Preface ........................................................ ix
1 Introduction ................................................. 1
1.1 Examples of multiphase flows ............................ 3
1.2 Computational modeling .................................. 7
1.3 Looking ahead .......................................... 18
2 Fluid mechanics with interfaces ............................. 21
2.1 General principles ..................................... 21
2.2 Basic equations ........................................ 22
2.3 Interfaces: description and definitions ................ 30
2.4 Fluid mechanics with interfaces ........................ 36
2.5 Fluid mechanics with interfaces: the one-fluid
formulation ............................................ 41
2.6 Nondimensional numbers ................................. 42
2.7 Thin films, intermolecular forces, and contact lines .. 44
2.8 Notes .................................................. 47
3 Numerical solutions of the Navier-Stokes equations ......... 50
3.1 Time integration ....................................... 51
3.2 Spatial discretization ................................. 55
3.3 Discretization of the advection terms .................. 59
3.4 The viscous terms ...................................... 61
3.5 The pressure equation .................................. 64
3.6 Velocity boundary conditions ........................... 69
3.7 Outflow boundary conditions ............................ 70
3.8 Adaptive mesh refinement ............................... 71
3.9 Summary ................................................ 72
3.10 Postscript: conservative versus non-conservative
form ................................................... 73
4 Advecting a fluid interface ................................. 75
4.1 Notations .............................................. 76
4.2 Adverting the color function ........................... 77
4.3 The volume-of-fluid (VOF) method ....................... 81
4.4 Front tracking ......................................... 84
4.5 The level-set method ................................... 87
4.6 Phase-field methods .................................... 90
4.7 The CIP method ......................................... 91
4.8 Summary ................................................ 93
5 The volume-of-fluid method ................................. 95
5.1 Basic properties ....................................... 95
5.2 Interface reconstruction ............................... 98
5.3 Tests of reconstruction methods ....................... 106
5.4 Interface advection ................................... 108
5.5 Tests of reconstruction and advection methods ......... 122
5.6 Hybrid methods ........................................ 128
6 Advecting marker points: front tracking .................... 133
6.1 The structure of the front ............................ 134
6.2 Restructuring the fronts .............................. 143
6.3 The front-grid communications ......................... 145
6.4 Advection of the front ................................ 150
6.5 Constructing the marker function ...................... 152
6.6 Changes in the front topology ......................... 158
6.7 Notes ................................................. 160
7 Surface tension ............................................ 161
7.1 Computing surface tension from marker functions ....... 161
7.2 Computing the surface tension of a tracked front ...... 168
7.3 Testing the surface tension methods ................... 177
7.4 More sophisticated surface tension methods ............ 181
7.5 Conclusion on numerical methods ....................... 186
8 Disperse bubbly flows ...................................... 187
8.1 Introduction .......................................... 187
8.2 Homogeneous bubbly flows .............................. 189
8.3 Bubbly flows in vertical channels ..................... 194
8.4 Discussion ............................................ 201
9 Atomization and breakup .................................... 204
9.1 Introduction .......................................... 204
9.2 Thread, sheet, and rim breakup ........................ 205
9.3 High-speed jets ....................................... 214
9.4 Atomization simulations ............................... 219
10 Droplet collision, impact, and splashing ................... 228
10.1 Introduction .......................................... 228
10.2 Early simulations ..................................... 229
10.3 Low-velocity impacts and collisions ................... 229
10.4 More complex slow impacts ............................. 232
10.5 Corolla, crowns, and splashing impacts ................ 235
11 Extensions ................................................. 243
11.1 Additional fields and surface physics ................. 243
11.2 Imbedded boundaries ................................... 256
11.3 Multiscale issues ..................................... 266
11.4 Summary ............................................... 269
Appendix A Interfaces: description and definitions ........... 270
A.1 Two-dimensional geometry .............................. 270
A.2 Three-dimensional geometry ............................ 272
A.3 Axisymmetric geometry ................................. 274
A.4 Differentiation and integration on surfaces ........... 275
Appendix В Distributions concentrated on the interface ....... 279
B.l A simple example ...................................... 281
Appendix С Cube-chopping algorithm ........................... 284
C.l Two-dimensional problem ............................... 285
C.2 Three-dimensional problem ............................. 286
Appendix D The dynamics of liquid sheets: linearized theory .. 288
D.l Flow configuration .................................... 288
D.2 Inviscid results ...................................... 288
D.3 Viscous theory for the Kelvin-Helmholtz instability ... 293
References .................................................... 295
Index ......................................................... 322
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