Seeds of phase transition to thermoacoustic instability

Tackling the problem of emissions is at the forefront of scientific research today. While industrial engines designed to operate in stable regimes produce emissions, attempts to operate them at ‘greener’ conditions often fail due to a dangerous phenomenon known as thermoacoustic instability. Hazardous high amplitude periodic oscillations during thermoacoustic instability lead to the failure of these engines in power plants, aircraft, and rockets. To prevent this catastrophe in the first place, identifying the onset of thermoacoustic instability is required. However, detecting the onset is a major obstacle preventing further progress due to spatiotemporal variability in the reacting field. Here, we show how to overcome this obstacle by discovering a critical condition in certain zones of the combustor, which indicates the onset of thermoacoustic instability. In particular, we reveal the critical value of the local heat release rate that allows us to distinguish stable operating regimes from hazardous operations. We refer to these zones as seeds of the phase transition because they show the earliest manifestation of the impending instability. The increase in correlations in the heat release rate between these zones indicates the transition from a chaotic state to a periodic state. Remarkably, we found that observations at the seeds of the phase transition enable us to predict when the onset occurs, well before the emergence of dangerous large-amplitude periodic acoustic pressure oscillations. Our results contribute to the operation of combustors in more environment-friendly conditions. The presented approach is applicable to other systems exhibiting such phase transitions.


Introduction
All combustion engines in automobiles, airplanes, ships etc produce emissions. But industrial systems produce more emissions such as oxides of nitrogen and sulphur due to high flame temperatures. To reduce these emissions and operate in 'greener' conditions, we need to lower the flame temperatures. Increasing the proportion of air to the fuel reduces the flame temperature. However, such 'greener' operations often result in the dangerous phenomenon of thermoacoustic instability, which results in large amplitude oscillations leading to catastrophic consequences. Controlling the occurrence of thermoacoustic instability creates opportunities to operate under 'greener' conditions. But the major problem is in identifying the onset of thermoacoustic instability. This is a problem that has been documented in various systems, but still remains a challenging one (Biggs 2009, Blomshield et al 1997, Sutton 2003, Lieuwen and Yang 2005, McManus et al 1993, Lieuwen 2002, Ducruix et al 2003, Juniper and Sujith 2018. Thermoacoustic instability occurs because of the coupling between the flame and the acoustics field within the combustion chamber (McManus et al 1993, Akkerman and Law 2013, Moon et al 2020 reinforced by emerging large coherent structures or vortices in the flow field (Poinsot et al 1987, Schadow et al 1989.
Turbulent flow systems are characterized by chaotic or disordered behaviour. However, at certain conditions they have organized flow structures (Kline et al 1967, Brown and Roshko 1974, Cantwell 1981. The emergence of such ordered spatiotemporal patterns is ubiquitous in ecology (Ball andBorley 1999, Meron 2016), climate (Rastogi et al 2016, Chang et al 2016, chemistry (Busse 1978, Croquette 1989, biology (Ball andBorley 1999, Kanders et al 2020), engineering (Cross andHohenberg 1993, Bayliss andMatkowsky 1991), etc. Such regularities in the spatiotemporal behaviour appear near a phase transition, when certain system parameters move beyond a critical point (Scheffer et al 2001, Yang et al 2012, Kanders et al 2020. This phase transition demarcates two contrasting regimes of the system. However, identifying the impending phase transition is quite often a challenge, since there is no visible change in the dynamics as the system approaches the critical point.
Recently, progress has been made towards the solution of identifying the phase transition, wherein some studies on spatiotemporal systems show that while a phase transition occurs for the whole system, there may be regions where such transitions emerge earlier. In a study on the Indian monsoon prediction, Stolbova et al (2016) uncovered two unique locations in the Indian subcontinent (referred to as tipping elements), where critical conditions originate and then propagate in a certain direction that results in the transition from pre-monsoon to monsoon; the regularities between the tipping elements allows a long term prediction of the monsoon onset. In short, it is necessary to locate such zones to identify the origin of the phase transition. In this paper, we focus on the identification of such zones in a turbulent combustion system. We utilize the phenomenon of critical growth of fluctuations (Surovyatkina 2004, Surovyatkina et al 2005 as an indicator of the onset of such a phase transition.
In particular, we investigate the spatiotemporal dynamics in a confined reactive turbulent system when it approaches the state of thermoacoustic instability (S TI ). In this system, flow perturbations cause heat release rate fluctuations from the flame, which subsequently generates sound waves. Next, these sound waves get reflected from the boundaries back to the source and further modify the heat release rate fluctuations. This process results in a positive feedback, when the acoustic pressure fluctuations due to the sound waves and the unsteady heat release rate become in phase (Rayleigh 1878). Such a positive feedback favours the growth of acoustic oscillations, leading to periodic temporal dynamics (McManus et al 1993, Lieuwen 2002, Ducruix et al 2003, Juniper and Sujith 2018 and the periodic emergence of spatially organized patterns (Candel 1992, Lieuwen 2012, Sujith and Unni 2020a, 2020b. Notably, the dangerous high amplitude periodic oscillations of acoustic pressure during S TI cause significant losses to the power and propulsion industry by causing structural failure, damaging the navigation and control system, overwhelming the thermal protection system, etc (Lieuwen 2002, 2012, Juniper and Sujith 2018. The onset of periodic oscillations (S TI ) occurs via a transition from a state of chaotic fluctuations (S C ), through the state of intermittency (S IN ) (Gotoda et al 2011, Delage et al 2017.
Different spatiotemporal patterns characterize each state of the system. It was recently discovered that S C , referred to as combustion noise, is chaotic (Nair et al 2013, Tony et al 2015. S C is also characterized by disorganized flow and flame dynamics (Mondal et al 2017, Raghunathan et al 2020. During S IN , bursts of periodic oscillations appear amidst epochs of aperiodic dynamics at apparently random intervals . Additionally, during S IN , macroscopic patterns at the size of the system geometry emerge during the regime of periodic oscillations, while disordered small-scale structures exist during the regime of aperiodic oscillations (Mondal et al 2017, Raghunathan et al 2020. On the other hand, S TI typically exhibits the coexistence of large-scale coherent flow structures or vortices, spatially and temporally organized reaction fields, distinct standing wave modes, large-amplitude periodic acoustic pressure oscillations etc . In fact, during the transition from S C to S TI via S IN , the disordered dynamics decreases continuously, while the ordered dynamics increases through the collective behaviour of small-scale structures , Raghunathan et al 2020. This continuous evolution makes it difficult to identify the critical point of the system parameter or the onset of S TI , which is crucial for designing safe operating regimes of such complex spatiotemporal systems. Further, whether the onset of S TI can be forewarned has been a vital research question that has been pursued in the recent decades , Gopalakrishnan et al 2016, Gotoda et al 2014, Kobayashi et al 2019, Sengupta et al 2020. The present study is the first attempt to identify the onset of thermoacoustic instability that is not based on user-defined thresholds. To that end, we take a step towards identifying zones where conditions for the phase transition originate. Following the approach of Stolbova et al (2016), we utilize the variance of fluctuations of system observables (flame fluctuations) at different locations across the spatial domain of the thermoacoustic system to find zones with maximum growth in the variance of fluctuations, unraveling the seeds of the phase transition. Furthermore, we identify emerging long-range correlations in the flame fluctuations between these zones well before S TI . Finally, we analyze the flame fluctuations at these zones to define the onset of S TI and predict it in advance. We conjecture that such an approach will be helpful to Figure 1. Bluff-body stabilized turbulent thermoacoustic instability system. The main components of the experimental system are a plenum chamber, a burner, a combustion duct and a decoupler. We simultaneously measure acoustic pressure fluctuations p (t), global heat release rateQ (t), and high speed CH * chemiluminescence field as a function of fuel-air mixture ratio. The inset shows the zoomed version of the combustion chamber with the bluff body that is used to stabilize the flame. identify the earliest manifestation of phase transitions and predict them in other engineering and natural systems as well.

Experimental setup
We perform experiments in a turbulent thermoacoustic system. The turbulent combustor consists mainly of a plenum chamber, a burner and a combustion chamber. The schematic of the experimental setup is shown in figure 1. The flame stabilizes inside the combustor on a flame holding device called a bluff-body shown as an inset. Air and liquefied petroleum gas (LPG: butane 60% and propane 40%) mix just before the burner. Most of the reaction occurs in the combustion chamber. Table 1 lists the dimensions of the turbulent combustor.
In this study, we vary the fuel-air mixture ratio to control the system characteristics. Mass flow controllers (MFCs) (Alicat Scientific, MCR Series) are used to control and measure the mass flow rates of air and fuel, with an uncertainty of ±0.8% of the flow rate reading in standard liters per minute (SLPM) +0.2% of the maximum flow rate of the air flow controller: 2000 SLPM. The fuel-air mixture ratio is varied linearly by increasing the air mass flow rate at a rate of 3 SLPM (with uncertainty ±0.11 SLPM) per second.
The MFCs that we utilized are proportional-integral-derivative (PID) based. The rise time of the MFCs is 100 ms. The set points of the MFC, determining the flow rates of air were regulated using voltage signals from an National Instruments Data Acquisition (NI DAQ) system. Since there is no manual operation here, repeatability of the experiments are ensured with respect to the change in flow rate of air. The mass flow rate of fuel is kept constant at 34 SLPM. An approximate thermal power for this mass flow rate of fuel is 49 kW assuming complete combustion.
We acquire p (t) using a piezoelectric pressure transducer (PCB103B02, uncertainty ±0.15 Pa). In order to record the maximum amplitude of the standing wave, the sensor is mounted at the antinode of the acoustic oscillations (25 mm distance from the backward-facing step of the combustor). The piezoelectric pressure transducer is mounted on a pressure port (T-joint) that is flush mounted on the combustor wall. To protect the transducer from excess heating from the combustor, a Teflon adapter was also used. Further, one shoulder of the T-joint was also provided with semi-infinite tubes of 10 m in length to prevent acoustic resonance within the ports and also ensure integrity of the measured signals. The mounting arrangement does not significantly affect the analysis that we perform in the study. The voltage signals from the piezoelectric transducer were recorded utilizing a 16 bit A-D conversion card NI-6143 with an input voltage range of 5 V and a resolution of 0.15 mV.
The global CH * chemiluminescence intensityQ(t) is captured using a photomultiplier (PMT, Hamamatsu H10722-01) mounted with a CH * filter (435 nm ± 12 nm full width at half maximum [FWHM]) in front of it. Both p (t) andQ(t) are acquired at a sampling rate of 10 kHz. The lowest values of signal-to-noise ratio of are 11 and 28 for pressure and global heat release rate measurements respectively. Pressure and global heat release rate data from all experiments do not show a large deviation due to random errors with respect to the transition to thermoacoustic instability. At low values ofṁ air , the standard deviation of the rms values across multiple experiments is 17 Pa for pressure and 0.003 arbitrary units (a.u.) for global heat release rate. At high values ofṁ air , the standard deviation is (135 Pa) for pressure and 0.015 a.u. for global heat release rate. These deviations across multiple experiments do not affect the major conclusions of the study. Further, systematic errors in these measurements do not affect the analysis of fluctuations.
The spatially resolved chemiluminescence intensity (q(x, t)) is captured using a Phantom V12.1 high-speed camera (of spatial resolution 800 × 600 pixels) equipped with a Zeiss 100 mm lens and a CH * filter. The images are captured at a sampling rate of 500 frames per second for a duration of 84 s. All these measurements are acquired simultaneously. Detailed descriptions of the experimental configuration, data recording systems and control systems can be found in George et al (2018).

Analysis of fluctuations in the global observables
To investigate the transition from S C to S TI via S IN , we perform experiments on a laboratory scale turbulent combustor (Komarek and Polifke 2010, Unni et al 2015, Silva et al 2017 and acquire spatiotemporal data. Our study analyses the transition between the dynamical states by continuously varying a system parameter, the air mass flow rate (ṁ air ) from low to high values (Gopalakrishnan et al 2016, Bonciolini and Noiray 2019, Pavithran and Sujith 2021. In all the experiments, we increaseṁ air from 537 ± 8 SLPM to 780 ± 10 SLPM linearly with time. This change inṁ air corresponds to an increase in the Reynolds number Re from 1.45 × 10 4 ± 428 to 2.05 × 10 4 ± 555 and a decrease in the approximate global equivalence ratio from 0.99 (±0.015) and 0.69 (±0.009). We measure two variables that represent the global state of the system: (i) the acoustic pressure fluctuations p (t) and (ii) the global heat release rateQ(t) at a data acquisition rate of 10 kHz. Meanwhile, we acquire spatiotemporal data of the local heat release ratė q(x, t) through high-speed imaging of CH * radical in the combustion zone at a data acquisition rate of 500 frames per second and a spatial resolution of 800 × 600 pixels. Furthermore, we perform seven independent experiments keeping the same conditions of system parameters.
At the outset, we analyze the acoustic pressure and global heat release rate fluctuations, p (t) andQ (t) respectively. As we increaseṁ air , p (t) andQ (t) change from aperiodic fluctuations to periodic oscillations as seen in figures 2(a) and (b) respectively. It is important to note that p (t) andQ (t) are variables that indicate the global state of the system and are only functions of time. At the beginning of the experiment (t = 0) p (t) andQ (t) are aperiodic and have low amplitude (see picture insets near t = 5 s). Such fluctuations show the chaotic temporal behaviour of the system at low values ofṁ air during the occurrence of S C (Nair et al 2013). In contrast, asṁ air increases, the fluctuations become periodic and grow in amplitude (picture insets at t = 70 s). The dominant frequency in p (t) for the large amplitude periodic oscillations, estimated by the fast Fourier transform (FFT) of a 1 s time window between t = 79 s and 80 s is 135 Hz (uncertainty of 0.61 Hz). Based on earlier experiments on the combustor of length 1100 mm with measurements of pressure along multiple locations of the combustor, we have found that the dominant frequency of 135 Hz is of a quarter-wave mode. For a combustor of longer length, Pawar et al (2017) have provided a detailed analysis on the evolution of the dominant frequency. However, neither p (t) norQ (t) show a demarcation between the aperiodic state and periodic state of the thermoacoustic system because of the continuous growth of periodicity.

Critical phenomena prior to the phase transition from S C to S TI
We utilize an indicator of phase transitions, the growth of variance of fluctuations (σ 2 ) to reveal critical phenomena in the local heat release rate fluctuations (q (x, t)). In this study,q (x, t) is the only variable where we have spatial information. In fact, the spatial integral ofq (x, t) gives the global measureQ (t). We calculate σ 2 ofq (x, t) at each pixel in the images (refer to appendix B for the calculation of σ 2 ). Following this, we estimate the composite averages of the σ 2 field by identifying the OOD for each experiment. We utilize the composite averages because the OOD need not occur at the same point in time across different  At the center of the images, the bluff body and its shaft are marked with a gray color. The bluff body creates a low-velocity region to stabilize the flame in the combustor. Behind the bluff body is the wake, a recirculation zone formed by the fluid moving past the bluff body. σ 2 increases at certain regions with a high magnitude, marked by black squares and referred to as zone 1 (Z1), zone 2 (Z2) and zone 3 (Z3). Z1 is above the bluff body, Z2 is at the corner between the bluff body and the shaft while Z3 is upstream of the bluff body and near the combustor wall. At Z1 and Z2, σ 2 increases before the OOD, while at Z3, σ 2 increases after OOD. (d) The growth of σ 2 , calculated as the difference of σ 2 between two-time instants at t = 5.5 s and t = 59.5 s ((d): (c) − (a)). The phase transition ceases to exist in the wake, as observed by the lack of growth in σ 2 in (d). We also mark a reference zone (Z4) by an orange square in (d) at a region between the wake of the bluff body and Z4. Thus, Z4 indicates a boundary between regions where the phase transition occurs (at Z1, Z2 and Z3) and where it does not (the wake).
experiments. More details on the calculation of σ 2 ofq (x, t), composite averages and the identification of OOD are described in appendix A. Figures 3(a)-(c) show the composite averages of σ 2 at various time instants with OOD as the time reference. The composite averages show a localized increase in σ 2 as the system approaches OOD.
The non-uniform distribution of σ 2 before OOD occurs due to non-uniform flow conditions associated with the geometry of the bluff body and the resultant non-uniform heat release rate. Figure 3(a) shows that The time series of p rms shows an exponential growth from t = 4 to t = 60. The accuracy of the exponential fit is given by R 2 = 0.984. (f) The time series of σ 2 ofq (t) at the Zs, the entire wake region and the full combustion zone. σ 2 for the full region shows little change in their value. The σ 2 at Z1 and Z2 increases dramatically before the OOD indicating the proximity to an imminent phase transition. This increase in σ 2 at these Z1, Z2 and Z3 occurs before any visible growth in p rms . σ 2 at Z3 increases after the OOD and matches with the growth of p rms as compared to Z1 and Z2. At Z4, σ 2 is high compared to the other regions at the beginning of the experiment and shows a linear increase prior to OOD. The OOD is denoted by a dotted vertical line at t = 26.5 s. For the time series ofq (t) and σ 2 we utilize the spatial mean ofq (t) over certain areas: for Z1, Z2 and Z3, we utilize an area of 5 × 5 pixels, for Z4: 20 × 20 pixels, for wake: 120 × 120 pixels. All the measures are then normalized with respect to the area. σ 2 at all regions is low much before the OOD. However, with an increase inṁ air and near OOD, σ 2 increases particularly at two zones, referred to as Z1 and Z2 as shown in figure 3(b). Additionally, at these locations, σ 2 remains high relative to other regions long after the OOD (see figure 3(c)). On the other hand, figures 3(a)-(c) exhibit that the recirculation zone behind the bluff body has low σ 2 because of aperiodic fluctuations (shown later in figure 4(d)). After the OOD, in figure 3(c) we observe a third zone, Z3, which exhibits an increase in σ 2 . In general, figure 3(d) reveals that Z1, Z2 and Z3 exhibit the highest growth of σ 2 . We find that prior to OOD the growth of variance of fluctuations at these locations we consider to be seeds of the transition is indeed statistically significant with p-value < 0.01 (see appendix E for details). We mark a region Z4 by an orange square between Z1 and the wake region in figure 3(d), which shows low growth in σ 2 . Due to the combustor's axisymmetric profile with respect to the shaft, we have similar dynamics below the shaft. Thus, the growth in σ 2 of the flame fluctuations allows us to locate Z1, Z2 and Z3, out of which Z1 and Z2 exhibit the earliest manifestation of the onset of thermoacoustic instability.
The increase of σ 2 within the confined reactive system indicates that critical phenomena occurs at selected zones of the flame enroute to S TI . The growth of σ 2 at these zones remain high for independent experiments conducted at different combustor lengths, position of bluff body, mass flow rates of fuel, rate of increase of mass flow rate of air etc. The geometry of the system gives special importance to different regions within the confined reactive system. For instance, flame stabilization occurs at the bluff body. The corner between the shaft and the bluff body where Z2 is located provides a low-velocity region, vital for flame stabilization (Lieuwen and Yang 2005). This region is directly in line with the incoming bulk flow of fresh reactants. Z1 is a location of high heat release rate fluctuations after the impingement of large-scale flow structures while Z3 is at the region where the largest size of the flow structure exists before impingement . Even though the flame is spatially extended in the confined reactive system, there may be certain zones where the increase in local fluctuations appear regularly due to the geometry of the system. Thus, the fact that criticality does not appear in all regions within the combustor is not surprising.

Local increase in variance of fluctuations as an indicator of the phase transition
The increase in the variance of fluctuations ofq (t) at Z1 and Z2 indicates an early manifestation of the phase transition from the aperiodic state to the periodic state. Figures 4(a)-(c) show the evolution from aperiodic dynamics to periodic oscillations at Z1, Z2 and Z3 respectively. At Z1 and Z2,q (t) has a low amplitude and is aperiodic before the OOD, while near the OOD, the amplitude ofq (t) increases at both Z1 and Z2. In fact, the increase in amplitude ofq (t) manifests as a localized emergence of periodic flame oscillations at these regions. This increase inq (t) is evident from the substantial increase in σ 2 at these locations, seen in figure 4(f). The highq (t) at Z2 can be attributed to the flapping of the shear layer which is one of the underlying mechanisms of S TI (Premchand et al 2019). The maximum value of σ 2 at Z1 and Z2 occurs near the OOD, indicating that they form a pair of zones wherein critical phenomena occurs first.
On the other hand, this critical phenomena is neither seen in global measures of p (t) andQ (t) nor in σ 2 at other regions of the combustor prior to the OOD. In particular, figures 4(e) and (f) indicate that both the p rms (root mean square of the acoustic pressure fluctuations) and σ 2 of the full combustion zone show a negligible increase before the OOD. Hence, there is no indication of a phase transition in the global dynamics. At the same time, local flame fluctuations at Z1 and Z2 indicate an early manifestation of the transition from S C to S TI . In contrast, σ 2 at the wake remains low for the entire duration of the experiment. The local flame fluctuations in the wake may be aperiodic or periodic with low strength of fluctuations. Figures 4(a), (b) and (d) indicate that at Z4, the growth ofq (t) is relatively small compared to that at Z1 and Z2.
Even though the importance of Z3 during S TI due to the emergence of large-scale coherent vortices is clear, Z3 does not show the earliest indication of the phase transition. The amplitude ofq (t) at Z3 grows gradually with increase inṁ air . In fact, figure 4(e) exhibits that σ 2 at Z3 closely matches the growth of p rms in comparison to σ 2 at Z1 and Z2. This close match of the variation of σ 2 at Z3 and p rms is expected because of the influence of impingement of large-scale coherent flow structures formed from spatial organization of small-scale vortices during S TI (Poinsot et al 1987, Yu et al 1991, Hong et al 2013. On the other hand, at S C , such spatial organization does not occur, resulting in low amplitude aperiodic flame fluctuations. Thus, althoughq (t) at Z3 is high during S TI , the growth in σ 2 is low before the OOD. However, the localized increase in periodic fluctuations ofq (t) at Z1 and Z2 may eventually enhance the mutual feedback between p (t) andq (x, t), pushing the state of the system from S C to S TI .

Teleconnections within the flame near the phase transition
Correlation analysis shows that Z1, Z2 and Z3 indicate a localized increase in long-range connections or teleconnections between them before the OOD. We utilize Pearson correlation (Lee Rodgers and Nicewander 1988) and Moran's spatial correlation (Moran 1950) to analyze the correlations within the combustion zone (more details in appendices C and D). Spatial order emerges in the flame and flow dynamics from S C to S TI . Pearson correlation coefficient ofq (t) (CCq ) between Z1, Z2 and Z3 shown in figure 5(a) reveals non-local and long-range connections between them near the OOD. In fact, we observe that CCq between Z1, Z2 and Z3 increases prior to the OOD. The correlation between these regions is statistically significant with p-values < 0.001 (see appendix F for details). Rocha et al (2018) showed that an interaction between tipping elements can result in cascading effects and a phase transition in the global dynamics. Clearly, the increase in CCq corresponds to the increase in σ 2 at both Z1 and Z2. Such an increase in CCq , especially before any appreciable change in the global measures of p rms andQ rms , indicates an emerging teleconnection between Z1, Z2 and Z3 before the OOD. In a recent study Godavarthi et al (2020) utilized non-identical Rössler oscillators and a chaotic Van der Pol oscillator to represent the spatially resolved heat release rate and the acoustic pressure respectively. They showed that with increase in the coupling strength, the acoustic pressure fluctuations changed from a chaotic state to a periodic state via intermittency. The increase in correlation of the local heat release rate near the zones Z1 and Z2 mirrors the change from asynchronized chaos to generalized synchronization observed by Godavarthi et al (2020).
On the contrary, beyond the OOD, the correlation between these zones decrease, when one may expect it to increase because the amplitude of the periodic oscillations continues to grow after the OOD (refer figure 2). But, we also observe that the spatial extent of the correlated behaviour increases as the system approaches S TI . In fact, in figure 5(b), we observe an increase in spatial correlation of the local heat release rate fluctuations Rq with increase inṁ air (more details on calculation of Rq is in appendix D). Rq fluctuates around 0.34 before exhibiting a continuous increase in Rq after t = 18 s. Thus, even though after the OOD the temporal correlation between the zones Z1, Z2 and Z3 decreases, the spatial correlation in the heat release rate fluctuations within the entire flame zone increases. The high value of (CCq ) between Z1, Z2 and Z3 just before the OOD indicates a non-local correlation (teleconnection) between Z1, Z2 and Z3. We also estimate that the increase in Pearson correlation between the Z1, Z2 and Z3 prior to OOD by utilizing the Mann-Kendall test between t = 10 s to t = 20 s, skipping 0.5 s is significant (p-values < 0.001). We evaluate the increase in Pearson correlation in this time period because it is prior to OOD and shows simultaneous increase in correlation for all the three cases. We note that for Z1-Z3, the correlation switches between high and low values between t = 0 and t = 10 s. However, it is only after t = 10 s that it increases continuously till t = 20 s. (b) Time series of Moran's spatial correlation of local heat release rate fluctuations Rq that exhibit the increase in spatial extent of local heat release rate fluctuations. The spatial correlation of local heat release rate increases from S C to S TI . Utilizing the Mann Kendall test, we find the p value obtained for the increase in spatial correlation from t = 10 s to t = 81 s, skipping 2 s is less than 0.001.

Identification of critical conditions and predicting the phase transition in advance
Analysis of the strength of the local flame fluctuations reveal a crucial temporal relationship between Z2 and Z4. As we utilize a sliding window approach to obtain the fluctuations of the heat release rate, the rms of these fluctuations:q rms becomes equivalent to the square root of σ 2 . Nevertheless, since the rms value has traditionally been used to define the strength of the flame fluctuations, we utilize the rms ofq (x, t) to define the relationship between Z2 and Z4. Notably, even though the utilization ofq rms is not new, phenomena of emerging correlations or synchronization of local heat release rate fluctuations has been investigated only recently (Mondal et al 2017. Further,q rms has traditionally been utilized to estimate only the mean field conditions of the system. It is only recently that σ 2 has been employed on temporal fluctuations to obtain early warning signals in thermoacoustic systems (Gopalakrishnan et al 2016). However, utilizing the increase in the rms (or σ 2 ) of spatial measures has never been explored to identify the phase transition nor provide early warning measures. Figure 6(a) shows thatq rms at Z2 abruptly shifts from 800 a.u. atṁ air = 580 SLPM (t = 14 s) to 4000 a.u. atṁ air = 610 SLPM (t = 24 s), indicating a high magnitude transition. At other locations, the transition is not as sharp as that at Z2. Such a high magnitude transition is a key feature of a phase transition from one state to another state. Further, as seen earlier in figure 3(d), the earliest manifestation of critical phenomena also occurs at Z1 and Z2 wherein Z2 shows the highest growth in σ 2 of the local flame fluctuations. These features of the phase transition at Z2 makes it an ideal candidate to demarcate the chaotic state (S C ) and the periodic state (S TI ). On the other hand,q rms at Z4 is slightly higher than at Z2 during the state of S C and increases withṁ air .q rms at Z4 and Z2 intersect before OOD near the mid point between the low values and high values ofq rms at Z2. This intersection ofq rms at Z2 and Z4 occurs for all the seven experiments prior to the OOD. For the experimental data shown in figure 6(a), the intersection occurs atṁ air = 597 SLPM, whereq rms = 1839 a.u. We denoteq rms at the point of intersection asq i rms . Indeed, there are some fluctuations near the intersection because of the turbulent flow field, which pushes the local dynamics back and forth between the chaotic state (lowq rms ) and large amplitude periodic state (highq rms ) due to its inherent stochasticity. Nevertheless, we consider the last intersection to denoteq i rms . We define the set of critical conditions under which a transition to thermoacoustic instability begins: critical zones and a critical value in the heat release rate fluctuations. There are two critical zones: Z2 and Z4. While Z2 exhibits the earliest manifestation of critical phenomena and the highest increase in fluctuations, Z4 shows a faster rate of growth ofq rms on the eve of the onset of S TI . The onset of S TI happens when the magnitudes ofq rms at these two zones reach the same level. Utilizing the point of intersectionq i rms , we define the critical value in the heat release rate fluctuations. We select four out of the seven experiments  1, 2, 3, 4). This median valueq c rms is 1555 a.u. The green band indicates the prediction horizon to the phase transition. The right border of the pink band indicates whenq rms at Z3 and Z4 intersect. The inset represents the sketch of the turbulent combustor and the zones with respect to the bluff body. The lower and upper limits ofṁ air corresponding to t = 0 s and t = 81 s are 537 SLPM and 780 SLPM respectively. Table 2. Utilizing the interconnected zones show high accuracy in predicting the phase transition. Key:ṁ a -mass flow rate of air at whichq rms of Z2 crosses Z4 for the experiment;ṁ p -predicated value of air mass flow rate in SLPM utilizing the regression line from the time series at Z4 and the critical valueq c rms ; δ-standard error in percentage and is given as (ṁ a − m p )/ṁ a . The discrepancy betweenṁ a anḋ m p is within the uncertainty limits of the controller at these flow rates (±9 SLPM). to make a training set. We choose the median ofq i rms from these four experiments to define the critical value of heat release rate fluctuations:q c rms = 1555 a.u. Once the strength of the local heat release rate fluctuations at Z2 increases beyondq c rms , these fluctuations do not return to their low amplitude chaotic state but continue to grow in amplitude and periodicity en route to S TI for all the experiments.  proposed a user-defined threshold of 0.1 for the Hurst exponent (H) of p (t) to demarcate the state of S TI from S C . Notably, H decreases to 0.1 at the intersection point of Z2 and Z4 (see appendix G for details). This feature further substantiates our view that the occurrence of the critical phenomena at Z2 and Z1 locally indicates the onset of thermoacoustic instability.
Subsequently, we utilize thisq c rms (marked by the gray dashed line in figure 6(a)) to predict the onset in the other three experiments. At Z4, theq rms increases linearly till the OOD that may be attributed to the gradual increase in the frequency of vortex shedding from the tip of the bluff body depending onṁ air . We attempt to predict the onset by making a linear fit ofq rms at Z4 for each of the three experiments and identify where the linear fit intersectsq c rms (see dashed black line in figure 6(a)). Table 2 shows the results from the prediction. The second column of table 2 represents the mass flow rate of airṁ a where the intersection betweenq rms at Z2 and Z4 occur. The third column shows the predicted air mass flow rate (ṁ p ) obtained from the linear fit andq c rms . The discrepancy betweenṁ a andṁ p is within 1%, for the three test cases. These discrepancies are within the uncertainty limits of the MFC. In general, this methodology predicts the earliest manifestation of the onset 11 s in advance, which corresponds to a change ofṁ air = 33 SLPM.
The choice of the reference zone Z4 to define the critical point and predict the phase transition is not arbitrary. Unlike the interconnected zones, theq rms at Z4 increases from the beginning of the experiment.
Once the phase transition occurs,q rms at Z4 saturates to a value close to 2000 a.u. On the other hand, at the wake, theq rms always remains low while regions within the area spanned by Z1 and Z2 show a phase transition near the OOD (not shown here). Meanwhile, at regions near Z3, the large magnitude phase transition occurs slower, as indicated by the slow increase inq rms at Z3. Hence, Z4, is a special location, representing a boundary between the zones wherein the earliest manifestation of phase transition occurs (Z1 and Z2) and where the phase transition ceases to exist (wake). The intersection point between Z2 and Z4 means an equalization of the strength of fluctuations at the stagnation point and the vortex shedding region. Changes in the area selected for Z4 (within the wake region) do not significantly affect the value oḟ q c rms . Large changes in the selection of Z4 would result in eitherq c rms being too high, which is inappropriate for the definition of critical value in this case, or too low, which may lie within the vicinity of local fluctuations at Z2.
Interestingly,q rms at Z3 and Z4 also intersect near the mid point between the low values and high values ofq rms at Z3. This intersection occurs atṁ a = 645 SLPM, marked by the right border of the vertical pink band. We observe that at this point, the amplitude of the pressure oscillations reach 25% of the maximum observed amplitude, giving us an insight to how early the onset is. In fact, this intersection occurs 16 s later than the earliest manifestation of the phase transition at Z1 and Z2. Z3 shows highq rms when large-scale coherent flow structures emerge. This shows clear evidence that the earliest manifestation of the phase transition occurs in advance to the emergence of large-scale coherent structures. Such features of Z4 and its relationship with Z2 motivates its use in defining the onset of thermoacoustic instability and predicting it.
We hypothesize that the features at Z1, Z2, Z3 and Z4 would remain the same irrespective of the changes in the experimental conditions such as different fuel flow rates, different rate of change of mass flow rate of air. The features persist due to following reasons. Firstly, at Z1, which is just above the bluff body, high heat release rate occurs after impingement of the large-scale flow structures . Secondly, at Z3, which is located upstream of the bluff body, large-scale flow structures emerging during the state of thermoacoustic instability is ubiquitous and has been reported in multiple studies in the same experimental setup , Raghunathan et al 2020. Finally, the divergent flow at Z2 (corner of bluff body and shaft) and the vortex shedding at Z4 (between the wake and Z1) are imposed by the geometry of the bluff body. Hence, we expect critical phenomena to occur locally at Z1 and Z2 before the phase transition, irrespective of changes in experimental conditions.

Discussion and conclusion
We have presented comprehensive experimental evidence on identifying two interconnected zones-seeds of the phase transition in a turbulent reactive flow system that point to the earliest manifestation of the phase transition. First, we have unraveled those zones, where the earliest manifestation of the onset of phase transition appears, utilizing the increase in the variance of local fluctuations of the flame. Second, the emergence of long-range correlations between these zones prior to the onset of the phase transition has been found. Next, we have identified the critical value of the heat release rate fluctuations at these interconnected zones. These findings enable us to identify the onset of a phase transition from a chaotic state to a periodic state, which in our case is the onset of thermoacoustic instability. Notably, all precursors which have been shown in recent decades are indicators of an impending thermoacoustic instability. To the best of the authors' knowledge, a prediction of when the onset will occur has not been shown in any experimental, numerical or analytical study to date. Our result opens promising perspectives for predicting the onset of transitions in complex systems wherein acquisition of spatiotemporal data is possible.
Previously, Stolbova et al (2016) revealed tipping elements in the Indian monsoon system by analyzing observational data of atmospheric variables, utilizing the time series at the tipping elements. These tipping elements allowed predicting the onset of the monsoon with large lead times. It has been proven to be successful retrospectively in the prediction of future monsoons for five consecutive years since its introduction in 2016 (Ludescher et al 2021). We have adapted the tipping elements approach for turbulent combustor and have developed it further for the case of thermoacoustic instability. This methodology allows to predict the earliest manifestation of the onset before it appears. Importantly, the methodology that we have used to identify the zones where the earliest manifestation of the onset appears do not need detailed information of the influential processes that drive the transition. In fact, the earliest manifestation of the phase transition is found only by utilizing the criterion, which is the growth in the variance of fluctuations of certain system variables. The evidence from both observational data analysis and our experimental results shows the applicability of the methodology for various complex systems and demonstrate the predictive power with sufficient lead times.
The developed methodology is applicable to multidimensional complex systems exhibiting phase transitions. The only prerequisite is to have spatial data of a system variable that shows the phase transition.
For more complex 3D flames, the identification of seeds of a phase transition may require acquisition of spatial data in x, y and z directions, which is more challenging. On the other hand, in simpler systems such as the Rijke tube, the onset is well defined by the Hopf point. Thus, the usefulness of the methodology in thermoacoustics would lie in more complex geometries with turbulent and non-compact flames wherein the acquisition of spatial information is of value and possible.
The suppression of the onset of the phase transition may open new perspectives for 'greener' operations. We hypothesize that the interconnected zones perform as bridges to ordered dynamics that initiate positive feedbacks within the system, necessary for the phase transition to occur. Suppressing the spatiotemporal organization in such zones should break a pathway to S TI . In general, most control strategies intend to destroy large-scale coherent flow structures that exist at S TI . However, we have revealed that the onset of S TI at the interconnected zones Z1 and Z2 occurs even before the emergence of these large-scale coherent structures. The growth of σ 2 at these zones remain high for independent experiments conducted at different combustor lengths, position of bluff body, mass flow rates of fuel, rate of increase of mass flow rate of air etc.
Our findings suggest that it may be necessary and useful to implement control strategies at this onset of S TI which appears much before the large amplitude oscillations of p occur. The revealed interconnections between these zones suggests a promising way to suppress the phase transition by disrupting the linkage between them and thus block a pathway to S TI . Further, the question of whether suppressing critical phenomena at the interconnected zones can suppress thermoacoustic instability also remains an open question. Pursuing this research question is important for developing passive control strategies. Preliminary experiments show that suppression of thermoacoustic instability close to 90% is possible by adopting such a strategy, but this is for future study. Thus, our study provides a very-much-needed understanding of experimental evidence showing the special role of the seeds of phase transition (Frenkel 2006) that mark the earliest manifestation of the transition to thermoacoustic instability, which allow us to define the onset of thermoacoustic instability. This is a crucial step for 'greener' operations. Figure A1. Transition from low amplitude chaotic fluctuations to high amplitude periodic oscillations in a turbulent thermoacoustic system. The time series of (a) p (t), (b)Q (t), (c) the phase difference between p (t) andQ (t): δθ, (d) first derivative of phase difference: dδθ dt , calculated using first order finite difference method. All time series are shown for continuous variation ofṁ air from 537 SLPM at t = 0 to 780 SLPM at t = 81 s. The insets in (a) and (b) show zoomed-in views of p (t) anḋ Q (t) during S C and S TI . The inset in (c) shows a zoomed-in view of the phase difference. The unwrapped phase difference δθ increases till t = 25 s and becomes constant. Utilizing a near zero value of dδθ dt , we mark the point of the OOD by a dashed vertical gray line. At the OOD,ṁ air and Re are 617 SLPM and 16 380 respectively. become periodic, their phase difference becomes constant (Pawar et al 2017, Chen et al 2021. Next, we calculate the first derivative of δθ: dδθ dt , which is shown in figure A1(d). As the phase difference becomes constant, the first derivative of the phase difference will approach zero. We mark the time instant wherein the dδθ dt reaches 99.99% of the difference between the largest positive value and zero, and refer to it as the OOD. For experimental data shown in figure A1, we identify the OOD at t = 26.5 s whereṁ air is 617 SLPM. The same approach is followed to obtain OOD for the other experiments. Subsequently, we estimate averages across the seven experiments with respect to OOD as the reference point from each experiment. These averages, referred to as composite averages are shown in figure 3 for σ 2 fields at different instants of time with respect to OOD. We utilize these composite averages to standardize σ 2 prior to the OOD.  Appendix G. Relation of the critical value of p with the Hurst exponent of p Figure G1(a) is similar to figure 6 but with the addition ofq rms at Z4. Figure G1(b) indicates that the Hurst exponent H decreases and becomes nearly constant after t = 41 (ṁ air = 660 SLPM). At the intersection point of Z2 and Z4, we find H as 0.1.