In part 5 of this series, a computer code was presented which can be used to design a propeller. The equations used in that design code were taken from Larrabee's report to the NFFS in 1979, entitled 'Propeller Design and Analysis for Modelers'. The particular section of that report entitled 'Arbitrary propeller performance' was used by me in novel ways, namely using an iteration procedure to make slip constant. The code is illustrative, but does not include the important correction for compressibility.
In this part 6 presentation, the same code is used but with corrections for compressibility incorporated. The phenomenon of compressibility was first encountered with propellers, as propeller tip speeds were much higher than the speed of early airplanes. So what is compressibility, and why is it a problem?
When a moving airfoil approaches the air, the air is forced to follow the airfoil contour. In doing so over the top surface of the airfoil, the air is speeded up, the pressure in the air drops and lift is thus engendered. When the wing is really going fast, the air has difficulty moving fast enough to get around the contour, to the point where the air not only gets speeded up but it gets compressed as well. The sort of speeds involved are those approaching the speed of sound. This compression changes appreciably the way the airfoil produces lift, which we need to know.
Note that propellers carry airfoils that more often than not are moving at speeds which are an appreciable fraction of the speed of sound. We can expect complications!!
Now that we have an idea of what is meant by compressibilty, we will move on to the aerodynamic effects that are actually observed.
Firstly, the lift on the airfoil is increased. So is the pitching moment. So is the angle of zero-lift.
With rigid propellers we are not too concerned with the pitching moment, but the change in lift and zero-lift angle are a fundamental part of propeller performance. We now need a genius to come along and tell us how to proceed. Enter Prandtl and Glauert. These guys were busy in the mid 1920's figuring out what to do. This is what they found. An airfoil section which works nicely at low speed will behave at high speed as if it is thicker, has more camber, and is operating at an increased angle of attack. Not only that, the behaviour is a function of Mach number: that function is given by the Prandtl-Glauert rule. The changes all follow the expression 1/(sqr(1-Mach^2)). This function is always greater than unity, going to infinity at Mach 1. Well, that seems a good reason not to go that fast! Looks like a sound barrier to me!
Imagine now that we want to apply the Prandtl-Glauert rule to a real propeller. Say we choose Clark-Y to go all the way from root to tip. Well, no problem with the innerparts of the blade, but out near the tip Clark-Y behaves most ungainly, being effectively thicker, having more camber and washed-in. Plainly we need the airfoil to be thinner at the tip, to have less camber and not to be washed in. The text books tell us to apply the rule to the force coefficients. I tried this, and what a mess I got into. It didn't seems to matter what I did, this wretched Prandtl-Glauert factor kept on appearing. What with Larrabee's zero-lift angle thrown in as well, I simply could not hack it.
Well, to shorten a long and sad story of failure, one night I was in bed having a read of Katz and Plotkin when I opened the page on Prandtl-Glauert rule. I was arrested by this phrase: 'the x-co-ordinate is being stretched as the Mach number increases'. Ye Gods and little fishes, are you telling me that the Prandtl-Glauert rule is a SCALING theorem? Well why am I bothering with modifying force coefficients?
All I need to do is take my root airfoil, like at low Mach, and stretch it to suit the rule? And the new airfoil will behave as though it has the same force coefficients as at low Mach? Yup, thats it. The Prandtl-Glauert rule is a scaling theorem that can be used to design all the prop airfoils, based on the root airfoil chosen.
Here is how to do it. Choose the root airfoil for its low Mach force coefficients. Set it at the angle-of -attack you want. Typically 6 degrees above the zero-lift angle. Now multiply all the x (abcissa) values by the the Prandtl-Glauert factor for your station Mach number. Look at it. What you see is now a stretched, thinner airfoil, with less camber, at a new, lower angle of attack. This new airfoil now has the same force coefficients as the root airfoil section. Except its chord is increased. Bring it back to the chord you want by scaling BOTH the x and y coordinates to your chosen chord. Job done. All the airfoils can now be designed by the scaling process, and all the force coefficients are known...they are the same as the root airfoil!! There is no need for the wretched force coefficient corrections to appear anywhere in the prop design code, once the airfoils are done.
OK, thats enough applause..stop clapping..stop.. please stop!
So now we can return to the prop design code with correction for compressibility. It is the same code as part 5. The only changes are that now the Mach number needs to be known at every station, while the airfoils must be designed in a subroutine that does the stretching and re-scaling of the root airfoil, as necessary, each time the angle-of-attack is iterated.
In fact, I did not quite do this. I already had a favourite airfoil I was using at 3/4 radius. So I reverse scaled it to give me the root airfoil. No big deal, but a bit messy.
So if you want to run this code, you need also load the data file for the favourite airfoil.
The code will run on any machine with operating system Windows 98 or earlier. You need to load both the code and the airfoil data file, GENERIC.DAT. The source code is labelled F2CDESGP.BAS.
Both files are text files. They should be stored in a directory C:\CNCMOULD\F2C\F2CDESGP
Oh, I forgot the caveat. Above Mach .7 we have to be wary. This correction does not predict the airfoil design when the local Mach number over the airfoil surface exceeds the speed of sound. It does, however, produce airfoils of a shape favourable to a rise in the critical Mach number.
You've stopped clapping? How ungrateful. Start again. Now!
Further reading:
1. Katz and Plotkin. Low Speed Aerodynamics, 2001
2. Godsey and Young. Gas Turbines for Aircraft, 1949
3. Hoerner. Aerodynamic Drag, 1951
4. Hoerner. Fluid-Dynamic Lift, 1975
5. Abbott and Von Doenhoff. Theory of Wing Sections, 1958
6. Supercool. Propeller Dynamics,1994
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