Investigating thermoacoustic instability mitigation dynamics with a Kuramoto model for flamelet oscillators

In this paper, we present experimental observations and phenomenological modeling of the intermittent dynamics that emerge while mitigating thermoacoustic instability by rotating the otherwise static swirler in a lean premixed, laboratory-scale combustor. Starting with a self-excited thermoacoustically unstable combustor, here we find that a progressive increase in swirler rotation rate does not uniformly decrease amplitudes of coherent, sinusoidal pressure or heat-release-rate oscillations. Instead, these oscillations emerge as high-amplitude bursts separated by low-amplitude noise in the signal. At increased rotational speeds, the high-amplitude coherent oscillations become scarce and their duration in the signal reduces. The velocity field from high-speed particle image velocimetry and simultaneous pressure and chemiluminescence data support these observations. Such an intermittent route to instability mitigation is reminiscent of the opposite transition implemented by changing the Reynolds number from a fully chaotic state to a fully unstable state. To model such dynamics phenomenologically, we discretize the swirling turbulent premixed flame into an ensemble of flamelet oscillators arranged circumferentially around the center body of the swirler. The Kuramoto model is proposed for these flamelet oscillators which is subsequently used to analyze their synchronization dynamics. The order parameter $r$, which is a measure of the synchronization between the oscillator phases, provides critical insights on the transition from the thermoacoustically unstable to stable states via intermittency. Finally, it is shown that the Kuramoto model for flamelet oscillator can qualitatively reproduce the time-averaged and intermittent dynamics while transitioning from the state of thermoacoustic instability to a state of incoherent noisy oscillations.

Figure 1

Three-dimensional model of the laboratory-scale combustor and the diagnostics tools.

Figure 2

Variation of first and second mode pressure amplitude $\left(\left|P\right(f\left)\right|\right)$ with different swirler rotation rates where $P\left(f\right)$ is the Fourier transform of $p^{\prime }\left(t\right)$.

Figure 3

Pressure signal acquired by the transducer for 5 s at (a) 0-, (b) 1800-, and (c) 2100-rpm swirler rotation rate.

Figure 4

Normalized pressure fluctuation $\left(\tilde{p}\right)$ and heat-release-rate fluctuation $\left(\tilde{q}\right)$ signals at swirler rotation rates of (a) 0 rpm, (b) 1800 rpm, and (c) 2100 rpm.

Figure 5

Short-time Fourier transform of the pressure signal is shown here for (a) 0-, (b) 1400-, (c) 1800-, and (d) 2100-rpm swirler rotation rates. $P\left(f\right)$ is the Fourier transform of pressure signal $\left[p^{\prime }\left(t\right)\right]$.

Figure 6

Cross wavelet transform (XWT) of $p^{\prime }$ and $q^{\prime }$ at (a) 0, (b) 1800, and (c) 2100 rpm. The color bar shows the common power between the two signals and the arrows represent the instantaneous relative phase lag between them.

Figure 7

HSPIV results of the $y$ component velocity $(v_y$ in m/s) for one complete cyclic pressure oscillation for (a) 0 rpm unstable, (b) 1800 rpm unstable, (c) 1800 rpm stable, and (d) 2100 rpm stable. Corresponding normalized pressure $\left(\tilde{p}\right)$ and spatially averaged chemiluminescence fluctuation $\left(\tilde{q}\right)$ data are shown alongside.

Figure 8

The schematic of flamelet oscillators in the experimental setup during unstable (0 rpm) and stable (2100 rpm) conditions. For clarity, only two of a large number of oscillators which would be circumferentially distributed are shown in each case.

Figure 9

Oscillator distribution $\left(N_f\right)$ at different frequencies $\left(f\right)$ as calculated from the Fourier transform of the 2100-rpm heat-release-rate signal.

Figure 10

Variation of time-averaged order parameter $\left(\overline{r}\right)$ for different swirler rotation rates.

Figure 11

Variation of time-averaged, normalized first mode amplitude $\left(\left|Q\left(f\right)\right|/|Q_0\left(f\right)|\right)$ of experimental and synthetic heat-release-rate signal with swirler rotation rate. The normalization is done with respect to the 0-rpm amplitude $\left(|Q_0\left(f\right)|\right)$. $Q\left(f\right)$ is the Fourier transform of fluctuating heat-release rate $q^{\prime }\left(t\right)$.

Figure 12

Comparison between normalized experimental and generated synthetic heat-release rate $\left(\tilde{q}\right)$ for 0, 1800, and 2100 rpm over 0.5-s windows. The leftmost column corresponds to the experimental data, the middle column shows the synthetic $C=3$ data, and rightmost column is for synthetic $C=2$ data.

Figure 13

Comparison between normalized experimental (on the left) and synthetic (on the right) heat-release rate $\left(\tilde{q}\right)$ for 1800 rpm over a 2-s window. The top row shows the time series, whereas the bottom row shows the short-time Fourier transform of the corresponding data set.

Figure 14

The fast Fourier transform of experimental and synthetic $\left(C=2\right)$ heat-release rate $\left(\tilde{q}\right)$ for 0, 1800, and 2100 rpm.