kinetic theory liouville bbgky lattice boltzmann bgk dissertation thesis pdf

Kinetic Theory Liouville Bbgky Lattice Boltzmann Bgk Dissertation Thesis Download Citation Copy

Kinetic Theory Liouville Bbgky Lattice Boltzmann Bgk Dissertation Thesis Download Citation Copy

E, Statistical physics, plasmas, fluids, and related interdisciplinary topics 56(6):6811-6817 DOI: 10.1103PhysRevE.56.6811 Authors: Xiaoyi He Independent Researcher Li-Shi Luo Old Dominion University Download full-text PDF Read full-text Download full-text PDF Read full-text Download citation Copy link Link copied Read full-text Download citation Copy link Link copied Citations (1,375) References (58) Abstract In this paper, the lattice Boltzmann equation is directly derived from the Boltzmann equation.

It is shown that the lattice Boltzmann equation is a special discretized form of the Boltzmann equation.

Various approximations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail.

The lattice Boltzmann models derived include the two-dimensional 6-bit, 7-bit, and 9-bit, and three-dimensional 27-bit models.

Various approxi- mations for the discretization of the Boltzmann equation in both time and phase space are discussed in detail.

A general procedure to derive the lattice Boltzmann model from the continuous Boltzmann equation is dem- onstrated explicitly.

S1063-651X 97 12512-0 PACS number s: 47.10.

Dd I. INTRODUCTION In the last few years, we have witnessed a rapid develop- ment of the method known as the lattice Boltzmann equation LBE 16.

Although only in its infancy, the LBE method has demonstrated its ability to simulate hydrodynamic sys- tems 15, magnetohydrodynamic systems 7, multiphase and multicomponent uids 8 including suspensions 9 and emulsions 10, chemical-reactive ows 11, and multi- component ow through porous media 12.

Together with modern computers of massively parallel processors, the LBE method has become a powerful computational method for studying various complex systems.

The obvious advantages of the LBE method are the parallelism of the method, the simplicity of the programming, and the capability of incor- porating model interactions.

However, a rigorous framework of the LBE method is still lacking in spite of the great inter- est in the method.

Historically, the models of the lattice Boltzmann equation directly evolve from the models of the lattice-gas automata LGA 13.

While the LGA models are Boolean ones, the LBE models are indeed the oating-number counterpart of the corresponding LGA modelsa particle in the LGA model represented by a Boolean number is replaced by the single-particle distribution function represented by a real number.

Even though the LBE models appear to be rather different from their LGA counterparts because various approximations, such as the linearization of the collision op- erator 2 and the Bhatnagar-Gross-Krook BGK 151 7 approximation 3, have been applied, the theoretical frame- work of the LBE method nevertheless rests upon the Chapman-Enskog analysis of the LGA method 14.

Al- though the connection between the LBE models and the con- tinuous Boltzmann equation are discussed in various places 18,19, so far there exists no rigorous result in this direction.

The lack of a thorough understanding of the LBE method has some immediate implications.

For instance, the LBE method has not been very successful in simulating thermohydrody- namic systems 192 2, nor has the LBE method been able to be implemented on arbitrary mesh grids 5,23, even though considerable effort has been applied in these direc- tions.

As we have shown recently 242 6, substantial progress can be made in the aforementioned areas once a better understanding of the LBE method is attained.

Further- more, through the derivation we can directly show the con- nection between the LBE method and other newly developed gas kinetic methods 273 3.

In this paper we will show that the lattice Boltzmann equation can be directly derived from the continuous Boltz- mann equation discretized in some special manner in both time and phase space.

Our analysis shows that theoretically the lattice Boltzmann equation is independent of the lattice- gas automata.

Kinetic Theory Liouville Bbgky Lattice Boltzmann Bgk Dissertation Thesis Pdf