SELF-CONVECTING FLOWS 



Marshall P. Tulin 



Hydvonautios 3 Incorporated 



Laurety Maryland 



and 



Josef Shwartz 



Hydronautias-Israeli Ltd. and 



Israel Institute of Technology 



ABSTRACT 



A theory for the motion of two-dimensional turbulent 

 vortex pairs in homogeneous media has been developed 

 based on separate velocity scaling of the internal and 

 external flow fields involved in the motion and taking 

 into account variations in volume, circulation, momen- 

 tum, and energy. Based on the results obtained from 

 this theory (I) a simiplified theory (II) is derived to deal 

 with the rising motion of turbulent vortex pairs in 

 stratLfled naedia. The theoretical results are compared 

 with systematic experimental observations. 



In theory (I) the ratio of internal to external velocity 

 scales, 4^, is introduced as an important variable and 

 the theory is specifically derived for the two limiting 

 cases of weak (i|j -*■ 1) and strong (vjj » 1) circulation. 

 The weak circulation theory leads to results similar to 

 those obtained in the past using theory based on com- 

 plete similarity and momentum conservation; i.e. , 

 z ~ t"*^. The strong circulation theory leads to results 

 which depend very much on the way in which vorticity 

 from the shear layer is ingested into the vortex pair. 

 When ingested so as to cause annihilation (cancellation) 

 of the ingested vorticity, the asymptotic trajectory is 

 z ~ t'/2. Under the same conditions the velocity ratio, 

 4i, increases toward an asymptotic value, and the 

 virtual momentunn coefficient for the motion tends 

 to zero. As a result, the asymptotic motion (assuming 

 vorticity annihilation) corresponds to a motion with 

 complete similarity and with energy conservation. 



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