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I've noticed a motionless kingfisher over a lake looking for prey and wondered what amount of energy does a bird, weighing 0.15kg, require to hover for 15s?

user9590
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    In the physics 101 sense it requires no work (i.e. energy expended) to hover. Think about it (and see Why does holding something up cost energy while no work is being done?). So what you've got here is a question about biomechanics. – dmckee --- ex-moderator kitten Jun 02 '12 at 20:19
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    Is this so? Certainly not in a more general sense. Say it wasn't a bird but a helicopter. Does a biomechanics analysis allow me to determine the amount of gasoline consumed. – user9590 Jun 02 '12 at 22:56
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    Understand that the amount of physics 101 "work" done holding the bird or helicopter or whatever in place is the same if it is hovering on wingpower or hovering rotor power or sitting on a pillar. And that is zero. This is not a fault in your understanding of how the world works, but a difference in the day to day meaning of "work" and the one where $W = \int \vec{F} \cdot \mathrm{d}\vec{x}$. Ron's answer shows the way out of this dilemma: analyze the forces on the thing doing the supporting. – dmckee --- ex-moderator kitten Jun 02 '12 at 23:05

2 Answers2

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If the mass of the bird is $M$, and it is modelled as a fan which is pushing air to velocity $v$ downward constantly and continuously, then in any unit of time $dt$ it must push an amount of air down on average to get $M\,g\,dt$ up-momentum. This means that the mass $dm$ of the air it pushes down to velocity $v$ in time $dt$ is such that its momentum is $dm\,v = M\,g\,dt$, so the amount of air pushed down per unit time is

$$ dm = {M\,g\over v} dt $$

The energy this air gets, assuming the air starts at rest is

$$ dm {v^2\over 2}$$

So the power consumption is

$$ {dE\over dt} = {M\,g\over v} {v^2\over 2} = {M\,g\,v\over 2} $$

This assumes that all the air accelerated by the bird dissipates its energy, so that the energy is lost forever. This is not accurate, and the above is a simple estimate. For a bird of mass 0.1 kg, gravitational acceleration g=10 m/s2, v=1 m/s (assuming the wing is 10 cm from top to bottom of the stroke and flaps 20 times a second), the power required is 1 watt.

The parameter $v$ is determined from the wing-speed, and the total mass of air you push per wing-flap is the area of the wing times the density of air times the period of a wing-flap. The gives a relation between the size of the bird and the wing-flap frequency. This is order of magnitude only, and it is more valid the more turbulent the air-flow is.

endolith
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Ron Maimon
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  • Thanks somuch, esp. for the wing-flap displacement explanation. Just to clarify my (remedial) understanding of the units, did you mean the energy required is 1 W or 1 Watt-second? Would this mean that a hovering time of 15s would require 15W-s of energy? I think of the Watt as a unit of power, not energy. Again, thanks. – user9590 Jun 03 '12 at 00:26
  • @user9590: Yes, stupid mistake, I'm sorry--- the power is 1W, and the energy in each second is 1 W-s. – Ron Maimon Jun 03 '12 at 05:35
  • So this is why human-powered helicopters have enormous surface area: Using a large surface area to move the same mass rate of air at a lower velocity requires lower power levels for the same payload weight. – endolith Aug 07 '14 at 03:16
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I agree with dmckee's comment above. I would just like to add that hovering motionless birds most probably use ascending air streams (unless when they descend slowly). Pretty much the same happens with gliders: they use ascending streams to go up or they descend slowly.

akhmeteli
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  • What do you mean by "ascending"? Do you mean "defleted down"? If so, it is a confusing way to say it. The bird pushes air down, and it's always pushing new air down, so it does a lot of work. – Ron Maimon Jun 02 '12 at 22:22
  • Many raptors hover by using the lift from a headwind instead of or as well as soaring in a glider like mode. – dmckee --- ex-moderator kitten Jun 02 '12 at 22:44
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    @Ron Maimon: I am afraid I am not a native English speaker, so I may have chosen a wrong word, but by "ascending" I meant "going upwards" (I looked up your word "defleted" and could not find it). And I don't quite agree that birds always do work to stay in air - if air is going up, they can just soar in it, without doing any work - a glider never does any work, but it can go up with the air stream. – akhmeteli Jun 02 '12 at 23:24
  • @dmckee: I don't quite understand that - birds would lose speed if they fly against horizontal wind, so such a flight would not be sustainable without birds doing work. It seems possible, however, to use a tail wind to soar. – akhmeteli Jun 02 '12 at 23:33
  • You're assuming that their wings produce lift only in the upward direction, but that is not necessary. I've seen kestrels, falcons and (on days with strong sustained winds) hawks hovering over one spot without flapping their wings. They do it by adjusting their wing position and shape such the the lift vector both holds them up and counters the drag from the headwind. Nor does soaring require a tailwind: most soaring birds go in circles. – dmckee --- ex-moderator kitten Jun 02 '12 at 23:39
  • @dmckee: I suspect that lift is only in the upward direction by definition :-) However, it is my understanding that a bird cannot move against horizontal wind without doing any work and without losing speed or altitude. – akhmeteli Jun 02 '12 at 23:45
  • Come out to the plains on a windy day and you can watch it. I've done so many times. Or ask google for some youtube help. – dmckee --- ex-moderator kitten Jun 02 '12 at 23:57
  • @dmcee: I respectfully disagree. Laws of physics rule supreme. And in the youtube clip the birds do waive their wings, thus doing some work. – akhmeteli Jun 03 '12 at 00:53
  • It helps if you can stand there in the same wind. The bird has to flap when the wind speed sags, and it has to make constant adjustments to the position and shape of its wings as the wind buffets it. But you can see the brids hovering still for seconds at a time in both those clips. They are shaping their wings to get a forward component to the lift, and the energy comes from the wind itself. They can do that because the laws of physics reign. – dmckee --- ex-moderator kitten Jun 03 '12 at 01:18
  • @dmckee: With all due respect, I cannot accept those clips as a proof: first, the birds there definitely wave their wings; second, we cannot be sure that, when they don't, they don't lose speed or altitude and that there is no upward wind component. If the wind is horizontal and the bird flies horizontally against the wind with a constant speed and maintains a constant wing profile (does not wave wings), the only forces are: in the horizontal direction - drag; in the vertical direction - lift and the force of gravity, so the bird will lose speed with respect to wind. – akhmeteli Jun 03 '12 at 06:37