Numerical Solution of the Incompressible Navier-Stokes Equations

296 Pages·1993·5.378 MB·English

by  L. Quartapelle (auth.)

ISNM
International Series of
Numerical Mathematics
Vol. 113
Edited by
K.-H. Hoffmann, Munchen
H. D. Mittelmann, Tempe
J.Todd,Pasadena
Numerical Solution
of the Incompressible
Navier-Stokes
Equations
L. Quartapelle
Springer Basel AG
Author
L. Quartapelle
Dipartimento di Fisica
Politecnico di Milano
Piazza Leonardo da Vinei, 32
1-20133 Milano
ltaly
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Quartapelle, Luigi:
Numerical solution of the incompressible Navier-Stokes
equations / L. Quartapelle. - Basel ; Boston; Berlin :
Birkhauser, 1993
(International series of numerical mathematics ; VoI. 113)
ISBN 978-3-0348-9689-4 ISBN 978-3-0348-8579-9 (eBook)
DOI 10.1007/978-3-0348-8579-9
NE:GT
This work is subject to copyright. AlI rights are reserved, whether the whole ar part of the material is concerned,
specifically the rights of translation, reprinting, re-use of illustrations, reeitation, broadcasting, reproduction an
microfIlms or in other ways, and storage in data banks. For any kind of use the permission of the copyright owner must
be obtained.
© 1993 Springer Basel AG
Originally published by Birkhăuser Verlag in 1993
Softcover reprint ofthe hardcover Ist edition 1993
Camera-ready copy prepared by the author
Prlnted on aeid-free paper produced from chlorine-free pulp
Cover design: Heinz Hiltbrunner, Basel
ISBN 978-3-0348-9689-4
987654321
a Giuliana
Contents
Preface ......................... . xi
1 The incompressible Navier-Stokes equations
1.1 Introduction . 1
1.2 Incompressible N avier-Stokes equations . 2
1.3 Organization of the book . 7
1.4 Some references 10
2 Nonprimitive variable formulations in 2D
2.1 Introduction. 13
2.2 Vorticity-stream function equations . .. 14
2.3 Biharmonic formulation 20
2.4 Coupled vorticity-stream function equations 21
2.5 Vorticity integral conditions 22
2.5.1 Green identities . 23
2.5.2 Vorticity integral conditions 23
2.5.3 What the integral conditions are not 25
2.6 Split vorticity-stream function equations 27
2.7 One-dimensional integral conditions . 29
2.8 Orthogonal projection operator 32
2.9 Factorized vorticity-stream function problem. 39
2.10 Numerical schemes: local discretizations 40
2.10.1 Boundary vorticity formula methods 41
2.10.2 Decomposition scheme 42
2.10.3 Glowinski-Pironneau method 44
2.10.4 Discretization of the nonlinear terms 47
2.11 Numerical schemes: spectral method 49
2.11.1 Modal equations 51
2.11.2 Influence matrix method 52
2.11.3 Integral conditions 53
2.11.4 Chebyshev spectral approximation 56
2.11.5 Numerical comparisons. 63
2.12 Higher-order time discretization 71
2.13 Rotationally symmetric equations 72
viii CONTENTS
3 Nonprimitive variable formulations in 3D
3.1 Introduction....... 75
3.2 Vorticity vector equation . . . . . . . . . . 77
3.3 <p-(-A formulation ............. 78
3.3.1 Equations and boundary conditions for velocity potentials 78
3.3.2 Governing equations . . . . . . . . . . 81
3.3.3 Integral conditions for vorticity vector 84
3.4 qs-(-'I/J formulation . . . . . . . 87
3.4.1 Surface scalar potential. 88
3.4.2 Governing equations . . 92
3.4.3 Split formulation . . . . 95
3.4.4 Time-discretization and orthogonal projection 98
3.4.5 Glowinski-Pironneau method 101
3.4.6 Vector elliptic equations .... 103
3.4.7 Pressure determination . . . . . 106
3.5 Irreducible vorticity integral conditions 107
3.5.1 Orthogonal decomposition of the projection space 108
3.5.2 Uncoupled formulation . . . . . . . . . . . . . . . 110
3.5.3 A representation of the irreducible projection space 113
3.6 (-</1 formulation . . . . . . . . . . . . . 116
3.6.1 Governing equations . . . . . . 117
3.6.2 An equivalent (-</1 formulation . 118
3.6.3 Uncoupled formulation 120
3.7 Conclusions . . . . . . . . . . . . 122
4 Vorticity-velocity representation
4.1 Introduction......... 125
4.2 Three-dimensional equations . 127
4.2.1 Governing equations . 127
4.2.2 Uncoupled formulation 130
4.3 Two-dimensional equations .. 132
4.3.1 Governing equations . 132
4.3.2 Uncoupled formulation 134
4.3.3 Glowinski-Pironneau method 135
4.3.4 Discussion....... 137
5 Primitive variable formulation
5.1 Introduction ........ . 139
5.2 Pressure-velocity equations 141
5.3 Pressure integral conditions 142
5.4 Decomposition scheme . . . 144
CONTENTS ix
5.5 Equations for plane channel flows 148
5.5.1 Uncoupled formulation. 148
5.5.2 Influence matrix method 151
5.5.3 Numerical comparison 153
5.6 Direct Stokes solver . . . . . . . 155
5.7 General boundary conditions. . 158
5.8 Extension to compressible equations. 160
5.8.1 Generalized Stokes solver. 161
5.8.2 Integral conditions ..... . 163
6 Evolutionary pressure-velocity equations
6.1 Introduction.......... 165
6.2 Unsteady Stokes problem ........ . 166
6.3 Space-time integral conditions . . . . . . 168
6.4 Drag on a sphere in nonuniform motion . 170
6.5 Pressure dynamics in incompressible flows 172
6.6 Comments.................. 174
7 Fractional-step projection method
7.1 Introduction ........... . 177
7.2 Ladyzhenskaya theorem .... . 178
7.3 Fractional-step projection method 180
7.3.1 Homogeneous boundary condition 181
7.3.2 Nonhomogeneous boundary condition. 181
7.4 Poisson equation for pressure 187
7.4.1 On higher-order methods .. 189
7.5 A finite element projection method 191
7.5.1 Variational formulation ... 192
7.5.2 Finite element equations . . 194
7.5.3 Discretized projection operator 196
7.5.4 Diagonalization of the mass matrix 200
7.5.5 Taylor~Galerkin scheme for advection-diffusion. 201
8 Incompressible Euler equations
8.1 Introduction .......... . 209
8.2 Incompressible Euler equations 211
8.2.1 Basic equations ..... 211
8.2.2 Fractional-step equations. 212
8.3 Taylor~Galerkin method . . . . . 214
8.3.1 Basic third-order TG scheme. 214
8.3.2 Two-step third-order TG scheme 217
8.3.3 Two-step fourth-order TG schemes 223
8.3.4 Vector advection equation 227
8.4 Euler equations for vortical flows ..... 230
x CONTENTS
8.5 VortiCity-velocity formulation . . 233
8.5.1 Basic equations ..... . 234
8.5.2 An equivalent formulation 235
8.6 Nonprimitive variable formulations 238
8.6.1 (-¢ formulation ...... . 238
8.6.2 qs-(-'IjJ formulation .... . 240
8.6.3 Vorticity-stream function equations 241
APPENDICES
A Vector differential operators
A.1 Orthogonal curvilinear coordinates 243
A.2 Differential operators . 244
A.3 Cylindrical coordinates ...... . 247
A.3.1 Definition ......... . 247
A.3.2 Gradient, divergence and curl 248
A.3.3 Laplace and advection operators . 249
A.4 Spherical coordinates . . . . . . . . . 250
A.4.1 Definition ............ . 250
A.4.2 Gradient, divergence and curl .. 251
A.4.3 Laplace and advection operators. 252
B Separation of vector elliptic equations
B.1 Introduction. . . . . . . . . . . . . . . 255
B.2 Polar coordinates . . . . . . . . . . . . 256
B.3 Spherical coordinates on the unit sphere 260
B.4 Cylindrical coordinates 261
B.5 Spherical coordinates . . . 264
C Spatial difference operators
C.1 Introduction ................ . 269
C.2 2D equation: four-node bilinear element 269
C.3 3D equation: eight-node trilinear element. 271
D Time derivative of integrals over moving domains
D.1 Circulation along a moving curve 276
D.2 Flux across a moving surface .. 277
D.3 Integrals over a moving volume 278
References . . . . . . . . . . . . . . . . 281