How to convert IAS/CAS to TAS up to Mach 3?

Question

Someone had recommended that I use this formula to convert KCAS/KIAS to KTAS:

https://www.experimentalaircraft.info/flight-planning/aircraft-navigation-speed.php

If you scroll down to "True Airspeed, TAS," you will see this formula:

=KIAS*SQRT(SEA LEVEL AIR DENSITY/AIR DENSITY OF DESIRED ALT)

So, enetered into an Excel spreadsheet cell, the formula would look like this:

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=100*SQRT(1.225/0.905)

100 is the KIAS, 1.255 is air density at sea level, and 0.905 is air density at 10,000 feet.

This formula gives us 116 knots for KTAS.

This formula works for low airspeeeds, but as airspeeds get higher, the KTAS that is given is not accurate. As I have learned, this formula does not take into account "compressibility," and therefore the KTAS will not be accurate at higher airspeeds.

I am interested in a formula that I can enter into my Excel spreadsheet that will give an accurate KTAS for all airspeeds up to Mach 3, and altitudes up to 100,000 feet.

Is there one particular formula that will work?

Answer 1

For higher airspeeds, TAS is always calculated based on the Mach number: $$ \mathrm{TAS} = M \cdot a_0 \sqrt{\frac{T}{T_0}} $$ where $ a_0 $ is the speed of sound at ground level ($ a_0 \approx 661 \, \mathrm{kt} $) and $ T_0 = 288.15 \, \mathrm{K} $ is the temperature at ground level. This formula is just based on the fact that the speed of sound in air is proportional to the square-root of the temperature.

How to calculate the Mach Number

Now, we need to distinguish between subsonic and supersonic flow. The pitot tube will give you the ratio of impact pressure $ q $ and static pressure $ p $. From this ratio, you calculate the the Mach number:

• subsonic flow: $$ M = \sqrt{5 \left( (q/p)^{5/7} - 1 \right)} $$
• supersonic flow: $$ M \approx 0.8813 \sqrt{\frac{q}{p} \left( 1 - \frac{1}{7 M^2} \right)^{5/2}} $$ This equation needs to be solved iteratively.

The following plot shows the Mach Number as a function of the $ p / q $ ratio (red for subsonic flow, blue for supersonic flow):

Calculate the True Air Speed

Now that we have the Mach number, we need to calculate the TAS using the formula from above. The last missing piece is the static air temperature $ T $, which can be calculated from the measured total air temperature $ T_\mathrm{tot} $:

$$ T = \frac{T_\mathrm{tot}}{1 + M^2 / 5} $$

This gives the final answer:

Unfortunately, there is no easy way to calculate this based on KIAS as an input. You will need the ratio $ q / p $ and the temperature $ T_\mathrm{tot} $ as measured by the aircraft.