4.3 Fundamentals of boundary conditions

The specification of boundary conditions is one of the most challenging tasks in setting up a CFD simulation. The range of possible boundary conditions is endless, to cover all of the potential applications and physics.

Setting boundary conditions is not an exact science but is guided by an basic specification using the fixed value (Dirichlet) and fixed gradient (Neumann) conditions introduced in Sec. 4.2 .

PICT\relax \special {t4ht=

The figure above shows the basic specification for eqn, eqn and eqn for incompressible subsonic flow, e.g. described by Eq. (2.47 ), Eq. (2.48 ) and Eq. (2.65 ).

The boundary conditions for eqn and eqn require particular attention since they are coupled. The conditions on eqn are independent of eqn and eqn and are representative of other transported scalar fields, e.g. turbulent kinetic energy eqn.

The specification for inlets and outlets summarises as:

  • zero gradient on eqn at an inlet, fixed value on other variables;
  • fixed value on eqn at an outlet, zero gradient on other variables.

Propagation of disturbances

A disturbance in a flow is simply any change from an equilibrium or steady solution. A disturbance at one location travels, or propagates, through the fluid.

The combination of boundary conditions at open boundaries, i.e. those excluding walls, relates to the propagation of disturbances. While disturbances are transported by advection with the flow, they propagate as waves at the speed of sound eqn.

PICT\relax \special {t4ht=

Sound waves can propagate disturbances against the direction of flow if it is subsonic, i.e. eqn. Disturbances must be able to propagate outwards through the inlet, which requires one variable not to be prescribed at the inlet. Similarly, they must be able propagate inwards from the outlet, which requires one variable to be prescribed at the outlet.

A pressure equation, combining mass and momentum conservation, describes wave propagation. For an incompressible fluid the wave speed eqn, since Eq. (2.48 ) contains no eqn term. A disturbance at any point influences the solution everywhere in the domain instantaneously, as discussed in Sec. 2.22 .

Pressure is then logically the variable on which to specify the boundary conditions to support wave propagation. Pressure is therefore: prescribed at the outlet, i.e. we specify fixed value condition; not prescribed at the inlet, i.e. we specify a fixed gradient condition, usually set to eqn.

Notes on CFD: General Principles - 4.3 Fundamentals of boundary conditions