Chichester: John Wiley & Sons Ltd, 2003. — 358 p. — ISBN 0471496669.In recent years there have been significant developments in the development of stable and accurate finite element procedures for the numerical approximation of a wide range of fluid mechanics problems. Taking an engineering rather than a mathematical bias, this valuable reference resource details the fundamentals of stabilised finite element methods for the analysis of steady and time-dependent fluid dynamics problems. Organised into six chapters, this text combines theoretical aspects and practical applications and offers coverage of the latest research in several areas of computational fluid dynamics.Contents Preface Introduction and preliminaries Finite elements in fluid dynamics Subjects covered Kinematical descriptions of the flow field Lagrangian and Eulerian descriptions ALE description of motion The fundamental ALE equation Time derivative of integrals over moving volumes The basic conservation equations Mass equation Momentum equation Internal energy equation Total energy equation ALE form of the conservation equations Closure of the initial boundary value problem Basic ingredients of the finite element method Mathematical preliminaries Trial solutions and weighting functions Compact integral forms Strong and weak forms of a boundary value problem Finite element spatial discretization Steady transport problems Problem statement Strong form Weak form Galerkin approximation Piecewise linear approximation in 1D Analysis of the discrete equation Piecewise quadratic approximation in ID Analysis of the discrete equations Early Petrov—Galerkin methods Upwind approximation of the convective term First finite elements of upwind type The concept of balancing diffusion Stabilization techniques The SUPG method The Galerkin/Least-squares method The stabilization parameter Other stabilization techniques and new trends Finite increment calculus Bubble Junctions and wavelet approximations The variational multiscale method Complements Applications and solved exercises Construction of a bubble function method One-dimensional transport Convection—diffusion across a source term Convection—diffusion skew to the mesh Convection—diffusion-reaction in 2D The Hemker problem Unsteady convective transport Introduction Problem statement The method of characteristics The concept of characteristic lines Properties of the linear convection equation Methods based on the characteristics Classical time and space discretization techniques Time discretization Galerkin spatial discretization Stability and accuracy analysis Analysis of stability by Fourier techniques Analysis of classical time-stepping schemes The modified equation method Taylor—Galerkin Methods The need for higher-order time schemes Third-order explicit Taylor—Galerkin method Fourth-order explicit leap-frog method Two-step explicit Taylor—Galerkin methods An introduction to monotonicity-preserving schemes Least-squares-based spatial discretization Least-squares approach for the 6 family of methods Taylor least-squares method The discontinuous Galerkin method Space—time formulations Time-discontinuous Galerkin formulation Time-discontinuous least-squares formulation Space-timeGalerkin/Least-squaresformulation Applications and solved exercises Propagation of a cosine profile Travelling wave package The rotating cone problem Propagation of a steep front Compressible Flow Problems Compressible Flow Problems Introduction Nonlinear hyperbolic equations Scalar equations Weak solutions and entropy condition Time and space discretization The Euler equations Strong form of the conservation equations The quasi-linear form of the Euler equations Basic properties of the Euler equations Boundary conditions Spatial discretization techniques Galerkin formulation Upwind-type discretizations Numerical treatment of shocks Introduction Early artificial diffusion methods High-resolution methods Nearly incompressible flows Fluid-structure interaction Acoustic approximation Nonlinear transient dynamic problems llustrative examples Solved exercises One-step Taylor—Galerkin solution of Burgers' equation The shock tube problem Unsteady convection—diffusion problems Introduction Problem statement Time discretization procedures Classical methods Fractional-step methods High-order time-stepping schemes Spatial discretization procedures Galerkin formulation of the semi-discrete scheme Galerkin formulation of 0 family methods Galerkin formulation of explicit Pade schemes Galerkin formulation of implicit multistage schemes Stabilization of the semi-discrete scheme Stabilization of multistage schemes Stabilized space-time formulations Solved exercises Convection-diffusion of a Gaussian hill Transient rotating pulse Steady rotating pulse problem Nonlinear propagation of a step Burgers' equation in 1D Two-dimensional Burgers' equation Appendix Least-squares in transient/relaxation problems Viscous incompressible flows Introduction Basic concepts Strain rate and spin tensors The stress tensor in a Newtonian fluid The Navier—Stokes equations Main issues in incompressible flow problems Trial solutions and weighting functions Stationary Stokes problem Formulation in terms of Cauchy stress Formulation in terms of velocity and pressure Galerkin formulation Matrix problem Solvability condition and solution procedure The LBB compatibility condition Some popular velocity—pressure couples Stabilization of the Stokes problem Penalty method Steady Navier—Stokes problem Weak form and Galerkin formulation Matrix problem Unsteady Navier—Stokes equations Weak formulation and spatial discretization Stabilized finite element formulation Time discretization by fractional-step methods Applications and solved exercices Stokes flow with analytical solution Cavity flow problem Plane jet simulation Natural convection in a square cavity
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