Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition
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1 Fluid Dynamics: Theory, Computation, and Numerical Simulation Second Edition C. Pozrikidis m Springer
2 Contents Preface v 1 Introduction to Kinematics Fluids and solids Fluid parcels and flow kinematics Coordinates, velocity, and acceleration Cylindrical polar coordinates Spherical polar coordinates Plane polar coordinates Fluid velocity Velocity vector field, streamlines and stagnation points Point particles and their trajectories Path lines Ordinary differential equations (ODEs) Explicit Euler method Modified Euler method Description in polar coordinates Streaklines Material surfaces and elementary motions Fluid parcel rotation Fluid parcel deformation Fluid parcel expansion Superposition of rotation, deformation, and expansion Rotated coordinates Flow decomposition Interpolation Interpolation in one dimension Interpolation in two dimensions Interpolation of the velocity in a two-dimensional flow Streamlines by interpolation 49 vii
3 Vlll 2 More on Kinematics Fundamental modes of fluid parcel motion Function linearization Velocity gradient tensor Relative motion of point particles Fundamental motions in two-dimensional flow Fundamental motions in three-dimensional flow Gradient in polar coordinates Fluid parcel expansion Fluid parcel rotation and vorticity Curl and vorticity Two-dimensional flow Axisymmetric flow Fluid parcel deformation Numerical differentiation Numerical differentiation in one dimension Numerical differentiation in two dimensions Velocity gradient and related functions Flow rate Areal flow rate and flux Areal flow rate across a line Numerical integration The Gauss divergence theorem in two dimensions Flow rate in a three-dimensional flow Gauss divergence theorem in three dimensions Axisymmetric flow Mass conservation Mass flux and mass flow rate Mass flow rate across a closed line The continuity equation Three-dimensional flow Rigid-body translation Evolution equation for the density Properties of point particles The material derivative The continuity equation Point particle acceleration Incompressible fluids and stream functions Mathematical consequences of incompressibility Stream function for two-dimensional flow Stream function for axisymmetric flow Kinematic conditions at boundaries Ill The no-penetration boundary condition Ill
4 IX Flow Computation based on Kinematics Flow classification based on kinematics Irrotational flow and the velocity potential Two-dimensional flow Incompressible fluids and the harmonic potential Three-dimensional flow Boundary conditions Cylindrical polar coordinates Spherical polar coordinates Plane polar coordinates Finite-difference methods Boundary conditions Finite-difference grid Finite-difference discretization Compilation of a linear system Linear solvers Gauss elimination A menagerie of other methods Two-dimensional point sources and point-source dipoles Function superposition and fundamental solutions Two-dimensional point source Two-dimensional point-source dipole Flow past a circular cylinder Sources and dipoles in the presence of boundaries Three-dimensional point sources and point-source dipoles Three-dimensional point source Three-dimensional point-source dipole Streaming flow past a sphere Sources and dipoles in the presence of boundaries Point vortices and line vortices The potential of irrotational circulatory flow Flow past a circular cylinder Circulation Line vortices in three-dimensional flow 161 Forces and Stresses Body forces and surface forces Body forces Surface forces Traction and the stress tensor Traction on either side of a fluid surface Traction on a boundary Symmetry of the stress tensor 170
5 X 4.3 Traction jump across a fluid interface Force balance at a two-dimensional interface Force balance at a three-dimensional interface Axisymmetric interfaces Stresses in a fluid at rest Pressure from molecular motions Jump in the pressure across an interface Constitutive equations Simple fluids Incompressible Newtonian fluids Viscosity Ideal fluids Significance of the pressure in incompressible fluids Pressure in compressible fluids Simple non-newtonian fluids Unidirectional shear flow Stresses in polar coordinates Cylindrical polar coordinates Spherical polar coordinates Plane polar coordinates Boundary conditions for the tangential velocity No-slip boundary condition Slip boundary condition Wall stresses in Newtonian fluids Shear stress Normal stress Interfacial surfactant transport Two-dimensional interfaces Axisymmetric interfaces Three-dimensional interfaces Hydrostatics Equilibrium of pressure and body forces Equilibrium of an infinitesimal parcel Gases in hydrostatics Liquids in hydrostatics Force exerted on immersed surfaces A sphere floating on a flat interface Archimedes' principle Net force on a submerged body Moments Interfacial shapes Curved interfaces 236
6 XI The Laplace-Young equation Three-dimensional interfaces A semi-infinite interface attached to an inclined plate Numerical method A meniscus between two parallel plates The shooting method A two-dimensional drop on a horizontal or inclined plane Drop on a horizontal plane A drop on an inclined plane Axisymmetric meniscus inside a tube Axisymmetric drop on a horizontal plane Solution space A sphere straddling an interface Spheroidal particle A three-dimensional meniscus Elliptic coordinates Finite-difference method Capillary force and torque Equation of Motion and Vorticity Transport Newton's second law of motion for a fluid parcel Rate of change of linear momentum Equation of parcel motion Two-dimensional flow Integral momentum balance Flow through a sudden enlargement Isentropic flow through a conduit Cauchy's equation of motion Hydrodynamic volume force Force on an infinitesimal parcel The equation of motion Evolution equations Cylindrical polar coordinates Spherical polar coordinates Plane polar coordinates Vortex force Summary of governing equation Accelerating frame of reference Euler's and Bernoulli's equations Boundary conditions Irrotational flow Steady irrotational flow Steady rotational flow 334
7 xn Flow with uniform vorticity The Navier-Stokes equation Pressure and viscous forces A radially expanding or contracting bubble Boundary conditions Polar coordinates Vorticity transport Two-dimensional flow Axisymmetric flow Three-dimensional flow Dynamic similitude and the Reynolds number Dimensional analysis Structure of a flow as a function of the Reynolds number Stokes flow Flow at high Reynolds numbers Laminar and turbulent flow Dimensionless numbers in fluid dynamics Channel, Tube, and Film Flow Steady flow in a two-dimensional channel Two-layer flow Multi-layer flow Power-law fluids Steady film flow down an inclined plane Multi-film flow Power-law fluids Steady flow through a circular tube Multi-layer tube flow Flow due to a translating sector Steady flow through an annular tube Multi-layer annular flow Steady flow in channels and tubes Elliptical tube Rectangular tube Triangular tube Semi-infinite rectangular channel Steady swirling flow Annular flow Multi-layer flow Transient channel flow Couette flow Impulsive motion of a plate in a semi-infinite fluid Pressure- and gravity-driven flow 406
8 xiii 7.8 Oscillatory channel flow Oscillatory Couette flow Rayleigh's oscillating plate Pulsating pressure-driven flow Transient and oscillatory flow in a circular tube Transient Poiseuille flow Pulsating pressure-driven flow Transient circular Couette flow More on Bessel functions Finite-Difference Methods Choice of governing equations Unidirectional flow; velocity/pressure formulation Governing equations Explicit finite-difference method Implicit finite-difference method Steady state Two-layer flow Unidirectional flow; velocity/vorticity formulation Boundary conditions for the vorticity Alternative set of equations Comparison with the velocity/pressure formulation Unidirectional flow; stream function/vorticity formulation Boundary conditions for the vorticity A semi-implicit method Two-dimensional flow; stream function/vorticity formulation Flow in a cavity Finite-difference grid Unsteady flow Steady flow Summary Velocity/pressure formulation Alternative system of governing equations Pressure boundary conditions Compatibility condition for the pressure Operator splitting and solenoidal projection Convection-diffusion step Projection step Boundary conditions for the intermediate velocity Flow in a cavity Computation of the pressure Staggered grids 485
9 XIV 9 Low Reynolds Number Flow Flow in narrow channels Governing equations Scaling Equations of lubrication flow Lubrication in a slider bearing Flow in a wavy channel Dynamic lifting Film flow on a horizontal or inclined wall Thin-film flow Numerical methods Multi-film flow on a horizontal or inclined wall Evolution equations Numerical methods Two-layer channel flow Flow due to the motion of a sphere Formulation in terms of the stream function Traction, force, and the Archimedes-Stokes law Point forces and point sources in Stokes flow The Oseen tensor and the point force Flow representation in terms of singularities A sphere moving inside a circular tube Boundary integral representation Two-dimensional Stokes flow Flow due to the motion of a cylinder Rotation of a circular cylinder Simple shear flow past a circular cylinder The Oseen tensor and the point force Local solutions Separation of variables Flow near a corner High Reynolds Number Flow Changes in the structure of a flow with increasing Reynolds number Prandtl boundary layer analysis Boundary-layer equations Surface curvilinear coordinates Parabolization Flow separation Blasius boundary layer on a semi-infinite plate Self-similarity and the Blasius equation Numerical solution 574
10 XV Wall shear stress and drag force Vorticity transport Displacement and momentum thickness Von Kärmän's approximate method Boundary layers in accelerating and decelerating flow Self-similarity Numerical solution Momentum integral method The von Kärmän-Pohlhausen method Pohlhausen polynomials Numerical solution Boundary layer around a curved body Instability of shear flows Stability analysis of shear flow Normal-mode analysis Finite-difference solution Turbulent flow Transition to turbulence Lagrangian turbulence Features of turbulent motion Decomposition into mean and fluctuating components Inviscid scales Viscous scales Relation between inviscid and viscous scales Fourier analysis Analysis and modeling of turbulent flow Reynolds stresses Prandtl's mixing length model Logarithmic law for wall-bounded shear flow Correlations Vortex Motion Vorticity and circulation in two-dimensional flow Point vortices Dirac's delta function in a plane Evolution of the point vortex strength Velocity of a point vortex Motion of a collection of point vortices Effect of boundaries A periodic array of point vortices A point vortex between two parallel walls A point vortex in a semi-infinite strip Two-dimensional flow with distributed vorticity Vortex patches with uniform vorticity 646
11 XVI Contour dynamics Gauss integration quadrature Representation with circular arcs Vorticity and circulation in three-dimensional flow Preservation of circulation Flow induced by vorticity Axisymmetric flow induced by vorticity Biot-Savart integral for axisymmetric flow Line vortex ring Vortex rings with a finite core Motion of a collection of vortex rings Vortex patch in axisymmetric flow Three-dimensional vortex motion Vortex particles Line vortices and the local induction approximation (LIA) Aerodynamics General features of flow past an aircraft Airfoils and the Kutta-Joukowski condition The Kutta-Joukowski theorem The Kutta-Joukowski condition Vortex panels From point vortices to vortex panels Vortex panels with uniform strength Vortex panel with linear strength density Vortex panel method Velocity in terms of the panel strength Point collocation Circulation and pressure coefficient Lift Vortex panel code Vortex sheet representation Thin airfoil theory Point-source-dipole panels Source-dipole panel method Source-dipole representation Solution of the interior problem Point-source panels and Green's third identity Source panels with constant density Green's third identity 725 A FDLIB Software Library 728
12 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
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