Numerical Solution of the Incompressible Navier-Stokes Equations

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 A substantial portion of the book contains new results and unpublished material.

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.- 2.6 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.- 2.11 Numerical schemes: spectral method.- 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.- 3.4 qs-Æ-? formulation.- 3.5 Irreducible vorticity integral conditions.- 3.6 Æ-? formulation.- 3.7 Conclusions.- 4 Vorticity-velocity representation.- 4.1 Introduction.- 4.2 Three-dimensional equations.- 4.3 Two-dimensional equations.- 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.- 5.6 Direct Stokes solver.- 5.7 General boundary conditions.- 5.8 Extension to compressible equations.- 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.- 7.4 Poisson equation for pressure.- 7.5 Afinite element projection method.- 8 Incompressible Euler equations.- 8.1 Introduction.- 8.2 Incompressible Euler equations.- 8.3 Taylor-Galerkin method.- 8.4 Euler equations for vortical flows.- 8.5 Vorticity-velocity formulation.- 8.6 Nonprimitive variable formulations.- 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.