True, Equivalent, and Calibrated Airspeeds

 

The energy equation for adiabatic flow of an ideal gas is

 

 

where v is the flow velocity and h is the specific enthalpy.  This implies that if a flow with free stream velocity v0 and enthalpy h0 is brought to rest adiabatically, and if h1 is the enthalpy of the resting gas, then

 

 

Also, for an ideal gas we have dh/dT = cp, where the specific heat cp is essentially constant for relatively small changes in temperature, so this equation can be written as

 

 

Now, the speed of sound in an ideal gas is given by  where g = cp/cv is the ratio of specific heats and R is the gas constant, so we can express any given temperature T in terms of the speed of sound at that temperature.  With these substitutions, the preceding equation becomes

 

 

Also, for an ideal we have R = cp - cv, so we can make this substitution, divide through by a02, and re-arrange terms to give

 

 

The ratio v0/a0 is, by definition, the Mach number M of the free stream, and since temperature is proportional to the square of the speed of sound, the ratio (a1/a0)2 equals the ratio of temperatures T1/T0.  Hence we have

 

 

Furthermore, using the ideal gas equation p = rRT we can substitute for the temperatures to give

 

 

As discussed in The Speed of Sound, for an adiabatic quasi-static (i.e., isentropic) process, the ratio p/rg is constant.  Thus if we re-write the above equation as

 

 

the first factor on the left side is unity, and we arrive at the formula for the isentropic pressure rise of an ideal fluid being brought to a stop from a Mach number M

 

 

Solving this equation for M and then multiplying through by the freestream speed of sound , we have the formula for the true airspeed vtrue = v0 as a function of the freestream static pressure p0, the total (stagnation) pressure p1, and the freestream static temperature T0

 

 

If the pressure ratio p1/p0 is known but the freestream static temperature T0 is not (as is sometimes the case with primitive instrumentation), we can agree by convention to simply use for T0 the standard sea level atmospheric temperature

 

Tstandard  =  518.67 R

 

When this is done, the result is called equivalent airspeed.  The factor of  in the above equation is then simply taken to be the speed of sound at standard sea level temperature, so the equation for equivalent airspeed is

 

 

where

astandard  =  1116 ft/sec  =  661.47 knots

 

Of course, the square root in this expression is simply the true Mach number, so equivalent airspeed can also be written as

 

 

Thus, noting that a0M is the true airspeed, and letting q denote the ratio T0/Tstandard, we have

 

 

Sometimes we lack not only the static temperature, but the pressure ratio as well.  With certain kinds of primitive instrumentation systems we can measure only the difference p1 - p0 between the total and static pressures.  (This difference is sometimes called the impact pressure, which is the same as the dynamic pressure 1rv2 for incompressible flow, but not for compressible flow.)  Even if we can re-write the ratio p1/p0 as (p1-p0)/p0 + 1, we would still need to know the static pressure p0 (in addition to the impact pressure) in order to compute equivalent airspeed.  However, lacking the static pressure, we can agree by convention to simply use the standard sea level static pressure in place of p0.  When this is done, we get the following formula for calibrated airspeed as a function only of the impact pressure

 

 

where

pstandard  =  14.696 psia

 

Given the definition of calibrated airspeed, we sometimes need to compute it based on the actual measured values of the static pressure p0 and freestream Mach number M.  (Of course, knowing p0 and M, we could compute equivalent airspeed, but convention may still force us to deal with calibrated airspeed.)  For convenience, we will set g = 1.4, which is the value for atmospheric air.  Then p1 is given in terms of p0 and Mach by the relation

 

 

Substituting this into the calibrated airspeed equation gives

 

 

where d denotes the pressure ratio p0/pstandard.  For values of M less than  this can be expanded into a convergent power series in M, the first few terms of which are

 

 

In terms of the true airspeed this is

 

 

For small Mach numbers (much less than 1) the zeroth order term of the expansion is often accurate enough for practical purposes, but for Mach numbers approaching 1 it is necessary to take account of the higher-order terms - or else simply use the exact analytical expression.

 

Return to MathPages Main Menu