Exact Treatment of the Helicoidal Wake in the Propeller Lifting-Surface Theory

Abstract

This study is fifth in a series concerned with the adaptation of lifting-surface theory to the marine propeller case. In the present investigation an accurate mathematical treatment of the helicoidal wake supersedes the deformation of the helicoidal integral path to a staircase path, introduced in the previous studies to simplify the solution. The kernel function is derived with correct directional derivatives normal to the helicoidal surface as well as with the simplified derivatives assumed in previous studies. The propeller blade is of sector form with arbitrary pitch, not necessarily low, and the loading distribution is taken to be that of a flat plate. The results of numerical calculations for the consistent mathematical model which treats the helicoidal wake exactly and uses the correct directional derivatives normal to the helicoid are found to compare favorably with the results obtained previously when a staircase approximation of the wake was used with the simplified, but physically consistent, directional derivatives in the axial direction. The closeness of the results for a propeller of not too low pitch indicates that the staircase approximation, which was introduced for low pitch angle, is a satisfactory model. This model can be used to avoid all complications arising from the rigorous treatment of the helicoidal wake, particularly when arbitrary blade form and sweep angle are incorporated in the problem.