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D'Arcy Thompson's Scaling Analysis of Flight

One of the problems D'Arcy Thompson's applied scaling analysis to was the minimum velocity required for flight of creatures of different scales. Here are the variables D'Arcy Thompson used to set up the analysis:

For the analysis it does not matter whether the creature is moving at a velocity V through still air or the creature is still, as in a wind tunnel, and the air flows past it at a velocity V. For the analysis below it is the latter picture which is being considered.

It helps to introduce into Thompson analysis the angle θ by which the horizontal air stream is deflected downward. The change in the momentum in the vertical direction of the air stream is then Msin(θ). This has to balance the weight of the creature. Thus

Msin(θ) = w

But the weight of the creature is proportional to the cube of the scale so

w = kl3

Momentum is simply given by

M = mV

It is the mass flow m that required some thought. The mass flow will depend upon the air velocity V and upon some cross section area of the wings. This cross section area will depend upon the square of the creature's scale; i.e., l². Thus

m = ql²V

Therefore

M = ql²V²
and
M = kl³
so
ql²V² = kl³
and hence
V² = (k/q)l.

Therefor the minimum velocity V required to keep the creature airborn is proportional to the square root of the scale of the creature; i.e.,

V = (k/q)½l½

This velocity V is also the speed at which a creature needs to be running to get airborn.

Thus if one bird has twice the scale of another the larger bird needs to be moving about 41 percent faster to get airborn. A wild duck is perhaps six times as long as a song bird and would have to be moving about two and a half times as fast to take flight. Small bird need on the ambient wind velocities and can just turn into the breeze to lift off.

Larger birds need a substantial amount of space for a running takeoff. An aviary for them with restricted horizontal dimensions does not need a top.

The work that a creature must do to stay aloft is then given by

W = MV = kl³(k/q)½l½
= Kl3.5

where K is determined by the other constants of proportionality.

The work per unit weight a creature has to do to get airborn is then proportional to the square root of its scale; i.e.,

W/w = (K/k)l½

Larger creatures are at an increasing disadvantage with respect to flight. An ostrich is about 25 times as large as a sparrow. It would require five times as much energy per unit mass to take flight as a sparrow. There is no way the ostrich's tissue could be five times more efficient in producing energy as the sparrow's.

The larger birds which can fly rely upon gliding to reduce the energy demands of flight. By perching in high places they can avoid the problem of a running take-off.

Thompson's equation for the minimum velocity for takeoff or maintaining flight can be derived from standard aerodynamic relations. The lift L for an airfoil is given by

L = CLρSV²

where CL is a coefficient which is a function of the shape of the airfoil, ρ is the air density, S is the area of the airfoil and V is velocity. At the miniumum velocity the lift would be equal to the weight of the creature and hence proportional to the scale cubed. The airfoil area would be equal to the scale squared. Therefore the velocity squared would be proportional to the scale and hence velocity would be proportional to the square root of scale.


Sources:
D'Arcy Wentworth Thompson, On Growth and Form.
L. Prandtl and O.G. Tietjens, Applied Hydro- and Aeromechanics, Dover Publications, New York, 1957.


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