Supercavitating Propeller Theory 



forecasting required radial characteristics to meet prescribed design condi- 

 tions, whereas phase (d) is concerned with the practical achievement of these 

 characteristics, recognizing that the blades possess finite chord length. For 

 inviscid flow calculations, the blade can be represented mathematically by a 

 lifting line for phase (c), whereas a lifting- surface representation is required 

 for phase (d). As already intimated for the supercavitating propeller phases 

 (c) and (d) should adequately recognize the self and mutual interference effects 

 of the blade cavities. 



The purpose of this paper is to derive and assemble the necessary equa- 

 tions for supercavitating propeller design theory, such that cavity interference 

 effects can be adequately accounted for at all stages of the design process. 

 The major task is to derive the necessary equations for the induced velocity 

 components, since they are directly or indirectly involved in every aspect of 

 propeller design calculations. The problem is formulated in Appendix A, and 

 the approach used is somewhat similar to that of Widnall [9] for the three- 

 dimensional supercavitating hydrofoil, but considers the more complicated case 

 of the screw propeller. Linearized equations of motion are used to define the 

 existence of a perturbation velocity potential and perturbation pressure, i.e., 

 acceleration potential, which satisfy the Laplace equation. Green's theorem 

 and linearized boundary conditions for the blades and cavities are used to de- 

 fine a mathematical model, which consists of a distribution of pressure doublets 

 over the linearized blade surfaces to represent loading, and a distribution of 

 pressure sources over the linearized blades and cavity surfaces to represent 

 the cavities. Lifting- surface equations are obtained for the induced velocity 

 components and pressure at any point relative to axes rotating with the propeller. 



Section 2 presents the lifting- surface equations for pressure and induced 

 velocity components at any point on the blade surface. The pressure doublets 

 are transposed into the more usual and convenient bound and free vorticity dis- 

 tribution. In conformity with normal practice for subcavitating propeller 

 theory, the radial component of induced velocity is ignored and a nonlinear re- 

 finement is incorporated into the pitch of the lifting surface, so as to extend 

 consideration to the case of moderate propeller loading. The solution of the 

 lifting- surface equations is discussed briefly, and it is pointed out that effec- 

 tive computation to determine axial and tangential induced velocities, as for 

 the subcavitating propeller case, requires a knowledge of lifting- surface pitch 

 and loading, i.e., induced advance ratio and bound vorticity distribution, respec- 

 tively. Finally it is hypothesized that for uniform propeller inflow, there ap- 

 pears to be little advantage in not assuming a constant -pitch lifting surface. 



Section 3 discusses the necessity of specifying a simplified mathematical 

 model, i.e., lifting line, for a propeller blade. Such a model is required for 

 two purposes, first for prediction of supercavitating propeller performance, 

 and secondly to provide necessary information for lifting- surface calculations. 

 Induced velocity and pressure equations are defined for such a model and de- 

 tailed consideration given to the case of uniform inflow and constant induced 

 advance ratio. The simplification of the definite integrals which arise in the 

 solution of the lifting- line equations is discussed in Appendix B. 



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