Dynamics of Water Surface Flows and Waves

Description

Dynamics of Water Surface Flows and Waves provides theoretical descriptions of the whole life of water surface waves through their birth, propagation, evolution and finally breaking. While initial capillary waves are created via instability at air-water interfaces, potential wave theories adequately describe interactions of waves with current, bathymetry and structure. In the final breaking stage, potential fluid motions in the waves rapidly evolve into vortical turbulent flows that disturb the surfaces, resulting in entrainment of air-bubbles and ejection of sea spray in bursting bubbles floating on the surface. All theories and analytical methods required to understand the series of wave processes, over diverse areas of subjects, including turbulence, diffusion, vortex and capillary dynamics, shallow water approach, and stability analysis, as well as the conventional potential wave theory, are comprehensively covered in this book. All of the mathematical formulas are consistently developed from theorems and linked with physics, which provides theoretical understanding and further interest in wave dynamics. This is an ideal graduate-level textbook or reference for engineers and researchers in the fields of fluid and wave mechanics, coastal and ocean engineering. Cover Half Title Title Page Copyright Page Contents Preface Author 1. Introduction of Fluid Mechanics 1.1. Mathematical representation of flows 1.1.1. Coordinates and vector calculus 1.1.1.1. Cartesian coordinates 1.1.1.2. Cylindrical coordinates 1.1.1.3. Spherical coordinates 1.1.2. Index notation 1.1.3. Trigonometric and hyperbolic functions 1.1.4. Stresses acting on fluid 1.1.4.1. Inertial force 1.1.4.2. Pressure and viscous stress 1.1.5. Reynolds transport theorem 1.2. Mass conservation 1.3. Momentum conservation 1.3.1. Kinematic energy 1.3.2. Dimensionless numbers 1.3.3. Wall boundary conditions 1.3.4. Boundary layer flows 1.3.4.1. Flow between two horizontal plates (Couette flow) 1.3.4.2. Hagen-Poiseuille flow 1.3.4.3. Flow above an oscillating plate 1.4. Velocity potential 1.5. Stream function 1.6. Potential flows 1.6.1. Uniform flow 1.6.2. Source and sink 1.6.3. Doublet 1.6.4. Uniform flow across a sphere 1.7. Bernoulli equation 1.7.1. Bernoulli equation on a streamline 1.7.2. Bernoulli equation for unsteady irrotational flow 1.8. Stokes drag 1.8.1. Stokes law 1.8.2. Fall/rise velocity of a spherical particle 2. Turbulence and Diffusion 2.1. Turbulence 2.1.1. Turbulent statistics 2.1.1.1. Kolmogorov's hypothesis of local isotropy 2.1.1.2. Kolmogorov's first similarity hypothesis 2.1.1.3. Kolmogorov's second similarity hypothesis 2.1.2. Reynolds decomposition 2.1.3. Turbulent energy and dissipation 2.1.4. Turbulent boundary layer 2.2. Diffusion 2.2.1. Molecular and convective diffusion 2.2.2. Solution of diffusion equation 2.2.2.1. Mass transfer across a spherical bubble 2.2.2.2. Higbie penetration model 2.2.3. Effects of turbulence 2.2.4. Turbulent mass boundary layer 2.2.5. Immiscible particles 2.2.5.1. Homogeneous turbulence 2.2.5.2. Turbulent boundary layer 3. Surface and Vorticity Dynamics 3.1. Mathematical descriptions of a surface form 3.1.1. Arbitrary surface forms 3.1.2. Surface tension 3.2. Boundary conditions at free-surface/interface 3.2.1. Kinematic boundary condition 3.2.2. Dynamic boundary condition 3.3. Vorticity 3.3.1. Vorticity equation 3.3.2. Vorticity in turbulence 3.3.3. Rankine vortex 3.3.4. Batchelor's vortex pair 3.4. Vorticity dynamics 3.4.1. Biot-Savart law 3.4.2. Point vortex 3.4.3. Pair vortices 3.4.3.1. Counter-rotating pair 3.4.3.2. Corotating pair 3.4.3.3. Rotational behavior 3.5. Surface–vortex interactions 3.5.1. Vorticity on curved surfaces 3.5.2. Formation of scars 3.5.3. Vortex ring 3.6. Bubbles, foams, and drops 3.6.1. Motion of a particle 3.6.2. Particle drag 3.6.3. Bubble swarms 3.6.4. Foam 3.6.4.1. Meniscus 3.6.4.2. Surface form around the floating bubble 3.6.4.3. Note on pressure in a bubble 4. Linear Wave Theory 4.1. The Laplace equation 4.2. Linear boundary conditions 4.3. Progressive waves 4.4. Waves on thin sheets of fluid 4.5. Standing waves 4.6. Planar wave propagation 4.7. Free oscillation of water surfaces in containers 4.7.1. Oscillation in a rectangle container 4.7.1.1. Two-dimensional tank 4.7.1.2. Three-dimensional tank 4.7.2. Oscillation in a cylindrical container 4.8. Evanescent waves 4.9. Edge waves 5. Shallow Water Equation 5.1. Derivation of shallow water equation 5.1.1. Continuity equation 5.1.2. Momentum equation 5.2. Linear shallow water waves 5.3. Method of characteristics 5.4. Planar shallow water waves 5.4.1. Seiching 5.4.2. Edge waves 5.5. Flows in a rotating system 5.5.1. Poincare waves 5.5.2. Kelvin waves 5.6. Tsunami 5.6.1. Shallow water equation in a spherical coordinate 5.6.2. Generation of tsunami 5.6.3. The 2011 Tohoku tsunami 6. Stability of Flows 6.1. Gas-liquid two-layer flow 6.1.1. Solution of the flow 6.1.2. Stability 6.2. Breakup of liquid sheets 6.2.1. Antisymmetric waves 6.2.2. Symmetric waves 6.3. Capillary instability on cylindrical jets 6.3.1. Solutions for perturbations 6.3.2. Linearized boundary conditions 6.3.3. Eigenvalue equation 6.4. Crown splash 6.4.1. Governing equation 6.4.2. Solutions for perturbations 7. Ocean Waves 7.1. Properties of ocean waves 7.1.1. Propagation of waves 7.1.2. Refraction 7.1.3. Approximation of the dispersion relation 7.1.4. Mass transport 7.1.5. Wave energy 7.1.6. Energy flux 7.1.7. Wave group 7.1.8. Shoaling 7.2. Evolution of wave group 7.3. Wave–current interactions 7.3.1. Refraction owing to currents 7.3.2. Wave height variation in a current field 7.4. Irregular waves 7.4.1. Statistical representation of wave height 7.4.2. Wave spectrum 7.4.3. Standard wave spectra 7.4.3.1. Pierson-Moskowitz spectrum 7.4.3.2. JONSWAP (Joint North Sea Wave Observation Project) spectrum 7.4.3.3. Bretschneider-Mitsuyasu spectrum 7.4.4. Prediction of wave spectrum 7.5. Stokes wave 7.5.1. Perturbation method 7.5.2. Second-order solutions 8. Breaking Wave Dynamics 8.1. Shoaling wave breaking 8.1.1. Breaker type 8.1.1.1. Surf similarity parameter 8.1.2. Breaker limits 8.2. Turbulence and vortices in breaking waves 8.2.1. Vortex structures 8.2.2. Surface–vortex interactions 8.2.3. Vortex-induced suspension of sediment 8.3. Bubbles, foams, and sea sprays 8.3.1. Generation and entrainment of bubbles 8.3.2. Bubble size distributions 8.3.3. Gas transfer from bubbles 8.3.4. Whitecapping 8.3.5. Sea spray and marine aerosol 8.4. Air–sea interactions 8.4.1. Friction at sea surfaces 8.4.2. Gas exchanges across sea surfaces A. Appendix A.1. Vector formulas A.2. Hyperbolic functions A.2.1. Definitions A.2.2. Derivatives A.2.3. Useful relations A.3. Gauss' divergence theorem A.4. Delta function A.5. Stokes' theorem A.6. Green function A.7. Leibniz rule A.8. Bessel functions A.8.1. Bessel function A.8.2. Modified Bessel function Bibliography Index