Mechanics of Fluids

496 Compressible flow of gases
Sonic velocity
Mach number
If the fluid is a gas obeying Boyle’s Law, then from eqn 11.4
� �
∂p
∂ρ
= dv
� �
∂p
dρ ∂v
= pγ
ρ
1
since ρ =
v
s
s
The speed of propagation of a very small pressure wave (relative to the fluid)
is therefore
B, I
a = � (pγ/ρ) B, I (11.19)
Alternatively, from the equation of state for a thermally perfect gas,
a = � (γ RT) TPG, I (11.20)
Sound is propagated by means of a succession of very small pressure waves
in which δp is alternately positive and negative. (The faintest sound that
the human ear can detect unaided corresponds to a pressure fluctuation of
about 3 × 10−5 Pa; the loudest that can be tolerated without physical pain
corresponds to a fluctuation of about 100 Pa.) The velocity represented by
eqns 11.13–11.20 is therefore known as the speed of sound or velocity of
sound or sonic velocity or acoustic velocity in the gas. As it is a function of
temperature it varies in general from point to point in the fluid. The validity
of the assumptions made in deriving the expressions is indicated by the
excellent agreement found with experimental determinations of the velocity
of sound. For air of moderate humidity γ = 1.4 and R = 287 J · kg −1 · K −1 ;
so at 15 ◦C �
�
a = 1.4 × 287 J · kg −1 · K −1 �
(273 + 15) K = 340 m · s −1
The velocity of sound is appreciably less at high altitudes because of the
lower temperature there.
It is important to note that the preceding expressions refer only to waves
in which the change of pressure is very small compared with the pressure
itself. Waves in which a comparatively large pressure change occurs will
be considered in Section 11.5. The assumption of constant entropy is not
justified for these larger waves, and they move at velocities greater than that
of sound.
When the velocity of the fluid at a particular point is less than the velocity
of sound there, small pressure waves can be propagated both upstream and
downstream. When, however, the velocity of the fluid exceeds the local sonic
velocity a, a small pressure wave cannot be propagated upstream. A velocity
equal to a thus sharply divides two essentially different types of flow.
We have seen already in Sections 1.5 and 5.3.4 that it is useful to express
the velocity of a gaseous flow in relation to that of the sonic velocity. The
Mach number, M, is defined by the relation
M = u/a
where u is the local velocity and a is the speed of sound. Fluid velocities less
than the sonic velocity are known as subsonic (M < 1), those greater than