Tulin and Shwartz 



In view of these facts, and for other reasons, it seems 

 desirable to attempt a more general theory of the motion of turbulent 

 vortex pairs, based on the assumption of separate velocity scaling 

 of the internal and external motions; i.e. allowing W/W to vary 

 continuously. Afterwards, a simplified theory pertaining to motions 

 in stratified media will be developed and the results compared to 

 experimental observations. 



II. THEORY (HOMOGENEOUS FLOWS) 



Separate Similarity of Internal and External Flows . We 

 visualize the vortex-pair motion to be divided into internal and ex- 

 ternal flow fields, separated by a thin region of high shear, which 

 also forms the boundary of the captured mass, see Fig. 3, We 

 assume that each flow field Is Itself self- similar. 



Internal. 



External. 



w.{x,y,t) = W.(t) . w. (|;X) (2) 



Wg(x,y,t) = Wg(t) . w,(| ;^) (3) 



and similarly for the other velocity components. 



We let ijj = Wj/Wg, where 4i Is , In general, not constant In 

 time as It Is In the case of complete similarity. 



We choose Wj as the clrcumferental velocity averaged over 

 the Inner boundary of one-half of the vortex pair and Wg the same 

 except averaged over the outer boundary, (The inner and outer 

 boundaries are separated by a thin shear layer,) 



Volume Changes. The volume of fluid comprising the vortex 

 pair Increases continuously with time due to entralnment Into It, 

 Because of the similarity assumed, the rate of entralnment of 

 volume must (In two dimensions) be proportional to a characteristic 

 velocity and a characteristic length. We take for the former, the 

 velocity difference Wj - Wg 



2 

 ii^^ = 2TrR(Wj - W.) • a(4;) (4) 



or. 



330 



