Store Is Closed
On Flight Dynamics: Influence on Living Systems
May 31 2011 1:33:00 pm EST
Regarding flight, D’Arcy Wentworth Thompson expresses the unique requirements to remain aloft, and the effect these requirements have on the structure of an organism. From On Growth and Form:
The bird’s case is of peculiar interest. In running, walking or swimming, we consider the speed which an animal can attain, and the increase of speed which increasing size permits of. But in flight there is a certain necessary speed—a speed (relative to the air) which the bird must attain in order to maintain itself aloft, and which must increase as its size increases. It is highly probable, as Lanchester remarks, that Lilienthal met his untimely death (in August 1896) not so much from any intrinsic fault in the design or construction of his machine, but simply because his engine fell somewhat short of the power required to give the speed necessary for its stability (41).
…a bird, in order to counteract gravity, must cause air to move downward and obtains an upward reaction thereby. But the air displaced downward beneath the wing accounts for a small and varying part, perhaps a third perhaps a good deal less, of the whole force derived; and the rest is generated above the wing, in a less simple way. For, as the air streams past the slightly sloping wing, as smoothly as the stream-lined form and polished surface permit, it swirls round the front or “leading” edge*, and then streams swiftly over the upper surface of the wing; while it passes comparatively slowly, checked by the opposing slope of the wing, across the lower side. And this is as much as to say that it tends to be compressed below and rarefied above; in other words, that a partial vacuum is formed above the wing and follows it wherever it goes, so long as the stream-lining of the wing and its angle of incidence are suitable, and so long as the bird travels fast enough through the air (43).
Speaking generally, the necessary or minimal speed of an aeroplane varies as the square root of its linear dimensions; if (ceteris paribus) we make it four times as long, it must, in order to remain aloft, fly twice as fast as before. If a given machine weighing, say, 500 lb. be stable at 40 miles an hour, then a geometrically similar one which weighs, say, a couple of tons has its speed determined as follows:
(W, the work done—all in unit time; w, the weight, and V, the velocity of the bird; I, a linear dimension, the form of the bird being supposed constant.)
W : w :: L³ : l³ :: 8 : l.
Therefore L : l :: 2 : l.
But V² : v² :: L : l.
Therefore V : v :: √2 : l = 1∙414 : 1.
That is to say, the larger machine must be capable of a speed of 40 X 1∙414, or about 56½, miles per hour (44-45).
The wing of a bird or insect, like the tail of a fish or the blade of an oar, gives rise at each impulsion to a swirl or vortex, which tends (so to speak) to cling to it and travel along with it; and the resistance which wing or oar encounter comes much more from these vortices than from the viscosity of the fluid. We learn as a corollary to this, that vortices form only at the edge of oar or wing—it is only the length and not the breadth of these which matters .A long narrow oar outpaces a broad one, and the efficiency of the long, narrow wing of albatross, swift or hawkmoth is so far accounted for. From the length of the wing we can calculate approximately its rate of swing, and more conjecturally the dimensions of each vortex, and finally the resistance or lifting power of the stroke; and the result shews once again the advantages of the small-scale mechanism, and the disadvantage under which the larger machine or larger creature lies.
A bird may exert a force at each stroke of its wing equal to one-half, let us say for safety one-quarter, of its own weight, more or less; but a bee or a fly does twice or thrice the equivalent of its own weight, at a low estimate. If stork, gull, or pigeon can thus carry only one-fifth, one-third, one-quarter of their weight by the beating of their wings, it follows that all the rest must be borne by sailing-flight between the wing-beats. But an insect’s wings lift it easily and with something to spare; hence sailing-flight, and with it the whole principle of necessary speed, does not concern the lesser insects, nor the smallest birds, at all; for a humming-bird can “stand still” in the air, like a hover-fly, and dart backwards as well as forwards, if it pleases (46-47).