Self-Conveoting Flows 



II = ^ (i|; - 1) . a{^\>) = (4; - i)a\^) (5) 



Note that the dependence of the proportionality constant a' on the 

 velocity ratio i|j has been left unspecified. 



Circulation Changes. The circulation T about one half of 

 the vortex pair is, on account of the inner similarity, proportional 

 to the length scale and inner velocity scale. On account of the way 

 in which Wj was defined, 



r « RWj (6) 



The fluid entrained into the vortex pair from the surrounding 

 high shear layer carries vorticity of opposite sign to that already 

 within the interior, see Fig. 3 and thus reduces it to the extent it is 

 not annihilated through mixing with vorticity being entrained in the 

 opposite lobe. The strength of entrained vorticity must, on account 

 of similarity, be proportional to the ratio of a pertinent velocity and 

 length scale. In particular: 



^(entrained) « <^e - ^i^ (7) 



The flux of entrained vorticity takes the form: 



Vorticity Flux ** Volume Flux • Y(entrained) 



« [ 2uR( Wj - W,) . a(^)] [ ^e^ ^' ] (8) 



Finally, the change in circulation may be related to the flux of 

 vorticity: 



^= - A, . 2Tra{^\>) • (Wj - W,)^ (9) 



dt 



or. 



ii^^^ = - A,a' . (Wj - Wg) • (4^ - i) (10) 



The value of the parameter A, will depend in part upon the 

 extent to which entrained vorticity from each side mixes together 



331 



