Modelling of a spherical deflagration at constant speed

Abstract

We build in this paper a numerical solution procedure to compute the flow induced by a spherical flame expanding from a point source at a constant expansion velocity, with an instantaneous chemical reaction. The solution is supposed to be self-similar and the flow is split in three zones: an inner zone composed of burnt gases at rest, an intermediate zone where the solution is regular and the initial atmosphere composed of fresh gases at rest. The intermediate zone is bounded by the reactive shock (inner side) and the so-called precursor shock (outer side), for which Rankine-Hugoniot conditions are written; the solution in this zone is governed by two ordinary differential equations which are solved numerically. We show that, for any admissible precursor shock speed, the construction combining this numerical resolution with the exploitation of jump conditions is unique, and yields decreasing pressure, density and velocity profiles in the intermediate zone. In addition, the reactive shock speed is larger than the velocity on the outer side of the shock, which is consistent with the fact that the difference of these two quantities is the so-called flame velocity, i.e. the (relative) velocity at which the chemical reaction progresses in the fresh gases. Finally, we also observe numerically that the function giving the flame velocity as a function of the precursor shock speed is increasing; this allows to embed the resolution in a Newton-like procedure to compute the flow for a given flame speed (instead of for a given precursor shock speed). The resulting numerical algorithm is applied to stoichiometric hydrogen-air mixtures. Key words. spherical flames, reactive Euler equations, Riemann problems burnt zone (constant state) intermediate zone (regular solution) unburnt zone (constant initial state) W b W 2 W 1 W 0 reactive shock, r = $\sigma$ r t. precursor shock, r = $\sigma$ p t. W = ($\rho$, u, p): local fluid state. Fig. 1.1. Structure of the solution. 1. Problem position. We address the flame propagation in a reactive infinite atmosphere of initial constant composition. The ignition is supposed to occur at a single point (chosen to be the origin of R 3) and the flow is supposed to satisfy a spherical symmetry property: the density $\rho$, the pressure p, the internal energy e and the entropy s only depend on the distance r to the origin and the velocity reads u = ur/r, where r stands for the position vector. The flame is supposed to be infinitely thin and to move at a constant speed. The flow is governed by the Euler equations, and we seek a solution with the following structure: