Wing Cube Loading

[Don Cooper]

The July issue of Model Aviation contains an article where "Wing Cubed Loading (WCL)" is discussed. Does anybody have any information on how the parameter was derived or any experience with its use? It would seem to me that the major differences between large and small planes are Reynolds Number and inertias, and it is not clear to me how they are factored into WCL.

[Richard Miller]

Don,

The great nay-sayer of aviation was one Simon Newcomb, an astonomer, head of the Smithsonian Institution, sort of the Carl Sagan of his time. Also a man, like many men, who knew a bunch of things that weren't so, and wasn't afraid to say them.

He said, for example, that powered flight was impossible, that it wouldn't happen, and when it did, retrenched, said yes, airplanes of limited size would take to the skies, but nothing of any substantial size.

Simon had science on his side, which brings to a couple of points. One is that there's science and there's science, and it depends on how deeply you see into things, and the other what that science was.

It went like this: The mass of any flying vehicle would scale to the 3rd power of its linear dimension. Make a Cub twice as long, twice as deep and twice as wide, a 2X Cub, and you have 2X2X2=8. While this is happening the surface of the wing is increasing to the second power, chord times span: 2X2=4. Plot those two exponential lines and you'll find they cross a little way up the graph. That's the point, Simon said, that marks the limit of how big an airplane can be.

End of Part 1

[Richard Miller]

...and then he wrote.

Human beings, particularly "scientific" human beings - and this is the larger picture - have trouble with things they cannot see nor directly sense. "Science" is based on ponderables, on what can be measured, on quantification. [Stop me before I rave on!]

What Simon didn't see, or "see", and what many since have failed to see, is that although the wing may, in one sense be considered a surface, and we typically call it that, it does have a vertical dimension. Some people got this far and endeavored to patch things up with that insight. Not, however, sufficient.

Now - finally - to the point that the wing, or anything that qualifies as a lifting surface, works on a QUANTITY OF AIR that has three dimensions. Would you imagine that if the depth of the air affected by that little RC airplane you fly was only 1/4" the same would be true for a full-scale airplane of the same configuration? Not very likely.

It's like this: The force generated by the engine [or gravity] turns a prop that generates a thrust force. This impells the wing through the air and affects a quantity of air, per unit time, proportional to the mass of the aircraft and that thrust force. You'll find, if you collect some figures on wingspan cubed and horse power, that they pretty well match.

So - Wing loading IS three dimension, not two-dimensional as old Simon, and many since, have imagined. The mass of the model, or airplane, directly reflects this, as does the lift necessary to make it fly and the force [thrust] necessary to generate that lift.

End of Part II

[Richard Miller]

...and finally...

We're not going to go so far as to get into quantum mechanics or anything like that, but just mention it as a way of getting to the point that wing loading tends to fall into a heirarchy of two-dimensional ranges. For me, when I began modeling, it was 3, later 4-oz. per 100 sq.in., and that was fine. It wasn't fine for the next stage up, giant RC scale, but as long as I stayed in my little 3-4-oz/100 I could understand what it said in those learned article in Model Airplane News and communicate with my fellow modelers in a satisfactory way.

Same for people designing, building and flying small personal airplanes. Those of a Cub, or a Luscomb, or a Taylorcraft, would, do all fall in segment of the total scale, as do light twins, business jets, and on up.

Got all that? What maybe helps most, at least helps me most, is to think of that mass of air, one span dimension, one chord dimension, and one vertical dimension roughly equal to the chord.
That is processed, which is to say moved, accelerated and decelerated, and it experiences friction in shearing, and that requires the force imparted to it by whatever is provicing the thrust and results in that final system force, lift.

-Richard

[Don Cooper]

Richard,

Thanks, I think. It is going to take me a while to digest your replies.

[Mike Stramba]

Found an interesting page when I searched google for 'wing cubed loading"

http://www.astroflight.com/scalespeed.html

[Richard Miller]

Mike,

That's a very attractive and informative site. Mr. Boucher, however, states that "Scale Wing Loading" is related to the surface area of the wing. The idea won't go away. I run across it all the time.

I say once more: What is visible and palpable is a surface having two - count 'em, two - dimensions. On that the consciousness fixes. What is invisible and impalpable is a mass of air having three, I say THREE dimensions, which mass must be activated, charged, motivated, in the process of generating lift.

It's pretty simple as I see it, and curious to me that others do not see it as well.

-Richard

[Dave Robelen]

Hello Richard,
You finally yanked my chain. The difficulty with using the volume of the wing to predict performance ignores the very large effect of airfoil performance. Depending on the specific model, it often turns out that a thinner airfoil will lift more, and with less drag than a thicker one.
The other element is the effect of scaling. If I make a 1/10 scale model of say, a Cessna with a wingspan of 35', and chord of 5', a weight of 2000 pds., and a 12% thick airfoil, the resulting model does not measure up to the volume predictions.
Given that the dimensions are reduced by the scale factor, the areas by the square of the scale, and the volumes and mass by the cube of that scale. The resulting model would have a span of 3.6', an area of 1.75 sq. ft. and a weight of 2.0 pds.
If we state the aircraft is flying at a speed of 100 feet/second, the factor for the model is the square root of the scale for velocity, giving a predicted speed of 31.6 feet/second, or about in the park flyer class. The problem is that a model with an area of 1.75 sq. ft. and a weight of 2.0 pds will actually need to fly much faster to stay level.
The answer here is the disparity in the Reynolds number. The full scale machine is operating at about 3,200,000, while the model would only be acheiving 100,900 which will reduce it's airfoil performance dramatically compared to the full scale machine.
We can bring the model closer to the "scale speed" if we substitute a thinner airfoil with more camber, but there will still be differences. This kind of situation is why I have real dificulty relating to a cubic wing loading. I do agree that the aircraft must be displacing a mass of air equal to it's own mass to fly level, but experience shows that this can occur with many different volumes of wing.
Regards, Dave

[Mike Stramba]

Originally posted by Richard Miller

Mr. Boucher, however, states that "Scale Wing Loading" is related to the surface area of the wing. The idea won't go away.

-Richard

Richard,

Are you saying that I should stop building my 1/4 scale SuperCub using Mr. Boucher's numbers ? :)

I didn't see the quote you are referring to, but anyway is there an error in his calculations ?

What

[Richard Miller]

Dave,

Sometimes anbiguities creep, or smash into what I write in these postings. I kind of charge ahead when it would be better to go slow and be more careful with my words.

That, I suppose, was the case with the idea of [physical] wing volume. This, as I see it, was a kind of half-way house to the idea of the 3-dimensional mass of air that I've been talking about. I've read commentaries on it here and there and always considered that the authors had not really thought the problem through.

It is the VOLUME OF AIR affected, as I see it, and the vertical dimension of that volume, the other two being very obvious, might best be thought of as an expression of the coefficient of lift. That, as anyone who's examined airfoil characteristics knows, is independent of actual section thickness. Does that settle that?

I disqualify myself from saying anything about scale speed. I see so far, that three-dimensional box/mass of air around the wing clearly because I've thought a lot about it for a long time. The velocity aspects never particularly interested me.

There are two factors that affect aerodynamics - at least two - when we scale down [or up]. One is that the boundary layer will tend to relate to the wetted area of the surface; thus it will scale two-dimensionally, to the disadvantage of the smaller. The other is the ratio of viscosity to mass, the Rn, which you already mentioned, also to the advantage of the small, except if you're so small you swim yourself through the viscous aspect of the air.

-Richard

[Richard Miller]

Mike,

"SCALE WING LOADING" was one of the centered headlines, 2nd on down I think, on Bob Boucher's site. He appears to have the same idea in his head as ol' Simon Newcomb, that loading is 2-dimensional.

The math is kind of easy. The scale unit = 4. That cubed =64. You 1/4-scale Cub should weigh, other factors equal, 1 pound for every 64 pounds of full-scale Cub. The power to fly it, in real-time thrust terms, should also be one HP for every 64 HP of that aircraft.

Does that work out?

-Richard

[Dave Robelen]

Hello Richard,
I too tend to jump in with a portion of my thoughts somewhere behind. No disagreement about the 3-dimensional concept of air affected for an airplane flying level at a given speed. That simply makes sense, and can be acheived with all manner of airfoils, some cambered differently from others, and some equipped with flaps to acheive the same effect.
When dealing with the scaling issue, the rate that the machine is moving that mass of air downward becomes a significant issue, and this is where the Reynolds number comes into play. Most any published airfoil data for a given section shows a drop in lift for a smaller size/speed, while the drag is increasing. The actual mechanism of why small, slow airfoils perform less well than their larger counterparts has been the subject of an incredible amount of research, and the jury is still out on some of the issues.
The effect of scaling can be quite important to properly interpret the flight results of a model when it is used to predict the flight characteristics of a larger mahine.
This whole business could become an issue for you if you are pursuing using a model to develop your RFD glider. Keeping the scale weight and moment of inertias is vital to evaluate the control characteristics, but a change in airfoil for the model to a higher lift section could bring the flying speed closer to scale, and this could be very beneficial as you evaluate the stability and control effects.
To slower flight, Dave
PS the thrust would scale to the third power, but the horsepower is more like the 3/2 power of the scale

[Dave Robelen]

Hi,
I blew it, the scale factor for horsepower is the scale factor to the 3.5 power.
Regards, Dave

[Don Cooper]

Thanks for the responses. They gave me an incentive to dust some of the 40 year accumulation of rust off my aerodynamics knowledge so I could understand them.

After thinking about WCL and trying a few examples, it appears to me that WCL is not generally applicable - it ignores too many parameters that effect how a plane flys and feels (e.g. Reynolds Number, Cd, L/D, moments of inertia, etc, etc.)
It might be useful in comparing designs which are 'similar' in some sense (I don't know what parameters define that similarity), but does not do so well when applied to basically different designs. For example, the article I referred to said that thermal gliders have a WCL of around 4. By chance I have, or have had, two planes with WCL in that range. One was a 26", high drag biplane slowflyer. The other is a Tiny with GWS power and electronics. Believe me, neither is a thermal glider. Also the flight characteristics of the bipe and the Tiny are significantly different.

[Dave Robelen]

Hello Don,
It does seem to get messy whenever someone tries to simplify the principles of model aviation. We are already at a disadvantage because most of the engineering tables on airfoils and such are meant for full scale. There was a gentleman, Brad Powers, who presented several outstanding articles in Model Aviation back in the 80's that presented the principles of scaling and comparing different sizes. Work like that is really hard to find nowadays. Good luck on the project.
Regards, Dave

[Wells]

The lift equation looks like this:

lift = coefficient * wing area * speed^2 * density / 2

I guess you could say that the first 2 terms (coefficient and area) make for a cube loading. However, for a given coefficient, it's only wing area that matters, so for a given model, wing loading as a 2-dimensional relation is just fine. All it really tells you, is how fast the plane needs to fly and how much field you'll need to land it! If you want a model to fly at scale speeds (for a given lift coefficient), you'll need to scale down the weight by 1/scale^4, since lift is the opposite force to weight. There's more on my website if interested...

[Dave Robelen]

Hello wells,
This concept of "cubic wing loading" goes back quite a ways. It has it's roots in the non engineering model design folks who feel that as long as the model is scaled (up or down) the volume of the wing is scaled and therefore the performance can be scaled. This would be super if there were no such thing as the Reynolds number, and it's frequently massive effect on the lift and drag coeeficients such as you show in your equation. Pretty much any text on airfoils will show the variation of performance with the Reynolds no. When you get down to the small and light models, the performance can take a tremendous hit.
The usual factor for scaling mass (or weight) is N to the 3rd power. Not sure where you derive the 4th power. Also, the statement "weight is the opposite force to weight" is not especially clear. Could you clarify?
Take care, Dave

[Richard Miller]

Wells,

A pyramid seen from a considerable distance has what appears to be smooth sides; as one approaches this [continua] morphs to step changes. If we examine wing loading from base to pyramidon we must, I believe, conclude it's three dimensional. If we take any element [quanta] as models of the scale dealt with on this site, or 1/4 scale, or the size of a Cub, or a light twin, we can content ourselves with the simplification of speaking of loading in two dimensions.

Way many years ago, in the 60's to be exact about it, when I wanted to get this matter straight in my head, I collected a large list of dimensionally similar things that flew and arranged them from smallest to largest with columns for one, two, and three dimensions. The smallest was very small, the largest very large.

When you do that, and examine the range of figures in each column, the three-dimensional nature of wing loading becomes quite apparent.

What is that website?

-Richard

[Wells]

Sorry Dave, my fault! I should've written that lift was the opposite force to weight.

From the lift equation, you can see that a 1/5 scale model, at 1/5 the speed, has 1/625 the lift. If you scale the weight down by the same factor, things stay in balance and the plane is able to fly at scale speeds. You're right about the Rn though. It's effects would probably vary, depending on the airfoil. I'm working on a P-38 model right now and I've decided to use scale airfoils, since in my CFD analysis, I'm only gonna lose 10% of the max lift coefficient due to lower Rn.

[Dave Robelen]

Hello Richard,
That pyramid analogy was fascinating. If our little world only stayed as linear through the range, I would buy off on it. It is the non-linear effect of reynolds no. at the lower values that prevent straight scaling from being a useful prediction tool. Possibly if you stay at fairly high Reynolds no's, the scaling would be more linear. Often it is the drag that takes the largest hit due to the thickening boundary layer. Wells, did you check the effect of drag on your P-38 scale job? sometimes this will push you toward more power than you might anticipate.
Regards, Dave

[Wells]

yeah, power, thrust and drag are a whole other ball of wax! On one hand, we tend to use props that are smaller than scale diameters and that turn higher rpms (more rotational losses), so we lose some thrust. On the other hand, we gain some thrust by having the model flying slower than the prototype. I like to use the merry-go-round analogy for that one. You know how when you first start spinning it, you can accelerate it pretty good, then once it's up to speed and your friend is about to puke, you can't seem to spin it any faster? That's because a given amount of kinetic energy produces less of a speed increase when the object is already moving. It's the same with the propeller and thrust. The faster the plane goes, the less the prop can accelerate the air mass and the less thrust is produced. I reckon that the net thrust increase (for a given power loading) is probably close to cancelling out the drag increase from lower Rn, so you can still use 1:1 power loadings for a model. At least, all of the scale models that I've flown so far seem very scale-like to me with that configuration Although, I suppose it wouldn't hurt to increase power a bit more to compensate for lack of superchargers and constant-speed props, for those planes that were so equipped.

[Dave Robelen]

Hi Wells,
I appreciate the insights and analogy. It is kinda rare to have situations in scale modeling where we can use that sort of approach to an explanation, but that one seems to fit. I wish you well on the P-38 project. What size model are you working up?
My career at NASA was using dynamicaly scaled models to study stability and control issues. We worked mostly with general aviation configurations with the outdoor R/C models. As the wing loading of the full scale went up, the models became real pigs with the available power. This, even after using more horsepower than scale. That nasty old Reynolds no. works on props too.
Take care, Dave

[Richard Miller]

Dave,

I'll begin by agreeing that R/n is going to skew the line. Me thinking out loud now: The wing moves air in two modes, the boundary layer, which varies with velocity [and of course area] and doesn't do anything for lift; the rest of the air affected by the wing. Move down the scale and there's more, proportionally, of the former until we get to wee things swimming in viscosity.

As for the other, the air that does something, THAT must have a direct dimensional relationship with the aircraft mass. Work backwards. That mass is sustained by an aerodynamic force, lift, equal to gravitational effects. THAT force is the a product of the mass of the affected air X the velocity changes effected by the field of the wing.

I'm stopping. I'm kind of satisfied, kind of dissatisfied. Late in the day.
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Tomorrow - this - morning, on which I declare that I have a couple of lucid hours in the early part of the day, sometimes, but things go to pieces later and I shift into my confused old man mode.

The point I was intent on making was that wing loading is three dimensional as distinct from two dimensional. I did not mean, although it seemed that way, that the side of the pyramid, the continua of loading across the total scale of flying objects was a straight line. It isn't. You're right about the boundary layer Dave.

However. Some years past I encountered an enthusiatic review, in New Scientist, of a new book on aerodynamics by a chap named Hank Tennekes. I'm always eager to see new books on aerodynamics and learn what I can.

I have this n' that to say about the book, but will limit myself to a quote:

"The role of the wing is straightforward: A wing's aerodynamic lift L is proportional to its surface area S."

So the idea of two-dimensional wing loading is alive and well at the highest levels - MIT and Cambridge, and will probably remain so for - well, a long time.

Here then, is what I want to say. Dynamic pressure is differentiated from velocity in all aerodynamic interactions. What was in the flow as velocity manifests, in the production of lift, as pressure. Bernoulli: Velocity + pressure = C[onstant].

Imagine a mill wheel. It produces a force operating against gravity, lifting water, equal to the pressure, extracted from the flow of the stream, by the resistance of the paddles. A quantity of water, equal to the paddle face, say one foot square, and of a given depth, resulting from the upstream propagation of pressure. Thus, a three-dimensional mass of water from which velocity is extracted in the service of the dynamic pressure that does the work. [Am I being redundant?]

Let's double everything, size of the mill, its wheel, the load that wheel labors under, which is to say the work it does; the linear dimension of everything is 2X, and as a consequence the paddle size is squared. Does the upstream dimension of the affected water remain the same as with the smaller wheel? Presumably, if that were the case, it would be the same, say one foot, for a paddle with 3-inch sides, or 3-ft. sides, for 10-ft. sides.

So of course not, and by the same logic, if such it is, the depth of the air about a functioning airfoil, other things equal, is proportional to the chord, thus more for bigger, less for smaller.

That's a little more like I wanted to say it. Not quite all, but a start at it.

-Richard