Untitled - Aerobib - Universidad Politécnica de Madrid

5.5. DETONATIONS 115
Of these two velocities, one is subsonic and the other supersonic. In fact, the
critical velocity v cr , corresponding to the point defined by the value ε of the degree of
advancement of the combustion, is given, in virtue of (5.29a), by
v 2 cr =
2 (γ − 1)
γ + 1
(e + qε) . (5.36)
This value, however, is exactly the product of roots v 1 and v 2 of equation (5.33).
Consequently
v 1 v 2 = v 2 cr, (5.37)
that is, of both velocities one is subcritical, that is to say subsonic, and the other supercritical,
that is to say, supersonic. Furthermore, the two values v 1 and v 2 correspond to
the velocities before and after a normal shock wave, since (5.37) is the Prandtl relation
for shock waves. 7
A
F
v
C
E
B
ε
D
Figure 5.1: Schematic diagram showing the two possible velocities resulting from the heat
addition.
Figure 5.1 represents the pair of values corresponding to Eq. (5.33) for variable
ε. Their corresponding curve is a parabola. The upper branch of this parabola, AC,
corresponds to the supersonic velocities, and the lower branch, BC, corresponds to the
subsonic velocities. At point C, where both branches join, the velocity of the gases
with respect to the wave is equal to the sound velocity.
7 See R. Courant and K. O. Friedrichs: Supersonic Flow and Shock Waves. Interscience Pub. Inc., New
York, 1948, p. 147.