Mach number calculation from calibrated airspeed and pressure altitude

In many performance analyses, the calculation of Mach number is required as a function of Vc (calibrated airspeed) and Hp (pressure altitude). While flight test textbooks include all the analysis steps in form of mathematical formulas, for this specific step the use of a chart is recommended. See below chart as given by USN TPS performance manual:

mach_USNThe problem with this chart, as with all similar charts is that it does not provide the student or analyst a convenient way to get high accuracy values of Mach number, and it can not be easily automated or implemented in a code, let alone the process of trying to draw straight lines on a paper or screen. In other words, its not an optimum engineering approach.

Before trying to approximate graphically the drawn curves with equations, going back to basics and using the basic thermodynamic laws, the standard atmosphere relations and the instrument correction equations, the following formula was derived, for use in the troposphere up to 36,089 ft (for higher altitudes the appropriate pressure variation with altitude should be used).

mach_equation

where assl is the standard speed of sound at sea level. In the formula above pressure altitude (Hp) is in ft and up to 36,089. Speed of sound and calibrated airspeed may have any same units as they cancel out.

The derivation process is found in the attached images below.

A quick superposition of the derived equation over the given chart shows that it provides a good approximation, but not a perfectly accurate agreement. (In the plot below, the color lines are the lines coming from the derived equation for the various altitudes).

Mymethod_vs_USNThis finding was encouraging, but also confusing. In order to cross-check the validity of the equation, it was compared to an other similar chart of M, Vc, Hp found in a technical paper. The agreement with those published charts was the found to be excellent. 

Mymethod_vs_paper

Lessons learnt:

1. Always try to use equations rather than graphical charts, if not provided, try to derive them. Many times, the textbook editor just recycles information without deeper examination.

2. Many graphs are drawn “by hand” rather than by analytic curves. Be aware as this can mesh up your data!

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4 Responses to Mach number calculation from calibrated airspeed and pressure altitude

  1. Robert M says:

    I believe that formula for Mach is only effective in the Troposphere. Air density follows an exp curve in the tropopause

    • aeroscience says:

      Right, it is actually applicable up to 36,089 ft, as the pressure ratio formula changes for higher altitudes due to temperature lapse rate change. I will add a clarification in the initial message. Thanks

  2. jant789 says:

    Which units are you using here for speed of sound, H_p, and airspeed?

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