Data from Se-Won Kim and Paul Yoon, with Jon Eales: 2006
Chang (2005)(1) measured the terminal velocities of a range of glass and steel balls falling in a tube of water. He showed that ,at terminal velocity, the balls fall in turbulent flow, and that to a first approximation the terminal velocity is proportional to the square root of the radius. Chang's data was obtained by analyzing 25 fps video by hand. The largest steel ball he used as 22 mm in diameter and the tube had a diameter of 15 cm. Chang's data was not sufficiently accurate to show a wall effect.
Chang's measurements were repeated by Kim and Yoon using a range of seven steel balls with diameters from 10 to 40 mm in a vertical tube with an internal diameter of 29 cm. Kim and Yoon were able to find the terminal velocities with the the direct video analysis function in Logger Pro 3.4.
Kim and Yoon's data plot in Logger Pro is shown in figure 2 and listed in table 1 below.
Table 1
Terminal velocity is plotted against the ball radius in Graph 1. The data is a good fit to a power law but the power is 0.37; not the expected 0.50 and the value of A is 10.9 not 20.4 as expected (see below).
Terminal velocity in open water
The terminal velocity for a falling ball in open water (in the turbulent flow regime) is expected to be given by....
Using Se-Won's value for the density of the second largest ball ( 8.20 g/cc) and the drag coefficient (cd) for a smooth sphere (0.47) gives A = 21.4.
Following Gibson and Jacobs the expected terminal velocity for balls falling in the laminar flow regime can be represented by the measured terminal velocity multiplied by a correction factor that accounts for the finite width of the tube. The suggested expression, omitting terms in (r/R)2 and higher, reads....
... where r is the radius of the falling ball, R is the internal radius of the tube, and B is a scale factor to be determined. The measured terminal velocity in the tube is then given by ...
Putting A equal to 21.4 and the radius of the tube equal to 0.145 m gives....
This expression has been fitted to the data points in Graph 2. The lower curve is a good fit to the data points. but the value of B is ~1.5, not the value of 2.4 given by Uselman at.... http://www.cord.edu/ ... , relying on the work of Gibson and Jacobs in 1920. The wall and end corrections Gibson and Jacobs developed apply to conditions of laminar flow, for which Stokes' law holds. The balls in this work were falling in turbulent flow in what is known as the Newtonian regime. The wall effect coefficient (determined for conditions of laminar flow) does not apply in this case.
Suggestions for further work
Since the modified correction factor gives a good fit to the data points it is suggested that more work be carried out with sets of balls of different densities, over a greater range of tube and ball radii, to establish whether or not this correction coefficient can be used to predict the wall effect for balls falling in the turbulent flow regime. The open water terminal velocity relationship could be tested, to verify the value of the constant A as 21.4.
Comments:
Se-Won and Paul made careful measurements and repeated the process ten times. The measured velocities varied by as much as ±15% from one trial to the next, but the averages were an excellent fit to a power law curve when plotted against the ball radii. In a case like this, when that much data is available, the standard deviations are the most appropriate errors to plot on the graph. Some uncertainty was probably introduced because the trials were repeated after a three second interval and some turbulence was present in the water before each ball was dropped. Some additional variations were noted because the density of the balls varied by ±3%. Some of the balls were less smooth than others, which may have affected the drag coefficient. Tests are required to quantify these effects. A set of steel balls with the same density and surface roughness has been ordered, along with a range of tube sizes.
Graphs 1-3 in this note have been plotted with just the one data set, shown in figure 1. Kim and Yoons original graphs are included. Nothing new is added by using the full data set. The original paper is available on request and will be published with additions in the ISB website when finished and reviewed.
Thank you to Jon Eales for drawing attention to the wall effect equations, and to his students for careful well presented measurements.
Reference
(1)Chang, Daniel Terminal velocity for balls falling in water 2005.