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For aircraft flying at higher airspeeds it is necessary to take into account the compressibility
of the air, which causes the air density to vary from one location to the
next. In this case, the energy law for adiabatic flow of an ideal gas is used [5]:
g
g − 1
p
r
+ 1
2V2 = constant (3.27)
where g = Cp
Cv is the ratio of the specific heats of a gas with constant pressure (Cp)
and constant volume (Cv), respectively. For air, g equals 1.4. It can be shown that the
pressure and density of an isentropic flow are related as follows:
ps
rg = constant (3.28)
If we now define the ‘total conditions’ to be the pressure and density where the flow
is brought to rest isentropically1, we find:
pt
ps
=
rt
r
g
=
Tt
T
g
g−1
(3.29)
Applying equation (3.27), the total pressure for a compressible airflow is thus found
to be:
pt = ps
1 + g − 1
2g
r
ps
V2
g
g−1
(3.30)
It is possible to re-write this relation using the Mach number M, which is defined as
the ratio of the speed of the airflow to the speed of sound:
M = V
a
(3.31)
1In an isentropic process, a gas is compressed and/or expanded gradually enough for the process
to be reversible, which implies that its entropy does not change.
3.5. Atmosphere and airdata variables 37
where V is the (local) airspeed and a is the speed of sound (in [ms−1]):
a =
p
gRT (3.32)
The Mach number is an important property that appears in many of the isentropic
flow equations; it is also used as main control variable for the velocity of jet aircraft
at high altitudes. Substituting these relations in equation (3.30) yields:
pt = ps
1 + g − 1
2
M2
g
g−1
(3.33)
Notice that equations (3.30) and (3.33) are only valid for Mach numbers smaller than
one, as they are based on the assumption of isentropic behaviour. If the aircraft flies
at supersonic speeds, shockwaves appear over which the air pressure, temperature,
and density rise abruptly, causing the process to become irreversible. In that case,
the isentropic relations are no longer valid and the flow is governed by supersonic
shock relations, which go beyond the scope of this report.
For the Beaver aircraft it is not necessary to take into account these compressibility
effects, although it can still be quite useful to know the value of the Mach number.
For instance, if one wants to compare flight-test results with the simulations it is useful
to compute the Mach-dependent ‘impact pressure’ qc and total temperature Tt
since those quantities can be measured in flight. The impact pressure is defined as
the total pressure minus the static pressure:
qc = pt − ps (3.34)
where pt is determined by equation (3.30) or (3.33), whichever is more convenient.
The impact pressure is the equivalent of the dynamic pressure from Bernoulli’s equation
(3.26), except in this case the compressibility of the flow is taken into account.
It can be measured using a pitot static system that subtracts the static pressure from
the total pressure that is measured in the stagnation point of the pitot tube.
The total air temperature Tt is the temperature that is measured when air is
brought to rest isentropically. This measured temperature is higher than the temperature
that would be measured when the air is completely in rest (relative to the
temperature sensor):
Tt = T
1 + g − 1
2
M2
(3.35)
Other important variables are the calibrated and equivalent airspeeds Vc and Ve, which
are determined by the equations:
Vc =
vuut
2g
g − 1
p0
r0
(
1 + qc
p0
g−1
g
− 1
)
(3.36)
Ve = V
r
r
r0
=
s
2qdyn
r0
(3.37)
Equation (3.36) can be derived easily from equations (3.34) and (3.30), by substituting
the conditions at sea level for r and ps. This relation is often used to calibrate
the airspeed indicators in the cockpit; the true airspeed differs from the indicated
airspeed due to the calibration for sea level conditions, hence the terminology. The
38 Chapter 3. The aircraft model
‘equivalent airspeed’ does the same assuming that the airflow is incompressible. Expression
(3.37) is used to calibrate less sophisticated flight instruments that can be
found in light aircraft that operate in airspeed regions where compressibility of the
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(23)
