2 Synchronization of coupled oscillators

Synchronization of coupled chaotic oscillators has been mainly categorized into phase synchronization (Rosenblum, Pikovsky & Kurths Reference Rosenblum, Pikovsky and Kurths1996), lag synchronization (Rosenblum, Pikovsky & Kurths Reference Rosenblum, Pikovsky and Kurths1997), complete synchronization (Fujisaka & Yamada Reference Fujisaka and Yamada1983) and generalized synchronization (Rulkov et al. Reference Rulkov, Sushchik, Tsingring and Abarbanel1995). In phase synchronization, both of the oscillators of a coupled system show perfect locking in their phases; however, their amplitudes remain uncorrelated. During lag and complete synchronization, both the amplitudes and the phases of the signals from the coupled oscillators display exactly the same behaviour. The only difference is that, in lag synchronization, the phases of the signals are lagged by a constant value of time, whereas, in complete synchronization, the signals are in phase with each other. In the case of generalized synchronization, there exists a functional relationship between the signals of coupled oscillators. In some cases, the state of perfect synchronization is preceded by a state of intermittent synchronization, wherein regions of synchronized and desynchronized oscillations are simultaneously present in the signal (Sungwoo, Park & Rubchinsky Reference Sungwoo, Park and Rubchinsky2011).

3 Experimental set-up

The experiments were conducted in a laboratory-scale combustor with a partially premixed turbulent flame, as shown in figure 1. The experimental set-up consists mainly of three parts: (i) a settling chamber, (ii) a burner and (iii) a combustor. The airflow is first passed through the settling chamber, which serves the purpose of reducing the effect of hydrodynamic fluctuations generated inside the combustor on the flow at the air inlet. In the burner, fuel (liquefied petroleum gas, propane 40 % and butane 60 % by volume) is partially mixed with the incoming airflow from the settling chamber at different equivalence ratios. This partially premixed fuel–air mixture then enters the main combustor section. The combustor is a rectangular duct which is 1400 mm long and 90 mm $\times$ 90 mm wide and has a backward-facing step (dump plane) at the inlet. A bluff body, a circular disc of diameter 47 mm and thickness 10 mm, is located at a distance of 4.5 mm from the inlet of the combustor. The bluff body aids in anchoring the flame in the high-velocity turbulent environment of the combustor. The air and fuel flow rates are controlled separately by using mass flow controllers (Alicat Scientific MCR 2000SLPM for air and MCR 100SLPM for fuel; the uncertainty is $\pm$ 0.8 % of the measured reading $+$ 0.2 % of the full scale reading) in the system. The fuel flow rate is maintained at a constant value of 25 slpm and the air flow rate is varied from a value of 400 slpm to 940 slpm such that the flow field in the system is turbulent ( $Re=1.09\times 10^{5}$ to $Re=2.12\times 10^{5}$ ) throughout the experiment. The estimated uncertainties in $Re$ are $\pm 1.97\times 10^{3}$ to $\pm 2.71\times 10^{3}$ . The equivalence ratios range from $0.95\pm 0.02$ to $0.46\pm 0.01$ . For the purpose of initial ignition, a spark plug (along with a 11 kV ignition transformer – National Engineering Corporation) fixed at the dump plane is used to ignite the combustible air–fuel mixture. Quartz windows of size 90 mm $\times$ 360 mm, located on both of the sidewalls of the combustor, provide the optical access required for the measurement of heat release rate fluctuations from the flame.

Figure 1. A schematic of the turbulent flame combustor. The partially premixed air–fuel mixture is burned in a turbulent flow field inside the combustor. The flame is stabilized using a bluff body located near the dump plane of the combustor.

We measure the acoustic pressure fluctuations ( $p^{\prime }$ ) from the combustor using a piezoelectric transducer (PCB Piezotronics PCB103B02, with a sensitivity of $223.4~\text{mV}~\text{kPa}^{-1}$ and an uncertainty of $\pm 0.15~\text{Pa}$ ). The pressure transducer is fixed on the top wall near the inlet step of the combustor. This position of the transducer corresponds to a near maximum amplitude of the acoustic pressure in the duct. It is an appropriate location for the measurement of the acoustic pressure in this combustor, as it always remains a pressure antinode for all of the acoustic modes of the duct. The unsteady heat release rate fluctuations ( $\dot{q}^{\prime }$ ) are captured by using a photomultiplier tube (PMT; Hamamatsu H10722-01). A $\text{CH}^{\ast }$ band-pass filter ( $\unicode[STIX]{x1D706}=432~\text{nm}$ and 10 nm full width at half maximum (FWHM)), which captures the $\text{CH}^{\ast }$ chemiluminescence intensity from the flame, is used to filter the input to the PMT. The chemiluminescence intensity thus recorded is a measure of the heat release rate from the flame (Ikeda, Kojima & Hashimoto Reference Ikeda, Kojima and Hashimoto2002; Guethe et al. Reference Guethe, Guyot, Singla, Noiray and Schuermans2012). The PMT is positioned at a distance of 500 mm normal to the combustor wall near the location of the bluff body. The signals of the pressure fluctuations and the heat release oscillations are acquired for 3 s at a sampling frequency of 10 kHz. A 16-bit analog to digital (NI-6143) card is used for the data acquisition. A frequency resolution of 0.33 Hz is used in the fast Fourier transform calculation. Since our main objective in the present study is to analyse the synchronization characteristics of $p^{\prime }$ and $\dot{q}^{\prime }$ , we directly use the raw (mean subtracted) voltage signals obtained from the pressure transducer and the PMT throughout our analysis. This further ensures that both $p^{\prime }$ and $\dot{q}^{\prime }$ fluctuations are represented in comparable scales of voltage. Welch’s average method (Welch Reference Welch1967) is used to smooth the periodogram of the signals. A Hanning window of 5000 data points with an overlap of 50 % is used in the analysis. This method improves the spectral representation of signals in the waterfall diagram (shown in figure 3) by reducing noise from the estimated power spectra.

4 Results and discussion

We investigate the synchronization behaviour of the acoustic pressure ( $p^{\prime }$ ) and the heat release rate ( $\dot{q}^{\prime }$ ) oscillations present in a turbulent combustor, as we vary the mean velocity of the flow ( $\bar{u}$ ). Because a turbulent flow generates sound, there will always be acoustic oscillations in the duct. Further, the measured heat release rate fluctuations incorporate the fluctuations from both chemical kinetics and that induced by the associated flow turbulence present in the system (Shanbhogue, Husain & Lieuwen Reference Shanbhogue, Husain and Lieuwen2009). As the resulting effects produced from the fluctuations in hydrodynamic and combustion processes in the combustor are difficult to separate, we consider them as a single oscillator, which we call ‘turbulent reactive flow’. An increase in $\bar{u}$ results in a decrease in the equivalence ratio of the combustible air–fuel mixture from a value close to stoichiometry ( $\unicode[STIX]{x1D719}=0.98$ ) to a value of fuel lean condition ( $\unicode[STIX]{x1D719}=0.49$ ). At lower values of $\bar{u}$ ( $\bar{u}=9.4~\text{m}~\text{s}^{-1}$ ), we observe low-amplitude aperiodic oscillations in both the acoustic pressure and the heat release rate signals (see figure 2 a). This state of combustion dynamics is traditionally referred to as combustion noise (Strahle Reference Strahle1978). Recent studies by Nair et al. (Reference Nair, Thampi, Karuppasamy, Gopalan and Sujith2013) and Tony et al. (Reference Tony, Gopalakrishnan, Sreelekha and Sujith2015) have shown that the dynamical features of combustion noise consist of high-dimensional chaos contaminated with white and coloured noise. With further increase in $\bar{u}$ ( $\bar{u}=11.9~\text{m}~\text{s}^{-1}$ ), we observe the emergence of bursts of periodic oscillations from a background of aperiodic oscillations (figure 2 b). These bursts appear to occur randomly, with no apparent pattern. Nair et al. (Reference Nair, Thampi and Sujith2014) referred to this state of combustion dynamics as intermittency. Visual comparison of the acoustic pressure and heat release rate signals (in the inset of figure 2 b) shows that both of the signals are locked in the burst regions of the periodic oscillations, while unlocked in the relatively silent regions of the aperiodic oscillations.

Figure 2. (a–d) The time series of the acoustic pressure ( $p^{\prime }$ – black colour) and heat release rate ( $\dot{q}^{\prime }$ – red colour) signals obtained from experiments at $\bar{u}=9.4~\text{m}~\text{s}^{-1}$ , $\bar{u}=11.9~\text{m}~\text{s}^{-1}$ , $\bar{u}=13.2~\text{m}~\text{s}^{-1}$ and $\bar{u}=17.2~\text{m}~\text{s}^{-1}$ respectively. Magnified views of the corresponding signals are shown in the insets above each panel. (a) A low-amplitude aperiodic state, (b) an intermittency, (c) a weakly periodic limit cycle state and (d) a strongly periodic limit cycle state of oscillations. (e) The variation of the root mean square value of the acoustic pressure ( $p_{rms}^{\prime }$ ) plotted against the mean velocity of the flow ( $\bar{u}$ varies from $9.4~\text{m}~\text{s}^{-1}$ to $18.1~\text{m}~\text{s}^{-1}$ ).

At sufficiently high values of $\bar{u}$ ( $\bar{u}=13.2~\text{m}~\text{s}^{-1}$ ), we notice a transition of the combustion dynamics from a state of intermittency to a state of weakly periodic oscillations (figure 2 c). In this state, we observe wide cycle-to-cycle variation in the amplitude of the pressure signal. This variation in the amplitude of the limit cycle oscillations has been postulated to be due to the forcing exerted by background noise (Lieuwen Reference Lieuwen2002, Reference Lieuwen2003b ; Noiray & Schuermans Reference Noiray and Schuermans2012). With further increase in $\bar{u}$ ( $\bar{u}=17.2~\text{m}~\text{s}^{-1}$ ), we observe a transition of the system dynamics to a state of strongly periodic oscillations (figure 2 d), wherein the amplitude of the pressure oscillations nearly remains the same. Figure 2(e) shows the variation of the root mean square value of the acoustic pressure ( $p_{rms}^{\prime }$ ) acquired for different values of the mean velocity of the flow ( $\bar{u}$ ). We notice a continuous increase in the amplitude of the pressure oscillations with an increase of $\bar{u}$ in the system. This plot suggests that the growth in the amplitude of the pressure oscillations is not a linear function of $\bar{u}$ , as the slope of the plot varies differently in different regions of the system dynamics.

Figure 3. (a) The variation of the PSD of both $p^{\prime }$ (black colour) and $\dot{q}^{\prime }$ (red colour) for a range of $\bar{u}$ from $9.4~\text{m}~\text{s}^{-1}$ to $17.2~\text{m}~\text{s}^{-1}$ . (b) The magnified view of the PSD for a velocity range from $\bar{u}=9.4~\text{m}~\text{s}^{-1}$ to $\bar{u}=11.1~\text{m}~\text{s}^{-1}$ . (c) The variation of the dominant frequency ( $f_{Dominant}$ ) of both $p^{\prime }$ and $\dot{q}^{\prime }$ obtained for a range of $\bar{u}$ in the system. The magnified view of the PSD shown in (b) is highlighted by a green colour in (c). In (b), the separate unlocked frequency bands of $p^{\prime }$ and $\dot{q}^{\prime }$ , obtained for an initial value of $\bar{u}=9.4~\text{m}~\text{s}^{-1}$ , are shown by green arrows, whereas the frequency corresponding to the first onset of frequency lock-in is shown by a blue arrow.

In order to investigate the coupled interaction between the acoustic pressure and the heat release rate oscillations in the system, we plot a waterfall diagram (see figure 3 a), which demonstrates the variation of the power spectral density (PSD) of the signals obtained at different values of $\bar{u}$ . Figure 3(a) shows the emergence of sharp dominant peaks in the three-dimensional plot of the PSD for both the acoustic pressure and the heat release rate oscillations at large values of $\bar{u}$ . The presence of sharp peaks in the PSD indicates the existence of periodic oscillations in the dynamics of the system. The variation of the dominant frequencies of both the acoustic pressure and heat release rate oscillations, for a range of $\bar{u}$ , is plotted in figure 3(c). This plot reveals the locking behaviour of the dominant frequencies of both of the signals, which occurs for the first time at $\bar{u}=11.1~\text{m}~\text{s}^{-1}$ . The variation of the dominant frequencies of these signals prior to the locking of frequencies (the highlighted portion of figure 3 c) is further explored by plotting a zoomed in view of figure 3(a) and is shown in figure 3(b). At low mean velocities of the flow ( $\bar{u}=9.4~\text{m}~\text{s}^{-1}$ ), the plots of the PSD for both $p^{\prime }$ and $\dot{q}^{\prime }$ show distinct shallow bands of dominant frequencies, around 130.7 Hz for the acoustic pressure signal and around 25.3 Hz for the heat release rate signal (marked by green arrows in figure 3 b). The dominant peak in the acoustic pressure spectrum corresponds to the fundamental mode of the acoustics in the combustor, which is around 125 Hz (the calculation is based on the assumption of the closed–open geometry of the combustor having a length of 1.4 m, and an approximate value of the speed of sound equal to $700~\text{m}~\text{s}^{-1}$ ). On the other hand, we observe that the dominant frequency corresponding to 25.3 Hz in the heat release rate signal is a consequence of the underlying turbulent hydrodynamic fluctuations present in the reaction zone of the combustor. When the large-scale fluid structures shed from the dump plane impinge on the bluff body and the sidewalls of the combustor, there is a sudden release of heat in the system (observations are based on the high-speed $\text{CH}^{\ast }$ chemiluminescence images of the flame; not shown in this paper). The frequency corresponding to these heat release fluctuations is close to 25.3 Hz, which indicates that hydrodynamics has a key role to play in the low-frequency oscillations observed in the power spectra of the heat release rate signal.

The existence of different dominant frequencies indicates that these signals are not locked during the stable operation of the combustor. A close inspection of figure 3(b) reveals the emergence of a secondary band of frequencies with increasing $\bar{u}$ , which is in between the disparate frequency bands that we had for $p^{\prime }$ and $\dot{q}^{\prime }$ during aperiodic oscillations. The magnitude of the secondary band frequencies gradually grows with $\bar{u}$ ; yet, the dominant frequencies of the pressure and heat release rate oscillations in this secondary band still remain different (see the highlighted portion of figure 3 c). The locking of these dominant frequencies occurs at $\bar{u}=11.1~\text{m}~\text{s}^{-1}$ (around $f_{lock}=86.6~\text{Hz}$ ). The presence of distinct frequency peaks for low mean velocities, and the locking of these peaks to a common frequency, which is in between their initial values, indicates the presence of mutual coupling (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003) between these two oscillators of a thermoacoustic system. We further note that such an observation of switching of the dominant frequencies of $p^{\prime }$ and $\dot{q}^{\prime }$ to a secondary frequency band, which is sufficiently away from the natural acoustic mode of the duct, is different from the problem of forced flame response, wherein the low frequency associated with heat release rate fluctuations is damped and a single peak arises at the forcing frequency. However, whether the phenomenon we observe can be termed as merging or adjustment of frequencies needs to be clarified through further studies.

Frequency analysis of the acoustic pressure and heat release rate oscillations (figure 3 c) reveals an important feature of the locking of dominant frequencies of these signals in our system: it occurs in the intermittency state ( $\bar{u}=11.1~\text{m}~\text{s}^{-1}$ ), long before the onset of limit cycle oscillations ( $\bar{u}=13.2~\text{m}~\text{s}^{-1}$ ) in the system. Possible physical mechanisms that bring about the locking of frequencies of $p^{\prime }$ and $\dot{q}^{\prime }$ during the onset of intermittency could be the locking of hydrodynamic and acoustic oscillators during periodic epochs of the intermittent oscillations. To further explore this proposition, we refer to the study performed by Unni & Sujith (Reference Unni and Sujith2015), where they described the behaviour of the flame during intermittency. They observed that during the periodic segments of intermittency, ring vortices are periodically shed from the dump plane and convected downstream along the outer shear layer. This causes periodic fluctuations in the flame surface area, and when the shed vortex impinges on the obstacles (the bluff body and the sidewalls), we correspondingly have periodic heat release in the system. On the other hand, during aperiodic epochs of the oscillations, this periodic shedding of vortices ceases, and the tip of the flame exhibits oscillations in an aperiodic manner. We further need to keep in mind that the vortex shedding itself might be influenced by the acoustics and combustion dynamics in the system. Hence, all three of these processes are likely to play a role in the mutual synchronization during the periodic epochs of intermittency. The dominant frequencies of the $p^{\prime }$ and $\dot{q}^{\prime }$ oscillators remain locked as the system dynamics changes from intermittency ( $\bar{u}=11.1~\text{m}~\text{s}^{-1}$ ) to limit cycle oscillations ( $\bar{u}=17.2~\text{m}~\text{s}^{-1}$ ). During the onset of limit cycle oscillations, we observe periodic shedding of large-scale vortices, and hence a periodic heat release rate, during every acoustic cycle. We further notice that this locked-in frequency of the signals continuously increases with $\bar{u}$ until the final state (see the non-highlighted portion of figure 3 c).

To ascertain the synchronization behaviour of the acoustic pressure and heat release rate oscillations, we use a statistical measure of synchronization based on the RPs. This is the same measure as we described previously in § 2.2: the probability of recurrence ( $P(\unicode[STIX]{x1D70F})$ ). We compute $P(\unicode[STIX]{x1D70F})$ for $p^{\prime }$ and $\dot{q}^{\prime }$ , and then compare the coincidence of the peaks of $P(\unicode[STIX]{x1D70F})$ obtained from these signals in order to analyse the synchrony between them (Romano et al. Reference Romano, Thiel, Kurths, Kiss and Hudson2005).

Figure 4. Plots of probability of recurrence ( $P(\unicode[STIX]{x1D70F})$ ) of a phase space trajectory at different values of the time lag ( $\unicode[STIX]{x1D70F}$ ) for $p^{\prime }$ (green colour) and $\dot{q}^{\prime }$ (red colour). (a–c) $P(\unicode[STIX]{x1D70F})$ corresponding to three states of synchronization, namely desynchronization (aperiodic oscillations), phase synchronization (periodic oscillations) and generalized synchronization (periodic oscillations), obtained at $\bar{u}=9.4~\text{m}~\text{s}^{-1}$ , $\bar{u}=13.2~\text{m}~\text{s}^{-1}$ and $\bar{u}=17.2~\text{m}~\text{s}^{-1}$ respectively. The parameters used are embedding dimension $=$ 12, time delay $=$ 2 ms and recurrence threshold $=$ 25 % of the maximum size of the attractor. The data set contains 3000 points (shown for 1200 points).

Figure 4(a) shows a plot of $P(\unicode[STIX]{x1D70F})$ as a function of the time lag $\unicode[STIX]{x1D70F}$ , for both the acoustic pressure and the heat release rate signals, obtained during the aperiodic state (as shown in figure 2 a). We see from the figure that there is no correspondence between the peaks of $P(\unicode[STIX]{x1D70F})$ of these signals. This confirms that the two oscillators are not in synchrony. On the other hand, when the system dynamics undergoes a transition to limit cycle oscillations (see figure 2 c,d), we see a perfect locking of the positions of the peaks of $P(\unicode[STIX]{x1D70F})$ of both of the signals, as shown in figure 4(b,c). The locking of the positions of the peaks confirms the synchronization between acoustic pressure and heat release rate oscillations. Furthermore, the absence of exact locking of the heights of these peaks in figure 4(b) indicates that we have phase synchronization (PS) in the system (Romano et al. Reference Romano, Thiel, Kurths, Kiss and Hudson2005). Figure 4(c) reveals the perfect coincidence of the height as well as the position of the peaks of the probability of recurrence obtained from the acoustic pressure and the heat release rate oscillations, suggesting the presence of generalized synchronization (GS) in the system (Romano et al. Reference Romano, Thiel, Kurths, Kiss and Hudson2005). Previous studies on synchronization have reported chaotic oscillators synchronizing in the chaotic state (Boccaletti et al. Reference Boccaletti, Kurths, Osipov, Valladares and Zhou2002) or periodic oscillators synchronizing in the periodic state (Blekhman et al. Reference Blekhman, Landa and Rosenblum1995). However, in our system, chaotic oscillators transition to the periodic state as they synchronize. This makes the study of such a synchronization transition quite interesting.

Figure 5. (a) Plots of time series of $p^{\prime }$ (green colour) and $\dot{q}^{\prime }$ (red colour) acquired during the intermittency state at $\bar{u}=11.6~\text{m}~\text{s}^{-1}$ . (b,c) Plots of $P(\unicode[STIX]{x1D70F})$ shown for periodic and aperiodic regions of both $p^{\prime }$ and $\dot{q}^{\prime }$ respectively. The parameters used in plotting (b,c) are embedding dimension $=$ 12, time delay $=$ 2 ms and recurrence threshold $=$ 25 % of the maximum size of the attractor. The data set contains 3000 points (shown for 1200 points).

We therefore analyse the synchronization behaviour of acoustic pressure and heat release rate oscillators during intermittency (see figure 5), as this is the intermediate state of the transition. Since the intermittent oscillations consist of alternating occurrences of high-amplitude periodic and low-amplitude aperiodic oscillations, we plot $P(\unicode[STIX]{x1D70F})$ for short windows of the periodic (figure 5 b) and aperiodic (figure 5 c) regions of the signals. The plot of $P(\unicode[STIX]{x1D70F})$ for a periodic window (figure 5 b), corresponding to both $p^{\prime }$ and $\dot{q}^{\prime }$ , shows locking of the positions of the peaks without locking of their heights, which confirms the phase synchronization of the two signals during the periodic epoch. On doing the same for the aperiodic region, we see a mismatch in the locations as well as the heights of the peaks; this depicts the desynchronized behaviour during the aperiodic phase. The alternate switching of the coupled oscillators between synchronized and desynchronized behaviour describes the property of intermittent synchronization in the system. Moreover, since the two oscillators in our system are phase synchronized in the periodic region and desynchronized in the aperiodic region, such a state of the system is referred to as intermittent phase synchronization (IPS). This state shows that the oscillators in our system gradually change their underlying dynamics from aperiodic to periodic as they synchronize. Recently, Mondal, Unni & Sujith (Reference Mondal, Unni and Sujith2017) investigated the spatio-temporal synchronization behaviour of the local heat release rate in the flame with the global acoustic field of the duct. During intermittency, their study demonstrated the coexistence of patches of both synchronized periodic oscillations and desynchronized aperiodic oscillations in the reaction zone. Furthermore, these patterns of spatial synchrony and desynchrony interchanged as the flow convected downstream, which they referred to as a breathing chimera-like state. Therefore, IPS is arguably an apt intermediate state that we observe during the transition of combustion dynamics from a state of completely desynchronized oscillations to a state of perfectly phase synchronized oscillations. Thus, it is evident from figures 4 and 5 that in the turbulent flame combustor examined in the present study, the desynchronized oscillations pave the way to PS through IPS, and subsequently to GS with an increase in the mean velocity of the flow $(\bar{u})$ .

Figure 6. (a,b) The temporal variation of the relative phase ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{p^{\prime },\dot{q}^{\prime }}$ ) between $p^{\prime }$ and $\dot{q}^{\prime }$ calculated using an analytical signal approach based on the Hilbert transform. (a) The PS ( $\bar{u}=13.2~\text{m}~\text{s}^{-1}$ ) and (b) the GS ( $\bar{u}=17.2~\text{m}~\text{s}^{-1}$ ) state of oscillations. (c) A plot depicting the variation of the mean value of the relative phase ( $\overline{\unicode[STIX]{x0394}\unicode[STIX]{x1D719}}_{p^{\prime },\dot{q}^{\prime }}$ ) with the mean velocities of the flow observed during the PS and GS states.

As discussed earlier, the unstable operation of a turbulent combustor has been traditionally perceived in terms of limit cycle oscillations (Lieuwen Reference Lieuwen2002). We, however, observe two different types of limit cycle in our system, which exhibit different dynamical characteristics. We call them weakly correlated (PS) and strongly correlated (GS) limit cycle oscillations. In order to identify the distinguishing features of these two states, we compare the temporal variation in instantaneous phase difference (relative phase) between the acoustic pressure and the heat release rate oscillations. The instantaneous phases of these signals are calculated using the Hilbert transform (explained in § 2.1). Figures 6(a) and 6(b) show the relative phase plots ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{p^{\prime },\dot{q}^{\prime }}$ ) of these signals obtained during PS and GS states respectively. The fluctuation of the relative phase around a constant phase shift (as shown in figure 6 a,b) suggests synchronization of the signals of the coupled oscillator system (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). The enlarged view of the relative phase for the PS state shows a noisy behaviour (figure 6 a), while that for the GS state shows an apparently periodic behaviour (figure 6 b). In figure 6(c), we plot the temporal mean value of the relative phase ( $\overline{\unicode[STIX]{x0394}\unicode[STIX]{x1D719}}_{p^{\prime },\dot{q}^{\prime }}$ ), in the interval $-180^{\circ }$ to $180^{\circ }$ , versus different values of the mean velocities of the flow. The plot shows the velocity range corresponding to PS and GS states alone. During the PS state, we notice that the mean value of the relative phase remains close to $50^{\circ }$ , whereas during the GS state (figure 6 b), it approaches a value closer to zero degrees (refer to figure 6 c). Thus, we observe a reduction in the mean phase angle between the acoustic pressure and the heat release rate fluctuations as the system dynamics transitions from the PS to the GS state.

Figure 7. (a) A portion of the time series of $p^{\prime }$ (black colour) and $\dot{q}^{\prime }$ (red colour) obtained from experiments during the GS state ( $\bar{u}=17.2~\text{m}~\text{s}^{-1}$ ). (b) A portion of the signal obtained from an assumed functional form for both acoustic pressure ( $P^{\prime }$ – black colour) and heat release rate ( $\dot{Q}^{\prime }$ – red colour) oscillations. (c,d) The temporal variation of the relative phase obtained from the Hilbert transform for the signals shown in (a) and (b) respectively.

It should be recalled that when two oscillators are in GS, there exists a functional relation between the dynamics of these oscillators (Rulkov et al. Reference Rulkov, Sushchik, Tsingring and Abarbanel1995). Say, for example, that we have two non-identical oscillators, namely $X(t)$ and $Y(t)$ . When these oscillators are in the state of GS, there exists a functional relationship between them, which can be expressed as $Y(t)=\unicode[STIX]{x1D6F7}(X(t))$ . This means that the response of $Y(t)$ can be uniquely determined from $X(t)$ if the functional form of their relation is known (Pyragas Reference Pyragas1998). However, the determination of the exact functional form for experimental signals is not straightforward.

In our system, we can obtain an approximate functional relation between the acoustic pressure ( $p^{\prime }$ ) and the unsteady heat release rate ( $\dot{q}^{\prime }$ ) during the GS state, using the knowledge obtained from experiments. During the GS state (see figure 7 a), visual inspection of the acoustic pressure signal hints at the features of a sinusoidal function, whereas the heat release rate signal shows the features of a spiky signal. By saying spiky, we mean that the signal is relatively flat for some time, then rises up, decays and becomes flat again. The behaviour of these signals can then be qualitatively mimicked by the following equations:

and

where $P^{\prime }$ and $\dot{Q}^{\prime }$ are the assumed functional forms of $p^{\prime }$ and $\dot{q}^{\prime }$ , $\unicode[STIX]{x1D714}$ is the angular frequency of the signals, $a_{1}$ , $a_{2}$ and $a_{3}$ are constants and $\emptyset _{1}$ and $\emptyset _{2}$ are the initial phases of $P^{\prime }$ and $\dot{Q}^{\prime }$ respectively. Here, $n_{p1},n_{p2},n_{q1}$ and $n_{q2}$ are Gaussian white noise terms added to the amplitude and phase parts of the signals.

The values of the constants in the assumed functional forms of $P^{\prime }$ and $\dot{Q}^{\prime }$ are derived from the statistical properties of the corresponding $p^{\prime }$ and $\dot{q}^{\prime }$ signals obtained from experiments. This is to ensure an approximate fit between the data generated from numerical results and the signals obtained from experiments.

The relative phase between the two signals can then be expressed as

where $G(t)$ is the phase noise present in the relative phase of both $P^{\prime }$ and $\dot{Q}^{\prime }$ .

The parameters selected in plotting figure 7(b) are $a_{1}=0.8$ , $a_{2}=0.35$ , $a_{3}=0.7$ , $\unicode[STIX]{x1D714}=120.2~\text{Hz}$ , $\emptyset _{1}=0~\text{rad}$ and $\emptyset _{2}=\unicode[STIX]{x03C0}/2$ rad, and the strength of noise is 0.05.

Figure 7(b) qualitatively simulates the features of the acoustic pressure and the heat release rate oscillations observed in experiments during the GS state (figure 7 a). The plot of the relative phase between the two signals obtained from the experiments (figure 7 c) is qualitatively captured by the proposed functional form for these signals (figure 7 d). These analogies between the experimental results and the assumed functional relation are completely drawn from the characteristics of the acoustic pressure and the heat release oscillations observed during the GS state. In reality, the functional relationship may be considerably more complicated, involving different combinations of sine and cosine functions along with inherent turbulent fluctuations present in the combustion system. Identification of the exact functional form of these signals is difficult, mainly because the complex interaction between acoustic pressure and heat release rate oscillations is still not completely known.

We now try to elucidate the possible physical reason for having separate PS and GS states in our turbulent flame combustor. Towards this purpose, we calculate the amplitude difference (figure 8 a) and the time difference (figure 8 b) between the consecutive cycle peaks of the acoustic pressure and the heat release rate oscillations respectively, and subsequently compare them for the PS (figure 8 c,d) and GS (figure 8 e,f) states. The time at which $\dot{q}^{\prime }$ reaches a local maximum is referred to as a kicking time (figure 8 b). The reason for using such a terminology is to relate it to the kicked oscillator model that we propose in § 5. The term kick refers to the release of heat in a very short time span when the shed vortex from the dump plane impinges on the bluff body. This sudden generation of heat leads to the spiky behaviour in $\dot{q}^{\prime }$ . Figures 8(d) and 8(f) show the difference in the impingement time (i.e. the kicking time) of two consecutive vortices (normalized by the period, $T=1/f_{Dominant}$ ) obtained during the PS and GS states respectively. We observe a wide variation in the kicking times of the vortex during PS compared with what is seen during GS.

Figure 8. The portion of (a) acoustic pressure and (b) heat release rate oscillations depicting the computation procedure for the amplitude difference and the kicking times respectively. (c,e) The temporal variation of the amplitude difference (normalized by signal global maxima) between consecutive peaks of the acoustic pressure oscillations obtained for PS and GS states respectively. (d,f) The temporal variation of the time difference (normalized by the signal period $T=1/f_{Dominant}$ ) between the consecutive peaks of the heat release rate oscillations, referred to as kicking times, obtained for PS and GS states respectively. The PS state corresponds to $\bar{u}=13.2~\text{m}~\text{s}^{-1}$ and the GS state corresponds to $\bar{u}=17.2~\text{m}~\text{s}^{-1}$ . (g) The variation of the cross-correlation coefficient ( $r$ ) between $p^{\prime }$ and $\dot{q}^{\prime }$ with different values of the mean velocity of the flow ( $\bar{u}$ ).

The variation in the kicking times is possibly reflected in the irregular nature of the amplitudes of the acoustic pressure oscillations observed during PS. Figures 8(c) and 8(e) show plots of the amplitude difference between consecutive cycles of the acoustic pressure ( $p^{\prime }$ ) signal, normalized by the global maximum $(p_{max}^{\prime })$ of the respective signal, for the PS and GS states respectively. We notice that the cycle-to-cycle variation in the amplitude of $p^{\prime }$ is comparatively much larger during the PS state than that during the GS state. The consequence of such cycle-to-cycle variation in the kicking times as well as in the amplitudes of the signals observed during PS and GS is further investigated by plotting a linear measure of correlation between $p^{\prime }$ and $\dot{q}^{\prime }$ for the whole range of $\bar{u}$ used in this study (see figure 8 g). The cross-correlation coefficient ( $r$ ) (or Pearson’s coefficient) is a measure of the strength of the linear relation between the two signals. It further helps in determining the synchronization of amplitudes of the signals obtained from a coupled oscillator system (Gonzalez-Miranda Reference Gonzalez-Miranda2002; Bove et al. Reference Bove, Boccaletti, Bragard, Kurths and Mancini2004), and is given by the following equation:

where $x_{i}$ and $y_{i}$ are the samples of the bivariate data obtained from the experiments and $n$ is the total number of samples in the signal.

The values of $r$ range from $-$ 1 to $+$ 1; $r=-1$ corresponds to strong negative correlation and $r=1$ corresponds to strong positive correlation between the signals. The value $r=0$ indicates zero linear correlation between the signals. In our system, we observe values of $r$ greater than zero, suggesting the presence of a positive correlation between the acoustic pressure and heat release rate oscillations (figure 8 g). We notice an increase in the value of the linear correlation as the system dynamics transitions from combustion noise to a final state of limit cycle oscillation. During PS, the weak correlation of the oscillators is reflected in relatively low values of $r$ , which are around 0.5–0.6. Although the amplitude of the acoustic pressure oscillations shows a significant increase for all of the values of $\bar{u}$ ( $\bar{u}=12.9~\text{m}~\text{s}^{-1}$ – $15.4~\text{m}~\text{s}^{-1}$ ) during PS (figure 2 e), the value of $r$ shows a marginal variation during this state (figure 8 g). The transition from PS to GS results in an increase in the value of $r$ from a value close to 0.5 to a value close to 1. Thus, the oscillators in GS ( $\bar{u}=16.1~\text{m}~\text{s}^{-1}$ – $18.5~\text{m}~\text{s}^{-1}$ ) in our system are very strongly correlated. Changes in the behaviour of both the kicking time and the cycle-to-cycle amplitude variation have a major impact on the linear correlation during the transition from the PS to the GS state.

5 Model

Figure 9. Schematic of the combustor model. The inlet end contains a step of height $d$ from which vortices are shed. These shed vortices carry reactant mixture at a fraction of the mean velocity of the flow ( $\bar{u}$ ). The reactant mixture burns when these vortices impinge on the bluff body, located at a distance $L_{c}$ from the inlet. The acoustic waves generated as a result of this burning affect the vortex shedding, and thus establish a closed loop interaction between hydrodynamic, combustion and acoustic processes.

Lastly, we introduce a phenomenological model that qualitatively captures the experimental results. The model takes into account the three important phenomena vital to the problem at hand – hydrodynamics, combustion of reactants and propagation of sound waves in the duct (see figure 9). For a set of values of the control parameter, the interactions between these subsystems intertwine in a manner that gives rise to self-sustained synchronized oscillations in a thermoacoustic system. Matveev & Culick (Reference Matveev and Culick2003) developed a reduced-order model that captures interactions between the vortex shedding, acoustic field and combustion processes of such systems. They modelled the interaction of the heat release rate with the acoustic field as a kicked oscillator system, where the vortices that impinge on the bluff body impulsively force the acoustic field. Their model was able to capture the stable and unstable operation of the combustor; however, it did not show intermittency prior to thermoacoustic instability. Nair & Sujith (Reference Nair and Sujith2015) extended Matveev’s model to capture the state of intermittency in the system dynamics. In order to capture the intermittent dynamics, they added a stochastic noise term in the convection velocity of the vortex. Seshadri, Nair & Sujith (Reference Seshadri, Nair and Sujith2016) used a deterministic approach to capture all of the dynamical states such as combustion noise, intermittency and limit cycle oscillations of a thermoacoustic system. They suggested that the feedback due to the acoustic waves, generated as a result of the instantaneous release of heat at the bluff body, affects the vortex shedding process at the dump plane of the combustor.

Here, we consider a simplified configuration of the experimental system for construction of the model (see figure 9) while retaining the key geometrical features of the real turbulent combustion system. The combustor is of length $L$ , with a step at the inlet of height $d$ . The combustion of the reactants is assumed to be localized at a distance $L_{c}$ from the inlet end (Matveev & Culick Reference Matveev and Culick2003). Vortices containing unburnt reactants are formed at the inlet end, which is due to the build-up of circulation at the step. This process of build-up of circulation ( $\unicode[STIX]{x1D6E4}$ ) is described by (5.1), where $u_{sep}=\bar{u}+u^{\prime }(L_{s},t)$ is the velocity at the step, $\bar{u}$ is the mean velocity, $u^{\prime }(L_{s},t)$ is the acoustic velocity at the step and $p_{L_{c}}^{\prime }$ is the acoustic pressure at the point of combustion (i.e. at $L_{c}$ ),

Here, $t_{j}$ is the time when the $j$ th vortex reaches the combustion location, $k$ is a proportionality factor (Seshadri et al. Reference Seshadri, Nair and Sujith2016), $\unicode[STIX]{x1D70F}_{a}$ is the time taken by the acoustic wave to reach the inlet from $L_{c}$ and $\unicode[STIX]{x1D70E}_{1}$ indicates the intensity of the Gaussian white noise $N(0,1)$ . The second term in the above equation describes the delayed feedback of acoustics on the vortex shedding process at the step. The form of (5.1) is similar to that proposed by Seshadri et al. (Reference Seshadri, Nair and Sujith2016), except that we have additionally incorporated a noise term; we will soon explain the reason for this.

When the circulation exceeds a critical value $(\unicode[STIX]{x1D6E4}_{cr}=\unicode[STIX]{x1D6E4}_{o}u_{sep})$ , a vortex is shed from the step, and the circulation is reset to zero (Matveev & Culick Reference Matveev and Culick2003). Here, $\unicode[STIX]{x1D6E4}_{o}$ is given as $d/(2S_{t})$ , where $S_{t}$ is the Strouhal number (Matveev & Culick Reference Matveev and Culick2003). In the experiments, the inlet is not a perfectly closed end, as opposed to what has been assumed in the previous models (note that for a perfectly closed end, $u_{sep}=\bar{u}$ ). Further, the turbulence of the flow at the inlet affects the circulation building up at the step. For this reason, we incorporate a noise term, which accounts for the variation in the velocity at the step (which appears in (5.1)), and correspondingly obtain $\unicode[STIX]{x1D6E4}_{cr}$ as $\unicode[STIX]{x1D6E4}_{cr}=\unicode[STIX]{x1D6E4}_{o}u_{sep}+\unicode[STIX]{x1D70E}_{2}N(0,1)$ . We note that the source of the noise is same for both $\text{d}\unicode[STIX]{x1D6E4}/\text{d}t$ and $\unicode[STIX]{x1D6E4}_{cr}$ , but it affects them with strengths $\unicode[STIX]{x1D70E}_{1}$ and $\unicode[STIX]{x1D70E}_{2}$ respectively. The repeated process of increase in the value of the circulation up to a critical value and its subsequent fall to zero depicts the oscillatory behaviour of the circulation. The vortex shed from the inlet convects downstream at a fraction of the mean velocity, and this is captured by (5.2) (Matveev & Culick Reference Matveev and Culick2003),

Here, $\unicode[STIX]{x1D6FC}$ gives the fraction of $\bar{u}$ with which the vortex is transported.

When these vortices reach the location of combustion ( $L_{c}$ ), a localized and instantaneous burning of the reactants in the vortex occurs, and, corresponding to this event, the generated heat release rate ( $\dot{q}^{\prime }$ ) is modelled as (Matveev & Culick Reference Matveev and Culick2003)

The heat release rate fluctuations ( $\dot{q}^{\prime }$ ) affect the acoustic oscillations ( $p^{\prime }$ and $u^{\prime }$ ) present in the duct at time instants $t_{j}$ , when the reactants in the vortices burn.

The acoustic pressure ( $p^{\prime }$ ) and velocity ( $u^{\prime }$ ) in the duct are decomposed into a product of time-varying components ( $\unicode[STIX]{x1D702}_{n},\dot{\unicode[STIX]{x1D702}}_{n}$ ) with the natural spatial modes ( $M$ ) of the duct as follows (Zinn & Lores Reference Zinn and Lores1971; Seshadri et al. Reference Seshadri, Nair and Sujith2016):

where $k_{n}=\unicode[STIX]{x1D714}_{n}/c$ .

Using the energy conservation equation, we obtain an oscillatory equation for the coefficients of the natural modes ( $\unicode[STIX]{x1D702}_{n},\dot{\unicode[STIX]{x1D702}}_{n}$ ),

where $b$ is a proportionality constant. Here, the damping term has been artificially introduced, and the damping coefficient $(\unicode[STIX]{x1D701}_{n})$ is modelled as $\unicode[STIX]{x1D701}_{n}=(2n-1)^{2}\unicode[STIX]{x1D701}_{1}$ (Nair & Sujith Reference Nair and Sujith2015; Seshadri et al. Reference Seshadri, Nair and Sujith2016).

When the acoustic oscillations are excited, the sound waves travel upstream and affect the circulation build-up at the step, after a time delay of $\unicode[STIX]{x1D70F}_{a}$ (Seshadri et al. Reference Seshadri, Nair and Sujith2016), which is evident from (5.1). This dependence of the circulation on the acoustic field closes the feedback loop of interaction between the acoustic oscillations in the duct, the vortex shedding and the combustion processes. We note that the heat release rate ( $\dot{q}^{\prime }$ ) in the system can be related to the circulation ( $\unicode[STIX]{x1D6E4}$ ) at the step as the shedding of a vortex is controlled by (5.1). Consequently, we study the synchronization of the acoustic pressure oscillations ( $p^{\prime }$ ) and the circulation ( $\unicode[STIX]{x1D6E4}$ ) obtained from the model. We remark that the acoustic oscillator described in the model is not a self-sustained oscillator and hence is not an ideal candidate for the synchronization study. The turbulent flow in the duct contributes to the production of sound in the actual system, and thus the acoustic variables would not decay even in the absence of a heat release rate. This can be captured in this model by adding a noise term to the evolution equation governing the acoustic modes, or through a more detailed modelling of the system incorporating turbulent flow. We nevertheless stick to the current model for studying the synchronization behaviour of acoustics and circulation, as we are only trying to compare a phenomenological model with the experiments.

Figure 10. The time series of the acoustic pressure $(p^{\prime })$ obtained from the model showing (a) low-amplitude aperiodic oscillations at $\bar{u}=9.2~\text{m}~\text{s}^{-1}$ , (b) an intermittency signal at $\bar{u}=10.2~\text{m}~\text{s}^{-1}$ , (c) low-amplitude periodic oscillations at $\bar{u}=10.41~\text{m}~\text{s}^{-1}$ and (d) high-amplitude periodic oscillations at $\bar{u}=10.6~\text{m}~\text{s}^{-1}$ . (a) The desynchronized state, (b) the IPS state, (c) the PS state and (d) the GS state of oscillations is observed when the acoustic pressure ( $p^{\prime }$ ) is compared with the circulation ( $\unicode[STIX]{x1D6E4}$ ) at the step. The data were acquired for 3 s and then undersampled to a frequency of 10 kHz.

Further, given that the model proposed by Seshadri et al. (Reference Seshadri, Nair and Sujith2016) captures the transition from combustion noise to thermoacoustic instability via the intermittency route, a question arises as to what prompted us to incorporate noise into the system. We here present the motivating reason for such a modification. We observed that the model by Seshadri et al. (Reference Seshadri, Nair and Sujith2016) showed desynchronized oscillations, IPS and GS, but we could not observe the state of PS (note that, in the absence of noise, the empirical parameters need to be appropriately chosen to obtain the intermittency route to thermoacoustic instability). In other words, the subsystems in the previous model are very strongly correlated during the limit cycle oscillations (thus giving rise to GS but not PS), and, therefore, some aspect that induces a weakly correlated limit cycle is being overlooked. A plausible reason for not observing PS in the previous model, as we have explained above, is that the inlet end of the duct is not perfectly closed (as opposed to what is assumed in that model), and the turbulence in the flow gives rise to small fluctuations that affect the circulation build-up at the inlet step. This is not captured in the model presented by Seshadri et al. (Reference Seshadri, Nair and Sujith2016). We incorporate these fluctuations as noise in the current model, paving the way to (5.1). We note that our intention is to effect minimal changes to the previous model so that a qualitative match with the experimental observation is obtained.

We used the Runge–Kutta method of order 4 for performing the integration of the deterministic part of the above equations, while the Euler–Maruyama scheme was employed in integrating the stochastic part. The values of the parameters used in the computations are $\text{d}t=10^{-5}$ , $\unicode[STIX]{x1D6FC}=0.53$ , $\unicode[STIX]{x1D6FE}=1.4$ , $\unicode[STIX]{x1D701}_{1}=29~\text{s}^{-1}$ , $L_{c}=0.045~\text{m}$ , $L=1.4~\text{m}$ , $d=0.025~\text{m}$ , $c=700~\text{m}~\text{s}^{-1}$ , $\bar{p}=101\,325~\text{Pa}$ , $M=10$ , $S_{t}=0.145$ (i.e. $\unicode[STIX]{x1D6E4}_{0}=0.086$ ), $k=12.9+6.2\ast S^{\ast }(\bar{u}-9.4)$ , $b=0.003$ , $\unicode[STIX]{x1D70E}_{1}=0.017$ and $\unicode[STIX]{x1D70E}_{2}=0.0937$ , where $S^{\ast }$ is the step function.

Figure 10 shows a plot of the time series of the acoustic pressure obtained at different states of synchronization from the theoretical model. At low values of the mean velocities of the flow ( $\bar{u}$ ), we notice low-amplitude aperiodic oscillations in the system, as shown in figure 10(a). With further increase in $\bar{u}$ , we observe the emergence of bursts of large-amplitude periodic oscillations from the background of low-amplitude aperiodic oscillations in the system (see figure 10 b). Since this type of oscillation consists of alternate occurrence of both periodic and aperiodic oscillations, it corresponds to the state of intermittency. At sufficiently large values of $\bar{u}$ , the system dynamics transitions to a state of low-amplitude limit cycle oscillations (see figure 10 c). With further increase in $\bar{u}$ , we notice a transition of the system dynamics to sufficiently large-amplitude limit cycle oscillations (see figure 10 d). It should be noted that the spikes observed in these acoustic pressure signals are the result of kicking of the acoustic modes due to vortex impingement on the bluff body (see (5.5)). Although such spikes are a feature of the model, they are not observed in the experiments. Further, in the model, the vortex impingement and the heat release rate (i.e. flame response) occur at the same time due to the assumption of a localized instantaneous heat release rate. These aspects of the model could be improved upon by incorporating equations that describe flame behaviour and its interaction with vortices and acoustics, and can be taken up in future studies of the system.

The synchronization features of the signals from the coupled oscillators, namely the acoustic pressure in the duct ( $p^{\prime }$ ) and the circulation at the inlet step ( $\unicode[STIX]{x1D6E4}$ ), obtained from the model are characterized by using the statistical measure of synchrony based on the RPs (Romano et al. Reference Romano, Thiel, Kurths, Kiss and Hudson2005).

Figure 11. (a–c) The probability of recurrence ( $P(\unicode[STIX]{x1D70F})$ ) of the phase space trajectory for the signals ( $p^{\prime }$ – green colour; $\unicode[STIX]{x1D6E4}$ – red colour) obtained from the numerical model at $\bar{u}=9.2~\text{m}~\text{s}^{-1}$ , $\bar{u}=10.41~\text{m}~\text{s}^{-1}$ and $\bar{u}=10.6~\text{m}~\text{s}^{-1}$ respectively. (a) The desynchronized state, (b) a PS state and (c) a GS state of oscillations. The parameters used in plotting are embedding dimension $=$ 12, time delay $=$ 2 ms and recurrence threshold $=$ 20 % of the maximum size of the attractor. The data window contains 4000 points (shown for 1000 points).

Figure 11 shows plots of the probability of recurrence ( $P(\unicode[STIX]{x1D70F})$ ) of the phase space trajectory in the embedded phase space at different values of the time lag ( $\unicode[STIX]{x1D70F}$ ). In the case of an aperiodic state of oscillations (figure 11 a), the plot of $P(\unicode[STIX]{x1D70F})$ for both of the signals shows a mismatch in the locations as well as heights of the peaks, confirming their desynchronized nature. When the system enters a state of low-amplitude limit cycle oscillations, we observe the locking of positions of peaks in the plot of $P(\unicode[STIX]{x1D70F})$ of acoustic pressure and circulation. However, the heights of the peaks do not match perfectly. This shows that the system is likely to be in a state of PS. At the onset of large-amplitude limit cycle oscillations in the system, we notice almost perfect locking of both the positions and the heights of the peaks of $P(\unicode[STIX]{x1D70F})$ , suggesting the presence of GS of the two oscillators in the system. These observations conform to what we observed in the experiments (refer to figure 4).

Figure 12 shows the synchronization features of the intermittency signal of $p^{\prime }$ and $\unicode[STIX]{x1D6E4}$ obtained from the model. The plots of the probability of recurrence $P(\unicode[STIX]{x1D70F})$ obtained during aperiodic and periodic epochs of the intermittent oscillations are shown in figures 12(b) and 12(c) respectively. In the model, we observe smaller durations of the aperiodic oscillations compared with those of the bursts of periodic oscillations during intermittency (see figure 12 a). During the periodic epochs of the signal (figure 12 c), we notice locking of the positions of the peaks of $P(\unicode[STIX]{x1D70F})$ without locking of their heights, which confirms their PS nature. On the other hand, during the aperiodic epochs of the signal (figure 12 b), locking of the positions as well as the heights of the peaks of $P(\unicode[STIX]{x1D70F})$ is absent, hinting at their desynchronized nature. This suggests that the intermittency in the model corresponds to the IPS state, which is the same as what we observed in the experiments (refer to figure 5).

Figure 12. (a) The time series of the acoustic pressure ( $p^{\prime }$ ) acquired during an intermittency state ( $\bar{u}=10.2~\text{m}~\text{s}^{-1}$ ) from the model. (b,c) Plots of $P(\unicode[STIX]{x1D70F})$ , obtained for $p^{\prime }$ (green colour) and $\unicode[STIX]{x1D6E4}$ (red colour), corresponding to the regions (highlighted by blue colour) of aperiodic and periodic oscillations respectively. The parameters used in plotting (b,c) are embedding dimension $=$ 12, time delay $=$ 2 ms and recurrence threshold $=$ 20 % of the maximum size of the attractor. The data window contains 4000 points (shown for 600 points).

As before, we study the properties of the limit cycle oscillations observed during PS and GS states in the model by examining the kicking times (see figure 13). The kicking time corresponds to the time instant at which a vortex shed from the step reaches the location of the bluff body (refer to figure 9). It should be recalled that on the impingement of a vortex, an acoustic wave travels upstream and affects the vortex shedding process, which affects the subsequent kicking times. Therefore, kicking times contain information about the dynamics of the system. Figure 13 illustrates the difference in two consecutive kicking times (normalized by the period of the respective signal) observed during PS and GS states in the system. In the case of PS (figure 13 a), we notice more distributed kicking times around the dominant time period as opposed to what is observed during GS (figure 13 b). This reasserts that when the kicking times are more ordered, i.e. the time of kicking matches the dominant period of the signal, the system exhibits large-amplitude limit cycle oscillations. This, in turn, ensures that the acoustic feedback is stronger. These observations are the same as what we observed in the experiments.

Figure 13. (a,b) Plots corresponding to the difference between consecutive kicking times, normalized by the respective signal period, obtained from the model. Each point in the plot corresponds to the difference in the kicking time due to two consecutively shed vortices. The time instant at which a shed vortex reaches the bluff body location is called the kicking time, as this event leads to impulsive forcing of the acoustic oscillator through the heat release rate. The plot of kicking times (a) corresponds to the PS state obtained at $\bar{u}=10.41~\text{m}~\text{s}^{-1}$ and the plot (b) corresponds to the GS state obtained at $\bar{u}=10.6~\text{m}~\text{s}^{-1}$ .