Page 83 - Aerodynamics for Engineering Students

66 Aerodynamics for Engineering Students

2.3.4 The incompressibility assumption
As a first step in calculating the stagnation pressure coefficient in compressible flow
we use Eqn (1.6d) to rewrite the dynamic pressure as follows:

(2.30)

where M is Mach number.
When the ratio of the specific heats, y, is given the value 1.4 (approximately the
value for air), the stagnation pressure coefficient then becomes

c --=- Po -P (2.31)
0.7pW 0.7M2 p
("" __ 1)
Now
1
E=[l+p4] (Eqn (6.16a))
112
2
P
Expanding this by the binomial theorem gives
-
7531
Po++- 7(1 -M2 ) +--- 751 (1 -M 2)2 + ---- (I -M2 )3 +
P 25 222! 5 2223! 5
+-
7M2 7M4 7M6 7M8
=1+-+7
10 400 +-+a 16 000
Then

10 7M2 7M4 7M6 7M8
-
-- +-+-+-+... 1
7M2 [w 40 400 16 000
iW? M4 M6
=I+-+-+-+.*, (2.32)
4 40 1600
It can be seen that this will become unity, the incompressible value, at M = 0. This is
the practical meaning of the incompressibility assumption, i.e. that any velocity
changes are small compared with the speed of sound in the fluid. The result given
in Eqn (2.32) is the correct one, that applies at all Mach numbers less than unity. At
supersonic speeds, shock waves may be formed in which case the physics of the flow
are completely altered.
Table 2.1 shows the variation of C,, with Mach number. It is seen that the error in
assuming C,, = 1 is only 2% at M = 0.3 but rises rapidly at higher Mach numbers,
being slightly more than 6% at M = 0.5 and 27.6% at M = 1.0.

Table 2.1 Variation of stagnation pressure coefficient with Mach numbers less than unity

M 0 0.2 0.4 0.6 0.7 0.8 0.9 1 .o
G o 1 1.01 1.04 1.09 1.13 1.16 1.217 1.276