Fluid Dynamics: Theory, Computation, and Numerical Simulation

790 Pages·2009·8.93 MB·English

by  Constantine Pozrikidis (auth.)

FLUID DYNAMICS:

THEORY, COMPUTATION,
AND NUMERICAL SIMULATION
Fluid Dynamics:
Theory, Computation,
and Numerical Simula tion
Second Edition

C. Pozrikidis
C. Pozrikidis
U niversity of MVassachusetts
(cid:65)(cid:109)(cid:104)(cid:101)(cid:114)(cid:115)(cid:116)(cid:44)(cid:32)(cid:77)(cid:65)
(cid:85)(cid:83)(cid:65)

ISBN: 978-0-387-95869-9 e-ISBN: 978-0-387-95871-2
DOI: 1 0.1007/978-0-387-95871-2

L ibrary of Congress Control Number: 2008943356

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Preface
Readyaccesstocomputershasdefinedanewerainteachingandlearning. The
opportunitytoextendthesubjectmatteroftraditionalscienceandengineering
curricula into the realm of scientific computing has become not only desirable,
butalsonecessary. Thankstoportabilityandlowoverheadandoperatingcost,
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
fromamodernperspectivethatincorporatesnumericalandcomputerprogram-
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-
structionsimmediatelyfollowingproblemformulations. Seamlesslyinterjecting
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
enthusiastswithageneralinterestincomputing. Thisbookshouldbeespecially
appealingtothosewhoaremakingafirstexcursionintotheworldofnumerical
computation and computational fluid dynamics (CFD) beyond the black-box
approach. Thisbookshouldbeanidealtextforanintroductorycourseinfluid
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 illustrativeMatlab codes are discussed in the
text. Second, numerical methods are introduced on a need-to-know basis and
inorderofincreasingdifficulty: functioninterpolation, functiondifferentiation,
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
compilationofintroductory, advanced, andspecializedtextsonfluiddynamics,
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.
Amajorfeatureofthisbookistheaccompanyingfluiddynamicssoftwareli-
braryFdlibdiscussedinappendixA.TheFortran77andMatlabprograms
of Fdlibexplicitlyillustratehowcomputationalalgorithmstranslateintocom-
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
laminarchannelflows,tovortexflows,toflowpastairfoils. TheMatlabcodes
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
compilerisrequired. Freecompilersareavailablethankstothegnufoundation.
Theinputdataiseitherenteredfromthekeyboardorreadfromdatafiles. The
output is recorded in output files in tabular form so that it can be read and
displayedusingindependentgraphics,visualization,andanimationapplications
on any computer platform, including Matlab.
Thesecondeditionincorporatessignificantimprovementsinsubstanceand
style. First, additional examples, solved problems, and newmaterial have been
introducedforamorecomprehensivetreatmentofthevarioustopics. Examples
includesurfactanttransportandabriefintroductiontocompressibleflow. Sec-
ond, sample Matlab programs integrating numerical computation and graph-
icsvisualizationarelistedanddiscussedinthetext. AMatlabprimerexplaining
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 IntroductiontoKinematics 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 MoreonKinematics 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
ix
3 FlowComputationbasedonKinematics 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 ForcesandStresses 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