## Abstract

We experimentally study thermoacoustic transitions in an annular combustor consisting of sixteen premixed, swirl-stabilized turbulent flames. We show the changes in the characteristics of bifurcations leading to the state of longitudinal thermoacoustic instability (TAI) when equivalence ratio and bulk velocity are systematically varied. Depending upon the bulk velocity, we observe different states of combustor operation when the equivalence ratio is varied. These states include combustion noise, intermittency, low-amplitude TAI, mixed-mode oscillations (MMO), and high-amplitude TAI. We closely examine the special case of MMO that is encountered during the transition from low-amplitude TAI to high-amplitude TAI. We also discuss the global and local flame dynamics observed during the state of MMO. We find that during epochs of low-amplitude oscillations of MMO, all the flames are partially synchronized, while during epochs of high-amplitude oscillations, all the flames are perfectly synchronized. Finally, we replicate the criticalities of bifurcation of the annular combustor in a phenomenological model containing sixth-order nonlinearities.

## 1 Introduction

Modern gas turbine combustors require stable operation over a wide range of operating conditions with severe constraints on their emissions. Consequently, annular combustors used in land-based engines in thermal power plants operate at fuel-lean conditions. However, in the limit of lean operations, these combustors are prone to the phenomenon of thermoacoustic instability (TAI), where high-amplitude pressure oscillations develop due to positive feedback between heat release rate (HRR) and acoustic pressure oscillations [1]. These high-amplitude oscillations are highly undesirable and lead to significant damages to the combustor.

The dynamical transition from stable combustor operation (otherwise known as combustion noise) to the unstable combustor operation (TAI) has attracted much interest in recent times [2–4]. Traditionally, the transition from combustion noise (CN) to the state of TAI following some change in a system parameter (*μ*), such as equivalence ratio ($\varphi $) or bulk-flow velocity ($\upsilon z$), has been thought of as a fixed point solution losing its stability and giving rise to a limit cycle solution. The transition of the acoustic subsystem to a limit cycle solution is then either due to a supercritical or subcritical Hopf bifurcation [2]. In supercritical Hopf bifurcation, when a control parameter *μ* is increased above some critical value (called the Hopf point, *μ _{H}*), the fixed point solution becomes linearly unstable, leading to limit cycle oscillations (LCO). In contrast, in subcritical Hopf bifurcation, the system is bistable in the range $\mu F<\mu <\mu H$, i.e., the fixed point solution is linearly stable to small-amplitude perturbations and unstable to perturbations above a threshold value. At $\mu =\mu H$, the system jumps from the stable fixed point to a limit cycle attractor with a large amplitude. For $\mu \u2265\mu H$, the fixed point solution is unstable, while the limit cycle is stable. For reverting to the stable fixed point solution,

*μ*must be reversed past

*μ*($\mu F<\mu H$), which is known as the fold point of the system. Thus, such systems exhibit hysteresis in the system dynamics [5].

_{F}Whether the transition occurs through a supercritical or subcritical Hopf bifurcation depends on the stabilizing or destabilizing nature of the dominant nonlinearities in the system when a parameter is varied [6]. The dependence of the nonlinearity on the critical parameter leading to a change in the nature of the bifurcation is referred to as *change in criticality* of a Hopf bifurcation [7]. In general, the source of nonlinearities in a thermoacoustic system is the acoustic damping and the acoustic driving [8]. Both of these quantities depend on the amplitude of $p\u2032$ and also the control parameters. Any variation in these quantities can change the balance between driving and damping and can lead to the usual case of supercritical or subcritical bifurcation to LCO. However, in specific scenarios, higher order nonlinearities in the system can destabilize a stable limit cycle solution, leading to a secondary fold bifurcation to a high-amplitude LCO [9]. Such a transition to high-amplitude LCO has been observed, for example, in aircraft flight dynamics called wing rock, where the aircraft exhibits high amplitude oscillatory rolling motion [10]. In thermoacoustic systems, the existence of secondary bifurcation to high-amplitude thermoacoustic instability (HA-TAI) was theoretically predicted by Ananthkrishnan et al. [11] and observed experimentally in a laminar system [12].

Turbulence plays a crucial role in determining how nonlinearities manifest in the dynamics of the system during the transition to TAI. Specifically, Nair et al. [4] observed that the presence of turbulence leads to an intermediate stable state during the transition from CN to TAI, known as intermittency. During intermittency, bursts of periodic pressure oscillations appear randomly amidst chaotic oscillations. Consequently, the characteristics of the Hopf bifurcation are modified. The transition to TAI shows a continuous increase in the amplitude of pressure oscillations as a result of a progressive increase in the duration of bursts of periodic oscillations until the state of full-blown TAI is reached.

In addition to the intermittency route, the transition to constant amplitude LCO can take place through a state, where pressure oscillations alternate between low-amplitude and high-amplitude periodic oscillations. These oscillations are referred to as mixed-mode oscillations (MMO) [13,14]. MMO appears in systems with characteristic slow time scales in addition to fast time scales [14,15]. Kasthuri et al. [15] observed MMO in a ducted matrix burner and modeled them using a slow time scale associated with the heat release rate in addition to the inherently fast acoustic time scales. Amplitude modulated oscillations, known as beats, with well-defined interference patterns due to linear superposition of modes with close by frequencies, have been reported in thermoacoustic systems [16–18]. However, the applicability of beats is restricted to linear systems. We show here that the framework of MMO is more suitable for describing the amplitude modulations in the present nonlinear annular system possessing large scale separation.

Flame–flame interactions in annular combustors during the state of TAI are complex and of crucial importance. In a geometry similar to the one considered in this study, Bourgouin et al. [19] analyzed the flame behavior during longitudinal TAI at a frequency *f *=* *252 Hz and amplitude 330 Pa and extracted the associated dynamic mode decomposition mode at different phases. They concluded that longitudinal TAI is still possible even though the flames showed some degree desynchrony. Worth and Dawson [20] showed that the net HRR response increased by $\u2248$10–15% through large-scale mixing when the separation distance between burners was decreased, leading to high gains in the amplitude of spinning azimuthal TAI. Han et al. [18] explored the phenomenon of beating in stratified swirl flames. Through a dynamic mode decomposition analysis, they revealed the presence of bulk oscillations associated with equivalence ratio fluctuations at *f *=* *198 Hz, whereas convective coherent structures associated with the mode at *f *=* *223 Hz. Recently, Roy et al. [21] considered the synchronization characteristics of flame-flame interactions during the transition to HA-TAI through intermittency and secondary bifurcation. They observed that the flames are weakly synchronized to each other and the acoustic pressure fluctuations during low-amplitude thermoacoustic instability (LA-TAI), similar to what was observed in Ref. [19]. In contrast, the flames were perfectly synchronized during HA-TAI.

This study focuses on quantifying various possible dynamical transitions from a state of stable combustor operation to the state of longitudinal TAI when control parameters are systematically varied in a turbulent annular combustor. In particular, we depict how varying the bulk velocity alters the characteristics of the bifurcations leading to TAI obtained by changing the equivalence ratio alone. We observe different states with distinct dynamics. These include intermittency, LA-TAI, HA-TAI, and MMO. We analyze the global and local flame dynamics during the state of MMO. Specifically, we compare the amplitude and phase evolution of neighboring flames during MMO. Finally, we describe a phenomenological model that captures the different states of the combustor operation observed in experiments. Secondary bifurcation from LA-TAI to HA-TAI is replicated by considering higher order nonlinearities.

The scheme of the paper is as follows. In Sec. 2, we discuss the experimental setup and measurement tools. In Secs. 3.1 and 3.2, we discuss the different routes to TAI. In Sec. 3.3, we show the presence of MMO during the transition from LA-TAI to HA-TAI and discuss the associated global and local flame dynamics. We replicate the observed criticality of bifurcation in a minimal model in Sec. 3.4. We present the conclusions of this study in Sec. 4.

## 2 Experimental Setup and Measurements

Experiments were performed on a laboratory-scale turbulent annular combustor. The combustor is shown in Fig. 1(a). The schematic of different parts of the experimental setup is shown in Figs. 1(b)–1(d). The design of the annular combustor is similar to the designs of Bourgouin et al. [19] and Worth and Dawson [22]. The setup consists of a settling chamber, burner tubes, and inner and outer ducts forming the annulus (Fig. 1(b)). Dehumidified air from a compressor was mixed with liquefied petroleum gas (LPG—40% propane and 60% butane by volume) in a mixing chamber to produce technically premixed LPG-air mixture. The LPG-air mixture enters through the bottom of the settling chamber through twelve inlet ports. Each has an internal diameter of 9.5 mm and mounted perpendicular to the axis of the combustion chamber. The diameter and length of the settling chamber are 400 mm and 440 mm, respectively. The settling chamber is equipped with a flow straightener, to arrest any transverse velocity fluctuations, followed by a flow divider to ensure uniform flow to all the burners.

The reactant mixture flows into the annular chamber through sixteen burner tubes mounted on the dump plane. The inner diameter and length of the burner tubes are 30 mm and 150 mm, respectively. A schematic of the dump plane with the arrangement of the burner tubes, pressure measurement ports, and the location of the pilot flame is shown in Fig. 1(c). A schematic of the cross section of the burner tube showing the position of the flame arrestor, swirler, and the convergent section at the exit is shown in Fig. 1(d). Sixteen circular disks of 10 mm thickness and 36 mm diameter with 179 holes (diameter of each hole is 1.5 mm) are mounted at the bottom of each burner tube to prevent flashback. Sixteen axial swirlers are mounted on top of each burner tube to impart solid-body counterclockwise rotation downstream of the swirler. Each swirler consists of six guide vanes mounted on a central shaft of diameter 15 mm and inclined at $\alpha =60\u2009deg$ with respect to the injector axis. The geometric swirl number is calculated as $S=2/3\u2009tan\u2009\alpha =1.15$ [23]. A converging section having a contraction area ratio of 2, height of 18 mm, and exit diameter *d *=* *15 mm connects the burner tube to the dump plane (Fig. 1(d)).

The dump plane consists of sixteen burner inlets, a port for a pilot flame, and three pressure measurement ports. The annular combustion chamber is made up of two cylindrical concentric ducts mounted on the dump plane. The flames are confined within the inner and outer stainless-steel ducts of diameter 300 mm and 400 mm, respectively [19]. The length of the inner and outer ducts is 200 mm and 400 mm, respectively. Mass flow controllers (Alicat Scientific, Tucson, AZ, MCR 2000 SLPM for air, and MCR 100 SLPM for LPG) were used for controlling the air and fuel flow rates. A pilot flame (Fig. 1(c)) was used to ignite all the sixteen burners at a nominal equivalence ratio of $\varphi =0.31$. The pilot flame was switched-off after flame stabilization. The equivalence ratio was then varied in the range of $\varphi \u2248$ 0.4–0.62 in the forward and reverse path in a quasi-static manner for different values of $\upsilon z$. The combustor was allowed to run for 5 s before acquiring data for 3 s at each value of the fuel flowrate. Finer step-size of $\varphi $ was used to capture points close to the transition, whereas coarser step-size was used after the transition to limit the overall time of combustor operation to 6 min to prevent any damage to the combustor from overheating. The mixture bulk flow velocity was varied in the range of $\upsilon z\u2248$ 5.5–12 m/s. The Reynolds number based on the burner exit diameter varies from Re $\u2248$ 0.56$\xd7104$ to 1.22 × $104$. The power of the combustor varies between 10 and 100 kW depending upon the values of $\varphi $ and $\upsilon z$. The maximum uncertainty in the values of $\varphi $ is $\xb11.6%$ and for $\upsilon z$ and Re is $\xb10.8%$.

The acoustic pressure fluctuations are recorded using three piezo-electric transducers (PCB Piezotronics, PCB-103B02, Depew, New York, sensitivity—217.5 mV/KPa, uncertainty: ±0.15 Pa), named as PC1, PC2, and PC3, are mounted on a waveguide (diameter 4 mm and length 3.2 m) at a distance of 75 mm from the combustor backplane (Fig. 1(c)). The pressure signals were acquired for 3 s at a sampling frequency of 10 kHz and digitized using a 16 bit analog-to-digital (A-D) conversion card (NI-6348, Austin, TX).

A high-speed CMOS camera (Phantom V 12.1, Wayne, NJ) with CH* filter (bandwidth of 435±10 nm) was used to capture chemiluminescence images of the flames. The camera was operated at a sampling frequency of 2 kHz and a pixel resolution of 1280 × 800. Combustor half-plane of size 400 mm$\xd7200$ mm consisting of eight flames was visualized with the aid of an air-cooled mirror placed overhead of the combustor. The position of eight burners, as imaged by the camera, has been highlighted in Fig. 1(c). The camera was outfitted with a Nikon AF Nikkor 70–210 mm $f/4$ to $f/5.6$ camera lens. A total number of 7417 images were acquired during the MMO. A pulse generated from a function generator (Tektronix AFG1022, China) was used to trigger the camera and the pheripheral component interconnect card to acquire measurements simultaneously.

## 3 Results and Discussions

To study the dependence of the characteristics of bifurcation to TAI of the annular combustor, we vary the bulk flow velocity in the range $\upsilon z=$ 5.5 to 12 m/s (Re $=0.56\u22121.22\xd7104$). For each value of $\upsilon z$, we vary $\varphi $ in the range of $0.4\u22120.62$.

### 3.1 Routes to Thermoacoustic Instability.

Figure 2(a) shows the variation of root-mean-square (rms) value of pressure oscillations ($prms\u2032$) as a function of $\varphi $ corresponding to $\upsilon z\u22486.09$ m/s. For $\varphi \u2264$ 0.47 (region I), we observe the state of CN with $prms\u2032\u223c20$ Pa. For $\varphi $ between 0.47 and 0.49 (region II), we see intermittency in pressure oscillations with $prms\u2032\u223c$ 60 Pa. For $\varphi >0.49$, we observe the state of LA-TAI with $prms\u2032\u223c100$ Pa. On decreasing $\varphi $ from 0.60 to 0.40, hysteresis is seen in the system dynamics. The amplitude of LA-TAI decreases gradually as $\varphi $ is lowered. The hysteresis and the difference in the amplitude of oscillations in the forward and reverse path possibly arise due to thermal inertia of the combustor walls [24].

In Fig. 2(b), we plot the variation of $prms\u2032$ as a function of $\varphi $ at $\upsilon z\u22488.51$ m/s. For $\varphi \u22640.47$ (region I), the amplitude of pressure fluctuations is very low, indicating the state of CN. Upon increasing $\varphi $ up to 0.50 (region II and III), we first notice the state of intermittency followed by LA-TAI. On increasing $\varphi $ past 0.50 (region IV), we observe an abrupt increase in the amplitude of $prms\u2032$ indicating a secondary bifurcation to HA-TAI. At the fold point ($\varphi =0.49$), the low-amplitude stable limit cycle loses stability and jumps to a secondary limit cycle, which has a higher amplitude. The bistable region (hatched region V in Fig. 2(b)) is obtained for $\varphi =$ 0.54 to 0.45. The difference of about 400 Pa between the forward and reverse path is possibly due to different boundary conditions of the combustor as a result of prolonged operations [24].

In Fig. 3, we plot the time series and the power spectrum of $p\u2032$ during different states of combustor operation for $\upsilon z\u22488.51$ m/s acquired in the forward direction. At $\varphi $ = 0.44, we see aperiodic oscillations having broadband spectrum, indicating the state of CN (Figs. 3(a) and 3(b)). At $\varphi $ = 0.48, there are intermittent bursts of high-amplitude periodic pressure oscillations amidst low-amplitude aperiodic pressure fluctuations (see insets), corresponding to the state of intermittency (Fig. 3(c)) [4]. In order to ensure that the state of intermittency is statistically stable, we show a long time series of 15 s. We also notice that the power spectrum narrows at $fn\u2248213$ Hz associated with the first longitudinal mode of the combustor with an intensity of 125 dB (Fig. 3(d)). At $\varphi =$ 0.49, $p\u2032$ is periodic with a dominant peak at $fn\u2248218$ Hz and intensity of 141 dB (Figs. 3(e) and 3(f)). The characteristics of LA-TAI are very similar to what was reported in Ref. [19]. We refer to this state as LA-TAI. Finally, at $\varphi =$ 0.50, $p\u2032$ is periodic with $prms\u2032\u22481425$ Pa corresponding to $fn\u2248227$ Hz and intensity of 157 dB (Figs. 3(g) and 3(h)). This state is referred to as HA-TAI. Similar HA-TAI has also been reported for a standing mode of transverse instability in a similar configuration [25], further exemplifying the relevance of the problem.

### 3.2 Change of Criticality of Bifurcation.

We now focus on the effect of variation of the control parameters on the characteristics of the bifurcations to TAI. Figure 4(a) shows $prms\u2032$ as a function of $\varphi $ and $\upsilon z$. For visual clarity, the bottom surface is plotted only till the secondary fold point where the oscillations abruptly transition from LA-TAI to HA-TAI (up arrow). The top surface is plotted till the fold point where the dynamics of pressure oscillations jumps from HA-TAI to CN (down arrows).

For low bulk velocities from $\upsilon z\u22486$ to 6.5 m/s, the transition from CN to LA-TAI of the acoustic subsystem takes place through a bifurcation similar to supercritical Hopf bifurcation [4] (see Fig. 2(a)). Since the flow is turbulent, the transition is associated with an intermittent state between the state of CN and TAI. This is different from a typical supercritical bifurcation where the dynamics are expected to change smoothly from CN to TAI. For $\upsilon z\u22487.3$ m/s (refer to Fig. 5), there is a transition from CN to HA-TAI through the state of intermittency, LA-TAI, and MMO. This route to HA-TAI is explained in more detail in Sec. 3.3. For $\upsilon z>7.5$ m/s, the transition from CN to HA-TAI takes the following route: CN to LA-TAI via intermittency followed by a secondary bifurcation from LA-TAI to HA-TAI (Fig. 2(b)).

The interpolated boundary between different dynamical states in the parametric $\varphi \u2212\upsilon z$ plane is shown in Fig. 4(b). A cubic spline has been used to interpolate and extrapolate the boundary between different regions. Since we do not observe secondary bifurcation to HA-TAI below $\upsilon z<6$ m/s, we have limited the ordinate to $\upsilon z\u22656$ m/s. The approximate boundary separating regions I and III marks the transition from CN to LA-TAI. This transition is always associated with the state of intermittency (indicated with “×” markers in region II). The boundary between III and IV indicates the boundary of the secondary fold bifurcation to HA-TAI. The small parametric region for which we observe MMO is also indicated in Fig. 4(b). The bistable region has been hatched, and the boundary between regions V and I shows the fold point of the system. Several salient features can be observed from the stability diagram. First, the onset of LA-TAI and HA-TAI takes place at a progressively lower value of $\varphi $ as $\upsilon z$ is increased. Second, the range of $\varphi $ over which we observe LA-TAI decreases, and HA-TAI increases on increasing $\upsilon z$. Finally, there is also an increase in the width of the bistable region with an increase of $\upsilon z$.

### 3.3 Mixed-Mode Oscillations: A Transition State Between Low-Amplitude Thermoacoustic Instability to High Amplitude Thermoacoustic Instability.

For $\upsilon z<7$ m/s, the combustor dynamics exhibits the state of LA-TAI, while for $\upsilon z>7.5$ m/s there is a secondary fold bifurcation of LA-TAI to HA-TAI (Fig. 4(a)). Figure 5 shows $prms\u2032$ as a function of $\varphi $ for $\upsilon z\u22487.3$ m/s. As before, we observe the state of CN (region I), intermittency (region II), LA-TAI (region III), and HA-TAI (region V). Of special interest here is the transition from LA-TAI to HA-TAI. The transition is not abrupt, but takes place smoothly with a monotonic increase in the value of $prms\u2032$ (Fig. 2(b)).

A representative time series of $p\u2032$ at $\varphi $ = 0.54 (region IV) is shown in orange in Fig. 6(a). The associated amplitude spectrum $|p\u0302(f)|$ is shown in Fig. 6(b). The peaks at *f _{n}* = 216 Hz and $fn+fs=223$ Hz are indicated. The low frequency modulation calculated from the envelope of $p\u2032$ is

*f*= 7 Hz. Next, we calculate the instantaneous HRR $q\u02d9k\u2032(t)$ from the chemiluminescence images by considering a region circumscribing the

_{s}*k*th burner and summing over all the intensity values. An example of such a region for burner four is shown in Fig. 6(c). The fluctuations in the local HRR oscillations $q\u02d94\u2032$ from burner 4 is shown in Fig. 6(a) (in blue) and the corresponding amplitude spectrum $|q\u03024(f)|$ in Fig. 6(b) (in blue). We observe a peak at

*f*= 216 Hz and

_{n}*f*= 7 Hz and their combination at $fn+fs$.

_{s}In combustion literature, amplitude-modulated limit cycle oscillations are usually referred to as beats [16–18]. The phenomenon of beats is associated with the linear superposition of acoustic waves with a very small frequency difference. Beats are characterized by the constructive and destructive interference pattern, as was observed in Refs. [17] and [18]. However, the applicability of beats is restricted only to linear systems. The amplitude-modulated oscillations in Fig. 6(a) are different from beats because of two key reasons. First, the oscillations do not show the well-known constructive and destructive pattern usually observed for beating (e.g., Fig. 12(*a*) in Han et al. [18]). Second, the parametric value ($\upsilon z=7.3$ m/s) for which we observe these amplitude-modulated oscillations separates the region for which we get LA-TAI and secondary bifurcation from LA-TAI to HA-TAI. Consequently, slow scale (*f _{s}* = 7 Hz) oscillations associated with $q\u02d9\u2032$ causes the parameter to increase past the bifurcation point effecting high-amplitude oscillations. When the parameter value decreases due to slow oscillations, it crosses the bifurcation point, and the system dynamics switch back to low-amplitude oscillations. The latter is a well-known mechanism associated with MMO [14] and have been discussed in the context of Rijke tube recently [26]. Thus, we refer to these oscillations as MMO.

#### 3.3.1 Global Flame Dynamics.

We have already discussed the local and global flame dynamics during the state of CN, intermittency, LA-TAI, and HA-TAI in another complementary study [21]. Here, we discuss the flame dynamics associated with the state of MMO in detail. Figures 6(c)–6(f) correspond to phase-averaged CH* images for pressure maxima (90 deg) and minima (270 deg) during epochs of high-amplitude oscillations are shown in Figs. 6(c) and 6(d), and for the low-amplitude oscillations in Figs. 6(e) and 6(f). The phase-averaged images ($\u27e8q\u02d9\u2032\u27e9$) were calculated from mean-subtracted chemiluminescence images. Hence, the phase-averaged images at pressure minima in Figs. 6(d) and 6(f) show negative values. Finally, we note that the colorbar is not centered around zero, which could be a consequence of the apparent nonlinear interaction between heat release rate and acoustic pressure fluctuations.

At the pressure maxima during epochs of high-amplitude oscillations (Fig. 6(c)), intense HRR oscillations manifest in the high intensity at the center of each burner in the phase-averaged images. The intense central region of the burners indicates the presence of a well-developed inner recirculation zone. In addition, all of the eight flames have almost the same intensity with a very similar flame structure. The flame intensity levels are low at the pressure minima (Fig. 6(d)). The behavior is remarkably different during the epochs of low-amplitude fluctuations of MMO. At pressure maxima (Fig. 6(e)), the intensity is much lower. The flame structure shows that the flames are stabilized along the central shear layer separating the inner and outer recirculation zones and indicate the absence of the vortex bubble at the burner centerline. At pressure minima (Fig. 6(f)), we notice very low intensities for all the flames. The global phase-averaged flame dynamics observed in Figs. 6(c)–6(f) are qualitatively similar to the phase-averaged flame dynamics observed during HA-TAI and LA-TAI [21].

#### 3.3.2 Local Flame Dynamics.

Next, we discuss the local flame behavior during MMO by analyzing the mutual interaction of adjacent flames. Figure 7(a) shows an enlarged portion of the acoustic pressure oscillations during the MMO consisting of alternate epochs of high-amplitude and low-amplitude oscillations.

*k*th burner $q\u02d9k\u2032$, we construct a complex analytic signal $\zeta (t)=q\u02d9k\u2032(t)+iH[q\u02d9k\u2032(t)]=Ak(t)exp(i\theta kt)$. Here $\theta k(t)$ is the instantaneous phase and $Ak(t)$ is the instantaneous amplitude of the analytic signal associated with $q\u02d9k\u2032$. The Hilber transform is defined as

evaluated at the Cauchy principal value. The normalized HRR can then be evaluated as: $q\u02d9k\u2032(t)/Ak(t)=sin\u2009\theta k$. We plot the temporal variation in the normalized amplitude of HRR from all of the burners in Fig. 7(b) and the instantaneous relative phase between different pairs of burners in Fig. 7(c).

We observe that during the epochs of high-amplitude oscillations (region I), all of the burners attain maxima and minima in the HRR concurrently with almost the same amplitude (Fig. 7(b)). In contrast, during epochs of low-amplitude oscillations (region II), we observe that all of the burners do not attain maxima and minima in the HRR at same time instances. Instead, there is some phase mismatch in the HRR oscillations associated with the different flames (Fig. 7(b)). This contrasting behavior is well captured from the instantaneous relative phase. During high-amplitude oscillations (region I), the burners are in-phase synchronized, and the phase difference between various pairs of burners always remains well-below 90 deg (Fig. 7(c)). However, during low-amplitude oscillations (region II), we observe that some burner pairs are in-phase while others are out-of-phase. For instance, burners 2–3 are initially in-phase and, after some time, go 180 deg out-of-phase, while burners 6–7 are 90 deg out-of-phase for almost the entirety of the duration of low-amplitude oscillations. Similar observations can be drawn for other pairs of burners.

*R*), which is defined as [30]

where *θ _{k}* is the instantaneous phase calculated from $q\u02d9k\u2032$ and

*N*is the total number of burners. The value of

_{b}*R*defines the degrees of spatial coherence of the oscillators over time. At any time instance,

*R*=

*0 indicates spatial desynchrony, while*

*R*=

*1 indicates spatial synchrony. Intermediate values of*

*R*indicate partial synchrony/desynchrony.

We plot the temporal variation of the Kuramoto order parameter in Fig. 7(d). We can observe that the value of *R* is near 1 for epochs of high-amplitude oscillations (region I). While, during low-amplitude pressure oscillations, the value of *R* fluctuates between 0.50 and 1 (region II). Thus, during the epochs of high-amplitude oscillations, all the flames get perfectly synchronized, while during epochs of low-amplitude oscillations, they are weakly synchronized. Similarly, the flames are weakly synchronized during the state of LA-TAI with $R\xaf=0.84$ and perfectly synchronized during the state of HA-TAI with $R\xaf=0.97$ [21].

We also note that the symmetry of flame response increases during high-amplitude oscillations. The symmetry here refers to the similar manner in which the amplitude and phase of the eight burners evolve during HA-TAI. Thus, during the transition from CN, where the degree of symmetry in flame response is minimal, symmetry is established during HA-TAI. The degree of symmetry increases smoothly as the number and duration of high-amplitude oscillations during the state of MMO increases with an increase in $\varphi $ till HA-TAI is established. This is in contrast to the secondary bifurcation from LA-TAI to HA-TAI where there is an abrupt transition from a partially symmetric flame response to symmetric response [21].

As already discussed in Fig. 7(c), we notice a difference in the relative phase between the pairs of burners 2–3 and 6–7 inside the region II. For a better understanding of the scenario, we further investigate the instantaneous interactions between burner pairs 2–3 and 6–7 and contrast the synchronous behavior during low and high-amplitude oscillations over one cycle of oscillations. Figure 8(a) shows one cycle of low-amplitude oscillations during MMO. Figures 8(b)–8(f) shows the corresponding instantaneous flame image of burner pairs 2–3 and 6–7 at various points over the cycle. In Fig. 8(b), we observe high-intensity levels for burner 6 while the rest of the burners have lower intensity levels. In addition, burner 6 is 100 deg out-of-phase with burner 7 at the beginning of the cycle and has a mean phase difference of $\u27e8\Delta \theta 67\u27e9=53deg$ over the cycle. In comparison, burner 2 is in-phase with burner 3 for all the points over the cycle with a mean phase difference of $\u27e8\Delta \theta 23\u27e9=16\u2009deg$ (Figs. 8(b)–8(f)).

In Fig. 8(g), we show the high-amplitude oscillation during MMO over a cycle along with the instantaneous images of the burner pairs 2–3 and 6–7 in Figs. 8(h)–8(l). Compared to the behavior of burner pairs over a cycle of low-amplitude pressure oscillation (Figs. 8(b)–8(f)), the intensity of heat release rate at each of the burner is almost the same over the cycle of high-amplitude pressure oscillation (Figs. 8(h)–8(l)). The averaged phase difference over the points of the cycle are $\u27e8\Delta \theta 23\u27e9=13\u2009deg$ and $\u27e8\Delta \theta 67\u27e9=5\u2009deg$. The flame structures at each of the burners are also quite similar over different points in the cycle.

### 3.4 Modeling the Criticality of Bifurcation.

*F*contains the effects of unsteady heat release rate and damping. We can then expand the solution using modal expansion along the natural acoustic modes [31]

_{H}*γ*is the ratio of specific heat capacities,

*M*is the Mach number. Substituting Eq. (4) into Eq. (3), and projecting the equation on the

*m*th mode basis function, we obtain the following set of ordinary differential equations:

where *μ _{i}* are the model parameters, for $i=0,2,4,6$ and $\xi (t)$ is Gaussian white noise. We start by considering $\mu 4=\mu 6=0$ and absence of noise. In such a case, $\mu 2\eta 2$ makes up the nonlinear damping term while

*μ*

_{0}controls the driving. Setting $\mu 2=1$ (for convenience), it is easy to see that $\mu 2\eta 2$ positively damps the system and counteracts the driving induced by $\mu 0>0$ for large

*η*. Thus, the system undergoes a supercritical Hopf bifurcation for $\mu 2=1$ when

*μ*

_{0}is increased above 0. For $\mu 2=\u22121$, a family of unstable limit cycle exists, which can be stabilized by introducing fourth-order nonlinearity. For $\mu 4>0$, the unstable limit cycle undergoes a fold bifurcation and jumps to a stable limit cycle solution. Thus, fourth-order nonlinearity is required for subcritical Hopf bifurcation [9]. The fold point of the subcritical system depends on the choice of

*μ*

_{4}. Laera et al. [32] considered a similar flame model reducible to Eq. (6) for $\mu 6=0$ to predict supercitical and subcritical bifurcation in longitudinal and annular combustors.

In order to capture the secondary bifurcation to HA-TAI (Fig. 2(b)), we then need to consider higher order nonlinearity such that there is a fold bifurcation on stable low-amplitude LCO to high-amplitude LCO. This form has been considered by Campa and Juniper [33] in modeling TAI in a Rijke tube by considering the flame response to depend on the mass flowrate through sixth-order nonlinearity.

We include delta-correlated Gaussian white noise $\xi (t)=\sigma 2\delta (t)$ to the governing equation to account for the noisy behavior associated with the state of CN and aperiodic epochs of intermittency. Here, $\delta (\xb7)$ is the Dirac–Delta function and $\sigma 2$ indicates the strength of white noise. We solve this equation numerically using stochastic Runge–Kutta method [34] for 1000 time steps with *dt *=* *0.01, $\sigma 2=0.001,\u2009\mu 2=1$, and $\mu 4=\u22121$ chosen for convenience from previous discussion. The frequency of limit cycle oscillations is set as $\omega =2\pi f=1$ rad/s. The initial conditions for the forward path are: $\eta (0)=0.10$ and $\eta \u02d9(0)=0$. For the reverse path, $\eta \u02d9(0)=0$ and $\eta (0)$ are chosen as the amplitude of limit cycle oscillations of $\eta (t)$ obtained for a given *μ*_{0} in the forward path.

Figure 9(a) depicts the variation of *η*_{rms} as a function of *μ*_{0} and *μ*_{6}. As with the experimental results, the bottom surface is plotted till the secondary fold bifurcation where the dynamics abruptly transitions from low to high-amplitude LCO, and the top surface is plotted till the fold point, where the dynamics transition from high-amplitude LCO to a fixed-point solution. For $\mu 6>0.25$, there is a supercritical-like bifurcation to LA-TAI through the state of intermittency. For $0.15<\mu 6<0.25$, there is a transition from the fixed point solution to low-amplitude LCO through intermittency, followed by secondary bifurcation to high-amplitude LCO. Figure 9(b) shows the stability diagram observed from the model. The different regions are serialized as per Fig. 4. We observe that Figs. 9(a) and 9(b) and Figs. 4(a) and 4(*b*) are qualitatively similar. We can observe the two different final states. First, there is a transition from a fixed point to low-amplitude LCO through intermittency. Second, there is a transition from a fixed point to intermittency to low-amplitude LCO followed by a secondary bifurcation to high-amplitude LCO. Moreover, the bistable region broadens when *μ*_{6} is decreased. The parameters *μ*_{0} and *μ*_{6} in the model capture the respective effects of $\varphi $ and $\upsilon z$.

Finally, we note that secondary bifurcation can occur only when stable limit cycle solution arising from supercritical Hopf bifurcation, due to a change in leading-order control *μ*_{0}, loses its stability and give rise to an unstable branch of limit cycle oscillations and stable limit cycle solution of significantly higher-amplitude. The loss of stability of limit cycle oscillations can only be effected when higher order nonlinearities become significant [9]. Thus, we infer that $\varphi $ and $\upsilon z$ influence the physical processes, which affect the higher order nonlinearities in the system, thereby leading to a secondary bifurcation. However, we also note the limitations of the model in predicting the exact nature of the boundaries separating different dynamical states (Figs. 4(b) and 9(b)). Also, we observe that the trend of bifurcation captured by *μ*_{6} is the opposite of what we observed when $\upsilon z$ is changed (Fig. 4(a)).

## 4 Conclusions

In this study on a premixed swirl-stabilized annular combustor, we demonstrate the effect that change of control parameters has on determining the characteristics of the bifurcations leading to thermoacoustic instability (TAI). We systematically vary the bulk flow velocity ($\upsilon z=5.5\u221212$ m/s) and equivalence ratio ($\varphi =0.40\u22120.62$) and measure the acoustic pressure fluctuations during different states of combustor operation. We observe different dynamical states depending upon the values of $\upsilon z$ and $\varphi $, namely, combustion noise, intermittency, low-amplitude thermoacoustic instability (LA-TAI), high-amplitude thermoacoustic instability (HA-TAI), and mixed-mode oscillations (MMO). Of particular interest is the secondary bifurcation where there is an abrupt jump in the amplitude of pressure oscillations from the state of LA-TAI to HA-TAI at high $\upsilon z$. For intermediate values of $\upsilon z$, we find that the transition from LA-TAI to HA-TAI happens through MMO, which leads to a continuous increase in the amplitude of pressure oscillations from LA-TAI till the state of HA-TAI is reached. We replicate these dynamics in a phenomenological model. The secondary bifurcation is modeled through the inclusion of a higher order source/sink.

We also focus on the flame dynamics associated with MMO. We find that the phase-averaged flame structure is very different for the epochs of low- and high-amplitude oscillations. Specifically, the flame structure shows an intense heat release rate along the inner recirculation zone during the epochs of high-amplitude oscillations. In contrast, during the epochs of low-amplitude oscillations, the flames are stabilized along the central shear layer of the burner with an absence of vortex breakdown bubble. We then compare the local flame-flame interaction of the neighboring flames during MMO. We quantify the degree of synchronization among the flames during the MMO using the Kuramoto order parameter (*R*). We find that the flames are perfectly synchronized for the duration of the high-amplitude oscillations possessing a very high value of *R*, while during low-amplitude oscillations, the flames are only weakly synchronized with a comparatively lower value of *R*.

## Acknowledgment

This work was supported by the Office of Naval Research Global (Contract Monitor: Dr R. Kolar) Grant no. N62909-18-1-2061. We gratefully acknowledge Mr S. Thilagaraj and Mr S. Anand for their contribution in the design and fabrication of the experimental setup and Mr P. R. Midhun for his help in conducting experiments. We would also like to thank Dr Samadhan Pawar and Ms Krishna Manoj for interesting discussions. Asalatha Nair is grateful to J C Bose Fellowship for funding her postdoctoral fellowship.

## Funding Data

Office of Naval Research Global (Grant No. N62909-18-1-2061; Funder ID: 10.13039/100007297).

## Nomenclature

- CN =
combustion noise

- HA-TAI =
high-amplitude thermoacoustic instability

- HRR =
heat release rate

- LA-TAI =
low-amplitude thermoacoustic instability

- LCO =
limit cycle oscillations

- MMO =
mixed-mode oscillations

- $p\u2032$ =
acoustic pressure fluctuations

- $q\u02d9k\u2032$ =
HRR fluctuations of kth flame

*R*=Kuramoto order parameter

- Re =
Reynolds number

- TAI =
thermoacoustic instability

*θ*=_{k}phase of HRR of the kth flame

- $\upsilon z$ =
nominal bulk flow velocity

- $\varphi $ =
equivalence ratio