FLUID DYNAMICS:

THEORY, COMPUTATION, AND NUMERICAL SIMULATION

Fluid Dynamics: Theory, Computation,

Second Edition

C. Pozrikidis

and Numerical Simulation

V

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 9 8 7 6 5 4 3 2 1

springer.com

All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connectionwith any form of information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

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

C. PozrikidisUniversity of Massachusetts

Library of Congress Control Number: 2008943356

Amherst, MA

Preface

Ready access to computers has defined a new era in teaching and learning. Theopportunity to extend the subject matter of traditional science and engineeringcurricula 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, andoccasionally the only alternative, to physical experimentation.

The new framework has necessitated the writing of texts and monographsfrom 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 motivatesand underlines the urgency of the new elements, but neither compromises noroversimplifies 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 andimplementation are kept complementary and presented in a sequential fashion.In another approach, the coupling involves deriving computational methodsand simulation algorithms, and translating equations into computer code in-structions immediately following problem formulations. Seamlessly interjectingmethods of scientific computing in the traditional discourse offers a powerfulvenue for developing analytical skills and obtaining physical insight.

The goal of this book is to offer an introductory course in traditional andmodern fluid mechanics, covering topics in a way that unifies theory, computa-tion, computer programming, and numerical simulation. The approach is trulyintroductory in that only a few prerequisites are required. The intended au-dience includes undergraduate and entry-level graduate students, as well as abroader class of scientists, engineers, fluid dynamics and computational scienceenthusiasts with a general interest in computing. This book should be especiallyappealing to those who are making a first excursion into the world of numericalcomputation and computational fluid dynamics (CFD) beyond the black-boxapproach. This book should be an ideal text for an introductory course in fluidmechanics and CFD.

The presentation of the material is distinguished by two features. First,solution procedures and algorithms are developed immediately after problemformulations are presented, and illustrative Matlab codes are discussed in thetext. Second, numerical methods are introduced on a need-to-know basis andin 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-

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tation and simulation to study the effect of various parameters determining aflow.

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 selectedcompilation of introductory, advanced, and specialized texts on fluid dynamics,calculus, numerical methods, and computational fluid dynamics are listed inappendix B. The reader who wishes to focus on a particular topic is directedto 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 programsof 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: fromlaminar channel flows, to vortex flows, to flow past airfoils. The Matlab codesof Fdlib combine numerical computation, graphics display, data visualizationand 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. Theoutput is recorded in output files in tabular form so that it can be read anddisplayed using independent graphics, visualization, and animation applicationson any computer platform, including Matlab.

The second edition incorporates significant improvements in substance andstyle. First, additional examples, solved problems, and new material have beenintroduced for a more comprehensive treatment of the various topics. Examplesinclude 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 explainingbasic programming procedures is presented in appendix C. Third, the revisedtext 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 . . . . . . . . . . . . . 31.3.1 Cylindrical polar coordinates . . . . . . . . . . . . . . 61.3.2 Spherical polar coordinates . . . . . . . . . . . . . . . 91.3.3 Plane polar coordinates . . . . . . . . . . . . . . . . . 13

1.4 Fluid velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.1 Velocity vector field, streamlines and stagnation points 18

1.5 Point particles and their trajectories . . . . . . . . . . . . . . 191.5.1 Path lines . . . . . . . . . . . . . . . . . . . . . . . . 201.5.2 Ordinary differential equations (ODEs) . . . . . . . . 201.5.3 Explicit Euler method . . . . . . . . . . . . . . . . . . 211.5.4 Modified Euler method . . . . . . . . . . . . . . . . . 231.5.5 Description in polar coordinates . . . . . . . . . . . . 261.5.6 Streaklines . . . . . . . . . . . . . . . . . . . . . . . . 27

1.6 Material surfaces and elementary motions . . . . . . . . . . . 281.6.1 Fluid parcel rotation . . . . . . . . . . . . . . . . . . 281.6.2 Fluid parcel deformation . . . . . . . . . . . . . . . . 291.6.3 Fluid parcel expansion . . . . . . . . . . . . . . . . . 301.6.4 Superposition of rotation, deformation, and expansion 311.6.5 Rotated coordinates . . . . . . . . . . . . . . . . . . . 321.6.6 Flow decomposition . . . . . . . . . . . . . . . . . . . 34

1.7 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.7.1 Interpolation in one dimension . . . . . . . . . . . . . 381.7.2 Interpolation in two dimensions . . . . . . . . . . . . 421.7.3 Interpolation of the velocity in a two-dimensional flow 451.7.4 Streamlines by interpolation . . . . . . . . . . . . . . 49

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2 More on Kinematics 54

2.1 Fundamental modes of fluid parcel motion . . . . . . . . . . . 542.1.1 Function linearization . . . . . . . . . . . . . . . . . . 552.1.2 Velocity gradient tensor . . . . . . . . . . . . . . . . . 572.1.3 Relative motion of point particles . . . . . . . . . . . 592.1.4 Fundamental motions in two-dimensional flow . . . . 602.1.5 Fundamental motions in three-dimensional flow . . . 622.1.6 Gradient in polar coordinates . . . . . . . . . . . . . 62

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

2.3 Fluid parcel rotation and vorticity . . . . . . . . . . . . . . . 662.3.1 Curl and vorticity . . . . . . . . . . . . . . . . . . . . 682.3.2 Two-dimensional flow . . . . . . . . . . . . . . . . . . 702.3.3 Axisymmetric flow . . . . . . . . . . . . . . . . . . . . 70

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

2.5 Numerical differentiation . . . . . . . . . . . . . . . . . . . . . 742.5.1 Numerical differentiation in one dimension . . . . . . 742.5.2 Numerical differentiation in two dimensions . . . . . . 762.5.3 Velocity gradient and related functions . . . . . . . . 78

2.6 Flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.6.1 Areal flow rate and flux . . . . . . . . . . . . . . . . . 872.6.2 Areal flow rate across a line . . . . . . . . . . . . . . 882.6.3 Numerical integration . . . . . . . . . . . . . . . . . . 892.6.4 The Gauss divergence theorem in two dimensions . . 902.6.5 Flow rate in a three-dimensional flow . . . . . . . . . 912.6.6 Gauss divergence theorem in three dimensions . . . . 922.6.7 Axisymmetric flow . . . . . . . . . . . . . . . . . . . . 92

2.7 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . 942.7.1 Mass flux and mass flow rate . . . . . . . . . . . . . . 942.7.2 Mass flow rate across a closed line . . . . . . . . . . . 942.7.3 The continuity equation . . . . . . . . . . . . . . . . . 952.7.4 Three-dimensional flow . . . . . . . . . . . . . . . . . 962.7.5 Rigid-body translation . . . . . . . . . . . . . . . . . 962.7.6 Evolution equation for the density . . . . . . . . . . . 97

2.8 Properties of point particles . . . . . . . . . . . . . . . . . . . 992.8.1 The material derivative . . . . . . . . . . . . . . . . . 1002.8.2 The continuity equation . . . . . . . . . . . . . . . . . 1012.8.3 Point particle acceleration . . . . . . . . . . . . . . . 102

2.9 Incompressible fluids and stream functions . . . . . . . . . . . 1062.9.1 Mathematical consequences of incompressibility . . . 1072.9.2 Stream function for two-dimensional flow . . . . . . . 1072.9.3 Stream function for axisymmetric flow . . . . . . . . 109

2.10 Kinematic conditions at boundaries . . . . . . . . . . . . . . . 1112.10.1 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 . . . . . . . . . . . 1173.2.1 Two-dimensional flow . . . . . . . . . . . . . . . . . . 1173.2.2 Incompressible fluids and the harmonic potential . . . 1193.2.3 Three-dimensional flow . . . . . . . . . . . . . . . . . 1203.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . 1213.2.5 Cylindrical polar coordinates . . . . . . . . . . . . . . 1223.2.6 Spherical polar coordinates . . . . . . . . . . . . . . . 1223.2.7 Plane polar coordinates . . . . . . . . . . . . . . . . . 123

3.3 Finite-difference methods . . . . . . . . . . . . . . . . . . . . 1243.3.1 Boundary conditions . . . . . . . . . . . . . . . . . . 1243.3.2 Finite-difference grid . . . . . . . . . . . . . . . . . . 1263.3.3 Finite-difference discretization . . . . . . . . . . . . . 1273.3.4 Compilation of a linear system . . . . . . . . . . . . . 128

3.4 Linear solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1383.4.1 Gauss elimination . . . . . . . . . . . . . . . . . . . . 1393.4.2 A menagerie of other methods . . . . . . . . . . . . . 140

3.5 Two-dimensional point sources and point-source dipoles . . . 1413.5.1 Function superposition and fundamental solutions . . 1413.5.2 Two-dimensional point source . . . . . . . . . . . . . 1413.5.3 Two-dimensional point-source dipole . . . . . . . . . 1443.5.4 Flow past a circular cylinder . . . . . . . . . . . . . . 1483.5.5 Sources and dipoles in the presence of boundaries . . 149

3.6 Three-dimensional point sources and point-source dipoles . . 1513.6.1 Three-dimensional point source . . . . . . . . . . . . 1513.6.2 Three-dimensional point-source dipole . . . . . . . . . 1523.6.3 Streaming flow past a sphere . . . . . . . . . . . . . . 1533.6.4 Sources and dipoles in the presence of boundaries . . 154

3.7 Point vortices and line vortices . . . . . . . . . . . . . . . . . 1553.7.1 The potential of irrotational circulatory flow . . . . . 1563.7.2 Flow past a circular cylinder . . . . . . . . . . . . . . 1573.7.3 Circulation . . . . . . . . . . . . . . . . . . . . . . . . 1583.7.4 Line vortices in three-dimensional flow . . . . . . . . 161

4 Forces and Stresses 163

4.1 Body forces and surface forces . . . . . . . . . . . . . . . . . . 1634.1.1 Body forces . . . . . . . . . . . . . . . . . . . . . . . . 1634.1.2 Surface forces . . . . . . . . . . . . . . . . . . . . . . 164

4.2 Traction and the stress tensor . . . . . . . . . . . . . . . . . . 1654.2.1 Traction on either side of a fluid surface . . . . . . . . 1684.2.2 Traction on a boundary . . . . . . . . . . . . . . . . . 1694.2.3 Symmetry of the stress tensor . . . . . . . . . . . . . 170

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4.3 Traction jump across a fluid interface . . . . . . . . . . . . . . 1714.3.1 Force balance at a two-dimensional interface . . . . . 1724.3.2 Force balance at a three-dimensional interface . . . . 1764.3.3 Axisymmetric interfaces . . . . . . . . . . . . . . . . . 179

4.4 Stresses in a fluid at rest . . . . . . . . . . . . . . . . . . . . . 1834.4.1 Pressure from molecular motions . . . . . . . . . . . . 1844.4.2 Jump in the pressure across an interface . . . . . . . 185

4.5 Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 1864.5.1 Simple fluids . . . . . . . . . . . . . . . . . . . . . . . 1884.5.2 Incompressible Newtonian fluids . . . . . . . . . . . . 1884.5.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . 1904.5.4 Ideal fluids . . . . . . . . . . . . . . . . . . . . . . . . 1924.5.5 Significance of the pressure in incompressible fluids . 1934.5.6 Pressure in compressible fluids . . . . . . . . . . . . . 193

4.6 Simple non-Newtonian fluids . . . . . . . . . . . . . . . . . . 1964.6.1 Unidirectional shear flow . . . . . . . . . . . . . . . . 197

4.7 Stresses in polar coordinates . . . . . . . . . . . . . . . . . . . 1994.7.1 Cylindrical polar coordinates . . . . . . . . . . . . . . 2004.7.2 Spherical polar coordinates . . . . . . . . . . . . . . . 2024.7.3 Plane polar coordinates . . . . . . . . . . . . . . . . . 204

4.8 Boundary conditions for the tangential velocity . . . . . . . . 2064.8.1 No-slip boundary condition . . . . . . . . . . . . . . . 2064.8.2 Slip boundary condition . . . . . . . . . . . . . . . . . 207

4.9 Wall stresses in Newtonian fluids . . . . . . . . . . . . . . . . 2084.9.1 Shear stress . . . . . . . . . . . . . . . . . . . . . . . 2084.9.2 Normal stress . . . . . . . . . . . . . . . . . . . . . . 209

4.10 Interfacial surfactant transport . . . . . . . . . . . . . . . . . 2104.10.1 Two-dimensional interfaces . . . . . . . . . . . . . . . 2104.10.2 Axisymmetric interfaces . . . . . . . . . . . . . . . . . 2144.10.3 Three-dimensional interfaces . . . . . . . . . . . . . . 216

5 Hydrostatics 218

5.1 Equilibrium of pressure and body forces . . . . . . . . . . . . 2185.1.1 Equilibrium of an infinitesimal parcel . . . . . . . . . 2205.1.2 Gases in hydrostatics . . . . . . . . . . . . . . . . . . 2225.1.3 Liquids in hydrostatics . . . . . . . . . . . . . . . . . 223

5.2 Force exerted on immersed surfaces . . . . . . . . . . . . . . . 2255.2.1 A sphere floating on a flat interface . . . . . . . . . . 226

5.3 Archimedes’ principle . . . . . . . . . . . . . . . . . . . . . . 2315.3.1 Net force on a submerged body . . . . . . . . . . . . 2335.3.2 Moments . . . . . . . . . . . . . . . . . . . . . . . . . 234

5.4 Interfacial shapes . . . . . . . . . . . . . . . . . . . . . . . . . 2355.4.1 Curved interfaces . . . . . . . . . . . . . . . . . . . . 236

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5.4.2 The Laplace-Young equation . . . . . . . . . . . . . . 2375.4.3 Three-dimensional interfaces . . . . . . . . . . . . . . 238

5.5 A semi-infinite interface attached to an inclined plate . . . . . 2395.5.1 Numerical method . . . . . . . . . . . . . . . . . . . . 241

5.6 A meniscus between two parallel plates . . . . . . . . . . . . . 2455.6.1 The shooting method . . . . . . . . . . . . . . . . . . 249

5.7 A two-dimensional drop on a horizontal or inclined plane . . 2535.7.1 Drop on a horizontal plane . . . . . . . . . . . . . . . 2535.7.2 A drop on an inclined plane . . . . . . . . . . . . . . 261

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

5.9 Axisymmetric drop on a horizontal plane . . . . . . . . . . . 2765.9.1 Solution space . . . . . . . . . . . . . . . . . . . . . . 278

5.10 A sphere straddling an interface . . . . . . . . . . . . . . . . . 2865.10.1 Spheroidal particle . . . . . . . . . . . . . . . . . . . . 296

5.11 A three-dimensional meniscus . . . . . . . . . . . . . . . . . . 2985.11.1 Elliptic coordinates . . . . . . . . . . . . . . . . . . . 2995.11.2 Finite-difference method . . . . . . . . . . . . . . . . 3005.11.3 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 . . . . . . . 3086.1.1 Rate of change of linear momentum . . . . . . . . . . 3096.1.2 Equation of parcel motion . . . . . . . . . . . . . . . 3096.1.3 Two-dimensional flow . . . . . . . . . . . . . . . . . . 310

6.2 Integral momentum balance . . . . . . . . . . . . . . . . . . . 3136.2.1 Flow through a sudden enlargement . . . . . . . . . . 3166.2.2 Isentropic flow through a conduit . . . . . . . . . . . 318

6.3 Cauchy’s equation of motion . . . . . . . . . . . . . . . . . . 3196.3.1 Hydrodynamic volume force . . . . . . . . . . . . . . 3206.3.2 Force on an infinitesimal parcel . . . . . . . . . . . . 3206.3.3 The equation of motion . . . . . . . . . . . . . . . . . 3226.3.4 Evolution equations . . . . . . . . . . . . . . . . . . . 3236.3.5 Cylindrical polar coordinates . . . . . . . . . . . . . . 3236.3.6 Spherical polar coordinates . . . . . . . . . . . . . . . 3256.3.7 Plane polar coordinates . . . . . . . . . . . . . . . . . 3256.3.8 Vortex force . . . . . . . . . . . . . . . . . . . . . . . 3266.3.9 Summary of governing equation . . . . . . . . . . . . 3266.3.10 Accelerating frame of reference . . . . . . . . . . . . . 326

6.4 Euler’s and Bernoulli’s equations . . . . . . . . . . . . . . . . 3276.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . 3286.4.2 Irrotational flow . . . . . . . . . . . . . . . . . . . . . 3296.4.3 Steady irrotational flow . . . . . . . . . . . . . . . . . 3316.4.4 Steady rotational flow . . . . . . . . . . . . . . . . . . 334

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

6.5 The Navier-Stokes equation . . . . . . . . . . . . . . . . . . . 3376.5.1 Pressure and viscous forces . . . . . . . . . . . . . . . 3386.5.2 A radially expanding or contracting bubble . . . . . . 3396.5.3 Boundary conditions . . . . . . . . . . . . . . . . . . 3406.5.4 Polar coordinates . . . . . . . . . . . . . . . . . . . . 341

6.6 Vorticity transport . . . . . . . . . . . . . . . . . . . . . . . . 3436.6.1 Two-dimensional flow . . . . . . . . . . . . . . . . . . 3436.6.2 Axisymmetric flow . . . . . . . . . . . . . . . . . . . . 3466.6.3 Three-dimensional flow . . . . . . . . . . . . . . . . . 347

6.7 Dynamic similitude and the Reynolds number . . . . . . . . . 3506.7.1 Dimensional analysis . . . . . . . . . . . . . . . . . . 352

6.8 Structure of a flow as a function of the Reynolds number . . 3556.8.1 Stokes flow . . . . . . . . . . . . . . . . . . . . . . . . 3566.8.2 Flow at high Reynolds numbers . . . . . . . . . . . . 3566.8.3 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 . . . . . . . . . . . 3607.1.1 Two-layer flow . . . . . . . . . . . . . . . . . . . . . . 3637.1.2 Multi-layer flow . . . . . . . . . . . . . . . . . . . . . 3657.1.3 Power-law fluids . . . . . . . . . . . . . . . . . . . . . 370

7.2 Steady film flow down an inclined plane . . . . . . . . . . . . 3737.2.1 Multi-film flow . . . . . . . . . . . . . . . . . . . . . . 3747.2.2 Power-law fluids . . . . . . . . . . . . . . . . . . . . . 375

7.3 Steady flow through a circular tube . . . . . . . . . . . . . . . 3777.3.1 Multi-layer tube flow . . . . . . . . . . . . . . . . . . 3807.3.2 Flow due to a translating sector . . . . . . . . . . . . 380

7.4 Steady flow through an annular tube . . . . . . . . . . . . . . 3837.4.1 Multi-layer annular flow . . . . . . . . . . . . . . . . . 387

7.5 Steady flow in channels and tubes . . . . . . . . . . . . . . . 3877.5.1 Elliptical tube . . . . . . . . . . . . . . . . . . . . . . 3887.5.2 Rectangular tube . . . . . . . . . . . . . . . . . . . . 3907.5.3 Triangular tube . . . . . . . . . . . . . . . . . . . . . 3937.5.4 Semi-infinite rectangular channel . . . . . . . . . . . . 393

7.6 Steady swirling flow . . . . . . . . . . . . . . . . . . . . . . . 3957.6.1 Annular flow . . . . . . . . . . . . . . . . . . . . . . . 3967.6.2 Multi-layer flow . . . . . . . . . . . . . . . . . . . . . 399

7.7 Transient channel flow . . . . . . . . . . . . . . . . . . . . . . 4007.7.1 Couette flow . . . . . . . . . . . . . . . . . . . . . . . 4007.7.2 Impulsive motion of a plate in a semi-infinite fluid . . 4037.7.3 Pressure- and gravity-driven flow . . . . . . . . . . . 406

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7.8 Oscillatory channel flow . . . . . . . . . . . . . . . . . . . . . 4097.8.1 Oscillatory Couette flow . . . . . . . . . . . . . . . . 4097.8.2 Rayleigh’s oscillating plate . . . . . . . . . . . . . . . 4117.8.3 Pulsating pressure-driven flow . . . . . . . . . . . . . 413

7.9 Transient and oscillatory flow in a circular tube . . . . . . . . 4157.9.1 Transient Poiseuille flow . . . . . . . . . . . . . . . . 4157.9.2 Pulsating pressure-driven flow . . . . . . . . . . . . . 4207.9.3 Transient circular Couette flow . . . . . . . . . . . . . 4227.9.4 More on Bessel functions . . . . . . . . . . . . . . . . 422

8 Finite-Difference Methods 424

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

8.2 Unidirectional flow; velocity/pressure formulation . . . . . . . 4258.2.1 Governing equations . . . . . . . . . . . . . . . . . . . 4268.2.2 Explicit finite-difference method . . . . . . . . . . . . 4268.2.3 Implicit finite-difference method . . . . . . . . . . . . 4298.2.4 Steady state . . . . . . . . . . . . . . . . . . . . . . . 4358.2.5 Two-layer flow . . . . . . . . . . . . . . . . . . . . . . 436

8.3 Unidirectional flow; velocity/vorticity formulation . . . . . . . 4438.3.1 Boundary conditions for the vorticity . . . . . . . . . 4448.3.2 Alternative set of equations . . . . . . . . . . . . . . . 4458.3.3 Comparison with the velocity/pressure formulation . 446

8.4 Unidirectional flow; stream function/vorticity formulation . . 4478.4.1 Boundary conditions for the vorticity . . . . . . . . . 4488.4.2 A semi-implicit method . . . . . . . . . . . . . . . . . 449

8.5 Two-dimensional flow;stream function/vorticity formulation . . . . . . . . . . . . . 4518.5.1 Flow in a cavity . . . . . . . . . . . . . . . . . . . . . 4518.5.2 Finite-difference grid . . . . . . . . . . . . . . . . . . 4528.5.3 Unsteady flow . . . . . . . . . . . . . . . . . . . . . . 4538.5.4 Steady flow . . . . . . . . . . . . . . . . . . . . . . . . 4548.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 460

8.6 Velocity/pressure formulation . . . . . . . . . . . . . . . . . . 4638.6.1 Alternative system of governing equations . . . . . . 4648.6.2 Pressure boundary conditions . . . . . . . . . . . . . 4658.6.3 Compatibility condition for the pressure . . . . . . . 465

8.7 Operator splitting and solenoidal projection . . . . . . . . . . 4668.7.1 Convection–diffusion step . . . . . . . . . . . . . . . . 4678.7.2 Projection step . . . . . . . . . . . . . . . . . . . . . . 4698.7.3 Boundary conditions for the intermediate velocity . . 4718.7.4 Flow in a cavity . . . . . . . . . . . . . . . . . . . . . 4718.7.5 Computation of the pressure . . . . . . . . . . . . . . 484

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

xiv

9 Low Reynolds Number Flow 494

9.1 Flow in narrow channels . . . . . . . . . . . . . . . . . . . . . 4949.1.1 Governing equations . . . . . . . . . . . . . . . . . . . 4959.1.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 4959.1.3 Equations of lubrication flow . . . . . . . . . . . . . . 4979.1.4 Lubrication in a slider bearing . . . . . . . . . . . . . 4979.1.5 Flow in a wavy channel . . . . . . . . . . . . . . . . . 5009.1.6 Dynamic lifting . . . . . . . . . . . . . . . . . . . . . 503

9.2 Film flow on a horizontal or inclined wall . . . . . . . . . . . 5059.2.1 Thin-film flow . . . . . . . . . . . . . . . . . . . . . . 5069.2.2 Numerical methods . . . . . . . . . . . . . . . . . . . 509

9.3 Multi-film flow on a horizontal or inclined wall . . . . . . . . 5119.3.1 Evolution equations . . . . . . . . . . . . . . . . . . . 5149.3.2 Numerical methods . . . . . . . . . . . . . . . . . . . 516

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

9.5 Flow due to the motion of a sphere . . . . . . . . . . . . . . . 5349.5.1 Formulation in terms of the stream function . . . . . 5359.5.2 Traction, force, and the Archimedes-Stokes law . . . . 539

9.6 Point forces and point sources in Stokes flow . . . . . . . . . 5419.6.1 The Oseen tensor and the point force . . . . . . . . . 5429.6.2 Flow representation in terms of singularities . . . . . 5449.6.3 A sphere moving inside a circular tube . . . . . . . . 5449.6.4 Boundary integral representation . . . . . . . . . . . . 547

9.7 Two-dimensional Stokes flow . . . . . . . . . . . . . . . . . . 5499.7.1 Flow due to the motion of a cylinder . . . . . . . . . 5499.7.2 Rotation of a circular cylinder . . . . . . . . . . . . . 5529.7.3 Simple shear flow past a circular cylinder . . . . . . . 5529.7.4 The Oseen tensor and the point force . . . . . . . . . 553

9.8 Local solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 5549.8.1 Separation of variables . . . . . . . . . . . . . . . . . 5559.8.2 Flow near a corner . . . . . . . . . . . . . . . . . . . . 557

10 High Reynolds Number Flow 562

10.1 Changes in the structure of a flowwith increasing Reynolds number . . . . . . . . . . . . . . . . 562

10.2 Prandtl boundary layer analysis . . . . . . . . . . . . . . . . . 56610.2.1 Boundary-layer equations . . . . . . . . . . . . . . . . 56810.2.2 Surface curvilinear coordinates . . . . . . . . . . . . . 56910.2.3 Parabolization . . . . . . . . . . . . . . . . . . . . . . 57010.2.4 Flow separation . . . . . . . . . . . . . . . . . . . . . 570

10.3 Blasius boundary layer on a semi-infinite plate . . . . . . . . 57110.3.1 Self-similarity and the Blasius equation . . . . . . . . 57110.3.2 Numerical solution . . . . . . . . . . . . . . . . . . . 574

xv

10.3.3 Wall shear stress and drag force . . . . . . . . . . . . 57610.3.4 Vorticity transport . . . . . . . . . . . . . . . . . . . 577

10.4 Displacement and momentum thickness . . . . . . . . . . . . 57910.4.1 Von Karman’s approximate method . . . . . . . . . . 581

10.5 Boundary layers in accelerating and decelerating flow . . . . . 58310.5.1 Self-similarity . . . . . . . . . . . . . . . . . . . . . . 58510.5.2 Numerical solution . . . . . . . . . . . . . . . . . . . 586

10.6 Momentum integral method . . . . . . . . . . . . . . . . . . . 58710.6.1 The von Karman-Pohlhausen method . . . . . . . . . 58910.6.2 Pohlhausen polynomials . . . . . . . . . . . . . . . . . 59010.6.3 Numerical solution . . . . . . . . . . . . . . . . . . . 59210.6.4 Boundary layer around a curved body . . . . . . . . . 595

10.7 Instability of shear flows . . . . . . . . . . . . . . . . . . . . . 59910.7.1 Stability analysis of shear flow . . . . . . . . . . . . . 60010.7.2 Normal-mode analysis . . . . . . . . . . . . . . . . . . 60110.7.3 Finite-difference solution . . . . . . . . . . . . . . . . 604

10.8 Turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . 61010.8.1 Transition to turbulence . . . . . . . . . . . . . . . . 61110.8.2 Lagrangian turbulence . . . . . . . . . . . . . . . . . 61310.8.3 Features of turbulent motion . . . . . . . . . . . . . . 61310.8.4 Decomposition into mean and fluctuating components 61510.8.5 Inviscid scales . . . . . . . . . . . . . . . . . . . . . . 61710.8.6 Viscous scales . . . . . . . . . . . . . . . . . . . . . . 61810.8.7 Relation between inviscid and viscous scales . . . . . 61810.8.8 Fourier analysis . . . . . . . . . . . . . . . . . . . . . 619

10.9 Analysis and modeling of turbulent flow . . . . . . . . . . . . 62310.9.1 Reynolds stresses . . . . . . . . . . . . . . . . . . . . 62310.9.2 Prandtl’s mixing length model . . . . . . . . . . . . . 62510.9.3 Logarithmic law for wall-bounded shear flow . . . . . 62710.9.4 Correlations . . . . . . . . . . . . . . . . . . . . . . . 628

11 Vortex Motion 631

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

11.2 Point vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . 63311.2.1 Dirac’s delta function in a plane . . . . . . . . . . . . 63411.2.2 Evolution of the point vortex strength . . . . . . . . . 63611.2.3 Velocity of a point vortex . . . . . . . . . . . . . . . . 63611.2.4 Motion of a collection of point vortices . . . . . . . . 63611.2.5 Effect of boundaries . . . . . . . . . . . . . . . . . . . 63711.2.6 A periodic array of point vortices . . . . . . . . . . . 63911.2.7 A point vortex between two parallel walls . . . . . . . 64111.2.8 A point vortex in a semi-infinite strip . . . . . . . . . 641

11.3 Two-dimensional flow with distributed vorticity . . . . . . . . 64511.3.1 Vortex patches with uniform vorticity . . . . . . . . . 646

xvi

11.3.2 Contour dynamics . . . . . . . . . . . . . . . . . . . . 64911.3.3 Gauss integration quadrature . . . . . . . . . . . . . . 65111.3.4 Representation with circular arcs . . . . . . . . . . . 652

11.4 Vorticity and circulation in three-dimensional flow . . . . . . 65711.4.1 Preservation of circulation . . . . . . . . . . . . . . . 65811.4.2 Flow induced by vorticity . . . . . . . . . . . . . . . . 660

11.5 Axisymmetric flow induced by vorticity . . . . . . . . . . . . 66111.5.1 Biot-Savart integral for axisymmetric flow . . . . . . 66311.5.2 Line vortex ring . . . . . . . . . . . . . . . . . . . . . 66611.5.3 Vortex rings with a finite core . . . . . . . . . . . . . 66811.5.4 Motion of a collection of vortex rings . . . . . . . . . 67211.5.5 Vortex patch in axisymmetric flow . . . . . . . . . . . 673

11.6 Three-dimensional vortex motion . . . . . . . . . . . . . . . . 67511.6.1 Vortex particles . . . . . . . . . . . . . . . . . . . . . 67611.6.2 Line vortices

and the local induction approximation (LIA) . . . . . 676

12 Aerodynamics 680

12.1 General features of flow past an aircraft . . . . . . . . . . . . 680

12.2 Airfoils and the Kutta-Joukowski condition . . . . . . . . . . 68212.2.1 The Kutta-Joukowski theorem . . . . . . . . . . . . . 68612.2.2 The Kutta-Joukowski condition . . . . . . . . . . . . 687

12.3 Vortex panels . . . . . . . . . . . . . . . . . . . . . . . . . . . 68712.3.1 From point vortices to vortex panels . . . . . . . . . . 68812.3.2 Vortex panels with uniform strength . . . . . . . . . . 68912.3.3 Vortex panel with linear strength density . . . . . . . 691

12.4 Vortex panel method . . . . . . . . . . . . . . . . . . . . . . . 69412.4.1 Velocity in terms of the panel strength . . . . . . . . 69812.4.2 Point collocation . . . . . . . . . . . . . . . . . . . . . 69912.4.3 Circulation and pressure coefficient . . . . . . . . . . 70012.4.4 Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70012.4.5 Vortex panel code . . . . . . . . . . . . . . . . . . . . 702

12.5 Vortex sheet representation . . . . . . . . . . . . . . . . . . . 70912.5.1 Thin airfoil theory . . . . . . . . . . . . . . . . . . . . 709

12.6 Point-source-dipole panels . . . . . . . . . . . . . . . . . . . . 71712.6.1 Source-dipole panel method . . . . . . . . . . . . . . 71812.6.2 Source-dipole representation . . . . . . . . . . . . . . 72012.6.3 Solution of the interior problem . . . . . . . . . . . . 721

12.7 Point-source panels and Green’s third identity . . . . . . . . . 72312.7.1 Source panels with constant density . . . . . . . . . . 72312.7.2 Green’s third identity . . . . . . . . . . . . . . . . . . 725

A FDLIB Software Library 728

xvii

B References 738

C Matlab Primer 741

C.1 Invoking Matlab . . . . . . . . . . . . . . . . . . . . . . . . 741

C.2 Matlab programming . . . . . . . . . . . . . . . . . . . . . . 742

C.3 Matlab Grammar and syntax . . . . . . . . . . . . . . . . . . 743

C.4 Precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744

C.5 Matlab commands . . . . . . . . . . . . . . . . . . . . . . . 744

C.6 Matlab examples . . . . . . . . . . . . . . . . . . . . . . . . . 747

C.7 Matlab functions . . . . . . . . . . . . . . . . . . . . . . . . 750

C.8 User-defined functions . . . . . . . . . . . . . . . . . . . . . . 751

C.9 Matlab graphics . . . . . . . . . . . . . . . . . . . . . . . . . 755

Index 763