Lattice Boltzmann Method
• Fluids represented by fictional “particles” with discrete velocities, occupying sites on a discrete lattice.
• Particles described by real-valued distribution function
• Density of single component given by
• Momentum of single component given by
)()( xx i
ifm
ii
ifm cu
Time Evolution
• Advection step: particles move to adjacent sites.
• Collision step: distribution function relaxes to equilibrium value via BGK operator.
(eq)1'
iiii ffff
• The equilibrium distribution is a function only of density and velocity at a given site.
• Weighting factors chosen to ensure isotropic hydrodynamics.
• Bounce-back boundaries give non-slip walls.
222
(eq)
21
s
ii
ss
iiii c
cc
c
uu
c
ccnTf
iT
n u
• Velocity is perturbed to take account of collisions with other components, and forcing term.
• Force term includes gravity and Shan-Chen immiscibility force.
• Immiscibility force proportional to single-component density gradient, repelling differing components.
iigrav nnngF ccxxzx )()(ˆ)(
Fuv
/
Spinodal decomposition
• Two fluids may mix above some temperature• A mixture quenched below this temperature
will separate into its component fluids: this process is called Spinodal Decomposition.
• Simulation performed of demixing process using periodic boundary conditions.
CT
Growth Regimes
• Very early time: exponential growth in structure factor as interfaces form according to a Cahn-Hilliard model.
• Early-time hydrodynamic regime dominated by viscosity.
• Late-time inertial hydrodynamic regime: domains grow as two-thirds power of time.
• Very late time turbulent mixing regime postulated but not seen.
Early-time Cahn-Hilliard Growth
• Circularly-averaged structure factor S(k) retains shape but grows exponentially in magnitude as domains form.
• Although the model does not define any free energies, it produces results in agreement with free-energy models of phase separation.
Breakdown of scaling• Many theories assume that a phase-separating system
contains a single length scale evolving in time.
• Simulation shows regime with multiple length scales due to competition between different growth mechanisms.