## Abstract

In this paper, we present combustor acoustics in a high-pressure liquid-fueled rich burn—quick quench—lean burn (RQL) styled swirl combustor with two separate fuel circuits. The fuel circuits are the primary (which has a pressure atomizer nozzle) and secondary (which has an air blast type nozzle) circuits. The data were acquired during two dynamical regimes—combustion noise, where there is an absence of large amplitude oscillations during the unsteady combustion process, and intermittency, where there are intermittent bursts of high amplitude oscillations that appear in a near-random fashion amidst regions of aperiodic low amplitude fluctuations. This dynamic transition from combustion noise to combustion intermittency is investigated experimentally by systematically varying the fuel equivalence ratio and primary-secondary fuel splits. Typical measures such as the amplitude of oscillations cannot serve as a measure of change in the dynamics from combustion noise to intermittency due to the highly turbulent nature. Hence, recurrence plots and complex networks are used to understand the differences in the combustor acoustics and velocity data during the two different regimes. We observe that the combustor transitions from stable operation to intermittency when the equivalence ratio is increased for a given primary fuel flowrate and conversely when the percentage of secondary fuel flowrate is increased for a given equivalence ratio. The contribution of this work is to demonstrate methodologies to detect combustion instability boundaries when approaching them from the stable side in highly turbulent, noisy combustors.

## Introduction

The occurrence of detrimental instabilities in aviation gas turbine combustors continues to hinder the development of modern combustors [1–3]. The acoustics of the confined combustor is driven by heat release from the combustion process adding energy to the turbulent flow. When there is a lock in between the hydrodynamic fluctuations and the acoustics, large amplitude oscillations are established [1,4]. Such combustion-driven oscillations are undesirable because they significantly reduce the lifetime of the combustor with increased wear and tear, and fatigue of the combustor walls due to enhanced heat transfer to the walls and thus can lead to structural damage to the combustor. They also reduce the operational space of the combustor and increase performance losses [5,6].

Unsteady combustion tends to be a noisy process even in the absence of large amplitude oscillations and this noise is termed combustion noise. It is generally attributed to two sources, namely, direct combustion noise—due to unsteady volumetric expansion in the heat release zone, and indirect combustion noise—produced by the hot products traversing a region of mean flow gradients. In many turbulent combustors, the transition from combustion noise to thermoacoustic instability can often be triggered by the unsteadiness in the flow and combustion process. The prediction of the amplitude and/or the frequency of such triggered oscillations and the stability margins of the combustor still remains a challenge to researchers due to the complex nature of the dynamics in the combustor amongst flow, heat release, and the chamber acoustics which can often couple with each other.

Between the stable (combustion noise) and unstable operation (thermoacoustic instability), there exists a stable dynamical regime known as intermittency distinct from the previously mentioned regimes. During intermittency, there are intermittent bursts of high-amplitude periodic oscillations that appear in a near random fashion amidst regions of low-amplitude aperiodic fluctuations. Shown in Fig. 1 are signals obtained during stable operation (combustion noise) and intermittency. Although reports of such possibly intermittent burst states exist in literature, their dynamics have not been investigated in detail for a turbulent combustor. Nair et al. [7] illustrated that the transition from combustion noise to thermoacoustic instability happens via intermittency for the bluff body and swirl combustors. Gotoda et al. [8] showed that the transition to periodic unstable oscillations from stochastic stable fluctuations when the equivalence ratio of the combustor is varied. Sujith et al. [9] showed that the dynamic transition in a thermoacoustic system that consists of ducted laminar premixed flame occurs via quasi-periodic oscillations and the intermittent large amplitude irregular fluctuations were seen in the unsteady pressure fluctuations before blow-off. Kasthuri et al. [10] observed the transition from small amplitude stable operation to large amplitude thermoacoustic instability occurs through intermittency in a multi-element liquid rocket combustor. They used recurrence and fractal-based analysis to differentiate between the different regimes of operation. Clavin et al. [11] showed that the oscillation bursts in the pressure fluctuations are an effect of a multiplicative noise term, which was used to model the effect of turbulence, in the vicinity of subcritical and supercritical Hopf bifurcation.

Typically, changes in the dynamics of the thermoacoustic system are captured by bifurcation diagrams usually constructed by tracking the peak in the measured signal and plotting them as a function of the control parameter. However, with the presence of turbulence, the amplitude is a stochastic quantity, even during thermoacoustic combustion, and bifurcations become “smeared out.” This can obscure the identification of bifurcations, especially in conditions where combustion instability amplitudes are low. To overcome this, a ratio of number of peaks can be deployed to get the bifurcation diagram as detailed in Ref. [7] and the bifurcation diagram constructed using this method is detailed in later sections of this paper.

While such a bifurcation diagram can enable us to infer the nature of the criticality of the change in the dynamics of the system, it cannot be used to determine the proximity of the system to an impending instability sufficiently in advance. The impending instability can be predicted using recurrence plots (RPs) and recurrence quantification analysis. Recurrence of state points in the phase space is a fundamental property of deterministic systems. Recurrence plots are used to visually identify the time instants at which the phase space trajectory of the system revisits roughly the same location in the phase space. Several derived parameters from the RPs such as recurrence rate, determinism, trapping time, etc., can be used to measure the proximity of the system to an impending instability. Excellent reviews on RPs can be found in Refs. [12] and [13].

The complex interactions between the acoustic field, the hydrodynamic field, and the unsteady combustion process resulting in different dynamical regimes varying from combustion noise to thermoacoustic instability in a thermoacoustic combustor suggest that it can be treated as a complex system [14,15]. In such a system, the interaction between the different components is nonlinear such that the collective behavior of the system is different from the sum of their individual contributions. Complex networks comprise nodes and links, where nodes represent the components of the system and links represent the interactions between these components. The topology and the measures derived from a network can also be used to characterize the qualitative and quantitative behavior of the complex system as it transitions between the dynamical regimes. Murugesan et al. [16,17] used visibility networks to analyze the dynamical regimes of the thermoacoustic system. While converting a time series into complex networks using a visibility algorithm, information related to geometry characteristics of the attractor and topology of the attractor is lost. Alternatively, *ε*-recurrence networks (RN) can be used which preserves the geometric characteristic of the attractor. Godavarthi et al. [18] constructed RN from the time series of the acoustic pressure obtained from a turbulent combustor. They demonstrated that the RN preserves the geometry of the attractor when constructed for the different dynamical regimes as the system transitions from thermoacoustic instability to blow-off.

This paper is organized as follows. The Materials and Methods section2 describes the experimental setup used to investigate the intermittent oscillations in turbulent combustion and gives a brief description of the construction of the bifurcation diagrams, phase portraits, recurrence plots, and recurrence networks that are used in this study. After which, the results obtained from these tools to characterize intermittency in the turbulent combustor are described. Finally, the conclusions of this study are summarized.

## Materials and Methods

### Experimental Setup.

Experiments were conducted in a high-pressure, liquid-fueled, single sector rich burn—quick quench—lean burn (RQL) style combustor with the capability of having different swirlers. The schematic of the setup is shown in Fig. 2, respectively. The key component of the facility includes an upstream flow conditioning section, a nozzle bulkhead where the swirler is located, an optically accessible pressure vessel, an exhaust section, and a modular combustion liner.

The combustion liner consists of optically accessible windows on the sides and has laser access through the top and an effusion bracket on the bottom. The top and bottom panels consist of dilution holes while the bottom panel also has a homogenous distribution of effusion holes upstream and downstream of the dilution holes. A more detailed description of the liner can be found in Refs. [19] and [20]. The pressure vessel in which the liner sits has optical access on all four sides, through quartz windows with dimensions 215 mm × 115 mm. The water-cooled exhaust section consists of a rectangular choked orifice at the exhaust exit to maintain a prescribed pressure in the combustion chamber.

The air feed includes two lines, namely, core flow and the quench flow, and is supplied from the same compressed air facility with preheat capabilities. The mass flow rates of the core and the quench flows are individually monitored using Vortex flowmeters (with an accuracy of ±1\% of reading) installed approximately 9 m from the inlet of the test section. Before entering the test section, the preheated, pressurized core flow passes through the swirler depicted in Fig. 3 with nozzle diameter *D* while the quench air flows around the liner and enters the combustion liner through the effusion holes and dilution holes present. The core air and quench air are maintained at similar preheat temperatures of around 420 K which was measured using a *K-*type thermocouple (with a typical accuracy of ±0.75\% of reading) located approximately 10 cm and 12 cm from the inlet of the test section, respectively. Their values remained ±5 K of the nominal value during a measurement. Static pressure transducers (with an accuracy of ± 0.5\% of reading) are also installed prior upstream of the inlets of the core flow and quench flow into the test section. The differential pressure across the swirler and liner is continuously monitored using differential pressure transducers.

The upstream end of the liner is the nozzle bulkhead, as shown in Fig. 3, where the modular swirler is fitted and contains a flush-mounted *K*-type thermocouple, a static pressure transducer feed, an acoustic pressure transducer feed, and the ignitor. The acoustic pressure transducer is located outside the combustor and connected to the nozzle bulkhead feed using a standoff tube and a special tee fitting that minimizes any cross-sectional area changes along the length of the tube. The acoustic waves travel from the combustor through the length of the tube and are measured by the transducer before continuing along approximately 12 m of loosely coiled tubing which is meant to attenuate the signal before it has the chance to reflect off the end of the tube and interfere with the measurement of the transducer.

The fuel circuits consist of two lines, namely, the primary and the secondary fuel circuits. The primary line feeds to pressure atomizer nozzle in the fuel injector while the secondary line feeds to six air blast nozzles located in the fuel injector. The fuel in the primary and the secondary lines were individually regulated using pressure regulators and were individually metered using a Coriolis flowmeter (with an accuracy of ±0.2\% of reading) on the primary and a turbine flowmeter (with an accuracy of ±0.25\% of reading) on the secondary line. A 150 L tank pressurized using Nitrogen is used to feed liquid fuel to the two lines. Jet A was burnt in these experiments to simulate a realistic aircraft engine.

### Bifurcation Diagrams.

Typically, bifurcation diagrams of experimental data are drawn by tracking the peaks in a measured signal and plotting them as a function of the control parameter. But given the highly turbulent processes that take place during combustion, the peak amplitude is shifted across a range of values even during combustion instability. Nair et al. [7] proposed an alternative method using a threshold-based analysis to create the bifurcation diagrams which are briefly described below.

*N*) for a time duration

*t*above a fixed threshold

*ε*, which would correspond to acceptable levels of amplitude for the system. For this study,

*ε*was chosen to be the maximum normalized acoustic pressure during the combustion noise case. If

*N*

_{tot}represents the total number of peaks that occur within that time, one can then assign the probability of the system attaining instability as

For all the cases considered, the number of samples considered for this threshold-based analysis is $4\u2009x\u2009104$ and the threshold ε was chosen to be 1\% of the temporal mean of the combustor pressure $(Pcomb)\xaf$ during this interval of the stable operation, which roughly corresponds to the maximum amplitude level during the stable operation.

### Phase Portraits.

Recurrence plots and recurrence networks are constructed using the knowledge of the distances of the different states in the phase portraits and hence, we briefly describe the construction of the phase portraits from the measured acoustic signal. Additionally, the state space representation of data is useful because it allows one to readily visualize and compare the oscillations over a large number of cycles [2].

The states space of an n degree-of-freedom system is characterized at each instant by the vector $[p\u2032(t)\u2009dp\u2032(t)/dt\u2026dn\u22121p\u2032(t)/dtn\u22121]$. Such a characterization of discretely sampled data where only a single variable is measured, i.e., $[p\u2032(t)]$, requires numerical differentiation of the data, which is very sensitive to noise and measurement errors. Consequently, experimental time series are typically characterized by time-delay embedding methods [22]. The basic idea is to construct the vector $P(d,\tau )=[p\u2032(t)\u2009p\u2032(t+\tau )\u2009p\u2032(t+2\tau )\u2026p\u2032(t+d\tau )]$. Here t is varied from 1 to n_{o} – (d – 1)τ, where *n*_{o} is the total number of data points in the time series, d is the embedding dimension and $\tau $ is the optimum delay. Each delay vector corresponds to a state point in the phase space and the combination of all these vectors constitutes the phase space trajectory. To perform an appropriate phase space reconstruction for a particular state of the system, we need to obtain the minimum embedding dimension and the optimum time delay. The minimum embedding dimension was obtained using Cao's method [22] and the optimum delay was obtained using the average mutual information of the time series [23]. Note that the phase space portrait of a purely sinusoidal wave of one mode is elliptic in shape.

### Recurrence Plots.

*d*). The recurrence plot of any time series is constructed by computing the pairwise distances between the state points of the reconstructed state space portraits. The recurrence matrix is given by the equation

where *ε* is the threshold to define the neighborhood of a state point in the reconstructed phase space and $||p\u2032i\u2212p\u2032j||$ is the Euclidean distance between any two pair of states *i* and *j* which range from 1 to n_{o} – (d – 1) τ. The recurrence matrix is a binary matrix, with black points in the matrix indicating the recurrence based on the predefined threshold while white points indicate nonrecurring pairs.

In the recurrence matrix, lines that are parallel to the mean diagonal represent such segments of the phase space trajectory which run parallel for some time and the vertical lines (or equivalently horizontal lines, since the recurrence matrix is symmetric) represent segments that remain in the same phase space region for some time. Hence, for a purely stochastic time series, the recurrence plot will be homogeneously grainy while for a purely deterministic signal such as a sinusoidal time series, the recurrence plot will contain only diagonal lines [12,13]. Several statistical measures such as recurrence rate (RR), determinism, trapping time (Τ), etc., can be used to study the recurrence behavior of the phase space trajectory.

_{o}– (d – 1) τ. The trapping time (Τ) is defined as shown in Eq. (5). The quantity Τ measures how long the system remains in a particular dynamical state, which is measured using the length of vertical lines (or equivalently the horizontal lines) in the RPs and is obtained by

where v is the length of a vertical line.

## Results and Discussions

In this study, we seek to find the boundary of the different stable dynamical behavior observed in the turbulent RQL-style combustor. As mentioned before, the Reynolds number at the exit of the swirler was fixed at $2.038\u2009x\u2009105$ and the head-end equivalence ratio was parametrically varied from 0.83 to 1.42. At each of the equivalence ratio values, two measurements were made, one with a primary equivalence ratio of 0.36 and the other at 0.56.

Figure 4 shows the bifurcation diagram constructed using the probability of the system attaining instability since the value of α is a measure of the proximity of a system to instability. In the figure, we have plotted values of α as a function of head-end equivalence ratio ($\phi tot$) and primary equivalence ratio ($\phi p$) starting from very low amplitude combustion noise to intermittency. The values of α along a given branch, i.e., given value of $\phi p$, seems to vary smoothly as the control parameter $\phi tot$ traverses from regions of combustion noise to the intermittent regime, where the pressure signals occasionally cross the threshold and lead to an increase in the value of α. These intermittent excursions increase as $\phi tot$ is increased. Comparing the two branches, i.e., between the $\phi p$ = 0.36 and 0.56 branches, we observe that the $\phi p$ = 0.36 has higher values of α compared to the other for a given value of $\phi tot$, which indicates that the pressure signal crosses the threshold more often in the branch with lower $\phi tot$ for a given value of $\phi tot$.

From the bifurcation plot, we observe that at a low head-end equivalence ratio $\phi tot$ with large percentage of the primary fuel, i.e., large $\phi p$, the pressure fluctuations have low amplitude and are aperiodic, for instance, ($\phi tot,\u2009\phi p$) = (0.83, 0.56). The time series of the acoustic pressure signal during this stable operation is shown on the left plot in Fig. 1. As the head-end equivalence ratio is increased or when the percentage primary in the total fuel is decreased, we observe some periodic part in the time series alternating with the aperiodic part. Generally, as $\phi tot$ is increased and/or as $\phi p$ is decreased the amplitude of the periodic part increases. The acoustic pressure time series obtained during the intermittent state with ($\phi tot,\u2009\phi p$) = (1.43, 0.36) is shown in the right plot in Fig. 1.

The power spectra (FFTs) corresponding to the pressure time series shown in Fig. 1 are shown in Fig. 5. We observe that there are no significant narrowband features observed in FFTs of the time series corresponding to the stable combustion case and on the contrary, we see several narrowband peaks in the intermittent case with the fundamental harmonic at St = 0.11. (Note that $St=fDU$, where *D* is the diameter of the swirler and *U* is the velocity at the exit of the swirler). For other cases, we observe similar narrowband features in the power spectra.

Next, consider the phase portraits for the different dynamical regimes. The phase portraits were constructed using the delayed vectors with minimum embedding dimension (*d*) and optimum delay (*τ*) obtained from Cao's algorithm [22] and average mutual information, respectively. The two parameters from Cao's algorithm E1 and E2, are evaluated for a range of embedding dimensions from 1 to 20. E1 measures the ratio of mean distances between two points in the phase space in two successive embedding dimensions and when a sufficient embedding dimension is attained, E1 attains a value close to 1 and remains constant for further increments in embedding dimension. The quantity E2 helps distinguish between a stochastic signal from a deterministic signal. The value of E2 remains unity irrespective of embedding dimension for a stochastic signal while for a deterministic signal, E2 varies for lower embedding dimensions and saturates beyond a certain embedding dimension. As an example, Fig. 6 shows the variation of E1 and E2 for the stable case. It can be observed from the variation in E2, that the acoustic pressure time series for the stable case is not completely stochastic. The first minimum of the average mutual information is chosen as the optimum value of the time delay [23].

With the minimum embedding dimension and optimum delay, we construct the phase portraits for the stable operation with ($\phi tot,\u2009\phi p$) = (0.83, 0.56) and for the intermittent case with ($\phi tot,\u2009\phi p$) = (1.43, 0.36). Figure 7 shows the state space representation for the respective cases.

From the phase portraits, we observe that the state space representation of the stable case is a clutter of points. The phase portrait has no distinct repeating patterns and corresponds to low-amplitude aperiodic oscillations. The size of the stable attractor looks small in comparison to the attractor of the intermittent case. The phase portrait of the intermittent case is seen to follow the low amplitude aperiodic fluctuation trajectory during the aperiodic epochs in the pressure time series and a limit cycle during the periodic epoch characteristic of thermoacoustic instability [24]. The limit cycle portion of the phase portrait of the intermittent case is not entirely elliptical and is distorted due to the presence of higher harmonics in the pressure time series as seen in Fig. 5. The limit cycle does not show repeatability with each cycle due to the presence of turbulent noise.

With the constructed phase portraits, we now construct the RPs as discussed in Sec. 2.4 to capture the time evolution of the system dynamics. The threshold ε for constructing the RPs was set to $\lambda /5$, where *λ* is the size of the attractor of the intermittent case with ($\phi tot,\u2009\phi p$) = (1.43, 0.36). The threshold was fixed to make comparisons between the different cases. The RPs for the stable case with ($\phi tot,\u2009\phi p$) = (0.83, 0.56) and intermittent case with ($\phi tot,\u2009\phi p$) = (1.43, 0.36) are shown for 40 instability cycles in Fig. 8.

The recurrence plot for the stable combustion (left in Fig. 8) case is seen to be mostly grainy. This is to be expected since the dynamics of the combustor during stable combustion are predominantly aperiodic with little repeatability in the patterns. On the other hand, the intermittent case shows black patches amidst white patches. The white patches represent higher amplitude periodic bursts as observed in the left figure in the right column of Fig. 9. Broken diagonal lines are also observed in the intermittent case implying deterministic behavior. The black patches represent the times when the system exhibits low amplitude aperiodic oscillations as seen in the right figure in the right column in Fig. 9. This corresponds to the trajectory trapped within a small region in phase space. This is a pattern typical of intermittent burst oscillations and thus, RPs can help visualize the route to instability in turbulent combustors.

### Recurrence Quantification Analysis.

From the RPs for the different cases, we calculate the RR and T as defined in Eqs. (5) and (6) for 40 instability cycles. The results of these statistical measures as a function of control parameter are plotted in Fig. 10.

The recurrence rate denotes the density of black points (instances when the distances between states in the phase portraits are below the fixed threshold), which measures the recurrence rate in the dynamics of the system. The density of the black points is seen to monotonically decrease as the system transitions from combustion noise to intermittency. Between the two branches plotted in Fig. 10, we observe that the branch with a lower percentage of primary fuel, i.e., $\phi p$ = 0.36, is seen to have consistently lower RR values as compared to the branch with $\phi p$ = 0.56 for a given value of $\phi tot$, suggesting that the occurrence of black points is lower for the branch with $\phi p$ = 0.36 as compared to $\phi p$ = 0.56 for a given value of $\phi tot$. In other words, the acoustic pressure signal, corresponding to the branch with lower percentage of primary fuel, tends to have more instances where the distances between states are beyond the threshold.

The trapping time measures the duration of a particular state to be at the same position in the state space representation. During combustion noise, we observe that the value of T is maximum and this value progressively decreases as $\phi p$ is lowered and/or when $\phi tot$ is increased as seen in Fig. 10. This is due to the fact that the acoustic signal contains more aperiodic epochs at lower $\phi tot$ and larger percentage of primary fuel, i.e., larger $\phi p$. As the system transitions to intermittency, we observe that the value of *T* decreases due to the increase in periodic epochs in the acoustic pressure signal.

### Variations in Degree Distribution With Changes in Dynamics.

Upon constructing the RPs, we construct the RNs for the different cases as described in Sec. 2.5. Degree distribution is the graph of $P(k)$ plotted as a function of $k$, where $P(k)$ is the probability of a node to have the degree $k$, i.e., $P(k)$ is given by $n(k)/N$, where $n(k)$ is the total number of nodes having the degree $k$ and degree $N$ is the total number of nodes. Plotted in Fig. 11 is the degree distribution for the stable and intermittent cases.

From the degree distribution of the stable case with ($\phi tot,\u2009\phi p$) = (0.83, 0.56), we observe that the peak lies closer to lower degree values and hence, lower degree values have a higher chance of occurring that the higher degree values, i.e., the phase space density at those locations is low over the attractor. On the other hand, we observe a totally different trend for the intermittent case when ($\phi tot,\u2009\phi p$) = (1.42, 0.36). We notice that the peak lies closer to degrees with higher values, suggesting that the phase space density at those locations is high over the attractor. Such a peak in the degree distribution concentrated at high degree corresponds to periodic part in the acoustic signal.

## Conclusions

The novelty of this study is to demonstrate methodologies to quantify combustion instability and intermittency in low amplitude, noisy data sets. This is useful in high Reynolds number situations where combustion instability is approached from the stable side of a bifurcation, and when the bifurcation must be identified while amplitudes remain low. The present study demonstrates this capability during combustion noise and intermittency in a single sector RQL style liquid fueled swirl combustor with two fuel circuits. The equivalence ratio and distribution of fuel between the two circuits are varied as control parameters. We observe that the combustor transitions from combustion noise to intermittency when the head-end equivalence ratio is increased and/or when the percentage of primary fuel in the fuel flow is decreased. We use spectra of the combustor acoustic measurements, phase portraits, and recurrence networks to differentiate the dynamics of the combustor during these two dynamical regimes. During the transition from combustion noise to intermittency, we observe that the topology of the attractor changes from a complex topology to a topology with some presence of limit cycle behavior. Recurrence plots and Recurrence Quantification Analysis were used to predict the onset of stability in the combustor. We also show that the degree distribution from the constructed recurrence networks shows contrasting behaviors between the two dynamical regimes as the topology of the attractor changes between the two dynamical regimes.

## Funding Data

National Aeronautics and Space Administration (Grant No. 80NSSC21M0068; Funder ID: 10.13039/100000104).

ULI University PIs: Dr. Karen Thole and Dt Timothy Charles Lieuwen.