13 



The faired curve of Figure 5 showing drag coefficient as a function of 

 Reynolds number was compared with the data obtained by previous experimenters. 

 This comparison is shown in Figure 9 and indicates the considerable scatter in 

 the results. Still, for Re greater than 400, the points cluster fairly well 

 around the curve obtained from TMB tests. 



As pointed out before, the term A used in Equation [2] to define the 

 drag coefficient of a body is generally the projected area. This formulation 

 facilitates comparison of the drag developed by different shapes. In order to 

 compare the drag measurements for bubbles with those obtained for solid bodies 

 of roughly similar shape, the data are presented in the more usual form in 

 Figure 10. The values of the parameters shown were computed from the experi- 

 mental data according to the definitions. 



u na 2 U 2 



Re,^- [10] 



n 



in which a is the over-all transverse diameter of the flattened bubble as 

 measured from the photographs. Although the value employed for each bubble 

 is an average of measurements from several frames of the film, there is still 

 considerable scatter in the computed values of the drag coefficient deter- 

 mined in this way. No values were computed in the range of Re from 4000 to 

 10,000 where, because of the irregular and extreme fluctuations in shape ac- 

 companying the rise of the bubbles, no reliable estimate of the transverse 

 dimension could be made. Not all of the bubble data are included. Instead, 

 typical bubbles of various sizes were selected. The data for spherical-cap 

 bubbles indicate an average of 0.86 for the drag coefficient CJ . The value 

 of CJ for a solid of roughly similar shape has been determined experimen- 

 tally 17 and is O.65 at a Reynolds number of 21,000. 



A description in geometrical terms of the bubble shape in the vari- 

 ous flow regions may be of interest. Since a bubble fluctuates in shape as it 

 rises, all bubble dimensions which were obtained from the photographs repre- 

 sent average values. For the bubbles which can be approximated by oblate 

 ellipsoids, a completely descriptive shape parameter is the ratio of the major 

 to minor axes, a/b. This is shown as a function of bubble size in Figure 11 

 using data obtained for typical bubbles. As the equivalent radius increases, 

 the bubble becomes flatter until a maximum ratio of about 2.7 is reached at an 



