The Fully developed flow boundary condition can be understood from the following figure:
The flow to the domain Ω is assumed to enter through a straight channel of length
L. The channel is a virtual extrusion of the inlet cross section and a pressure constant pressure
Pinl is applied on the inlet of the virtual channel.
The exact value of L is somewhat arbitrary as long as it is not too high or too low.
L is therefore set to ten times the inlet edge length in 2D and to ten times the square root of the inlet area in 2D axisymmetry and in 3D.
The fact that the velocity profile is not prescribed, but rather the solution of a projected weak contribution, means that the actual velocity profile that is obtained on the inlet can deviate from the analytical fully developed flow profile (in cases such an analytical solution exists) if required by the solution inside the computational domain, Ω. This is most notably if the flow is strongly curved just downstream of the inlet. This can, for example, be the case if an object is positioned just downstream of the inlet or if the adjacent boundaries are not orthogonal to the inlet. The analytical solution can in these cases be recovered by including a little bit of the inlet channel in the computational domain.
The inlet pressure, Pinl, must be solved for and the its equation is a discrete algebraic equation (DAE) for
Pinl. When, for example, the average velocity is specified, the DAE reads
where <⋅> denotes the average over the inlet. Since the equation for
Pinl is a DAE (the equation for
Pinl does not contain
Pinl), it must be solved coupled to Navier–Stokes and it must be treated by a Vanka presmoother and postsmoother if iterative solvers are used.