Open boundaries
---------------

.. contents:: This section contains some details of open boundary conditions
	      available to drive channel-like flows.
   :depth: 2
   :local:
   :backlinks: none

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 :math:`x`-direction.

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

.. code-block:: none

  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.


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
:math:`x`-direction. Inflow conditions are indicated by
:math:`x=0`, and outflow by :math:`x=L_x`.
Gernealisations to the
:math:`y`-direction or :math:`z`-direction may be made is
a straightforward manner.


Type ``rhou`` inflow
""""""""""""""""""""

The following section:

.. code-block:: none

  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
:math:`u(0,y,z) = u_0`. The density is
:math:`\rho(0,y,z) = \rho(1,y,z)`. The incoming distributions
are then computed as :math:`f_i = f^{eq}_i (\rho, u)`. The velocity
specified at the inflow should be in the positive :math:`x`-direction.


Type ``rhou`` outflow
"""""""""""""""""""""

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

.. code-block:: none

  lb_bc_outflow_type         rhou
  lb_bc_outflow_rhou_rho0    1.0

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


Open boundaries for composition
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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

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

.. code-block:: none

  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
"""""""""""""""""""""

.. code-block:: none

  phi_bc_outflow_type          fixed
  phi_bc_outflow_fixed_phib    -1.0

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

The boundary condition sets all values of :math:`\phi` in the boundary
region (up to the extent of the parallel halo region). This influences
the calculation of the order parameter gradients
:math:`\partial_\alpha \phi` and :math:`\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
"""""""""""""""""""""

.. code-block:: none

  phi_bc_outflow_type          free


This outflow boundary condition sets
:math:`\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.


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.