格子boltzmann和ppt10

2 3
f7 f4


ρ 0 v = ρ 0 v + 2 f 6 − 2 f 8 − ( f1 − f 3 ) ⇒
2 3 f8 = f 6 − 1 ( f1 − f 3 ) − 1 ρ 0v. 2 6
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]); • }
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
f 7 = f5 + 1 ( f1 − f3 ) − 1 ρ0v. 2 6
2 3
Pressure/Density Boundaries
ρ 0v = f 2 − f 4 + f 5 + f 6 − f 7 − f 8 ⇒ ρ 0 v = f 2 − f 2 − ρ 0 v + f 5 + f 6 − ( f1 − f 3 + f 5 − f 6 + f 8 ) − f 8 ⇒
Pressure/Density Boundaries
• Dirichlet boundary conditions constrain the pressure/density at the boundaries • Solution is closely related to that for velocity boundaries • A density ρ0 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 = RTρ 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.
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
ρ0 + ρ0v = f0 + f1 + f2 + f3 + f5 + f6 + f2 + f5 + f6 ⇒
v = −1+ f0 + f1 + f3 + 2( f2 + f5 + f6 )
ρ0
.
Pressure/Density Boundaries
f4 = f2 − f
2 3
eq 2
+f
Choices
• Specify density (i.e., pressure via EOS)
– Velocity computed – Dirichlet
• Specify velocity
– Density/pressure computed – Von Neumann
• Lots of temporal/spatial flexibility
Unknown after streaming
2 ρv0 = f2 − f2 − ρv0 + f5 + f6 − ( f1 − f3 + f5 − f6 + f8 ) − f8 ⇒ 3 f
7
f4
2 ρv0 = ρv0 + 2 f6 − 2 f8 − ( f1 − f3 ) ⇒ 3 1 1 f8 = f6 − ( f1 − f3 ) − ρv0. 2 6
eq 2
= f4 − f
eq 4
Constant Concentration
Velocity/Flux BCs
Unknown after streaming
• Two equations have the directional density unknowns f4, f7 and f8 in common, so rewrite them with those variables on the left hand side:
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.
f 4 + f 7 + f8 = ρ 0 − f 0 − f1 − f 2 − f 3 − f 5 − f 6
f 4 + f 7 + f8 = f 2 + f 5 + f 6 − ρ 0v
ρ0 − f 0 − f1 − f 2 − f 3 − f 5 − f 6 = f 2 + f 5 + f 6 − ρ0v
f0 + f1 + f3 + 2( f2 + f5 + f6 ) ρ= . 1+ v0
Constant Concentration
Velocity/Flux BCs
f4 = f2 − f
f 4eq − f 2eq =
eq 2
Unknown after streaming
• Solving the bounceback equation for f4:
Constant Concentration
Velocity/lux BCs
Unknown after streaming
• Finally, we assume bounceback of f perpendicular to boundary for a fourth equation:
f2 − f
f 4 + f 7 + f8 = ρ − f 0 − f1 − f 2 − f 3 − f 5 − f 6
f 4 + f 7 + f8 = f 2 + f 5 + f 6 − ρv0
Constant Concentration
Velocity/Flux BCs
• Equating them gives:
Unknown after streaming
ρ − f 0 − f1 − f 2 − f 3 − f 5 − f 6 = f 2 + f 5 + f 6 − ρv0
• Solving for ρ ρ:
ρ + ρv0 = f0 + f1 + f2 + f3 + f5 + f6 + f2 + f5 + f6 ⇒
+f
eq 4
2 = f 2 − ρv0 3
• In detail, part of this is:
1 1 2 1 1 2 2 ρ + ρ ( −1 ⋅ v0 ) + ρv0 − ρ (u0 + v0 ) 3 2 6 9 1 1 2 1 2 1 2 2 − ρ + ρ (1 ⋅ v0 ) + ρv0 − ρ (u0 + v0 ) = − ρv0 3 2 6 3 9
D2Q9 BCs
• For example: Out In
– f(4,7,8) = function of f(1,2,3,5,6) and BC type
Constant Concentration
Velocity/Flux BCs
Unknown after streaming
• Need to solve for ρ, f4, f7, f8 • Need 4 equations • The macroscopic density formula is one equation: