A Few More LBM Boundary Conditions

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.

Choices Specify density (i.e., pressure via EOS) Specify velocity Velocity computed Dirichlet Johann Peter Gustav Lejeune Dirichlet, 13 February 1805 – 5 May 1859, German mathematician Specify velocity Density/pressure computed Neumann Carl Gottfried Neumann, May 7, 1832 - March 27, 1925, German mathematician Lots of temporal/spatial flexibility

D2Q9 BCs For example: Out In f(4,7,8) = function of f(1,2,3,5,6) and BC type Out In

Velocity/Flux BCs Need to solve for r, f4, f7, f8 Need 4 equations The macroscopic density formula is one equation:

Velocity/Flux BCs The macroscopic velocity formula gives two equations: x-direction: y-direction: Components of ea are all unit vectors Assuming ux = 0

Velocity/Flux BCs Finally, we assume bounceback of non-equilibrium part of f perpendicular to boundary for a fourth equation:

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:

Velocity/Flux BCs Equating them gives: Solving for r:

Velocity/Flux BCs Solving the bounceback equation for f4: In detail, part of this is:

Velocity/Flux BCs Solving …:

Velocity/Flux BCs Solving …:

Velocity/Flux BCs // Zou and He velocity BCs on north side. for( i=0; i<ni; i++) { fi = ftemp[nj-1][i]; rho0 = ( fi[0] + fi[1] + fi[3] + 2.*( fi[2] + fi[5] + fi[6])) / ( 1. + uy0); ru = rho0*uy0; fi[4] = fi[2] - (2./3.)*ru; fi[7] = fi[5] - (1./6.)*ru + (1./2.)*( fi[1]-fi[3]); fi[8] = fi[6] - (1./6.)*ru + (1./2.)*( fi[3]-fi[1]); }

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.

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

Pressure/Density Boundaries

Pressure/Density Boundaries

Pressure/Density Boundaries // Zou and He pressure boundary on north side. for( i=0; i<ni; i++) { fi = ftemp[nj-1][i]; uy0 = -1. + ( fi[0] + fi[1] + fi[3] + 2.*( fi[2] + fi[5] + fi[6])) / rho0; ru = rho0*uy0; fi[4] = fi[2] - (2./3.)*ru; fi[7] = fi[5] - (1./6.)*ru + (1./2.)*( fi[1]-fi[3]); fi[8] = fi[6] - (1./6.)*ru + (1./2.)*( fi[3]-fi[1]); }

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