Numerical Methods for Two-Phase Incompressible Flows

by Gross, Sven; Reusken, Arnold

• ISBN13: 9783642196850
• ISBN10: 3642196853
• Format: Hardcover
• Copyright: 2011-06-01
• Publisher: Springer Verlag

Summary

This book is the first monograph providing an introduction to and an overview of numerical methods for the simulation of two-phase incompressible flows. The Navier-Stokes equations describing the fluid dynamics are examined in combination with models for mass and surfactant transport. The book pursues a comprehensive approach: important modeling issues are treated, appropriate weak formulations are derived, level set and finite element discretization techniques are analyzed, efficient iterative solvers are investigated, implementational aspects are considered and the results of numerical experiments are presented. The book is aimed at M Sc and PhD students and other researchers in the fields of Numerical Analysis and Computational Engineering Science interested in the numerical treatment of two-phase incompressible flows.

Table of Contents

Introduction	p. 1
One-and two-phase flow models in strong formulation	p. 1
Navier-Stokes equations for one-phase flow	p. 2
Navier-Stokes equations for two-phase flow	p. 6
Two-phase flow with transport of a dissolved species	p. 9
Two-phase flow with transport of a surfactant on the interface	p. 10
Modeling of interfacial phenomena	p. 12
Initial and boundary conditions	p. 19
Examples of two-phase flow simulations	p. 20
Numerical simulation of a rising droplet	p. 22
Numerical simulation of a droplet with surfactant transport	p. 24
Overview of numerical methods	p. 26
One-phase incompressible flows
Mathematical models	p. 33
Introduction	p. 33
Weak formulation	p. 35
Function spaces	p. 36
Oseen problem in weak formulation	p. 39
Time dependent (Navier-)Stokes equations in weak formulation	p. 42
Finite element discretization	p. 51
Multilevel tetrahedral grid hierarchy	p. 51
Multilevel triangulation	p. 51
A multilevel refinement method	p. 53
Hood-Taylor finite element spaces	p. 60
Simplicial finite element spaces	p. 60
Hood-Taylor finite element discretization of the Oseen problem	p. 64
Matrix-vector representation of the discrete problem	p. 66
Hood-Taylor semi-discretization of the non-stationary (Navier-)Stokes problem	p. 67
Numerical experiments	p. 73
Flow in a rectangular tube	p. 73
Flow in a curved channel	p. 74
Discussion and additional references	p. 76
Time integration	p. 83
Introduction	p. 83
The ¿-scheme for the Navier-Stokes problem	p. 91
Fractional-step ¿-scheme for the Navier-Stokes problem	p. 94
Numerical experiments	p. 95
Discussion and additional references	p. 97
Iterative solvers	p. 99
Linearization method	p. 99
Iterative solvers for symmetric saddle point problems	p. 101
Preconditioned MINRES	p. 103
Inexact Uzawa method	p. 109
Iterative Solvers for Oseen problems	p. 116
Preconditioners	p. 120
Introduction	p. 120
Multigrid preconditioner	p. 121
Preconditioners for the Schur complement	p. 148
Numerical experiments	p. 152
Stokes case	p. 152
Oseen case	p. 154
Navier-Stokes case	p. 155
Discussion and additional references	p. 155
Two-phase incompressible flows
Mathematical model	p. 161
Introduction	p. 161
Interface representation	p. 169
Explicit interface representation: interface tracking	p. 169
Volume tracking based on the characteristic function	p. 171
Volume tracking based on the level set function	p. 177
Phase field representation	p. 180
Weak formulation	p. 191
Finite element discretization of two-phase flow model	p. 197
Introduction	p. 197
Discretization of the level set equation	p. 197
Introduction to stabilization	p. 198
Discretization of the level set equation by the streamline diffusion finite element method	p. 201
Construction of an approximate interface ¿h	p. 205
Error in approximation of ¿ by ¿h	p. 207
Corrections of the level set function	p. 212
Re-initialization	p. 212
Mass conservation	p. 218
Numerical experiments with the level set equation	p. 220
Discretization using the SDFEM	p. 221
Re-initialization by the Fast Marching Method	p. 226
Discretization of the surface tension functional	p. 227
Treatment of general surface tension tensors	p. 230
Analysis of the Laplace-Beltrami discretization	p. 231
Preliminaries	p. 231
Error bounds for discrete surface tension functionals	p. 237
Numerical experiments with the Laplace-Beltrami discretization	p. 242
XFEM discretization of the pressure	p. 245
Approximation error for standard FE spaces	p. 246
Extended finite element method (XFEM)	p. 248
Modifications and implementation issues	p. 251
Analysis of XFEM	p. 254
Numerical experiment with XFEM	p. 261
Numerical experiments for a Stokes problem	p. 263
A stationary Stokes test problem	p. 263
Test case A: Pressure jump at a planar interface	p. 267
Test case B: Static droplet	p. 271
Finite element discretization of two-phase flow problem	p. 276
Spatial finite element discretization	p. 276
Numerical experiment with a two-phase flow problem	p. 279
Time integration	p. 283
A generalized ¿-scheme	p. 283
Case I: B independent of time	p. 283
Case II: B may depend on time	p. 288
An implicit Euler method with decoupling	p. 291
Numerical experiments	p. 292
Iterative solvers	p. 297
Decoupling and linearization	p. 297
Numerical experiment	p. 306
Iterative solvers for linear saddle point problems	p. 308
Schur complement preconditioners	p. 308
Numerical experiments	p. 317
Stationary Stokes case	p. 318
Generalized Stokes case	p. 320
Mass transport
Mathematical model	p. 327
Introduction	p. 327
Weak formulation: stationary interface	p. 329
Weak formulation: non-stationary interface	p. 333
Preliminaries	p. 333
Space-time weak formulation	p. 340
Well-posedness of the space-time weak formulation	p. 341
Finite element discretization	p. 345
Nitsche-XFEM method	p. 345
Analysis of the Nitsche-XFEM method	p. 349
Time discretization	p. 357
Numerical experiments	p. 359
Test problems	p. 359
Numerical results	p. 360
Discretization in case of a non-stationary interface	p. 364
Rothe's method combined with Nitsche-XFEM	p. 365
Nitsche-XFEM space-time discretization	p. 368
Numerical experiment: mass transport coupled with fluid dynamics	p. 375
Surfactant transport
Mathematical model	p. 385
Surfactant transport on a stationary interface	p. 385
Surfactant transport on a non-stationary interface	p. 387
Finite element methods for surfactant transport equations	p. 391
Finite element methods based on Lagrangian interface tracking	p. 392
Finite element methods based on Eulerian interface capturing	p. 396
An extension-based Eulerian finite element method	p. 397
Eulerian surface finite element method for a stationary interface	p. 401
Numerical experiments	p. 407
Discretization error analysis	p. 412
Eulerian space-time surface finite element method for a non-stationary interface	p. 422
Numerical experiments	p. 428
Appendix
Appendix A: Results from differential geometry	p. 433
Results for a stationary surface	p. 433
Results for an evolving surface	p. 437
Appendix B: Variational formulations in Hilbert spaces	p. 439
Variational problems and Galerkin discretization	p. 439
Application to elliptic problems	p. 441
Application to saddle point problems	p. 443
A Strang lemma for saddle point problems	p. 447
Schur complement preconditioning for parameter dependent saddle point problems	p. 449
Preliminaries	p. 449
Schur complement preconditioner	p. 453
Application to a generalized Stokes equation	p. 456
References	p. 459
Index	p. 473
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