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2.6 Boundary Conditions

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  sum 
  f \special {t4ht=, 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 f
  b  \special {t4ht=
  • We can simply substitute fb  \special {t4ht= in cases where the discretisation requires the value on a boundary face ff  \special {t4ht=, e.g. in the convection term in Equation 2.16.
  • In terms where the face gradient ( \~/ f)
     f  \special {t4ht= is required, e.g. Laplacian, it is calculated using the boundary face value and cell centre value,
    S  •( \~/ f)  = |S |fb---fP-
 f       f     f    |d |
           \special {t4ht=
    (2.38)

Fixed gradient
The fixed gradient boundary condition gb  \special {t4ht= is a specification on inner product of the gradient and unit normal to the boundary, or
     (        )
       S--
gb =   |S |• \~/ f
                f
      \special {t4ht=
(2.39)

  • When discretisation requires the value on a boundary face ff  \special {t4ht= we must interpolate the cell centre value to the boundary by
    ff = fP + d •( \~/ f)f

   = fP + |d |gb
           \special {t4ht=
    (2.40)

  • f
 b  \special {t4ht= can be directly substituted in cases where the discretisation requires the face gradient to be evaluated,
    Sf •( \~/ f)f = |Sf|gb
           \special {t4ht=
    (2.41)

2.6.1 Physical boundary conditions

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**]