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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...

Transonic Airfoils for Propellers

This months hot topic is propeller tips, literally. Normally the shape and airfoil of propeller tips is not of much interest, as they don't much affect performance. However, in some racing classes the propeller tip speeds approach that of sound. This introduces a whole new realm of aerodynamics and the profile of the airfoil section becomes very critical indeed. These classes include F2A and F3D, with F2A the most critical of all.

There is one piece of jargon that cannot be avoided, that being the Mach number, denoted by M. If we have a situation where the speed is 340 m/s, then the Mach number is defined as 1. Since this is the speed of sound, an aeroplane flying at this speed is doing Mach 1. At half this speed, say 170 m/s, the Mach number is, by simple proportion, Mach .5.

To get anywhere, we need to be able to calculate the propeller tip Mach number. Firstly we need the speed of sound, Mo, which is a function of air temperature alone. With the air temperature denoted by T in degrees Centigrade:

Mo = .594 * T + 325.56 , the units being metres/second.

The propeller tip speed Vr due to rotation alone is given by:

Vr = .00010472 * RPM * R m/s

Here, RPM is revs per minute and R is the propeller radius measured in millimetres. When the airplane is at full speed V (m/s), the speed of the air over the propeller tip is increased above that due to rotation alone. The "helical" tip speed in flight Vtip is obtained by adding the airspeed V to the rotation tip speed Vr using the rule of vector addition, viz:

Vtip = SQR( Vr ^ 2 + V ^ 2 )

where SQR means square root and ^ 2 means squared.
The tip Mach number is then just:

M = Vtip / Mo

, with no units at all, ie, M is non-dimensional. Consider an example from F2A, with a 75mm radius propeller of unmentionable origin. Lets say you do 300 k/hr (83.333 m/s) at 38000 RPM on a nice sunny day in WA, when T = 40 degrees Centigrade.

Then:

Mo = .594 * 40 +325.56
= 349.3 m/s

Also,

Vr = .00010472 * 38000 * 75
= 298.452 m/s

and

Vtip = SQR( 298.452 ^ 2 + 83.333 ^ 2 )
= 309.87 m/s

Finally,

M = 309.87 / 349.3
= .887

The question now is whether a propeller tip speed of M = .887 is something to give us pause. If the Mach number was less than M = .7, we could just forget it, as with most useful airfoils the performance is OK. But above M = .7, awful things start to happen. For a start, the noise produced by the airfoil at the tip starts to rise very rapidly. But much worse, the lift may fall, also very rapidly. If the airfoil thickness-to-chord ratio (t/c) is above 15%, then at M = .887 the lift of the tip airfoil can actually be negative. That is, the tip is actually pushing backwards ! This is weird, demanding explanation, at least if you want to go fast. By reducing t/c to something like 7%, this problem is overcome, the tip airfoil again providing satisfactory lift.

There is something about t/c that is important at high Mach numbers. You cannot use wing type airfoils for high Mach number propeller tips and expect to get good results. Clark Y, for example, is no good unless thinned right out.
The aerodynamic problem is that, above M = .7, shock waves start to form and become stronger as speed increases further. Accompanying the shock wave are increased drag and reduced lift, due to turbulent airflow behind the shock. You may have seen these shocks yourself, when travelling in a jet airliner. Modern jets have best range when flying at a speed which produces a mild shock wave on the wing surface.

If you look out the window, you should be able to see a shadow on the wing surface, just like a thin line reaching from root toward the tip for a few metres. The line may be a centimetre wide, and be moving erratically fore-and-aft a few centimetres as turbulence affects the velocity over the wing. The shock exists at a point where the airflow reduces below the speed of sound. Forward of the shock, the airflow has high velocity and low pressure, while to the rear of the shock the airflow has reduced velocity and increased pressure. The shock wave itself is a thin planar surface, less then .1 mm thick.

 Flow pattern around an airfoil section at low speed. No shockwave present.

 Flow pattern at  high speed, about M=.7. Air speeding over the curvature of the wing has exceeded M=1  in front of the shockwave, and drops below M=1 behind the shockwave. Also shown in blue are reduced pressure regions   at the leading edge, and a yellow region at the leading edge showing compression at the stagnation point.

 At still higher speeds, a shock wave also forms on the lower surface. Again, the flow in front of the shock is supersonic,   and behind, subsonic.

 Even faster, and the shockwaves have moved rearward. Lift is lost due to shock-induced flow separation on the upper  surface near the trailing edge.

The point is, that once shock waves form they dominate the characteristics of the airfoil. Most airliners have very thick wing roots, to provide strength and somewhere to store the undercarriage: you would expect shock waves to form on these thick wing roots, as the air must be speeded up considerably to get over them, and hence be solidly supersonic. However, the designers dodge this in 2 ways.

Firstly, they add an extension to the trailing edge at the wing root, so that the inboard wing planform is almost a delta: there may be little or no sweep on the inboard trailing edge. This trick reduces the thickness-to-chord ratio, thereby limiting the airflow velocity increase over the roots, and avoiding the shocks.
Secondly, they build the airfoil upside down. No joke, look for yourself next time you're at the airport. We noted above that that some airfoils lift downward at high Mach numbers: it only stands to reason that they will lift upwards if you turn them upside down !

Designers only started doing this in the early 1970's, even though data were available in the 1950's which suggested this course of action. To cover their a...s's, they called the new type wings "supercritical wings" and the airfoils "supercritical sections".

There is of course nothing "critical" about them at all. This word arises from the "critical Mach number", which is the Mach number at which the shock wave starts to form on a given airfoil. The associated rapid change in characteristics is called "force divergence".

To be fair, there is some additional sculpting of the section to spread the lift force over most of the chord, and to delay the upper surface shock so that it occurs at the same speed as the lower surface shock. This raises the critical Mach number even higher, yielding more speed and better range .
We must now return to propeller sections. In F2A and F3D, the tip speeds routinely exceed the critical Mach number of conventional lifting sections, often by a considerable margin. This is disastrous. Propeller efficiency falls hopelessly: all that engine power you worked so hard to get is wasted just by overcoming the tip drag caused by shock waves.

 Comparison of "thick" and "thin" airfoil sections at the same Mach number. The thick airfoil forms shockwaves before the thin section.  Thus thin section have lower drag at high speed. The thin section is said to have a "higher critical Mach number"  In crossing the shock wave, the air becomes heated, and its pressure increases. The energy lost in this manner is not available to produce lift   and is thus wasted. This means that a propeller tip running at high Mach number wastes a lot of energy that would otherwise go into making thrust.

 When the angle of attack is increased, shock waves occur earlier. Thus at high Mach numbers the amount of lift available  from the airfoil is limited.

It is the fashion these days to rake the blade tip over the last few millimetres into a scimitar shape. This certainly raises the critical Mach number,in just the same way as does sweepback: but it does little for the lift, or, more importantly, for the lift-to-drag ratio. The problem is that the resultant very narrow tips have lower Reynolds numbers, producing poor flow characteristics that reinforce the poor high-Mach airfoil performance.

You just can't beat wide chords for good airfoil performance.
So where does this all leave us? For M > .7, you need:

1. Squared off tip planforms
2. Supercritical airfoil sections

In Britain, in the early 70's, the Aeronautical Research Association produced a new family of airfoils specifically for propeller use, and called them ARA-D. Likewise, in the early 80's, Grumman Aerospace, using advanced computational aerodynamics, developed their M series sections. The ARA-D sections are a single parameter family, based solely on t/c. Since, as we have seen earlier, t/c must be chosen to suit the Mach number, it follows that the ARA-D family depend also on Mach number. See the attached drawings.

Insofar as propeller tip airfoils are concerned, the ARA-D section for M = .95 is very thin (3% t/c) and highly cambered (5%). The leading edge is well rounded and the trailing edge cut off-square. Compared to most model use, this is radical. The camber high point is well forward at 10% for low Mach, moving back to 30% for high Mach.

Empirical model propeller tip sections are thin (6%), have low camber (are symmetrical) with sharp leading and trailing edges. The high point is commonly at 40%, irrespective of Mach. It is probable that these sections are quite inappropriate for F2A and F3D.
The problems are:

1. Low camber sections have low lift to drag ratios. That is fine for a wing in a dive, but no good for a propeller section which must always produce high lift.
2. Sharp leading edges promote flow breakaway with rapid changes in angle of attack. During a single rotation of a propeller, angle of attack changes occur rapidly due to inflow variations caused by the presence of the airframe behind the propeller, manoeuvering and engine induced vibration.
3. Sharp trailing edges do not enhance flow re-attachment at high Mach numbers.
4. Rearward high points produce lower maximum lift.

The ARA-D sections overcome these objections. In addition, they delay shock formation, featuring high lift-to-drag ratios at high mach numbers.
Model-wise, they are difficult to reproduce, and have high variation in zero-lift angle, requiring greater care in setting the pitch angles. These latter objections are overcome only by using sophisticated CAD/CAM production methods. Despite this, it pays to make the section as thin as possible and maintain some camber.

However, it is quite likely that significant performance gains may be had from use of sections. Chances are, what you'll see is essentially a cambered flat plate, similar to the airfoils shown here.

Finally, you may wonder how we do as well as we do if all the foregoing is true. There is end of the blade, it is affected by the 3-dimensional flow associated with the tip vortex. This type of flow delays the formation of the shock wave. As a rule of thumb, airfoils at propeller tips think the tip speed is .05 Mach less than it actually is. Thus, if we computed above that M = .887, then the tip airfoil thinks it sees M = .837, which is a bit better situation.

Stuart,

Interesting article. My comments are the following:

1) I haven't seen an airfoil that produces negative lift at positive angle of attack. As Mach number is increased, the lift-curve slope (lift vs angle-of-attack) actually increases up to the critical Mach number, then drops off. After Mcr, all sorts of weird things happen that are highly dependant on the specific airfoil, so I wouldn't discount the idea entirely.

But even though lift increases with M, drag goes up a LOT (thus lower L/D).

2) Thick airfoil sections can reach Mcr even at M=.6, or below. We design our airfoils to operate at a Maximum of Cl=.6 or .7 because of the lower Reynolds numbers. At low Rn, max. lift decreases; airfoils that stall at normal Rn at CL=1.3 stall instead around 0.8 at low Rn.

3) Airlines utilize wing sweep to delay Mcr. In fact, the l.e. of the root sections sometimes fair into the fuselage such that the sweep is higher than the main wing. Root sections have always been a pain-in-the-a*s for aerodynamicists. Low wing aircraft require fancy work at the root to make up for their inefficiencies - mainly the effect of the fuselage boundary layer on the top of the wing. High wings are much better in this respect. But in terms of shock waves, this is exactly the reason that Witcomb used the 'area rule' - to decrease the shock wave interaction.

4) Maybe I'm too conservative, but its probably best to avoid M>.7 entirely instead of trying to design around it. Lower the rpm (if possible), and increase the blade area.