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, 1591-1598.

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

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

5. Velocity/Flux BCs Need to solve for , f 4, f 7, f 8 Need 4 equations The macroscopic density formula is one equation:

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

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

8. Velocity/Flux BCs Two equations have the directional density unknowns f 4, f 7 and f 8 in common, so rewrite them with those variables on the left hand side:

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

10. Solving the bounceback equation for f 4 : In detail, part of this is: Velocity/Flux BCs

11. Solving …: Velocity/Flux BCs

12. Solving …: Velocity/Flux BCs

13. // 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]); } Velocity/Flux BCs

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  0 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 = RT  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, f 4, f 7 and f 8

16. Pressure/Density Boundaries

18. // 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]); }

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