This open access book allows the reader to grasp the main bulk of fluid flow problems at a brisk pace. Starting with the basic concepts of conservation laws developed using continuum mechanics, the incompressibility of a fluid is explained and modeled, leading to the famous Navier-Stokes equation that governs the dynamics of fluids. Some exact solutions for transient and steady-state cases in Cartesian and axisymmetric coordinates are proposed. A particular set of examples is associated with creeping or Stokes flows, where viscosity is the dominant physical phenomenon. Irrotational flows are treated by introducing complex variables. The use of the conformal mapping and the Joukowski transformation allows the treatment of the flow around an airfoil. The boundary layer theory corrects the earlier approach with the Prandtl equations, their solution for the case of a flat plate, and the von Karman integral equation. The instability of fluid flows is studied for parallel flows using the Orr-Sommerfeld equation. The stability of a circular Couette flow is also described. The book ends with the modeling of turbulence by the Reynolds-averaged Navier-Stokes equations and large-eddy simulations. Each chapter includes useful practice problems and their solutions.
The book is useful for engineers, physicists, and scientists interested in the fascinating field of fluid mechanics.
Author(s): Michel O. Deville
Publisher: Springer
Year: 2022
Language: English
Pages: 333
City: Cham
Foreword
Preface
Acknowledgments
Contents
1 Incompressible Newtonian Fluid Mechanics
1.1 Introduction
1.1.1 Circular Couette Flow
1.1.2 Flow Around a Cylinder
1.2 Fluid Kinematics
1.2.1 Material and Spatial Descriptions
1.2.2 Velocity, Material Derivative and Acceleration
1.2.3 Jacobian
1.2.4 Reynolds Transport Theorem
1.3 Velocity Gradient and Associated Tensors
1.4 Mass Conservation
1.5 Equation of Motion
1.6 Equation of Energy
1.7 Constitutive and State Equations
1.8 Incompressible Navier–Stokes Equations
1.9 Boundary and Initial Conditions
1.9.1 No Slip Wall
1.9.2 Interface
1.9.3 Laminar Free Surface
1.9.4 Perfect Fluid
1.10 Thermodynamics Considerations and Incompressibility
1.10.1 Compressible Fluid and Compressible Navier–Stokes Equations
1.10.2 Incompressibility
1.10.3 Boussinesq Approximation for Weakly Dilatable Fluids
1.11 The Method of Control Volume
Exercises
3 Exact Solutions of the Navier–Stokes Equations
3.1 Plane Stationary Flows
3.1.1 Plane Couette Flow
3.1.2 Plane Poiseuille Flow
3.1.3 Flow of an Incompressible Fluid on an Inclined Plane
3.2 Axisymmetric Stationary Flows
3.2.1 Circular Couette Flow
3.2.2 Circular Poiseuille Flow in a Cylindrical Pipe
3.2.3 Helical Flow Between Two Circular Cylinders in Relative Motion
3.3 Plane Transient Flows
3.3.1 Transient Flow in a Semi-infinite Space
3.3.2 Flow on an Oscillating Plane
3.3.3 Channel Flow with a Pulsatile Pressure Gradient
3.4 Axisymmetric Transient Flows
3.4.1 Starting Transient Poiseuille Flow
3.4.2 Pulsating Flow in a Circular Pipe
3.5 Plane Periodic Solutions
3.6 Pipe Flow
3.6.1 Polynomial solutions
3.6.2 The Rectangular Pipe
Exercises
4 Vorticity and Vortex Kinematics
4.1 Kinematic Considerations
4.2 Dynamic Vorticity Equation
4.2.1 General Equation
4.2.2 Physical Interpretation of Vorticity Dynamics for the Incompressible Perfect Fluid
4.2.3 The Vorticity Number
4.3 Vorticity Equation for a Viscous Newtonian Fluid
4.4 Circulation Equation
4.5 Vorticity Equation for a Perfect Fluid
4.6 Bernoulli's Equation
4.7 Vorticity Production on a Solid Wall
4.8 Flow Behind a Grid
4.9 Taylor–Green Vortex
Exercises
5 Stokes Flow
5.1 Plane Creeping Flows
5.1.1 Flow in a Corner
5.2 Two-Dimensional Corner Moffatt Eddies
5.2.1 Real Solutions for lamdaλ (alpha greater than 73.15 Superscript degreesα>73.15°)
5.2.2 Complex Solutions for lamdaλ (alpha less than 73.15 Superscript degreesα<73.15°)
5.3 Stokes Eigenmodes
5.3.1 Periodic Stokes Eigenmodes
5.3.2 Channel Flow Stokes Eigenmodes
5.4 Parallel Flow Around a Sphere
5.4.1 Oseen's Improvement
5.5 Parallel Flow Around a Cylinder
5.6 Three-Dimensional Stokes Solution
Exercises
6 Plane Irrotational Flows of Perfect Fluid
6.1 Complex Velocity
6.2 Complex Circulation Γ
6.3 Elementary Complex Potential Flows
6.3.1 Parallel Homogeneous Flow
6.3.2 Vortex and Source
6.3.3 Complex Potential in Power of zz
6.4 Flow Around a Circular Cylinder
6.4.1 Flow Without Circulation Around a Cylinder
6.4.2 Flow with Circulation Around a Cylinder
6.5 Blasius Theorem: Forces and Moment
6.6 The Method of Conformal Transformation
6.6.1 A Few Properties of the Conformal Transformation
6.6.2 Application to Potential Flows
6.7 Schwarz-Christoffel Transformation
6.7.1 Mapping of a Semi-infinite Strip
6.7.2 Mapping of a Plane Channel
6.7.3 Schwarz-Christoffel Transformation of a Converging Channel
6.8 Joukowski Transformation
6.8.1 Flow over a Flat Plate
6.8.2 Joukowski Profiles
Exercises
7 Boundary Layer
7.1 The Equations of the Laminar Boundary Layer
7.1.1 Dimensional Analysis
7.1.2 Prandtl's Equations
7.2 Boundary Layer on a Flat Plate
7.2.1 Solution of Prandtl's Equations
7.2.2 Boundary Layer Thicknesses
7.2.3 Friction and Drag Coefficients
7.3 von Kármán Integral Equation
7.4 von Kármán-Pohlhausen Approximate Method
Exercises
8 Instability
8.1 Transition
8.2 Orr-Sommerfeld Equation
8.3 Stability of the Circular Couette Flow
8.3.1 Rayleigh's Criterion
8.3.2 Linear Stability of Viscous Circular Couette Flow
8.3.3 Non-linear Axisymmetric Taylor Vortices
Exercises
9 Turbulence
9.1 General Considerations
9.2 General Equations of Incompressible Turbulence
9.3 Kinetic Energy
9.4 Dynamic Equation of the Reynolds Tensor
9.5 Structures and Scales of Homogeneous Turbulence
9.6 Homogeneous Turbulence
9.6.1 Correlations and Spectra
9.6.2 Velocity Correlations and Associated Spectra
9.6.3 Correlations and Spectra in Isotropic Turbulence
9.7 Fourier Spectral Solution
9.8 Linear Turbulence Models
9.8.1 Zero Equation Model
9.8.2 Turbulent Flow in a Plane Channel
9.8.3 The Logarithmic Velocity Profile
9.9 The One-Equation Model: the MathID371K-ell Model
9.10 The Two-Equation Models
9.10.1 MathID396K-ε Model
9.10.2 MathID418K-ω Model
9.11 Non-linear Turbulence Models
9.11.1 Anisotropy Tensor
9.11.2 Dynamic Equation for the Anisotropy Tensor
9.12 Reynolds Stress Tensor Representation Using Integrity Bases
9.13 Large Eddy Simulation
9.13.1 Definitions
9.13.2 LES Equations
9.13.3 The Smagorinsky Model
9.13.4 The Dynamic Model
9.13.5 The Dynamic Mixed Model
9.13.6 The Approximate Deconvolution Method
9.14 Concluding Remarks
Exercises
10 Solutions of Exercises
10.1 Chapter One
10.2 Chapter Two
10.3 Chapter Three
10.4 Chapter Four
10.5 Chapter Five
10.6 Chapter Six
10.7 Chapter Seven
10.8 Chapter Eight
10.9 Chapter Nine
Appendix A Cylindrical Coordinates
Appendix B Spherical Coordinates
Appendix C Bessel Functions
Appendix D Fortran Programme for the Orr–Sommerfeld Equation
Appendix E Figures Credits
Appendix References
Index
