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## Transonic airfoils Part 2: Dance of the shock waves

By Joe Supercool

Well folks, hope you have read Part 1. Just in case you have not, here is the story so far.

Propeller tip speeds can easily exceed Mach 0.7. When they do, the air flowing over them does weird things. I mean even weirder than below Mach 0.7, so that is pretty weird indeed. The main weird thing that happens above Mach 0.7 is that shock waves start to form on the airfoil upper surface: at even higher speeds they appear also on the lower surface.

These phenomena give rise to something that appears to be complete nonsense. The airfoils actually work better if they are upside down. No joke, they really do, and props I have made with this feature (negative camber) actually do work better.

Now it turns out that while turning the airfoil upside down is good, we can do better. The extra improvement is achieved by modifying the way the shock waves form. So first we had better observe these weird waves a bit more closely.

Above, a biconvex airfoil is operating below Mach 0.7, at a low angle of attack. The blue region at the leading edge shows low pressure, and yellow is high pressure. Make a mental note now of the yellow region below the trailing edge. We are later going to try and enhance the size of this region in an effort to generate extra lift when the flow goes transonic: such an enhancement we will call “rear loading”.

Now let us look at the same airfoil (below), this time with the flow strongly in the transonic region. This time a strong shock wave has formed. Behind this wave, the flow has suddenly slowed below Mach 1: the pressure and temperature have increased suddenly and the flow may be turbulent. If you have ever lap-swum in the turbulent wake of another swimmer, then you will know that this air is not much use for making lift.

At the leading edge, the blue areas again indicate low pressure. This low pressure arises as the air from the stagnation point (in yellow, at the leading edge) increases in velocity as it wraps around the leading edge. Unfortunately, it then goes supersonic from there all the way to the shock wave, which means there is little or no lift produced in that region.

Things are not looking good for this airfoil, but they can get worse! Below, the flow has speeded up some more: the flow on the undersurface has also gone supersonic, with the result that we now have a lower surface shock as well! As far as using this airfoil for a propeller tip, we might as well give up now and go home.

But just for the moment, lets assume this airfoil was giving some useful lift. Then we might want to run it at a higher angle of attack. Let’s see what that does.

In the figure below, we can compare the shock formation at low and high angles of attack. Unfortunately, all we have done by increasing the angle of attack is make the shock that much stronger. There is a possible gain in lift from the leading edge low pressure, but the drag rise from the strong shock we could do without.

This is all getting rather depressing. Is there nothing we can do to fix this horrid set of conditions? Well, lets try a thinner airfoil and see if that helps.

Here (above) we have 2 airfoils, one thick and one thin. Or, as they say in New Zealand, land of the lost vowels, one thuck and one thun. At a Mach number less than 0.7, they seem to be behaving. Now we will speed up the flow to the transonic region.

Perhaps that was not a good idea. The thick airfoil has gone berserk, while the thin airfoil seems to be unaffected by the speed increase. Aha, perhaps this is why Abbott and Von Doenhoff conclude that transonic airfoils should be thin and symmetrical!

Or maybe not, how would I know what other people think. But perhaps we can say something along these lines below.

Shock waves form on the upper surface because the speed of the air over that surface exceeds Mach 1. If we could slow that upper-surface air down a bit, we could reduce the strength of the shock. As seen on the thin airfoil, reducing the curvature of the upper surface (thereby reducing the speed of the air) reduces the strength of the shock.

Make a mental note now. We are going to make an airfoil that is nearly flat on top. That is what happened in Part 1 when we turned NACA 3409 upside down to make the eggfoil -3409.

OK. Now where are we? Well, we’ve had a look at some shock waves and how they affect the transonic region between Mach 0.7 and Mach 1.

Also we have made 2 mental notes. In case you have forgotten already, the first was that we were going to modify the airfoil to produce lift at the trailing edge under-surface by “rear loading”.
The second is that we are going to flatten off the upper surface to reduce the strength of the upper surface shock, which appears to be the main bogey man.

Now for a little abstraction. Below the speed at which shock waves form (the so-called “critical Mach number”), lift on an airfoil arises from reducing the pressure on the upper surface (suction side), whilst also increasing pressure on the lower side (pressure side). Most of the lift from this arrangement is generated on the suction side, the pressure lift being much less.
Now we have just seen 2309 that this set-up does not work above the “critical Mach number”, where shock waves start to mess up our lovely air-flow.

Where we are heading is this. We are going to give up on the suction side of the airfoil: just make it pretty flat, and hope to eliminate the nasty upper surface shock. On the lower surface, we are going to do 2 things. The first is to increase pressure lift at the rear undersurface of the airfoil: the second is to weaken the lower surface shock

This is quite radical. Switching from suction lift to pressure lift is really quite novel. As far as I can tell, it was only in the early 1970’s that airfoil designers started to think this way for the transonic region. This despite the revealing data already known to NACA in the late 1940’s and published widely by Abbott and Von Doenhoff in “Theory of Wing Sections”.

OK. Strike up the band, its time to enter the dance floor. Now we really start to swing to the tune of “Dance of the shock waves”.

Focus on the diagram above. Use the left eye to examine the diagram, the right eye to follow this text.

Running from top to bottom, the trailing edge is progressively drooped, what I call a “cusp”. As this happens, the upper surface shock moves back toward the trailing edge.
The lower surface shock moves forward and disappears.

Now ain’t that cute! The dance of the shock waves! I guess we should make a stab at just what weirdnesses are happening here.

Well I guess the effect of the cusp is to speed up the air over the upper surface, thereby moving the shock back.

Likewise, the effect of the cusp is to slow the air on the undersurface, causing the lower surface shock to move forward and TO DISAPPEAR altogether! Now we are cooking with gas! We didn’t want that lower surface shock, and here we have it disappearing altogether. Mate!!! What is more, that subsonic air under the cusp has increased in pressure, giving us “rear loading” effect, thereby boosting lift on the pressure side! Eureka!

What with having already reduced the strength of the upper surface shock by flattening out the upper surface, we have got ourselves a really nice transonic airfoil.
But is this all true? Or just another Supercool aberration, what with the war going on between his left lobe, right lobe and imminent senility?

First off, where on earth did that diagram come from? Well I would have given you photos from my transonic wind tunnel (fantasy) but I couldn’t capture good frames from my video. The tunnel set-up was an aileron going up and down rather fast at Mach .9, so while I could see the dance of the shock waves taking place, I just couldn’t get clear pics. A secondary shock forming at the aileron hinge point didn’t help either. So I have faked up the above diagram to make things clear.

What isn’t faked are test flights we have done with these airfoils in F3D and F2A models. Both these classes have propeller tip speeds close to Mach 0.95, so we are really in a very bad way if we get the airfoils wrong. Fancy tip plan-forms won’t fix this problem!

Well, in F3D we picked up 15 kph in the straights, and 5 kph in the turns. In F2A we picked up 11 kph, to jump from 284 to 295! Change your undies Paramon, here we come!

Finally, Supercool has these props for sale at a premium price to selected rich guys. Price on application. Cheers.