Untitled - Aerobib - Universidad Politécnica de Madrid

118 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
The study of the aerodynamic field subsequent to a detonation wave can be performed
by applying the Gas Dynamic methods. 10 For instance, Zeldovich [10] has calculated
the aerodynamic field following a detonation wave that propagates unchanged
from the closed end of a tube, initially full of detonant gas at rest. By neglecting the
friction and the heat lost through the walls of the tube, a solution is obtained which
propagates with the Chapman-Jouguet velocity. The detonation wave is followed by
an expansion region with an approximate length of 50% of the total length covered
by the detonation wave. The remaining burnt gases (approximately 40% of the burnt
mass) are at rest. By including the effect of friction and heat losses through the walls
of the tube, a stable Chapman-Jouguet wave is obtained, followed by an expansion
in which the direction of the motion of the burnt gases in the tube is reversed. Following
the expansion there also exists in this case a region at rest in which the burnt
gases have cooled down to ambient temperature. There is photographic evidence of
the existence of this reversal motion in the expansion region.
The aerodynamic field that follows a spherical detonation wave has been studied
by G.J. Taylor [11] in England, and by Zeldovich in U.S.S.R. [12]. The calculations
also demonstrate the existence of a solution which propagates unchanged with
the Chapman-Jouguet velocity. Immediately after the detonation wave there is a strong
expansion region, followed by a central nucleus at rest. Within the expansion region
more than 90% of the burnt mass is in motion. The spherical detonation waves have
been experimentally observed, for example, by Manson and Ferrié [13]. A strong detonation
can occur, for example, when the detonation propagates inside a tube, starting
from an end closed by a piston, that moves after the wave with a subsonic velocity
with respect to the wave. In such a case the intensity of the strong detonation would
be determined by the compatibility condition obtained by expressing that the velocity
of the burnt gases with respect to the wave must equal the velocity of the piston.
Then, when friction and heat losses through the walls of the tube are neglected, it
results that the strong detonation wave propagates unchanged throughout the mass of
unburnt gases.
The relation between pressure and density within the wave can be obtained
from Eqs. (5.1.a) and (5.28.a) by elimination of v. Thus resulting
p + m2
ρ
= i, (5.38)
or else, introducing the state (p 1 , ρ 1 ) of the unburnt gases
( 1
p − p 1 = m 2 − 1 )
. (5.39)
ρ 1 ρ
10 See Courant and Friedrichs, ib., pp. 218 and 416 for plane and spherical waves, respectively.