Gravity Currents and Intrusions: Analysis and Prediction

Description

The flow of gravity currents and intrusions is a subject of active research and engineering application. Currently, there are no formal teaching courses for this topic. Materials and information available in the market are scattered and dated. Researchers and engineers face difficulties in acquiring the 'state-of-the-art' knowledge. The book bridges this gap between the need and supply of the relevant insight and know-how.Written by a renowned author who is a recognized authority in the field, this unique compendium assembles the relevant knowledge into a systematic and unified framework. The presentation is gradual from the elementary to the frontier, and accessible to readers with only a basic background in fluid mechanics and applied mathematics. This will facilitate the systematic acquirement and application of available knowledge to both practical problems and further research.This must-have volume is a useful monograph -- that can also serve as a textbook in advanced courses -- for researchers, students, engineers and applied mathematicians in the fields of civil engineering, hydraulic engineering, mechanical engineering, ocean engineering and environmental engineering. Contents Preface About the Author Copyright acknowledgment Abbreviations 1. Introduction 1.1 The gravity current (GC) 1.2 Classification 1.3 Approximate models 1.3.1 Some words of caution 2. Theoretical formulation 2.1 The Navier–Stokes (NS) equations 2.2 General thin-layer balances 2.2.1 Motion of the interface and the continuity equation 2.2.2 Hydrostatic pressure and buoyancy-driving term 2.2.2.1 Buoyancy-driving term for constant ρa and ρc 2.2.2.2 Non-constant ρa and/or ρc PART I : NON-STRATIFIED AMBIENT 3. Shallow-water (SW) analysis of the inertial GC: The one-layer model 3.1 Governing equations 3.1.1 Volume and momentum 3.1.2 Nose jump conditions 3.1.3 Some remarks 3.1.4 Scaling 3.1.5 Summary 1 3.1.6 A useful transformation 3.1.7 Summary 2 3.1.8 Input parameters and the Boussinesq simplification 3.2 The full behavior by numerical solution 3.3 Dam-break and slumping stages 3.3.1 The gate and reservoir (lock) 3.3.2 Constant h = hN domain 3.4 Similarity solution 3.4.1 Fixed volume V 3.4.2 Extensions for V = qtα 3.4.3 Local similarity 3.4.4 Non-Boussinesq systems 3.5 Effective Ree of the SW solution and transition xV 3.5.1 Non-Boussinesq extension 3.6 Reservoir drainage from an edge 4. Jumps in steady-state streams and control-volume matching conditions 4.1 General considerations 4.2 Internal jump solution 4.2.1 Major result 4.2.2 Vorticity balance 4.2.3 Small amplitude jump h/hu → 1 4.2.4 General cross-section 4.3 Front jump and Benjamin’s analysis 4.3.1 Major result Fr(a) 4.3.2 Energy considerations: The amax restriction for Fr 4.3.3 Summary of Benjamin’s solution 4.3.4 Jump condition 4.3.5 Generalization of Benjamin’s analysis 4.3.5.1 Fixed top 4.3.5.2 Open top 4.3.6 “Critical” nose: The acrit restriction for Fr 4.3.6.1 Fixed top 4.3.6.2 Open top 4.3.7 How to assess a given Fr formula 4.4 Internal jump in one-layer model 4.5 Jump reflected by a vertical boundary 5. Box models for 2D geometry 5.1 Fixed-volume current with inertial–buoyancy balance 5.2 Inflow volume change V = qtα 5.2.1 Inertial–buoyancy balance 5.2.2 Viscous–buoyancy balance 5.2.3 Transition length xV 5.2.4 Critical α 5.2.5 Comparisons and experimental verification 5.2.6 Non-Newtonian power-law viscosity 5.3 Extension to non-rectangular cross-section 6. Two-layer SW model 6.1 Introduction 6.2 The governing equations 6.3 Boussinesq system in dimensionless form 6.3.1 “Critical” nose acrit restriction for H < 1.25 6.4 Jumps of interface for H 0 11.2.2.1 Fixed volume 11.2.2.2 Fixed influx, downslope current 11.2.2.3 Summary 11.3 Upslope currents γ 1 cases 13.7 Concluding remarks 13.8 Asymptotic behavior for R → 0 13.8.1 Heavy-fluid domain 13.8.2 Light-fluid domain (air cavity) 13.9 Tailwaters current 13.9.1 The nose condition 13.9.2 Method of solution 13.9.3 Some results 13.9.4 One-layer reduction 14. Lubrication theory formulation for viscous currents 14.1 2D geometry 14.1.1 The governing equations 14.1.1.1 Balances 14.1.1.2 Boundary conditions at the nose 14.1.1.3 Scaling consideration 14.1.2 Similarity solution 14.1.3 Summary 14.1.4 Some comparisons 14.1.5 Extension to surface currents and intrusions 14.2 Axisymmetric current 14.2.1 The governing equations 14.2.2 Similarity solution 14.2.3 Summary 14.2.4 Some comparisons 14.3 Non-Newtonian power-law viscosity 14.3.1 2D case 14.3.1.1 Experiments 14.3.2 Axisymmetric case 14.3.2.1 Experiments 14.4 Two-layer model for viscous GC 14.5 Non-rectangular cross-section 14.6 Current in a porous medium 15. Vegetation effects 15.1 Theoretical model 15.2 Similarity solutions 15.3 One-layer solutions 15.3.1 α = 2 : Linear H(y) and constant uN 15.3.2 Fixed volume (α = 0) with linear U(y) 15.3.3 Constant h(x = 0) current (δ = 0) 15.3.4 Solutions for α ≠ 2, 0 and δ ≠ 0 15.4 The Boussinesq lock-exchange problem 15.4.1 λ = 1, β = 1/2 15.4.2 λ = 2, β = 2/3 15.5 The axisymmetric case, one-layer model 15.6 A remark about c and cD PART II : STRATIFIED AMBIENT CURRENTS AND INTRUSIONS 16. Continuous density transition 16.1 Introduction 16.2 The SW formulation 16.2.1 The nose condition 16.3 SW results and comparisons with experiments and simulations 16.4 Dam break 16.5 Critical speed and nose–wave interaction 16.6 Similarity solution 16.7 The validity of the inertial (inviscid) approximation 16.8 Double stratification (in both current and ambient) 16.8.1 SW formulation 16.8.1.1 Summary 16.8.2 Dam-break results 16.8.3 Time-dependent SW results 16.8.4 Concluding remarks 17. Axisymmetric and rotating cases 17.1 SW formulation 17.1.1 Steady-state lens (SL) 17.1.1.1 Analytical approximations for the SL 17.1.1.2 Analytical approximations for the SL with S = 1 17.1.1.3 Numerical solution of the SL 17.1.2 The energy of the SL 17.2 SW and NS finite-difference results 17.2.1 Sustained rotating intrusion 17.3 The validity of the inertial (inviscid) approximation 18. The steady-state current 18.1 Steady-state flow pattern 18.2 Results 18.2.1 Small S (small γ) 18.2.2 Large γ. Validity–stability and criticality 18.2.3 The effective g′ and Fr 18.2.4 Energy dissipation 18.3 Comparisons and conclusions 19. Intrusions in 2D geometry 19.1 Introduction 19.2 Two-layer stratification 19.3 Linear transition layer 19.3.1 Formulations 19.3.2 SW equations and boundary conditions 19.4 Rectangular lock configurations 19.4.1 Part-depth transition layer and full-depth lock 19.4.2 Fully linearly-stratified tank, part-depth locks 19.4.2.1 Comparisons of SW, NS, and experiments 19.4.2.2 Nose–wave interaction 19.5 Cylindrical lock in a fully linearly-stratified tank 19.6 Similarity solution 19.6.1 Local similarity 19.7 The validity of the inertial (inviscid) approximation 19.8 Non-symmetric intrusions 19.8.1 Equilibrium intrusion 19.8.2 Full-depth lock 19.8.3 Continuous linear stratification 19.8.4 Summary 20. Intrusions in axisymmetric geometry 20.1 Introduction 20.2 Two-layer stratification 20.3 Fully linearly-stratified tank, part-depth locks 20.3.1 Formulation and SW approximations 20.3.2 Similarity solution 20.3.3 Local similarity 20.3.4 SW results and comparisons with NS 20.3.4.1 The tail and variables stretching at long time 20.3.4.2 Comparison to NS simulation 20.3.5 The validity of the inertial approximation 20.3.6 Summary 21. Box models for 2D geometry 21.1 Fixed volume and inertial–buoyancy balance 21.2 S = 1, inflow volume change V = qtα 21.2.1 Inertial–buoyancy balance 21.2.2 Viscous–buoyancy balance 21.2.3 Critical α 22. Box models for axisymmetric geometry 22.1 Fixed volume and inertial–buoyancy balance 22.2 S = 1, inflow volume change V = qtα 22.2.1 Inertial–buoyancy balance 22.2.2 Viscous–buoyancy balance 22.2.3 Critical α 22.2.4 Some experimental support 23. Hybrid model for sustained axisymmetric intrusion 23.1 Non-rotating system 23.1.1 The steady-state domain (ri < r < rJ(t)) 23.1.2 The head region (rJ(t) < r < rN(t)) 23.1.2.1 Long-time asymptotes 23.1.3 Comparison of hybrid model to SW results 23.2 Rotating frame 23.2.1 SW components 23.2.1.1 Steady states 23.2.1.2 Temporal evolution: Numerical SW computations 23.2.2 The Coriolis Hybrid model 23.2.2.1 The head “box” 23.2.2.2 First stage 23.2.2.3 Second stage 23.2.2.4 Third stage: The Coriolis lens 23.2.3 Comparisons of hybrid model to SW results 23.2.4 Conclusions 24. Lubrication theory for viscous GCs with S = 1 24.1 2D geometry 24.1.1 Summary 24.2 Axisymmetric geometry 24.2.1 Summary 24.2.2 Some comparisons 25. Energy 25.1 Introduction 25.2 2D geometry 25.2.1 SW formulation 25.2.1.1 Energy and work 25.2.2 NS considerations 25.2.3 Results 25.3 Axisymmetric geometry 25.3.1 SW model 25.3.1.1 Energy and work 25.3.2 NS considerations 25.3.3 Results PART III : EXTENSIONS 26. Non-rectangular cross-section area 26.1 Introduction 26.2 Two-layer SW model for fixed top 26.2.1 Formulation 26.2.2 Fr interlude 26.2.3 Dimensionless variables and Re 26.2.4 The components of the flow-field solution 26.2.4.1 Characteristics and types of flow field 26.2.4.2 Internal jump 26.2.5 Critical nose speed and height 26.2.6 Remark on shapes, types, and open top 26.2.7 Example of dam-break solution 26.2.8 The reflected bore 26.2.9 Top current 26.3 One-layer Bq models 26.3.1 Formulation 26.3.2 The slumping stage 26.3.3 The general time-dependent flow 26.3.4 Similarity solution 26.3.4.1 The constant-influx case 26.3.5 Particle-driven current 26.4 Viscous regime 26.4.1 Typical flow 26.4.2 Transition xV 26.4.3 Influx V = qtδ and critical δ 26.5 Experimental support 26.6 Concluding remarks 26.7 Box model 26.7.1 Constant ρc current 26.7.2 Particle-driven current 27. SW model extensions: Flux and open top 27.1 Throughput flux in channel with fixed top 27.1.1 Dimensionless variables 27.1.2 The essentials of the flowfield solution 27.1.3 Some results 27.2 Open upper surface 27.2.1 Lock-release two-layer flow 27.2.2 One-layer reduction 27.3 Conclusion Appendix A Scaling suggestions Appendix B SW equations: Characteristics and finite-difference schemes B.1 Characteristics B.2 Numerical solution of the SW equations Appendix C NS numerical simulations C.1 Formulation C.2 A finite-difference code C.3 Other codes Appendix D Some useful formulas D.1 Leibniz’s Theorem D.2 Vectors and coordinate systems D.2.1 Cartesian coordinates x, y, z D.2.2 Cylindrical coordinates r, θ, z Notation guide Bibliography Index