Self-similar problems of combustible gas mixture flow past bodies

Abstract

The recently initiated study of steady-state supersonic flows of combustible gaseous mixtures with the occurrence of detonation waves and slow-combustion fronts is the result of several factors. We shall consider the problem of fuel combustion in a supersonic stream, and also the utilization of stationary detonation waves for certain processes in chemical technology and for the study of the kinetics of chemical reactions at high temperature. The experimental study of stationary detonation waves presents considerable difficulty, largely because of the complexity of obtaining steady-state flows of combustible gas mixtures with velocities exceeding the propagation velocity of the detonation waves in the mixtures (as is known, this velocity amounts to 2 km/sec or more) and with sufficiently high temperature. A stationary detonation wave was observed in [1] during the exhaust from a nozzle of an overexpanded hydrogen-air jet. The normal and oblique detonation waves in the same mixture within the working section of a wind tunnel were studied in [2]. The stationary flow of a combustible mixture past a body was investigated in [3, 4] by shooting the body into the medium at rest. In this case it was found, in particular, that the detonation wave which forms ahead of the body decomposes at some distance from the body into a conventional adiabatic shock and a slow-combustion front which propagates through the unburned gas behind the shock. In spite of the interesting information obtained, the experimental study of stationary detonation waves and slow-combustion fronts in a supersonic stream is still in the embryonic stage of accumulation and systematization of the facts. The results of the theoretical analysis of multi-dimensional stationary gas flows with detonation waves and combustion are also still very limited even in the simplest case of the representation of these waves in the form of a zero-thickness surface with specified heat release. The polar equation was obtained in [5] for an exothermal shock, generalizing to this case the equation of the shock polar for conventional adiabatic shocks. In the reference cited the polar equation was analyzed only for the case of supersonic velocity ahead of the shock, i.e., only detonation waves were considered. The authors of [6], using the equation of the detonation polar, gave the solution of the problem of combustible mixture flow past a wedge with a detonation wave attached to the apex of the wedge. A solution was obtained in [7] of the problem of flow past a cone with attached detonation wave and flow past a cone with attached detonation wave and flow past a point ignition source which leads to detonative combustion of the gas. The author of [8], on the basis of consideration of the simplest internal structure of the detonation wave in the form of a conventional shock wave and an infinitely-thin flame front with a space between them corresponding to the ignition delay, associated the decomposition of the detonation wave noted in [3, 4] with the influence of the magnitude of the ratio of the characteristic dimension of this space to the dimension of the body in the flow. The present paper presents the solutions of several self-similar problems on plane and axisymmetrical supersonic flows of combustible gas mixtures with detonation waves and slow-combustion fronts. For completeness, the previously studied case of flow about a wedge [6] and cone [7] with attached detonation wave is included; in so doing an inaccuracy in the description of possible flow regimes about the cone [7] is corrected. The new solutions relate to the cases of flows with the formation of adiabatic shocks with subsequent combustion of the mixture in slow-combustion fronts. In particular, solutions are presented of the problems for these cases of flow past the wedge and cone and for line and point ignition sources which lead to the appearance of slow-combustion fronts. The problem is also considered of the degeneration of the detonation wave into an adiabatic shock. The more general case of decomposition of the detonation wave into an adiabatic shock and a slow-combustion front is presented in [9]. All the problems are solved under the assumption of zero thickness of the detonation front and slow combustion. We shall consider in greater detail than in [5, 6] the relationships on the surface of the exothermal shock. As in these references, for simplicity we shall consider that the gas on both sides of the shock is perfect with constant specific heats, and the gas specific heats do not change with passage through the shock.