Wing Cube Loading and the P-51 - RC Groups
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Oct 15, 2017, 05:22 PM
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Wing Cube Loading and the P-51


BMathews was kind enough to post a link to Francis Reynolds' 1989 article on Wing Cube Loading (WCL). You can find it here:

http://theampeer.org/CWL/reynolds.htm

Reynolds argument for WCL comes down to this (quoted from the above link):

Quote:
Wing loading is a lousy way to compare models with each other and with full-scale airplanes, because wing loading varies with the size of the plane. The problem is that we are dividing weight, a cubic-like function (weight is proportional to volume which we measure in cubic feet) by area, a squared function measured in square feet. We should be, and many modelers are, comparing planes by their wing cube loading, which is independent of size because both the numerator and the denominator are cubic.
For scale models, this is "going a long way about the bush to set fire to it". If you want to know the "scale" model weight you might just as well divide the full-scale weight by cube of the scale ratio and get the answer directly.

Thinking about this, I thought it might be fun to compare a full-scale P-51D with my Ares model. The original has a span of 37 ft 0 in, wing area of 235 sq ft, and an empty weight of 7635 lb (I think empty weight is most representative since RC models typically don't leave the landing pattern). Engine rated power is 1490 HP, stall speed 100 MPH. Using these specs you can calculate:

Wing loading: 32.5 lb/sq ft
Power loading: 0.195 HP/lb
WCL: 2.12 lb/cu ft

The Ares Mustang 350 has a 29.5 in wingspan, 147.5 sq in wing area, and tips the scales at 12.8 oz. With the standard 3S 600MAh battery I measure about 100W continuous power. Stall speed based on wing loading (assuming a CL_max of 1.0) works out to 17.5 MPH. We can calculate:

Wing loading: 12.5 oz/sq ft
Power loading: 0.000655 HP/lb
WCL: 0.77 lb/cu ft

Now, what about a "scale weight" Mustang the same size? First we find the scale ratio. You could take the ratio of the wingspans, but to allow for slightly non-scale shapes I used the square root of the wing area ratio. I get:

Scale ratio = 15.1

Applying this ratio I get:

Wingspan: 29.3 in (about a quarter inch less than the Ares)
Weight: 35.2 oz (2.75 times the Ares!)
Power: 82.2 W (vs. 100 for the Ares)
Stall speed: 25.7 MPH (based on full-scale and probably unrealistic without flaps)
Wing loading: 34.3 oz/sq ft
Power loading: 0.000196 HP/lb
WCL: 2.12 lb/cu ft

Looking at the numbers, a "scale" weight Mustang this small would be quite a dog. It would probably need a catapult instead of a hand launch to get flying speed, and would need a long, slow climb before attempting any maneuvers. That said, it would probably be pretty insensitive to turbulence .

I think it's fun to go the other way, and see what kind of full-scale performance the Ares model represents. Scaling up, we get the following:

Empty weight: 2780 lb (36.4% of full scale)
Power: 1814 HP (20% higher than full scale - Griffon engine?)
Stall speed: 68 MPH (based on CL_max = 1.0, so no flaps needed)
Wing loading: 11.8 lb/sq ft (36.4% of full scale)
Power loading: 0.652 HP/lb) (3.34 times full scale)
WCL: 0.77 lb/cu ft

So this "ultra Mustang" would have a little more engine power than an original, but just over a third of the weight. Some structural simplification could be expected from eliminating flaps, but I don't think even modern composite materials could deliver adequate strength with that kind of weight reduction. Besides, the pilot would be uncomfortable with those enormous servo arms flailing around !
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Oct 15, 2017, 08:09 PM
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The scaling rules used in the first post are based on "dynamic scaling". The guiding principle is that accelerations are invariant with scale. That is, an n "G" maneuver by the model corresponds to an n "G" maneuver by the full size plane.

Acceleration has units of length/time*time. Length varies with the scale ratio, so for dynamic scaling, time varies with the square root of the scale ratio.

Velocity is length/time. Therefore, velocity also varies with the square root of the scale ratio.

Density is mass/volume. Volume varies with the cube of the scale ratio. Density of the atmosphere doesn't change, so mass also varies with the cube.

Weight is mass*gravity. Gravity doesn't change (we don't built scale Earths for our model planes), so weight also varies with the cube.

Weight is a force, and since every action has an equal and opposite reaction, other forces (lift and drag) will also vary with the cube.

Power is force*length/time. Therefore, power varies with the scale ratio to the 3.5 power.

Some object to "dynamic scaling" because it produces heavily loaded models which fly too fast. We could use another kind of scaling law; call it "true scaling". The principle here is that time is invariant. If the real plane goes X distance, the model should cover a scaled down distance in the same amount of time. Therefore:

Velocity will vary directly proportional to the scale ratio.

Stall speed varies with the square root of the wing loading. Therefore, if the model can achieve full-scale CL_max (big assumption at small scales!) the wing loading varies with the square of the scale ratio.

Wing loading is weight/area, and area varies with the square of the scale ratio. Therefore, weight varies as the fourth power of the scale ratio.

Power is force*length/time. Therefore, power varies with the fifth power of the scale ratio.

Let's apply these rules to the P-51, using the same 15.1 scale ratio. We get:

Weight: 2.32 oz (less than 1/5 of the Ares)
Stall speed: 6.6 MPH
Wing loading: 2.27 oz/sq ft
Power: 1.4 W

To get to these numbers, you'd need a lightly built tissue-covered balsa frame, or maybe microfilm.

Conclusion: Indefinite -- but maybe "dynamic scale" and "true scale" serve some purpose as boundaries for reasonable flight weights for scale models?
Oct 16, 2017, 07:01 AM
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Very interesting. We all know that certain things do not even closely "scale up", for ex. going from sub-atomic physics to macro. That is because despite our best efforts "reality" is far more complex and in many ways indeterminate and thus escape our essential simplifications no matter how hard we try (or can ever, an essential element in the ancient idea in the Tao). Being a realist I use practical evidence, such as Bruce noted to make models lighter, or as have you, to just let them fly faster.

My simple mind associates both of these solutions (which are not mutually exclusive) with Re, and we know from measurement that the L/D values go increasingly, I would say exponentially, off a cliff as Re becomes smaller and smaller. The simple solution I have used for decades is to compare apples with apples, and when scaling from a full-size aircraft to a model, simply observe the performance of various scaled models (incl. actual speed and cubic wing-loading) and go from there.

However it gets easier when going from one model size to another, depending of course on the Re range/s. So taking a 1m span model and reducing it to 0.5m, I reduce the cubic wing loading by approx. 15%. And similarly when going from 1m to 2m I increase the cubic wing-loading by ca. 15%. This is obviously quite a rough approximation, but it gives reasonable ballpark results. And when scaling down from say 6m to 3m, since the Re curve is only approaching the dastardly L/D drop-off, I might consider only reducing the cubic wing-loading by maybe 5%.

In any of the above cases I "check my work" by comparing my calculated results to the behavior of similar models with the same cubic wing-loading. And also make compromises between model performance and scale flight appearance (visual speed: body-lengths/sec.). Both the 20cm span F-22 and Su-47 toys I made (which also involved cheating re: camber, wing-twist and power loading) fly in speed envelopes from "scale" approx. 150 mph landing speed to just over Mach 1 full power. Close enough for horseshoes (no "t" in that word)
Oct 16, 2017, 05:26 PM
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Quote:
Originally Posted by xlcrlee
Both the 20cm span F-22 and Su-47 toys I made (which also involved cheating re: camber, wing-twist and power loading) fly in speed envelopes from "scale" approx. 150 mph landing speed to just over Mach 1 full power.
Unfortunately there is no easy way to scale down Mach number, since the speed of sound is a function of the absolute temperature. So although your 20cm jets may fly at "scale" Mach 1 you don't get to make sonic booms with them.
Oct 16, 2017, 06:42 PM
B for Bruce
BMatthews's Avatar
Even using an online calculator the number for WCL comes out to something crazy. Using THIS CALCULATOR the Mustang comes in at 33.9 even for the empty weight. Never mind adding some fuel just for some close in flying.

On the other hand the numbers for a Piper Cub comes back with a rather proper sounding 8.1. And that puts it neatly into the lighter sort of range for trainers and light sport models.

So what constitutes a "racer" as noted in the ranges for suitable WCL's from the link above? I ran the numbers for the F3D event (max 34dm^2 and 2250 to 3000gm) and it gave me back a value of 11.5. So that puts the Mustang WAY up out of the range of even a racing model aircraft. Although it sounds a lot like an F3D is going to be nicer to fly than some scale models.

So clearly the whole WCL thing fails miserably once the size and weight start climbing into the full size world. Yet the results for the Piper Cub dovetail neatly into the range of values.

What the heck.... the 787 comes in at 31.1 So the same as the Mustang.... There's something to ponder! ! ! !
Oct 16, 2017, 06:55 PM
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richard hanson's Avatar
What are you going to do with the information?
That is what I say to myself whenever I see an exhaustive study on scaling models.
Having built thousands of models , I have yet to see an exception to the golden rule "build it as light as possible".
The exception is where the model is gravity propelled (glider)
If you want to do a scale model which is intended to fly, no matter what scale, LEARN TO BUILD LIGHT.
Forget about scale airfoils unless you can see it at a glance because you will never be able to prove how the full scale works once it is scaled down.
I know the math is fun to do so go for it
I just don't see where it helps.
Oct 16, 2017, 08:20 PM
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Quote:
Originally Posted by richard hanson
Having built thousands of models , I have yet to see an exception to the golden rule "build it as light as possible".
What does "possible" mean to you?

I could build a stick and tissue Mustang the same size, but could it handle a 100W motor and 3S power? What would happen to it in a power dive if I pulled out too quick?

I like the Ares model, which is about three times lighter than "dynamic scale" weight. Wing loading (not cube) is 12.5 oz/sq ft, which is heavier than most E-flite UMX planes, but still low enough for a reasonable stall speed. It has plenty of power and can sustain a near vertical climb. Plenty of roll rate, little or no adverse yaw and does aerobatics very well IMO.

And, this is one foamie with enough "meat" in the structure to be durable and repairable; as demonstrated after a couple of bad hand launches .

I just don't see that there's anything to gain by making a lighter model in this case.




Quote:
Originally Posted by richard hanson
Forget about scale airfoils unless you can see it at a glance because you will never be able to prove how the full scale works once it is scaled down.
I will agree with you here. The original Mustang had a laminar flow airfoil that was notable for an abrupt stall even at full scale. The Ares model has an undercambered one.
Oct 16, 2017, 08:38 PM
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richard hanson's Avatar
my idea of possible is apparently not the same as yours.
My models are built to a simple profile:
the power and airframe are matched for the best performance.
As a kid I found I could outrun 400 cu in engines with a puny 283 design.
Same idea - you build the airframe to do the job needed and power it as necessary.
Repairability is important of course but it can be done without adding weight .
Oct 16, 2017, 10:09 PM
B for Bruce
BMatthews's Avatar
I actually feel that a 4oz, 30inch span P51 would be a lot of fun in its own way to fly. But the overall character would certainly be wildly different from the 13'ish oz Ares version.

And a 30 inch P51 that weighed a true scale amount of 35oz? That would indeed be a crash looking for a place to occur. And I strongly suspect it would occur pretty close to where it left the owner's hands or lifted off Terra Firma. OR... it would need a long paved runway and fly at some crazy amount of speed ALL the time! ! ! !
Oct 16, 2017, 11:32 PM
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richard hanson's Avatar
When you are playing in the 12 ounce down to 4 ounce weight models , it is amazing how an ounce or two changes performance and when you are in the one ounce model arena, a change of 2 grams is quite noticeable
Our pattern planes for the 1980's were 8 pounds and the same design slimmed by improved construction ,elimination of trike gear etc. pulled almost a pound which made a huge performance envelope change .
same plane just lighter.
You could fly the pattern SLOWER and at a more constant speed which made it easier to do accurate maneuvers , much easier to setup spins etc..
Oct 17, 2017, 06:59 AM
Registered User
Quote:
Originally Posted by jruley
Unfortunately there is no easy way to scale down Mach number, since the speed of sound is a function of the absolute temperature. So although your 20cm jets may fly at "scale" Mach 1 you don't get to make sonic booms with them.
As I had assumed anyone would understand that we are talking VFR RC here and VISUAL scale speed, and so the distance from the remote pilot would put the tiny toy max 100m away and not too high above sea level (Mach 1 then being ca. 340 m/s), say, I do assume and accept that you are making a silly joke, right?

Mach 1 at that speed, altitude and comfortable temperature would scale down (0.2m/14m span) to ca. 11 mph, and since that toy can easily fly in 15 mph winds, with an airspeed I guess of say 20 mph .... hey it can fly at VISUAL "scale" Mach 1.8 (varying with "scale" Mach 1-speed winds)

I am impressed!
Last edited by xlcrlee; Oct 17, 2017 at 12:00 PM.
Oct 19, 2017, 08:11 AM
Registered User
jruly wrote: "The original Mustang had a laminar flow airfoil that was notable for an abrupt stall even at full scale."

I'd like to see a reference for such an assumption. at ANY scale.
Oct 19, 2017, 08:39 AM
Registered User
Quote:
Originally Posted by packardpursuit
jruly wrote: "The original Mustang had a laminar flow airfoil that was notable for an abrupt stall even at full scale."
I'd like to see a reference for such an assumption. at ANY scale.
I can't find the paper at the moment, but an evaluation was performed on several restored WWII aircraft, including a P-51. The pilot noted that the aircraft had a tendency to drop a wing abruptly at the stall, and IIRC he attributed that to the airfoil.

It's been awhile since I read it, and the problem may have only been in certain kinds of stalls rather than all stalls (accelerated vs. wings level).

If you're familiar with the original airplane, what's your experience?
Oct 19, 2017, 01:42 PM
Registered User
Found the following in "AAF Manual 51-127-5, "Pilot Training Manual for the P-51 Mustang", 15 Aug 1945"

Quote:
STALLS

A stall in the P-51 is comparatively mild. The airplane does not whip at the stall, but rolls rather slowly and has very little tendency to drop into a spin. When a complete stall is reached, a wing drops. After that, if you continue to pull back on the stick, the airplane falls off into a steep spiral.

When you release the stick and rudder, the nose drops sharply and the airplane recovers from the stall almost instantly.

You'll generally be warned of an approaching stall by a buffeting at the elevators. In a power-off stall the buffeting is slight, becoming noticeable at 3 or 4 mph above stalling speed. Violence of the elevator buffet increases with the speed of the stall.

The speeds at which stalling occurs vary widely, depending on the gross weight and the external loading of the airplane. Lowering the flaps and landing gear, of course, reduces stalling speed considerably.

A power stall either with wheels and flaps up or with wheels and flaps down is much more violent than a power-off stall.

Notice that while in a stalling attitude the rudder remains sensitive well after the ailerons have lost their efficiency. You can see, therefore, why a sudden application of power in making a landing will aggravate a wing-low condition.

Recovery from any stall is entirely normal. Apply opposite rudder to pick up the dropping wing and release the back pressure on the stick.
So nothing much to be concerned about, at least by 1945 standards.

I'm still looking for that paper. Either the modern evaluation pilot was more sensitive to wing drops, or he was talking about power-on stalls.
Oct 19, 2017, 06:57 PM
Registered User
Quote:
Originally Posted by jruley
The scaling rules used in the first post are based on "dynamic scaling". The guiding principle is that accelerations are invariant with scale. That is, an n "G" maneuver by the model corresponds to an n "G" maneuver by the full size plane.

Acceleration has units of length/time*time. Length varies with the scale ratio, so for dynamic scaling, time varies with the square root of the scale ratio.

Velocity is length/time. Therefore, velocity also varies with the square root of the scale ratio.

Density is mass/volume. Volume varies with the cube of the scale ratio. Density of the atmosphere doesn't change, so mass also varies with the cube.

Weight is mass*gravity. Gravity doesn't change (we don't built scale Earths for our model planes), so weight also varies with the cube.

Weight is a force, and since every action has an equal and opposite reaction, other forces (lift and drag) will also vary with the cube.

Power is force*length/time. Therefore, power varies with the scale ratio to the 3.5 power.

Some object to "dynamic scaling" because it produces heavily loaded models which fly too fast. We could use another kind of scaling law; call it "true scaling". The principle here is that time is invariant. If the real plane goes X distance, the model should cover a scaled down distance in the same amount of time. Therefore:

Velocity will vary directly proportional to the scale ratio.

Stall speed varies with the square root of the wing loading. Therefore, if the model can achieve full-scale CL_max (big assumption at small scales!) the wing loading varies with the square of the scale ratio.

Wing loading is weight/area, and area varies with the square of the scale ratio. Therefore, weight varies as the fourth power of the scale ratio.

Power is force*length/time. Therefore, power varies with the fifth power of the scale ratio.

Let's apply these rules to the P-51, using the same 15.1 scale ratio. We get:

Weight: 2.32 oz (less than 1/5 of the Ares)
Stall speed: 6.6 MPH
Wing loading: 2.27 oz/sq ft
Power: 1.4 W

To get to these numbers, you'd need a lightly built tissue-covered balsa frame, or maybe microfilm.

Conclusion: Indefinite -- but maybe "dynamic scale" and "true scale" serve some purpose as boundaries for reasonable flight weights for scale models?
Try out my free software, it could help
http://store.laser-design-services.c...roducts_id=102
Bob


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