Untitled - Aerobib - Universidad Politécnica de Madrid

3.10. STATIONARY, ONE-DIMENSIONAL MOTION OF IDEAL GASES WITH HEAT ADDITION 81
1.2
6
1.0
0.8
b
5
4
b
0.6
3
M 0
0.4
2
0.2
M 0
1
0
0
0.0 0.5 1.0 1.5 2.0 2.5
*
M 0
Figure 3.2: M 0 and b (Eq. (3.92)) as a function of M0 ∗ .
4) Likewise, the temperature is expressed in dimensionless form by
θ = b 2 T
T s0
. (3.97)
If these variables are taken into the previous system and this system is solved
for v ∗ and θ, one obtains
(v ∗ − 1) 2 = 1 − 2x
γ + 1 , (3.98)
θ = x − γ − 1 v ∗2 . (3.99)
2
The values of v ∗ and θ are represented in the diagram Fig. 3.3 where the law
of variation of the Mach number M of the motion is also included. M is given by the
expression
M = v∗
√
θ
. (3.100)
It can easily be verified that for v ∗ and θ to be real, x cannot exceed the value
x max = γ + 1 , (3.101)
2
to which corresponds a maximum value n max of n given by the expression
n max = γ + 1
2b 2 . (3.102)