Studies on the Motion of Viscous Flows— I 



I — Fundamental Properties of Eddies 



■ '. ^ Kirit Yajnik .; . ; .. 



- ;- Institute of Technology .-!■ 



:/ ;-/■;; r^'.. Kan frur, India •• 



. 7r , and 



Paul Lieber 



University of California 

 ; .. : Berkeley, California ■:- ,;■ - 



.7"-.:: r - -, . ABSTRACT o . ■ : ■ :.•: ; ■ . ' -; 



A new concept of eddy is rigorously introduced here, by stating it in 

 precise mathematical definition. This definition originated in a de- 

 scription of certain general differential-geometrical features of flows 

 which are based on a rotation tensor of the third rank. It is used here 

 to investigate mathematically some fundamental and general properties 

 of eddies. Because any criterion based on the distribution of vorticity 

 alone does not distinguish eddies from other regions (rigid-body rota- 

 tion and plane Couette flow, for example), new descriptions of rotation 

 having more information than vorticity are needed. The definition pre- 

 sented here is formulated in termis of angular velocities of differential 

 elements of nnaterial curves and surfaces. A characteristic kinematic 

 property of an eddy is the positive value of a certain invariant of the 

 velocity gradient. Although vorticity cannot be zero in an eddy, exam- 

 ples show that nonzero vorticity is not sufficient and that concentration 

 of vorticity need not occur in eddies of a real fluid. Other kinematic 

 properties include connections of whirling or closed streamilines with 

 the presence of eddies in plane flows or axisymmetric flows without 

 tangential component of velocity, and with the presence of convex 

 streamilines within the eddies. The connection of eddies of Newtonian 

 fluids of constant properties with low-pressure regions is indicated by 

 the characteristic dynamic property that V^p - pV • F is greater than a 

 certain datum determined by the local three-dimensional character of 

 the flow. The values of the datum for plane flows and axisymmetric 

 flows without tangential component of velocity are zero and - ( 3/0/2)(Uj./r) 2. 

 The no- slip condition on a stationary solid boundary is shown to lead to 

 large interference of even thin rods with eddies, and experinnental evi- 

 dence given here and elsewhere supports this conclusion. A highly ef- 

 fective miethod of vortex control can be devised from the conclusion. 



INTRODUCTION 



Eddies have been extensively investigated on account of their role in fluid 

 motion. The earliest model of an eddy was a potential vortex and its study led 

 to many conclusions about the dynamical behavior of rectilinear vortices and 

 vortex rings. The stability of the vortex street and its associated drag received 

 considerable attention following the successes of the pioneering work of von 



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