Basic concepts of Lattice Boltzmann Methods

Overview of LBM

• Overview of Lattice Boltzmann Methods
  • Continuous Boltzmann equation
  • Discretization of velocity space
  • Lattice Boltzmann Equation (LBE)
  • Equilibrium distribution function
  • Explicit algorithm
  • Chapman-Enskog expansion
  • Variable change in LBE with a source term
  • References

Equilibrium distribution functions

• Equilibrium distribution functions
• Equilibrium distribution functions for incompressible Navier-Stokes equations
  • Standard \(f_i^{eq}\) for single-phase incompressible NS
  • \(f_i^{eq}\) for incompressible two-phase flows
  • References
• Equilibrium distribution functions for Transport equations
  • \(g_i^{eq}\) for ADE with source term
  • \(g_i^{eq}\) for Cahn-Hilliard equation
  • \(g_i^{eq}\) for Conservative Allen-Cahn equation

Variable change for source and force terms

• Variable change in LBE for source term
  • Equivalence
  • Bibliography

Chapman-Enskog expansion

• Chapman-Enskog expansion for Navier-Stokes equations
  • Starting point
  • Objectives
  • Conservative quantities and moments of equilibrium distribution functions
  • Chapman-Enskog expansion
  • Second part: viscosity coefficient and impulsion balance
  • Récapitulatif
  • Démonstration de l’égalité eq:Equivalence
  • Bibliography