21 



SUMMARY AND CONCLUSIONS 



The motion of air bubbles in a liquid can best be characterized by 

 the use of three dimensionless parameters, the Reynolds number, the drag co- 

 efficient and a third parameter M which for a specific liquid is proportional 

 to the pressure gradient . 



Tests on air bubbles at their terminal velocity in water indicate 

 that for Reynolds numbers of less than 70, bubbles behave like rigid spheres. 

 At greater values of the Reynolds number, the drag coefficients of the bubbles 

 are considerably less than for rigid spheres even though the bubbles are still 

 spherical in shape. This may be due to the development of slip at the bound- 

 aries. For Reynolds numbers from 400 to 5000 the hydrodynamic and surface- 

 tension forces are both important in determining the shape and consequently 

 the drag coefficient of the bubbles. Beyond this range, hydrodynamic forces 

 almost exclusively determine the shape of the bubble, resulting in a spherical 

 cap. The test results are shown by means of the curve of Figure 5 giving the 

 drag coefficient as a function of the Reynolds number and Table 1 describing 

 the motion and shape of the bubbles. 



In the tests at the Taylor Model Basin, the slight change in volume 

 resulting from the change in pressure as the bubble rose in the field of the 

 camera resulted in no appreciable change in the terminal velocity. However, 

 the bubble fluctuated in shape as it rose. Therefore it was difficult to as- 

 sign any one set of dimensions to a particular bubble. An average was ob- 

 tained. Using these average dimensions, it was shown that spherical caps are 

 geometrically similar. Also, their drag coefficients are independent of bub- 

 ble size, having a value of about 2.6. 



The rate of rise of a spherical cap is a relatively simple function 

 of the radius of curvature of the nose, the relation determined experimentally 

 being given by: 



U = 0.645 VgR 



The effect of the parameter M on the relation between the drag co- 

 efficient and Reynolds number is uncertain since there are only very incom- 

 plete data available. The results indicate that M influences the relation be- 

 tween the other parameters in the region in which the bubbles no longer act 

 like rigid spheres and have not yet attained the shape of spherical caps. The 

 value of M affects the Reynolds-number range for which this zone exists and 

 also the minimum value of C D . As M increases, transitions in shape occur at 

 lower Reynolds numbers and the minimum value for C D increases, approaching 

 closer to rigid spheres. 



