Control of growth of local heat release rate fluctuations to suppress thermoacoustic instability

First, we analyze the results obtained for experiments with a combustor length of 1100 mm, operating at $\dot{m}_{fuel}$ of 34 SLPM with the bluff-body positioned 45 mm from the dump plane and where $\dot{m}_{air}$ changes at a rate of 3 SLPM/s. We name this set of operating conditions and experimental configurations used in [15] as the reference experimental configuration (REC) for the purpose of comparing the results. For all the subsequent figures, instead of discussing the plots with respect to $\dot{m}_{air}$, we describe the figures using the global equivalence ratio $\phi$. Even though the flame is only partially premixed, we use $\phi$ so that we can compare the results at different thermal power ratings.

Decreasing $\phi$ results in a continuous growth in the amplitude of acoustic pressure ($p^{\prime}$) and global heat release rate oscillations ($\dot{Q}^{\prime}$), as shown in Fig. 2a. The short-time Fourier transform (STFT) plot in figure 2b shows a dominant mode emerging near 100 Hz at $\phi$ $<$ 0.87 for $p^{\prime}$. As $\phi$ is reduced, the frequency of the dominant mode increases and reaches an asymptotic state for $\phi$ $<$ 0.67. At high values of $\phi$ (0.99 $\geq\phi\geq$ 0.88), $p^{\prime}$ and $\dot{Q}^{\prime}$ have dissimilar dominant modes with shallow bands (Pawar et al. [16]).

Next, we analyze the local growth of heat release rate fluctuations $\dot{q}^{\prime}(x,y)$ as $\phi$ is reduced. The spatial distribution of the root mean square (mean is calculated for 1 second, which means 500 data points) of local heat release rate fluctuations $\dot{q}^{\prime}_{rms}$ is shown at three values of equivalence ratio, $\phi$ = 0.97 (Fig. 2c), $\phi$ = 0.88 (Fig. 2d) and $\phi$ = 0.69 (Fig. 2e). $\dot{q}^{\prime}_{rms}$ shows high strength around the bluff body and near the combustor walls.

In the framework of phase transition theory, it is possible to analyze critical phenomena as the transition is approached by estimating the growth in variance of fluctuations ($\sigma^{2}$) of a system variable termed as pre-bifurcation noise amplification [17]. We estimate the growth of $\sigma^{2}$ of the local heat release rate fluctuations from the difference between $\sigma^{2}$ at $\phi$ = 0.97 and $\phi$ = 0.88 (Fig. 2f). We use these two values of $\phi$ (also for subsequent sections) to calculate the growth of $\sigma^{2}$ prior to the occurrence of large-amplitude pressure oscillations (refer Fig. 2a) for the REC configuration.

We mark two regions that have high growth of $\sigma^{2}$ using black squares in Fig. 2f. Z1 is a location of high heat release rate during the occurrence of thermoacoustic instability because of the impingement of large-scale coherent flow structures on the combustor wall. Z2 represents a stagnation point that stabilizes the flame because of the low velocity region at the corner between the shaft and the bluff-body. Both Z1 and Z2 exhibits critical phenomena prior to the transition to thermoacoustic instability, illustrated by the high growth in $\sigma^{2}$ (Fig. 2f). In our previous study, we referred to these zones as “seeds” of the phase transition due to critical phenomena occurring at these locations, well before the appearance of large-amplitude pressure oscillations. After the critical transition in the local heat release rate fluctuations occurs at these zones, the strength of pressure oscillations ($p_{rms}$) increases rapidly as $\phi$ is reduced for REC as shown in Fig. 2g.

We now characterize the transition using the temporal measurements of $p^{\prime}$ for changes in the bluff-body position, and $\dot{m}_{fuel}$. As the bluff-body position from the dump plane is reduced to 40 mm from 45 mm (REC), it appears that large-amplitude oscillations occur earlier compared to REC (See Fig. 2g). But at $\phi$ $<$ 0.6, the pressure oscillations with 45 mm bluff-body position (REC) is stronger than those with 40 mm.

We also investigate changes in the transition with respect to the thermal power rating: $\dot{m}_{fuel}$. A lower $\dot{m}_{fuel}$ does not affect the $p^{\prime}_{rms}$ values with respect to REC till $\phi$ = 0.7 (Fig. 2g). For $\phi<$ 0.7, as expected $p^{\prime}_{rms}$ is lower in comparison to REC.

The high growth in $\sigma^{2}$ at Z1 and Z2 (Fig. 2f) occurs for all the operating conditions that we have described above. In the current parametric study on the suppression of the phase transition to thermoacoustic instability, we will target Z2 and modify the flame distribution around the bluff-body. Utilizing the IS bluff-body design, we introduce passages on the bluff-body in the form of slots. These slots are introduced between a radius of 8 mm and 15 mm. Since the shaft radius is also 8 mm, we eliminate the stagnation zone at the corner between the bluff-body and the shaft in order to reduce the growth of local heat release rate fluctuations at Z2. We first analyze results with the IS bluff-body at REC.

Figures 3a-c shows the variation in the spatial distribution of $\dot{q}^{\prime}_{rms}$ as the equivalence ratio is reduced. The IS configuration results in higher strength of the local heat release rate oscillations downstream of the bluff-body (see Figs. 3b, c) in comparison to the BL configuration. In comparison to the BL bluff-body at REC, we observe that the strength of the heat release rate oscillations ($\dot{q}^{\prime}_{rms}$) have reduced at the stagnation point (refer Figs. 2d, e and 3b, c). The highest strength of $\dot{q}^{\prime}$ oscillations are observed near the combustor walls (see Fig. 3c).

Thus, the inner slot (IS) bluff-body design redistributes the heat release rate fluctuations around the bluff-body. This redistribution results in a reduction of growth of $\sigma^{2}$ at Z2 (Fig. 3d). This low growth corresponds with a reduction of $\dot{q}^{\prime}_{rms}$ at Z2. On the other hand, near Z1, the growth of $\sigma^{2}$ is still high. Further, the wake region of the IS bluff-body also indicates a growth of $\sigma^{2}$. But importantly, the growth of $\sigma^{2}$ is not as high as the growth at Z2. It is crucial that other regions do not show high growth of $\sigma^{2}$ due to the change in the bluff-body design. Other regions showing high growth of $\sigma^{2}$ may mean the existence of other pathways for the transition to thermoacoustic instability.

The plot of $p^{\prime}_{rms}$ at REC shows the suppression of the global transition to thermoacoustic instability (Fig. 3e). $p^{\prime}_{rms}$ remains low as $\phi$ is reduced to low values. In fact, at low values of $\phi$, there is 88$\%$ decrease in the $rms$ value of pressure oscillations in comparison to the $rms$ value obtained with BL bluff-body at REC. However, $p^{\prime}_{rms}$ at low values of $\phi$ with IS bluff-body is 0.45 kPa that is approximately 21$\%$ higher than the strength of pressure oscillations ($p^{\prime}_{rms}$ = 0.37 kPa) at high values of $\phi$ (combustion noise) with the BL bluff-body. The power spectral density of $p^{\prime}$ for REC at $\phi$ = 0.69 shows a shallow band centered around 106.2 Hz (Fig. 3f). This dominant mode obtained using IS bluff-body at REC is at a considerably lower frequency than the dominant mode observed for the BL bluff-body at REC.

Irrespective of the changes in the thermal power loading ($\dot{m}_{fuel}$), there is a suppression of thermoacoustic instability when the IS bluff-body is used (shown in Supplementary Fig. S1). However, a higher reduction of the strength of pressure oscillations is observed for higher $\dot{m}_{fuel}$. Further, as the bluff-body is brought closer to the dump plane from 45 mm to 40 mm, the reduction in the strength of oscillations is not as large as that observed with 45 mm bluff-body position. For this bluff-body position of 40 mm, the PSD of $p^{\prime}$ at $\phi$ = 0.69 displayed in Fig. 3h shows a shallow band centered around 116 Hz, which is higher when compared to the band corresponding to REC with IS bluff-body. However, this frequency is still lower than the narrow band observed for REC with BL bluff-body (Refer Fig. 2b).

We investigate the effect of changes in the area of the passage by the four-hole (4H) bluff-body. Analysis of the local heat release rate fluctuations prove that for the 4H bluff-body, at Z1 and Z2, i) higher $\dot{q}^{\prime}_{rms}$ exists compared to IS bluff-body (Figs. 4a-c) and ii) high growth of $\sigma^{2}$ occurs (Fig. 4d). In addition, we observe a small growth of $\sigma^{2}$ in the wake of the 4H bluff-body due to secondary flames, as observed with the IS bluff-body.

When comparing the experimental results at REC between 4H and BL bluff bodies, we do not observe any large changes in the transition from low-amplitude pressure fluctuations to high-amplitude pressure oscillations (Fig. 4e). A gradual growth in $p^{\prime}_{rms}$ occurs when using REC (Fig. 4e). The strength of $p^{\prime}$ at low $\phi$ is also similar in magnitude with the strength obtained for BL bluff-body. However, the power spectral density of $p^{\prime}$ at $\phi$ = 0.69 shows a sharp peak centered around 115.7 Hz (Fig. 4f), which is significantly lower than the the dominant mode observed with the BL bluff-body (Fig. 2b) for the same combustor length. Thus, from the results obtained with both IS and 4H bluff bodies, it is evident that the introduction of slots has shifted the dominant frequency to a lower value for the experiments performed at REC.

When operating at a higher rate of change of $\dot{m}_{air}$, the change in the transition is negligible in comparison to the $p^{\prime}_{rms}$ for REC (Fig. 4e). The dominant frequency at 0.69 is at 114.7 Hz (Fig. 4g). Thus, multiple experiments with the 4H bluff-body show that the onset of thermoacoustic instability is not suppressed. The reduction in the amplitude of pressure oscillations for the 4H configuration is negligible with respect to the BL bluff-body.

We now investigate the effect of change in the distribution of the passages on the bluff-body. To that end, we utilize the MH bluff-body. In Figs. 5a-c, we observe high strength of $\dot{q}^{\prime}_{rms}$ at Z1 and downstream of the MH bluff body for REC. As seen in Fig. 5d, with the MH bluff-body, the growth of $\sigma^{2}$ still exists at Z2, although it is low. Higher growth of $\sigma^{2}$ downstream of the bluff-body can be attributed to the large heat release rate fluctuations of the multiple secondary flames anchored on the bluff-body surface.

Using the MH bluff-body, we observe a similar transition from combustion noise to thermoacoustic instability to that of BL bluff body till $\phi$ = 0.75 (refer Figs. 5e and 2g). The $p^{\prime}_{rms}$ for REC (679 Pa) is slightly higher compared to the $p^{\prime}_{rms}$ obtained with the BL bluff-body at REC (644 Pa). The power spectral density of $p^{\prime}$ at $\phi$ = 0.69 show the dominant frequency at 128 Hz (Fig. 5f). This dominant frequency is still lower than the dominant frequency obtained with the BL bluff-body (Fig. 2b) for the same combustor length, but closest to it as compared to the values obtained with IS and 4H bluff bodies. Thus, suppression of the transition does not occur for the MH bluff-body.

Interestingly, as the equivalence ratio is reduced further to less than 0.66, an additional very sharp transition occurs as seen in Fig. 5e. The increase in the strength of pressure oscillations is rather catastrophic, increasing to almost 300 $\%$ of the $p^{\prime}_{rms}$ value after the primary bifurcation. This secondary bifurcation is observed only for the MH bluff body. The dominant frequency increases to 150 Hz for the secondary bifurcation at $\phi$ = 0.62 (Fig. 5g).

This secondary bifurcation highlights the challenge in suppressing the onset of thermoacoustic instability without introducing other routes for the dangerous transition to occur. Although the reason behind introducing the slots on the bluff-body is to reduce the growth of local heat release rate fluctuations upstream of the bluff-body at the stagnation zone, not all designs result in a reduction of fluctuations and hence the suppression of thermoacoustic instability. In the case of the MH bluff-body, the highest growth of local heat release rate fluctuations is not observed at the stagnation point, but rather downstream of the bluff-body as a result of the secondary flames (Fig. 5h). The appearance of new regions with high growth of heat release rate fluctuations indicates new pathways through which thermoacoustic instability emerges. Thus, the MH configuration does not suppress thermoacoustic instability but in fact enhances the strength of pressure oscillations at certain conditions.