[gurps] Dragon Aerodynamics

[Susan Koziel]

For winged flight look into the aerodynamics of bats (no feathers to mess with your calcs)... if you want a really fantasy like critter.
Or really large birds (geese) if you want something much more realistically lizard like... since birds are much more closely realated to lizards in body form then they are to mammals.
 
All flying animals have oddnesses to the basic aerodynamics principles then what you'd expect allowing them to get away with things that seem impossible at first glance. Things like the fact humming birds and bees share a type of nerve that allows it's wing to flap 2X or more per synapse fire (therefore faster flapping and more lift then expected). 
Geese shouldn't get off the ground but have some weirdness in their feathers that helps (I don't recall specifics).
Albatross can sustain longer flights - and have longer periods then normal between flapping... I'm not sure if this is a flying wing design or something.
 
And frankly "a dragon that's 10 hexes long, half of which is tail, and is probably no more than 8' at the shoulder" would be enough to scare me spit-less. Ever been hiking and met a golden eagle on your path? The thing was sitting on a farmers fence post and decided to stretch - it's one wing was the length to the next post. It may have only stood half my size but while flying it was huge with claws and a huge beak.... and I felt very very small suddenly. It was similar to encountering a Grizzly Bear (but I never got that close to the Bear - and I wasn't about to stop and watch either).
-Sue

----- Original Message ----
From: Anthony Jackson <ajackson at iii.com>
To: The GURPSnet mailing list <gurpsnet-l at sjgames.com>
Sent: Thursday, November 8, 2007 6:22:56 PM
Subject: [gurps] Dragon Aerodynamics

Having just been reading a little about aerodynamics, I decided to amuse 
myself by trying to apply them to dragons (yes, this is slightly 
pointless). Since I'm applying rather basic principles to the rather 
complex subject of winged flight, this may not be a perfect translation, 
but it's good enough to amuse me, so I thought I'd share it and also see 
if anyone has helpful comments:

Applying Aerodynamics to Dragons

It is generally obvious that a dragon, if made of normal materials and 
flying by normal principles, cannot fly. The usual solution taken by 
people who want dragons anyway is to give them some exotic method of 
flight, such as magical levitation. However, just as it's obvious that a 
dragon can't fly, we also know that a giant made of normal flesh and 
bone cannot stand up, so supposing that dragons are not made of normal 
flesh and bone is hardly strange.

So, if we accept dragons which aren’t made of normal flesh, what can we 
compute about a dragon? We'll start with a dragon that's 30' long. For 
the body, we'll give it about the same build as a komodo dragon, which 
for an average specimen
might be 8' and 150 lb. Scaling up to 30' we get about 8,000 lb. Note 
that this corresponds to about 4' of head/neck, 11' of body, 15' of 
tail. Body width is assumed to peak at about 4'6" wide and 3' tall, for 
an area of 10.6 sf or about 1 square meter.

Now, for the wings, we'll go with a wingspan of 40' (12m) with an aspect 
ratio of 8, which seems like a fair approximation of typical dragon 
wings. Including some lost area in the body and variation in shape, the 
maximum width of the wing is probably 8' or so.

Now, let's start with just how strong the wings are. For simplicity, let 
us assume that wing flapping is somewhere between the level of effort of 
flapping your arms (up and down) and doing that with a 5 lb weight in 
your hand.  This corresponds to a torque of somewhere up to about 15 lb*ft.

Now, we'll set the dragon wings as a structure five times larger than a 
human arm/shoulder combination, making it somewhat larger (as compared 
to the torso, which is only about 4x larger) than a human shoulder. The 
maximum torque of a rod is proportional to thickness^3, so the natural 
maximum sustained torque is about 1,875 ft*lb. Since the dragon has a 
weight of 8,000 lb (4,000 lb per wing) with the an average separation 
(due to flapping mechanics) of about 15', the average torque is about 
60,000 lb per wing, or 32x greater. Flapping is probably actually more 
stressful than raising and lowering the arm (unless you do it very 
fast), so we might want to double it.

If we assume the same numbers for other limbs, an 7.5' komodo dragon 
probably has a bite ST of 10, which means our 30' dragon would have a 
basic bite ST of 40, multiplied by sqrt(32) for stronger materials, or 
226. Dragon arms usually seem to be shorter and smaller, compared to the 
torso size, than human arms, so we'll multiply by 0.75 for a limb ST of 170.

However, flapping is still pretty slow movement relatively speaking, so 
it might be better to assume a 32x Power, rather than 32x Force. This 
changes the scaling from X^1/2 to X^1/3, and gives us a slightly more 
reasonable figure of 130 bite ST, 95 limb ST.

The speed of an action with a limb is proportional to 
sqrt(force/weight*length). We're assigning our dragon arms 3x length, 
27x mass, and 90x the force allowed for a human arm, so the dragon can 
strike about as quickly as a human would punch -- though the velocity of 
the arm is three times greater. This is broadly compatible with the 
damage the dragon does, though it's quite a bit faster than any real animal.

Going onward to actual aerodynamics, the lift equation is:
L = Cl * S * 1/2rho * V^2, where L is lift, Cl is the coeffient of lift, 
S is wing area, rho is atmospheric density, V is velocity.  For our 
sample dragon, S is a value of 18 m^2, rho is 1.225 kg/m^3, and the 
remaining values can vary.

Ordinary wings have a maximum Cl that is typically 1.5 to 2, but to 
handle the additional lift that flapping can grant, we're going to treat 
the wings as allowing a Cl of up to 3. Setting L to 36,000N and solving 
for V^2, we discover a stall speed of 33 m/s or Move 36.

Now, we want to know drag. Looking at some handy formula:
Di = kL^2/(1/2*rho*V^2*S*pi*Ar), where Di is induced drag, k indicates 
the degree to which the wing is worse than elliptical (probably 
significant for a dragon wing, but we'll set it to 1.1), Ar is aspect 
ratio (8, as above), and other terms are as before. At stall speed, it 
is equal to
1.1*35584^2/(0.5*1.225*33^2*18*3.1416*8) or 4600N.
Ds = 0.5*rho*V^2*Cd*A, where Cd is the coefficient of drag and A is 
frontal area. Neither of these numbers is particularly easy to compute 
for a dragon, but for now we'll set Cd*A at 0.2 square meters. At stall, 
this gives us a drag of 0.5 * 1.225 * 33^2 * 0.2 or 133N.

Now, the basic power requirement for movement is equal to Drag*Velocity, 
with some additional inefficiency caused by the mechanics of propulsion 
that is fairly hard to calculate, and is probably fairly small for 
wings, as they are quite large. Thus, the basic power requirement is 
equal to 4800N*33m/s or 160 kilowatts (210 horsepower). As a dragon is 
about six times more massive than a horse, this is 35x the power to 
weight ratio, which is reasonably comparable to our strength ratio 
(derived above) and thus agrees with the idea that the strength might be 
relatively slow pull muscle.

Other than stall speed, there are two other numbers of special interest. 
These are the speed at drag (and thus work per unit distance) is 
minimized (ideal cruising speed) and the speed at which power is 
minimized (ideal loiter speed). Drag is minimized at the speed where 
induced and static drag are equal; power is minimized at a speed equal 
to the minimum drag speed / sqrt(sqrt(3)) (this will not be derived 
here). We'll also give it a sprint speed that is 25% faster than 
cruising speed. As it happens, I have a program which is solving this, 
which gives us:
Stall Speed          :  32.8 m/s, Drag  4804N, Power 1.576e+05W
Minimum Power/Dist  : 80.04 m/s, Drag  1569N, Power 1.256e+05W
Minimum Power        : 60.81 m/s, Drag  1812N, Power 1.102e+05W
Sprint Speed        :  100 m/s, Drag  1728N, Power 1.728e+05W

There is one more category of interest: hovering. The lift for a column 
of air is equal to V^3*A*rho, where V is the speed of the air, A is the 
cross-section of the column, and rho is air density. Assuming some 
sculling motion to allow a cross-section wider than wing area, but also 
some loss due to flapping, an effective area of 25 square meters results 
in an air speed of 10.5 meters per second and would require a power of 
190 kW. As the airspeed is actually a bit unbalanced, the actual power 
requirement is more on the order of 250 kW. It is possible that a dragon 
can maintain that for a few instants, but it probably cannot hover in a 
sustained way.

We are also ignoring one key effect: body lift. Depending on body 
structure, this could be substantial, especially if the body and tail 
can be flattened or made concave (model after a flying snake). However, 
while the body has a potentially fairly large area (6 square meters is 
not hard to imagine), it has a very poor aspect ratio. If we treat the 
belly as half as effective as ordinary wings, we reduce stall speed but 
don't affect other numbers:
Stall Speed          : 30.37 m/s, Drag  5564N, Power 1.69e+05W
This is mostly useful because it means the dragon can keep its tail up 
without resorting to extremes of strength.

All of this results in a dragon that, while not grossly large (30' long, 
4 tons) is terrifyingly powerful, at least by low tech standards (230 
horsepower in a 4 ton car isn't very impressive; 230 horsepower in a 4 
ton helicopter would be very low). If we figure a komodo dragon has DR 
1, and make the scale materials stronger in the same ratio as we made 
its muscles, net DR is 1(base)*4(4x bigger)*32^1/3) or 13, and bite 
damage is on the order of 13d (DR may be low; realistically humans 
probably have DR 1, and weapon damages have been adjusted to compensate. 
That would give an actual DR of 25).  This may well be beyond the 
reasonable limits for fantasy PCs, and discovering that a dragon _runs_ 
twice as fast as a racehorse will likely strike PCs as a bit odd (it's 
also unlikely for something with this sort of build; the strength ratios 
only support a peak running speed of about 30 mph.

So, what numbers can we tweak to make this a bit less absurd? We don't 
have much in the way of options to change the shape, but the weight is 
probably subject to change. In general, multiplying weight by X 
multiplies the speed of stall, minimum drag, and minimum power by 
sqrt(X). It multiplies power consumption at each of those speeds by 
X^1.5. It also multiplies calculated ST by either sqrt(X) or X^1/3, 
depending on which assumptions were being used on muscle. Thus, if we 
take our 8,000 lb dragon and cut it down to 2,000 lb, we halve all 
velocities and reduce ST to 80 body, 60 limb; quickness is unaffected; 
DR remains at a constant of about ST/10(8) (or, with the 'humans are DR 
1' theory, go with ST/5-1, or 15).  The new stats are as follows:
Stall Speed          :  16.4 m/s, Drag  1201N, Power 1.97e+04W
Minimum Power/Dist  : 40.02 m/s, Drag 392.4N, Power 1.57e+04W
Minimum Power        : 30.41 m/s, Drag 453.1N, Power 1.378e+04W
Sprint Speed        :    50 m/s, Drag 431.9N, Power 2.16e+04W
Again, with the body lift assumptions:
Stall Speed          : 15.18 m/s, Drag  1391N, Power 2.112e+04W

We still have 18x the power to weight and 5x the strength to weight of a 
human, but this is at least probably closer to something that is playable.

Of course, a dragon that's 10 hexes long, half of which is tail, and is 
probably no more than 8' at the shoulder, isn't all that impressive. So, 
how do these numbers change for a dragon of a different size? It turns 
out that there is a very simple scaling: if you multiply size by X, 
multiply weight by X^3, speed (all) by X^1/2, power consumption by 
X^3.5, and ST/DR by X^3/2.  Thus, an immense 100' dragon would be 37 
tons, stall 30 m/s, minimum P/D 73, minimum P 55, sprint 90, ST 370/490, 
DR 49, and uses 1.35 megawatts on takeoff.
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