Front roughening of flames in discrete media

The morphology of flame fronts propagating in reactive systems composed of randomly positioned, pointlike 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 $200\times 200$ 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 nondimensional 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 root mean square deviation of the ignition temperature contour from the average flame position] grew following a power-law dependence that was in excellent agreement with the Kardar-Parisi-Zhang (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.

Figure 1

Visualization of a small domain ($50\times 50$) with ${\theta }_{\mathrm{ign}}=0.5$, ${\tau }_{\mathrm{c}}=1$, showing the color contours of temperature field, with the flame front (marked by the black-white line), and randomly distributed ignited (black dots) and unignited (white dots) sources. The uniform Cartesian grid over which the temperature field is calculated for front roughness measurement is illustrated as the blue lines. See Supplemental Material 1 [28].

Figure 2

Temperature fields of flames propagating rightward in systems of discrete sources. Average intersource spacing is unity, and size of each plot window is chosen to be $50\times 50$. Each system corresponds to a different set of nondimensional ignition temperature ${\theta }_{\mathrm{ign}}$ (increasing from left to right) and discreteness parameter ${\tau }_{\mathrm{c}}$ (increasing from bottom to top). The color bars above the top row correspond to the column's ignition temperature. The flame front is near the middle of each plot window. See Supplemental Material 2 [28].

Figure 3

Temperature profiles plotted against spatial coordinate in direction of flame propagation, i.e., $\theta \left(x\right)$, for ignition temperature ${\theta }_{\mathrm{ign}}=0.2$ and discreteness parameter (a) ${\tau }_{\mathrm{c}}=100$, (b) ${\tau }_{\mathrm{c}}=1$, (c) ${\tau }_{\mathrm{c}}=0.1$. Corner inset: Corresponding temperature contour plots. Thin dashed curves in temperature profile are extracted from corresponding lines shown on the temperature contours. The red line corresponds to width-averaged temperature profile, and the black line corresponds to the analytic solution based on a continuum flame model.

Figure 4

Ensemble-averaged flame front roughness $W$ as a function of time $t$ for ${\theta }_{\mathrm{ign}}=0.2$ and ${\tau }_{\mathrm{c}}$ from 0 to 100 using (a) linear scales and (b) logarithmic scales. Each curve is plotted for flame propagation in a $200\times 200$ domain ($40000$ discrete sources, beginning from the time at which at least 20 sources have been triggered). The measured $\beta$ is plotted as a function of domain width $L$ in (c). The reference line in (b) and (c) shows $\beta =1/3$. Error bars show the $95\%$ confidence interval on the slope measurement.

Figure 5

Ensemble-averaged flame front roughness $W$ as a function of time $t$ for ${\tau }_{\mathrm{c}}=0$ and ${\theta }_{\mathrm{ign}}=0.025$ to 0.5 using logarithmic scales. Each curve is plotted for flame propagation in a $200\times 200$ domain ($40000$ discrete sources). The roughening behavior influenced by the initiation of the flame and the finite domain width are indicated on the curve of ${\theta }_{\mathrm{ign}}=0.5$.

Figure 6

Simulation results categorized according to the flame front roughening behavior and summarized as a map in ${\theta }_{\mathrm{ign}}\text{−}{\tau }_{\mathrm{c}}$ parameter space. Each color-shaded region represents a category of front roughening behavior; each symbol represents an ensemble of simulations with a specific set of ${\theta }_{\mathrm{ign}}$ and ${\tau }_{\mathrm{c}}$.