Amount of energy required to hover.

Question

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?

Answer 1

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.

Answer 2

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.