This book presents different formulations of the equations governing incompressible viscous flows, in the form needed for developing numerical solution procedures. The conditions required to satisfy the no-slip boundary conditions in the various formulations are discussed in detail. Rather than focussing on a particular spatial discretization method, the text provides a unitary view of several methods currently in use for the numerical solution of incompressible Navier-Stokes equations, using either finite differences, finite elements or spectral approximations. For each formulation, a complete statement of the mathematical problem is provided, comprising the various boundary, possibly integral, and initial conditions, suitable for any theoretical and/or computational development of the governing equations. The text is suitable for courses in fluid mechanics and computational fluid dynamics. It covers that part of the subject matter dealing with the equations for incompressible viscous flows and their determination by means of numerical methods. A substantial portion of the book contains new results and unpublished material.

Inhalt

1 The incompressible Navier-Stokes equations.- 1.1 Introduction.- 1.2 Incompressible Navier-Stokes equations.- 1.3 Organization of the book.- 1.4 Some references.- 2 Nonprimitive variable formulations in 2D.- 2.1 Introduction.- 2.2 Vorticity-stream function equations.- 2.3 Biharmonic formulation.- 2.4 Coupled vorticity-stream function equations.- 2.5 Vorticity integral conditions.- 2.5.1 Green identities.- 2.5.2 Vorticity integral conditions.- 2.5.3 What the integral conditions are not.- 2.6 Split vorticity-stream function equations.- 2.7 One-dimensional integral conditions.- 2.8 Orthogonal projection operator.- 2.9 Factorized vorticity-stream function problem.- 2.10 Numerical schemes: local discretizations.- 2.10.1 Boundary vorticity formula methods.- 2.10.2 Decomposition scheme.- 2.10.3 Glowinski-Pironneau method.- 2.10.4 Discretization of the nonlinear terms.- 2.11 Numerical schemes: spectral method.- 2.11.1 Modal equations.- 2.11.2 Influence matrix method.- 2.11.3 Integral conditions.- 2.11.4 Chebyshev spectral approximation.- 2.11.5 Numerical comparisons.- 2.12 Higher-order time discretization.- 2.13 Rotationally symmetric equations.- 3 Nonprimitive variable formulations in 3D.- 3.1 Introduction.- 3.2 Vorticity vector equation.- 3.3 Æ-?-A formulation.- 3.3.1 Equations and boundary conditions for velocity potentials.- 3.3.2 Governing equations.- 3.3.3 Integral conditions for vorticity vector.- 3.4 qs-Æ-? formulation.- 3.4.1 Surface scalar potential.- 3.4.2 Governing equations.- 3.4.3 Split formulation.- 3.4.4 Time-discretization and orthogonal projection.- 3.4.5 Glowinski-Pironneau method.- 3.4.6 Vector elliptic equations.- 3.4.7 Pressure determination.- 3.5 Irreducible vorticity integral conditions.- 3.5.1 Orthogonal decomposition of the projection space.- 3.5.2 Uncoupled formulation.- 3.5.3 A representation of the irreducible projection space.- 3.6 Æ-? formulation.- 3.6.1 Governing equations.- 3.6.2 An equivalent Æ-? formulation.- 3.6.3 Uncoupled formulation.- 3.7 Conclusions.- 4 Vorticity-velocity representation.- 4.1 Introduction.- 4.2 Three-dimensional equations.- 4.2.1 Governing equations.- 4.2.2 Uncoupled formulation.- 4.3 Two-dimensional equations.- 4.3.1 Governing equations.- 4.3.2 Uncoupled formulation.- 4.3.3 Glowinski-Pironneau method.- 4.3.4 Discussion.- 5 Primitive variable formulation.- 5.1 Introduction.- 5.2 Pressure-velocity equations.- 5.3 Pressure integral conditions.- 5.4 Decomposition scheme.- 5.5 Equations for plane channel flows.- 5.5.1 Uncoupled formulation.- 5.5.2 Influence matrix method.- 5.5.3 Numerical comparison.- 5.6 Direct Stokes solver.- 5.7 General boundary conditions.- 5.8 Extension to compressible equations.- 5.8.1 Generalized Stokes solver.- 5.8.2 Integral conditions.- 6 Evolutionary pressure-velocity equations.- 6.1 Introduction.- 6.2 Unsteady Stokes problem.- 6.3 Space-time integral conditions.- 6.4 Drag on a sphere in nonuniform motion.- 6.5 Pressure dynamics in incompressible flows.- 6.6 Comments.- 7 Fractional-step projection method.- 7.1 Introduction.- 7.2 Ladyzhenskaya theorem.- 7.3 Fractional-step projection method.- 7.3.1 Homogeneous boundary condition.- 7.3.2 Nonhomogeneous boundary condition.- 7.4 Poisson equation for pressure.- 7.4.1 On higher-order methods.- 7.5 A finite element projection method.- 7.5.1 Variational formulation.- 7.5.2 Finite element equations.- 7.5.3 Discretized projection operator.- 7.5.4 Diagonalization of the mass matrix.- 7.5.5 Taylor-Galerkin scheme for advection-diffusion.- 8 Incompressible Euler equations.- 8.1 Introduction.- 8.2 Incompressible Euler equations.- 8.2.1 Basic equations.- 8.2.2 Fractional-step equations.- 8.3 Taylor-Galerkin method.- 8.3.1 Basic third-order TG scheme.- 8.3.2 Two-step third-order TG scheme.- 8.3.3 Two-step fourth-order TG schemes.- 8.3.4 Vector advection equation.- 8.4 Euler equations for vortical flows.- 8.5 Vorticity-velocity formulation.- 8.5.1 Basic equations.- 8.5.2 An equivalent formulation.- 8.6 Nonprimitive variable formulations.- 8.6.1 ?-? formulation.- 8.6.2 qs-?-? formulation.- 8.6.3 Vorticity-stream function equations.- APPENDICES.- A Vector differential operators.- A.1 Orthogonal curvilinear coordinates.- A.2 Differential operators.- A.3 Cylindrical coordinates.- A.3.1 Definition.- A.3.2 Gradient, divergence and curl.- A.3.3 Laplace and advection operators.- A.4 Spherical coordinates.- A.4.1 Definition.- A.4.2 Gradient, divergence and curl.- A.4.3 Laplace and advection operators.- B Separation of vector elliptic equations.- B.1 Introduction.- B.2 Polar coordinates.- B.3 Spherical coordinates on the unit sphere.- B.4 Cylindrical coordinates.- B.5 Spherical coordinates.- C Spatial difference operators.- C.1 Introduction.- C.2 2D equation: four-node bilinear element.- C.3 3D equation: eight-node trilinear element.- D Time derivative of integrals over moving domains.- D.1 Circulation along a moving curve.- D.2 Flux across a moving surface.- D.3 Integrals over a moving volume.- References.