Untitled - Aerobib - Universidad Politécnica de Madrid

116 CHAPTER 5. STRUCTURE OF THE COMBUSTION WAVES
Let A and B be the two representative points of the two possible initial states
(ε = 0), compatible with the assumed values for m, i and e. Velocity is supersonic
in A and subsonic in B. The jump from A to B occurs through a shock wave, as
aforesaid. Since, as previously seen in the preceding chapter, the propagation velocity
of a detonation is supersonic, the representative point for the initial state of the mixture
is A. When ε increases, that is, when the reaction occurs, two alternatives are possible
either the representative point of the intermediate states of the mixture throughout the
wave moves along the upper branch AC (see Fig. 5.1), or else a shock wave produces
first which changes the velocity from A to B and then, as the combustion progresses,
the representative point moves along the lower branch BC. A decision between both
alternatives can not be taken from purely hydrodynamic considerations. Let us analyze
both cases separately.
Suppose that the point moves on the upper branch, starting from A. This means
that the combustion initiates in the state of the unburnt gases. But in this state, the
temperature is small and the reaction velocity cannot be sufficiently large to burn
the gases with the speed required by the detonation wave. Therefore, the detonation
must be initiated by a shock wave which, by making the gases pass from the state
represented by point A to the one represented by point B, compresses and heats the
gases, taking them to a state in which the reaction velocity can be sufficiently large to
burn them with the required speed.
The ratio of the thickness of the shock wave to the thickness of the detonation
wave is measured by the ratio of the thermodynamic characteristic time τ t to the chemical
characteristic time τ c . Therefore, the thickness of the shock wave is very small
with respect to the thickness of the detonation wave. 8 This means that while the gases
pass through the shock wave, the burnt fraction is insignificant. Thus the detonation
wave appears as formed by the shock wave, followed by a combustion wave.
The need of a shock wave to initiate the chemical reaction in the detonation
wave has always been acknowledged. This has been the stand-point, for instance, for
Vieille [5] and Jouguet [6] in 1900. 9 These authors, however, have consider the detonation
always as a discontinuity, in which not only the shock wave is instantaneously
produced, but also the subsequent reaction. The ideas developed in the present study,
concerning the structure of the wave, belong to the modern theories on detonation
8 The study of the dynamic transformations within the shock wave cannot be performed with the system
obtained by neglecting the action of viscosity and thermal conductivity, whose action is essential within the
wave. See for example M. Roy: Structure de l’onde de choc et des flammes deflagrantes. ONERA, Paris,
1952.
9 In the fundamental work of Jouguet [6] data can be found concerning the historical evolution of the
ideas relative to the classical theory of the detonation waves.