Prescribing Inlet and Outlet Conditions
The Navier-Stokes equations can show large variations in mathematical behavior, ranging from almost completely elliptic to almost completely hyperbolic. This has implications when it comes to prescribing admissible boundary conditions. There is also a discrepancy between mathematically valid boundary conditions and practically useful boundary conditions. See Inlet and Outlet for the node settings.
Inlet Conditions
An inlet requires specification of the velocity components. The most robust way to do this is to prescribe a velocity field using a Velocity condition.
A common alternative to prescribing the complete velocity field is to prescribe a pressure and all but one velocity component. The pressure cannot be specified pointwise because this is mathematically over-constraining. Instead the pressure can be specified via a stress condition:
(13-13)
where ∂un/∂n is the normal derivative of the normal velocity component. Equation 13-13 is prescribed by the Pressure condition in the Inlet and Outlet features and the Normal stress condition in the Open Boundary and Boundary Stress features. Equation 13-13 is mathematically more stringent compared to specifying the pressure pointwise and at the same time cannot guarantee that p obtains the desired value. In practice, p is close to Fn, except for low Reynolds number flows where viscous effects are the only effects that balance the pressure. In addition to Equation 13-13, all but one velocity component must be specified. For low Reynolds numbers, this can be specified by a vanishing tangential stress condition:
which is what the Normal stress condition does. Vanishing tangential stress becomes a less well-posed inlet condition as the Reynolds number increases. The Pressure condition in the Inlet feature therefore requires a flow direction to be prescribed, which provides a well-posed condition independent of Reynolds number.
Outlet Conditions
The most common approach is to prescribe a pressure via a normal stress condition on the outlet. This is often accompanied by a vanishing tangential stress condition:
where ∂ut/∂n is the normal derivative of the tangential velocity field. It is also possible to prescribe ut to be zero. The latter option should be used with care since it can have a significant effect on the upstream solution.
The elliptic character of the Navier-Stokes equations mathematically permit specifying a complete velocity field at an outlet. This can, however, be difficult to apply in practice. The reason being that it is hard to prescribe the outlet velocity so that it is consistent with the interior solution at each point. The adjustment to the specified velocity then occurs across an outlet boundary layer. The thickness of this boundary layer depends on the Reynolds number; the higher the Reynolds number, the thinner the boundary layer.
Normal Stress Boundary Condition