Numerical Solution of the Incompressible Navier-Stokes Equations

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.