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 twodimensional 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 twodimensional flow Fundamental motions in threedimensional flow Gradient in polar coordinates Fluid parcel expansion Fluid parcel rotation and vorticity Curl and vorticity Twodimensional 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 threedimensional 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 Threedimensional flow Rigidbody 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 twodimensional flow Stream function for axisymmetric flow Kinematic conditions at boundaries Ill The nopenetration boundary condition Ill
4 IX Flow Computation based on Kinematics Flow classification based on kinematics Irrotational flow and the velocity potential Twodimensional flow Incompressible fluids and the harmonic potential Threedimensional flow Boundary conditions Cylindrical polar coordinates Spherical polar coordinates Plane polar coordinates Finitedifference methods Boundary conditions Finitedifference grid Finitedifference discretization Compilation of a linear system Linear solvers Gauss elimination A menagerie of other methods Twodimensional point sources and pointsource dipoles Function superposition and fundamental solutions Twodimensional point source Twodimensional pointsource dipole Flow past a circular cylinder Sources and dipoles in the presence of boundaries Threedimensional point sources and pointsource dipoles Threedimensional point source Threedimensional pointsource 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 threedimensional 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 twodimensional interface Force balance at a threedimensional 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 nonnewtonian fluids Unidirectional shear flow Stresses in polar coordinates Cylindrical polar coordinates Spherical polar coordinates Plane polar coordinates Boundary conditions for the tangential velocity Noslip boundary condition Slip boundary condition Wall stresses in Newtonian fluids Shear stress Normal stress Interfacial surfactant transport Twodimensional interfaces Axisymmetric interfaces Threedimensional 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 LaplaceYoung equation Threedimensional interfaces A semiinfinite interface attached to an inclined plate Numerical method A meniscus between two parallel plates The shooting method A twodimensional 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 threedimensional meniscus Elliptic coordinates Finitedifference 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 Twodimensional 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 NavierStokes equation Pressure and viscous forces A radially expanding or contracting bubble Boundary conditions Polar coordinates Vorticity transport Twodimensional flow Axisymmetric flow Threedimensional 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 twodimensional channel Twolayer flow Multilayer flow Powerlaw fluids Steady film flow down an inclined plane Multifilm flow Powerlaw fluids Steady flow through a circular tube Multilayer tube flow Flow due to a translating sector Steady flow through an annular tube Multilayer annular flow Steady flow in channels and tubes Elliptical tube Rectangular tube Triangular tube Semiinfinite rectangular channel Steady swirling flow Annular flow Multilayer flow Transient channel flow Couette flow Impulsive motion of a plate in a semiinfinite fluid Pressure and gravitydriven flow 406
8 xiii 7.8 Oscillatory channel flow Oscillatory Couette flow Rayleigh's oscillating plate Pulsating pressuredriven flow Transient and oscillatory flow in a circular tube Transient Poiseuille flow Pulsating pressuredriven flow Transient circular Couette flow More on Bessel functions FiniteDifference Methods Choice of governing equations Unidirectional flow; velocity/pressure formulation Governing equations Explicit finitedifference method Implicit finitedifference method Steady state Twolayer 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 semiimplicit method Twodimensional flow; stream function/vorticity formulation Flow in a cavity Finitedifference 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 Convectiondiffusion 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 Thinfilm flow Numerical methods Multifilm flow on a horizontal or inclined wall Evolution equations Numerical methods Twolayer channel flow Flow due to the motion of a sphere Formulation in terms of the stream function Traction, force, and the ArchimedesStokes 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 Twodimensional 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 Boundarylayer equations Surface curvilinear coordinates Parabolization Flow separation Blasius boundary layer on a semiinfinite plate Selfsimilarity 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 Selfsimilarity Numerical solution Momentum integral method The von KärmänPohlhausen method Pohlhausen polynomials Numerical solution Boundary layer around a curved body Instability of shear flows Stability analysis of shear flow Normalmode analysis Finitedifference 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 wallbounded shear flow Correlations Vortex Motion Vorticity and circulation in twodimensional 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 semiinfinite strip Twodimensional 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 threedimensional flow Preservation of circulation Flow induced by vorticity Axisymmetric flow induced by vorticity BiotSavart 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 Threedimensional vortex motion Vortex particles Line vortices and the local induction approximation (LIA) Aerodynamics General features of flow past an aircraft Airfoils and the KuttaJoukowski condition The KuttaJoukowski theorem The KuttaJoukowski 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 Pointsourcedipole panels Sourcedipole panel method Sourcedipole representation Solution of the interior problem Pointsource 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 Userdefined functions 751 C.9 MATLAB graphics 755 Index 763
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