On the response of a flame ball to oscillating velocity gradients

Abstract

We consider the dynamics of a flame-ball combustion mode of light deficient fuels, in situations where the gas-velocity components around the flame ball are assumed to be time-dependent linear functions of the Cartesian coordinates attached to its centre. This may be viewed as a mathematical attempt to describe flame balls in prescribed, turbulent-like flows.

Using the Arrhenius law for the reaction rate, in combination with asymptotic methods in the limit of large Zel'dovich numbers, we derive an integro-differential, nonlinear evolution equation for the flame-ball radius. This equation involves memory kernels in which the velocity gradients intervene via the determinant of a matrix that is in principle (but generally only numerically) deducible from the prescribed ambient velocity field.

The evolution equation is first specialized to shear flows, for which the aforementioned kernels are available in closed form, and then to harmonic shearing intensities. The equation is then exploited analytically and numerically. In this particular case we show that fluctuating velocity gradients with zero time average can, if of appropriate amplitude and frequency, stabilize the flame-ball radius around a finite, nonzero value. On average they indeed act effectively like volumetric losses. Next we study pseudo-periodic shearing flows analytically and/or numerically, and then more general velocity gradients: we found no counterexample to the trends identified with shear flows.

Hints of physical generalizations and potential implications of the results on the existence of a new regime of turbulent combustion are finally evoked.