4.5 Inlets and outlets

A basic set of boundary conditions was introduced in Sec. 4.3 for incompressible subsonic flow.

PICT\relax \special {t4ht=

At an inlet, fields are generally specified as fixed value. From a “physical” perspective, this is justified by the fact that disturbances propagate in the direction of flow so must be specified at the upstream boundary.

Advection requires interpolation of variables from cell centres (eqn) to faces (eqn). Some degree of upwind interpolation, e.g. as part of a limited scheme, is generally required. At a face at an inlet patch there is no upwind cell, so values eqn must be specified at inlet faces instead. From a “numerical” perspective, this justifies the fixed value boundary condition at an inlet.

For incompressible, subsonic flow, eqn is the exception. A fixed gradient condition allows disturbances to propagate upstream through the inlet as sound waves. The gradient condition is further justified since there is no advection of eqn, see Eq. (2.48 ), so no upwind interpolation is required at the inlet patch faces.

At an outlet, the converse is then true: fields are specified as fixed gradient condition, with the exception of eqn which is fixed value. The outlet conditions ultimately dictate the traction force combining Eq. (2.16 ), Eq. (2.33 ), Eq. (2.46 ) and Eq. (2.41 ) as follows:

t = (rnu + run) pn : \relax \special {t4ht=

PICT\relax \special {t4ht=

The standard condition applied to eqn is eqn. By Eq. (4.6 ), this results in a uniform normal traction force corresponding to the outlet pressure eqn. The traction force tangential to the outlet eqn, where eqn is the tangential direction and eqn is the normal component of eqn.

Supersonic conditions

PICT\relax \special {t4ht=

If the fluid is compressible and the flow speed is supersonic at the inlet, i.e. eqn, waves can no longer propagate outwards through the inlet boundary. When this occurs, a fixed value eqn must be specified at the inlet.

Similarly if the flow is supersonic at an outlet, all disturbances propagate through the outlet. In that case, eqn cannot be specified, i.e. a condition on the normal gradient eqn is applied.

Some flow domains combine subsonic flow at the inlet and supersonic flow at the outlet. When this occurs, a gradient condition is required for eqn at both boundaries, leaving eqn under-specified. This problem is best overcome by moving the outlet boundary sufficiently downstream for the flow to expand to subsonic speed. This allows eqn to be specified through a fixed value condition.

Notes on CFD: General Principles - 4.5 Inlets and outlets