Studies on the Motion of Viscous Flows— II 



sophisticated hydrodynamical structure that includes qualitatively essentially 

 all the observable structural features of actual flows (including zones of high 

 dissipation and eddy structure), and also the fact that this viscous flow structure 

 which strongly differs from the potential flow from which it is obtained by a 

 single iteration, is connected to it by a linear relation. This relation is an as- 

 pect of the linear substructure noted above, which I believe is an intrinsic fea- 

 ture of all actual flows. Moreover, the agreement of the quantitative results 

 obtained from the first iteration with the measurements is striking. 



The application of higher order iterations directed toward improving agree- 

 ment between quantitative results and measurements, necessarily calls for the 

 inclusion of higher harmonics that correspond to the higher order terms of a 

 Fourier series representation of the flow field. These higher harmonics as- 

 sume a fundamental role in the development of eddy structure and become in- 

 creasingly significant in our mathematical representation of the viscous flow 

 fields as the Reynolds number increases. We envisage these higher harmonics 

 as seeds of turbulence, which are highly attenuated in very low Reynolds number 

 flows, but which must be used in increasingly refining flow fields at lower Reyn- 

 olds numbers by successively applying increasingly higher order iterations. We 

 may think of these higher harmonics as the grindstones on which the iteration 

 process works and by which it progressively sharpens by higher iterations a 

 finer hydrodynamical structural detail. 



On the other hand, if higher order iterations are applied without correspond- 

 ingly introducing higher harmonics, even at lower Reynolds numbers, then we 

 cannot expect the higher iterations to improve the flow fields previously calcu- 

 lated by lower order iterations. As nonlinear effects grow in intensity, the 

 higher order iterations become increasingly necessary, as do the higher har- 

 monics to which they are applied, and these will increase in amplitude as the 

 Reynolds number increases. 



We are presently engaged in extending the application of our algorithm to 

 the calculation of actual time -dependent viscous flow fields. In so doing we have 

 come to realize that its application to the calculation of higher Reynolds number 

 flows demands that we give explicit consideration, and representation in the ac- 

 tual calculation, to the historical process of their development. 



The concepts and considerations presented here on the physical aspects of 

 evolution displayed in classical hydrodynamical systems, has motivated the con- 

 ception and design of simple hydrodynamical experiments, by which we expect 

 to identify experimentally the process of evolution in hydrodynamics as a proc- 

 ess which accords with, but is not implied by, the Navier-Stokes equations. The 

 evolutionary aspects in classical hydrodynamical systems are strongly linked 

 with the features of flows which we now try to examine from the standpoint of 

 hydrodynamical stability theory, without taking cognizance of their historical 

 development. Hydrodynamical stability theory is accordingly basically limited, 

 because in its deeper meaning the stability of a hydrodynamical system is inex- 

 tricably linked with the process of its historical evolution, and therefore cannot 

 be separated from it. At higher Reynolds number the flow fields corresponding 

 to a particular boundary condition which is maintained in space and time, depend 

 not only on the Navier-Stokes equations, but also upon the way the boundary 



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