Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer
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
Vlll 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 Ill 2.10.1 The no-penetration boundary condition Ill
IX 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 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
XI 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
xn 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 Kärmän'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 Kärmän-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
XVI 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
xvii 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 763