1. A Few More LBM Boundary Conditions

2. Key paper:
Zou, Q. and X. He, 1997, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. Fluids 9,

3. 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, March 27, 1925, German mathematician
Lots of temporal/spatial flexibility

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

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

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

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

8. 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:

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

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

11. Velocity/Flux BCs
Solving …:

12. Velocity/Flux BCs
Solving …:

13. 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]); }

14. 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.

15. 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

16. Pressure/Density Boundaries

17. Pressure/Density Boundaries

18. Pressure/Density Boundaries
// Zou and He pressure boundary on north side.
for( i=0; i<ni; i++) {
fi = ftemp[nj-1][i];
uy0 = ( 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]); }

19. 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