Potential and Viscous Parts of the Thrust Deduction and Wake Fraction for an Ellipsoid of Revolution

Expressions are developed for wake fraction and thrust deduction due to the potential flow and to the boundary-layer effects for a fully-submerged prolate ellipsoid of revolution. The functional dependence of wake fraction and thrust deduction on axial-propeller clearance, body slenderness, after body geometry, and Reynolds number (scale effect) are exhibited for both potential and viscous-flow cases. Closed-form expressions are derived for the potential-flow case by representing the body by a line source-sink distribution and the propeller action by a sink disk. The boundary-layer effect is determined by Lighthill's method of equivalent sources distributed on the surface having strength proportional to the displacement thickness and its derivative. The wake is replaced by a cylinder of diameter equal to twice the displacement thickness at the stern. Although in practice the propeller is usually fully submerged in the wake of the hull, in this case the substitute cylinder has been shown by computation to be no wider than the hub diameter and thus the propeller is operating in a potential field. This consideration is fundamental to the construction of a possible mathematical model having the surface sources mentioned and an equivalent sink on the longitudinal axis whose position is determined on the basis of the velocity distribution in the wake. Computational work is carried out for a modification of the airship Akron. Four different methods, with various degrees of accuracy, are used for the evaluation of the boundary-layer growth in order to ascertain the degree of sensitivity of the thrust deduction and wake fraction to the boundary-layer development.