4.1 Spatiotemporal evolution of acoustic power sources during combustion noise and thermoacoustic instability

We first examine the spatiotemporal evolution of the acoustic power sources during combustion noise (stable combustion) and thermoacoustic instability (unstable combustion). The spatial distributions of the node strengths (3.1) of the acoustic power and the vorticity networks at different time instants during combustion noise are shown in figure 3. In the acoustic power network (figure 3 b), the blue colour represents the local acoustic power sinks. The colourbar indicates the magnitude of node strength, which corresponds to the local acoustic power production in the turbulent reactive flow field. The spatial distribution of the node strength reveals that the acoustic power production happens in small fragmented clusters during combustion noise. The distribution of the local acoustic power sources resembles small islands scattered in an ocean of acoustic power sinks. In the vorticity network (figure 3 c), the node strength gives the average vorticity in the immediate neighbourhood of a node. Velocity vectors obtained from the analysis of PIV are superimposed onto the vorticity network to understand the flow field better. High values of positive node strength (denoted by red regions) indicate the presence of a train of small discrete vortices in the outer shear layer. High values of negative node strength (denoted by blue regions) in the inner shear layer represents the small vortices shed from the tip of the bluff body. The grey colour mask in the subplots (figures 3 b and c) represents the bluff body. The bluff body shape in figures 3(b) and (c) looks dissimilar due to the fact that in figure 3(b) the nodes are the pixels whereas in figure 3(c), the nodes are the interrogation windows used in the analysis of PIV. The size of the interrogation windows is 32 pixels $\times$ 32 pixels (Unni et al. Reference Unni, Krishnan, Manikandan, George, Sujith, Marwan and Kurths2018). Hence, the area that a node occupies in figure 3(c) is bigger than that in figure 3(b). Also, in figure 3(c), the mask includes the region where laser reflection is high.

Figure 3.

The spatial distribution of the node strengths of (b) the acoustic power network and (c) the vorticity network at the corresponding time instants (A, B, C and D) marked on (a) the time series of fluctuating pressure during combustion noise. The blue region in the spatial distribution of the node strength of the acoustic power ( $p^{\prime }\dot{q}^{\prime }$ ) network represents local acoustic power sinks. The colourbar in (b) represents the magnitude of the node strength, which indicates the average acoustic power produced in the neighbourhood of a node. The spatial distribution of the node strength shows that acoustic power production happens in small disconnected fragments during combustion noise. In the vorticity network (c), the node strength indicates the average vorticity in the neighbourhood of a node. Positive values of node strength represent the vorticity distribution in the anticlockwise direction, whereas negative node strength indicates the vorticity distribution in the clockwise direction. Velocity vectors are superimposed onto the vorticity network to understand the flow field better. Small vortices are present in the outer shear layer as well, as they are being shed from the tip of the bluff body. The grey rectangles in the subplots represent the bluff body.

Figure 4.

The spatial distribution of the node strengths of (b) the acoustic power network and (c) the vorticity network at the corresponding time instants (A, B, C, D and E) marked on (a) the time series of oscillating pressure during thermoacoustic instability. The spatial distribution of the node strength of the acoustic power network indicates that during thermoacoustic instability, the acoustic power production occurs in large clusters. The high values of positive node strength of the vorticity network coincide with the centre of the coherent structure formed downstream of the dump plane. The maximum acoustic power production occurs when the coherent structure impinges on the top wall of the combustor (C). The magnitude of acoustic power production during thermoacoustic instability is almost three orders of magnitude higher than that during combustion noise, as indicated by the node strength of the acoustic power network.

In figure 4, the spatial distributions of the node strengths of the acoustic power and the vorticity networks at different time instants during a periodic cycle of thermoacoustic instability are shown. Unlike the situation during combustion noise, the acoustic power production happens now in large clusters during thermoacoustic instability (figure 4 b). The acoustic power production during thermoacoustic instability is around three orders of magnitude higher than that during combustion noise, as indicated by high values of node strength. High values of positive node strength in the vorticity network (at time instants A and B in figure 4 c) coincide with the centre of the coherent structure being formed downstream of the bluff body. The coherent structure begins to form at the dump plane, when the acoustic pressure fluctuation is near local minima (E). As the coherent structure convects downstream it grows in size. The formation of large coherent structures at the onset of thermoacoustic instability has been reported in earlier studies, as discussed in § 1. The maximum acoustic power production occurs when the large coherent structure impinges on the top combustor wall.

As the coherent structure convects towards the top of the bluff body (from time instants A to C in figure 4 c), the region between the outer and inner shear layer is compressed against the rear end of the bluff body. This increases the shear strain, which leads to local extinction (Hult et al. Reference Hult, Meier, Meier, Harvey and Kaminski2005). This may be the reason for almost no reaction to take place just upstream of the bluff body, leading to the presence of local acoustic power sinks in the rear end of the bluff body, above the bluff body shaft. As the coherent structure passes over the top of the bluff body, separation of the inner shear layer from the tip of the bluff body happens. This suggests the frequency of formation of the coherent structure downstream of the dump plane and the large vortex from the tip of the bluff body to be almost identical. High values of negative node strength in the vorticity network indicate the shedding of these large vortices from the tip of the bluff body. As the large vortex convects downstream of the bluff body, the region between the outer and the inner shear layer expands. This is accompanied by a flow reversal at the bluff body due to volume expansion because of intense heat release, as well as the reversal of acoustic velocity. This inhibits the flow of fresh reactant mixture into the combustion zone. This is the reason why we have large clusters of acoustic power sources in the flow field when the acoustic pressure fluctuation is negative (see time instants E and A in figure 4 b). When the pressure inside the combustion chamber becomes very low, the flow surges back into the combustion chamber. As the acoustic mode is reinforced by the intense in-phase heat release rate, the velocity fluctuation amplitude increases. This induces the formation of a large coherent structure downstream of the dump plane and the cycle continues.

4.2 Spatiotemporal evolution of acoustic power sources during intermittency

Next, we examine the spatiotemporal evolution of the acoustic power sources during intermittency. Intermittency can be viewed, in simple terms, as a tug of war between the acoustic driving and the acoustic damping present in the system. As the acoustic driving wins over the damping, the growth of the acoustic pressure oscillations occurs. As the damping overtakes the driving, these high-amplitude acoustic pressure oscillations start to decay. It is this competition between the acoustic driving and the damping that leads to the emergence of bursts of high-amplitude periodic oscillations in the background of low-amplitude aperiodic pressure oscillations during intermittency.

In this section, we will see how spatially continuous large clusters of acoustic power sources emerge at the onset of thermoacoustic instability from the small fragmented clusters of acoustic power sources observed during combustion noise. To that end, we investigate the spatial and temporal evolution of acoustic power sources during an aperiodic epoch of intermittency and a periodic epoch of intermittency, the growth of low-amplitude aperiodic pressure fluctuations to high-amplitude periodic oscillations and the decay of high-amplitude periodic pressure oscillations to low-amplitude aperiodic fluctuations. In figure 5, we show the spatiotemporal evolution of acoustic power sources during an aperiodic epoch of intermittency. In this case, similar to the one observed during combustion noise, the acoustic power production happens in small fragmented clusters or islands scattered in an ocean of acoustic power sinks.

Next, we examine the spatiotemporal evolution of acoustic power sources during a periodic epoch of intermittency, as shown in figure 6. We observe coherent structures downstream of the dump plane, as seen during thermoacoustic instability. However, there are a few points to be noted here. Firstly, the size and strength of the coherent structure are not as high as in that observed during thermoacoustic instability. George et al. (Reference George, Unni, Raghunathan and Sujith2018) quantitatively measured the size of the coherent structures during the different stages of combustor operation using a vortex detection technique. Also, the strength of the coherent structure, which is reflected in the magnitude of the positive node strength, is lower than that during thermoacoustic instability. More significantly, the coherent structure impinges on the rear-end tip of the bluff body (see time instant C in figure 6 c) and not on the top wall of the combustor. After impingement, the coherent structure breaks down into small vortices. Since the area of impingement of the coherent structure is lower, the heat release rate that follows is not as intense as in the case of thermoacoustic instability. It is to be noted that these coherent structures are not consistently observed during the periodic epochs of intermittency. They are mostly observed during the high-amplitude peaks of the periodic epochs of intermittency. Here also, we observe the shedding of large vortices from the tip of the bluff body. It is to be noted that the size of the clusters of the acoustic power sources is not as large as that observed during thermoacoustic instability.

Figure 5.

The spatial distribution of the node strengths of (b) the acoustic power network and (c) the vorticity network at the corresponding time instants (A, B, C and D) marked on (a) the time series of fluctuating pressure during the aperiodic regime of intermittency. The acoustic power production happens in small fragmented clusters, as in the case of combustion noise.

Figure 6.

The spatial distribution of the node strengths of (b) the acoustic power network and (c) the vorticity network at the corresponding time instants (A, B, C, D and E) marked on (a) the time series of fluctuating pressure during the periodic regime of intermittency. The acoustic power production occurs in large clusters as in the case of thermoacoustic instability. However, unlike the case during thermoacoustic instability, the coherent structure impinges the rear-end tip of the bluff body. After impingement, the coherent structure disintegrates into small vortices that convect over the top of the bluff body.

In order to understand how large clusters of acoustic power sources emerge in the flow field during the periodic epoch of intermittency, we examine the spatiotemporal evolution of the acoustic power sources during the growth of low-amplitude aperiodic fluctuations to high-amplitude periodic oscillations, as shown in figure 7. As the pressure oscillations grow, we find the emergence of acoustic power sources at a greater number of spatial locations in the turbulent reactive flow field. We refer to this phenomenon as the nucleation of acoustic power sources in the flow field. Also, as the acoustic pressure amplitude grows, the small fragmented clusters of acoustic power sources coalesce to form large clusters, as reflected in the spatial distribution of the node strength of the acoustic power network (figure 7 b). The mechanism behind the formation of these large clusters can be deduced by examining the vorticity network (figure 7 c). As the acoustic pressure grows, the magnitude of the positive as well as the negative node strength increases; this denotes the increase in the strength of the vorticity field upstream and downstream of the bluff body, respectively.

Figure 7.

The spatial distribution of the node strengths of (b) the acoustic power network and (c) the vorticity network at the corresponding time instants (A, B, C, D, E and F) marked on (a) the time series of fluctuating pressure during the growth of pressure oscillations observed during intermittency. As the thermoacoustic system transits from low-amplitude aperiodic oscillations to high-amplitude periodic oscillations during intermittency, the acoustic power sources nucleate, coalesce and grow in size and the strength of acoustic power production also increases.

During the aperiodic epoch of intermittency, a train of small discrete vortices is observed in the outer shear layer downstream of the dump plane. At the same time, small vortices are shed downstream of the bluff body as the inner shear layer gets separated from the tip of the bluff body. As the perturbations near the dump plane increase, these small vortices in the outer shear layer merge to form large vortices. Merging of vortices leads to an increase in fine scale mixing that accelerates combustion (Schadow & Gutmark Reference Schadow and Gutmark1992), thus causing the nucleation of acoustic power sources in the flow field. In addition, as the small vortices merge, coalescence of small fragmented clusters of acoustic power sources occurs. These large vortices convect downstream of the dump plane and impinge on the rear-end tip of the bluff body. Upon impingement, a spike occurs in the heat release rate due to the sudden increased fine scale mixing of the reactants and hot products. This spike in the heat release rate adds energy to the acoustic field, thus acting as a kicked oscillator (Seshadri, Nair & Sujith Reference Seshadri, Nair and Sujith2016). Since the heat release rate occurs in phase with the acoustic pressure oscillation, the amplitude of the acoustic pressure increases, satisfying Rayleigh criteria. This reinforcement of the acoustic mode leads to an increase in the amplitude of velocity fluctuations at the dump plane, thus inducing the formation of a new vortex, and the aforementioned cycle continues. When the perturbation in the flow field is high enough, a large coherent structure forms downstream of the dump plane, gets convected and impinges on the top wall of the combustor. As the large coherent structure carries with it a large pocket of unburnt reactants, upon impingement, the heat release rate is higher. Hence, the energy added to the acoustic field overcomes the damping present in the combustor and the onset of thermoacoustic instability occurs. We note that the convection of the large vortex towards the rear-end tip of the bluff body/top wall of the combustor perturbs the inner shear layer more and hence the size of the vortices shed downstream of the bluff body also increases.

Finally, we examine the spatiotemporal evolution of acoustic power sources during the decay of high-amplitude periodic pressure oscillations to low-amplitude aperiodic pressure fluctuations, as shown in figure 8. During the high-amplitude periodic pressure oscillations, the acoustic power production occurs in large clusters, as evident at time instant A. As the acoustic damping overcomes the acoustic driving in the combustor, the amplitude of the pressure oscillation decays. As the thermoacoustic system transits from high-amplitude periodic oscillations to low-amplitude aperiodic fluctuations, the strength of acoustic power production diminishes and the large clusters disintegrate into small fragmented clusters.

The decay of the acoustic pressure amplitude during intermittency begins to occur once the damping in the turbulent combustor starts to outweigh the acoustic driving. As the damping overcomes the driving, the roll-up of the shear layer into a large vortex, observed during the periodic epochs of intermittency, fails to happen, thus ceasing the formation of large clusters of acoustic power sources. Hence, the amount of energy added to the acoustic field reduces. Henceforth, the periodic epochs of intermittency fail to persist and decay, thus preventing the onset of full-blown thermoacoustic instability.

Figure 8.

The spatial distribution of the node strengths of (b) the acoustic power network and (c) the vorticity network at the corresponding time instants (A, B, C, D, E and F) marked on (a) the time series of fluctuating pressure during the decay of pressure oscillations observed during intermittency.

Thus, we observe that the emergence of large clusters of spatially coherent acoustic power sources at the onset of thermoacoustic instability happens via the phenomena of nucleation, coalescence and growth of acoustic power sources. The physical mechanisms behind these phenomena are the merging of small-scale vortices and the formation of larger vortices as the perturbations to the shear layer increase. These findings can be generalized to turbulent thermoacoustic systems involving large-scale vortex shedding. The formation of spatially coherent regions of acoustic power sources at the onset of thermoacoustic instability has also been reported in swirl stabilized combustors (Paschereit, Gutmark & Weisenstein Reference Paschereit, Gutmark and Weisenstein1999; Lee & Santavicca Reference Lee and Santavicca2003; Huang & Ratner Reference Huang and Ratner2009; Tachibana et al. Reference Tachibana, Kanai, Yoshida, Suzuki and Sato2015; Okuno et al. Reference Okuno, Small and Gotoda2015; Kheirkhah et al. Reference Kheirkhah, Geraedts, Saini, Venkatesan and Steinberg2016; Pawaret al. Reference Pawar, Mondal, George and Sujith2018). Hence, we can hypothesize that the phenomena of nucleation, coalescence and growth of acoustic power sources would be happening during intermittency. However, the physical mechanisms leading to the phenomena of nucleation, coalescence and growth may be different as the flow field in a swirl stabilized combustor is different from that observed in a bluff body stabilized combustor.

4.3 Statistical analysis of the clusters of acoustic power sources

In this section, we perform a statistical analysis of the clusters of acoustic power sources as the thermoacoustic system transits from a state of combustion noise to thermoacoustic instability via intermittency. The use of complex network theory enables us to quantify the spatiotemporal evolution of the clusters of acoustic power sources. To that end, we calculate the link density ( $\unicode[STIX]{x1D70C}$ ), the size of the giant cluster ( $S$ ) and the number of clusters ( $N$ ) at different equivalence ratios as the combustor transits from combustion noise to thermoacoustic instability.

Link density refers to the fraction of the links or edges present in the network. If $n$ represents the total number of nodes and $e$ represents the actual number of links in the network, then link density is given by

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D70C}=\frac{2e}{n(n-1)}.\end{eqnarray}$$

The value of $\unicode[STIX]{x1D70C}$ lies between 0 and 1. In simple terms, $\unicode[STIX]{x1D70C}$ reflects the number of connections among the nodes of the network. A high value of $\unicode[STIX]{x1D70C}$ implies that the network is densely connected and a low value implies that the network is sparsely connected. In the present analysis, a high value of $\unicode[STIX]{x1D70C}$ implies that acoustic power production happens in a large number of spatial locations in the flow field as we do not consider acoustic power sinks in the network construction, as explained in § 3.

In graph theory, a connected component of an undirected graph refers to a subgraph in which any two nodes are connected to each other by a path, i.e. a sequence of links (Newman Reference Newman2010). In the present study, the connected components refer to islands or clusters of acoustic power sources. We track the number of such clusters ( $N$ ) as well as the size of the giant cluster ( $S$ ); i.e. the largest connected component of a graph, at each instant of time. In this manner, we obtain, for a given equivalence ratio, $\unicode[STIX]{x1D70C}$ , $N$ and $S$ at every instant of time. Thus, for each equivalence ratio, we construct 2000 acoustic power networks, corresponding to a duration of 1.0 s, as described in § 3.

Figure 9.

Half cross-section of the combustion chamber showing the ROI, in which the acoustic power network is constructed to compute the network properties. The ROI is chosen such that it coincides with the location of formation of coherent structures downstream of the dump plane. The pressure transducer used for the acoustic pressure measurement is located 40 mm downstream of the dump plane.

Since most of the interesting flow dynamics leading to the coalescence/disintegration of clusters happens upstream of the bluff body, we choose a rectangular ROI upstream of the bluff body for the acoustic power network construction. The chosen ROI, at a distance of 15 mm downstream of the dump plane and 10 mm above the bluff body shaft, has physical dimensions of 25 mm in length and 20 mm in breadth (figure 9). The ROI has a resolution of 225 pixels $\times$ 180 pixels. Since the calculation of network properties with the original resolution would be computationally very intensive, we downsampled the resolution to 112 pixels $\times$ 90 pixels (50 % of the original resolution) using bi-cubic interpolation (Zhang et al. Reference Zhang, Zhao, Zhang, Xiong and Gao2011). Hence, the number of nodes in the network reduces from 40 500 to 10 080. Also, a small threshold ( $\unicode[STIX]{x1D716}$ ) was introduced for the connections among the nodes to prevent the electronic noise of the high-speed camera, used for chemiluminescence measurements, from creeping into the calculation, especially during combustion noise. We choose the threshold for network link connectivity as $\unicode[STIX]{x1D716}=1.0$ (around 5 % of the maximum acoustic power production during combustion noise). Hence, two neighbouring nodes are connected if the values of $p^{\prime }\dot{q}^{\prime }$ at these nodes are greater than 1.0 instead of 0.0 (without any threshold). The sensitivity of the network properties for different thresholds of network link connectivity ( $\unicode[STIX]{x1D716}$ ) and the convergence of the network properties with respect to the number of images/networks considered for their calculation are discussed in appendix A (see figures 12 and 13). In this study, we use the NetworkX Python package to calculate the network properties (Hagberg, Swart & Chult Reference Hagberg, Schult, Swart, Varoquaux, Vaught and Millman2008).

Figure 10.

Variation of the temporal mean of the network properties: (a) the link density ( $\unicode[STIX]{x1D70C}$ ), (b) the size of the giant cluster ( $S$ ) and (c) the number of clusters ( $N$ ), along with the variation of the root mean square (r.m.s.) value of the acoustic pressure as the thermoacoustic system transits from combustion noise to thermoacoustic instability via intermittency. The increase in the link density indirectly indicates the increase in the number of locations in the flow field where acoustic power production occurs. The increase in the size of the giant cluster along with the decrease in the number of clusters show that the acoustic power sources coalesce to form large clusters at the onset of thermoacoustic instability. The dashed vertical lines divide the parameter space, i.e. the equivalence ratio $(\unicode[STIX]{x1D719})$ , into regimes of combustion noise, intermittency and thermoacoustic instability, respectively.

Figure 10 illustrates the variation of the temporal mean of $\unicode[STIX]{x1D70C}$ (figure 10 a), $S$ (figure 10 b) and $N$ (figure 10 c) of the acoustic power sources at seven different equivalence ratios that cover the transition of the thermoacoustic system from combustion noise to thermoacoustic instability via intermittency. The dashed vertical lines in the plot separate the combustor operational regime into combustion noise, intermittency and thermoacoustic instability, respectively. The variation of the root mean square (r.m.s.) value of the acoustic pressure is also plotted along with the variation of the network properties. As the thermoacoustic system transits from combustion noise to thermoacoustic instability, the average $\unicode[STIX]{x1D70C}$ increases sevenfold. This implies that the average number of links during thermoacoustic instability is around seven times that during combustion noise (4.1). A higher value of the average number of links during thermoacoustic instability implies that the acoustic power production occurs at a large number of locations in the flow field when compared to that during combustion noise. This reflects the phenomenon of nucleation of acoustic power sources occurring in the turbulent reactive flow field. The very low value of the average $\unicode[STIX]{x1D70C}$ (the maximum value is around 0.0004) is due to the fact that only the nearest neighbours of a node are considered when establishing a link, as mentioned in § 3.

During combustion noise, the acoustic power production happens mostly in a large number of fragmented clusters or islands in an ocean of acoustic power sinks. This explains the presence of a large number of clusters of acoustic power sources during the occurrence of combustion noise in figure 10(c). Mondal et al. (Reference Mondal, Unni and Sujith2017) attributed this spatial incoherence in the acoustic power production to the random distribution of instantaneous phase between the pressure oscillations and the heat release rate oscillations during the occurrence of combustion noise. Also, the average $S$ (in figure 10 b) is very small during combustion noise, as the acoustic power production occurs in small fragmented clusters. The average $S$ during thermoacoustic instability is approximately six times the value observed during combustion noise. Additionally, we observe only half the average $N$ at the onset of thermoacoustic instability when compared to that during combustion noise. These observations clearly show that the large number of fragmented clusters of acoustic power sources observed during combustion noise coalesce to form large clusters at the onset of thermoacoustic instability.

Figure 11.

Variation of the standard deviation of the network properties: (a) the link density ( $\unicode[STIX]{x1D70C}$ ), (b) the size of the giant cluster ( $S$ ) and (c) the number of clusters ( $N$ ), along with the variation of the r.m.s. value of the acoustic pressure as the thermoacoustic system transits from combustion noise to thermoacoustic instability via intermittency. The dashed vertical lines divide the parameter space, i.e. the equivalence ratio $(\unicode[STIX]{x1D719})$ , into regimes of combustion noise, intermittency and thermoacoustic instability, respectively.

Figure 11 shows the variation of the standard deviation of $\unicode[STIX]{x1D70C}$ (figure 11 a), $S$ (figure 11 b) and $N$ (figure 11 c) of the acoustic power sources, along with the variation of the r.m.s. value of the acoustic pressure. During combustion noise, there are instances where large clusters of acoustic power sources are present in the flow field upstream of the bluff body. However, these large clusters having positive values of $p^{\prime }\dot{q}^{\prime }$ occur mostly when the acoustic pressure fluctuation is negative. During combustion noise, most of the reaction occurs in the recirculation zone downstream of the bluff body. In the ROI considered for the calculation of the network properties, which is upstream of the bluff body, not much reaction happens most of the time. Hence, when the acoustic pressure fluctuation ( $p^{\prime }$ ) and the local heat release rate fluctuation ( $\dot{q}^{\prime }$ ) become negative simultaneously, we observe large clusters of acoustic power sources (positive values of $p^{\prime }\dot{q}^{\prime }$ ) upstream of the bluff body. This explains the high value of the standard deviation for $\unicode[STIX]{x1D70C}$ , $S$ and $N$ during combustion noise. However, we observe high values of standard deviation only for $\unicode[STIX]{x1D70C}$ and $S$ during intermittency. This can be attributed to the fact that we observe small fragmented clusters of acoustic power sources during the aperiodic epochs of intermittency and large clusters of acoustic power sources during the periodic epochs of intermittency. The standard deviation for the number of clusters of acoustic power sources is low during thermoacoustic instability compared to that during combustion noise (figure 11 c). As discussed earlier, during thermoacoustic instability, the acoustic power production happens mostly in large clusters. Hence, the average number of clusters (see figure 10 c) as well as the standard deviation (figure 11 c) is low during thermoacoustic instability. During one acoustic cycle, when the coherent structure convects out of the ROI towards the top of the bluff body, there will not be many acoustic power sources inside the region of interest. This is the reason for the high value of the standard deviation for link density (figure 11 a) and the size of the giant cluster (figure 11 b) during thermoacoustic instability.

We note that the gradual change in the average network properties happens much before the rise in the r.m.s. of the acoustic pressure as the thermoacoustic system transits from combustion noise to thermoacoustic instability via intermittency (figure 10). This suggests that the transition from small fragmented clusters to large clusters of acoustic power sources is not abrupt. In statistical physics, percolation theory is used in lattice models to study the phenomenon of phase transition. The variation of the size of the giant cluster is used as a marker to study the critical behaviour near the percolation threshold (Newman Reference Newman2010; Barabási Reference Barabási2016). Using complex network theory, researchers have reported percolation-like phase transitions in brain networks (Gallos, Sigman & Makse Reference Gallos, Sigman and Makse2012; Fornito, Zalesky & Bullmore Reference Fornito, Zalesky and Bullmore2016), protein networks (Deb, Vishveshwara & Vishveshwara Reference Deb, Vishveshwara and Vishveshwara2009; Weber & Pande Reference Weber and Pande2015) and recently in climate networks (Lu, Yuan & Fu Reference Lu, Yuan and Fu2016). In the present study, the variation of the average size of the giant cluster (see figure 10 b) reveals a continuous percolation-like phase transition at the onset of thermoacoustic instability in a turbulent combustor. However, a detailed analysis is required to confirm whether this is the case in turbulent combustors with other flame stabilization mechanisms.