Mixing And Dispersion In Flows Dominated By Rotation And Buoyancy (cism International Centre For Mechanical Sciences)

The book presents a state-of-the-art overview of current developments in the field in a way accessible to attendees coming from a variety of fields. Relevant examples are turbulence research, (environmental) fluid mechanics, lake hydrodynamics and atmospheric physics. Topics discussed range from the fundamentals of rotating and stratified flows, mixing and transport in stratified or rotating turbulence, transport in the atmospheric boundary layer, the dynamics of gravity and turbidity currents eventually with effects of background rotation or stratification, mixing in (stratified) lakes, and the Lagrangian approach in the analysis of transport processes in geophysical and environmental flows. The topics are discussed from fundamental, experimental and numerical points of view. Some contributions cover fundamental aspects including a number of the basic dynamical properties of rotating and or stratified (turbulent) flows, the mathematical description of these flows, some applications in the natural environment, and the Lagrangian statistical analysis of turbulent transport processes and turbulent transport of material particles (including, for example, inertial and finite-size effects). Four papers are dedicated to specific topics such as transport in (stratified) lakes, transport and mixing in the atmospheric boundary layer, mixing in stratified fluids and dynamics of turbidity currents. The book is addressed to doctoral students and postdoctoral researchers, but also to academic and industrial researchers and practicing engineers, with a background in mechanical engineering, applied physics, civil engineering, applied mathematics, meteorology, physical oceanography or physical limnology. Preface 6 Contents 8 Contributors 9 Effects of Rotation and Stratification: An Introduction 10 1 Effects of Rotation 10 1.1 Flow on a Spherical Planet 12 1.2 Some Horizontal Balances 14 1.3 How to Create Vortices in the Lab 16 1.4 The Ekman Layer 17 1.5 The Shallow-Water Approximation 20 2 Density Effects 22 2.1 Density Currents 22 2.2 Stability 23 2.3 Baroclinic Generation of Vorticity 24 2.4 Boussinesq Approximation 25 2.5 Waves in a Stratified Fluid 26 2.6 Vortices in a Stratified Fluid 29 3 Concluding Remarks 32 References 32 Turbulence and Mixing in Flows Dominated by Buoyancy 34 1 Introduction 34 2 Governing Equations 36 2.1 Mass and Momentum Conservation 36 2.2 Boussinesq Approximation 36 2.3 Scaling 38 3 Richardson Number 41 4 Turbulent Kinetic Energy Equation 42 5 Stirring Versus Mixing 44 5.1 Batchelor Scale 45 6 Stratified Bottle 45 7 Mixing Efficiency 46 8 Boundary Layer Similarity Theory 48 8.1 Unstratified Boundary Layer 48 8.2 Stratified Boundary Layer 49 9 Layers and Interfaces 52 9.1 Energetics 52 10 Stratified Mixing Box Experiment 53 11 Mixing Efficiency from Available Potential Energy 55 11.1 Tall Rayleigh-Taylor Instability 55 11.2 Mixing Efficiency in Lock Exchange Gravity Currents 57 12 Concluding Remarks 67 References 68 Mixing in Stratified Lakes and Reservoirs 70 1 Introduction 70 2 Surface and Bottom Boundary Layers in Lakes 71 2.1 The Role of Boundary Layers in Natural Waters 71 2.2 The Turbulent Kinetic Energy Balance 73 2.3 Vertical Structure of Wind-Induced Surface Boundary Layers 75 2.4 Vertical Structure of the Bottom Boundary Layer 80 2.5 Convectively-Induced Turbulence and Mixing in the Boundary Layers 84 3 Stratified Turbulence in Lakes Interior 88 4 Wind Set-up, Internal Standing Waves and Modes 92 5 Concluding Remarks 96 References 96 Energy Balance in Stably-Stratified, Wall-Bounded Turbulence 98 1 Introduction 98 2 Description of the DNS Database 99 2.1 Case C: Open-Channel Flow 100 2.2 Case E: Capped Ekman Layer 100 2.3 Flow Initialization 101 3 Results 102 3.1 Time Evolution and Turbulence Collapse 102 3.2 Energy Budgets 104 4 Conclusions 107 References 107 Some Aspects of Lagrangian Dynamics of Turbulence 109 1 Introduction 109 2 Basic Multi-scale Properties of Lagrangian Turbulence 112 2.1 A Brief Recall of K41 Ideas in the Eulerian Framework 112 2.2 Lagrangian K41 Phenomenology 114 2.3 Lagrangian Dispersion 114 3 On the Validity of Kolmogorov's Local Isotropy Hypothesis 116 3.1 The Local Isotropy Hypothesis 116 3.2 The Local Isotropy Hypothesis in Large-Scale Anisotropic Flows 118 3.3 The Influence of Increasing Reynolds Number 119 4 Turbulent Dispersion of Tracer Particles 121 4.1 The Turbulent Pair Dispersion Problem 121 4.2 Batchelor and Richardson Regimes for Pair Dispersion 122 4.3 A Simple Ballistic Phenomenology of Turbulent Superdiffusion 124 4.4 Practical Implementation of the Ballistic Phenomenology 128 4.5 About the Time Irreversibility of Turbulent Relative Dispersion 129 5 Conclusion 133 References 133 Gravity and Turbidity Currents: Numerical Simulations and Theoretical Models 136 1 Introduction 136 2 Direct Numerical Simulations of Turbidity Currents 138 2.1 Problem Description and Modeling Approach 138 2.2 Numerical Method 141 2.3 General Current Properties 142 2.4 Structure of the Current 146 2.5 Fate of Particles 154 3 Vorticity-Based Models of Stratified Flows 162 3.1 Introduction 162 3.2 Gravity Currents 163 3.3 Circulation Based Models for Boussinesq Gravity Currents 165 3.4 Two-Layer Bores 166 3.5 Circulation-Based Model 168 3.6 Gravity Currents Propagating into Shear 170 3.7 Gravity Currents Thicker Than Half the Channel Height 182 4 Concluding Remarks 183 References 184 Transport Phenomena in Rotating Turbulence 188 1 Introduction 188 1.1 Homogeneous Isotropic Turbulence 190 1.2 What Is the Effect of Rotation on Turbulence? 192 1.3 Rotating Turbulence: A Regime Diagram 194 2 Taylor-Proudman Theorem and Inertial Waves 195 2.1 Navier-Stokes Equation in a Rotating Frame of Reference 195 2.2 Taylor-Proudman Theorem 196 2.3 Inertial Waves 198 3 Quasi-Two-Dimensional Flows 200 3.1 The Turbulence Triangle 203 4 Rotating Turbulence: Eulerian and Lagrangian Approach 206 4.1 Eulerian Velocity Measurements in Rotating Turbulence 207 4.2 Lagrangian Velocity Measurements in Rotating Turbulence 209 4.3 Lagrangian Acceleration Statistics in Rotating Turbulence 211 5 Rotating Turbulent Convection 212 5.1 The Boussinesq Equations in a Rotating Frame of Reference 213 5.2 Ekman Boundary Layers 214 5.3 Transition Between Turbulent States at Critical Rossby 216 5.4 Eulerian and Lagrangian Statistics in (R)RB Turbulence 219 6 Concluding Remarks 222 References 222 Front Matter ....Pages i-ix Effects of Rotation and Stratification: An Introduction (GertJan F. van Heijst)....Pages 1-24 Turbulence and Mixing in Flows Dominated by Buoyancy (Paul F. Linden)....Pages 25-60 Mixing in Stratified Lakes and Reservoirs (Damien Bouffard, Alfred Wüest)....Pages 61-88 Energy Balance in Stably-Stratified, Wall-Bounded Turbulence (Oscar Flores, James J. Riley)....Pages 89-99 Some Aspects of Lagrangian Dynamics of Turbulence (Mickaël Bourgoin)....Pages 101-127 Gravity and Turbidity Currents: Numerical Simulations and Theoretical Models (Eckart Meiburg, Mohamad M. Nasr-Azadani)....Pages 129-180 Transport Phenomena in Rotating Turbulence (Herman J. H. Clercx)....Pages 181-218