That pesky clearance problem

I have received quite a few questions over the last year or two about wing clearance during takeoff in pterosaurs.  This seems to be a sticking point for some, as evidenced by the problem rearing it's ugly head again with the recent Chatterjee et al. GSA conference spectacular (see earlier posts below).  It would seem prudent to lay out some of the issues surrounding this problem - or, more specifically, to explain why this isn't really such a huge problem after all.

Because of the way that flying animals scale, larger, long-winged species with greater flight speeds flap with lower amplitudes than smaller species (on average, that is).  Interestingly enough, this means that the amount of clearance required by large flyers is comparatively small, so long as they can get up a good bit of speed on takeoff.  To examine this issue more closely and quantitatively for giant pterosaurs, we can look at something call the Strouhal number.

Strouhal Number is a dimensionless parameter that describes the "gait" of a flapping flyer (or really, anything that is oscillating its propulsion system in a fluid).  As it turns out, because of vortex shedding efficiency constraints, animals are remarkably constrained with regards to their Str during cruising flight: it only varies from about 0.2 to 0.4 including everything from insects to large birds.  There is a great explanation of this number, and its application to flying animals, here (I've shared that link elsewhere to good effect).

Str for a flapping flyer can be calculated as the ratio of flapping amplitude to the product of frequency and velocity.  The largest pterosaurs probably flapped at a rate just over 1 hertz in cruising flight, and likely had minimum steady state speeds near 12 m/s and a cruising speed a good bit greater, say around 20 m/s or more. 

Now, during launch, the animal probably only gets up near steady state stall speed (incidentally, it doesn't have to, contrary to what you often read in basic biology textbooks), and the Str can rise above the 0.4 mark that we might expect during cruising.  Let's let the Str rise to 0.50 and constrain the launch velocity to the min steady state stall speed above.  That still gives us an amplitude for the very tip of the wing in Quetzalcoatlus of 5.6 meters.  Of that total arc, about 40% of it is upstroke, so that leaves a required glenoid height at the end of the launch phase of 3.4 meters or so.  Given that Quetzalcoatlus had a glenoid height of about 2.5 meters while standing, it turns out that very little leaping is required at all for sufficient clearance (less than 1 meter).  The animal still needs to jump, but nothing extraordinary is required.