Self-Conveoting Flows 



> 



0.5 



RADIUS, R/R, 



Fig, 5, The vertical velocity vs. radius of a vortex- 

 pair moving in a homogeneous fluid 



Most significant, we found in our experiments and from the 

 data of Richards [ 1965] that the measured variation of vertical 

 velocity and pair radius (two- dimensions) conformed more closely 

 to the law W ~ R' or z ~ t'''^ than to the law derived in the past by 

 others and which is based on complete similarity and momentum 



D-2 



1^ 



conservation; i. e. , W ~ R or z~t'. A test of the simple con- 

 servation of momentum, W ~ R" , using a typical trajectory is 

 illustrated in Fig. 5 and, similarly, in Fig. 6 it is shown that the 

 trajectories, so far as they have been observed experimentally, 

 conform more closely to the asymptotic law derived earlier for the 

 case of strong circulation, utilizing a small value of A, 

 (0 < A, < 0.2). 



In the case of strong circulation, the radius grows in a linear 

 fashion asymptotically, but the theory predicts that during the initial 

 phases of the motion the quantity dR/dz is less than its asymptotic 

 value. A similar behavior was observed in our experiments, see 

 Fig. 4. The matching up of these observed trajectories with the 

 theory offers an opportunity to determine some of the constants of 

 the theory. For this purpose we assume to begin with that A, = 0, 

 since the comparison between observed and theoretical trajectories 



337 



