In chapter 2 the concept of a distribution function is
considered and the derivation and
theory of the classical Boltzmann equation
are discussed. Different forms of the collision
function and their effect on the equilibrium distribution function
are also considered. A brief outline of the derivation of the
equations of motions is also given.
In chapter 3 the concept of a lattice gas model is explained
and a full description of the technique is given. A square grid model is
described briefly, as are the three hexagonal grid models which have
been used most commonly in two-dimensional simulations.
A summary of Frisch's
derivation [8]
of the equations of the model is presented and
their implications on the lattice gas model considered. The development of
multi-fluid models is also considered. The strengths and weaknesses of the
lattice gas approach are discussed.
In chapter 4 an account of the development of the
lattice Boltzmann model, from the lattice gas technique, is given and
the similarity with the classical Boltzmann equation is shown.
A lattice Boltzmann model for a totally isotropic and
Galilean invariant fluid, from the literature, is presented which
satisfies the exact continuity equation and Navier-Stokes equation
correct to second-order in the expansion parameter and the velocity.
Multi-fluid lattice Boltzmann models are also discussed briefly
and the model which will be used in the subsequent chapters is
described in detail. At the end of the chapter a number of
simulation results for the chosen model, which have been performed
elsewhere, are repeated to highlight some of the important
features of the model.
In chapter 5 the inclusion of a body force in the lattice Boltzmann
scheme is investigated. A number of methods suggested in the literature are
considered and a new method is proposed which includes the body force
directly in the Boltzmann equation.
This method is compared with the other techniques and is tested to ensure it
is independent of the underlying grid orientation.
The immiscible binary fluid of Orlandini et al. [9]
with a body force incorporated is tested for Galilean invariance.
In chapter 6 the equations describing interfacial wave motion are
presented. These are all well established in the literature.
In chapter 7 the immiscible, binary fluid model of
Orlandini et al. [9] is combined with the
gravitational interactions described in chapter 5. A method
for initialising standing interfacial waves is described and the results obtained
from numerous simulations are presented. The results are compared with the
theoretical predictions given in chapter 6.
In chapter 8 the model implemented in chapter 7 is used to
simulate progressive interfacial waves. The progressive waves are
initialised using the information obtained in chapter 7 about the
density and velocity profiles. The resulting progressive wave
simulations are presented and compared with experimental results
obtained elsewhere [10]. A brief discussion of the experimental technique
is also given.