  The morphology of flame fronts propagating in reactive systems comprised of
randomly positioned, point-like sources is studied. The solution of the
temperature field and the initiation of new sources is implemented using the
superposition of the Green's function for the diffusion equation, eliminating
the need to use finite difference approximations. The heat released from
triggered sources diffuses outward from each source, activating new sources and
enabling a mechanism of flame propagation. Systems of $40000$ sources in a
$200$ by $200$ two-dimensional domain were tracked using computer simulations,
and statistical ensembles of $120$ realizations of each system were averaged to
determine the statistical properties of the flame fronts. The reactive system
of sources is parameterized by two non-dimensional values: The heat release
time (normalized by interparticle diffusion time) and the ignition temperature
(normalized by adiabatic flame temperature). These two parameters were
systematically varied for different simulations to investigate their influence
on front propagation. For sufficiently fast heat release and low ignition
temperature, the front roughness (defined as the RMS deviation of the ignition
temperature contour from the average flame position) grew following a power law
dependence that was in excellent agreement with the KPZ universality class
($\beta = 1/3$). As the reaction time was increased, lower values of the
roughening exponent were observed, and at a sufficiently great value of
reaction time, reversion to a steady, constant-width thermal flame was observed
that matched the solution from classical combustion theory. Deviation away from
KPZ scaling was also observed as the ignition temperature was increased. The
features of this system that permit it to exhibit both KPZ and non-KPZ scaling
are discussed.
