



Discussion
Wing Cube Loading and the P51
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:
Thinking about this, I thought it might be fun to compare a fullscale P51D 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 nonscale 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 fullscale 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 fullscale 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 ! 







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 fullscale 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 P51, 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 tissuecovered 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? 





Very interesting. We all know that certain things do not even closely "scale up", for ex. going from subatomic 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 fullsize aircraft to a model, simply observe the performance of various scaled models (incl. actual speed and cubic wingloading) 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 wingloading 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 dropoff, I might consider only reducing the cubic wingloading 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 wingloading. And also make compromises between model performance and scale flight appearance (visual speed: bodylengths/sec.). Both the 20cm span F22 and Su47 toys I made (which also involved cheating re: camber, wingtwist 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) 





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.






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! ! ! ! 





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. 





Quote:
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 Eflite 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. 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. 






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 . 





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! ! ! ! 





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.. 





Quote:
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 1speed winds) I am impressed! 


Last edited by xlcrlee; Oct 17, 2017 at 12:00 PM.





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. 





Quote:
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? 






Found the following in "AAF Manual 511275, "Pilot Training Manual for the P51 Mustang", 15 Aug 1945"
Quote:
I'm still looking for that paper. Either the modern evaluation pilot was more sensitive to wing drops, or he was talking about poweron stalls. 






Quote:
http://store.laserdesignservices.c...roducts_id=102 Bob 



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