Forcing of a flame by a periodic flow in a Hele-Shaw burner

The parametric forcing of premixed flames is observed here in the geometry of a Hele-Shaw burner. Using a vibroacoustic coupling, we are able to study the thresholds of the parametric flattening (where the flame wrinkles are suppressed) and the parametric instability (where the flame wrinkles oscillate at twice the acoustic period) and compare the measured values to a low-frequency theory. It is shown that the dependency of the parametric threshold with equivalence ratio is lower than predicted, suggesting that the Markstein lengths are frequency dependent and that the flattening threshold is independent of the Markstein number and of the nondimensional forcing frequency, in agreement with the Bychkov limit.

Figure 1

Experimental apparatus. (a) The flame propagates downwards in a Hele-Shaw burner composed of two borosilicate glass plates oriented vertically and separated by a 5-mm gap. The burner cavity is closed on the sides and at the bottom and open at the top. The eigenmodes of one of the plates are forced using a shaker, and the resulting vibrations are recorded using two accelerometers positioned on the plate (see [23] for details). (b) The oscillations of the plates generate an oscillating flow in the burner cavity. Here we plot the vertical component of the flow speed $u_a$ (on the horizontal axis) in terms of the vertical position in the Hele-shaw burner $y$ (on the vertical axis where the zero corresponds to the top end of the burner). Moving downwards in the burner, we observe a succession of areas with a low amplitude oscillating flow (nodes) and areas with a large amplitude oscillating flow (antinodes). These nodes and antinodes of the flow speed correspond to nodes and antinodes of the plate vibrations [23]. (c) When propagating downwards, the flame undergoes a periodic oscillating flow with different acoustic amplitude depending on the position in the tube, and exhibits parametric restabilization and/or parametric destabilization.

Figure 2

Stability diagrams for different sets of parameters and their corresponding regimes of propagation. For a given set of parameters where ${\tilde{u}}_I^{*}<{\tilde{u}}_{II}^{*}$ [see (a) and (d) for example], three different scenarios of propagation can be observed depending on the acoustic amplitude ${\tilde{u}}_a$: (i) if ${\tilde{u}}_a<{\tilde{u}}_I^{*}$, the flame is DL unstable and exhibits characteristic wrinkling with cusps pointing towards the burnt gases [see (d)]; (ii) if ${\tilde{u}}_I^{*}<{\tilde{u}}_a<{\tilde{u}}_{II}^{*}$, all the wave numbers are stable and parametric flattening is observed [see(f)]; (iii) if ${\tilde{u}}_a>{\tilde{u}}_{II}^{*}$, a typical parametric instability is observed with patterns pulsating at half the forcing frequency ${\tilde{\omega }}_a$. On frame (g) the evolution of a parametrically excited front is represented over one full acoustic period (from top to bottom). We note that at the end of the period the minima and maxima of the front have exchanged positions. After another acoustic period, the minima and maxima would get back to their original position. (The period of the pattern is twice the acoustic period.). On the other hand, if for a given set of parameters ${\tilde{u}}_{II}^{*}<{\tilde{u}}_I^{*}$, there is no parametric restabilization zone [see (b) and (c)]. Instead, DL instability and parametric instability may coexist for a certain range of ${\tilde{u}}_a$ [see (h), where one can note the DL (large wavelength) and the parametric (small wavelength) instabilities].

Figure 3

Dependence of the restabilization and parametric thresholds to Ma. (a) ${\tilde{u}}_I^{*}$ and ${\tilde{u}}_{II}^{*}$ vs Ma at small forcing frequency. (b) ${\tilde{u}}_I^{*}$ and ${\tilde{u}}_{II}^{*}$ vs Ma at high forcing frequency. (c) ${\tilde{k}}_I^{*}$ and ${\tilde{k}}_{II}^{*}$ vs Ma at small forcing frequency. (d) ${\tilde{k}}_I^{*}$ and ${\tilde{k}}_{II}^{*}$ vs Ma at high forcing frequency.

Figure 4

Comparison between experimental and theoretical threshold values with respect to the nondimensional forcing frequency ${\tilde{\omega }}_a$. Theoretical data for different Markstein number values are reported in shades of red, and experimental data for different equivalence ratios $\phi$ are reported in shades of gray. (a) Comparison of the restabilization threshold $u_I^{*}$. (b) Comparison of the restabilization threshold $u_{II}^{*}$. (c) Comparison of the last unstable reduced wave number $k_I^{*}$ before restabilization. (d) Comparison of the first unstable reduced wave number $k_{II}^{*}$ above the parametric threshold.

Figure 5

Comparison of experimental thresholds (gray) with those predicted by Searby & Rochwerger theory (red). For the three cases presented here the forcing frequency is $f_{\mathrm{shaker}}=400\text{Hz}$. A. Comparison of the restabilization threshold for propane-air flame with different equivalence ratios $\phi$. (b) Same for parametric thresholds. (c) Comparison of the last unstable wave number at the restabilization threshold. (d) First unstable wave number at the parametric threshold.