Fundamentals of Fluid Mechanics: For Scientists and Engineers

Preface
Acknowledgements
About This Book
Contents
Part I Fluid Properties and Flow Models
1 Physical Concepts and Mathematical Tools
1.1 The Different Description Levels of a Fluid Motion
1.2 The Macroscopic Concept and Its Consequences
1.2.1 Validity of the Continuous Assumption
1.2.2 The Fluid Particle Concept
1.2.3 The Functions of the Flow's Macroscopic Description
1.3 The Physical Macroscopic Properties of a Fluid
1.3.1 The Notion of Fluid
1.3.2 Viscosity
1.3.3 Thermal Conductivity
1.3.4 Compressibility
1.3.5 Thermal Expansion and the Specific Heat Coefficients
1.4 The Mathematical Description of the Fluid Motion
1.4.1 Lagrangian Variables
1.4.2 Eulerian Variables
1.5 General Principles for the Study of a Fluid Motion
1.5.1 The Eulerian Viewpoint and the Control Volume
1.5.2 The Lagrangian Viewpoint and the Material Domain
1.6 The Material Derivation
1.6.1 Definition
1.6.2 The Material Derivative of a Scalar Function
1.6.3 The Material Derivative of a Vector Function
1.6.4 The Material Derivative of a Fluid Line Element
1.6.5 The Material Derivation of a Fluid Line Integral
1.6.6 The Material Derivative of the Circulation
1.6.7 The Material Derivative of a Volume Integral
1.6.8 The Material Derivative Related to Mass-Weighted Balances
1.6.9 The Transport Theorems of Reynolds-Rayleigh
1.6.10 The Case of a Stationary Motion
2 Flow Kinematics
2.1 Specific Lines and Surfaces of a Flow Field
2.1.1 The Streamline and Streamtube
2.1.2 The Pathline
2.1.3 The Streakline
2.1.4 The Fluid-line
2.2 Fluxes and Flow Rates
2.2.1 The General Flux Notion
2.2.2 Volume and Mass Flow Rates
2.2.3 The Volume Flow Rate and Stream Function
2.3 Small Displacement Kinematics
2.3.1 The Space Variation Analysis of a Vector Field
2.3.2 The Infinitesimal Motion Components of an Elementary Fluid Body
2.4 Vorticity and Vortex
2.4.1 The Vorticity Vector, the Vorticity Field
2.4.2 The Vorticity Line, Tube
2.4.3 The Vorticity Intensity. Circulation. Stokes' Theorem
2.4.4 The Vortex—Eddy
2.4.5 The Vortex Persistence
2.4.6 Helmholtz's Theorems
2.4.7 The Vortex Stretching
2.4.8 The Biot-Savart Law
2.5 The Application to Flow Visualization
2.5.1 General Information on Flow Visualization Techniques
2.5.2 Examples of Visualization by a Tracer
2.5.3 Examples of Visualization by Property
3 The Fundamental Balances of a Fluid Motion
3.1 Mass Conservation
3.1.1 The Global Formulation
3.1.2 The Local Expression
3.1.3 The Isovolume Condition and the Incompressible Fluid Assumption
3.2 The Motion Equations
3.2.1 The Momentum Equation in Integral Form
3.2.2 The Surface Force Analysis
3.3 The Energy Balance Equations
3.3.1 The Kinetic Energy Theorem
3.3.2 The First Law of Thermodynamics in a Fluid Motion
3.3.3 The Internal Energy Transport Equation
3.3.4 The Entropy Transport Equation
3.3.5 The Second Law of Thermodynamics in a Moving Fluid
3.4 The Eulerian Formulation of the Global Balance Equations
3.4.1 Mass Conservation Through a Stream Tube
3.4.2 The Global Momentum Balance: Euler's Theorem
3.4.3 The Kinetic Energy Balance and Head Loss
4 Fluid Motion Models
4.1 Chapter Objectives
4.2 The Current State in the Equation Formulation
4.3 The Newtonian Behaviour Schemes
4.3.1 The Dynamic Behaviour Scheme
4.3.2 The Thermal Behaviour Scheme
4.3.3 The Thermodynamic Equation of State
4.4 The Ideal Fluid Concept
4.4.1 The Intrinsic Irreversibilities in a Newtonian Fluid Motion
4.4.2 The Ideal Fluid Concept
4.5 The General Navier-Stokes Model
4.5.1 The Governing Equations
4.5.2 Initial and Boundary Conditions
4.6 The Simplified Models
4.6.1 The Concept of Restricted Models
4.6.2 The Real Compressible Fluid Flow Model
4.6.3 The Ideal Compressible Fluid Flow Model
4.6.4 The Real Incompressible Fluid Flow Model
4.6.5 The Ideal Incompressible Fluid Flow Model
4.7 The Characteristic Numbers of a Fluid Motion
4.7.1 The Dimensionless Formulation of the Local Equations
4.7.2 The Dimensionless Continuity Equation
4.7.3 The Dimensionless Momentum Equation
4.7.4 The Dimensionless Energy Equation
4.7.5 The Physical Interpretation of the Characteristic Numbers
4.7.6 Applications of the Dimensionless Analysis
4.7.7 The Vaschy-Buckingham or `π' Theorem
4.8 The Incompressible Flow Models According to the Reynolds Number
4.8.1 The Creeping Motion: Stokes' Model
4.8.2 The Infinite Reynolds Number Flow: Euler's Model
4.8.3 The Thin Shear Layer Laminar Flow: Prandtl's Model
5 Flow Classes
5.1 Introduction
5.2 The Irrotational Flow Class
5.2.1 Definition
5.2.2 The Velocity Potential
5.2.3 The Acceleration Potential
5.2.4 The Velocity Circulation Expression
5.2.5 The Barotropy Relation in an Ideal Fluid
5.3 The Isovolume Flow Class
5.3.1 Definition
5.3.2 The Velocity Vector Potential
5.3.3 The Two-Dimensional Case Restriction
5.4 The Stationary Flow Class
5.4.1 Definition
5.4.2 The Mass-Weighted Velocity Vector Potential
5.4.3 The Two-Dimensional Restriction
5.4.4 Flow Examples Defined by a Stream Function
5.5 The Irrotational and Isovolume Flow Class
5.5.1 Definition
5.5.2 The Kinematic Properties
5.5.3 The Dynamical Properties
5.5.4 The Energetic Properties
5.6 The Fundamental Local Theorems
5.6.1 Bernoulli's First Theorem for an Irrotational Flow
5.6.2 The Ideal Fluid Theorems
5.6.3 Summary
5.7 Examples of Local Theorem Application
5.7.1 Toricelli's Formula
5.7.2 Pitot and Prandtl Probes
5.7.3 Venturi's Flow Meter
5.7.4 The Isentropic Ejection Velocity of a Compressed Gas
Part II Ideal Fluid Motions
6 Irrotational 2D-Plane Motions of an Incompressible Fluid
6.1 The General Assumptions and Their Consequences
6.1.1 The Complex Potential
6.1.2 The Initial Condition
6.1.3 The Boundary Conditions
6.2 Some Mathematical Aspects of the Potential Flow Theory
6.2.1 The Direct and Inverse Problem
6.2.2 The Materialization Principle
6.2.3 The Superposition Method
6.2.4 The Hydrodynamic Image Method
6.2.5 The Conformal Mapping
6.3 Examples of Complex Potentials
6.3.1 Elementary Flows
6.3.2 Superposition Flow Examples
6.3.3 The Method of Images
6.4 Efforts of a Potential Flow on an Immersed Solid
6.4.1 The General Action of a Moving Fluid on a Closed Contour
6.4.2 Application to a Closed Streamline: Blasius' Formulas
6.4.3 Kutta-Joukowsky's Theorem for the Circular Cylinder
6.4.4 The General Theorem for a Logarithmic Singularity
6.4.5 Physical Interpretations
6.5 The Theory of Sharp Trailing Edge Airfoils
6.5.1 The Airfoil Shape Design
6.5.2 The Kutta Condition
6.5.3 The Adapted Circulation Around an Airfoil
6.5.4 The Lift of an Infinite Span Airfoil
6.5.5 The Kutta-Joukowsky Theory Relevance
7 Sound Wave Propagation and the Shock Phenomenon
7.1 The Propagation of Small Pressure Disturbances
7.1.1 The Propagation Speed in a Still Environment
7.1.2 Energetic Considerations
7.1.3 The Propagation from a Moving Source
7.2 Finite Amplitude Waves, Shock
7.2.1 The Propagation of Successive Compression Waves
7.2.2 The Successive Expansion Wave Propagation
7.3 The Plane Shock Wave
7.3.1 The Study Assumptions
7.3.2 The Local Equations for a Normal Shock Crossing
7.3.3 Prandtl's Formula for the Normal Shock
7.3.4 The Mach Number Relation on Either Side of a Shock
7.3.5 Static Value Variations Through a Normal Shock
7.3.6 Hugoniot's Relation Through a Normal Shock
7.3.7 The Entropy Variation and Stagnation Conditions
7.3.8 The Normal Shock Wave Characteristics for the Air
7.4 Oblique Shock Waves
7.4.1 The Physical Evidence
7.4.2 The Local Equations
7.4.3 Prandtl's Formula for the Oblique Shock
7.4.4 The Relations on Either Side of the Shock
7.4.5 The Shock Angle as a Function of the Deflection and Upstream Mach Number
7.4.6 Mach Numbers on Either Side of the Shock
7.4.7 Summary
7.5 Practical Considerations
7.5.1 The Shock Formation in External Aerodynamics
7.5.2 Wave Reflection—Shock-Boundary Layer Interaction
8 One-Dimensional Steady Flows of an Ideal Compressible Gas
8.1 Ideal Gas Flows in Slowly Varying Cross-Section Ducts
8.1.1 The Situation Under Consideration
8.1.2 The Isentropic Motion Equations
8.2 Flow Variation Laws with the Cross-Section Area
8.2.1 The Logarithmic Differential Formulation
8.2.2 Discussion and Physical Interpretation
8.3 The Isentropic Flow Laws
8.3.1 The Stagnation Condition Concept
8.3.2 Expressions Based on Stagnation Conditions
8.3.3 Expressions Based on Critical Conditions
8.3.4 Physical Discussion
8.4 Examples of Ideal Compressible Fluid Flows
8.4.1 The De Laval Nozzle
8.4.2 The Compressible Flow in a Venturi Nozzle
8.4.3 Pitot Probing in the Supersonic Regime
8.5 Ideal Gas Flow with Friction and Heat Transfer in a Constant Cross-Section Duct
8.5.1 The Adiabatic Flow with Friction: Fanno's Flow
8.5.2 The Frictionless Duct Flow with Heat Transfer, Rayleigh's Flow
Part III Real Fluid Motions
9 The Incompressible Navier-Stokes Model
9.1 Introduction
9.2 A Return to the Model Equations
9.2.1 The Velocity-Pressure Formulation
9.2.2 The Vorticity Equation
9.2.3 The Velocity-Potential Equation
9.2.4 The Two-Dimensional Plane Flow Case
9.2.5 Poisson's Equation for the Pressure
9.3 The Time-Based Analysis of the Momentum Balance
9.3.1 The Characteristic Time Scales of the Momentum Equation
9.3.2 Characteristic Numbers as Time Scale Ratios
9.3.3 The Advection/Diffusion Comparison: New Reynolds Number Interpretations
9.4 The Rotational Properties of a Fluid Motion
9.4.1 The Vortical Nature of the Flow on a Solid Body
9.4.2 The Elementary Mechanisms of the Vorticity Dynamics
9.4.3 The Vorticity-Velocity Interaction
9.5 Energy Properties
9.6 Flows as an ``Exact Solution'' of the Navier-Stokes Equations
9.6.1 The Plane Poiseuille Flow
9.6.2 Poiseuille's Flow in a Pipe
9.6.3 Couette's Axisymmetric Flow
9.6.4 The General Flow Between Two Parallel Plates
9.6.5 The Unsteady Flow on a Plane in Translation, Stokes' First Problem
9.7 Basic Flow Stability Concepts
9.7.1 Presenting the Problem
9.7.2 Some Examples of Flow Instabilities
9.8 Transition and Turbulence
9.8.1 The Flow Regime Transition and Turbulence
9.8.2 Some Specific Features of the Turbulent Regime
10 Very Low Reynolds Number Flows
10.1 Introduction
10.2 Stokes' Model
10.2.1 The Assumptions
10.2.2 The Velocity-Pressure Formulation of the Equations
10.2.3 The Pressure-Vorticity Equations
10.2.4 The Main Motion Properties
10.3 The Validity of Stokes' Model
10.4 Examples of Creeping Motions
10.4.1 The Flow in a Hele-Shaw Cell
10.4.2 Lubrication, Viscous Film Motion and Fluid Bearing
10.4.3 Stokes' Flow Past a Sphere
11 The Laminar Boundary Layer. Dynamic and Thermal Concepts
11.1 High Reynolds Number Flows of Real Fluids
11.1.1 The High Reynolds Number Flow Class
11.1.2 The Viscous Effect Localization in Incompressible High Reynolds Number Flows
11.1.3 The Advection-Diffusion Time-Equilibrium in an Isothermal Boundary Layer
11.1.4 The Convection-Diffusion Time-Equilibrium in the Thermal Boundary Layer
11.2 The Boundary Layer's Characteristic Parameters
11.2.1 Thicknesses
11.2.2 The Viscous Friction
11.2.3 The Heat Transfer
11.3 The Isovolume Boundary Layer Equations
11.3.1 The Specification of the Boundary Layer Configuration
11.3.2 The Dimensionless Form of the Local Equations
11.3.3 Discussion: Prandtl's Assumptions
11.3.4 Prandtl's Model
11.3.5 The Viscous–Inviscid Matching
11.3.6 The Boundary Layer Separation
11.4 Integral Equations
11.4.1 The Cross-Wise Velocity at the Boundary Layer's Free Edge
11.4.2 Von Kármán's Equation
11.4.3 The Global Mass Balance and Entrainment Velocity
11.4.4 The Global Momentum Balance
11.5 The Thermal Boundary Layer
11.5.1 The Outline of the Problem
11.5.2 Basic Assumptions for the Thermal Boundary Layer
11.5.3 The Thermal Boundary Layer Equations
11.5.4 Some Typical Thermal Boundary Layer Configurations
12 Boundary Layer Type Flows: Calculation Methods and Examples
12.1 The Analytical Calculation Methods for the Isovolume …
12.1.1 Solving the Local Equations
12.1.2 The Integral Method Based Calculation
12.2 Calculation Examples Related to the Isovolume Dynamic Boundary Layer
12.2.1 Flows as Exact Solutions of the Local Equations
12.2.2 Examples of Boundary Layer Calculations Using the Integral Method
12.3 The Thermal Boundary Layer on a Flat Plate
12.3.1 Free Convection on a Heated Vertical Flat Plate
12.3.2 Forced Convection in Low Speed Flows
12.3.3 Forced Convection in High Speed Isovolume Flows
12.3.4 Some Basics on the Compressible Boundary Layer at Pr = 1
12.4 The Two-Dimensional Thin Shear Layers and Free Shear Flows
12.4.1 The Thin Shear Layer Approximation in Free Flows
12.4.2 The Plane Free Jet
12.4.3 The Plane Wake
Appendix A Thermodynamics
A.1 General Background
A.2 The State Variables and Functions
A.3 The Equation of State
A.4 The Pressure, Temperature and Characteristic Coefficients
A.5 The First Law
A.6 The Second Law
A.7 The Irreversibility Sources
A.7.1 The Extrinsic Irreversibility
A.7.2 The Intrinsic Irreversibility
A.8 The Differential Relation to the Internal Energy and Entropy Specific Values
A.9 The Ideal Gas Assumption
A.10 The Incompressibility Assumption
Appendix B The Tenso-Vectorial Notations and Operators
B.1 General Conventions
B.2 Some Usual Symbols
B.3 The Main Operators
B.4 Some Expressions of Tenso-Vectorial Operators in Index Notation
Appendix C Vector Calculus Identities
Appendix D Vector Operator Expressions in Projection
D.1 Definitions and Notations
D.2 The Expression of the Various Operators
D.2.1 The Divergence of a Vector
D.2.2 The Laplacian of a Scalar Field
D.2.3 The Gradient of a Scalar Function
D.2.4 The Curl of a Vector Field
D.2.5 The Divergence of a Symmetrical Second-Order Tensor
D.2.6 The Gradient of Vector Field
D.2.7 The Material Derivative of a Scalar Function
Appendix E The Projections of the Navier-Stokes Equations in Various Coordinate Systems
E.1 The Cartesian Coordinates
E.2 The Cylindrical Coordinates
E.3 The Spherical Coordinates
Appendix List of Exercises
Appendix References
Index