Gas Nuclei Trajectories and Cavitation Inception 



c* + c = ^-^ . (4) 



P V / a X 1 / 2 ' 



CT.W 



3V3 1 + 

 where C* is the critical pressure coefficient. 



BUBBLE TRAJECTORY 



The calculation of water droplet trajectories in air as they approach a body 

 has been carried out in Ref. 3, However, the trajectories of bubbles in a liquid 

 are entirely different from those of water droplets in air because (a) the bubble 

 is lighter than water, so that the pressure gradient produced by any obstacle in 

 the flow field may significantly affect its trajectory, and (b) the bubble size is 

 changing along its path as its surrounding pressure varies. By considering all 

 the forces acting upon a spherical bubble in a two-dimensional flow, the equation 

 of motion of the bubble is 



1 4 "^^b 1 



y X ^77R' V -JT = TP(*-*h) Iw-w.| CnTTR' ^ 



2 3 ""' ^ dt ' 2 



- f - |77R'3Vp+ 277pR'2(w-w^) ^ , (5) 



where 



Wj^ = iu^+ jv^ = bubble velocity vector 



v - iu' + jv' = fluid particle velocity vector 



Cj3 = drag coefficient of bubble in liquid 



P = liquid density 



R' = radius of bubble 



p = pressure surrounding the bubble 



t = time. 



In Eq. (5) the mass of the gas inside the bubble has been neglected, since it is 

 small compared to that of the added mass of the fluid. Since the spherical 

 bubble under consideration is in an accelerating field, Taylor (5) showed that 

 the pressure force acting on the bubble is 3/2 of its volume times the pressure 

 gradient. The last term in the right side of Eq. (5) was estimated to be negli- 

 gible throughout the entire path of the bubble except immediately prior to the 

 occurrence of instability and therefore was not included in subsequent trajec- 

 tory computations. 



The drag of small, spherical gas bubbles in a liquid was found by Haberman 

 (4) to coincide with those of corresponding rigid spheres. Thus, the experimen- 

 tal drag coefficient curve for a rigid sphere can be used, or 



167 



