Boundary conditions are required to complete the problem we wish to solve. We
therefore need to specify boundary conditions on all our boundary faces.
Boundary conditions can be divided into 2 types:

Dirichlet

prescribes the value of the dependent variable on the boundary
and is therefore termed ‘fixed value’ in this guide;

Neumann

prescribes the gradient of the variable normal to the boundary
and is therefore termed ‘fixed gradient’ in this guide.

When we perform discretisation of terms that include the sum over faces
, we need to consider what happens when one of the faces is a boundary
face.

Fixed value

We specify a fixed value at the boundary

We can simply substitute in cases where the discretisation
requires the value on a boundary face , e.g. in the convection
term in Equation 2.16.

In terms where the face gradient is required, e.g. Laplacian, it
is calculated using the boundary face value and cell centre
value,

(2.38)

Fixed gradient

The fixed gradient boundary condition is a specification on
inner product of the gradient and unit normal to the boundary,
or

(2.39)

When discretisation requires the value on a boundary face
we must interpolate the cell centre value to the boundary
by

(2.40)

can be directly substituted in cases where the discretisation
requires the face gradient to be evaluated,

The specification of boundary conditions is usually an engineer’s interpretation of
the true behaviour. Real boundary conditions are generally defined by some
physical attributes rather than the numerical description as described of the
previous Section. In incompressible fluid flow there are the following physical
boundaries

Inlet

The velocity field at the inlet is supplied and, for consistency, the
boundary condition on pressure is zero gradient.

Outlet

The pressure field at the outlet is supplied and a zero gradient
boundary condition on velocity is specified.

No-slip impermeable wall

The velocity of the fluid is equal to that of
the wall itself, i.e. a fixed value condition can be specified. The pressure
is specified zero gradient since the flux through the wall is zero.

In a problem whose solution domain and boundary conditions are symmetric
about a plane, we only need to model half the domain to one side of the symmetry
plane. The boundary condition on the plane must be specified according
to

Symmetry plane

The symmetry plane condition specifies the component
of the gradient normal to the plane should be zero. [Check**]