Desai and Lieber 



Navier- Stokes equations, no ideas of parameter or coordinate perturbations are 

 invoked. Instead, a fundamental role for potential flows is, a priori, asserted 

 and the iterative process is used as a method of obtaining solutions to the Navier- 

 Stokes equations. In this, we differ fundamentally from the existing theories. 



It has been our desire to understand the flow around a circular cylinder as 

 it evolves, because the understanding gained would also be gained for flows around 

 other obstacles as well. The existing theories do not shed light on the continuous 

 evolution of a flow field. This means that we must abandon well-trodden paths 

 and examine the nature of this flow with a fresh outlook. Well- chosen and criti- 

 cally performed experiments with actual flows provide information on which we 

 are building a theoretical structure, an image of reality which is also a reflec- 

 tion of our understanding of it. This information has accumulated over a period 

 of years, but there has been no theoretical structure which embraces all or even 

 a large part of it. Experience with small-perturbation theories and comparison 

 between them and actual experimental results lead us to believe that the Navier- 

 Stokes equations contain implicitly essential theoretical information about flows 

 of a class of actual fluids. Computer experiments [43,44,45,46] performed by 

 using the Navier-Stokes equations vividly demonstrate that the equations do 

 have this information implicitly. Consequently, the present work considers the 

 Navier-Stokes equations as embodying the essential theoretical information im- 

 plicitly and differs from other theories insofar as it endeavors to make explicit 

 as large a part of this information as it can, without setting a priori limits to 

 what is possible. 



Because a steady-flow situation actually exists in nature and because from 

 an analytical point of view it is the most appropriate one to study first, we have 

 directed our efforts to obtain concrete results for this aspect of the flow field 

 after obtaining the general information contained in the Symmetry Theorem 

 about the conditions under which the flow can be time-dependent. Fromm and 

 Harlow in their fine work [45] on the nonsteady problem of vortex street devel- 

 opment have used numerical techniques based on a method of iterations. They 

 have observed the following: 



All examples started at time t = 0, with the walls and fluid im- 

 pulsively accelerated to this velocity, and the first cycle iteration 

 procedure immediately adjusted the configuration to the nonvis- 

 cous laminar flow solution. Advancement of the configuration 

 through subsequent time cycles resulted in a gradual transition to 

 the viscous steady- state solution whose most prominent feature is 

 an eddy pair just behind the plate. Since the solution procedure 

 preserves symmetry to approximately one part in 10 5, the steady 

 solution persists for long times, even for large values of R. Thus 

 we found it desirable to introduce a perturbation, accomplished by 

 artificially increasing the value of <^ by a small amount at three 

 mesh points just in front of the plate; this was done at a time when 

 the double eddy pattern was well established. In all cases, the 

 perturbation was small enough that no immediate change was 

 visible in the flow pattern; nevertheless, such a small perturba- 

 tion was always effective in starting the vortex shedding process 

 within a fairly short time, provided, of course, that R was 



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