2.8. Open boundaries

2.8.1. Open boundary conditions and walls

Open boundary conditions may be used with a number of provisos:

  1. An inflow must be have a corresponding outflow, and vice-versa;

  2. The non-flow directions should have plane walls;

  3. We assume that the inflow is at the lower, or ‘left-hand’, edge of the domain, and the outflow is at the upper, or ‘right-hand’ end;

  4. All three dimensions must be non-periodic.

At present there is an additional constraint when running in parallel that the flow direction must be the \(x\)-direction.

A suitable set of input keys would therefore be, e.g.:

boundary_walls       0_1_1
periodicity          0_0_0
lb_bc_open           yes

The switch lb_bc_open introduces open boundary conditions of a type which is specified by further keys. It must be present for open boundaries to be recognised.

2.8.2. Hydrodynamic open boundaries

Hydrodynamic boundary conditions mean specifying appropriate density and velocities boundary conditions so that appropriate incoming distributions can be computed at both inflow and outflow. Outgoing distributions are lost.

In the following, we assume that the flow direction is the \(x\)-direction. Inflow conditions are indicated by \(x=0\), and outflow by \(x=L_x\). Gernealisations to the \(y\)-direction or \(z\)-direction may be made is a straightforward manner. Type rhou inflow

The following section:

lb_bc_inflow_type          rhou
lb_bc_inflow_profile       uniform
lb_bc_inflow_rhou_u0       0.001_0.0_0.0

will request an inflow condition which imposes a uniform velocity \(u(0,y,z) = u_0\). The density is \(\rho(0,y,z) = \rho(1,y,z)\). The incoming distributions are then computed as \(f_i = f^{eq}_i (\rho, u)\). The velocity specified at the inflow should be in the positive \(x\)-direction. Type rhou outflow

Corresponding outflow conditions are introduced via, e.g.:

lb_bc_outflow_type         rhou
lb_bc_outflow_rhou_rho0    1.0

Here the outflow density \(\rho(L_x,y,z) = \rho_0\), and the velocity \(u(L_x,y,z) = u(L_x-1,y,z)\); incoming distributions are again computed via \(f_i = f^{eq}_i(\rho_0, u)\).

2.8.3. Open boundaries for composition

Open boundary conditions (with the same provisos as mentioned above) are available for composition variable \(\phi\), as used in symmetric and Brazovskii free energy models.

Inflow and outflow conditions are set, e.g.,:

phi_bc_open                yes        # Use open boundaries for phi
phi_bc_inflow_type         fixed      # inflow type
phi_bc_inflow_fixed_phib   +1.0       # inflow parameter

phi_bc_outflow_type        free       # outflow type

These would typically be used with the corresponding hydrodynamic inflow/outflow conditions discussed above. Type fixed inflow

phi_bc_outflow_type          fixed
phi_bc_outflow_fixed_phib    -1.0

A fixed boundary condition sets \(\phi(x=0,y,z) = \phi_b\) where a uniform value \(\phi_b\) is specified via the key phi_bc_inflow_fixed_phib.

The boundary condition sets all values of \(\phi\) in the boundary region (up to the extent of the parallel halo region). This influences the calculation of the order parameter gradients \(\partial_\alpha \phi\) and \(\partial^2_\alpha \phi\) and hence the value of the chemical potential. The advective flux in the Cahn-Hilliard equation also responds to the boundary value appropriately in conjunction with the imposed inflow velocity. Type free outflow

phi_bc_outflow_type          free

This outflow boundary condition sets \(\phi_b (x = L+1, y, z) = \phi (L,y,z)\), again to the extent of the parallel halo region. There are no additional parameters associated with this boundary condition.

2.8.4. Restarting with open boundary conditions

At present, there may be some small differences in results introduced by a restart (compared with the same run ‘straight through’). This relates the need for density values in the open boundary regions. This should not impair the overall reliability of the results.