17 - Unsteady forces on two-dimensional sections and hydrofoils of finite span in gusts

Summary

As a preparation for determining unsteady forces on propellers in ship wakes, we first consider two-dimensional sections beset by travelling gusts. Our development of the unsteady force on such sections differs from that given in the seminal work of von Kármán & Sears (1938) by adopting a procedure which is easily extended to wings and propellers. Their formula for unsteady sectional lift is recovered, being that of lift at an effective angle of attack which varies with the parameter k = ωc/2U, the “reduced” frequency. Turning to hydrofoils of finite span, we derive results for low aspect ratio in steady flow. For wings in gusts there is no analytical inversion of the integral equation which involves a highly singular kernel function. Graphical results are given from numerical solutions for a range of aspect ratios which reveal diminishing unsteady effects with decreasing aspect ratio.

Corresponding reduced frequencies for propeller blades in terms of expanded- blade-area ratio are shown to be high relative to aerodynamic experience. This indicates that two-dimensional, unsteady section theory cannot be applied to wide-bladed (low-aspect-ratio) propellers.

TWO–DIMENSIONAL SECTIONS

The blades of a propeller orbit through the spatially non-uniform flow of the hull wake and consequently experience cyclic variations in the flow normal to their sections. For blades of small chord-to-radius, this is analogous to the case of a two-dimensional section moving at constant speed through a stationary, cyclic variation in cross flow distributed as a standing wave along the course of the moving section.