A Few More LBM Boundary Conditions

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A Few More LBM Boundary Conditions
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Key paper

• Zou, Q. and X. He, 1997, On pressure and velocity
  boundary conditions for the lattice Boltzmann BGK
  model, Phys. Fluids 9, 1591-1598.

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Choices

• Specify density (i.e., pressure via EOS)
• Velocity computed
• Dirichlet
• Specify velocity
• Density/pressure computed
• Von Neumann
• Lots of temporal/spatial flexibility

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D2Q9 BCs

• For example
• f(4,7,8) function of f(1,2,3,5,6) and BC type

Out
In
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Velocity/Flux BCs

• Need to solve for r, f4, f7, f8
• Need 4 equations
• The macroscopic density formula is one equation

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Velocity/Flux BCs

• The macroscopic velocity formula gives two
  equations
• x-direction
• y-direction

Components of ea are all unit vectors

Assuming ux 0
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Velocity/Flux BCs

• Finally, we assume bounceback of f perpendicular
  to boundary for a fourth equation

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Velocity/Flux BCs

• Two equations have the directional density
  unknowns f4, f7 and f8 in common, so rewrite them
  with those variables on the left hand side

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Velocity/Flux BCs

• Equating them gives
• Solving for r

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Velocity/Flux BCs

• Solving the bounceback equation for f4
• In detail, part of this is

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Velocity/Flux BCs

• Solving

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Velocity/Flux BCs

• Solving

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Velocity/Flux BCs

• // Zou and He velocity BCs on north side.
• for( i0 iltni i)
• fi ftempnj-1i
• rho0 ( fi0 fi1 fi3
• 2.( fi2 fi5 fi6)) / ( 1.
  uy0)
• ru rho0uy0
• fi4 fi2 - (2./3.)ru
• fi7 fi5 - (1./6.)ru (1./2.)(
  fi1-fi3)
• fi8 fi6 - (1./6.)ru (1./2.)(
  fi3-fi1)

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Pressure/Density Boundaries

• Dirichlet boundary conditions constrain the
  pressure/density at the boundaries
• Solution is closely related to that for velocity
  boundaries
• A density r0 is specified and velocity is
  computed
• Specifying density is equivalent to specifying
  pressure since there is an equation of state
  (EOS) relating them directly
• For single component D2Q9 model, the relationship
  is simply P RTr with RT 1/3.
• More complex EOS applies to single component
  multiphase models
• We assume that velocity tangent to the boundary
  is zero and solve for the component of velocity
  normal to the boundary.

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Pressure/Density Boundaries

• Assume that velocity tangent to the boundary is
  zero and solve for the component of velocity
  normal to the boundary
• Need to solve for v, f4, f7 and f8

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Pressure/Density Boundaries
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Pressure/Density Boundaries
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Pressure/Density Boundaries

• // Zou and He pressure boundary on north side.
• for( i0 iltni i)
• fi ftempnj-1i
• uy0 -1. ( fi0 fi1 fi3
• 2.( fi2 fi5 fi6))
  / rho0
• ru rho0uy0
• fi4 fi2 - (2./3.)ru
• fi7 fi5 - (1./6.)ru (1./2.)(
  fi1-fi3)
• fi8 fi6 - (1./6.)ru (1./2.)(
  fi3-fi1)

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Exercise

• Create a new version of your code that includes
  constant pressure boundaries at x 0 and x Lx.
  
• Plot the observations and expected Poiseuille
  velocity profile on the same graph