The Lattice Boltzmann Equation for Fluid Dynamics and Beyond

by Succi, Sauro

• ISBN13: 9780198503989
• ISBN10: 0198503989
• Format: Hardcover
• Copyright: 2001-08-30
• Publisher: Clarendon Press

Summary

In recent years, certain forms of the Boltzmann equation--now going by the name of "Lattice Boltzmann equation" (LBE)--have emerged which relinquish most mathematical complexities of the true Boltzmann equation without sacrificing physical fidelity in the description of complex fluid motion. This book provides the first detailed survey of LBE theory and its major applications to date. Accessible to a broad audience of scientists dealing with complex system dynamics, the book also portrays future developments in allied areas of science where fluid motion plays a distinguished role.

Table of Contents

I THEORY
Kinetic theory	3	(14)
Atomistic dynamics	3	(4)
Relaxation to local equilibrium	7	(2)
H-theorem	9	(1)
Length scales and transport phenomena	10	(1)
Chapman-Enskog procedure	11	(2)
The Navier-Stokes equations	13	(2)
Bhatnagar-Gross-Krook model equation	15	(1)
Exercises	15	(2)
Lattice Gas Cellular Automata	17	(23)
Fluids in Gridland: the Frisch-Hasslacher-Pomeau automaton	17	(2)
Fluons in action: LGCA microdynamic evolution	19	(8)
From LGCA to Navier-Stokes	27	(3)
Discrete local equilibria	28	(2)
Practical implementation	30	(3)
Lattice gas disease and how to cure them	33	(5)
Statistical noise	34	(1)
Low Reynolds number	34	(2)
Exponential complexity	36	(2)
Spurious invariants	38	(1)
Summary	38	(1)
Exercises	38	(2)
Lattice Boltzmann models with underlying Boolean microdynamics	40	(11)
Nonlinear LBE	40	(4)
Lattice quantum fluids	43	(1)
The quasilinear LBE	44	(2)
The scattering matrix Aij	46	(3)
Numerical experiments	49	(1)
Exercises	50	(1)
Lattice Boltzmann models without underlying Boolean microdynamics	51	(14)
LBE with enhanced collisions	51	(4)
Hydrodynamic and ghost fields	55	(6)
Field-theoretical analogies	58	(1)
Dimensional compactification	59	(1)
Removing ghost fields	60	(1)
The route to Navier-Stokes: adiabatic assumption	61	(1)
The mirage of zero viscosity	62	(1)
Numerical experiments	63	(1)
Exercises	64	(1)
Lattice Bhatnagar-Gross-Krook	65	(12)
Single-time relaxation	65	(1)
LBGK equilibria	66	(2)
LBGK versus LBE	68	(2)
Relation to continuum kinetic theory	70	(2)
Relation to discrete velocity models	72	(1)
LBE genealogy	72	(1)
Warm-up code	73	(1)
Exercises	73	(4)
II FLUID DYNAMICS APPLICATIONS AND ADVANCED THEORY
Boundary conditions	77	(20)
General formulation of LBE boundary conditions	77	(1)
Survey of various boundary conditions	78	(12)
Periodic boundary conditions	79	(3)
No-slip boundary conditions	82	(2)
Free-slip boundary conditions	84	(2)
Frictional slip	86	(1)
Sliding walls	87	(1)
The Inamuro method	88	(1)
Moving walls	89	(1)
Open boundaries	90	(1)
Complex (misaligned) boundaries	91	(3)
Staircased boundaries	92	(1)
Extrapolation schemes	92	(1)
The picky stuff	92	(1)
The surfel method	93	(1)
Exactly incompressible LBE schemes	94	(2)
Exercises	96	(1)
Flows at moderate Reynolds number	97	(13)
Moderate Reynolds flows in simple geometry	97	(2)
LBE implementation	99	(3)
Boundary conditions	102	(1)
Flows past obstacles	103	(3)
More on the pressure field: Poisson-freedom	106	(3)
Exercises	109	(1)
LBE flows in disordered media	110	(14)
Flows through porous media	110	(2)
LBE flows through porous media	112	(2)
Setting up the LBE simulation	114	(5)
Deposition algorithm	119	(2)
Numerical simulations	121	(1)
Synthetic matter and multiscale modeling	122	(1)
Exercises	123	(1)
Turbulent flows	124	(18)
Fluid turbulence	124	(3)
Two-dimensional turbulence	126	(1)
Turbulence and kinetic scales	126	(1)
LBE simulations of two-dimensional turbulence	127	(7)
Seeing the invisible: sub-grid scales and numerical stability	131	(3)
Three-dimensional turbulence: parallel performance	134	(2)
Three-dimensional channel flow turbulence	136	(1)
Sub-grid scale modeling	137	(4)
Two-equation models	139	(1)
Non-local eddy viscosity models	140	(1)
Wall-turbulence interactions	140	(1)
Summary	141	(1)
Exercises	141	(1)
Out of Legoland: geoflexible lattice Boltzmann equations	142	(13)
Coarse-graining LBE	142	(1)
Finite volume LBE	143	(6)
Piecewise-constant streaming	144	(1)
Piecewise-linear streaming	145	(1)
Piecewise-linear collision operator	146	(2)
Piecewise-parabolic interpolation	148	(1)
Finite difference LBE	149	(1)
Interpolation-supplemented LBE	149	(1)
Finite element LBE	150	(1)
Native LBE schemes on irregular grids	151	(1)
Implicit LBE schemes	151	(1)
Multiscale lattice Boltzmann scheme	152	(1)
Summary	153	(1)
Exercises	154	(1)
LBE in the framework of computational fluid dynamics	155	(24)
LBE and CFD	155	(17)
Causality	156	(2)
Accuracy	158	(1)
Stability	159	(3)
Consistency	162	(7)
Efficiency	169	(1)
Flexibility	170	(2)
Link to fully Lagrangian schemes	172	(3)
LBE in a nutshell	175	(1)
Exercises	176	(3)
III BEYOND FLUID DYNAMICS
LBE schemes for complex fluids	179	(35)
LBE theory for generalized hydrodynamics	179	(2)
LBE schemes for reactive flows	181	(5)
Reactive LBE applications	183	(3)
LBE schemes for multiphase flows	186	(15)
Surface tension and interface dynamics	188	(2)
Numerical methods for flows with interfaces	190	(1)
Chromodynamic models	190	(2)
The pseudo-potential approach	192	(2)
The free energy approach	194	(3)
Finite density models	197	(2)
Miscellaneous multiphase LBE applications	199	(2)
LBE schemes for flows with moving objects	201	(1)
Colloidal flows	201	(8)
The fluctuating LBE	202	(1)
Solid-fluid moving boundaries	203	(2)
Numerical tests	205	(3)
Computational cost of the fluctuating LBE	208	(1)
Thermal equilibrium	209	(1)
Polymers in LBE flows	209	(2)
Snow transport and deposition	211	(1)
A new paradigm for non-equilibrium statistical mechanics?	212	(1)
New vistas	213	(1)
Exercises	213	(1)
LBE for quantum mechanics	214	(19)
Quantum mechanics and fluids	214	(1)
The fluid formulation of the Schrodinger equation	214	(6)
Relativistic quantum mechanics: the Dirac equation	216	(2)
Dirac to Schrodinger: the adiabatic approximation	218	(2)
The interacting case	220	(1)
The quantum LBE	220	(4)
Extended operator splitting 3 = 1 + 1 + 1	221	(1)
Quantum LBE: move, turn and collide	222	(1)
Time marching	222	(2)
Numerical tests	224	(4)
Free particle motion	224	(2)
Harmonic oscillator	226	(1)
Scattering over a rectangular barrier	226	(2)
The quantum N-body problem	228	(4)
Quantum lattices for quantum computers	229	(1)
Bits, trits and q-bits	230	(1)
Quantum LBE and density functional theory	231	(1)
Exercises	232	(1)
Thermohydrodynamic LBE schemes	233	(21)
Isothermal and athermal lattices	233	(2)
Thermodynamic equilibria and multi-energy lattices	235	(1)
Extended parametric equilibria	236	(4)
Thermal LBE models without nonlinear deviations	240	(6)
Reduced thermohydrodynamic schemes	246	(2)
Attempts to rescue thermal LBE	248	(3)
Tolerance to realizability violations	248	(1)
The kinetic closure approach	248	(1)
Non space-filling lattices	249	(1)
Models with internal energy	249	(2)
The Digital Physics approach	251	(1)
Fake temperature schemes	252	(1)
Summary	252	(1)
Exercises	253	(1)
Finale: Who needs LBE?	254	(3)
DontUse class	254	(1)
CanUse class	254	(1)
ShouldUse class	255	(1)
MustUse class	255	(2)
Appendices	257	(6)
Integer LBE	257	(1)
The pseudospectral method	258	(1)
A primer on parallel computing	259	(2)
Fraction of parallel content	260	(1)
Communicativity	260	(1)
Load balancing	261	(1)
From lattice units to physical units	261	(2)
Length	261	(1)
Time	262	(1)
Mass	262	(1)
References	263	(18)
Index	281