In the present section we propose to go into various aspects of axi-symmetric 

 problems in greater detail. 



a. Exact Theory. The calculation of the conditions of flow in axi-symmetric 

 problems is much more difficult than the corresponding problem in two dimensions. 

 This is largely because in the axi-symmetric field we have no technique corresponding 

 to the theory of conformal representation of a function of a complex variable. Attempts 

 have been made to overcome this difficulty, with some success in specific problems. 



The Rankine method of using sources and sinks in a uniform stream was a 

 first attempt to solve this problem. Although this model gives a superficial represen- 

 tation of the flow, and may be adequate for many engineering problems for certain 

 contours, the source-sink technique often fails to give the finer detail, which is desirable 

 to obtain an accurate representation of conditions near flow singularities, such as at 

 pointed noses and at separation. 



Several developments of the last decade have had the object of improving on 

 the Rankine approach. Some of these researches have made use of mathematical 

 singularities of a higher order: thus the simple sources and sinks of Rankine, which 

 were distributed arbitrarily along the axis of symmetry, are replaced by vortex sheets 

 situated along the surface of the body and along the free streamline surfaces. This 

 type of model was proposed by J. W. Fisher in a war time report of 1944 and was 

 later developed in mathematical terms by Vandrey (1951) when working at the A.R.L. 

 in England. If, for example, this method is used to investigate the Riabouchinsky model 

 for the cavitating flow around a head contour (Fig. 6), then the method involves the 



M 



Figure 6. Riabouchinsky model for axi-symmetric cavitating flow. 



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