Page 84 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 67

It is often convenient to regard the effects of compressibility as negligible if the
flow speed nowhere exceeds about 100 m s-l. However, it must be remembered that
this is an entirely arbitrary limit. Compressibility applies at all flow speeds and,
therefore, ignoring it always introduces an error. It is thus necessary to consider, for
each problem, whether the error can be tolerated or not.
In the following examples use will be made of the equation (1.6d) for the speed of
sound that can also be written as

a = m

For air, with y = 1.4 and R = 287.3 J kg-'K-' this becomes
a = 20.05em s-' (2.33)

where Tis the temperature in K.

Example 2.1 The air-speed indicator fitted to a particular aeroplane has no instrument errors
and is calibrated assuming incompressible flow in standard conditions. While flying at sea level
in the ISA the indicated air speed is 950 km h-' . What is the true air speed?
950 km h-' = 264 m s-' and this is the speed corresponding to the pressure difference applied
to the instrument based on the stated calibration. This pressure difference can therefore be
calculated by
1
Po - P = AP = 5 PO4

and therefore
1
po -p = - x 1.226(264)' = 42670NmP2
2
Now

In standard conditionsp = 101 325Nm-'. Therefore
po - 42670 + 1 = 1.421
p 101325
Therefore
1
1 + - M2 = (1.421)2'7 = 1.106
5
1
- M2 = 0.106
5
M' = 0.530
M = 0.728

The speed of sound at standard conditions is
a = 20.05(288)4 = 340.3 m s-'