Direct Numerical Simulations of Gas-Liquid Multiphase Flows

DIRECT NUMERICAL SIMULATIONS OF GAS–LIQUID MULTIPHASE FLOWS......Page 2
Title......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 10
1 Introduction......Page 12
1.1 Examples of multiphase flows......Page 14
1.2.1 Simple flows (Re = 0 and Re = ∞)......Page 18
1.2.2 Finite Reynolds number flows......Page 22
1.3 Looking ahead......Page 29
2.1 General principles......Page 32
2.2.1 Mass conservation......Page 33
2.2.2 Momentum conservation......Page 35
2.2.3 Energy conservation......Page 36
2.2.4 Incompressible flow......Page 37
2.2.5 Boundary conditions......Page 40
2.3 Interfaces: description and definitions......Page 41
2.4 Fluid mechanics with interfaces......Page 47
2.4.1 Mass conservation and velocity conditions......Page 48
2.4.2 Surface tension......Page 49
2.4.3 Momentum conservation with interfaces......Page 50
2.4.4 Free-surface flow......Page 51
2.5 Fluid mechanics with interfaces: the one-fluid formulation......Page 52
2.6 Nondimensional numbers......Page 53
2.7.1 Disjoining pressure and forces between interfaces......Page 55
2.7.2 Contact line statics and dynamics......Page 57
2.8.1 Fluid and interface mechanics......Page 58
2.8.2 Thin films and contact lines......Page 59
3 Numerical solutions of the Navier–Stokes equations......Page 61
3.1 Time integration......Page 62
3.2 Spatial discretization......Page 66
3.3 Discretization of the advection terms......Page 70
3.4 The viscous terms......Page 72
3.5 The pressure equation......Page 75
3.6 Velocity boundary conditions......Page 80
3.7 Outflow boundary conditions......Page 81
3.8 Adaptive mesh refinement......Page 82
3.9 Summary......Page 83
3.10 Postscript: conservative versus non-conservative form......Page 84
4
Advecting a fluid interface......Page 86
4.1 Notations......Page 87
4.2 Advecting the color function......Page 88
4.3 The volume-of-fluid (VOF) method......Page 92
4.4 Front tracking......Page 95
4.5 The level-set method......Page 98
4.6 Phase-field methods......Page 101
4.7 The CIP method......Page 102
4.8 Summary......Page 104
5.1 Basic properties......Page 106
5.2.1 Convergence order of a reconstruction method......Page 109
5.2.2 Evaluation of the interface unit normal......Page 110
5.2.3 Determination of α......Page 115
5.3.2 Reconstruction accuracy tests......Page 117
5.4 Interface advection......Page 119
5.4.1 Geometrical one-dimensional linear-mapping method......Page 120
5.4.2 Related one-dimensional advection methods......Page 125
5.4.3 Unsplit methods......Page 127
5.5 Tests of reconstruction and advection methods......Page 133
5.5.1 Translation test......Page 134
5.5.2 Vortex-in-a-box test......Page 136
5.6 Hybrid methods......Page 139
6 Advecting marker points: front tracking......Page 144
6.1 The structure of the front......Page 145
6.1.1 Structured two-dimensional fronts......Page 146
6.1.2 Unstructured fronts......Page 149
6.2 Restructuring the fronts......Page 154
6.3.1 Locating the front on the fixed grid......Page 156
6.3.2 Interpolation and smoothing......Page 158
6.4 Advection of the front......Page 161
6.5 Constructing the marker function......Page 163
6.5.1 Constructing the marker function from its gradient......Page 164
6.5.2 Construction of the volume fraction from the front location......Page 166
6.6 Changes in the front topology......Page 169
6.7 Notes......Page 171
7.1.1 Continuous surface force method......Page 172
7.1.2 Continuous surface stress method......Page 175
7.1.3 Direct addition and elementary smoothing in the VOF method......Page 176
7.1.4 Weighted distribution in the VOF method: kernel smoothing......Page 177
7.2 Computing the surface tension of a tracked front......Page 179
7.2.1 Two-dimensional interfaces......Page 180
7.2.2 Three-dimensional interfaces......Page 182
7.2.3 Smoothing the surface tension on the fixed grid......Page 184
7.3.1 Static case: spurious currents......Page 188
7.3.2 Dynamic case......Page 190
7.4.1 Direct addition with pressure correction......Page 192
7.4.2 CSF method with better curvature: PROST......Page 194
7.4.3 Numerical estimate of the curvature from the volume fractions: the HF method......Page 195
7.5 Conclusion on numerical methods......Page 197
8.1 Introduction......Page 198
8.2 Homogeneous bubbly flows......Page 200
8.3 Bubbly flows in vertical channels......Page 205
8.4 Discussion......Page 212
9.1 Introduction......Page 215
9.2.1 The Plateau–Rayleigh jet instability......Page 216
9.2.2 Film and thread breakup......Page 217
9.2.3 The Taylor–Culick rim......Page 220
9.2.4 Rims leading to droplets and fingers......Page 222
9.3.1 Structure of the atomizing jet......Page 225
9.3.2 Mechanisms of droplet formation......Page 226
9.3.3.1 Elementary Kelvin–Helmholtz analysis......Page 227
9.3.3.2 Orr–Sommerfeld analysis......Page 229
9.4.2 Two-dimensional spatially developing simulations......Page 230
9.4.3 Three-dimensional calculations......Page 235
10.1 Introduction......Page 239
10.3 Low-velocity impacts and collisions......Page 240
10.4 More complex slow impacts......Page 243
10.5.1 Impacts on thin liquid layers......Page 246
10.5.2 Three-dimensional impacts......Page 250
11.1 Additional fields and surface physics......Page 254
11.1.2 Electrohydrodynamics......Page 255
11.1.3 Mass transfer and chemical reactions......Page 257
11.1.4 Boiling......Page 259
11.1.5 Cavitation......Page 265
11.2.1 The immersed boundary method of Peskin......Page 267
11.2.2 Solid boundaries......Page 268
11.2.3 Solidification......Page 273
11.3 Multiscale issues......Page 277
11.4 Summary......Page 280
A.1 Two-dimensional geometry......Page 281
A.2 Three-dimensional geometry......Page 283
A.3 Axisymmetric geometry......Page 285
A.4 Differentiation and integration on surfaces......Page 286
Appendix B Distributions concentrated on the interface......Page 290
B.1 A simple example......Page 292
Appendix C Cube-chopping algorithm......Page 295
C.1 Two-dimensional problem......Page 296
C.2 Three-dimensional problem......Page 297
D.2.1 The general dispersion relation......Page 299
D.2.3 The Kelvin–Helmholtz instability......Page 302
D.2.4 Effect of thick boundary layers in the inviscid framework......Page 303
D.3 Viscous theory for the Kelvin–Helmholtz instability......Page 304
References......Page 306
Index......Page 333