Noise influence on pole solutions of the Sivashinsky equation for planar and outward propagating flames

The dynamics of planar and outward propagating cylindrical flames has been studied in terms of exact solutions of the Sivashinsky equation with a random force term. The force term models the computational roundoff errors or a variety of perturbations of physical origins. In contrast to noiseless conditions, the number of poles in the system does not conserve and new poles appear due to the external forcing. It was found that modification of the pole solutions taking into account the appearance of new poles captures the features typical for the hydrodynamically unstable flames, which cannot be detected by the pole solutions with a fixed number of poles. Investigations based on the pole solutions make it possible to exclude the uncontrolled numerical noise that is always present in direct computations of the Sivashinsky equation, and to examine the interplay between noises and hydrodynamic instability. The study clearly demonstrates that the presence of noises is a necessary condition for flame acceleration.

Figure 1

Time dependency of the velocity of the flame propagating in the duct with $L=90$ and $\gamma =2$, disturbed by adding a pole at the position $(x_{\mathrm{add}},y_{\mathrm{add}})$.

Figure 2

Evolution of a flame upwardly propagating in the duct with $L=90$ and $\gamma =2$. The steadily propagating flame was perturbed by addition of one new pole at the position $x_{\mathrm{add}}=5$ and $y_{\mathrm{add}}=30$.

Figure 3

Velocity of a flame propagating in the duct versus diameter of the duct $L$. The noise effect is modeled by a periodic appearance of the $N_g$ new poles at random positions in the rectangle $[0,L]\times [y_{\min },y_{\max }]$. $\gamma =2$, $N_g∕L=0.02$, $T=50$.

Figure 4

Time dependency of the outwardly propagating flame velocity calculated by exact pole solutions with a fixed number of poles $N_p$ for $\gamma =2$.

Figure 5

Time dependency of the outwardly propagating flame velocity calculated by exact pole solutions for $\gamma =2$ and (1) $P_N=0.005$, $T=100$; (2) $P_N=0.002$, $T=100$; (3) $P_N=0.002$, $T=200$. Solid and dashed lines correspond to the calculations in the sector $\phi ∊[0,2\pi ∕6)$ and in the whole circle, respectively. Marked lines are the results of direct computation of Eq. (1) with zero (solid marks) and random (open marks) force terms.

Figure 6

Temporal evolution of the average flame front velocity calculated for (a) noiseless condition with $\gamma =2$; (b) periodic noise with $T=100$, $P_N=0.002$, and $\gamma =0$; (c) periodic noise with $T=100$, $P_N=0.002$, and $\gamma =2$.

Figure 7

Evolution of the outwardly propagating cylindrical flame with $\gamma =2$ and $P_N=0.002$.