considerably smaller percentagewise than the error in the computed height above the 

 separation point. The principal source of error in finite difference calculations is 

 probably the infinite curvature of the free streamline at separation, for small errors in 

 shape here are reflected in relatively large errors in the downstream position of the 

 free streamline — without necessarily incurring large velocity fluctuations downstream. 

 Good accuracy at detachment requires careful refinement of the mesh or special han- 

 dling of the flow at separation. The importance of truncation error has still to be 

 clarified. 



Although several approximate methods have been applied to cavity flows with 

 axial symmetry [13], calculations based on exact theory are quite rare. Recently, using 

 the scheme of dimensional interpolation outlined above, Garabedian obtained the value 

 C D z= .827 for the drag coefficient of the circular disk in infinite cavity flow. The 

 extrapolated experimental data favors a value between .80 and .81, but its spread 

 includes .827 at the upper end. In his treatment of the Riabouchinsky cavity, for which 

 the preceding method is unsuitable, Garabedian [14] has developed an iteration pro- 

 cedure based on linearization of the original boundary value problem. Namely, he 

 shows that on any curve r which is displaced normally by an amount Sn from the 

 free streamline, the correct stream function satisfies the condition 



1 ty K 



+ -* 



y dn y 



1 



(20) 



within an error of the order (8n) 2 , where k is the curvature of r. An approximate 

 solution of the flow problem is thus provided by a stream function ty satisfying (20) 

 on the starting approximation V, the next approximation to the free streamline — and 

 the starting point of the next cycle — is the curve on which xp r= 0. In the calculation 

 of flow past a disk, Garabedian finds C D (a)/(l + a) = .865 when CT = .2235. This 

 is a few percent higher than the experimental value, but confirms the observed increase 

 in slope of the C D (o) curve. His figure 2.30 for the ratio of cavity to disk diameter 

 at this value of a is in close agreement with observation. 



Comparison methods 



A fruitful source of qualitative results has been the comparison method that 

 was initiated by Lavrentieff and later extended by Gilbarg and Serrin [15]. This method 

 features simple geometric arguments in the physical plane, which are often equally 



Figure 6. 



289 



