Alex,
1. On aircraft with Air Data Computers the ASI displays CAS
2. The Air Data Computer calculates CAS using variations of Saint Venant's formulae which assume compressible flow.
Those are equivalent statements and are, so far as I'm aware, required by SAE AS 8019 (and its successor 8019A) which is referenced by
ETSO-C2d and US
TSO-C2d. I don't have a copy of the SAE Aerospace Standard but I believe the USAF counterpart is
MIL-PRF-27197E; table II uses impact pressure.
Because if #2 were true, the ADC would output what I know as EAS, CAS corrected for compressibility, although it still seems to be called CAS.
The second statement does not mean the ADC outputs EAS. If EAS is displayed as the primary airspeed by some ADC (fitted in a civil aircraft) then please provide a reference. The Saint-Venant solutions for compressible flow use impact pressure only. Constant CAS is equivalent to constant impact pressure. The Euler-Bernoulli solution for incompressible flow uses dynamic pressure only. Constant EAS is equivalent to constant dynamic pressure. The term "compressibility correction" is contentious in my opinion because it may suggest that EAS is more accurate than CAS. EAS clearly makes flight mechanics equations more tractable but it should be used with caution in the transonic regime.
KayPam,
Nothing prevents you from calculating a mach number, from speed and temperature, even in an incompressible flow.
The equation you would use was derived for a compressible medium and the only value Mach number can take in an incompressible flow is zero. Full understanding of the theory underlying the speed of sound proved surprisingly challenging in the history of science, defying such brilliant minds as Newton and Euler (it will be 2033 before all of Euler's work has been translated to English), and credit ultimately went to Laplace (
Sur la Vitesse du Son dans l'air et dans l'eau, Annales des Chimie et de Physique 3, 238–241 (1816)). An excellent summary of the struggle to resolve this problem is given in
Laplace and the speed of sound by
Bernard Finn who was, like John D Anderson, a curator at the Smithsonian Institution. Anderson provides a straight-forward derivation in Fundamentals of Aerodynamics, section 8.3, which is very similar to this
this two-page derivation plucked liberally from a search for "derivation of the speed of sound."
Nothing would prevent you from building a machmeter which would would exactly compute the mach from the following formula
An incompressible flow would prevent it. It isn't clear how you would propose to "correct" a zero Mach number. Such a Mach meter belongs in Eric Laithwaite's
Multiplcation of Bananas by Umbrellas.
If you measure a CAS of ...
... then you've used an airspeed indicator using Saint-Venant's solution for an isentropic flow. That is,
dynamic pressure cannot be measured in flight, and any incompressible flow solutions using a measured isentropic total pressure will be in error (including the fabled Mach meter you've proposed). The proposal amounts to misinterpreting useful information (impact pressure) as an approximation (dynamic pressure), followed by a "correction" to recover the original information. Such a scheme may work at low speed and low altitude but it is inconsistent and unnecessarily restrictive. Indeed it would have been simpler to have used Saint-Venant's solution directly in the first place to obtain CAS and Mach number, which is what modern ASIs and Mach meters do.
It is however very easy for modern ADC to use formulas as complicated as required thanks to IT.
That has already been achieved in purely mechanical instruments. There are numerous US patents outlining basic operations such as subtraction, division, and exponents can be dealt with by taking logarithms. You raise a very good question on precisely how Saint-Venant's solution can be implemented in analog ASIs and Mach meters.