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 Propeller Dynamics Essential reading for model aircraft contest fliers. This is the only book on the market explaining propeller theory in non-mathematical terms. A rattling good read, I know, I wrote it. Read more...

## Analytical Design of Propellers - part 1

By Joe Supercool

Like the title? Disregard it. It is a lie. It is about the empirical design of propellers. Also it is really about how Supercool discovered the late E. Eugene Larrabee, and came to hate his equations for propeller analysis. “Analytic” means you start off with nothing, throw in lots of equations, and end up with something cool. Try reading a book on Fluid Dynamics. Talk about fanciful. Those guys don’t even get out of their armchairs. But I’m not sure that they are at all cool.

An example: I need to understand how airfoil “downwash” affects the “induced” angle of attack. After reading a dozen authors, I’m none the wiser. Some seem to say the opposite of the others. Some are on the dark side of the moon.

Once again, Supercool has to invent his own reality. Refer to “Analytic Design of Propellers: Part3. Induced angle of attack”.

Another example: I have spent quite literally a thousand hours over 20 years trying to get Larrabee’s equations to design a realistic propeller for model aeroplanes, and failed. It may well be that the induced angle-of-attack problem mentioned above may have been my Nemesis. Despite this, Larrabee remains my “Om mani padme hum”. (Jewel in the Lotus).

After all this failure, where do we go from here? Well, while writing a turbine code for my son, I was struck by a rare insight. Very rare. All my life I have admired the handful of genius’s who created modern Maths and Physics: never have I had a new or original thought. But at last I have at least had a derivative thought, and I plan to milk it until it bleeds!

By way of background, my propellers are CNC machined to aerodynamic known parameters of diameter, blade angle, and airfoil. Further, from Doppler analysis I know their performance in the air, by way of airspeed and RPM. This is pretty neat: all I don’t know about them is whether the design is optimal. Maybe a change in airfoil would make them better, or a change in blade angle? But what new value should I try? No idea. So some sort of theory is really necessary.

Now back to the insight. What do we have? Well, we’ve got a pretty good idea how fast the air is entering the propeller disc from in front, because we know the airspeed. We know how fast the air is entering the prop disc from the side (well tangentially, anyway), because we know the RPM. We also know the advance per revolution, sometimes called the “effective pitch”.

But we also know the geometric pitch, because we know the blade angle. The problem is, the effective pitch is less than the geometric pitch by some strange amount. This is due to the aforementioned “downwash”. If we regard this “downwash” as an actual velocity of the air, at right angles to the actual inflow to the airfoil, then we complete a vector diagram of airflow components with 3 sides and a resultant.

Now this is incredibly cool. Because the “interference factors” are just the axial and torque components of the down wash, we know these factors without using any fluid dynamics at all! You see; the prime problem in propeller vortex theory is to determine these interference factors. We know them already without lifting a finger!

All that is unknown to us is the angle the airfoil is actually making with the inflow of air to the blade. So we resort to an equation originating with Joukowski, which lets us calculate the airfoil lift coefficient for our particular flight condition.
Refer to “Analytic Design of Propellers: Part2. Determination of c1”.

Well, it would if we knew the airfoil angle of attack, and right now we don’t.

Suppose we guess a value of lift coefficient which might apply to our airfoil, and call this the “design” lift coefficient. This will vary along the blade due to compressibility, but that is secondary to our discussion here.

We now have 2 lift coefficients, one for our guessed angle of attack and one we want to have, as being reasonable, like C1 = .5 for Clark Y.

This is not too good, we only want one value, not two, and we want to know the angle of attack. Well, that is OK. Lift coefficient and angle of attack are dependent variables: know one, know the other. This relationship is called the “lift slope”, and we have a pretty good idea of what that is.

All we have to do is guess new values for angle of attack until the Cl we got from Joukowski is the same as the “design” Cl, and we are done!

Yes folks, given that we already know the propeller design and its performance, we can now predict its performance retrospectively! How is that for a once-in-a-lifetime revelation! You are cool, Supercool, very definitely!

Now what use can we make of all this? Can we show that the design Cl is the right one? What can we do to improve the propeller?

Well, we might want the propeller to have “minimum induced loss”, whatever that is. Sounds good, so lets have some of that. Well, we have “minimum induced loss” when the axial component of the slipstream velocity (which makes thrust) is uniform across the prop disc.

This point can be understood if we remember that thrust results from the momentum change of the air passing through the prop disc, this thrust being directly proportional to induced velocity. By contrast, the energy required to produce this induced velocity is proportional to the square of that velocity, so it comes at some cost. If there is some annulus in the slipstream with a higher induced velocity, then more energy is required, which is thus disproportionate to the other annuli. On average over the whole prop disc, more energy is required to produce the same thrust, and hence efficiency is lost .

Returning to our problem: how can we make the axial slipstream uniform over the prop disc? Our good friend Joukowski again provides a clue. He reckons that the downwash velocity depends on the blade chord. So all we have to do is vary the blade chord from root tip until all the slipstream annuli have the same induced axial velocity. As Biggles would say: piece of cake!!

By now, our original prop design is lost in the noise. We have a new prop design. Regrettably, it does not match our original, as we would have hoped. Something in our physical model of the propeller has gone astray. The chord is wrong, the pitch is wrong, we have new blade angles.

There are important factors we have neglected. However, we may use our method to find them.

Prime among these deviant factors is the axial inflow to the propeller. The speed of the plane is not necessarily the inflow velocity seen by the prop. The spinner, and the presence of the fuselage/wing behind the propeller disc all modify the axial inflow velocity. The net effect is to reduce the axial inflow velocity, with the greatest reduction occurring at the root and the least at the tip. This reduction may be of the order of 20% at the root, dependent on configuration.

In our calculation, we must introduce this variation to our vector diagram.
Since the prop needs less pitch to match the reduced inflow velocity, our original calculation of pitch will have been too high. Matching the blade angles in this way provides a means of estimating the reduction in inflow velocity.

Care is needed, as there are other variables. The change in lift with angle of attack (the lift slope) is also important. Propellers nearly always end up running high Mach numbers near the tips. In models, anything from .6 to .9 is quite common. The effect of compressibility on the lift slope cannot be ignored. Fortunately this change can be estimated by using the Glauert-Prandtl rule, which is very useful for our purposes.

The theoretical value of the lift slope is 0.1 per degree for thin airfoils. However, variations from this value can be quite large, depending on the shape of the airfoil section, in particular, the leading edge. Sharp leading edges can give lift slopes of .12 or more for small angles of attack.

The drag of the airfoil section depends on the airfoil thickness and Mach number, although this is not such a critical factor as the foregoing effects.

The airfoil section itself is tricky to choose. There is not much in the way of high-Mach low-Reynolds number data. The safest approach is to use the thinnest airfoils your materials permit, especially at the tips. Also avoid blunt leading edges. The NACA 4-digit sections tend to be too blunt. I have had success modifying the upper surface forward of x/c = .3 with a simple circular curve.

More interesting is a comment by Weick. He experimented with circular sections, flat-bottomed model-sized sections (24”) in the wing tunnel and found them to be superior to conventional sections. This plus the whirling arm test results of Suzuki lead me to try these sections, and so far they have been very good!

Shock wave formation for M>.85 is a concern. About all you can do is use thin, sharp, flat-bottomed airfoils with the high point back past x/c = .5

The rest is technicalities. I attach the Quick Basic source code, which is heavily commented. Good luck with it!