Preface ......................................................... v
1 Introduction to Kinematics ................................... l
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
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 .................... 1ll
2.10.1 The no-penetration boundary condition .......... 111
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
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
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
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
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
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
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 Karman'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 Kàrmàn-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
11.3.2 Contour dynamics ............................... 649
11.3.3 Gauss integration quadrature ................... 651
11.3.4 Representation with circular arcs .............. 652
11.4 Vorticity and circulation in three-dimensional flow ... 657
11.4.1 Preservation of circulation .................... 658
11.4.2 Flow induced by vorticity ...................... 660
11.5 Axisymmetric flow induced by vorticity ................ 661
11.5.1 Biot-Savart integral for axisymmetric flow ..... 663
11.5.2 Line vortex ring ............................... 666
11.5.3 Vortex rings with a finite core ................ 668
11.5.4 Motion of a collection of vortex rings ......... 672
11.5.5 Vortex patch in axisymmetric flow .............. 673
11.6 Three-dimensional vortex motion ....................... 675
11.6.1 Vortex particles ............................... 676
11.6.2 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 ............ 682
12.2.1 The Kutta-Joukowski theorem .................... 686
12.2.2 The Kutta-Joukowski condition .................. 687
12.3 Vortex panels ......................................... 687
12.3.1 From point vortices to vortex panels ........... 688
12.3.2 Vortex panels with uniform strength ............ 689
12.3.3 Vortex panel with linear strength density ...... 691
12.4 Vortex panel method ................................... 694
12.4.1 Velocity in terms of the panel strength ........ 698
12.4.2 Point collocation .............................. 699
12.4.3 Circulation and pressure coefficient ........... 700
12.4.4 Lift ........................................... 700
12.4.5 Vortex panel code .............................. 702
12.5 Vortex sheet representation ........................... 709
12.5.1 Thin airfoil theory ............................ 709
12.6 Point-source-dipole panels ............................ 717
12.6.1 Source-dipole panel method ..................... 718
12.6.2 Source-dipole representation ................... 720
12.6.3 Solution of the interior problem ............... 721
12.7 Point-source panels and Green's third identity ........ 723
12.7.1 Source panels with constant density ............ 723
12.7.2 Green's third identity ......................... 725
A FDLIB Software Library ..................................... 728
В References ................................................. 738
С Matlab Primer .............................................. 741
C.l 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 ......................................................... 3
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