Yajnik and Lieber 



Karman in 1911 and 1912 (1). Despite its successes, certain limitations of the 

 model were recognized. The physical impossibility of velocities being arbitrar- 

 ily large, and associated difficulties such as infinite angular momentum and ki- 

 netic energy led to the modification of Rankine which requires a rigidly turning 

 core with a matching potential flow outside. The new model of steady flow was 

 still not suitable for describing the diffusive and dissipative action of viscosity, 

 particularly near solid boundaries. The unsteady exact solutions of Navier- 

 Stokes equations obtained by G. I. Taylor in 1918 (2), Oseen in 1911 (3), Hamel 

 in 1916 (4), and Rouse and Hsu in 1951 describe the growth and decay of a recti- 

 linear vortex away from a solid boundary and in the absence of neighboring 

 vortices. The three-dimensional behavior of certain eddies has also been de- 

 scribed by the exact solutions of N. Rott in 1958 (5) and others. 



The picture of an eddy or vortex which has emerged from these studies is 

 that it is essentially a region of concentrated vorticity surrounded by a region 

 of negligible vorticity (Kuchmann, 1965) (6). The axisymmetric or two- 

 dimensional character of some of the exact solutions is a mathematically con- 

 venient assumption meant to render tractable the problem of integration of 

 Navier -Stokes equations. 



The conceptual difficulties created by this picture are many. The picture 

 refers to vorticity distribution, with the inference that if two flows have identi- 

 cal distribution of vorticity and one flow can be called a vortex, the other can be 

 also. But the vorticity distribution of a rigidly turning fluid is the same as the 

 plane Couette flow, so that if a rigidly rotating core of a Rankine vortex is 

 called an eddy, so should the plane Couette flow. However, it is clear that the 

 plane Couette flow does not possess the same whirling character as the rigidly 

 rotating fluid. 



Another dilemma created by the picture is that there can be no eddy in a 

 fluid undergoing creeping motion for which vorticity satisfies Laplace's equa- 

 tion, since vorticity or its magnitude cannot be a maximum at any interior point. 

 However, Moffatt in 1964 (7) obtained such flows possessing whirling regions as 

 indicated by closed streamlines. 



Even the area of vortex streets is not free from difficulties (Wille, 1960) 

 (8). The earlier model of a line vortex of ideal fluid led to a constant spacing 

 ratio, whereas it was observed that the street becomes wider downstream from 

 the bluff body. Second-order analysis always predicted instability, whereas the 

 vortices preserve their regular pattern for a considerable distance. Theories 

 based on superposition of the vortices of Oseen and Hamel lead to the conclusion 

 that the street becomes narrower downstream. 



The work of Michalke in 1964 (9) reveals that the interval between regions 

 of closed streamlines is half that between the locations of maximum vorticity in 

 the case of an unstable free -jet boundary. 



These anomalies have led the authors to question whether the whirling 

 property can in principle be described by vorticity distribution. The whirling 

 property is identified by the common operational test of closed streamlines in 

 two-dimensional flows. Hence the authors questioned whether there existed any 



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