Coherence resonance in a thermoacoustic system

We experimentally investigated the noise-induced dynamics of a prototypical thermoacoustic system undergoing a subcritical Hopf bifurcation to limit cycle oscillations. The study was performed prior to the bistable regime. Analysis of the characteristics of pressure oscillations in the combustor and fluctuations in the heat release rate from the flame—the two physical entities involved in thermoacoustic coupling—at increasing levels of noise indicated precursors to the Hopf bifurcation. These precursors were further identified to be a result of coherence resonance.

Figure 1

Premixed flame combustor. Noise is introduced through the acoustic actuators driven by a white-noise input voltage signal.

Figure 2

Subcritical Hopf bifurcations in the noise-free thermoacoustic system. The acoustic pressure amplitude, $\mathrm{rms}\left[p^{\prime }\right]$, as a function of the control parameter, $\varphi$. The solid line connects experimental points obtained while increasing $\varphi$, while the dashed line indicates translation in the reverse direction.

Figure 3

Amplitude of $p^{\prime }$ fluctuations in response to varying levels of noise in the subthreshold region: $D=5.7$ Pa (squares), $D=9.9$ Pa (circles), and $D=14.2$ (triangles). The shaded region indicates the bistable and the thermoacoustically unstable zones. Base noise levels (horizontal dotted lines) and the forward direction of bifurcation analysis (asterisks) are included for reference.

Figure 4

Amplitude of ${\dot{q}}^{\prime }$ fluctuations in response to varying levels of noise in the subthreshold region: $D=5.7$ Pa (squares), $D=9.9$ Pa (circles), and $D=14.2$ (triangles). The shaded region indicates the bistable and the thermoacoustically unstable zones. The forward direction of bifurcation analysis (asterisks) is included for reference.

Figure 5

Power spectra from $p^{\prime }$ for different equivalence ratios and nonreacting flow (bottom-right). Curves correspond to the three noise levels: $D=5.7$ Pa (squares), $D=9.9$ Pa (circles), and $D=14.2$ Pa (triangles).

Figure 6

Coherence factor, $\beta$, as a function of the noise level for different equivalence ratios. A maximum occurs for an optimum noise level for each curve, indicating coherence resonance. As exemplified by the Lorentzian fits (gray bands), the optimum value decreases with an increase in $\varphi$.

Figure 7

Autocorrelation from $p^{\prime }$ for $\varphi$ = 0.708 at three noise levels. Decay in the autocorrelation is the slowest for $D=12.7$ Pa. Insets: Sections from the corresponding time series.

Figure 8

Amplitude distribution for $p^{\prime }$ at three noise levels. The distribution changes from a unimodal to a bimodal and back to a unimodal distribution with increasing noise intensity.

Figure 9

Iterations of the bifurcation experiments and the obtained critical parameter values. The shaded region marks the hysteresis zone for each iteration.