Environmental Engineering Reference
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(Mgh) by 10-25 W/kg of M. Level flight requires at
least that much power, but because muscles do not sus-
tain more than 20% efficiency, the metabolic cost of flap-
ping flight should be most commonly 50-120 W/kg.
The hovering flight of hummingbirds averages 98 W/
kg, and it can reach 133 W/kg before aerodynamic fail-
ure (Chai and Dudley 1995).
For animals between 1 g and 1000 g, flying is always
cheaper than running. COT in insects is more than 20
J/g km, in largest birds just a few J/g km. The aerody-
namic power needed for flight is the sum of induced,
profile, and parasite drag components. Induced power
generates lift and declines with speed (it is highest while
hovering). The standard understanding of bird and insect
flight had to be revised with the discovery of leading-
edge vortices: whereas the flow around the arm-wings
remains attached (according to conventional aerody-
namic principle), hand-wings can induce airflow separa-
tion resulting in leading-edge vortices and generating lift
(Videler, Stamhuis, and Povel 2004). Profile power (fric-
tion drag on the wings) and parasite power (drag on the
body) go up with speed (Norberg 1990).
Two dimensionless parameters inform about the per-
formance of fliers and swimmers. The Strouhal number
(St) divides the product of stroke frequency (f ) and am-
plitude (A) by forward speed (v),
ciency. Reynolds (Re) numbers express the relation be-
tween viscous and inertial forces,
Re ¼ rlv
m
;
where r is the density of the fluid (air or water), l is a
characteristic body length (of wing or fin), and m is the
fluid's viscosity. Low Reynolds numbers are produced
by small, slow fliers: for insects (with speeds 1-10 m/s)
they are mostly 10 2 -10 3 , and for birds (with speeds
10-30 m/s) 10 4 -10 6 (fig. 4.6).
Larger birds must fly faster to generate sufficient lifts:
the expected exponent for wing loadings is M 0 : 33 , actual
mean is M 0 : 28 . Because the power needed for flight goes
up faster than the available power (a multiple of RMR),
there must be a maximum size for flying animals. But
power scaling shows how much we still do not know
about bird flight. Metabolic power, a product of weight
ð M 1 Þ and speed ð M 1 = 6 Þ , should scale as M 1 : 17 , but
measurements of birds flying in wind tunnels show that
it scales much less steeply, as M 0 : 78 in starlings (Ward
et al. 2001) and only as M 0 : 35 in red knots (Kvist et al.
2001). The best explanation is that its conversion into
mechanical output rises with fuel load. Even small shore-
birds (red knots, weight 110-190 g) can thus make non-
stop flights of 4000 km despite the high metabolic power
input. Measurements also show that flight muscle effi-
ciency is not, as has usually been assumed, 23% but rather
17%-19% in starlings and 8%-14% in red knots.
There is also a structural limit: flight muscles make up
about 17% M (only in hummingbirds are they@30% M),
and hence birds heavier than about 13-16 kg (African
kori, great bustards) cannot generate enough lift to sup-
port their weight, particularly during low takeoff speeds.
St ¼ fA
v
;
and propulsive efficiency is high within a narrow range,
0 : 2 < St < 0 : 4 (fig. 4.6). Taylor, Nudds, and Thomas
(2003) showed that dolphins, sharks, and bony fish as
well as cruising birds, bats, and insects converge on this
narrow St range in order to achieve high power effi-
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