On the Edge Singularity of the Actuator Disk Model

Abstract

AbstractIn this technical brief, the classic actuator disk theory is revisited with a view to shed some light on the singularity of the flow at the edge of the disk where the vortex tube starts and where vorticity is generated. The study is carried out using small perturbation assumption in two-dimensions and simplified boundary conditions in all cases. The problem of the two-dimensional thin cambered plate with constant vorticity distribution is solved and the leading edge singularity is analyzed as it is believed to be relevant to the axisymmetric flow at the actuator disk edge. Next, the velocity components induced by the cylindrical vortex tube of constant vorticity are obtained via the Biot–Savart law and the near edge behavior is investigated. It is shown that the velocity components behavior is consistent with that of the thin cambered plate with constant loading, thus reinforcing the notion that the axisymmetric slip-line behaves as r − R ∝ −xlnx near the disk edge.