The bounce-back boundary condition

The treatment for a curved boundary is a combination of the bounce-back scheme and interpolations (Bouzidi et al.,2002). For the sake of simplicity, this boundary condition is illustrated in Fig. 6.7 for an idealized situation in two dimensions. Consider a wall located at an arbitrary position rW

between two grid sites rA and rS, with rS situated in the nonfluid region –

the shaded area depicted in Fig. 6.7. The parameter q defines the frac- tion of a grid spacing intersected by the boundary lying in fluid region, i.e.

conditions place the wall somewhere beyond the last fluid node (Ginzbourg and d’Humi`eres,1996,2003), and for the MRT-GLBM model with appropri- ate choice of the relaxation rates, it is about one-half grid spacing beyond the last fluid node, i.e. q = 1/2, as shown in Fig. 6.7(a). That is, even though the bounce-back collision occurs on the node rA, the actual position of the

wall is located at rW, which is about one-half grid spacing beyond the last

fluid node rA. Thus one could intuitively picture the bounce-back boundary

as follows: the particle with velocity e1, starting from rA, travels from left to

right, hits the wall at rW, reverses its momentum, then returns to its start-

ing point rA. This imaginary particle trajectory is indicated by the thick

bent arrow in Fig.6.7(a). The total distance traveled by the particle is one grid spacing ∆x during the bounce-back collision. Therefore, one can imag- ine that the bounce-back collision either takes one time step or no time at

Fig. 6.7. Illustration of the boundary conditions for a rigid wall located arbitrarily between two grid sites in one dimension. The thin solid lines are the grid lines, the dashed line is the boundary location situated arbitrarily between two grids. Shaded disks (•) are the fluid nodes, and the disks (•) are the fluid nodes next to boundary. Circles (◦) are located in the fluid region but not on grid nodes. The square boxes (2) are within the nonfluid region. The thick arrows represent the trajectory of a particle interacting with the wall, described in equations (6.66a) and (6.66b). The distribution functions at the locations indicated by disks are used to interpolate the distribution function at the location marked by the circles (◦). (a) q := |r_A − r_W|/∆x = 1/2. This is the perfect bounce-back condition – no interpolations needed. (b) q < 1/2. (c) q≥ 1/2.

Lattice Boltzmann Methods 185 all, corresponding to two implementations of the bounce-back scheme: the so-called “link” and “node” implementations (Ladd, 1994a,b), respectively. The difference between these two implementations is that the bounced-back distribution (e.g. f3in Fig.6.7a) at the boundary nodes in the “link” imple-

mentation is one time step behind that in the “node” implementation. This difference vanishes for steady state calculations, although these two imple- mentations of the bounce-back boundary conditions have different stability characteristics, because the “link” implementation destroys the parity sym- metry on a boundary node, whereas the “node” implementation preserves it. However, one cannot determine a priori the advantage of one implemen- tation over the other, because the features of the two implementations also strongly depend on the precise location of the boundary and the local flow structure. It is also important to stress that the actual boundary location is not affected by the two different implementations of the bounce-back scheme (Ginzbourg and d’Humi`eres,1996; Bouzidi et al.,2002).

Assuming that the picture for the simple bounce-back scheme is correct, let’s first consider the situation depicted in Fig.6.7(b) in detail for the case of q < 1/2. At time tn the distribution function of the particles with ve- locity pointing to the wall (e1 in Fig. 6.7) at the grid point rA (a fluid

node) would end up at the point rO located at a distance (1− 2q)∆x away

from the grid point rA after the bounce-back collision, as depicted by the

thin bent arrow in Fig. 6.7(b). Because rO is not a grid point, the value

of f3 at the grid point rA needs to be reconstructed. However, noticing

that f1 starting from point rO would become f3 at the grid point rA af-

ter the bounce-back collision with the wall, we can construct the value of f1 at the point rO by, for instance, a quadratic interpolation involving

values of f1 at the three locations: f1(rA), f1(rB) = f1(rA − e1∆t), and

f1(rC) = f1(rA− 2e1∆t). In a similar manner, for the case of q ≥ 1/2 de-

picted in Fig.6.7(c), we can construct f3(rA) by a quadratic interpolation

involving f3(rO) that is equal to f1(rA) before the bounce-back collision,

and the values of f3 at the nodes after collision and advection, i.e. f3(rB),

and f3(rC). Therefore the interpolations are applied differently for the two

cases:

• For q < 1/2, interpolate before propagation and bounce-back collision. • For q ≥ 1/2, interpolate after propagation and bounce-back collision. We do so to avoid the use of extrapolations in the boundary conditions for the sake of numerical stability. This leads to the following interpolation formulas which combine the advection and bounce-back together in one step

(Lallemand and Luo, 2003): f¯a(rA, t) = q (1 + 2 q)fa(rA+ ea∆t, t) + (1− 4 q2)fa(rA, t) −q (1 − 2 q)fa(rA− ea∆t, t) + 3wa(ea· uw), q < 1 2, (6.66a) f¯a(rA, t) = 1 q (2 q + 1)fa(rA+ ea∆t, t) + (2 q− 1) q fa¯(rA− ea∆t, t) −(2 q− 1) (2 q + 1)f¯a(rA− 2ea∆t, t) + 3wa q (2 q + 1)(ea· uw), q ≥ 1 2, (6.66b) where the coefficients{w_a} are model dependent (He and Luo,1997a,b), f¯a

denotes the distribution function of the velocity e¯a := −ea, and uw is the

velocity of the wall at the point rW in Fig. 6.7.

It should be mentioned that for linear GLBM, the multiple-reflection scheme (Ginzbourg and d’Humi`eres, 2003) can set the boundary location for quadratic flows (such as the Poiseuille flow) exactly one half grid spacing beyond the last fluid node, regardless of the flow direction relative to the un- derlying lattice. It should be emphasized that equation (6.66a) only involves the position of a boundary relative to the computational mesh of Cartesian grids, and therefore it can be readily applied in 3D (d’Humi`eres et al.,2001,

2002; Yu et al.,2002).

The interpolations only improve the geometric representation of a curved boundary, which is independent of the LBM. It can be analytically shown that the exact location where the no-slip boundary conditions are satisfied depends on the value of the relaxation parameter τ and the specific im- plementation of the BBBCs for the LBGK models (He et al., 1997c). The

τ -dependence of the boundary location is particularly severe when τ > 1 for

the LBGK equation (He et al.,1997). This deficiency of the LBGK equation can be easily overcome by using the MRT-LBM with two relaxation times:

se= 1 τ_∗ = 2 1 + 6ν_∗, so = 8 (2− se) (8− se) , (6.67)

where ν_∗ = ν/(e2∆t) is the dimensionless viscosity, and se and so are

the relaxation rates for the (nonconserved) even-order moments and odd- order moments, respectively. With two-relaxation times (Ginzbourg et al.,

2004; Ginzbourg, 2005), the viscosity-dependence of boundary location is completely eliminated for the linear LBM models valid for Stokes flows, i.e. the nonlinear terms in terms of j in equilibria are neglected. For the full MRT-LBM models with the nonlinear terms and the relaxation rates given by equation (6.67), the effect due to the viscosity dependence of