From the asymptotic properties of elliptic integrals as k' — > one obtains 

 (6) and (7) again, and also 



d 4 /2 + a t 



+ - ) (14) 



I 4 + 7T \ a 4 



h 



I 4 + 7T 



2 + cr\ 2 /2 + a 

 -Klog4( 



(15) 



These values are accurate within .7 percent at a = 1 and improve rapidly for smaller <r. 

 (6) is a special case of the formula 



C D (<r)~C D (0)(l + v), (16) 



which seems to be generally valid for both plane and axially symmetric cavities having 

 fixed separation points [4], and is in good agreement with observation. The error term 

 in (16) grows more rapidly with a in the axially symmetric case. Experimental results 

 indicate that the formula C D { a ) <-> C D (0)(l + a <j) should replace (16) in flows past 

 bodies having variable points of separation (such as the cylinder and sphere). 



Formulas analogous to (14) and (15) for the axially symmetric Riabouchinsky 

 flow past a body of diameter / are 



d 



I 



VC D (a) h IC D (a) 1 



and -~-<d- -log-. (17) 



o" I * a' 2 a 



From these it is apparent that the plane cavity has the larger length-width ratio for 

 fixed cavitation number, whereas the axially symmetric cavity is the flatter of the two 

 (as measured by d/l). The derivation of (17) is based on a convincing but heuristic 

 comparison between cavities and flows past ellipsoids [5]. 



In the limiting case a = 0, when the cavity is infinite, its asymptotic shape 

 is given by 



y ~ Cx* in plane flow; 



Cxi 



y ~ in axially symmetric flow [6]. (18) 



(log a;)* 



The drag is proportional to C~ in the former case and to C 4 in the latter. 



The theory of cavitational flow past smooth obstacles — that is, obstacles without 

 corners or sharp edges — still has important gaps. The problem here is that the separa- 

 tion points of the free streamlines are not known a priori and must be determined as 

 part of the solution; at the same time, it is not entirely clear what conditions are needed 

 to determine a unique and physically acceptable solution. It is well known that a free 

 streamline at detachment has either infinite curvature or the same curvature as the 

 body from which it separates. The latter type of detachment will be called smooth. 

 A flow will be considered physically acceptable if the free streamlines do not intersect 

 the body or themselves and if the maximum speed occurs on the free boundary (as 

 required by cavitation) . Since a free streamline in cavitating flow is convex, its detach- 

 ment (from a smooth body) must evidently be smooth if the flow is to be physically 

 acceptable, but this condition is certainly not sufficient, as one sees by counterexample. 



It is known [7] that within the class of bodies for which physically acceptable 

 infinite cavity flows always exist are those having non-decreasing curvature on both 

 sides of the forward stagnation point (Fig. 3). However, it is a simple matter to 

 exhibit other convex bodies for which physically acceptable solutions of this type do not 

 exist. In this connection the flow past an asymmetric wedge is instructive. (The 

 wedge can be looked upon as an approximation to a convex obstacle with large curva- 

 ture at the vertex.) The various possibilities when a — are the following: i. The 



285 



