It would be nice to see an answer to the question originally posed. So here is one. I must say I don't find the suggested HAW-Hamburg paper very helpful for answering the question, but then the authors were explaining something they felt was new, not basic aero to be found in textbooks.
Originally Posted by TedUnderwood
I'm trying to determine what happens to Critical Mach in relation to altitude. I cannot find any sources and I've tried reading thru High Speed Aerodymics in Aerodynamics for Naval Aviators but haven't had much luck.
The coefficient of pressure, Cp, is defined as the increment of pressure at that point on the airfoil (p.local-p.freestream) divided by the freestream dynamic pressure. Since it is a function of point x, let us denote it Cp(x). It follows from the compressible Bernoulli equation that Cp(x) is an arithmetic function (that is, a function which is a composition of addition, subtraction, multiplication, division, and exponent) of Mach number at that point on the airfoil, M.local(x), freestream Mach number, M.freestream, and gamma, a constant with value of about 1.4 for air. The only varying quantities for flight are M.local(x) and M.freestream.
Now, there is some point x0 on any airfoil at which Cp at that point, Cp(x0), is a minimum. Call that minimum simply Cp.0, to conform with convention. x is also a point of maximum velocity on the airfoil surface. There may be more than one x0 (indeed, for a constant-form airfoil there is going to be a line of them).
You can find out x0, and Cp.0, by experiment in incompressible conditions if you wish. Indeed, let Cp(y).0 denote Cp(y) in incompressible conditions for any point y on the airfoil.
According to the Prandtl-Glauert approximation, Cp at any point y is going to vary with freestream Mach number as Cp(y).0 / sqrt(1 - (M.freestream)^2). Now, Prandtl-Glauert is experimental (derived by curve-fitting on experimental data), but is pretty good up to M=0.7 or 0.8 or so, so I understand. Above that, there are more complicated formulas, but they are still arithmetic functions of Cp and M.freestream.
So x0 is going to be the first point at which M.local = 1, as M.freestream increases. That is because Cp(x0).0 is lower (or equal to) Cp(y).0 for any other point y, and when you divide them both by sqrt(1-(M.freestream)^2), that "less than or equal to" relation is preserved. That point at which M.local(x0)=1 is going to occur at a specific value of M.freestream. That value is known as M.crit.
No mention of altitude, everything is an arithmetic function of M.freestream. That may be why you find no mention of altitude in calculations of M.crit.
I can see why Hurt didn't help much, looking at the couple of sentences on p215 in which he talks about M.crit. Try Introduction to Flight, John D. Anderson Jr., Section 5.9, Sixth Edition, McGraw-Hill 2008. It is a standard reference, and Anderson devotes 12 pp to it, which means you get a lot more in the way of careful derivation and explanation.
I make no claim that this explanation is particularly intuitive, because that depends on one's favorite kind of intuition. Please feel free to find a more intuitive explanation than this one.
PBL