Can a Bumblebee Really Not Fly, According to the Maths?
― What failed was the calculation that treated the wing like an aeroplane's
"A round body and tiny wings. By the calculations of science, the bumblebee shouldn't be able to fly, yet it does." Have you heard this? In short, it's a myth. The bee isn't wrong. The old calculation was, because it treated a flapping wing as a "motionless aeroplane wing".
On a clear autumn day, a round bee covered in black and yellow fur arrives at the cosmos flowers in a flower bed. With a low buzzing hum, it drifts heavily from bloom to bloom.
Look closely and the wings are tiny for the size of the body, and so thin they are almost see-through. You start to wonder whether they can really lift that body.
"Scientists did the calculation and found it couldn't fly." This story has been told again and again in magazines and talks. So where did the calculation go wrong?
The calculation missed just two things
The force a wing exerts on the air depends on how fast the wing moves through it. Even if the bee's body moves slowly, its wings beat well over a hundred times a second, cutting through the air far faster than the body does.
When a wing is swung at a steep angle, a small vortex forms along its front edge. This vortex stays on top of the wing and produces more force than a motionless-wing theory predicts.
Both are parts that drop out when you apply aeroplane-wing thinking as it stands. Let's look at them in turn.
The force lifting the wing grows with the square of the speed
When you move a board through air, the air pushes back on it. Think of waving a paper fan slowly, then briskly. The faster you wave, the heavier it feels.
This force becomes four times as large when the speed doubles, because it is proportional to the square of the speed. So if you underestimate even a little how fast the wing meets the air, the calculated force drops a lot.
A bumblebee's wings are said to beat about 150 to 200 times a second. The middle of the wing moves several centimetres with each beat. Put together, the wing cuts through the air at roughly 5 metres per second. This holds whether the bee is flying slowly or hovering in mid-air.
Look at Figure 1. When you change the wing speed with the slider, the length of the force bars changes. At around 1 metre per second, neither bar comes close to the dotted body-weight line. Raise the speed to 5 metres per second and only the flapping-wing bar passes the dotted line.
A vortex "stays put" on top of the wing
An aeroplane wing produces the most force when it meets the air at a shallow angle. Tilt it too steeply and the air peels away from the surface, so the force drops suddenly. This is called a stall.
Yet insect wings are swung at around 40 degrees, a steep angle that would stall an aeroplane. The force still doesn't drop, because a vortex born at the front edge of the wing stays stuck to its upper surface for a while.
The air pressure is low at the centre of a vortex. With that low-pressure patch sitting on top of the wing, the wing is sucked upwards. The right side of Figure 2 shows this. On a motionless wing, as on the left, the vortex peels away and drifts backwards, and the force doesn't last.
Flapping is a back-and-forth motion, so the wing stops and changes direction on every stroke. The next swing starts before the vortex has fully peeled away, so the process repeats without a break.
A French book on insect flight, published in 1934, is said to contain a passage saying that a calculation done with an engineer concluded the insect "shouldn't be able to fly". It was a rough estimate that treated the wing like an aeroplane's. It is thought to have grown into the tale that "even science can't explain it" and then spread.
The front-edge vortex was clearly confirmed only in the 1990s, which is surprisingly recent. Researchers watched a large moth fly through smoke and measured forces on a big robotic wing moved through oil. Only then was "why the force is enough" explained with numbers.
Summary
"A bumblebee can't fly, by the maths" was never a mystery about the bee. It was a mistake in the assumptions of the calculation. The speed at which the wing really cuts the air was underestimated, and the fact that a vortex produces force even at a steep angle was left out. With the right assumptions, the bee's wings can support its weight just fine.
It was the calculation that couldn't fly, not the bee.
Tiny wings lift the body with speed and a vortex.
Why an aeroplane's wing lifts is explained in "Why do aeroplanes fly?". How a seed spins as it falls, using the same front-edge vortex, is covered in "Why do maple seeds spin as they fall?". To see how air and water feel to small creatures, try "Why does water feel as sticky as honey to a water flea?" too.
- Wave a paper fan slowly, once a second, and note how it feels. Then wave it twice a second over the same distance. The resistance should feel not twice as heavy, but much heavier.
- Hold a plastic sheet in your hand and push it quickly through the air, once at a nearly flat angle and once at a steep angle of about 40 degrees. The steep angle gives a bigger resistance for a short time.
- On a sunny day, look for a bumblebee in a flower bed and watch the moment it hovers to choose a flower. Even though the body is still, the wings look like a blur, showing they keep moving fast.
Bumblebees are gentle, but they are still bees. Please don't touch them or go near their nests. Watch from a little distance.
For those who want to know more ― terms, equations and links to textbooksWe state clearly what level each topic sits at, from middle-school science to university courses
- Middle schoolCovered in middle-school science
- High schoolCovered in high-school physics
- High school+Advanced high-school material, or treated as a textbook sidebar
- UniversityUniversity-level courses not taught in high school (fluid dynamics, biophysics)
- ResearchNot yet taught as settled fact even at university; being studied by researchers right now
Middle schoolTerms: this phenomenon has names
- Lift: the force an object moving through a flow feels at right angles to its direction of motion. It is the force that lifts a wing.
- Stall: when a wing's angle becomes too steep, the air separates from the surface and lift drops suddenly.
- Leading-edge vortex: a vortex born at the front edge of the wing that stays on its upper surface. It is the source of the large lift of flapping insects.
Middle schoolHigh schoolChecking with an equation: how "good at making force" must the wing be?
Using the lift equation, we find how efficiently the wing must produce force to support the bee's weight. The number that expresses this efficiency is called the lift coefficient, with the symbol CL. The values are rough figures for a worker bumblebee.
| In symbols | L ＝ ½ × ρ × v² × S × CL (to float, L ＝ m × g) |
| In words | Lift ＝ ½ × air density × wing speed squared × wing area × lift coefficient |
| Where it comes from | It comes from the change in momentum of the air that the moving wing pushes aside each second. ½ × ρ × v² is called dynamic pressure, and it is proportional to the square of the speed. |
| Symbol | Meaning and unit |
|---|---|
| L | Lift (N, newtons) |
| ρ | Air density (kg/m³) |
| v | Speed at which the wing cuts the air (m/s) |
| S | Wing area (m²) |
| CL | Lift coefficient (no unit; set by wing shape, angle and flow) |
| Body weight (typical worker) | about 0.2 g ＝ 0.0002 kg |
| Gravitational acceleration | 9.8 m/s² |
| Air density | 1.2 kg/m³ |
| Total area of the four wings (rough) | about 1 cm² ＝ 0.0001 m² |
| Speed at which mid-wing cuts the air (rough) | about 5 m/s |
| Weight (force to be supported) | 0.0002 × 9.8 ＝ 0.00196 N |
| Speed squared | 5 × 5 ＝ 25 |
| ½ × air density | 0.5 × 1.2 ＝ 0.6 |
| Dynamic pressure | 0.6 × 25 ＝ 15 Pa |
| Dynamic pressure × wing area | 15 × 0.0001 ＝ 0.0015 N |
| Required lift coefficient | 0.00196 ÷ 0.0015 ≒ 1.3 |
The required lift coefficient is about 1.3. Treat the tiny wing as a motionless wing and the usable lift coefficient is only around 0.8, which is not enough. This is one part of what lies behind the "can't fly" calculation. A flapping wing with a leading-edge vortex, on the other hand, has been shown in model experiments to reach nearly 2, which is plenty.
| Dynamic pressure at 1 m/s | 0.6 × 1 ＝ 0.6 Pa |
| Dynamic pressure × wing area | 0.6 × 0.0001 ＝ 0.00006 N |
| Required lift coefficient | 0.00196 ÷ 0.00006 ≒ 33 |
A lift coefficient of 33 is impossible for any wing. Underestimating the speed by a factor of 5 inflates the required value 25 times. Whether you put in the wing speed correctly decides whether the answer is "flies with room to spare" or "absolutely cannot fly".
High schoolHigh school+Speed squared, and the pressure of a vortex
High schoolThe force a fluid exerts on an object is proportional to the square of the speed because both the amount of air hitting it each second and the speed of that air are proportional to the speed. It is a product of quantity and speed, so it comes out squared.
High school+The drop in pressure at the centre of a vortex can be explained by Bernoulli's principle. The faster the air swirls, the lower the pressure. A leading-edge vortex makes a low-pressure region on the wing's upper surface and sucks the wing upwards.
UniversityUnsteady aerodynamics and the stability of the leading-edge vortex
The field that deals with wings moving back and forth, where the flow changes with time, is called unsteady aerodynamics. In insect flight, several effects are thought to combine: delayed stall from the leading-edge vortex, wing rotation at the moment of stroke reversal, and recapture of the vortex left by the previous stroke. At insect scale, the Reynolds number, which describes the character of the flow, is a few hundred to a few thousand. That is far smaller than the several million for an aeroplane, so the effect of stickiness cannot be ignored.
📖 Derivations and further reading: Lift (Wikipedia, in Japanese) ／ Reynolds number (Wikipedia, in Japanese)
ResearchWhat is still not clear
- Why can the vortex stay put without peeling away? One explanation is that a flow running from the wing base towards the tip carries the vortex's strength outward, but the fine details are still debated.
- The role of flexible wings. Real wings twist and bend with every beat. How this affects force production and energy saving is still being studied.
- Flying in wind and rain. Outdoors, bees meet gusts and raindrops. How a bee recovers its posture then is under active research.
So this article, too, is "an explanation of what is known today". Research on tiny flying machines modelled on insect wings is also advancing, and this sometimes deepens our understanding of how insects fly in return.
Links to textbooks (by level)
| Level | Subject and unit | Where in this article |
|---|---|---|
| Middle school | Science, "How forces act" | The wing pushes the air and is pushed back |
| High school | Physics, "Momentum and impulse" | Why force is proportional to speed squared |
| High school+ | Physics (advanced), Bernoulli's principle | Pressure falling at the centre of a vortex |
| University | Fluid dynamics, biophysics | Unsteady aerodynamics, Reynolds number |
| Research | Studies of biological flight | Stability of the leading-edge vortex, effect of flexible wings |
| ― | Link to daily life | If told "the calculation says it's impossible", doubt the assumptions behind the calculation |
- Ellington, C. P., van den Berg, C., Willmott, A. P. (1996) Leading-edge vortices in insect flight. Nature 384
- Dickinson, M. H., Lehmann, F.-O., Sane, S. P. (1999) Wing rotation and the aerodynamic basis of insect flight. Science 284
- Magnan, A. (1934) Le vol des insectes. Hermann.
- Lift - Wikipedia (in Japanese; 揚力 - ウィキペディア)
※This article is a general-audience science explanation. The figures given are rough guides to help you understand the mechanism. A bee's weight, wing size and beat rate vary greatly with species and individual.