Page 27 - Aerodynamics for Engineering Students

10 Aerodynamics for Engineering *dents

by combining the density p and the dynamic viscosity p according to the
equation
P
y=-
P
and has the dimensions L2T-l and the units m2 s-l.
It may be regarded as a measure of the relative magnitudes of viscosity and inertia
of the fluid and has the practical advantage, in calculations, of replacing two values
representing p and p by a single value.

1.2.7 Speed of sound and bulk elasticity

The bulk elasticity is a measure of how much a fluid (or solid) will be compressed by
the application of external pressure. If a certain small volume, V, of fluid is subjected
to a rise in pressure, Sp, this reduces the volume by an amount -SV, i.e. it produces a
volumetric strain of -SV/V. Accordingly, the bulk elasticity is defined as

(1.6a)

The volumetric strain is the ratio of two volumes and evidently dimensionless, so the
dimensions of K are the same as those for pressure, namely ML-1T-2. The SI units
are NmP2 (or Pa).
The propagation of sound waves involves alternating compression and expansion
of the medium. Accordingly, the bulk elasticity is closely related to the speed of
sound, a, as follows:
a=6 (1.6b)

Let the mass of the small volume of fluid be M, then by definition the density,
p = M/V. By differentiating this definition keeping M constant, we obtain

Therefore, combining this with Eqns (l.6ayb), it can be seen that

a=& (1.6~)

The propagation of sound in a perfect gas is regarded as an isentropic process.
Accordingly, (see the passage below on Entropy) the pressure and density are related
by Eqn (1.24), so that for a perfect gas

(1.6d)

where y is the ratio of the specific heats. Equation (1.6d) is the formula usually used
to determine the speed of sound in gases for applications in aerodynamics.