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## Transonic Airfoils Part 4: Local Mach number

By Joe Supercool

All this transonic flow stuff gets a bit confusing at times. When that confusion happens, it is best to carefully dot the t’s and cross the i’s, else, before you know it, you have answers to questions you don’t even know about. Which just means you haven’t understood what’s going on. At least, that is how it works for me.

Being the author of this series, I have the right to fudge and dissemble. That is exactly what I did in the previous discussion. Remember that graph with no less than 2 vertical axes? One was pressure ratio P/Po and the other “Local Mach#”: at the time, I said that because you can draw 2 vertical axes, the 2 quantities involved must be related. Well, that was true then, as it is now. However, I did not have a clue why! Like I say, an answer with no question.

Recall that the pressure ratio P/Po is the ratio of the local surface pressure P to the stagnation pressure of the air stuck at the leading edge, the air there not knowing which way to flow, over or under! Thus low values of P/Po represent “suction”.

Below is that figure again. Forget the shock stuff, we are only looking at the vertical axes (ordinates) this time. At “Local Mach#” of 1.0, I had drawn a red line across to a value of P/Po slightly above 0.5. I really did that just to show more clearly where the surface flow velocity went supersonic. At the time, I wondered about this point P/Po being near 0.5. Why on earth did it hit the axis at just that point: not just there, but in all the figures, irrespective of free-stream Mach number. Gotta be something weird going there. And there is!

We need to look again at just what is meant by Mach #.
Well, its starts off nice and simple. Call Mach # “M”, call the speed of the airplane “V” an the speed of sound “c”, then:

M = V/c

How simple is that? Would you believe that in my book, “Propeller Dynamics”, self-published in 1994, that I got that upside down? My excuse is that I was dying at the time, so I was in rather a hurry to build my pyramid. But then daughter Remy saved my life, and here I am still writing this bumpf.
OK. Now I said this was simple, or even obvious. Since I am usually lying when I say “simple” or “obvious”, regular readers will know that the game is afoot!

Return to that great period of really bad fighter designs, World War 2. Sir Sydney Camm of Hawkers produced a real monster of a fighter, the Typhoon. Starting life with a massive 24-cylinder RR Vulture engine, then progressing to the equally massive Napier Sabre, also of 24 cylinders, the Typhoon made the Spitfire look like a toy. But it had a serious problem (more than one actually, but only one of relevance here). The wing was 18% thick. That is good for keeping down the wing weight, or storing big cannons and shells, but lousy for flying at high altitude!

You see, with more than 2000 HP up front, this aircraft was intended to go fast. It didn’t. The problem was our old friend, shock waves, resulting from high local Mach numbers over that great thick wing. This problem was not so severe at low altitudes, which meant the Typhoon was great for hedge-hopping and firing rockets at tanks. But at high altitude in a dive, the machine wanted to keep on going straight down, or even tuck-under, despite the poor frightened pilot’s attempts to pull back on the stick. If he had the courage to ride the plane down to lower altitudes, then slowly the elevator would start responding in the normal way. Recovery into level flight then occurred (assuming (1) that the tail hadn’t fallen off and (2), that he wasn’t holding full back stick, in which case the wings ripped off)

This disconcerting problem suggests that at a lower, warmer altitude, the shock wave problem went away. You see, the speed of sound varies with temperature. It is cold at high altitudes (like -40, Centigrade or Fahrenheit, take your pick), so the speed of sound is much slower at 30000’ than at 10000’. That is to say, Mach number varies with height, simply because the speed of sound has changed.

So that is our first complication arising with our Mach number formula. We need to consider temperature of the air. But it gets worse. The speed of sound in air is also affected by pressure and density. Since we have seen that pressure is also affected by the speed of the air, it would seem that the speed of sound, instead of being a nice steady 340 m/s, can actually vary enough to upset the way an airplane flies.

Or even worse, it can upset our beloved propeller tips when they go transonic. But before I move on, a little irrelevancy. The quantities temperature, pressure and density are fundamental to the study of air. They are called “state variables”. The study of air involving these “state variables” comes under the heading of “thermodynamics”. If you think we are in a muddle already with shock waves, just try following some thermodynamic analyses!

OK, now what is it that we were supposed to be working on? Oh yes, the 2 vertical axes and the weird value of P/Po near 0.5 in our pressure diagram above. Consider that axis labelled “Local Mach#”. Whitford really meant that. He didn’t mean the speed of the airplane: he meant the speed of the air flowing locally over the surface of the wing. In our diagram above, that speed is considerably above the free-stream Mach number of unity.

So here is a problem. The speed of sound in the air rushing over the airfoil surface is affected by the speed of the air, which has lowered the pressure of the air. Where does that leave our definition of Mach#?. Well, we need a new one. Let us say that “Local Mach#” is written M*, and the speed of sound in air at the surface of the airfoil is c*. Then:

M* = V / c*

When M* is less than unity, we say that the flow is subsonic.
When M* is greater than unity, we say that the flow is supersonic
Remember, M* is the local Mach#, which is to say, the Mach number of the air rushing over the airfoil surface. We want that, for the pressure drop it causes produces lift. But we will come back to that later.

Now we don’t know the value of c*, but it turns out that we don’t have to. Thanks be to Edward R. C. Miles and his excellent book “Supersonic Aerodynamics”, 1950, all the messy thermodynamic reasoning has been done for us. I quote from page 19.

And

Recall that has the value 1.4, being the ratio of specific heats at constant pressure and at constant volume. That is the thermodynamic stuff I was talking about earlier. It has to be a really neat theory, because c* has disappeared altogether!

Now in case your mental arithmetic is as bad as mine, this is what Miles has to say about these formulae.

From the first formula, at Mach 1, ie M = 1, then M* also is unity. That’s nice. Our red line on the graph is the same for both M and M*. We can forget that. Consider it done!

When M* is subsonic, P/Po will be less than 1 but never less than 0.528
When M* is supersonic, P/Po can less than 0.528 but not greater,
When P/Po is really small, approaching zero, M* cannot exceed 6^.5

Wow, what rubbish: I cannot keep that in my head at all.

Lets put this in terms we can follow, we who love propellers.

If we have supersonic air rushing over the top of the airfoil (green line), that is good, because it lowers the air pressure on that surface and gives us lift. If it weren’t for the spoil-sport high drag of the shockwave which comes with the supersonic flow, we would be on a winner.

If we have subsonic air rushing over the lower surface (blue line), then its pressure will always be greater than over the top surface, and we again have good lift.

To finish. Remember that red line at Local Mach# 1? It was near 0.5 on the P/Po axis. Well, from our formula, the value was really 0.528! That number represents the dividing line between 2 quote different flow behaviours of air: that between subsonic and supersonic flow. Fascinating stuff, wouldn’t you agree?