Tulin and Shwartz 



captured mass. The theory takes into consideration variations in 

 volume, circulation, nnomentum, and energy in the flow field. The 

 ratio of Internal to external velocity scales, il"* is Introduced as an 

 important variable. The virtual momentum coefficient Is shown to 

 be linear In •I', of the form K| - K24'. 



The theory Is specifically derived for the two limiting cases 

 of weak and strong circulation. In the former case, ij; — ^ 1 and 

 the entralnment Is weak; the asymptotic behavior of the trajectory 

 Is z ~ t'''^ just as predicted by the usual theory based on complete 

 similarity and momentum conservation. 



In the case of strong circulation, i|j » 1 , the asymptotic 

 behavior of the trajectory depends very much on the way in which 

 vortlclty from the shear layer Is Ingested Into the vortex pair. In 

 the case where the shear layer from opposite sides Is Ingested In 

 such a way as to cause annihilation of the ingested vortlclty, then 

 the asymptotic trajectory Is z ~ t^ . Under the same conditions, 

 the velocity ratio, v]j, Increases toward the asymptotic value 

 K /K2 so that the virtual momentum coefficient tends to zero. As 

 a result, the asymptotic motion assuming vortlclty annihilation 

 corresponds to a motion with complete similarity and with energy 

 conservation. The ratio of growth of the pair radius with height Is 

 shown to Increase, approaching a linear relation asymptotically. 



Systematic experiments have been carried out, and the results 

 for rise versus time and radius versus height are compared with the 

 theory. They lend strong support to the strong circulation theory and 

 further suggest that Ingested vortlclty Is to a large degree annihilated. 



Based on these findings for the case of homogeneous flows , a 

 simplified theory is derived for the rising motion of vortex pairs In 

 stratified media. The assumptions of the theory are: (I) the motion 

 Is determined by conservation of volume, mass, and energy (neg- 

 lecting vortlclty and momentum); (II) complete similarity (dR/dz = P, 

 a constant). General laws of motion In stratified media have been 

 derived and solutions given; particularly Interesting cases are dis- 

 cussed In detail. 



Motions In stratified media were shown to depend on four 

 non-dlmemsional parameters. Two of these depend upon the Initial 

 conditions of the motion and the stratification of the media. The 

 other two are Inherent In the details of the motion and had to be 

 determined from experiments; one of these, the dissipation param- 

 eter D = Cp/pK was found to be 0.2 while the other, the ratio of 

 virtual kinetic and potential energy coefficients K/k was found to be 

 4. On the basis of these numbers it may be concluded that the dissi- 

 pation rate Is small and that the contribution of Internal motions to 

 the overall kinetic energy Is large. 



The experiments confirmed the environmental scaling param- 



356 



