Studies on the Motion of Viscous Flows — III 



sufficiently large. For R < 40 we found that the steady- state 

 flow pattern never visibly changed after introduction of the 

 perturbation. 



The above observation shows two things. One is that the evolution in time 

 of a viscous steady- state configuration takes place from an initial nonviscous 

 laminar flow configuration, i.e., a potential flow configuration. Consequently, 

 the viscous steady- state configuration may naturally be conceived of as a devia- 

 tion from a basic potential flow configuration. This is, indeed, the view basic 

 in the present work. Second, is that the steady- state motion is preserved so 

 long as symmetry is preserved, and it is destroyed only by the artificial intro- 

 duction of small perturbations. This is exactly what the Symmetry Theorem 

 asserts. Grove, Shair, Petersen, and Acrivos in their experimental work [47] 

 observe the following: "By artificially stabilizing the steady wake, this system 

 was studied up to Reynolds numbers R considerably larger than any previously 

 attained, thus providing a much clearer insight into the asymptotic character of 

 such flows at high Reynolds numbers." The first part of this statement is fac- 

 tual, whereas the second part is an interpretation of the significance of the first 

 part. From the point of view of the present work, the significance of the factual 

 part lies in the fact that it demonstrates the validity of the Symmetry Theorem. 

 The wake was stabilized, i.e., made steady by them, by the introduction of a 

 splitter plate along the line of symmetry in the wake. This device, in essence, 

 forces a symmetry, with the result that from all possible configurations — 

 symmetric and asymmetric — only a symmetric configuration emerges. This 

 symmetric configuration is a steady- state configuration. Thus, with forced 

 symmetry, a steady- state emerges — a result consistent with the Symmetry 

 Theorem. That Allen and Southwell [48] could calculate through relaxation 

 methods flow fields at R = 100 and R = 1000 which display steady- state con- 

 figuration, is to be considered a consequence of the Symmetry Theorem, be- 

 cause they started with equations and conditions which do not involve time as a 

 variable. However, for reasons which Kawaguti [49] has already pointed out, 

 their streamline fields and the pressure distributions over the surface of the 

 cylinder are suspect to some sort of error. Kawaguti and Apelt both find in 

 their numerical solutions that steady- state solutions are possible for somewhat 

 higher Reynolds number, even though they may not exist in nature. This is 

 again consistent with the Symmetry Theorem. 



Southwell and Squire [50] have used the potential solutions instead of a 

 uniform-flow solution to obtain governing equations for the flow past a plate 

 and a circular cylinder. They also point out that other authors, e.g., Zeilon, 

 Burgers, Boussinesq, Russel, and King have worked along similar lines. This 

 approach leads to their equation (no. 16) and conditions (no. 10) which naturally 

 correspond to our base flow and first iteration equations and conditions taken 

 together. However, their approach in obtaining the equations is technical in 

 spirit and does not recognize or assert the fundamental role which we have as- 

 signed to the potential flow as a base flow that is valid in the whole domain, 

 including, of course, the points near the wall and in the wake, for all Reynolds 

 numbers. To show this is the case, we quote the following from their work. 

 "Now we know from experiment that the undisturbed velocities u, v are approxi- 

 mately irrotational in parts of the field which are not very near to the solid 



581 



