Studies on the Motion of Viscous Flows— III 



thereby facilitating the numerical calculation of higher Reynolds -number flow 

 by available computer facilities. 



4. Another important result of this work concerns the asymptotic behavior 

 of the solutions obtained when the location of the surface joining the strictly po- 

 tential outside flow with the rotational flow inside, and in terms of which Desai 

 introduced his concept of physical infinity, is extended away from the cylinder. 

 This is illustrated in the graphical presentation of the results of numerical 

 calculation. 



5. A fundamental and striking result obtained in the present work concerns 

 a detailed field description of the evolution of flow fields with Reynolds number, 

 including eddies distinguished by the closure of streamlines which obtains from 

 the superposition of harmonics, in terms of which the solutions are here devel- 

 oped according to the linear substructure. This reveals the remarkable fact 

 that the superposition of two harmonics of the solution produces eddy structure 

 whose distinguishing feature is the closure of streamlines, a feature derived 

 from the linear substructure, and which was anticipated though not analytically 

 deduced from it. 



6. Flow separation from the circular cylinder predicted by the present 

 theory agrees favorably with available measurements. 



7. Though the drag calculated here for Reynolds numbers greater than 5 

 differs from measurements, this discrepancy has helped us recognize that har- 

 monics higher than the second must be included even at very low Reynolds num- 

 bers in order to apply effectively the higher order iterations needed to obtain 

 increasing accuracy in the calculation of the flow field, and in particular, of the 

 drag. This means that higher harmonics, which may be envisaged as represent- 

 ing the nuclei of turbulence which increase in strength with Reynolds number, 

 are significant aspects of viscous flows, even at very low Reynolds number. 

 Another reason for the discrepancy noted, is that for higher Reynolds number, 

 we must locate the surface of physical infinity that joins the rotational and 

 strictly potential regimes very close to the body, in order to work numerically 

 within the limitations of the available computers. We can free ourselves of this 

 restriction by applying another scale transformation. The close proximity of 

 this interface (physical infinity) relative to the body in the cases where a dis- 

 crepancy between calculated and measured drag was found, has the effect of 

 artificially restricting and thus deforming the eddy structure of the rotational 

 regime, thereby increasing the calculated drag over the actual value. 



8. The analytical representation of flow fields developed here is also found 

 to imply certain necessary relations between time -dependent motion and the 

 symmetry properties of flow fields, and correspondingly to reinforce the prin- 

 ciple of minimum dissipation by a theorem presented here. 



9. A very detailed field description of the pressure is obtained here and 

 presented for all Reynolds numbers for which actual flow solutions have been 

 calculated. 



499 



