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 (and, in some cases, for a reaction heat large enough), 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 between 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.

我们在本文中建立了一个数值求解程序，用于计算由点源以恒定膨胀速度膨胀的球形火焰引起的流动，并具有瞬时化学反应。该解决方案应该是自相似的，并且流动被分成三个区域：由静止的燃烧气体组成的内部区域，溶液规则的中间区域和由静止的新鲜气体组成的初始大气。中间带以反应激波（内侧）和所谓的前兆激波（外侧）为界，为此写出了 Rankine-Hugoniot 条件；该区域的解由两个数值求解的常微分方程控制。我们表明，对于任何可接受的前体冲击速度（并且在某些情况下，对于足够大的反应热），将此数值分辨率与跳跃条件的利用相结合的构造是独一无二的，并且在中间区域产生降低的压力、密度和速度剖面。此外，反应激波的速度大于激波外侧的速度，这与这两个量的差值就是所谓的火焰速度，即激波发生时的（相对）速度是一致的。化学反应在新鲜气体中进行。最后，我们还从数值上观察到火焰速度作为前驱冲击速度的函数的函数正在增加；这允许将分辨率嵌入牛顿类程序中，以计算给定火焰速度（而不是给定前体冲击速度）的流量。