Yajnik and Lieber 



Hence the characteristic feature of the distribution of pressure and body 

 force is such that (v^p- pv • F) is greater than a certain datum which is deter- 

 mined by the local three-dimensional character of the flow, i.e., - 3p (2N^)^ '^. 

 The datum value can be readily calculated in certain flows such as plane flows 

 and axisymmetric flows without tangential velocity, because elements of certain 

 planes are known to have zero angular velocity. In any event, a fluid particle is 

 in an eddy if V^p - pv • f is positive. When gravitational force is the only body 

 force, this sufficient condition reduces to positive v-^p . Regions where mini- 

 mum pressure occurs will particularly have eddies. This is fully supported by 

 observation. 



Eddies in plane flows have interesting properties. Because a surface ele- 

 ment parallel to the plane of motion has zero angular velocity, the characteris- 

 tic kinematic property of an eddy is positive swirl or the absence of a line ele- 

 ment in the plane of motion having zero angular velocity. The component of 

 vorticity normal to the plane cannot be zero in an eddy, and the circulations 

 along any two closed curves in the plane within an eddy have the same sign, 

 this property can be used to distinguish a clockwise eddy from a counterclock- 

 wise eddy. Also, Theorem I implies that a point of zero velocity in an eddy is a 

 whirlpoint. The results in the previous section of this paper further imply that 

 the curvature cannot vanish in an eddy. The center of curvature of a streamline 

 in an eddy remains on one side, as the curvature is continuous and finite if the 

 velocity is differentiable and nonzero. Convex streamlines are thus to be ex- 

 pected in eddies. Indeed, the computations of Yih in 1959 and 1960 (21,22) and 

 Michalke in 1964 (9) show such streamlines. 



If velocity vanishes at two points in the plane of motion, 3v/3x vanishes at 

 an intermediate point according to the mean-value theorem, the x axis being 

 parallel to the line joining the points. The zero angular velocity of a line ele- 

 ment at the intermediate point parallel to the line ensures zero or negative 

 swirl. Hence if the two-dimensional cross section of the eddy is convex, veloc- 

 ity cannot be zero at two points in the cross section. 



The intimate relation between the closed streamlines and plane eddies of an 

 incompressible fluid is brought out by Theorem II. If a particle in an eddy has 

 zero velocity, it is a vortex point. Conversely, a vortex point is in an eddy if 

 the derivatives 3u/bx , 3u/By , Bv/bx, and 3v/3y do not vanish simultaneously. 



The dynamic characteristic property of a plane eddy of a Newtonian fluid of 

 constant properties is readily obtained from Eq. (38) and from the observation 

 that N is zero everywhere. Thus a fluid particle in such flow is in an eddy if, 

 and only if, v2p_ pV • F is positive. When gravity is the only body force, super - 

 harmonic character of pressure is then the essential feature. If pressure is 

 minimum at a point within a region in the plane of motion, then there is an eddy 

 in the region, because v^p is positive somewhere in the region. The optical 

 method of identifying eddies by locating low-pressure spots in waterflows is 

 justified by the above argument. 



Eddies in axisymmetric flows without a tangential velocity component have 

 analogous properties, although the different form of the continuity equation 



454 



