ABSTRACT 



A linearized theory is developed for steady, two-dimensional cavity flows 

 about slender symmetric bodies. The theory is applied to the cases of zero and 

 nonzero (positive) cavitation numbers. It is shown that, for the case of finite 

 cavities, the linearized theory avoids the necessity for choosing an artificial cav- 

 itation model as must be done in any exact theory attempts. The problem of calcu- 

 lating cavity shapes and drags for arbitrary slender bodies is reduced to one of 

 quadratures. As an example, calculations are made for the family of wedge profiles 

 and results are shown to be in good agreement with "exact" theory results for suf- 

 ficiently slender bodies. In particular, the example demonstrates that the linear- 

 ized theory is a valid first order theory. 



INTRODUCTION 



The problem of practical importance being considered here is that of finding the flow 

 characteristics, and in particular the cavity shape and body drag which result when a two- 

 dimensional body, s\Tnmetric with respect to the flow direction, is immersed in a uniform^ in- 

 finite, steady stream for which it is assumed that cavitation occurs for a certain sufficiently 

 low fluid pressure. The slender body is considered to be of almost arbitrary shape, it only 

 being specified, for the sake of reality, that the body not be of a shape such that the velocity 

 at any forward point on the body exceed that at the cavity separation point. It has been found 

 experimentally that the flow about a cavitating body under the above circumstances involves 

 a trailing cavity of essentially constant interior pressure whose length is dependent upon that 

 pressure and that Froude and Reynolds effects are very often of second order of importance. 

 The pertinent hydrodynamic problem which might be expected to have a physically meaningful 

 solution would be stated thus: To find a (or the) closed (in the finite plane or at infinity) 

 symmetric streamline(s) whose foreshape is given and on whose after part (called the free 

 streamline) the flow velocity or, equivalently, the fluid pressure is a given constant, the flow 

 field exterior to the symmetric closed streamline being time independent, irrotational, incom- 

 pressible, and single valued. Mathematical investigations of this problem have led to the fol- 

 lowing important result; Only in the case where the streamline is closed at infinity (called a 

 Helmholtz flow) does a solution to the above problem exist. The existing solution is unique 

 and for it the velocity on the free streamline must equal the velocity of the uniform stream at 

 infinity. Birkhoff ^ has called the nonexistence part of tiiis result Brillouin's Paradox. The 

 fact that the flow conditions at the rear of a finite cavity are not ideal as described in the 

 problem leads to a resolution of the paradox. 



References are listed on page 20. 



