  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:
