Title:

Studies in the theory of detonation

The work of this thesis consists of two separate problems concerning the motion of detonation waves in inviscid nonheatconducting gases. A detonation wave is assumed to ho a surface of discontinuity moving through the medium. On crossing this wave the gas particles instantaneously release a certain amount of heat energy. Whereas shook waves are only of the compressive type, waves involving such an energy release can he eitherompressive (detonations) or expansive (deflagrations). Part I is concerned with the onedimensional motion of such waves through a uniform gas contained in a straight tube, which is closed at the end where the waves are Initiated, There are t\TO accepted models for the motion of a comhustihlo gas in this case. In the first a shook wave moves uniformly into the gas ahead of a deflagration v;ave which travels uniformly at subsonic speed relative to the stationary gas behind it. There is a region of uniform motion between the t\70 waves. The second concerns sonic (ChapmanJouguet) deflagrations and differs from the previous case in that there exists a pointcentred, simple rarefaction wave immediately behind the deflagration. For a certain value of the shock speed of the latter system the speeds of the deflagration and shook coincide, resulting in a single detonation wave front. In the present work the stability of these two systems is Investigated by considering the effect on the flow of certain types of small disturbances. The subsonic model is found to be nonevolutionary, i.e., the problem of introducing specified small disturbances into the system has no subsonic solution, A selfgenerating, unstable solution is found and is calculated in the case of the shock being uniformly accelerated. The existence of such a solution suggests that the subsonic model is unstable. The problem of small disturbances introduced into the ChapmanJouguet model is shown to be evolutionary, provided the ChapmanJouguet condition is relaxed for the disturbances. For two particular numerical cases it is shown that there is no selfgenerating solution in which the perturbation of the shook speed can be expressed as a povrer series in time, t. The solution for the perturbation on the system due to the gas being not quite at rest initially is found in terms of the initial velocity distribution along the tube. Part II is an investigation into the mathematical solution for a spherically or axially symmetric detonation front travelling into uniform gas in the direction towards the centre or axis of symmetry. It is known that there is no solution corresponding to a constant speed ChapmanJoixguet front for this case, unlike the expanding radially symmetric wave whose solution is analogous to the Ohap DianJouguet detonation in one dimension. However, examination of Guderley's similarity solution f03. a radially symmetric contracting shook front, valid near the centre or axis of symmetry, suggests a method of solving the present problem Guderley's solution shows that the shook accelerates towards the origin, where the solution is singular and the shook speed, particle velocity, and pressure are infinite. Since the addition of a heat release term across this front can only have a finite effect on the energy of the flow behind the front, it follows that the detonation problem can be considered as a small perturbation on the shook solution. This perturbation is of precisely the same form as the correction due to taking into account the sound speed of the stationary gas, which is neglected in Gudorley's solution, The equations of motion of this basic solution Involving a shook front reduce to a single nonlinear, first order, ordinary differential equation owing to the similarity assumption, which also permits the equations governing the perturbations to be written as a set of three simultaneous, linear, ordinary differential equations, the solution of the former single equation appears in the coefficients of the latter set of equations. Hence it is necessary to recompute Guderley's solution and his results are extended to higher values of the assumption that the flow is regular on a certain characteristic, on which the flow may be singular, ensures that there is a unique solution to both the basic and detonation problems. The equations are integrated numerically using a specially devised method which makes use of the power series expansions in the vicinity of this characteristic, which becomes a point in terms of the redefined variables, the method used has no difficulty in dealing with the solution in the neighborhood of this point, where the derivatives of three of the four variables, as given by the differential equations, are indeterminate (but are actually finite due to the regularity conditions).
