The Edge Singularity of an Actuator Disc With a Constant Normal Load

All rotor and propeller design methods using momentum theory are based on the concept of the actuator disc, formulated by Froude. In this concept, the rotor load is represented by a uniform pressure jump. This pressure jump generates infinite pressure gradients at the edge of the disc, leading to a velocity singularity. The subject of this paper is the characterization of this velocity singularity assuming inviscid flow. The edge singularity is also the singular leading edge of the vortex sheet emanating from the edge. The singularity is determined as a simple bound vortex of order O(1), carrying an edge force F_edge = −ρ V_edge × Γ. The order of F_edge equals the order of V_edge. This order is determined by a radial momentum analysis. The classical momentum theory for actuators with a constant, normal load Δp appears to be inconsistent: the axial balance provides a value for the velocity at the actuator, with which the radial balance cannot be satisfied. The only way to obtain consistency is to allow the radial component of F_edge to enter the radial balance. The analysis does not resolve on the axial component of F_edge. A quantitative analysis by a full flow field calculation has to assess the value of F_edge for the various actuator disc flow states. Two other solutions for the edge singularity have been published. It is shown that both solutions do not comply with the governing boundary conditions.