Introductory Fluid Mechanics for Physicists and Mathematicians E-Book

Geoffrey J. Pert

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 691

Veröffentlichungsjahr: 2013

Beschreibung

This textbook presents essential methodology for physicists of the theory and applications of fluid mechanics within a single volume.  Building steadily through a syllabus, it will be relevant to almost all undergraduate physics degrees which include an option on hydrodynamics, or a course in which hydrodynamics figures prominently.

Leseprobe

Table of Contents

Series page

Title Page

Copyright

Dedication

Preface

Chapter 1: Introduction

1.1 Fluids as a State of Matter

1.2 The Fundamental Equations for Flow of a Dissipationless Fluid

1.3 Lagrangian Frame

1.4 Eulerian Frame

1.5 Hydrostatics

1.6 Streamlines

1.7 Bernoulli's Equation: Weak Form

1.8 Polytropic Gases

Chapter 2: Flow of Ideal Fluids

2.1 Introduction

2.2 Kelvin's Theorem

2.3 Irrotational Flow

2.4 Irrotational Flow–Velocity Potential and the Strong Form of Bernoulli's Equation

2.5 Incompressible Flow–Streamfunction

2.6 Irrotational Incompressible Flow

2.7 Induced Velocity

2.8 Sources and Sinks

2.9 Two-Dimensional Flow

2.10 Applications of Analytic Functions in Fluid Mechanics

2.11 Force on a Body in Steady Two-Dimensional Incompressible Ideal Flow

2.12 Conformal Transforms

Appendix 2.A Drag in Ideal Flow

2.A.1 Helmholtz's Flow and Separation

2.A.2 Lines of Vortices

Chapter 3: Viscous Fluids

3.1 Basic Concept of Viscosity

3.2 Differential Motion of a Fluid Element

3.3 Strain Rate

3.4 Stress

3.5 Viscous Stress

3.6 Incompressible Flow–Navier–Stokes Equation

3.7 Stokes’ or Creeping Flow

3.8 Dimensionless Analysis and Similarity

Appendix 3.A Buckingham's Theorem and the Complete Set of Dimensionless Products

Chapter 4: Waves and Instabilities in Fluids

4.1 Introduction

4.2 Small-Amplitude Surface Waves

4.3 Surface Waves in Infinite fluids

4.4 Surface Waves with Velocity Shear Across a Contact Discontinuity

4.5 Shallow Water Waves

4.6 Waves in a Stratified Fluid

4.7 Stability of Laminar Shear Flow

4.8 Nonlinear Instability

Chapter 5: Turbulent Flow

5.1 Introduction

5.2 Fully Developed Turbulence

5.3 Turbulent Stress–Reynolds Stresses

5.4 Similarity Model of Shear in a Turbulent Flow–von Karman's Hypothesis

5.5 Velocity Profile near a Wall in Fully Developed Turbulence–Law of the Wall

5.6 Turbulent Flow Through a Duct

5.I.i Blasius wall stress correlation

Appendix 5.A Prandtl's Mixing Length Model

Chapter 6: Boundary Layer Flow

6.1 Introduction

6.2 The Laminar Boundary Layer in Steady Incompressible Two-Dimensional Flow–Prandtl's Approximation

6.3 Laminar Boundary Layer over an Infinite Flat Plate–Blasius's Solution

6.4 Laminar Boundary Layer–von Karman's Momentum Integral Method

6.5 Boundary Layer Instability and the Onset of Turbulence–Tollmein–Schlichting Instability

6.6 Turbulent Boundary Layer on a Flat Smooth Plate

6.7 Boundary Layer Separation

6.8 Drag

6.9 Laminar Wake

6.10 Separation in the Turbulent Boundary Layer

Appendix 6.A Singular Perturbation Problems and the Method of Matched Asymptotic Expansion

Chapter 7: Convective Heat Transfer

7.1 Introduction

7.2 Forced Convection

7.3 Heat Transfer in a Laminar Boundary Layer

7.4 Heat Transfer in a Turbulent Boundary Layer on a Smooth Flat Plate

7.5 Free or Natural Convection

Chapter 8: Compressible Flow and Sound Waves

8.1 Introduction

8.2 Propagation of Small Disturbances

8.3 Reflection and Transmission of a Sound Wave at an Interface

8.4 Spherical Sound Waves

8.5 Cylindrical Sound Waves

Chapter 9: Characteristics and Rarefactions

9.1 Mach Lines and Characteristics

9.2 Characteristics

9.3 One-Dimensional Time-Dependent Expansion

9.4 Steady Two-Dimensional Irrotational Expansion

Chapter 10: Shock Waves

10.1 Introduction

10.2 The Shock Transition and the Rankine–Hugoniot Equations

10.3 The Shock Adiabat

10.4 Shocks in Real Gases

10.5 The Hydrodynamic Structure of the Shock Front

10.6 The Shock Front in Real Gases

10.7 Shock Tubes

10.8 Shock Interaction

10.9 Oblique Shocks

10.10 Adiabatic Compression

Appendix 10.A An Alternative Approach to the General Conservation Law Form of the Fluid Equations

Chapter 11: Aerofoils in Low-Speed Incompressible Flow

11.1 Introduction

11.2 Two-Dimensional Aerofoils

11.3 Generation of Lift on an Aerofoil

11.4 Pitching Moment about the Wing

11.5 Lift from a Thin Wing

11.6 Application of Conformal Transforms to the Properties of Aerofoils

11.7 The Two-Dimensional Panel Method

11.8 Three-Dimensional Wings

11.9 Three-Dimensional Panel Method

Appendix 11.A Evaluation of the Principal Value Integrals

Appendix 11.B The Zhukovskii Family of Transformations

Chapter 12: Aerofoils in High-Speed Compressible Fluid Flow

12.1 Introduction

12.2 Linearised Theory for Two-Dimensional Flows: Subsonic Compressible Flow around a Long Thin Aerofoil—Prandtl–Glauert Correction

12.3 Linearised Theory for Two-Dimensional Flows: Supersonic Flow about an Aerofoil—Ackeret's Formula

12.4 Drag in High-Speed Compressible Flow

12.5 Linearised Theory of Three-Dimensional Supersonic Flow—von Karman Ogives and Sears–Haack Bodies

Chapter 13: Deflagrations and Detonations

13.1 Introduction

13.2 Detonations, Deflagrations and the Hugoniot Plot

Chapter 14: Self-similar Methods in Compressible Gas Flow and Intermediate Asymptotics

14.1 Introduction

14.2 Homogeneous Self-similar Flow of a Compressible Fluid

14.3 Centred Self-similar Flows

14.4 Flow Resulting from a Point Explosion in Gas—Blast Waves

14.5 Adiabatic Collapse of a Sphere

14.6 Convergent Shock Waves—Guderley's Solution

Problems

Solutions

Bibliography

Index

Series page

Frontispiece The first observation of shock waves generated by the passage of a bullet through air by Mach and Salcher (1887). Behind the primary shock can be seen the following rarefaction. Note also the turbulet wake leaving the rear of the bullet. Since the bullet was blunt the shock is slightly detached, although this cannot be clearly seen. The two vertical lines are for timing. The photograph was taken using a shadowgraph technique.

This edition first published 2013

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This book is dedicated with grateful thanks and appreciation to my doctoral supervisor John Pain who introduced me to the science of shock waves and fluid dynamics and who provided help and support in times of stress (even when we attacked the Tirpitz)

Preface

Every physicist is familiar with the two revolutions in thought which took place at the turn of the nineteenth and twentieth centuries. These revolutions in modern physics in the guise of relativity and quantum physics form essential components of today's undergraduate physics courses. Despite the fact that it has played a critical role in the development of the modern world, underlying the design of aircraft and industrial plants, the third contemporary revolution in fluid dynamics is less familiar, yet it was also necessary to eliminate paradoxes inherent in nineteenth-century physics. Fluid mechanics has become a neglected subject in the modern physics curriculum. The reason for this is twofold. Firstly, the subject is an area of ‘classical physics’, now generally regarded by students as ‘old fashioned’, yet one area, namely turbulence, remains one of the most intractable problems in physics at the forefront of complexity. Secondly, much of the field was developed by engineers and applied mathematicians. Until recently physicists have been less involved. In recent years this has changed as environmental physics, and plasma physics, have increasingly required a working knowledge of fluid dynamics. More generally, when physicists are expected to act as ‘troubleshooters’ in many fields of applied physics, a basic knowledge of fluid mechanics is needed.

The dominant problem in nineteenth-century fluid mechanics was drag (the resistance of the moving flow on a stationary body) in the flow of fluids with low viscosity. Stokes (1851) had solved the problem of the drag force experienced by a sphere in a flow dominated by viscosity. However, it was well known that fluids of low viscosity also generated drag, which had been measured and analysed notably by Froude in 1871, who measured the drag due to a ship's motion. The revolution in thought referred to above followed two seminal papers. The first by Reynolds (1883) introduced turbulence, namely that fluid motion could be disorganised and chaotic. The second was by Prandtl (1904), in which he introduced not only the concept of the boundary layer, but also flow separation (in only 10 pages!). These two papers opened the way to the resolution of this central problem of nineteenth-century fluid mechanics: that is, the drag force exerted by a moving fluid on a body immersed in it.

The development of the theory of compressible flow also dates from the late nineteenth-century, most importantly with the introduction of characteristics by Riemann (1860). Waves of discontinuity (shock waves) were propounded by Rankine (1870) and Hugoniot (1887) between 1870 and 1890, but were generally discounted on theoretical grounds despite observations, e.g. by Mach and Salcher (1887) and Boys (1893), in back-lit photographs of the motion of supersonic projectiles. It was not until the work of Rayleigh (1910), Taylor (1910), and more fully Becker (1922), demonstrated a narrow ‘shock layer’ supported by dissipation in a thin zonebetween two regions of dissipationless flow, that it became accepted that shock waves in compression existed, stabilised by thermal conduction and viscosity.

Both the problems associated with the boundary layer and shock wave stemmed from a lack of understanding of the role of viscosity in nearly inviscid fluids. It was assumed that because the viscosity of most fluids was known to be small, the flow could be accurately described by the ‘ideal fluid’ equations in which the viscosity is equal to zero. In fact, even though the viscosity is small, viscous forces are strong in regions of large velocity gradient, respectively adjacent to the surface of a solid body and in the shock ‘discontinuity’. The equations for an ideal fluid are therefore an ‘asymptote’ of those for a real fluid in the limit when the viscosity tends to zero. The singularities in the inviscid fluid equations, which are associated with the difference between zero viscosity and infinitesimally small viscosity, consequently disappear in a full treatment. More recently, new mathematical methods, based on the generalisation of the theory of functions through distributions, have allowed the singularities in ideal flow to be formally treated directly (Lax, 1954).

One further development in physics at the turn of the nineteenth to the twentieth century has played a major role in fluid mechanics. This was the evolution of dimensional analysis by Rayleigh (18999) and later Buckingham (1914). This led to the development of the methods of similarity and modelling, which are widely used to analyse experimental measurements.

This book is intended to give a pedagogical summary of the physics of fluid flow. Thus it builds on several classic texts, which cover specific aspects of the field in more detail than is possible here. In appropriate places the reader is referred to these for further study. The book includes both the applied mathematical development, which underlies much of the subject, and results from the more empirical engineering approach. Although lacking the rigour of the former, the latter are of equal importance to the working physicist. Unfortunately limitations of space have also prevented the discussion of two topics, which should have been included, were the book to be inclusive. These are, firstly, flows in a rotating environment, important in geophysics and meteorology, and, secondly, computational fluid dynamics. A full discussion of modern developments in turbulence was also not deemed appropriate, and a more engineering type of approach has been adopted.

Experimental fluid mechanics is a very visual subject. Many beautiful illustrations of effects described in this book have been published. Rather than extensive reproductions of these photographs, the reader is referred to relevant pictures in van Dyke (1982) or the classic photographs of Prandtl and Tietjens (1957). The ‘internet’ also provides an easily accessible source of much illustrative material.

Ideal inviscid fluid flow provides the basis for solving many problems of practical importance involving waves and instabilities, and underlying boundary layer theory. Its mathematical tractability allows it to be used for many of the methods of computational fluid dynamics. It is therefore important to give some background tothe analytic treatment of ideal fluid mechanics, which occupies Chapters 1 and 2. Viscosity is introduced in Chapter 3, together with a brief account of dimensional analysis. Waves and instabilities, which underlie many aspects of fluid dynamics, are discussed in Chapter 4. Turbulence is introduced in Chapter 5 and is treated predominately phenomenologically, with only a brief nod to modern analysis. Boundary layers in Chapter 6 cover the basic Prandtl approach, treating only simple problems, and development into separation and drag. Chapter 7 gives a brief account of the engineering approach to heat transfer.

Compressible fluids and the characteristic problems associated with them fill the remainder of the book. Chapter 8 gives a brief introduction to sound waves. Rarefaction flow, treated by the method of characteristics, occupies Chapter 9. Compression and shock waves are introduced in Chapter 10 studying both the foundations of the theory of shock waves and their application. The behaviour of fluid flow around aerofoils and wings occupies Chapters 11 and 12, the first for subsonic and the second for transonic and supersonic flight. Detonation and deflagration associated with flames and explosives are treated in Chapter 13. The book is concluded with Chapter 14 describing the application of self-similar methods applied to example problems in compressible flow.

It is expected that the book will be used by final year undergraduates and postgraduates in physics and applied mathematics. A good working knowledge of vector calculus and the functions of a complex variable is therefore assumed. Dimensional analysis is an essential tool of the fluid dynamicist, and some knowledge is expected. A brief introduction to Buckingham's Π theorem is given for those who have not previously met this approach.

The basic development of the subject forms the main text. Sections in small type are intended for reference rather than parts of the main development of the text. However, examples of applications are included either as case studies or as problems, whose solutions may be found at the end of book. Some specific points of mathematical development or of historical interest appear as appendices.

Chapter 1

Introduction

1.1 Fluids as a State of Matter

A standard dictionary definition of a fluid is

a substance whose particles can move about with freedom–a liquid or gas.

Whilst this formulation encapsulates our general concept of a fluid, it is not entirely satisfactory as a scientific basis for the understanding of such materials. More formally within the context of fluid mechanics the fluid is seen as an isotropic, locally homogeneous, macroscopic material whose particles are free to move within the constraints established by the dynamical laws of continuum physics. The requirement that the fluid be a continuum implies that if a volume of fluid is successively subdivided into smaller elements, each element will remain structurally similar to its parent, and that this process of subdivision can be carried out down to infinitesimal volumes. Under these conditions several useful macroscopic concepts may be defined:

Fluid particle a fictitious particle fixed within the fluid continuum and moving with the velocity of the flow, and representing an average over a large number of microscopic particles.

Fluid point fixed in the fluid moving with the flow velocity. A fluid particle is always situated at the same fluid point.

Infinitesimal volume within the continuum of the fluid, and large compared with microscopic scales, but small compared with macroscopic ones.

In fact of course the fluid is not a continuum in the strict mathematical sense used above. The fluid is made up of discrete microscopic particles, namely molecules, which are distributed randomly with a distribution of velocities characteristic of the fluid in thermal equilibrium, typically given by the Maxwell–Boltzmann distribution in a gas. Fortunately, at the densities at which most experiments are conducted, theintermolecule separation is extremely small and very much less than the laboratory scale. It is therefore possible to average over small volumes which contain a very large number of particles, yet are very small on the laboratory scale, and allow us to recover the continuum approximation. In this manner we obtain terms which characterise the fluid as a bulk material. Typical of these average quantities are:

Density number or mass of particles per unit volume.

Temperature average energy of the random motion per particle in thermal equilibrium.

Pressure average momentum flow associated with the random motion per unit area.

Flow velocity mean velocity of the molecules averaging out the random motion.

The role of collisions amongst the particles plays an important role in defining irreversibility through the loss of correlation between the particles. Particles collide on average after a distance equal to the mean free path, and time after the collision interval. Since fluid mechanics assumes the fluid particles are in thermal equilibrium and randomly distributed, this condition requires that spatial and temporal averages be taken to include a large number of collisions, i.e. the laboratory-scale length is large compared with the mean free path and time to the collision interval. In practice this is not normally a restrictive condition. The effects of the collisions on fluid transport (momentum and energy) are thereby averaged over the thermal distribution to yield bulk properties of the material, namely viscosity and thermal conduction respectively. Consequently (ideal) fluid motion without viscosity or thermal conduction is dissipationless, entropy generation being due to viscosity and thermal conduction.

Within the continuum theory it is implicitly assumed that locally the fluid is in thermal equilibrium, although the temperature may vary globally through the flow. As a result the thermodynamics of bulk materials may be applied locally in the flow to calculate the pressure from the density and temperature (say) using the equation of state of the fluid. More generally the quantities and relations of equilibrium thermodynamics may be applied in the flow.

The flow of a basic fluid may be calculated using Newtonian mechanics, classical thermodynamics and the values of viscosity and thermal conductivity. From the above discussion, the conditions under which this theory may be applied are:

Laboratory-scale lengths must be large compared with the intermolecule separation and mean free path.

Characteristic laboratory times must be large compared with the collision interval.

The fluid must locally be in thermal equilibrium.

The theory may be readily extended to relativistic mechanics and also to include additional dissipative terms, e.g. due to radiation. However, under normal laboratory conditions these are not required. Astrophysical systems provide examples of flows where more general approaches may be required.

Provided the above conditions are met, it is relatively straightforward to show that the fluid dynamical equations (to be obtained later) may be directly derived from the governing kinetic theory of the molecules.

1.2 The Fundamental Equations for Flow of a Dissipationless Fluid

The basic equations of fluid mechanics stem from simple concepts of conservation applied to mass, momentum and energy. These are completed by the thermodynamic equation of state of the material, in which the flow is to be calculated. The equations are, of course, complemented by the boundary conditions in an appropriate form, depending on the nature of the problem. In any problem, we seek to find five variables: three velocity components () and two thermodynamic state variables, e.g. density and pressure , as functions of space and time . In many problems the actual number of variables required is reduced, either by symmetry to a restricted number of spatial dimensions or by a specified thermodynamic state, e.g. constant entropy or constant temperature. The problem is often further simplified by the restriction to , when there is no time variation.