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.

http://naca.central.cranfield.ac.uk/reports/1920/naca-tn-4.pdf

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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A year or so ago I built a PC with the idea that i wanted it to last, therefore, I built it to go as quick as I could.

It's got a bog standard i7 920, but i got lucky. With some pricey dominator RAM and full board water cooling I can run it at almost 100% overclock (it will run prime95 for hours at 4.85ghz!!) - and then theres the 35% clocked crossfire ATI5890's. When i first put it together the highest score in the world for the 3d mark benchtest was 29800 using a i7 920. My first run was 36400! Needless to say i was quite chuffed with that, given that it was mainly down to getting 'lucky' silicon in my i7.

Anyway, to my point, I'm very busy with work at the moment but if you'd like to make use of my hardware I'd be happy to do runs for you, the PC is on 24 hours as it also runs my security system, it flies through anything I throw at it, for example arduino can compile the Arducopter code in about a fifth of the time my new i3 laptop can :)

Way over in the upper left.  Typical disk loading is in the 1 to 3 pounds/square foot and hovering efficiencies are directly related - 5 to 15 pounds per horsepower.

Yup count me in! Work is very hectic atm but from May 1st I will have time on my hands.

quick one to add to the mix, was wondering if anyone had done some simulation around ducted vs non ducted for similar set of props ( i.e. i am thinking of adding protection to arducopter thus reducing risk of injuries and understanding impact on perfomance)

Most of the claims I've seen average around a 3% performance improvement if done correctly.  However, they'll likely add significant parasitic drag in forward flight.  I've seen shrouds that are basically just barrier rings around the circumference of the propeller disks, which should reduce the risk of accidental strikes.

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:

http://www.mh-aerotools.de/airfoils/javafoil.htm

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.

The highest L/D ratio this section can achieve is around 50 at 200K Re and 8 degrees AoA.

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.

The Clark-Y's highest L/D is 71.4 at 200K Re and an AoA of 6.5.

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.

Brad, so do I understand correctly, based on a summary of the information presented so far, that for a given (model) aircraft such as a helicopter or multi-rotor, the lift duration decreases rapidly as payload increases?  ie: it's not a 1:1 ratio.  For example, the first pound of payload requires X power to lift.  Reducing flight time a certain amount.  But then for every extra pound added, the power required would be something like (Y^2)X, where Y is some number, probably between 1 and 2.

So theoretically, if one tried to extend the range or hover duration of an aircraft by adding more and more batteries, at first you would observe that each additional unit of battery capacity you added, will not increase the flight time as much as the unit before.  And eventually, you could even cross a point where adding more batteries REDUCES flight time?!

Yes, this is quite true.  Increasing the disk loading will reduce the hovering efficiency of the system. 

But that's just the first of the "triple whammy" here with electric power.  You'll also be placing a higher load on your motor(s), which will have the effect of reducing their energy conversion efficiency as well.  In virtually any electrical system, I^2R losses are the dominant waste heat mechanism, and the more load you place on the motor, the lower its impedance will drop and the more current will flow through it. 

In addition, the actual capacity of the battery will drop, too, as the load current goes higher.  Known as "Peukert's Law", this basic concept applies to virtually all chemical-to-electrical energy storage media.  Not only that, but there will be an additional I^2R loss across the battery's own internal resistance, causing a heating effect which can also drop the battery capacity.

To go to the ludicrous extreme, it's certainly possible to add so much battery weight that the copter won't fly at all.  If you go to the other extreme and make the disk loading too light, the controllability and stability of the craft in turbulent air will be compromised.  That's where the engineering comes in - balancing the variables in such a way as to provide the intended utility.

I sort of wish you hadn't stuck this in on page one.  Almost missed it.

Great info.

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, I was ignoring the electrical side of this for now.

Interesting you mention that the disk loading has to be engineered for stability.  I've seen pro R/C AP pilots say that they prefer not to have too light of a disk loading.  For stability.

Does anyone here make their own propellers or rotors?

I made a prop from a piece of oak, and it seems pretty nice, but I haven't flown with it yet.  I wonder how much efficiency there is to be had with a fancy plastic design.  My prop is kind of like those wooden hand spinner toys, the ones you rub your palms together and they take off.

It's airfoil shaped with the leading tips rounded out, but doesn't have the twist and kind of "S" shape.

Speculate now and I'll post back when I get a chance to try it.

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...

This attempt to neutralize torque forces has a price, however, as the maximum L/D of this airfoil is around 40.

Airfoil shape data is courtesy of "World of Krauss":

World of Krauss Airfoil Database

There are some interesting airfoils in there. 

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