Incompressible Fluid Dynamics

Cover
Incompressible Fluid Dynamics
Copyright
Dedication
Preface
Contents
Prologue
1: Elementary Definitions, Some Simple Kinematics, and the Dynamics of Ideal Fluids
	1.1 Elementary Definitions
		1.1.1 What is the Mechanical Definition of a Fluid?
		1.1.2 Fluid Statics and One Definition of Pressure
		1.1.3 Different Categories of Fluid and of Fluid Flow
	1.2 Some Simple Kinematics
		1.2.1 Eulerian Versus Lagrangian Descriptions of Motion
		1.2.2 The Convective Derivative
		1.2.3 Mass Conservation and the Streamfunction
	1.3 The Dynamics of an Ideal (Inviscid) Fluid of Uniform Density
		1.3.1 Euler’s Equation for a Fluid of Uniform Density
		1.3.2 Bernoulli’s Equation and Mechanical Energy Conservation
		1.3.3 Some Applications of Bernoulli’s Equation
		1.3.4 Inviscid Momentum Conservation in Integral Form
		1.3.5 Examples of the Use of Momentum Conservation to Calculate Pressure Forces
		1.3.6 More on Momentum Conservation: Inviscid Flow Through a Cascade of Blades
	1.4 Examples of the Failure of Ideal Fluid Mechanics
		1.4.1 The Borda–Carnot ‘Head Loss’ in a Sudden Pipe Expansion
		1.4.2 The Hydraulic Jump
	Exercises
	References
2: Governing Equations and Flow Regimes for a Real Fluid
	2.1 Viscosity, Viscous Stresses, and the No-slip Boundary Condition
	2.2 More Kinematics: Characterizing the Deformation and Spin of Fluid Elements
		2.2.1 Two Things that Happen to a Fluid Element as it Slides down a Streamline
		2.2.2 The Rate-of-strain Tensor and the Deformation of Fluid Elements
		2.2.3 Vorticity: the Intrinsic Spin of Fluid Elements
	2.3 Dynamics at Last: the Stress Tensor and Cauchy’s Equation of Motion
	2.4 The Navier–Stokes Equation
		2.4.1 Newton’s Law of Viscosity
		2.4.2 The Navier–Stokes Equation and the Reynolds Number
		2.4.3 Navier–Stokes as an Evolution Equation for the Velocity Field
		2.4.4 The Viscous Dissipation of Mechanical Energy
	2.5 The Momentum Equation for Viscous Flow in Integral Form
	2.6 The Role of Boundaries and Prandtl’s Boundary-layer Equation
		2.6.1 The Need for Boundary Layers at High Reynolds Number
		2.6.2 Changes in Flow Regime as the Reynolds Number Increases
	2.7 From Linear to Angular Velocity: Vorticity and its Evolution Equation
		2.7.1 The Biot–Savart Law Applied to Vorticity: an Analogy with Magnetostatics
		2.7.2 The Vorticity Evolution Equation
		2.7.3 Where does the Vorticity come from?
		2.7.4 Enstrophy and its Governing Equation
		2.7.5 A Glimpse at Potential (Vorticity Free) Flow and its Limitations
	2.8 Summing up: Real versus Ideal Fluid Mechanics
	Exercises
	References
3: Some Elementary Solutions of the Navier–Stokes Equation
	3.1 Some Simple Laminar Flows
		3.1.1 Planar Viscous Flow
		3.1.2 The Boundary Layer near a Two-dimensional Stagnation Point
	3.2 The Diffusion of Vorticity from a Moving Surface
		3.2.1 The Impulsively Started Plate: Stokes’ First Problem
		3.2.2 The Oscillating Plate: Stokes’ Second Problem
	3.3 The Navier–Stokes Equation in Cylindrical Polar Coordinates
		3.3.1 Moving from Cartesian to Cylindrical Polar Coordinates
		3.3.2 Hagen–Poiseuille Flow in a Pipe
		3.3.3 Rotating Couette Flow
		3.3.4 The Diffusion of a Long, Thin, Cylindrical Vortex
		3.3.5 A Thin Film on a Spinning Disc
		3.3.6 The Azimuthal–poloidal Decomposition of Axisymmetric Flows
	Exercises
	References
4: Flows with Negligible Inertia: Stokes Flow, Lubrication Theory, and Thin Films
	4.1 Motion at Low Reynolds Number: Stokes Flow
		4.1.1 The Governing Equations at Low Reynolds Number
		4.1.2 Flow past a Sphere at Low Reynolds Number
		4.1.3 The Oseen Correction for Flow over a Sphere at Low Re
		4.1.4 The Uniqueness and Minimum Dissipation Theorems for Low-Re Flows
		4.1.5 Two-dimensional Flow in a Wedge at Low Reynolds Number
		4.1.6 Suspensions
		4.1.7 The Subtleties of Self-propulsion at Low Reynolds Number
	4.2 Lubrication Theory
		4.2.1 The Approximations and Governing Equations of Lubrication Theory
		4.2.2 Reynolds’ Analysis of the Slipper Bearing
		4.2.3 Sommerfeld’s Analysis of the Journal Bearing
		4.2.4 Rayleigh’s Analysis of the Stepped Bearing
	4.3 Thin Films with a Free Surface
		4.3.1 Approximations and Governing Equations
		4.3.2 The Gravity-driven Spreading of a Circular Pool
		4.3.3 A Film on an Incline
		4.3.4 A Thin Film on a Rotating Disc (Reprise)
	Exercises
	References
5: Laminar Flow at High Reynolds Number: Boundary Layers
	5.1 Prandtl’s Boundary Layer and a Revolution in Fluid Dynamics
	5.2 The Archetypal Boundary Layer: a Flat Plate Aligned with a Uniform Flow
	5.3 A Generalization of Prandtl’s Boundary Layer to Other Physical Systems
		5.3.1 A Popular Model Problem and the Concept of Matched Asymptotic Expansions
		5.3.2 Prandtl’s Generalization of the Boundary Layer: Another Model Problem
	5.4 The Effects of an Accelerating External Flow on Boundary-layer Development
		5.4.1 The Falkner–Skan Solutions for Flow over a Two-dimensional Wedge
		5.4.2 The Boundary Layer near the Forward Stagnation Point of a Circular Cylinder
	5.5 Jeffery–Hamel Flow in a Convergent or Divergent Channel
	5.6 Boundary-layer Separation and Pressure Drag
	5.7 Thermal Boundary Layers
		5.7.1 Forced Convection
		5.7.2 Free Convection
	5.8 Submerged Laminar Jets
		5.8.1 The Two-dimensional Jet
		5.8.2 The Axisymmetric Jet
	Exercises
	References
6: Potential Flow Theory with Applications to Aerodynamics
	6.1 Some Elementary Ideas in Potential Flow Theory
		6.1.1 The Physical Basis for, and Dangers of, Potential Flow Theory
		6.1.2 The Retrospective Application of Newton’s Second Law: Bernoulli Revisited
		6.1.3 Some Simple Examples of Two-dimensional Potential Flow
		6.1.4 D’Alembert’s Paradox
	6.2 The Kinematics of Two-dimensional Potential Flow
		6.2.1 The Complex Potential
		6.2.2 Some Elementary Examples of the Complex Potential
		6.2.3 Flow Normal to a Flat Plate of Finite Width
		6.2.4 A Not so Simple Example: the Intake to a Submerged Duct
		6.2.5 The Method of Images for Plane and Cylindrical Boundaries
	6.3 The Lift Force Exerted on a Body by a Uniform Incident Flow
		6.3.1 Two-dimensional Flow over a Cylinder with Circulation: an Illustrative Example
		6.3.2 Flow over a Planar Body of Arbitrary Shape: the Kutta–Joukowski Lift Theorem
		6.3.3 Kelvin’s Circulation Theorem
		6.3.4 The Role of Boundary-layer Vorticity in Establishing Circulation round an Aerofoil
		6.3.5 The Lift Generated by a Slender Aerofoil
	Exercises
	References
7: Surface Gravity Waves in Deep and Shallow Water
	7.1 The Wave Equation and Dispersive versus Non-dispersive Waves
		7.1.1 The Wave Equation and d’Alembert’s Solution
		7.1.2 Two Classes of Waves: Dispersive versus Non-dispersive Waves
	7.2 Two-dimensional Surface Gravity Waves of Small Amplitude
		7.2.1 Surface Gravity Waves on Water of Arbitrary Depth
		7.2.2 Shallow-water and Deep-water Waves
		7.2.3 Particle Paths, Stokes Drift, and Energy Density in Deep-water Waves
		7.2.4 Wave Drag in Deep Water
	7.3 The General Theory of Dispersive Waves
		7.3.1 Dispersion, Wave Packets, and the Group Velocity
		7.3.2 The Energy Flux in a Wave Packet
	7.4 The Dispersion of Small-amplitude Surface Gravity Waves
		7.4.1 The Group Velocity and Energy Density for Waves on Water of Arbitrary Depth
		7.4.2 Waves Approaching a Beach
		7.4.3 The Influence of Surface Tension on Dispersion
	7.5 Finite-amplitude Waves in Shallow Water
		7.5.1 The Inviscid Shallow-water Equations
		7.5.2 Finite-amplitude Waves and Non-linear Wave Steepening
		7.5.3 The Solitary Wave 1: Rayleigh’s Solution
		7.5.4 Solitary Waves 2: The KdeV Equation
		7.5.5 More General Solutions of the KdeV Equation: Cnoidal Waves
		7.5.6 The Hydraulic Jump Revisited
	Exercises
	References
8: Vortex Dynamics: Classical Theory and Illustrative Examples
	8.1 Vorticity and its Evolution Equation (Revisited)
	8.2 Inviscid Vortex Dynamics
		8.2.1 The Classical Theories of Helmholtz and Kelvin
		8.2.2 Helicity and its Conservation
		8.2.3 Steady, Axisymmetric Flows and the Squire–Long Equation
		8.2.4 Viscous versus Inviscid Vortex Dynamics
	8.3 A Qualitative Overview of some Simple Isolated Vortices
		8.3.1 The Interaction of Line Vortices
		8.3.2 A Glimpse at Vortex Rings
		8.3.3 Vortices due to Boundary-layer Separation
		8.3.4 Columnar Vortices in the Atmosphere and Oceans
	8.4 Viscous Vortex Dynamics I: the Prandtl–Batchelor Theorem
		8.4.1 The Physical Origins of the Prandtl–Batchelor Theorem
		8.4.2 A Proof of the Theorem
	8.5 Viscous Vortex Dynamics II: Burgers’ Vortex
		8.5.1 A Dilemma in Turbulence: Finite Energy Dissipation for Vanishing Viscosity
		8.5.2 Burgers’ Axisymmetric Vortex
		8.5.3 The Robust Nature of Burgers’ Vortex
	8.6 More Axisymmetric Vortices (both Viscous and Inviscid)
		8.6.1 Hill’s Spherical Vortex
		8.6.2 The Velocity Field and Kinetic Energy of a Thin Vortex Ring
	8.7 Viscous Vortex Dynamics III: the Impulse of Localized Vorticity Fields
		8.7.1 The Far field of a Localized Vorticity Distribution
		8.7.2 The Spontaneous Redistribution of Momentum in Space
		8.7.3 Conservation of Linear Impulse and its Relationship to Linear Momentum
		8.7.4 Conservation of Angular Impulse and its Relationship to Angular Momentum
		8.7.5 Axisymmetric Examples of Impulse and Vortex Rings Revisited
	Exercises
	References
9: Waves and Flow in a Stratified Fluid
	9.1 The Boussinesq Approximation and a Second Definition of the Froude Number
	9.2 The Suppression of Vertical Motion: a Simple Scaling Analysis
	9.3 The Phenomenon of Blocking
	9.4 Lee Waves
		9.4.1 Linear Lee Waves in Two Dimensions
		9.4.2 Finite-amplitude Lee Waves in Two Dimensions
	9.5 Internal Gravity Waves of Small Amplitude
		9.5.1 Linear Theory and Simple Examples
		9.5.2 The Reflection of Internal Gravity Waves
	9.6 Generalized Vortex Dynamics: Bjerknes’ Theorem and Ertel’s Potential Vorticity
	Exercises
	References
10: Waves and Flow in a Rotating Fluid
	10.1 Rayleigh’s Stability Criterion for Inviscid, Swirling Flow
	10.2 The Equations of Motion in a Rotating Frame of Reference
		10.2.1 The Coriolis Force and the Rossby Number
		10.2.2 Rapid Rotation: the Taylor–Proudman Theorem and Drifting Taylor Columns
	10.3 Inertial Waves of Small Amplitude
		10.3.1 Their Dispersion Relationship, Group Velocity, and Spatial Structure
		10.3.2 The Formation of Transient Taylor Columns by Low-frequency Waves
		10.3.3 The Spontaneous Focussing of Inertial Waves and the Formation of Columnar Vortices
		10.3.4 Helicity Generation and Helicity Segregation by Inertial Waves
		10.3.5 Finite-amplitude Inertial Waves
	10.4 Rossby Waves
	10.5 Ekman Boundary Layers and Ekman Pumping
		10.5.1 Confined Swirling Flows: the Solutions of Kármán, Bödewadt, and Ekman
		10.5.2 Ekman Layers as a Mechanism for Energy Dissipation
	10.6 Tropical Cyclones
		10.6.1 The Anatomy of a Tropical Cyclone
		10.6.2 A Simple Model of a ‘Dry’ Cyclone
	Exercises
	References
11: Instability
	11.1 The Centrifugal Instability
		11.1.1 Rayleigh’s Inviscid Criterion for Axisymmetric Disturbances
		11.1.2 Two-dimensional Inviscid Disturbances (Rayleigh again)
		11.1.3 Viscous Instability and Taylor’s Analysis
		11.1.4 The Experimental Evidence
	11.2 The Stability of a Fluid Heated from Below
		11.2.1 Rayleigh–Bénard Convection
		11.2.2 Rayleigh’s Stability Analysis
		11.2.3 Slip Boundaries Top and Bottom: an Artificial but Informative Case
		11.2.4 No-slip Boundaries
	11.3 The Stability of Parallel Shear Flows
		11.3.1 Rayleigh’s Inflection Point Theorem for Inviscid, Rectilinear Flow
		11.3.2 The Subtle Effects of Viscosity
	11.4 The Kelvin–Helmholtz Instability
		11.4.1 The Instability of an Inviscid Vortex Sheet
		11.4.2 The Inviscid Instability of a Layer of Vorticity of Finite Thickness
	11.5 The Stability of Continuously Stratified Shear Flow
		11.5.1 The Taylor–Goldstein Equation for Fluctuations in a Stratified Shear Flow
		11.5.2 The Richardson Number Criterion for the Stability of a Stratified Shear Flow
		11.5.3 An Interpretation of the Stability Criterion in terms of Energy
	11.6 The Kelvin–Arnold Variational Principle for Inviscid Flows
		11.6.1 A Statement of the Theorem
		11.6.2 A Derivation of the Theorem
		11.6.3 Some Simple Applications of the Theorem
	11.7 A Variational Principle for Inviscid Flows based on the Lagrangian
	11.8 The Stability of Pipe Flow: a Qualitative Discussion
	Exercises
	References
12: The Transition to Turbulence and the Nature of Chaos
	12.1 Some Common Themes in the Transition to Turbulence
	12.2 A Definition of Turbulence
	12.3 The Nature of Chaos: the Logistic Map as an Example
	12.4 Landau’s Inspired (but Incomplete) Vision of the Transition to Turbulence
	Exercises
	References
13: An Introduction to Turbulence and to Kolmogorov’s Theory
	13.1 Elementary Properties of Turbulence: a Qualitative Overview
		13.1.1 The Need for a Statistical Approach and the Problem of Closure
		13.1.2 The Various Stages of Development of Freely Decaying Turbulence
		13.1.3 Richardson’s Energy Cascade
		13.1.4 The Rate of Destruction of Energy and an Estimate of Kolmogorov’s Microscales
	13.2 A Digression into the Kinematics of Homogeneous Turbulence
		13.2.1 Two Useful Diagnostic Tools: Correlation Functions and Structure Functions
		13.2.2 The Simplifications of Isotropy and the Taylor Scale
		13.2.3 Scale-by-scale Energy Distributions in Fourier Space: the Energy Spectrum
		13.2.4 Relating Real-space and Spectral-space Estimates of the Energy Distribution
		13.2.5 A Common Error in the Interpretation of Energy Spectra
	13.3 Kolmogorov’s Universal Equilibrium Theory of the Small Scales (K41)
		13.3.1 Does Small-scale Turbulence have a Universal, Isotropic Structure at Large Re?
		13.3.2 Kolmogorov’s Universal Equilibrium Theory: the Two-thirds and Five-thirds Laws
		13.3.3 The Kármán–Howarth Equation
		13.3.4 Kolmogorov’s Four-fifths Law
		13.3.5 Obukhov’s Constant Skewness Closure Model
	13.4 Subsequent Refinements to K41
		13.4.1 Landau’s Objection to K41 Based on Large-scale Intermittency of the Dissipation
		13.4.2 Kolmogorov’s 1961 Refinement of K41 based on Inertial-range Intermittency
	13.5 The Probability Distribution of the Velocity Field
		13.5.1 The Skewness and Flatness Factors
		13.5.2 The Flatness Factor as a Measure of Intermittency
		13.5.3 The Skewness Factor as a Measure of Enstrophy Production
	Exercises
	References
14: Turbulent Shear Flows and Simple Closure Models
	14.1 Reynolds Stresses, Energy Budgets, and the Concept of Eddy Viscosity
		14.1.1 Reynolds Stresses and the Closure Problem (Reprise)
		14.1.2 The Eddy Viscosity Model of Boussinesq, Taylor, and Prandtl
	14.2 The Transfer of Energy from the Mean Flow to the Turbulence
	14.3 Turbulent Jets
		14.3.1 The Plane Jet
		14.3.2 The Round Jet
	14.4 Turbulent Flow near a Smooth Boundary: the Log-law of the Wall
		14.4.1 The Log-law of the Wall in Channel Flow
		14.4.2 The Log-law and Viscous Sublayer for Other Smooth-walled Flows
		14.4.3 Inactive Motion: a Problem for the Universality of the Log-law?
		14.4.4 Energy Balances and Structure Functions in the Log-law Layer
		14.4.5 Coherent Structures and Near-wall Cycles
		14.4.6 Turbulent Heat Transfer near a Surface and the Log-law for Temperature
	14.5 The Influence of Surface Roughness and Stratification on Turbulent Shear Flow
		14.5.1 The Log-law for Flow over a Rough Surface
		14.5.2 The Atmospheric Boundary Layer, Stratification, and the Flux Richardson Number
		14.5.3 Prandtl’s Weak-shear Model of the Atmospheric Boundary Layer
		14.5.4 The Monin–Obukhov Theory of the Atmospheric Boundary Layer
	14.6 Closure Models for Turbulent Shear Flows: the k-" Model as an Example
		14.6.1 The Basis of the k-" Closure Model
		14.6.2 The k-" Model applied to Some Simple Turbulent Flows
	Exercises
	References
Appendices
	1: Dimensional Analysis
		References
	2: Vector Identities and Theorems
	3: Navier–Stokes Equation in Cylindrical Polar Coordinates
	4: The Fourier Transform
		References
	5: The Physical Properties of Some Common Fluids
Index