me to supplement his fine presentation with a brief discussion of certain currently 

 important cavity flow problems. 



Flows involving lifting surfaces of one configuration or another are now 

 especially of interest. For two-dimensional foils with sharp edges (which fix the cavity 

 separation), adequate, although approximate, theory for the prediction of lift, drag, 

 pitching moment, and cavity shape exist. In practice, however, supercavitating or 

 ventilating foils will often be operating with time varying angles of attack, as wings 

 of finite aspect ratio, and often in the proximity of free surfaces; implied is the possible 

 use of supercavitating foils for propellers, for alighting gear on water based aircraft, 

 and for sustention of high speed hydrofoil boats. 



In an excellent earlier talk, Mr. John Parkinson of the NACA has given us 

 some idea of trends in water based aircraft alighting gear development. We can sur- 

 mize from his remarks that designers of water based aircraft in addition to hydrofoil 

 boat designers would like to know how the loads on a ventilated (or supercavitating) 

 foil fluctuate during operations in a seaway, and whether, even in smooth water, these 

 foils might flutter; they would also like to be able to predict effects of finite span and 

 free surface proximity. Ability to design high speed foils which won't suffer large 

 changes in lift due to free surface effects is obviously essential. 



Water tunnel research people, working to provide needed experimental informa- 

 tion, would like to know how the presence of solid walls or a free jet influence the 

 flow over a supercavitating foil, particularly for low cavitation numbers when the cavity 

 becomes very large. The usefulness of their information depends upon adequate 

 knowledge of interference effects. 



The current problems indicated above together with those mentioned by Dr. 

 Maccoll, Mr. Parkinson and Mr. Nicolaides in their earlier talks and discussions are 

 few in number compared to those which designers will have to face in the future as 

 speeds of ships, aircraft, and underwater ordnance vehicles increase. The demands of 

 technology in the field of cavity flows deserve to be met and influenced by pertinent 

 mathematical treatments. 



/. Imai 



I would like to point out one thing about the determination of the free-streamline 

 separation point on the body. Professor Gilbarg told us that there are three possibilities 

 for the free streamline curvature at the separation point, that is, negative (inward) 

 curvature, positive (outward) infinite curvature, and finite curvature. He ruled out the 

 negative curvature from the geometrical viewpoint and the positive infinite curva- 

 ture by an intuitive argument to avoid "infinity", concluding that the finite curvature is 

 the only possibility. 



However, I think that the same conclusion could be drawn most naturally by 

 considering the separation phenomenon on the basis of the boundary layer theory. If 

 we assume that the streamline leaves the body at a certain point with infinite curvature, 

 it can be proved by the inviscid potential flow theory that the pressure gradient there 

 should be infinite. This in turn means on the boundary layer theory, that the separa- 

 tion should have occurred earlier somewhere upstream of that assumed separation 

 point. This is a contradiction. Therefore, on the basis of the boundary layer theory 

 we can determine the separation point as such a point that the free streamline leaves 

 the body with finite curvature. It can also be shown by the inviscid flow theory that 

 the condition of finite streamline curvature necessarily leads to the conclusion that the 

 finite curvature of the free streamline should be equal to the curvature of the body 

 at the separation point. 



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