Water Launch: Some Nuts and Bolts

I've been asked about the nuts and bolts of my pterosaur launch models at quite a few meetings.  I have a few papers that should be out later this year which include more of the computational details than prior publications (which focused on the comparative differences in maximum load potential, rather than specific performance).  In the meantime, though, here are a few of the bit and pieces that go into the water launch model (similar for the terrestrial launch model):

Flapping frequency

Based on the expectations from Pennycuick (2008) flapping frequency varies roughly as body mass to the 3/8 power, gravitational acceleration to the ½ power, span to the -23/24 power, wing area to the -1/3 power, and fluid density to the -3/8 power:

f = m^3/8g^1/2b^-23/24S^-1/3p -3/8

The result is the expected flapping frequency in hertz; taking the inverse gives the expected flapping time in seconds.  This provides a framework for estimating wing motion speed.  The wings would have begun launch folded, in the water.  As a result, the model begins using the density of saltwater and the folded span and area.  I then transition the model through a time-step series in which most of the animal is raised above the water (air drag), but the wing finger pivot and feet still experience subaqueous drag. 

Minimum Launch Speed

There are a couple of possibilities for the speed at the end of the launch cycle.  The first model forces the launch to provide horizontal speed equivalent to the stall speed (V_min=2*WL*(1/(1.23*C_LMax))^0.5) thereby propelling the modeled pterosaur to steady state from the launch alone.  The second approach allowed a flapping burst once airborne, and allows the pterosaur in question to accelerate to steady state within its anaerobic window. These estimates can be checked against expectations of Strouhal Number limitations (see Reconstructing the Past post from yesterday).  Burst flapping after launch should produce a relatively high Strouhal number (for the size regime of the animal in question), but is still constrained.  As a rough rule of thumb, a Str up to 200% of the optimal cruising value is pretty realistic (higher freq, higher amplitude, lower speed). Launch acceleration is calculated as the simple average over the course of the launch cycle.  The launch time can then be varied from the starting values to obtain a range of possible launch accelerations, and therefore a range of potential power requirements. 

Power estimates are used with ballistic motion model to determine the initial acceleration, velocity, and height during launch.  My model species for water launch has been Anhanguera santanae. The contact areas were estimated by mapping muscles onto a laser surface scan of AMNH 22555, which is a particularly nice uncrushed specimen.

Anhanguera input parameters

Flight muscle fraction: 20-26%

Hindlimb muscle fraction: 10-12%

Wingspan: 4.01m

Anaerobic Power: 300-400 W/kg

Taking the above input parameters, taking a conservative estimate of muscle power from living archosaurs, using the motion speed from part 1, and then adding in the reconstructed contact value and an estimate of flat plate drag coefficient for propulsive force efficiency (not actually that difficult as flat plate coefficients are measured for a wide variety of shapes) yields an estimate of potential acceleration.  

In my next post, I will briefly discuss what sort of results this yields for Anhanguera.