Wake Forces Implied in the Theodorsen and Goldstein Theories of Propellers

Abstract

This study examines wake edge forces and normal forces implied, but overlooked, respectively, in the Theodorsen and Goldstein theories of the optimum efe ciency of propellers. It is argued that the postulated trailing vortex sheets must coincide with real rigid surfaces to support forces exerted by the e uid. In the rigidized Theodorsen model, edge suction forces are tilted backward and circumferentially in the region of contraction. These are implicitly included in the predictions of propeller thrust and torque. The corrections are estimated herein. The resulting change in efe ciency, a gain, is normally less than 1%, but can reach over 2% in extreme examples. In the rigidized Goldstein model, the wake does not contract, but the streamlines do. The crosse ow component, acting on the bound trailing vortices, generates lift; hence, thrust and torque components. These corrections to the Goldstein-based predictions (using Theodorsen’ s formulas ) are found to be compatible with the contraction correction of the Theodorsen theory: the Goldstein theory, when rigidized, appears equivalent to the Theodorsen theory. In the Theodorsen theory, the required wake forces are seen to be so small that their neglect (by neglect of wake rigidity ) is of conceptual signie cance only. Nomenclature A = cross-sectional area of one turn of helicoid wake at z = ` projected on Trefftz plane, 2 pR `