Wing surface area for large flying creatures

Question

These creatures are a few feet longer and about a foot thicker than your average cow. Their skin is covered in densely packed bristles to trap air, and thus heat, close to the skin in cold climates. The 'wings' are large flaps of skin stretched from a foreleg to a rear leg, much like the flying squirrel.

How large would these wings have to be to produce enough lift, when flapped, to keep this creature airborne with a human riding it? Also, if there are other aspects of this animal that need to be changed/corrected, please notify me.

Answer 1

Cows vary in size (and mass) quite considerably. Given that you want to scale up a typical cow, let's use a relatively higher number for that mass of, say, 1000kg. For argument's sake, let's assume that includes the human rider. Let's also assume acceleration due to gravity and air density are similar to Earth.

The largest flying animal I know of was Teratornis merriami, (see comments for discussion on pterosaurs, also), which was about 15kg with a wingspan of 4m. Clearly, we will need a much larger wing surface area (and hence, wingspan) for a 1000kg cow + rider (especially considering the lack of optimized cow aerodynamics, requiring even more lift to compensate).

Your cows would probably be around 3m long, maybe 1m wide. So, the "flying squirrel" skin flaps clearly aren't going to cut it. Your cows are going to need real wings, and large ones at that, to even stand a chance at staying airborne, much less lifting off in the first place.

How would your cows take off, anyway? Cattle aren't known for their high speed on land, so unless your cows are sprinters, they will need yet more wing surface area to be able to take off at slower speeds. If they just, say, jump off a cliff and glide, this bit is less of a problem, but the above issues still exist.

It would be difficult to actually calculate what this surface area/wingspan would be (and what sort of mass that would add to the cow, especially given the significant muscle mass that would be required to hold the wings up, never mind flap them at a significant frequency). At least, it's beyond my limited abilities in aerodynamics.

The best I could find was this calculator, which gives a wingspan of about 9.5 meters. I don't know that this calculator had cows in mind, so I wouldn't trust that number without some more independent research.

However, it is probably a reasonable lower bound for a sanity check: since you wanted your cows to have a flight profile like a flying squirrel, are you now OK with cows that have wingspans as wide as a three-storey building is tall?

Edit to address cliff-gliding follow-up comment

The question of "what would it take to allow the cows to launch off of a cliff" (paraphrased) was raised in the comments. Let's see if we can come up with a reasonable approximation of wing surface area required to "glide" 1000kg into a safe descent.

The fundamental concept here is wing loading. The idea of gliding essentially means there will be zero lift, which allows a higher wing loading (that is, smaller wings), as you'd intuitively expect. But how big are the wings?

Since we're mostly looking for level flight, the following formula applies:

$\frac{L}A = \frac{Mg}A = \frac{1}2 v^2\rho{}C_{L}$

Where:

• $\rho = 1.2845 kg/m^3$ (rho, air density at sea level, 15C)
• $g = 9.81 m/s^2$ (acceleration due to gravity, Earth average @ sea level)
• $C_{L} = 1.5$ (lift coefficient, value of "typical" small aircraft wing)
• $M = 1000kg$ (mass of your aircow)
• $v = 100km/h$ (arbitrarily chosen "slow" airspeed gained from gliding off of a tall cliff)
• $A$ (wing area. This is the quantity we're looking for.)

We're interested in A, the wing area. So, some simple rearranging gets us:

$A = \frac{2Mg}{v^2\rho{}C_{L}}$

$A = 13.19m^2$

You can try the Wolfram|Alpha calculation for yourself if you'd like to play with the numbers.

This is consistent with my earlier (rough) estimate of a ~10m wingspan (assuming wings not much more than a meter from front to back, on average).

Note that with this "minimal" number, you would only be gliding. No lift (except perhaps a bit from the odd updraft), turning would be very slow, and it would be easy to stall and crash (ground beef, anyone?).

With a required wing area of 13.2 square meters, even if the entire underside of your cow was a big square-ish "wing" that had a lift profile as good as a small airplane, a cow's underside isn't big enough--the wing (or wings) would have to protrude from the cow somewhat.

I'm trying hard to find a way to make your cows fly, Wick. Now that you have a formula you can play with, my advice would be to consider taking some plausible artistic license with the mass of the cow (perhaps more like the size of a calf, perhaps imagine lighter bones and a slimmer body?), then perhaps consider lighter but still Earth-like gravity (say, $0.7g$) and more dense air ($\rho$). At this point, my hope is that I've given you the tools you need to tweak the parameters in a way that will best allow you to tell your story.