Global introduction .......................................... xiii
PART I PHYSICS OF WEAKLY COMPRESSIBLE FLUIDS
1 Lagrangian and Hamiltonian mechanics ....................... 3
1.1 Introduction ............................................... 3
1.2 Least action principle ..................................... 4
1.2.1 Inertial frames and principle of relativity ......... 4
1.2.2 Generalized coordinates. Lagrangian of a system ..... 6
1.2.3 Least action principle .............................. 8
1.2.4 Lagrange equations ................................. 10
1.3 Mechanics of a system of particles ........................ 14
1.3.1 Lagrangian of an isolated particle ................. 14
1.3.2 Lagrangian of a particle system .................... 16
1.3.3 Newton's second law ................................ 20
1.4 Conservation laws ......................................... 22
1.4.1 Energy and Hamiltonian ............................. 23
1.4.2 Hamilton equations ................................. 27
1.4.3 Linear momentum and angular momentum ............... 33
1.4.4 Investigating a generic case: the simple pendulum .. 37
1.5 Composite systems ......................................... 41
1.5.1 Centre of mass. Inertia tensor ..................... 41
1.5.2 Internal and external forces ....................... 49
1.5.3 Internal energy .................................... 51
1.5.4 Pressure and density ............................... 55
2 Statistical mechanics ..................................... 60
2.1 Introduction .............................................. 60
2.2 Statistical behaviour of large systems .................... 61
2.2.1 Probability densities .............................. 61
2.2.2 Liouville's theorem. Boltzmann's equation .......... 63
2.2.3 Entropy ............................................ 70
2.3 Thermodynamical quantities ................................ 75
2.3.1 Boltzmann distribution ............................. 75
2.3.2 Heat and work ...................................... 82
2.3.3 State equation ..................................... 87
2.4 Dissipative systems ....................................... 91
2.4.1 Kinetic coefficients ............................... 91
2.4.2 Dissipative forces ................................. 97
2.4.3 Particle friction ................................. 102
2.5 Further considerations on dissipation .................... 108
2.5.1 Dissipative Hamilton equations .................... 108
2.5.2 Mirror systems .................................... 116
2.5.3 Thermodynamical fluctuations ...................... 121
3 Continuous media and viscous fluids ...................... 132
3.1 Introduction ............................................. 132
3.2 Continuum kinematics ..................................... 133
3.2.1 Mesoscopic particles and continuous fields ........ 133
3.2.2 Strain of a medium ................................ 136
3.2.3 Continuity equation ............................... 140
3.3 Balances and fluxes ...................................... 143
3.3.1 Flux of a physical quantity ....................... 143
3.3.2 Cauchy stress tensor .............................. 146
3.3.3 Energy balances ................................... 149
3.4 Viscous fluids ........................................... 153
3.4.1 Behaviour law of a viscous fluid .................. 153
3.4.2 Navier-Stokes equations ........................... 158
3.4.3 Similarity. Orders of magnitude ................... 164
3.4.4 Pressure and incompressibility .................... 173
3.4.5 Surface tension ................................... 177
3.5 From Boltzmann to the continua ........................... 182
3.5.1 Evolution of statistical averages ................. 182
3.5.2 Balances and fluxes through Boltzmann ............. 184
3.5.3 Viscous stresses through Boltzmann ................ 188
3.6 Variational principles for fluids ........................ 193
3.6.1 Generals. Rigid media ............................. 193
3.6.2 Strained media .................................... 198
3.6.3 Incompressible flows .............................. 201
3.6.4 Irreversible processes ............................ 207
4 Turbulent flows .......................................... 210
4.1 Introduction ............................................. 210
4.2 Chaos dynamics ........................................... 211
4.2.1 Example: the Van der Pol pendulum ................. 212
4.2.2 Strange attractors ................................ 214
4.2.3 Scales and spectra ................................ 221
4.3 One-point statistical models ............................. 223
4.3.1 Reynolds-averaging ................................ 224
4.3.2 Reynolds equations ................................ 226
4.3.3 Reynolds stresses ................................. 229
4.4 First-order closure ...................................... 233
4.4.1 Boussinesq model .................................. 234
4.4.2 Turbulent kinetic energy .......................... 238
4.4.3 Kolmogorov analysis ............................... 242
4.4.4 Closures for eddy viscosity ....................... 249
4.4.5 Generic case study: the infinite open channel ..... 254
4.4.6 Turbulent boundary conditions ..................... 260
4.5 Advanced averaged models ................................. 264
4.5.1 First-order model deficiencies .................... 264
4.5.2 Second order closure .............................. 267
4.5.3 Explicit algebraic models ......................... 271
4.6 Non-averaged models ...................................... 280
4.6.1 Large eddy simulation ............................. 280
4.6.2 The a model ....................................... 287
4.6.3 Stochastic models ................................. 293
PART II THE SPH METHOD IN HYDRAULICS
5 Principles of the SPH method ............................. 303
5.1 Introduction ............................................. 303
5.2 Lagrangian interpolation ................................. 304
5.2.1 Kernel and continuous interpolation ............... 304
5.2.2 Kernel examples ................................... 311
5.2.3 Particles and discrete interpolation .............. 320
5.2.4 SPH operators ..................................... 328
5.2.5 Renormalization ................................... 334
5.3 Discrete forms of the fluid equations .................... 340
5.3.1 Discrete continuity equation ...................... 341
5.3.2 Discrete Euler equation ........................... 345
5.3.3 SPH through Lagrange .............................. 349
5.3.4 Discrete conservation laws ........................ 355
5.4 Temporal integration ..................................... 359
5.4.1 Numerical integrators ............................. 360
5.4.2 Symplectic integrators ............................ 366
5.4.3 Discrete Lagrange equations ....................... 375
5.4.4 Numerical stability ............................... 382
6 Advanced hydraulics with SPH ............................. 393
6.1 Introduction ............................................. 393
6.2 Discrete viscosity and diffusion ......................... 394
6.2.1 Second-order SPH operators ........................ 394
6.2.2 Discrete Navier-Stokes equations .................. 400
6.2.3 Stabilization by viscosity ........................ 405
6.2.4 Discrete scalar and energy balances ............... 410
6.2.5 Discrete incompressible scheme .................... 412
6.3 Boundary treatment ....................................... 421
6.3.1 Discrete surface tension .......................... 421
6.3.2 Solid wall modelling .............................. 426
6.3.3 Mobile rigid bodies ............................... 436
6.4 Discrete turbulence: basic models ........................ 440
6.4.1 Discrete Reynolds equations ....................... 440
6.4.2 One- and two-equation discrete models ............. 443
6.4.3 Stability of the two-equation turbulent models .... 448
6.5 Discrete turbulence: advanced models ..................... 452
6.5.1 Discrete LES ...................................... 452
6.5.2 a-XSPH model ...................................... 455
6.5.3 Discrete stochastic models ........................ 465
7 SPH method validation .................................... 470
7.1 Introduction ............................................. 470
7.2 Steady flows ............................................. 473
7.2.1 Periodic hill ..................................... 473
7.2.2 Lid-driven cavity flow ............................ 477
7.2.3 Infinite open channel ............................. 481
7.3 Collapse of a water column ............................... 484
7.3.1 Collapse onto a dry bottom ........................ 484
7.3.2 Collapse onto a wet bottom ........................ 489
7.3.3 Collapse onto a motionless obstacle ............... 492
7.4 Immersed bodies .......................................... 497
7.4.1 Motionless body in a channel ...................... 497
7.4.2 Mobile body within an enclosure ................... 500
7.4.3 Sphere in a turbulent flow ........................ 503
8 SPH applied to hydraulic works ........................... 511
8.1 Introduction ............................................. 511
8.2 Wave action upon waterworks .............................. 512
8.2.1 Coastal work design elements ...................... 512
8.2.2 Numerical replication of overtopping .............. 517
8.2.3 Numerical study of the setup phenomenon ........... 520
8.3 Fish pass ................................................ 522
8.3.1 Theoretical calculation of a pool fish pass ....... 523
8.3.2 Simulation of a pool fish pass .................... 527
8.4 Floating oil spill containment boom ...................... 532
8.4.1 Oil spill instability criterion ................... 532
8.4.2 Simulation of a floating boom ..................... 536
8.5 Dam spillway ............................................. 540
8.5.1 Dam release monitoring ............................ 540
8.5.2 Simulation of the Goulours spillway ............... 544
8.6 Conclusion: what future for SPH? ......................... 547
Appendix A: Tensorial formalism ............................... 549
Appendix B: Fourier transform ................................. 560
Bibliography .................................................. 570
Index ......................................................... 585
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