Lappa received his M.Sc. and Ph.D. in Aerospace Engineering from the University of Naples "Federico II", Italy. He spent time as a post-doc at the University of Naples as well as at the Institute of Advanced Material Study, Fukuoka, Kyushu University in Japan. From 2002-2008 he was a Senior Researcher at the Microgravity Advanced Research and Support Center (MARS). From 2005-present he has served as Editor-in-chief of the International Scientific Journal Fluid Dynamics & Materials Processing.His research interests include: Buoyant flows; Thermocapillary (Marangoni) flows; Materials Processing — CFD; Control of flow patterns and their stability; Multi phase flows; Methods of numerical analysis in Computational Fluid; Dynamics and Heat/Mass Transfer; High Performance Computing; Biological fluid dynamics; Tissue Engineering and CFD.

• Preface xiiiAcknowledgements xvii1 Equations, General Concepts and Nondimensional Numbers 11.1 The Navier-Stokes and Energy Equations 11.1.1 The Continuity Equation 21.1.2 The Momentum Equation 21.1.3 The Total Energy Equation 21.1.4 The Budget of Internal Energy 31.1.5 Closure Models 31.2 Some Considerations about the Dynamics of Vorticity 51.2.1 Vorticity and Circulation 51.2.2 Vorticity in Two Dimensions 71.2.3 Vorticity Over a Spherical Surface 81.2.4 The Curl of the Momentum Equation 101.3 Incompressible Formulation 101.4 Buoyancy Convection 131.4.1 The Boussinesq Model 131.4.2 The Grashof and Rayleigh Numbers 141.5 Surface-Tension-Driven Flows 141.5.1 Stress Balance 151.5.2 The Reynolds and Marangoni Numbers 161.5.3 The Microgravity Environment 181.6 Rotating Systems: The Coriolis and Centrifugal Forces 191.6.1 Generalized Gravity 201.6.2 The Coriolis, Taylor and Rossby Numbers 211.6.3 The Geostrophic Flow Approximation 221.6.4 The Taylor–Proudman Theorem 231.6.5 Centrifugal and Stratification Effects: The Froude Number 231.6.6 The Rossby Deformation Radius 241.7 Some Elementary Effects due to Rotation 251.7.1 The Origin of Cyclonic and Anticyclonic flows 251.7.2 The Ekman Layer 261.7.3 Ekman Spiral 281.7.4 Ekman Pumping 281.7.5 The Stewartson Layer 302 Rayleigh-Benard Convection with Rotation 332.1 Rayleigh-Benard Convection with Rotation in Infinite Layers 342.1.1 Linear Stability Analysis 352.1.2 Asymptotic Analysis 362.2 The Kuppers-Lortz Instability and Domain Chaos 382.3 Patterns with Squares 412.4 Typical Phenomena for Pr=  2.4.1 Spiral Defect Chaos and Chiral Symmetry 422.4.2 The Interplay between the Busse Balloon and the KL Instability 452.5 The Low-Pr Hopf Bifurcation and Mixed States 482.5.1 Standing and Travelling Rolls 502.5.2 Patterns with the Symmetry of Square and Hexagonal Lattices 522.5.3 Other Asymptotic Analyses 552.5.4 Nature and Topology of the Bifurcation Lines in the Space of Parameters (Pr) 562.6 Laterally Confined Convection 582.6.1 The First Bifurcation and Wall Modes 602.6.2 The Second Bifurcation and Bulk Convection 632.6.3 Square Patterns Driven by Nonlinear Interactions between Bulk and Wall Modes 642.6.4 Square Patterns as a Nonlinear Combination of Bulk Fourier Eigenmodes 672.6.5 Higher-Order Bifurcations 692.7 Centrifugal Effects 712.7.1 Stably Thermally Stratified Systems 712.7.2 Interacting Thermogravitational and Centrifugally Driven Flows 742.7.3 The Effect of the Centrifugal Force on Domain Chaos 842.8 Turbulent Rotating RB Convection 862.8.1 The Origin of the Large-scale Circulation 872.8.2 Rotating Vortical Plumes 892.8.3 Classification of Flow Regimes 912.8.4 Suppression of Large-scale Flow and Heat Transfer Enhancement 982.8.5 Prandtl Number Effects 102  3 Spherical Shells, Rossby Waves and Centrifugally Driven Thermal Convection 1073.1 The Coriolis Effect in Atmosphere Dynamics 1073.1.1 The Origin of the Zonal Winds 1073.1.2 The Rossby Waves 1103.2 Self-Gravitating Rotating Spherical Shells 1143.2.1 Columnar Convective Patterns 1153.2.2 A Mechanism for Generating Differential Rotation 1193.2.3 Higher-Order Modes of Convection 1213.2.4 Equatorially Attached Modes of Convection 1263.2.5 Polar Convection 1273.3 Centrifugally Driven Thermal Convection 1284 The Baroclinic Problem 1354.1 Energetics of Convection and Heuristic Arguments 1364.2 Linear Stability Analysis: The Classical Eady’s Model 1394.3 Extensions of the Eady’s model 1484.4 Fully Developed Nonlinear Waveforms 1544.5 The Influence of the Prandtl Number 1624.6 The Route to Chaos 1664.7 Hybrid Baroclinic Flows 1724.8 Elementary Application to Atmospheric Dynamics 1754.8.1 Spiralling Eddy Structures 1764.8.2 The Baroclinic Life-Cycle and the ‘Barotropization’ Mechanism 1774.8.3 The Predictability of Weather and Climate Systems 1795 The Quasi-Geostrophic Theory 1835.1 The Potential Vorticity Perspective 1835.1.1 The Rossby-Ertel’s Potential Vorticity 1835.1.2 The Quasi-Geostrophic (QG) Pseudo-Potential Vorticity 1845.2 The Perturbation Energy Equation 1895.3 Derivation of Necessary Conditions for Instability 1915.3.1 The Rayleigh’s Criterion 1925.3.2 The Charney–Stern Theorem 1935.4 A Generalization of the Potential Vorticity Concept 1955.4.1 The Origin of the Sheets of Potential Vorticity 1965.4.2 Gradients of Potential Vorticity in the Interior 1995.5 The Concept of Interlevel Interaction 2015.6 The Counter-Propagating Rossby-Wave Perspective on Baroclinic Instability 2055.6.1 The Heuristic Interpretation 2065.6.2 A Mathematical Framework for the ‘Action-at-a-Distance’ Dynamics 2085.6.3 Extension and Rederivation of Earlier Results 2115.7 Barotropic Instability 2155.8 Extensions of the CRW Perspective 2185.9 The Over-reflection Theory and Its Connections to Other Theoretical Models 2225.10 Nonmodal Growth, Optimal Perturbations and Resonance 2255.11 Limits of the CRW Theory 2296 Planetary Patterns 2316.1 Jet Sets 2326.2 A Rigorous Categorization of Hypotheses and Models 2366.3 The Weather-Layer Approach 2376.4 The Physical Mechanism of Vortex Merging 2386.4.1 The Critical Core Size 2406.4.2 Metastability and the Axisymmetrization Principle 2416.4.3 Topology of the Streamline Pattern and Its Evolution 2426.5 Freely Decaying Turbulence 2466.5.1 Two-dimensional Turbulence 2466.5.2 Invariants, Inertial Range and Phenomenological Theory 2476.5.3 The Vortex-Dominated Evolution Stage 2506.6 Geostrophic Turbulence 2546.6.1 Relationship with 2D Turbulence 2546.6.2 Vortex Stretching and 3D Instabilities 2566.7 The Reorientation of the Inverse Cascade into Zonal Modes 2586.7.1 A Subdivision of the Spectrum: Rossby Waves and Turbulent Eddies 2586.7.2 Anisotropic Dispersion and Weak Nonlinear Interaction 2596.7.3 The Stability of Zonal Mean Flow 2626.8 Baroclinic Effects, Stochasting Forcing and Barotropization 2626.9 Hierarchy of Models and Scales 2646.9.1 The Role of Friction 2646.9.2 The One-Layer Perspective and the Barotropic Equation 2656.9.3 Classification of Models 2666.9.4 Characteristic Wavenumbers 2676.10 One-Layer Model 2686.10.1 Historical Background 2686.10.2 The Wavenumber Sub-space 2766.11 Barotropicity, Baroclinicity and Multilayer Models 2786.11.1 Eddy Variability and Zonally Averaged Properties 2796.11.2 Polygonal Wave Structures 2836.12 The Ocean–Jupiter Connection 2866.13 Wave–Mean-Flow Dynamics 2876.13.1 The Barotropic Instability of Rossby Waves 2886.13.2 The Transition from Inflectional to Triad Resonance Instability 2916.13.3 Destabilization of Mixed Rossby–Gravity Waves 2966.13.4 Relaxation of the Triad Resonance Condition 2996.13.5 Interaction with Critical Lines 3006.14 Solitary Vortex Dynamics 3026.14.1 The Zoo of Vortex Instabilities on the f-Plane 3026.14.2 Free Vortices on the β Plane 3096.14.3 Gyres and Rossby-Wave-Induced Gradual Vortex Decay 3116.14.4 The Influence of Zonal Flow on Vortex Stability 3176.15 Penetrative Convection Model 3236.15.1 Limits of the Shallow Layer Approach 3236.15.2 Differential Rotation and Deep Geostrophic Convection 3246.16 Extension and Unification of Existing Theories and Approaches 3306.16.1 The Classical Bowl-Based Experiment 3306.16.2 Models with B Sign Reversal 3336.16.3 Models with Scaling Discontinuities 3376.16.4 Open Points and Future Directions of Research 3427 Surface-Tension-Driven Flows in Rotating Fluids 3457.1 Marangoni–Benard Convection 3467.1.1 Classical Patterns and Theories 3467.1.2 Stationary and Oscillatory Flows with Rotation 3477.2 The Return Flow 3527.3 The Hydrothermal Instability 3547.3.1 A LSA Including the Effect of Rotation 3567.4 The Annular Pool 3607.4.1 Liquid Metals and Semiconductor Melts 3637.4.2 Travelling and Stationary Waves 3657.4.3 Transparent Organic Liquids 3667.4.4 Modification of the Fundamental Hydrothermal Mechanism 3688 Crystal Growth from the Melt and Rotating Machinery 3718.1 The Bridgman Method 3728.2 The Floating Zone 3828.2.1 The Liquid Bridge 3838.2.2 Rotating Liquid Bridge with Infinite Axial Extent 3858.2.3 Rotation, Standing Waves and Travelling Waves 3868.2.4 Self-Induced Rotation and PAS 3908.3 The Czochralski Method 3948.3.1 Spoke and Wave Patterns 3968.3.2 Mixed Baroclinic-Hydrothermal States 3998.3.3 Other Effects, Cold Plumes and Oscillating Jets 4068.3.4 Geostrophic Turbulence 4118.4 Rotating Machinery 4138.4.1 The Taylor–Couette Flow 4138.4.2 Cylinders with Rotating Endwalls 4229 Rotating Magnetic Fields 4319.1 Physical Principles and Characteristic Numbers 4329.1.1 The Hartmann, Reynolds and Magnetic Taylor Numbers 4329.1.2 The Swirling Flow 4349.2 Stabilization of Thermo-gravitational Flows 4389.3 Stabilization of Surface-Tension-Driven Flows 4429.4 Combining Rotation and RMF 44610 Angular Vibrations and Rocking Motions 44910.1 Equations and Relevant Parameters 45010.1.1 Characteristic Numbers 45310.1.2 The Mechanical Equilibrium 45410.2 The Infinite Layer 45410.2.1 The Stability of the Equilibrium State 45510.2.2 Combined Translational-Rotational Vibrations 46010.3 The Vertical Coaxial Gap 46210.4 Application to Vertical Bridgman Crystal Growth 467References  473Index509  509 509

“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)