[gurps] Dragon Aerodynamics

[midnightwind at comcast.net]

Here's irony-- I just finished the linear kinetics chapter of the human biomechanics textbook I am reading-- and it included the equations you used for lift and drag (more as digressions on the section on air resistance vis-a-vis athletic competition such as running, swimming, skiing...)... it's actually on the order of 10+% that those body suits can reduce the drag effects of air for runners et. al..  I used to swim competitively, and we used to shave (don't even think it....) I guess it wasn't all psychological, after all.  Might be interesting to think about streamlining effects on dragons... especially sea dragons...  And if sea dragons had wings... would all your stats be different, and/or would the dragon have the capability of both air and sea "flight"?

-vk
 -------------- Original message ----------------------
From: Anthony Jackson <ajackson at iii.com>
> 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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