Setf-Conveating Flows 



t Time 



t Non-dimensional time, t = Wot/zg 



tj^g^ Time at which the maximum height of rise is reached 



u,v,w Velocity components, see Fig. 14 



W Vertical velocity of rising mass 



W Non-dim.ensional vertical velocity of rising mass, "W = W/Wq 



W Vertical velocity of ideal vortex-pair 



z Height of rising mass center above its virtual origin 



z Non-dimensional height of rising mass center, "z = z/z- 



z^g^ Maximum height reached by rising mass 



P Modified entrainment coefficient 



Y Local vorticity, Eq. (7) 



r Total circulation about a single vortex 



1,11,^ Coordinate system, see Fig. 14 



p Local density inside convected mass 



p Density of surrounding fluid 



p. Average density of convected nnass, Eq. (43) 



Ap Density difference, Ap = pj - pe 



i|j Velocity ratio, Wj /Wg 



2 



|j. Non-dimensional vertical momentum, (M2/p)/Wj R 



Subscripts 



( )q Initial conditions 



( )j Internal 



( )g External 



I. INTRODUCTION 



Ideal Vortex-Pairs. Flow visualization studies carried out 

 by Scorer [1957] , Woodward [ 1959] and Richards [ 1965] , Indicate 

 that the shear layer which is formed between a moving isolated mass 

 of fluid and the stationary surrounding medium tends to roll up and 

 create a flow field which resembles (in two dimensions) the one 

 associated with two line vortices of equal strength but opposite sign, 

 separated by a distance 2b, so-called "vortex-pairs. " The possi- 

 bility of vortex-pair motions in an inviscid fluid was considered and 

 analyzed over 100 years ago by Sir W. Thomson [ 1867] . His analysis 



3 23 



