flow velocity. By assuming that the same similarity law holds for cavitating flows 

 (for which the cavity is approximately ellipsoidal) it is possible to calculate the flow 

 about cavitating ellipsoidal head shapes. Fig. 9 shows the similarity relationship for 

 this case. 



If Q; 



1 + 8 = 



2-6 



2-4 



2-2 



20 



1-8 



1-6 



1-2 



1-0 



0-2 



04 0-6 



A /B OR B /A 



0-8 



1-0 



Figure 9. Similarity relationship between plane and axi-symmetric flow for ellipsoids. 



To obtain the plane flow past curved surfaces it is, however, first necessary to 

 determine the separation position. This type of problem was solved by Brodetsky 

 (1922) and Schmieden (1929) using the Levi-Civita method. The separation positions 

 obtained are in poor agreement with experiment, and it seems possible that the boundary 

 layer plays an important part. The Brodetsky solution for a circular cylinder gives, 

 for instance, separation at 55 degs. whereas experimental observations give values 10-20 

 degs. greater. A possible explanation of this is afforded by Fig. 10, which shows the 

 calculated distribution of wall shear stress in laminar flow using the Brodetsky velocity 

 distribution: the wall shear stress is high at the calculated separation position, and 

 may inhibit cavitation inception, causing the separation point to move downstream. 



There are also some analytical doubts as to the validity of the Brodetsky and 

 Schmieden solutions near the separation point since the calculated position turns out 

 to be a singular point of the head profile for derivatives of the curvature. Complete 

 removal of the infinity is only possible if the free streamline coincides with the body. 



Such considerations have to be borne in mind when any attempt is made to 

 obtain solutions of any steady cavity problem by means of high speed digital com- 

 puters. The application of relaxation or similar techniques is liable to conceal the 



224 



