This discussion thread is a follow-on to several conversations I've had with people in the forums who are particularly interested in the aerodynamics of vertical take off and landing (VTOL) aircraft.  Much of the dialog in these forums appropriately surrounds the mechanisms for robotic automation of VTOL aircraft, and in those contexts, I am much more a listener than a contributor.  There are some brilliant engineers, code smiths, and experimenters who frequent these hallowed pages.  The group effort to yield such a marvel as the APM platform is nothing short of astounding.

However, I think we can all agree that the primary functionality of anything that flies is related to how it generates forces to oppose gravity.  Much of the focus here has been on the control system, for a myriad of reasons.  Seemingly ignored is the aerodynamics of propeller thrust, but fairly speaking, it is unromantic as having been largely figured out 90 years ago.  In fact, here's a link to the NACA (forerunner to NASA) original paper entitled "The Problem of the Helicopter", dated 1920.  It is of interest to note that we widely applaud Sikorsky for inventing the modern helicopter, but his contribution was one of a control scheme; he gave us cyclic pitch variation for thrust vectoring coupled with a variable pitch tail rotor to counterbalance torque.

If technical papers like that make your eyes glaze over, perhaps an essential basic treatise is in order.

We go back to Newton's basic laws here, and one in particular: Force=Mass X Acceleration, or F=MA.  In order for our craft to fly, we need it to generate a force equal to and directly opposing the force of gravity.  To produce this force, we normally take the air around our craft as our readily available mass, (except in the case of the rocket and to some degree, the jet engine, where the mass is a product of combustion), and accelerate it (add to its velocity) toward the ground.  Yes, rotors, wings, and propellers all do this, and they all rely on the same principles.

However, there is another factor to consider.  While this particular law is not attributable to Newton, it is still a primary expression: energy is equal to half the mass times the velocity squared, or E= 1/2M X V^2.  So while the lifting force is linearly proportional to mass and acceleration, the energy required to perform the acceleration increases exponentially with the change in velocity.  It naturally follows, then, that taking a lot of air and accelerating it a little takes a lot less energy than taking a little air and accelerating it a lot.  This is why heavy-lift helicopters have such large rotor spans, and their technically analogous cousins, sailplanes, have long wings.  (I drive some people in the pseudo religion of ducted fan technology crazy by pointing out that all their purported efficiency gains can be had by merely making the propeller blade longer...ah, but I digress...) 

In the final analysis we must be concerned about lifting efficiency.  The basic expression for us in comparing efficiencies of different designs can be simplified to merely the number of watts (power) it takes to produce a pound of thrust (mass).  Of course, we cannot simply make our rotors infinitely large and fly with no power expended at all.  There are therefore some engineering compromises which must be made in a VTOL aircraft design.  I hope you can see now why aerodynamic designers first examine the ratio of lifting surface area to the weight lifted as an indicator of potential efficiency.  In the rotary wing world, this ratio is called disk loading, and it is expressed as so many pounds per square foot of total rotor swept area.

Disk loading is a basic predictor of hovering efficiency, but it is by no means the only one.  In my next message, I'll get into evaluating basic rotor (or propeller) blade design criteria.

I hope you've enjoyed this little introduction, and yes, I do plan to eventually show that electric multicopters can be a very viable solution for large payloads compared with conventional helicopters.  However, we need to "level set" on the concepts.  Let the discussions begin.

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I started this message thread with a promise to make the case for large scale electric multicopters.  Most of you have already conceded that quadcopters are far less mechanically complex and much easier to construct than regular single main rotor helicopters.  However, I haven't really even mentioned that fact until now - not because it's patently obvious, but because it is an excellent segue to a discussion of the aerodynamic compromises involved.  But before I tell the story behind airfoil polars, I have to discuss why they're important.

The basic question is: what is the difference between a helicopter rotor and an airplane propeller?  They both represent time-tested methods of converting torque forces to aerodynamic thrust.  Why are helicopter rotors long and skinny, whereas propellers are short, thick (especially near the hub), obviously twisted along their span, and usually tapered?   Most people guess that airplane propellers are usually operated much closer to the ground, so they're made thicker at the hub to increase their strength to mitigate the damage in the event of an object strike.  This is actually true to an extent, but it is certainly not even close to the whole story.  (you regular chopper guys know this stuff already I'm sure).

If you were intellectually intrepid enough to read The Problem of the Helicopter paper, then you have become familiar with the two basic realities staring Igor Sikorsky in the face, and why he is remembered for his inventiveness.  With 1940's technology, merely aiming one or more aircraft propellers at the sky yielded monumental challenges with elemental axes control (pitch, roll, and yaw).  Then there were the engines; they failed often enough that the ability for an aircraft to glide to a landing was of paramount importance.  Even if this meant crashing, the risk was far less than merely dropping out of the sky to certain death.

Sikorsky solved these two primary issues by making a large (mindful of disk loading) propeller with individually articulating blades.  Through some rather complex mechanical linkages, these blades could vary their pitch throughout their entire arc of rotation.  We'll cover this in greater detail in the airfoil polar section, but for now just know that the lift of a wing changes with the angle at which it encounters the air (angle of attack), and this is usually proportional to the pitch.  If Sikorsky wanted his helicopter to move forward, he applied pressure to a control stick which caused both rotor blades to reduce their pitch as they swung toward the front of the craft, and increase their pitch as they rotated toward the rear.  This "cycling of pitch" created a lifting force differential from front to back and caused the helicopter to tilt forward, vectoring some of the total lifting force to lateral motion.  If Sikorsky wanted more lifting thrust, he merely increased the pitch of all the blades "collectively".  This cyclic and collective mixing gave him complete control of thrust, pitch and roll.  To counteract the torque of the main rotor, he added a variable pitch tail rotor.  Viola!

There's still the issue of engine failure to be addressed.  Sikorsky added a clutch which would disconnect the rotors should the engine quit.  He also made sure the rotor blades were heavy enough to continue to rotate long enough to give the pilot time to react to the engine failure, lowering the collective pitch on the blades to initiate a sort of "windmill" effect.  This is known as autorotation in the helicopter world, and in essence, it represents the helicopter's ability to "glide" to a landing if there is enough altitude and forward speed to do so.

Those of you familiar with classical mechanics can readily see the conflicts arising from the attributes "heavy", "rotating" and "change" when applied to the same object.  Given the 600 RPM head speed of most full size helicopters and the 300 pound typical mass of each blade, the centrifugal force alone is rather...excruciating.  Now imagine those whirling blades doing a metaphorical dolphin "nose up, nose down" cycle motion 10 times per second, and you can begin to understand the issues involved.  The mass of the blades needs to be concentrated very near to the axis of this pitch rotation, or the forces on the control linkages, swashplates, etc. would be insurmountable.  A wide blade (long chord) and the associated increase in rotational inertial moment is simply not an option.

Given the mechanical challenges in a conventional helicopter design and the virtual perfectionism required for safety, it's no wonder that the adjective "excruciating" can also be applied to conventional, full-size helicopter maintenance costs. "You couldn't afford one if it was free," is the adage often jokingly tendered.

Now you know why helicopter blades are long and skinny - they have to be for the mechanics to work.  Does this compromise their aerodynamic efficiency?  Can a propeller blade not required to bob up and down like a porpoise be made to do better?  That, my friends, is the subject of my next message.

What is working for him is the patience of his investors.  :-)

I wonder (have wondered for a long time) if switching Moller's designs over to gigantic outrunner motors and generating the power with gas turbine generators could result in a controllable/practical solution for this sort of design. Hopefully that's one of the possible solution cases you'll be discussing eventually...

It seems like his Wankels don't respond fast enough for good multi-rotor control and are also gas hogs.

Great thread Brad.

I always thought that helicopter blades were long and thin because that is the most efficient wing profile, similar to the way sailplane airfoils are long and thin.  I thought the reason for that was that longer wings tend to be more efficient because there is less... what's the term... when the airflow wraps around the tip of the wing from the bottom back to the top.  Wing-tip vortex?  

Ah yes, here it is:

Wingtip vortices affect only the portion of the wing closest to the tip. Thus, the longer the wing the smaller the affected fraction of it will be. As well, the shorter the chord of the wing the less opportunity air will have to form vortices. This means that, for an aircraft to be most efficient, it should have a very high aspect ratio. This is evident in the design of gliders. It is also evident in long-range airliners, where fuel efficiency is of critical importance. However, increasing thewingspan reduces the maneuverability of the aircraft, which is why combat and aerobatic planes usually feature short, stubby wings despite the efficiency losses.

If what you say about the importance of Re is true, then why don't gliders all have delta wings or some other long chord wing design?  Doesn't skin drag across a wide chord propeller eventually come into the efficiency equation?

These are all questions for discussion, not challenges or statements of fact.


Full-scale gliders still have reasonably large Reynolds numbers. Long wings are still required to reduce induced drag, but if it forces your chord to be very short, like a rotor blade, or a propellor blade, or even a model glider, then the Reynolds number becomes more significant.

When you are above a million or two, which is where most full-size aircraft are, things are pretty predictable.  When you stray below the 1 million, or into the hundred-thousand range, things seem to become very difficult indeed.  Whereas an aerofoil section might get you a max Cl of 1.5 or more in full-scale, at miniature sizes, you're lucky to breach 1 in a predictable fashion.

I am beginning to wonder what a fully Re-optimised model glider might actually look like.  A lot less like a scaled down ASH26 than they currently do.  I suspect aspect ratios would drop significantly.

Is anyone tried props like this yet?

Whale power might make quads the underwater platform of the future.

That's an interesting technology Jake - thanks for posting about it.

Here's a CFD video I've found about it.  I believe what it's doing is suppressing the formation of the laminar separation bubble that forms at low Reynolds numbers at very low AoA's and which creates much of the extra drag compared with megaRe flow.

you can do CFD on a desktop PC these days Monroe, if you drink enough coffee between runs...

@ Robert: Thanks for your sentiments.  Sailplane wings were perhaps an inappropriate metaphor for me to use earlier, particularly because they don't rotate.  :-) 

It's not about having a high Re, but rather the best Re and AoA for the chosen airfoil section.  The airspeed changes with radius too, as does the inflow velocity (lift being the ^2 of speed).  Tip vortexes (vortices?) themselves are the stuff of PhD dissertations and ducted fan acolyte sermons.   

Speaking of PhD's, Paul Pounds does an excellent job of making the mathematical case for blade taper and twist in his X-4 paper.  It really is worth a read.

To make a great FM rotor disk we need a great airfoil (blade cross-section).  What is a great airfoil, exactly?

Propulsion force = Mass of the air moved * resulting velocity imparted.  As a propeller blade slices through the air, it separates it into two streams: upper and lower.  If the angle at which the blade parts the air is sufficiently positive, the lower air stream is forced toward the ground with no choice in the matter whatsoever.  The upper airflow, however, does not have a solid plane forcing it to go anywhere.  It has to be "coaxed" over the top of the wing to be added to the downward flow.  The only way to do this is to make the upper flow not change direction too fast by adding a gentle curve for it to follow.  Why does the upper stream follow this curve at all?  Well, believe it or not, that is the subject of some heated debate even now.  In my opinion, Bernoulli is given far more posthumous credit for total lifting force than he is due.  It suffices as an explanation of why the upper flow stays attached, or doesn't, depending on the airfoil's shape and angle of attack.

For our purposes, it suffices to say that wings and propeller blades do not get sucked up by differences in air pressure induced by curved surfaces opposed from flat ones.  If that were true, flat wings would produce no lift, and inverted flight would be impossible.  Nor is there any magical "temporal distortion field" effect at work here; the "flow arrival time" theory is absolutely ridiculous.  Serious analysis of lifting force is best done with the Navier-Stokes equations.  Here's a link to NASA's fine site:

NASA on Navier-Stokes

The only reason I told you that story is to be able to tell you this one; many, if not most, helicopter blades, regardless of scale, actually have the same curve on both the upper and lower surfaces.  In other words, their airfoil sections are symmetrical.  At zero pitch, and therefore zero angle of attack, they produce no lift whatsoever.  However, they do produce drag, as does anything moving through the air.  Symmetrical airfoils are almost twice as thick as their flat-bottom counterparts, and because of the greater frontal area they impose, their drag is greater, especially at lower Re.  This means that their lift-to-drag ratio will always be lower than a similar, asymmetric airfoil.  So, why are they used?  In some cases, particularly in models (fixed or rotary wing), the determining factor is inverted flight performance.  As you can imagine, flat-bottom sections often don't work well at negative attack angles.  But there is something else to consider in the case of the helicopter.  Remember the cyclic pitch changes required for thrust vectoring, and the concomitant control force mitigation required?

The overriding issue with helicopter blade design is minimizing pitch axis rotational forces.  They can be inertial moments, which is why the blade planform is long and narrow.  But there are also aerodynamic moments.  A helicopter airfoil must be chosen carefully so as to absolutely minimize any aerodynamic twisting effects which might place additional burdens on the cyclic control system.  Any attempts to give the blade section a higher lift-to-drag ratio might induce a pitch moment, which is a highly undesirable phenomenon.  It is simplest to just make the airfoil symmetrical, and deal with the lower lift-to-drag ratio - and obviously lower FM - by just increasing the power.  To be fair, there are some asymmetrical helicopter rotor blade designs in use, both full-scale and model-sized, which have better L/D ratios.  But the requirement to maintain a neutral pitch moment across all anticipated regimes of flight compromises the ultimately attainable FM by significantly limiting the airfoil selections available.  Virtually all of the highest L/D airfoils have pronounced negative pitch moments, precluding their use in cycled-pitch helicopters.

This is a limitation non-cycled-pitch VTOL aircraft simply do not have.  In fact, pitch moment can be virtually ignored.  Airfoil selection, twist (adjusting the AoA based on radius), and planform geometry can, for all practical purposes, be completely optimized without regard to radial axis forces.

In my next message, I shall compare some airfoil section data to reinforce this point.  The average full-scale production helicopter FM is 75, with Re's in the millions.  Dr. Pounds made a blade for a Re range of less than 100K and got an FM of 77!  What can be done with a blade operating in the Re range of 200K - 500K?  Could we see 90?  I'll provide some data to help you imagine.  :-)

I really intended to refer to Kutta-Joukowski, not Navier-Stokes.

Here's the the NASA link:

NASA on Kutta-Joukowski

Very interesting discussion.

Do you know about any measurements made on multicopters (tri, quad, hexa, etc.) regarding hover lift efficiency?

Where can be positioned   multicopters  on this diagram. 

(the diagram can be found  here )



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