...Several authorities have mentioned that props with diameter to pitch ratios less than 2 are subject to pitch stall making take offs difficult but don't effect flying performance (e.g. Motocalc)

From : Don Stackhouse

This is another rule of thumb. The phenomenon of blade stall does exist, but whether or not it occurs at diameter to pitch ratios of less than 2:1 is a gross oversimplification. There are a great many factors involved, not just that ratio.

Could you illuminate what is pitch stall (and how does the D/P ratio determine it). , and why is it not a problem when flying?

The angle of attack of a particular location on a blade is related to the local pitch angle, but most definitely not equal to it. To determine the local angle of attack, we first have to determine the direction of the local relative wind at that spot on the blade. This involves a bit of vector addition.

For those of you whose eyes glaze over when you hear the word "vector", it's really not that difficult. In engineering there are essentially two kinds of measurements: "scalars" and "vectors". Scalars are just an amount, such as volume or mass or surface area. Vectors have both an amount and a direction, like acceleration, force, or (in this case) velocity.

We represent vectors in graphical analysis (i.e.: solving math problems with drawings instead of equations) with arrows. The direction of the arrow corresponds to the direction of the vector, and the length of the arrow corresponds to the amount, according to whatever convenient scale we chose.

For example, we might decide that one inch equals 200 feet per second, so we would represent an upward velocity of 500 feet per second as an arrow pointing upwards in our drawing, with a length of (500/200), or 2.5 inches long.

We add a bunch of vectors together by joining them up nose-to-tail while keeping each of them pointed in its original direction. If we then draw an arrow from the beginning of the first vector to the end of the last vector in the chain, that new arrow's direction and length is equal to the "vector sum" of all the vectors in the chain. We call this result of adding the group of vectors the "resultant" (I know, not a very creative name, but it gets the idea across, and yes, that really is what engineers call it).

Now let's look at that question of the relative wind at a particular place on the propeller blade. There are two main categories of velocities that go into determining the speed and angle of the relative wind at that section of the blade: the velocity due to the prop's rotation (which we call the tangential velocity since it is tangential to the circular path of that blade section as it goes around the prop disk) and the inflow velocities due to the forward motion of the airplane plus the acceleration the prop imparts to the air going through it in the process of making thrust.

The tangential velocity is easy. If we know the radius of that segment of the blade from the center, and the prop's rpm (actually rps, or revolutions per second, works better for this calculation), we just take the radius times 2 times that weird little number Pi (3.141592...whatever; "Pi" is the ratio between any circle's diameter and its circumference) and we have the circumference of that blade section's path around the prop disk. Since we generally work in feet per second on this side of the pond, we want to use the radius in feet, so we get a circumference in feet. Next we multiply by revolutions per second and we get the tangential velocity (the velocity parallel to the plane of the prop disk) in feet per second.

Note that the tangential velocity varies linearly along the blade, from a value of zero at the propshaft center to a maximum at the blade tip. As we will see later, this variation is responsible for most (but NOT all) of the requirement for twist in the blade.

The vectors that define the relative wind for any given spanwise location on the blade form the legs of a right triangle. The tangential velocity is one of those legs. The hypotenuse of the triangle is the resultant, the relative wind we're trying to find. What we still need to determine the angle and length of that hypotenuse is the length of the other leg. However, it is composed of at least two other vectors, and determining some of them precisely will be extremely difficult.

The first piece of that other leg of the triangle is the forward velocity of the plane. If that was all there is to it, designing a prop would be easy. This might be altered locally by blockage effects from engine cowlings and blunt fuselage noses or windshields (typically the full-scale single-engined general aviation aircraft are among the worse in this regard), or increased by the flow accelerating around the shape of the fuselage or nacelle. Also, blockage and skin friction can make the inflow velocity near the hub slower than further out on the hub. Distortions in the flow due to nearby objects (such as the effects of a nearby fuselage and wing on a typical twin-engined aircraft's nacelle) also need to be considered, and the effects of those might vary in different parts of the prop's rotation. Pushers are often among the worst in these regards, with large local distortions in the inflow's direction as well as its speed. However, calculating all of this is still a fairly straightforward task in most cases.

The other component is the tricky one. Props make thrust by accelerating air, so we need to increase the predicted inflow velocity (the other leg of the triangle) to account for this extra "induced inflow" due to the acceleration. However, this acceleration depends on how much power the prop is absorbing and how that power is distributed along the blade. Unfortunately, in order to figure that we need to know the lift coefficient at each location along the blade, which means we need the local angles of attack, and to calculate them we need the local inflow velocities. It's a "Catch 22"; without the local angles of attack we can't calculate the total inflow velocity, and without the local inflow velocities we can't calculate the local angles of attack.

There are some methods that take a simplified view of this issue and come up with a reasonably good answer, such as the one on Martin Hepperle's website: http://www.mh-aerotools.de/airfoils/index.html

However, if you want to do a really thorough analysis without the simplifying assumptions, it normally requires an iterative approach. The computer starts with an assumed load distribution on the blade, calculates the inflow, power absorbed, etc., checks to see if the results are consistent with the inputs (which they almost certainly will not be), makes a new guess at the input conditions, reruns the analysis and rechecks the results against the inputs, etc., until the solution "converges" on an answer that makes sense.

One of the parameters that comes out of this analysis is the efficiency, and another is the thrust. This lays the foundation then for the next step - optimization. The computer can make small changes in the blade's airfoil thicknesses, cambers, chord distribution or (most important of all) the twist distribution, then re-run the analysis and check if the efficiency got better or worse. This process can continue until the computer finds the best combination of the input parameters to achieve the highest efficiency for that particular flight condition (i.e.: combination of power, rpm, airspeed and altitude, using that airplane's airframe characteristics). This tends to be an interactive process, since the computer left to its own decisions tends to paint itself into mental corners during this process, coming up with supposedly optimum designs that are not entirely valid in the real world for various reasons. There is still some "art" involved here, even with today's technology; computers cannot replace humans quite yet. They are great at chewing through mountains of numbers, but humans are still far better at making final decisions than the computers are.

The optimization typically is performed for a variety of different flight conditions, such as takeoff, climb at various altitudes, high speed cruise, economy cruise, perhaps some aerobatic flight conditions where appropriate, etc.. Typically the prop that performs best at one condition will not be the best at the other conditions. The real skill in prop design comes in juggling these different solutions to come up with a final prop design that comes as close to optimum at the highest priority flight condition, while giving up as little performance as possible at the other flight conditions. It's not easy to do, and it's something that still requires quite a bit of experience to do well.

Going on at the same time as all of this is the structural analysis of these different iterations of the blade design, making sure that their strength is sufficient everywhere on the blade, and (usually far more important) that the vibrational characteristics are properly tuned for the application, so that there are no resonant frequencies that coincide with an operating rpm. Normally that means making the blade stiff enough that its natural frequencies all fall comfortably above the normal operating range.

However, props with four or more blades, or props with any number of very long, flexible blades, also has "reactionless mode" frequencies to consider. These are where the vibrations in the blade tend to cancel each other's reaction forces against the hub, so that even though the blades could be flogging each other literally to death out there, no vibration from this will be felt in the airframe. The tricky thing is that the reactionless mode frequencies tend to be very low compared to the other resonances. If you stiffen the blade to drive the other frequencies up above the normal operating rpms, you might also raise the reactionless mode frequencies enough to pull them up INTO the normal operating range. The usual technique is to stiffen the blade just enough to put the reactionless frequencies just below the operating range and the other frequencies just above it, but achieving that in actual practice can be very difficult.

The thing that truly amazes me is how the old timers back in WW II and earlier, before the days of the calculator and computer, would do all of this with just pencil, paper and slide rule. Of course they did not optimize to nearly the degree we do today, and even the basic analysis tended to be very coarse, just to keep the number of sections along the blade involved in the analysis down to a manageable number.

OK, so now you know far more about the prop design process than you ever wanted to, and your brain still hurts from all those pointy little vector arrows it just had to digest. What does all of this have to do with Ted's question about blade stall?

If we know the diameter and rpm (so we can calculate the tangential velocities along the blade) and the inflow and induced velocities (obviously from the above discussion that's a really big "IF"), the vector sum of the local velocities is the relative wind coming at the blade at any given point along the span of the blade. The angle between the local relative wind and the local blade pitch angle is the angle of attack of the blade at that location. From that we can determine the local lift coefficient. If the local angle of attack is too high, then we can also see that this location on the blade is stalled.

Near the hub, the radius is low, so the tangential velocity is low. The inboard portions of the blade have trouble making very much thrust efficiently, so on a well-designed prop the induced velocities on the inboard portions of the blade tend to be low, but the inflow velocities due to the plane's forward motion are still about the same as everywhere else on the blade. As a result, the triangle formed with the tangential velocity to determine the local relative wind tends to be a tall, narrow one, and the required pitch angle therefore tends to be high. Further out on the blade, the tangential velocities are greater but the inflow stays about the same, so the required blade angle is lower. This is why the blade has twist. In addition, we need to consider what distribution of thrust along the blade results in the highest efficiency, and therefore what angles of attack we need to add to the angles of the local relative winds to get the correct local lift coefficients.

Now, let's say that we have a blade designed for very efficient cruise. On a piston engine, that generally means going for low rpm and max torque, which requires high pitch angles all along the blade. If the airplane is relatively fast, the pitch is even higher. Now, if we consider what happens to that cruise prop at takeoff conditions, we will see that the combination of high pitch and low forward speed results in relatively high angles of attack on the blade. While it is true that the high power at takeoff will cause more induced velocity, which will tend to reduce the angles of attack somewhat, there is only so much that this effect can overcome. If the pitch angles are too high for the flight condition, the blade will have some stalled zones on it.

In addition, the cruise-optimized high pitch angles will put a lot of load on the engine. This may keep the engine from turning up to its full rated rpm, even if the blades don't stall. Since power is proportional to torque times rpm, and there is only so much torque that an engine can make, the result of less than full rated rpm is a loss of power.

I heard about a case of this when I was working on the Lear Fan project. The Lear Fan used a single propeller driven by two motors driving into a combining gearbox. Because they still wanted to certify the plane as a multi-engined airplane, they had to accept some compromises in the propeller design. On an ordinary twin, if there's a failure of almost anything in one of the powerplants, you just feather that one and fly to the nearest airport with the remaining engine and prop.

However, if there is only one prop, you have to be careful to minimize any failures of the prop and gearbox, since your supposedly twin-engined airplane has only one of each of those. In the case of the Lear Fan, they had no beta system, and a hard stop in the prop hub's pitch control mechanism for the flight idle blade angle. In addition, to protect from a governor failure in cruise, they set the flight idle blade angle high enough that the rpm and drag in cruise would stay low enough to at least make sure the blades stayed attached to the hub if the governor failed and put the prop on the flight idle stop. This was a much higher blade angle than the normal criteria used in most twins of windmilling drag on final approach would dictate.

They did most of the testing at Reno, where the altitude was about 5000' msl (i.e.: the altitude measured from "Mean Sea Level", the level of the oceans after averaging out the tides). Props make thrust by accelerating air aft, and if the mass of the air is lower because of high altitude, a greater volume of it must be accelerated to add up to the same mass and therefore absorb the same horsepower. Thus, the prop pitch is related to true air speed, not the indicated air speed. The tangential velocity is still related to the rpm and diameter, but both the inflow and induced velocities are greater because of the thin air, therefore calling for a higher pitch. At the high altitude such as at Reno, the cruise-dictated high pitch of the Lear Fan's prop wasn't too terribly mismatched with the takeoff requirements, and so the takeoff performance wasn't too badly compromised.

However, when they took it down to Santa Barbara for some sea-level performance tests, the thicker air caused problems. The prop's excessively high pitch may have caused some stalling because of excessive angles of attack along the blade. More importantly, the high load on the engines held the rpm down to much less than 100% during the beginning of the takeoff run. Power is related to torque times rpm, the torque was limited to the max the engines were rated for, and therefore the reduced rpm also meant reduced power during the first part of takeoff, until the plane accelerated to a speed where the prop could develop max rpm. It was exactly as if they were taking off with only partial throttle. The required takeoff run was over twice as long. The Lear Fan is one of the few planes I know of where the sea level takeoff distance was greater than the high altitude takeoff distance.

I saw another example of something similar, and in this case I'm pretty sure it was a case of blade stall. There is a truly delightful little 4-seat low-wing light aircraft called the Grumman-American "Cheetah". It has a 145 horspower engine, nearly the same engine as its close competitors, the Piper Cherokee Warrior and the Cessna 172. However, the Cheetah is a very clean airplane with a laminar-flow wing and a cruise prop (at least on the one that I flew), and it cruises WAAYYY faster than its two competitors.

A good deal of that faster speed (ridiculously fast for a 145 horsepower four seat, fixed-gear, fixed-pitch prop airplane) is thanks to the cruise prop with its higher than normal pitch. This makes for some interesting takeoff characteristics, rather like a Republic F-105, especially when taking off on a grass runway. Open the throttle on a C-172 or a Cherokee and you get a reasonably brisk acceleration leading to a reasonably quick liftoff. However, when you open the throttle on a Cheetah, not much happens except for a big increase in the noise level. The pitch is too high at that low airspeed, and the blades are stalled from the excessive angle of attack. The wheels begin to roll very slowly, the airplane starts to gain speed, but appalling amounts of real estate are going by at an alarming rate and still the plane is nowhere near ready to transition from wheels to wings. Finally, about halfway through the takeoff run, the inflow velocity has increased enough due to the forward speed of the plane that the local angles of attack decrease to reasonable values, the prop begins to unstall and the acceleration improves quite noticeably. This isn't just some theoretical parameter, you can really feel the plane's acceleration improve.

From that point on the takeoff performance is comparable to its two competitors, and the climb performance is excellent. Once it gets to altitude it takes a long time to accelerate from climb speed to cruise speed (one of the side effects of getting very high cruise speeds through aerodynamic finesse instead of through brute horsepower), but that cruise speed is about 15 to 20 knots faster than its competition. One common trick with this airplane is to climb about 100-200 feet above your intended cruise altitude, then leave climb power on while descending back to your cruise altitude. This obviously accelerates the plane to cruise speed much more quickly.

Cruise props are wonderful things if used properly, but you have to be prepared to deal with the compromises, such as slow acceleration in the first half of the takeoff run. The airport I flew this plane from had a 3100' grass runway, with telephone lines on poles at the east end. We did not do touch and goes with the Cheetah when taking off to the east unless there was a lot of wind. If you got too slow during the "touch" part, you might get into that stalled-blade realm and get an extended "go" portion of the maneuver, with the possibility of having to do some experiments right after liftoff regarding the "conductivity" of airframes for telephone wires.

As far as why blade stall might be related to the ratio of diameter to pitch, the diameter largely determines the rpm (with very few exceptions such as the Russian "Bear" turboprop, we are careful to keep the tips of the blades' local relative winds comfortably subsonic throughout the entire operating envelope), and therefore the tangential velocity component. The sum of the induced and inflow velocities determines the other leg of the triangle, and the angle that forms with the tangential velocity, subtracted from the pitch angle at that location on the blade, gives you the local angle of attack. If the pitch angle is large, and the inflow velocity is small (such as during the beginning of the takeoff run), then the angle of attack is likely to be large. However, the thickness, camber and general airfoil shape and its aerodynamic characteristics at that location will help determine whether the airfoil is stalled or not at that angle. A thick, highly cambered section with a well rounded leading edge might stay unstalled at that angle, while a thin airfoil section with a sharper leading edge might stall. The Reynolds numbers of the various locations along the blade also influence this, which means the blade planform is an issue. In addition, the characteristics of a stall on a propeller blade are not the same as on a wing (the flow tends to move outboard and reattach, so the effective lift coefficient tends to go constant above stall, rather than decreasing like it does on a wing). Also, it isn't a question of whether the blade is stalled or not, it's a question of how much of the blade is stalled and what effect that has on performance. It could be a small area with little effect on thrust and efficiency, or it could be nearly the entire blade.

In any case, there are enough other factors involved that I personally don't put much stock in that 2:1 rule of thumb. If there's one thing I've learned about props, it's that each individual case must be evaluated on its own merits.

Don Stackhouse
DJ Aerotech