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.
Very interesting perspective and I'm looking forward to reading more on this..
Brad, very interesting thoughts indeed - especially in the case of ducted fans!
These days I was considering that if amperage is our "fuel capacity", than efficiency could be calculated based upon it. The common power/mass is a good expression, but how much air time can we achieve with the same amount of amperage? In a petrol world, a more km/l (mpg) way of thought, instead of displacement(cc)/power.
I would not go into equations with you, but I have faith in yours. :)
In my previous posting I tried to convey how important it is to have a large lifting area for a given VTOL aircraft weight. The next step is to take a look at some practical examples to show how much this can affect the amount of power required to produce lift. Just as a definition of terms, I tend to use "rotor" and "propeller" interchangeably, just as lift and thrust are the same if you're hovering in a VTOL aircraft.
A basic Newtonian physics analysis of a rotor blade thrust is known as "disk actuator" theory. Such a disk is modeled as merely a round, infinitely thin plane producing some change in velocity of an air column equivalent to its diameter. Using simple math we can easily calculate exactly how much power a "perfect" rotor disk of a given diameter would need to produce a certain amount of thrust in average density air. This "ideal power" is the theoretical measure upon which the actual aerodynamic performance of a rotor disk can be compared using a standard ratio called a Figure of Merit.
FM = ideal power/actual power
Remember that FM is really only applicable to efficiency if comparisons are made to similar disk loadings. In other words, FM is a relative expression of the quality of the aerodynamic design of the rotor blade system, and is not an indicator of efficiency of the entire aircraft. In the end, you have to look at both disk loading and FM to derive actual efficiency, although it's much easier to look directly at total watts of input divided by measured thrust output. Watts-per-pound is where the proverbial "rubber meets the road", so to speak.
@Marcos: You're absolutely on the right track, but don't forget to multiply your amps by the voltage under load to get watts. Battery manufacturers rely on amperage as a rating because in a certain family of cell chemistry, the voltage is considered a constant.
I know everybody hates math story problems, so I'll try to keep this short. How about an example close "to home"?
We have two quadcopters, each with a 4 pound total mass. One has 9" props and the other, 11" props. Because it's a quad, each prop has to produce 1 pound of thrust.
9" prop copter: square root of (1 pound/2 * 0.00238 * 0.442sqft.) = 21.8 feet/second ^-1 and times our thrust is 21.8 pound feet/sec ^-1. Dividing by 550 yields an ideal power of 0.0396 horsepower or * 746 = 29.54 watts.
11" prop copter: square root of (1 pound/2 * 0.00238 * 0.660sqft.) = 17.8 feet/second ^-1 and times our thrust is 17.8 pound feet/sec ^-1. Dividing by 550 yields an ideal power of 0.0324 horsepower or * 746 = 24.2 watts.
So, moving to an 11" prop from a 9" prop should cut your power consumption by 18%. Disk loading matters. It's the difference between 24 watts per pound and nearly 30 watts per pound in this example.
I know what you're thinking. I have to be making this stuff up Nobody, but nobody here has a quad that comes even close to consuming only 100 watts in hover (that's a 4C LiPoly battery delivering 133 watts to 75% efficient BLDC motors or a paltry 9.5 amps total current to all four motors combined).
I encourage you to check my math. I refer you to page 47 (2002 edition) of my favorite technical reference, J. Gordon Leishman's Principles of Helicopter Aerodynamics. He even shows a worked example for your edification:
It is my hope that you're starting to understand the importance of FM, and more generally, how dreadfully awful these model airplane propellers we use really are. But the fact is, they don't have to be so bad. More later.
My last message demonstrated the effects of disk loading on hovering efficiency and showed how ideal power is calculated. I also introduced the concept of the Figure of Merit (FM) as the ratio of Ideal Power to Real Power.
In the given example of the 11" prop quadcopter, the craft total ideal power is 24.2 X 4 or 96.8 watts at the propeller disks in hover. Don't forget that you're going to need at least a 30% greater capacity than that for control headroom, and the typical BLDC motor is only 75% efficient. But remember, too, that this is an ideal power - an absolute threshold that a propeller can never exceed. As a point of fact, the typical model airplane propeller is not very good at converting power to static (not moving forward, or hovering) thrust. Far from being ideal, they average in the 40 - 60 range, with some as low as 30, depending on the RPM.
A FM of 30 means for a calculated 100 watt ideal power, the motor(s) will really have to produce 333 watts to hover. If you add in the 75% efficiency of the motors and the 30% control head room ((333 * 1.3)/0.75), then your real-world battery needs the capability to supply 577 watts, or a non-trivial 41 amps from a 4S lithium polymer (3.5V/cell) pack. The ideal rotor ship needs only ((100 * 1.3)/0.75) or 12.4 amps from that same pack.
The difference is ALL in the rotors. And while disk loading matters, FM matters MORE.
Just as a frame of reference, the average FM for a full size conventional helicopter is around 75. Let's look at some real data of that 11 X 4.7 APC Slow-Flyer propeller that so many hobby shops sell as the default quadcopter part. These FMs are calculated from actual windtunnel data gathered from the University of Illinois. Note that the FM varies according to RPM (angular velocity):
RPM Ct Cp Calculated FM
1666 0.0971 0.0392 54.5875205
2018 0.0989 0.0391 56.2559251
2271 0.1007 0.0396 57.0689143
2556 0.1025 0.0401 57.8751349
2875 0.1043 0.0407 58.5305552
3144 0.1057 0.0411 59.1318205
3423 0.1074 0.0418 59.5498581
3728 0.1097 0.0426 60.3185555
3994 0.111 0.0432 60.54124
4290 0.1133 0.0442 61.0201381
4585 0.1155 0.0452 61.4165097
4853 0.1172 0.046 61.6856544
5175 0.119 0.0468 62.0333421
5450 0.121 0.0477 62.4036895
5710 0.1229 0.0486 62.6963337
6021 0.1239 0.0493 62.562001
Frankly, these FMs aren't bad at all. Also note that the FM keeps going up with RPM, until past the maximum RPM recommended by APC (5909). I suspect the fact that FM starts to dip at the final data point may be due to geometric distortion (i.e. don't try this at home).
From basic aerodynamics we know that lift increases with the square (^2) of velocity, but the power required to overcome the resulting drag increases with the cube (^3) of the velocity. This greater exponential influence of drag must catch up and eventually surpass the gains made in lift - marking a point of diminishing returns where the FM starts to go down with increasing RPM. Yet we do not see that here at all, even past the maximum RPM rating! Even so, while this propeller's performance looks good when compared with its model airplane peers, it still can't approach the average for full-size helicopters. Why is this so?
The answer is Reynolds Number, or Re. My next message will be about the effects of Re and how they can be addressed.
@Everyone: Thank you for your kind words of support.
The Reynolds number is an expression of the behavior of a fluid. Air is classified as a fluid for the purposes of dynamic analysis, but as a mixture of gases, it doesn't seem much like one. Gases literally do change their behavior depending on the scale of observation, starting out as a viscous fluid and, after a while, morphing into a collection of molecules with inertial tendencies. The Re number is a "dimensionless" expression for the ratio of viscosity to Newtonality (yes, I invented that word, but is seems to fit) in a given scenario.
If you want a more complete explanation, here's the Wiki link:
For aircraft, the Re varies according to the scale; the larger the scale, the larger the number. For regular air, the numbers are all rather large. The wing on a 747 will have a Re in the 10 million range, whereas the wing on a small balsa glider might only be around 10,000. This variation in ratio is important because it significantly changes the nature of drag forces. A viscous goop of cohesive molecules opposes anything that attempts to move through it, and the thicker the object, the more the viscosity opposes the motion. A Newtonian-dominated kinetic collection of particles merely wants to preserve inertia, and if the moving object is shaped properly, it can minimize the deflective impedance. It therefore follows that once the air is separated into the lower and upper flows over a wing surface, the longer you keep it flowing over the wing without creating turbulence, the more it starts to act kinetically, raising the Re. This is why the chord (or flow distance) is a factor in calculating the Re.
It follows, then, that really thin wings (and therefore, propeller blades) are best for low Re applications. Unfortunately, very thin propeller blades have other obvious issues. In a scale model hobbyist market, ideally-shaped low Re propellers would not look like their full size counterparts. Because they're thin, they would be more fragile, and they would be much better at slicing through almost anything with high viscosity, like your finger. :-) That's why high-performance, low Re propeller blades are very difficult to find (impossible).
Dr. Paul Pounds, now at Yale university, decided to make his own low Re quadcopter blades. Calculating that his X-4 Flyer would have a 75% blade radius (most stats on a propeller blade are given at the 3/4s of the length point) Re of about 70K, he quickly realized that there were no commercial options that would give him the efficiency required. He had his own carbon-fiber layup molds machined, and based his design on a low Re airfoil with a nearly-optimal taper and twist. The result of his effort yielded an astounding FM of 77!
However, there are issues with using thin airfoils, as I alluded to above. But there is big one unrelated to aesthetics or product liability litigation; the angle of attack range can be very limited. This means that there's a very narrow range of speeds (advance ratios) in which such blades can operate. To explain that, however, I need to introduce the whole concept of an "airfoil polar", which I shall do in my next message. Oh, and if you still believe that old dogma about lift being produced by the Bernoulli Principle sucking wings upward, prepare to get your proverbial paradigm shifted.
Here's a link to Dr. Pounds' original X-4 paper, including a picture of his custom blade:
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.
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:
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:
Here is some simulated airfoil data from Martin Hepperle's marvelous JavaFoil program. I've found it faithfully agrees with comparison data I've found, both from other simulations and actual testing. If you're into aerodynamics at all, you must visit this site regardless:
First let's look at a garden variety symmetrical airfoil section, the NACA 0012. The left plot below shows the coefficient of lift (Cl) plotted against the coefficient of drag (Cd) for 5 different Re's. On the right is a plot of Cl versus the angle of attack. Note that the drag is always greater at lower Re. Notice too that the Cl only goes so high before the Cd goes crazy; this is due to upper flow detachment (stall) because high AoA's try to make it turn that corner too fast. It also happens sooner at lower Re. However, because the upper surface and lower surface convergence angles at the trailing edge are the same and opposing with respect to the chord, there is no significant pitch moment. This is why symmetric airfoils are so often used in helicopter blades.
Now let's look at one of my favorite flat bottom airfoils, the venerable Clark-Y, which has been in use since the 1920's (and is probably guilty of spawning those erroneous theories of lift). If you're making your own blades, flat bottoms are easier to fabricate. However, the the convergence angles at the trailing edge begin to produce a pitching moment.
Do you want to see what's possible with a truly engineered high-lift airfoil? Take a look at this NACA 63-1211:
The L/D ratio of this little marvel is a whopping 171! (Re 200K, AoA 5) The relatively radical shape of the trailing edge is almost singularly responsible for this phenomenon (otherwise it looks almost the same as the Clark-Y). That same trailing edge would produce a pitch torque force mightily lethal if used in a cyclic helicopter blade (well, it would be a lot).
To be fair, I need to search out some recent asymmetrical helicopter blade airfoils and run them through JavaFoil. But rest assured, the best of them might approach the Clark-Y in L/D, and none could compare to the NACA 63-1211.
I sort of wish you hadn't stuck this in on page one. Almost missed it.
About pitching moments and helicopter blades. A clean-sheet helicopter blade designer can set the pivot point of the blade anywhere on the chord that he wants. Doesn't this solve the problem? Or is that pitching moment variable for different AoA?
Yes, this whole "thing" should have maybe been in the "blog" section, but who would have seen it there? :-P
The pitching axis location and the magnitude are different things, but I can't see the former diverging from the ultimate center of lift without other crazy things starting to happen.
Here's the whole airfoil design business explained in my usual, grossly oversimplified manner.
Thickness is a primary factor in drag, especially in lower Re. However, you need some thickness to support a larger-radius leading edge, which gives you a wider AoA range and more gentle stall characteristics. The upper surface contour is the most important shape because of the aforementioned necessity to "coax" the upper airflow to converge with the lower flow in a downward direction. Therefore, if you want to keep the section thin, the lower surface has to basically follow the contour of the upper, which is generally not an issue.
The key to pitch moment magnitude is the trailing edge geometry, and how the two flow vectors sum. Helicopter blade airfoil designers use a trick called "reflexing", which is really nothing more than giving the trailing edge a little upward curve to point the flow vector straight away from the center of the pitch axis.
Here's an example: the Hughes Aircraft HH-02...
There are some interesting airfoils in there.
I just meant it would be better to stick new material at the back of this thread, not the first page. ;)
I don't think this answered my question, or at least, not in simple enough terms that I understood it.
The pitching moment is created by a seperation between the pitch-axis, and the center of lift, yes?
The center of lift is fixed, yes?
Can't a heli blade designer simply match the pitch axis to the center of lift, resulting in zero pitch moment? I'm just not sure why this is an issue.