Fluid dynamics: theory, computation, and numerical simulation

FLUID DYNAMICS:

THEORY, COMPUTATION,

AND NUMERICAL SIMULATION

Fluid Dynamics:

Theory, Computation,

Second Edition

C. Pozrikidis

and Numerical Simulation

USA

ISBN: 978-0-387-95869-9 e-ISBN: 978-0-387-95871-2

© Springer Science+Business Media, LLC 2009

Printed on acid-free paper

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All rights reserved. This work may not be translated or copied in whole or in part without the written

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10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection

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The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are

not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject

to proprietary rights.

DOI: 10.1007/978-0-387-95871-2

C. Pozrikidis

University of Massachusetts

Library of Congress Control Number: 2008943356

Amherst, MA

Preface

Ready access to computers has defined a new era in teaching and learning. The

opportunity to extend the subject matter of traditional science and engineering

curricula into the realm of scientific computing has become not only desirable,

but also necessary. Thanks to portability and low overhead and operating cost,

experimentation by numerical simulation has become a viable substitute, and

occasionally the only alternative, to physical experimentation.

The new framework has necessitated the writing of texts and monographs

from a modern perspective that incorporates numerical and computer program￾ming aspects as an integral part of the discourse. Under this modern directive,

methods, concepts, and ideas are presented in a unified fashion that motivates

and underlines the urgency of the new elements, but neither compromises nor

oversimplifies the rigor of the classical approach.

Interfacing fundamental concepts and practical methods of scientific com￾puting can be implemented on different levels. In one approach, theory and

implementation are kept complementary and presented in a sequential fashion.

In another approach, the coupling involves deriving computational methods

and simulation algorithms, and translating equations into computer code in￾structions immediately following problem formulations. Seamlessly interjecting

methods of scientific computing in the traditional discourse offers a powerful

venue for developing analytical skills and obtaining physical insight.

The goal of this book is to offer an introductory course in traditional and

modern fluid mechanics, covering topics in a way that unifies theory, computa￾tion, computer programming, and numerical simulation. The approach is truly

introductory in that only a few prerequisites are required. The intended au￾dience includes undergraduate and entry-level graduate students, as well as a

broader class of scientists, engineers, fluid dynamics and computational science

enthusiasts with a general interest in computing. This book should be especially

appealing to those who are making a first excursion into the world of numerical

computation and computational fluid dynamics (CFD) beyond the black-box

approach. This book should be an ideal text for an introductory course in fluid

mechanics and CFD.

The presentation of the material is distinguished by two features. First,

solution procedures and algorithms are developed immediately after problem

formulations are presented, and illustrative Matlab codes are discussed in the

text. Second, numerical methods are introduced on a need-to-know basis and

in order of increasing difficulty: function interpolation, function differentiation,

function integration, solution of algebraic equations, finite-difference methods,

etc. Computer problems at the end of each section require performing compu￾v

vi

tation and simulation to study the effect of various parameters determining a

flow.

In concert with the intended usage of this book as a stand-alone introduc￾tory text and as a tutorial on numerical fluid dynamics and scientific comput￾ing, only a few references are provided in the discussion. Instead, a selected

compilation of introductory, advanced, and specialized texts on fluid dynamics,

calculus, numerical methods, and computational fluid dynamics are listed in

appendix B. The reader who wishes to focus on a particular topic is directed

to these resources for further details.

A major feature of this book is the accompanying fluid dynamics software li￾brary Fdlib discussed in appendix A. The Fortran 77 and Matlab programs

of Fdlib explicitly illustrate how computational algorithms translate into com￾puter instructions. The codes of Fdlib range from introductory to advanced,

and the topics span a broad range of applications discussed in this text: from

laminar channel flows, to vortex flows, to flow past airfoils. The Matlab codes

of Fdlib combine numerical computation, graphics display, data visualization

and animation.

To run the Fortran 77 codes of Fdlib, a Fortran 77 or Fortran 90

compiler is required. Free compilers are available thanks to the gnu foundation.

The input data is either entered from the keyboard or read from data files. The

output is recorded in output files in tabular form so that it can be read and

displayed using independent graphics, visualization, and animation applications

on any computer platform, including Matlab.

The second edition incorporates significant improvements in substance and

style. First, additional examples, solved problems, and new material have been

introduced for a more comprehensive treatment of the various topics. Examples

include surfactant transport and a brief introduction to compressible flow. Sec￾ond, sample Matlab programs integrating numerical computation and graph￾ics visualization are listed and discussed in the text. A Matlab primer explaining

basic programming procedures is presented in appendix C. Third, the revised

text refers to the latest version of Fdlib. These improvements should ren￾der the book an accessible introductory computational fluid dynamics (CFD)

resource.

The book Internet address is: http://dehesa.freeshell.org/FD2

I acknowledge with appreciation insightful comments by Keiko Nomura,

Siggi Thoroddsen, and Mark Blyth on the manuscript of the second edition.

C. Pozrikidis

Spring, 2009

Contents

Preface v

1 Introduction to Kinematics 1

1.1 Fluids and solids . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Fluid parcels and flow kinematics . . . . . . . . . . . . . . . . 2

1.3 Coordinates, velocity, and acceleration . . . . . . . . . . . . . 3

1.3.1 Cylindrical polar coordinates . . . . . . . . . . . . . . 6

1.3.2 Spherical polar coordinates . . . . . . . . . . . . . . . 9

1.3.3 Plane polar coordinates . . . . . . . . . . . . . . . . . 13

1.4 Fluid velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.1 Velocity vector field, streamlines and stagnation points 18

1.5 Point particles and their trajectories . . . . . . . . . . . . . . 19

1.5.1 Path lines . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5.2 Ordinary differential equations (ODEs) . . . . . . . . 20

1.5.3 Explicit Euler method . . . . . . . . . . . . . . . . . . 21

1.5.4 Modified Euler method . . . . . . . . . . . . . . . . . 23

1.5.5 Description in polar coordinates . . . . . . . . . . . . 26

1.5.6 Streaklines . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6 Material surfaces and elementary motions . . . . . . . . . . . 28

1.6.1 Fluid parcel rotation . . . . . . . . . . . . . . . . . . 28

1.6.2 Fluid parcel deformation . . . . . . . . . . . . . . . . 29

1.6.3 Fluid parcel expansion . . . . . . . . . . . . . . . . . 30

1.6.4 Superposition of rotation, deformation, and expansion 31

1.6.5 Rotated coordinates . . . . . . . . . . . . . . . . . . . 32

1.6.6 Flow decomposition . . . . . . . . . . . . . . . . . . . 34

1.7 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.7.1 Interpolation in one dimension . . . . . . . . . . . . . 38

1.7.2 Interpolation in two dimensions . . . . . . . . . . . . 42

1.7.3 Interpolation of the velocity in a two-dimensional flow 45

1.7.4 Streamlines by interpolation . . . . . . . . . . . . . . 49

vii

viii

2 More on Kinematics 54

2.1 Fundamental modes of fluid parcel motion . . . . . . . . . . . 54

2.1.1 Function linearization . . . . . . . . . . . . . . . . . . 55

2.1.2 Velocity gradient tensor . . . . . . . . . . . . . . . . . 57

2.1.3 Relative motion of point particles . . . . . . . . . . . 59

2.1.4 Fundamental motions in two-dimensional flow . . . . 60

2.1.5 Fundamental motions in three-dimensional flow . . . 62

2.1.6 Gradient in polar coordinates . . . . . . . . . . . . . 62

2.2 Fluid parcel expansion . . . . . . . . . . . . . . . . . . . . . . 65

2.3 Fluid parcel rotation and vorticity . . . . . . . . . . . . . . . 66

2.3.1 Curl and vorticity . . . . . . . . . . . . . . . . . . . . 68

2.3.2 Two-dimensional flow . . . . . . . . . . . . . . . . . . 70

2.3.3 Axisymmetric flow . . . . . . . . . . . . . . . . . . . . 70

2.4 Fluid parcel deformation . . . . . . . . . . . . . . . . . . . . . 71

2.5 Numerical differentiation . . . . . . . . . . . . . . . . . . . . . 74

2.5.1 Numerical differentiation in one dimension . . . . . . 74

2.5.2 Numerical differentiation in two dimensions . . . . . . 76

2.5.3 Velocity gradient and related functions . . . . . . . . 78

2.6 Flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

2.6.1 Areal flow rate and flux . . . . . . . . . . . . . . . . . 87

2.6.2 Areal flow rate across a line . . . . . . . . . . . . . . 88

2.6.3 Numerical integration . . . . . . . . . . . . . . . . . . 89

2.6.4 The Gauss divergence theorem in two dimensions . . 90

2.6.5 Flow rate in a three-dimensional flow . . . . . . . . . 91

2.6.6 Gauss divergence theorem in three dimensions . . . . 92

2.6.7 Axisymmetric flow . . . . . . . . . . . . . . . . . . . . 92

2.7 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . 94

2.7.1 Mass flux and mass flow rate . . . . . . . . . . . . . . 94

2.7.2 Mass flow rate across a closed line . . . . . . . . . . . 94

2.7.3 The continuity equation . . . . . . . . . . . . . . . . . 95

2.7.4 Three-dimensional flow . . . . . . . . . . . . . . . . . 96

2.7.5 Rigid-body translation . . . . . . . . . . . . . . . . . 96

2.7.6 Evolution equation for the density . . . . . . . . . . . 97

2.8 Properties of point particles . . . . . . . . . . . . . . . . . . . 99

2.8.1 The material derivative . . . . . . . . . . . . . . . . . 100

2.8.2 The continuity equation . . . . . . . . . . . . . . . . . 101

2.8.3 Point particle acceleration . . . . . . . . . . . . . . . 102

2.9 Incompressible fluids and stream functions . . . . . . . . . . . 106

2.9.1 Mathematical consequences of incompressibility . . . 107

2.9.2 Stream function for two-dimensional flow . . . . . . . 107

2.9.3 Stream function for axisymmetric flow . . . . . . . . 109

2.10 Kinematic conditions at boundaries . . . . . . . . . . . . . . . 111

2.10.1 The no-penetration boundary condition . . . . . . . . 111

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3 Flow Computation based on Kinematics 115

3.1 Flow classification based on kinematics . . . . . . . . . . . . . 115

3.2 Irrotational flow and the velocity potential . . . . . . . . . . . 117

3.2.1 Two-dimensional flow . . . . . . . . . . . . . . . . . . 117

3.2.2 Incompressible fluids and the harmonic potential . . . 119

3.2.3 Three-dimensional flow . . . . . . . . . . . . . . . . . 120

3.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . 121

3.2.5 Cylindrical polar coordinates . . . . . . . . . . . . . . 122

3.2.6 Spherical polar coordinates . . . . . . . . . . . . . . . 122

3.2.7 Plane polar coordinates . . . . . . . . . . . . . . . . . 123

3.3 Finite-difference methods . . . . . . . . . . . . . . . . . . . . 124

3.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . 124

3.3.2 Finite-difference grid . . . . . . . . . . . . . . . . . . 126

3.3.3 Finite-difference discretization . . . . . . . . . . . . . 127

3.3.4 Compilation of a linear system . . . . . . . . . . . . . 128

3.4 Linear solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.4.1 Gauss elimination . . . . . . . . . . . . . . . . . . . . 139

3.4.2 A menagerie of other methods . . . . . . . . . . . . . 140

3.5 Two-dimensional point sources and point-source dipoles . . . 141

3.5.1 Function superposition and fundamental solutions . . 141

3.5.2 Two-dimensional point source . . . . . . . . . . . . . 141

3.5.3 Two-dimensional point-source dipole . . . . . . . . . 144

3.5.4 Flow past a circular cylinder . . . . . . . . . . . . . . 148

3.5.5 Sources and dipoles in the presence of boundaries . . 149

3.6 Three-dimensional point sources and point-source dipoles . . 151

3.6.1 Three-dimensional point source . . . . . . . . . . . . 151

3.6.2 Three-dimensional point-source dipole . . . . . . . . . 152

3.6.3 Streaming flow past a sphere . . . . . . . . . . . . . . 153

3.6.4 Sources and dipoles in the presence of boundaries . . 154

3.7 Point vortices and line vortices . . . . . . . . . . . . . . . . . 155

3.7.1 The potential of irrotational circulatory flow . . . . . 156

3.7.2 Flow past a circular cylinder . . . . . . . . . . . . . . 157

3.7.3 Circulation . . . . . . . . . . . . . . . . . . . . . . . . 158

3.7.4 Line vortices in three-dimensional flow . . . . . . . . 161

4 Forces and Stresses 163

4.1 Body forces and surface forces . . . . . . . . . . . . . . . . . . 163

4.1.1 Body forces . . . . . . . . . . . . . . . . . . . . . . . . 163

4.1.2 Surface forces . . . . . . . . . . . . . . . . . . . . . . 164

4.2 Traction and the stress tensor . . . . . . . . . . . . . . . . . . 165

4.2.1 Traction on either side of a fluid surface . . . . . . . . 168

4.2.2 Traction on a boundary . . . . . . . . . . . . . . . . . 169

4.2.3 Symmetry of the stress tensor . . . . . . . . . . . . . 170

x

4.3 Traction jump across a fluid interface . . . . . . . . . . . . . . 171

4.3.1 Force balance at a two-dimensional interface . . . . . 172

4.3.2 Force balance at a three-dimensional interface . . . . 176

4.3.3 Axisymmetric interfaces . . . . . . . . . . . . . . . . . 179

4.4 Stresses in a fluid at rest . . . . . . . . . . . . . . . . . . . . . 183

4.4.1 Pressure from molecular motions . . . . . . . . . . . . 184

4.4.2 Jump in the pressure across an interface . . . . . . . 185

4.5 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 186

4.5.1 Simple fluids . . . . . . . . . . . . . . . . . . . . . . . 188

4.5.2 Incompressible Newtonian fluids . . . . . . . . . . . . 188

4.5.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 190

4.5.4 Ideal fluids . . . . . . . . . . . . . . . . . . . . . . . . 192

4.5.5 Significance of the pressure in incompressible fluids . 193

4.5.6 Pressure in compressible fluids . . . . . . . . . . . . . 193

4.6 Simple non-Newtonian fluids . . . . . . . . . . . . . . . . . . 196

4.6.1 Unidirectional shear flow . . . . . . . . . . . . . . . . 197

4.7 Stresses in polar coordinates . . . . . . . . . . . . . . . . . . . 199

4.7.1 Cylindrical polar coordinates . . . . . . . . . . . . . . 200

4.7.2 Spherical polar coordinates . . . . . . . . . . . . . . . 202

4.7.3 Plane polar coordinates . . . . . . . . . . . . . . . . . 204

4.8 Boundary conditions for the tangential velocity . . . . . . . . 206

4.8.1 No-slip boundary condition . . . . . . . . . . . . . . . 206

4.8.2 Slip boundary condition . . . . . . . . . . . . . . . . . 207

4.9 Wall stresses in Newtonian fluids . . . . . . . . . . . . . . . . 208

4.9.1 Shear stress . . . . . . . . . . . . . . . . . . . . . . . 208

4.9.2 Normal stress . . . . . . . . . . . . . . . . . . . . . . 209

4.10 Interfacial surfactant transport . . . . . . . . . . . . . . . . . 210

4.10.1 Two-dimensional interfaces . . . . . . . . . . . . . . . 210

4.10.2 Axisymmetric interfaces . . . . . . . . . . . . . . . . . 214

4.10.3 Three-dimensional interfaces . . . . . . . . . . . . . . 216

5 Hydrostatics 218

5.1 Equilibrium of pressure and body forces . . . . . . . . . . . . 218

5.1.1 Equilibrium of an infinitesimal parcel . . . . . . . . . 220

5.1.2 Gases in hydrostatics . . . . . . . . . . . . . . . . . . 222

5.1.3 Liquids in hydrostatics . . . . . . . . . . . . . . . . . 223

5.2 Force exerted on immersed surfaces . . . . . . . . . . . . . . . 225

5.2.1 A sphere floating on a flat interface . . . . . . . . . . 226

5.3 Archimedes’ principle . . . . . . . . . . . . . . . . . . . . . . 231

5.3.1 Net force on a submerged body . . . . . . . . . . . . 233

5.3.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.4 Interfacial shapes . . . . . . . . . . . . . . . . . . . . . . . . . 235

5.4.1 Curved interfaces . . . . . . . . . . . . . . . . . . . . 236

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5.4.2 The Laplace-Young equation . . . . . . . . . . . . . . 237

5.4.3 Three-dimensional interfaces . . . . . . . . . . . . . . 238

5.5 A semi-infinite interface attached to an inclined plate . . . . . 239

5.5.1 Numerical method . . . . . . . . . . . . . . . . . . . . 241

5.6 A meniscus between two parallel plates . . . . . . . . . . . . . 245

5.6.1 The shooting method . . . . . . . . . . . . . . . . . . 249

5.7 A two-dimensional drop on a horizontal or inclined plane . . 253

5.7.1 Drop on a horizontal plane . . . . . . . . . . . . . . . 253

5.7.2 A drop on an inclined plane . . . . . . . . . . . . . . 261

5.8 Axisymmetric meniscus inside a tube . . . . . . . . . . . . . . 273

5.9 Axisymmetric drop on a horizontal plane . . . . . . . . . . . 276

5.9.1 Solution space . . . . . . . . . . . . . . . . . . . . . . 278

5.10 A sphere straddling an interface . . . . . . . . . . . . . . . . . 286

5.10.1 Spheroidal particle . . . . . . . . . . . . . . . . . . . . 296

5.11 A three-dimensional meniscus . . . . . . . . . . . . . . . . . . 298

5.11.1 Elliptic coordinates . . . . . . . . . . . . . . . . . . . 299

5.11.2 Finite-difference method . . . . . . . . . . . . . . . . 300

5.11.3 Capillary force and torque . . . . . . . . . . . . . . . 306

6 Equation of Motion and Vorticity Transport 308

6.1 Newton’s second law of motion for a fluid parcel . . . . . . . 308

6.1.1 Rate of change of linear momentum . . . . . . . . . . 309

6.1.2 Equation of parcel motion . . . . . . . . . . . . . . . 309

6.1.3 Two-dimensional flow . . . . . . . . . . . . . . . . . . 310

6.2 Integral momentum balance . . . . . . . . . . . . . . . . . . . 313

6.2.1 Flow through a sudden enlargement . . . . . . . . . . 316

6.2.2 Isentropic flow through a conduit . . . . . . . . . . . 318

6.3 Cauchy’s equation of motion . . . . . . . . . . . . . . . . . . 319

6.3.1 Hydrodynamic volume force . . . . . . . . . . . . . . 320

6.3.2 Force on an infinitesimal parcel . . . . . . . . . . . . 320

6.3.3 The equation of motion . . . . . . . . . . . . . . . . . 322

6.3.4 Evolution equations . . . . . . . . . . . . . . . . . . . 323

6.3.5 Cylindrical polar coordinates . . . . . . . . . . . . . . 323

6.3.6 Spherical polar coordinates . . . . . . . . . . . . . . . 325

6.3.7 Plane polar coordinates . . . . . . . . . . . . . . . . . 325

6.3.8 Vortex force . . . . . . . . . . . . . . . . . . . . . . . 326

6.3.9 Summary of governing equation . . . . . . . . . . . . 326

6.3.10 Accelerating frame of reference . . . . . . . . . . . . . 326

6.4 Euler’s and Bernoulli’s equations . . . . . . . . . . . . . . . . 327

6.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . 328

6.4.2 Irrotational flow . . . . . . . . . . . . . . . . . . . . . 329

6.4.3 Steady irrotational flow . . . . . . . . . . . . . . . . . 331

6.4.4 Steady rotational flow . . . . . . . . . . . . . . . . . . 334

xii

6.4.5 Flow with uniform vorticity . . . . . . . . . . . . . . 335

6.5 The Navier-Stokes equation . . . . . . . . . . . . . . . . . . . 337

6.5.1 Pressure and viscous forces . . . . . . . . . . . . . . . 338

6.5.2 A radially expanding or contracting bubble . . . . . . 339

6.5.3 Boundary conditions . . . . . . . . . . . . . . . . . . 340

6.5.4 Polar coordinates . . . . . . . . . . . . . . . . . . . . 341

6.6 Vorticity transport . . . . . . . . . . . . . . . . . . . . . . . . 343

6.6.1 Two-dimensional flow . . . . . . . . . . . . . . . . . . 343

6.6.2 Axisymmetric flow . . . . . . . . . . . . . . . . . . . . 346

6.6.3 Three-dimensional flow . . . . . . . . . . . . . . . . . 347

6.7 Dynamic similitude and the Reynolds number . . . . . . . . . 350

6.7.1 Dimensional analysis . . . . . . . . . . . . . . . . . . 352

6.8 Structure of a flow as a function of the Reynolds number . . 355

6.8.1 Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . 356

6.8.2 Flow at high Reynolds numbers . . . . . . . . . . . . 356

6.8.3 Laminar and turbulent flow . . . . . . . . . . . . . . . 357

6.9 Dimensionless numbers in fluid dynamics . . . . . . . . . . . 357

7 Channel, Tube, and Film Flow 360

7.1 Steady flow in a two-dimensional channel . . . . . . . . . . . 360

7.1.1 Two-layer flow . . . . . . . . . . . . . . . . . . . . . . 363

7.1.2 Multi-layer flow . . . . . . . . . . . . . . . . . . . . . 365

7.1.3 Power-law fluids . . . . . . . . . . . . . . . . . . . . . 370

7.2 Steady film flow down an inclined plane . . . . . . . . . . . . 373

7.2.1 Multi-film flow . . . . . . . . . . . . . . . . . . . . . . 374

7.2.2 Power-law fluids . . . . . . . . . . . . . . . . . . . . . 375

7.3 Steady flow through a circular tube . . . . . . . . . . . . . . . 377

7.3.1 Multi-layer tube flow . . . . . . . . . . . . . . . . . . 380

7.3.2 Flow due to a translating sector . . . . . . . . . . . . 380

7.4 Steady flow through an annular tube . . . . . . . . . . . . . . 383

7.4.1 Multi-layer annular flow . . . . . . . . . . . . . . . . . 387

7.5 Steady flow in channels and tubes . . . . . . . . . . . . . . . 387

7.5.1 Elliptical tube . . . . . . . . . . . . . . . . . . . . . . 388

7.5.2 Rectangular tube . . . . . . . . . . . . . . . . . . . . 390

7.5.3 Triangular tube . . . . . . . . . . . . . . . . . . . . . 393

7.5.4 Semi-infinite rectangular channel . . . . . . . . . . . . 393

7.6 Steady swirling flow . . . . . . . . . . . . . . . . . . . . . . . 395

7.6.1 Annular flow . . . . . . . . . . . . . . . . . . . . . . . 396

7.6.2 Multi-layer flow . . . . . . . . . . . . . . . . . . . . . 399

7.7 Transient channel flow . . . . . . . . . . . . . . . . . . . . . . 400

7.7.1 Couette flow . . . . . . . . . . . . . . . . . . . . . . . 400

7.7.2 Impulsive motion of a plate in a semi-infinite fluid . . 403

7.7.3 Pressure- and gravity-driven flow . . . . . . . . . . . 406

xiii

7.8 Oscillatory channel flow . . . . . . . . . . . . . . . . . . . . . 409

7.8.1 Oscillatory Couette flow . . . . . . . . . . . . . . . . 409

7.8.2 Rayleigh’s oscillating plate . . . . . . . . . . . . . . . 411

7.8.3 Pulsating pressure-driven flow . . . . . . . . . . . . . 413

7.9 Transient and oscillatory flow in a circular tube . . . . . . . . 415

7.9.1 Transient Poiseuille flow . . . . . . . . . . . . . . . . 415

7.9.2 Pulsating pressure-driven flow . . . . . . . . . . . . . 420

7.9.3 Transient circular Couette flow . . . . . . . . . . . . . 422

7.9.4 More on Bessel functions . . . . . . . . . . . . . . . . 422

8 Finite-Difference Methods 424

8.1 Choice of governing equations . . . . . . . . . . . . . . . . . . 424

8.2 Unidirectional flow; velocity/pressure formulation . . . . . . . 425

8.2.1 Governing equations . . . . . . . . . . . . . . . . . . . 426

8.2.2 Explicit finite-difference method . . . . . . . . . . . . 426

8.2.3 Implicit finite-difference method . . . . . . . . . . . . 429

8.2.4 Steady state . . . . . . . . . . . . . . . . . . . . . . . 435

8.2.5 Two-layer flow . . . . . . . . . . . . . . . . . . . . . . 436

8.3 Unidirectional flow; velocity/vorticity formulation . . . . . . . 443

8.3.1 Boundary conditions for the vorticity . . . . . . . . . 444

8.3.2 Alternative set of equations . . . . . . . . . . . . . . . 445

8.3.3 Comparison with the velocity/pressure formulation . 446

8.4 Unidirectional flow; stream function/vorticity formulation . . 447

8.4.1 Boundary conditions for the vorticity . . . . . . . . . 448

8.4.2 A semi-implicit method . . . . . . . . . . . . . . . . . 449

8.5 Two-dimensional flow;

stream function/vorticity formulation . . . . . . . . . . . . . 451

8.5.1 Flow in a cavity . . . . . . . . . . . . . . . . . . . . . 451

8.5.2 Finite-difference grid . . . . . . . . . . . . . . . . . . 452

8.5.3 Unsteady flow . . . . . . . . . . . . . . . . . . . . . . 453

8.5.4 Steady flow . . . . . . . . . . . . . . . . . . . . . . . . 454

8.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 460

8.6 Velocity/pressure formulation . . . . . . . . . . . . . . . . . . 463

8.6.1 Alternative system of governing equations . . . . . . 464

8.6.2 Pressure boundary conditions . . . . . . . . . . . . . 465

8.6.3 Compatibility condition for the pressure . . . . . . . 465

8.7 Operator splitting and solenoidal projection . . . . . . . . . . 466

8.7.1 Convection–diffusion step . . . . . . . . . . . . . . . . 467

8.7.2 Projection step . . . . . . . . . . . . . . . . . . . . . . 469

8.7.3 Boundary conditions for the intermediate velocity . . 471

8.7.4 Flow in a cavity . . . . . . . . . . . . . . . . . . . . . 471

8.7.5 Computation of the pressure . . . . . . . . . . . . . . 484

8.8 Staggered grids . . . . . . . . . . . . . . . . . . . . . . . . . . 485

xiv

9 Low Reynolds Number Flow 494

9.1 Flow in narrow channels . . . . . . . . . . . . . . . . . . . . . 494

9.1.1 Governing equations . . . . . . . . . . . . . . . . . . . 495

9.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 495

9.1.3 Equations of lubrication flow . . . . . . . . . . . . . . 497

9.1.4 Lubrication in a slider bearing . . . . . . . . . . . . . 497

9.1.5 Flow in a wavy channel . . . . . . . . . . . . . . . . . 500

9.1.6 Dynamic lifting . . . . . . . . . . . . . . . . . . . . . 503

9.2 Film flow on a horizontal or inclined wall . . . . . . . . . . . 505

9.2.1 Thin-film flow . . . . . . . . . . . . . . . . . . . . . . 506

9.2.2 Numerical methods . . . . . . . . . . . . . . . . . . . 509

9.3 Multi-film flow on a horizontal or inclined wall . . . . . . . . 511

9.3.1 Evolution equations . . . . . . . . . . . . . . . . . . . 514

9.3.2 Numerical methods . . . . . . . . . . . . . . . . . . . 516

9.4 Two-layer channel flow . . . . . . . . . . . . . . . . . . . . . . 523

9.5 Flow due to the motion of a sphere . . . . . . . . . . . . . . . 534

9.5.1 Formulation in terms of the stream function . . . . . 535

9.5.2 Traction, force, and the Archimedes-Stokes law . . . . 539

9.6 Point forces and point sources in Stokes flow . . . . . . . . . 541

9.6.1 The Oseen tensor and the point force . . . . . . . . . 542

9.6.2 Flow representation in terms of singularities . . . . . 544

9.6.3 A sphere moving inside a circular tube . . . . . . . . 544

9.6.4 Boundary integral representation . . . . . . . . . . . . 547

9.7 Two-dimensional Stokes flow . . . . . . . . . . . . . . . . . . 549

9.7.1 Flow due to the motion of a cylinder . . . . . . . . . 549

9.7.2 Rotation of a circular cylinder . . . . . . . . . . . . . 552

9.7.3 Simple shear flow past a circular cylinder . . . . . . . 552

9.7.4 The Oseen tensor and the point force . . . . . . . . . 553

9.8 Local solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 554

9.8.1 Separation of variables . . . . . . . . . . . . . . . . . 555

9.8.2 Flow near a corner . . . . . . . . . . . . . . . . . . . . 557

10 High Reynolds Number Flow 562

10.1 Changes in the structure of a flow

with increasing Reynolds number . . . . . . . . . . . . . . . . 562

10.2 Prandtl boundary layer analysis . . . . . . . . . . . . . . . . . 566

10.2.1 Boundary-layer equations . . . . . . . . . . . . . . . . 568

10.2.2 Surface curvilinear coordinates . . . . . . . . . . . . . 569

10.2.3 Parabolization . . . . . . . . . . . . . . . . . . . . . . 570

10.2.4 Flow separation . . . . . . . . . . . . . . . . . . . . . 570

10.3 Blasius boundary layer on a semi-infinite plate . . . . . . . . 571

10.3.1 Self-similarity and the Blasius equation . . . . . . . . 571

10.3.2 Numerical solution . . . . . . . . . . . . . . . . . . . 574

xv

10.3.3 Wall shear stress and drag force . . . . . . . . . . . . 576

10.3.4 Vorticity transport . . . . . . . . . . . . . . . . . . . 577

10.4 Displacement and momentum thickness . . . . . . . . . . . . 579

10.4.1 Von K`arm`an’s approximate method . . . . . . . . . . 581

10.5 Boundary layers in accelerating and decelerating flow . . . . . 583

10.5.1 Self-similarity . . . . . . . . . . . . . . . . . . . . . . 585

10.5.2 Numerical solution . . . . . . . . . . . . . . . . . . . 586

10.6 Momentum integral method . . . . . . . . . . . . . . . . . . . 587

10.6.1 The von K`arm`an-Pohlhausen method . . . . . . . . . 589

10.6.2 Pohlhausen polynomials . . . . . . . . . . . . . . . . . 590

10.6.3 Numerical solution . . . . . . . . . . . . . . . . . . . 592

10.6.4 Boundary layer around a curved body . . . . . . . . . 595

10.7 Instability of shear flows . . . . . . . . . . . . . . . . . . . . . 599

10.7.1 Stability analysis of shear flow . . . . . . . . . . . . . 600

10.7.2 Normal-mode analysis . . . . . . . . . . . . . . . . . . 601

10.7.3 Finite-difference solution . . . . . . . . . . . . . . . . 604

10.8 Turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . 610

10.8.1 Transition to turbulence . . . . . . . . . . . . . . . . 611

10.8.2 Lagrangian turbulence . . . . . . . . . . . . . . . . . 613

10.8.3 Features of turbulent motion . . . . . . . . . . . . . . 613

10.8.4 Decomposition into mean and fluctuating components 615

10.8.5 Inviscid scales . . . . . . . . . . . . . . . . . . . . . . 617

10.8.6 Viscous scales . . . . . . . . . . . . . . . . . . . . . . 618

10.8.7 Relation between inviscid and viscous scales . . . . . 618

10.8.8 Fourier analysis . . . . . . . . . . . . . . . . . . . . . 619

10.9 Analysis and modeling of turbulent flow . . . . . . . . . . . . 623

10.9.1 Reynolds stresses . . . . . . . . . . . . . . . . . . . . 623

10.9.2 Prandtl’s mixing length model . . . . . . . . . . . . . 625

10.9.3 Logarithmic law for wall-bounded shear flow . . . . . 627

10.9.4 Correlations . . . . . . . . . . . . . . . . . . . . . . . 628

11 Vortex Motion 631

11.1 Vorticity and circulation in two-dimensional flow . . . . . . . 631

11.2 Point vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . 633

11.2.1 Dirac’s delta function in a plane . . . . . . . . . . . . 634

11.2.2 Evolution of the point vortex strength . . . . . . . . . 636

11.2.3 Velocity of a point vortex . . . . . . . . . . . . . . . . 636

11.2.4 Motion of a collection of point vortices . . . . . . . . 636

11.2.5 Effect of boundaries . . . . . . . . . . . . . . . . . . . 637

11.2.6 A periodic array of point vortices . . . . . . . . . . . 639

11.2.7 A point vortex between two parallel walls . . . . . . . 641

11.2.8 A point vortex in a semi-infinite strip . . . . . . . . . 641

11.3 Two-dimensional flow with distributed vorticity . . . . . . . . 645

11.3.1 Vortex patches with uniform vorticity . . . . . . . . . 646