Desai and Lieber 



[26], and other authors are significant. Of particular importance is the work of 

 Taneda [82]. When investigation is carried out for large values of Re, then the 

 literature based on the boundary layer theory becomes significant. However, 

 this latter literature cannot provide any understanding of the wake. The litera- 

 ture based on free- streamline theories which display infinite wake and on 

 theories which lead to finite wakes or no wakes at all, show attempts to give 

 the solutions to the Euler equations a central place. Boundary layer theory 

 definitely gives an important place to the potential solution. In the present 

 scheme we have attempted to unify the picture by asserting the fundamental 

 role of the potential flow as a base flow for actual flows under all conditions. 



CONCLUDING REMARKS 



The conclusions based on the general aspects of the theory and investiga- 

 tion of the steady flow in the range < Re < 20 are as follows: 



1. The potential flow solution does play a fundamental role inasmuch as it 

 has lead us to results which are in good agreement with the experimental re- 

 sults for Re < 4, and which shed light on the evolution of the vortex structure. 

 Moreover, a theoretical value of the Reynolds number at which separation be- 

 gins is obtained, in agreement with experiments. 



2. The results support the existence of a linear substructure underlying 

 the Navier- Stokes equations in the present case. 



3. The time-independent subsidiary equations and their solutions for the 

 first and second iterations show by induction that the coefficients of the sub- 

 sidiary equations for all iterations will be analytic with infinite radius of con- 

 vergence, leading to corresponding solutions which also have infinite radius of 

 convergence (Refs. 1 and 2). 



4. The analytical solutions for the first two iterations contain implicitly 

 all the information about the structure of vortices and the wake insofar as they 

 give rise to the streamline field around the circular cylinder. 



5. Nonsymmetric wakes and the evolution of vortices which are distin- 

 guished by closed streamlines are the result of the same process of superposi- 

 tion of the harmonics of the streamline field, and consequently one might view 



a vortex structure to be inherent in the flow field even at the smallest Reynolds 

 numbers, although explicitly identifiable closed streamline structures may not be 

 then manifest in the field. 



6. The discrepancy in the drag at higher Reynolds numbers is attributable 

 to the fact that in these cases the boundary at which the flow is assumed to be- 

 come potential is situated close to the cylinder to avoid numerical breakdown of 

 the computations. And to maintain this situation there, a highly rotational flow 

 is required to be constrained within a smaller domain than is physically 

 admissible. 



588 



