I’d like to extend special thanks to Graham Taylor of the University of Oxford Zoology Department, one of the authors of the Nature article on Strouhal numbers1 that inspired both of my articles, who writes in with a slightly revised figure:
As a dedicated Grail fan myself, I’ve been planning to do something along the lines of your Python-inspired calculations myself for my more lighthearted research talks. I was surprised by the low Strouhal number you found, as I ran some calculations based on the Lund paper2 myself a while ago and came up with a figure close to 0.2.
I re-ran things using the same method and selection criteria we applied in the paper — that is, pick on the single set of measurements cited as being from the most efficient, or lowest power speed. Assuming that this corresponds to the flight speed giving the minimum wingbeat frequency, which is reasonable if there is no other way given of estimating this, we have U as 8.89 m/s for Swallow 1 and 8.86 m/s for Swallow 2, with frequencies of 6.95Hz and 7.07Hz, respectively. The corresponding amplitudes (theta) are approximately 95 and 90 degrees, respectively. Using the formula we give in our paper1 A=b*sin(theta/2) yields estimates of A=0.23m for both swallows, based on spans (b) of 0.318 and 0.328m, respectively. This leads to Strouhal numbers of 0.18 and 0.19, respectively, for the two swallows at the reported minimum power speed, which would fit nicely in the figure in our paper (especially as birds are usually on the low side of the St=0.2-0.4 range anyway).
These findings correspond with an average speed for the two European Swallows in the Lund wind tunnel experiment of 8.8 meters per second, or 20 miles per hour, which is 4 miles per hour slower than my original estimate.