Gas Nuclei Trajectories and Cavitation Inception 



the X and y direction at any point in the flow field can be expressed by 



1 X 



u = 1 + 



1 



x" + y 



3x 



3C, 



X - y 



(x2 + y2)' 



_P _ 3 



y 2x + — 



(9a) 

 (9b) 



(9c) 

 (9d) 



(x^ + y2)2 • 



From Eq. (7), the bubble motion in the x and y direction are expressed by 



(10) 



dx 



dr 



18 ^b ^D 



R2 Rj 24 



1.1 



X' + y 



18 



dy _ 



dr r2 R 



24 



^1- 



W + y 



where 



dx 

 = u. 



dr b 



dy 



dr 



Rb = RRf 



1.1 



(11) 



(12) 

 (13) 



(14) 



For static equilibrium of gas bubbles, the trajectories can be determined by 

 solving Eqs. (10), (11), (12), (13), and (14) together with Eqs. (3a), (3b), and (3c) 

 for a given initial condition of the bubble and the flow field; that is, Rq, x^, y,, , 

 ^'vf ^j ^> P> '^ > y- Numerical methods of solving these equations were used. 



Example of Bubble Trajectories 



Theoretically, the trajectory of a bubble should be calculated from infinity 

 ahead of the body. Since far upstream of the body, the pressure gradient pro- 

 duced by the body has negligible effect on the trajectory of the bubble, the bubble 

 will just follow the streamline, and the streamline itself is parallel to the flow 



169 



