Mechanics of Fluids

518 Compressible flow of gases
The influence of compressibility is best illustrated, however, by using
the Mach number. Rearranging eqn 11.13 and noting that a2 = γ RT =
(γ − 1)cpT, we obtain
T0
T
= 1 + u2
2cpT
� �
γ − 1
= 1 + M
2
2
and then from eqn 11.6,
p0
p =
� � �
γ − 1
1 + M
2
2
�γ /(γ −1)
SF, A, PG (11.44)
SF, I, PG (11.45)
For subsonic flow (γ − 1) M 2 /2 < 1 and so the right-hand side of eqn 11.45
may be expanded by the binomial theorem to give
whence
p0 − p =
p0
p
pγ M2
2
= 1
2 ρu2
γ
= 1 +
2 M2 + γ
8 M4 γ(2 − γ)
+
48 M6 +···
�
�
1 + M2
4 +
1 + M2
4 +
� 2 − γ
24
� 2 − γ
24
�
M 4 �
+···
�
M 4 �
+···
SF, I, PG (11.46)
Comparing eqn 11.46 with the result for a fluid of constant density (Section
3.7.1) we see that the bracketed quantity, sometimes known as the
compressibility factor, represents the effect of compressibility. Table 11.1
indicates the variation of the compressibility factor with M when γ = 1.4.
It is seen that for M < 0.2 compressibility affects the pressure difference
by less that 1%, and the simple formula for flow at constant density is then
sufficiently accurate. For larger values of M, however, the compressibility
must be taken into account.
Table 11.1 Variation of
the compressibility factor
for air
M
p 0 − p
1
2 ρu2
0.1 1.003
0.2 1.010
0.3 1.023
0.4 1.041
0.5 1.064
0.6 1.093
0.7 1.129
0.8 1.170
0.9 1.219
1.0 1.276