Lattice Boltzmann Methods in
Interfacial Wave Modelling

James Maxwell Buick

Doctor of Philosophy
The University of Edinburgh
1997

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• Contents
• List of Figures
• List of Tables
• Introduction
  • The Study of Fluid Motion
  • Numerical Methods in Fluid Study
    • Numerical Solutions of the Navier-Stokes Equation
    • Molecular Dynamics
    • Lattice Gas Modelling
    • Lattice Boltzmann Models
  • Wave Modelling
  • Aims
  • Notation
  • Preview
• The Boltzmann Equation
  • The Classical Boltzmann Equation
    • The Conservation Equations
    • The Collision Function
  • Boltzmann's H-Theorem
  • The Chapman-Enskog Method
  • The Single Relaxation Model
  • The Boltzmann Equation for a High Density Fluid
  • Summary
• The Lattice Gas Model
  • Definition of a Lattice Gas Model
  • Development of the Lattice Gas Model
    • The HPP Model
    • The FHP Models
      • The FHP-I Model
      • The FHP-II Model
      • The FHP-III Model
      • Other FHP Models
    • Three-Dimensional Models
      • The Multi-Speed Model on a Cubic Lattice
  • Boundaries in the Model
  • Updating the Lattice
  • Equations for the Lattice Gas Model
    • Definitions
    • Microdynamical Equations
    • Macrodynamical Equations
    • Isotropy of the Model
  • The Navier-Stokes Equation
  • Units of Measurement in a Lattice Gas Model
  • Obtaining Macroscopic Quantities
  • Binary Fluid Models
    • Introduction to Models with Coloured Particles
    • Properties of Colour Models
    • Surface Tension in a Binary Fluid Model
      • The Reactive Model
      • The Colour-Field Model
  • Liquid-Gas Lattice Gas Models
    • The Interactions
    • Implementation of the Interactions
  • Lattice Gas Simulations
    • Drawbacks of the Lattice Gas Approach
      • Lack of Galilean Invariance
      • Noisy Results
      • Large Collision Matrices
      • Non-Physical Phase Separation
  • Summary
    • Exclusion Principle
    • Galilean Invariance
    • Conservables
    • Isotropy
    • Results for the FHP Models
• The Lattice Boltzmann Model
  • Development of the Lattice Boltzmann Model
    • Lattice Boltzmann Model for the Ensemble Averaged Distribution Function
    • The Linear Collision Operator
    • The Enhanced Collision Rules
    • The Single Relaxation Time Lattice Boltzmann Model
  • An Isotropic, Galilean Invariant BGK Model
    • The Equilibrium Distribution
    • The Conservation Equations
      • Useful Relations
      • Basic Model
    • Chapman-Enskog Expansion
  • Boundaries in a Lattice Boltzmann Model
    • Bounce Back Boundary Conditions
    • Higher-Order Boundary Conditions
      • A Finite Difference Method
      • Dirichlet Boundary Conditions
  • Binary-Fluid and Liquid-Gas Lattice Boltzmann Models
    • Colour Model
    • Miscible Binary Fluid
    • The Local Interaction Model
    • The Free Energy Model
      • Binary Fluid Model
      • Liquid-Gas Model
    • The Distribution Functions and the Equations of Motion for a Binary Fluid
      • Solving for the Distribution Functions
      • The Equations of Motion for a Binary Fluid
    • Model Selection
  • Implementation of the Free Energy Binary Model
    • Fluid Separation
    • Bubble Equilibrium
    • The Density and Order Parameter at the Interface
    • Isotropy
    • Galilean Invariance
  • Summary
• Gravity in a Lattice Boltzmann Model
  • Introducing Gravity
    • The Classic Boltzmann Equation
    • Combining the Gravity Term and the Pressure Tensor
    • Adding a Force Term to Equation ()
    • Calculating the Equilibrium Distribution with an Altered Velocity
    • Adding an Additional Term to the Boltzmann Equation
    • Review of Methods
  • Model Implementation
    • Density Gradient
    • Model Comparison
    • Grid Orientation
      • Results
    • Gravitational Strength
      • Single Species Model
      • Immiscible Binary Fluid
  • Galilean Invariance
  • Summary
• The Equations of Internal Wave Motion
  • The Potential Density
  • Inviscid Wave Equations
    • The Two-Layer Model
    • Continuous Density Variation
  • Waves in a Viscous Fluids
    • Frequency and Damping Parameter
    • Wave Velocities
  • Standing Waves
  • Summary
• Interfacial Standing Wave Simulations
  • The density Gradient, the Potential Density, the Relative Density and the Gravitational Strength
  • Standing Wave Initialisation
  • Standing Wave Simulations
  • The Wave Period and the Damping Parameter
    • The Curve Fitting Process
      • The Contribution to e from the Discrete Lattice
      • Relating the Error Parameter to an Error in the Fitted Parameters
      • Curve Fitting Results
    • Comparison with Theory
    • Continuously Varying Density at the Interface
  • Velocities
    • Velocity Variation Across a Vertical Cross-Section
    • Velocity Variation Across a Horizontal Cross-Section
    • Boundary Layer at the Solid Boundaries
    • Peak Horizontal Velocity
  • Summary
• Interfacial Progressive Wave Simulations
  • Progressive Wave Initialisation
  • Progressive Interfacial Waves
  • Experimental Investigations into Progressive Interfacial Waves
  • Comparison Between Interfacial Wave Simulations and Experimental Results
    • Wave Parameters
      • Density Profile
      • Dimensional Parameters
      • Dimensionless Parameters
      • Comparison of Dimensionless Parameters
    • Numerical Comparisons
  • Summary
• Conclusion
  • Body Forces
  • Interfacial Wave Modelling
  • Comparison of Results with Theory and Experiment
  • Model Strengths
• References
• Notation
• FHP-III Collisions
  • Boolean Equations
  • Lookup Tables
• Publications
• About this document ...