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# Data-driven model-based instability prediction in an annular combustor relying on a flame response mapping of the operating domain

Véranika Latour, Daniel Durox, Antoine Renaud, Sébastien Candel

► To cite this version:

Véranika Latour, Daniel Durox, Antoine Renaud, Sébastien Candel. Data-driven model-based instability prediction in an annular combustor relying on a flame response mapping of the operating domain. Combustion and Flame, 2024, 270, pp.113782. 10.1016/j.combustflame.2024.113782. hal-05233882

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https://centralesupelec.hal.science/hal-05233882v1

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# Data-driven model-based instability prediction in an annular combustor relying on a flame response mapping of the operating domain

Véranika Latour<fnref n="1" />ᵃ, Daniel Duroxª, Antoine Renaudª, Sébastien Candelª

ªLaboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3 rue Joliot Curie, 91190 Gif-sur-Yvette, France

## Abstract

At a stage where new architectures and alternative fuels are being proposed to tackle the environmental challenges, it is important to be able to deal with combustion dynamics issues that may arise in these new developments. Reduced order models are generally considered for that purpose but their capacity to predict combustion instabilities is still not fully demonstrated. One advantage of these models is that they mainly rely on flame transfer or describing functions (FTFs or FDFs) representing the flames' response to incoming disturbances. Recent measurements indicate that FDFs exhibit gain and phase variations with fuels, fuel blends, injector characteristics, but also with operating conditions. However, FTFs and FDFs are generally documented only for a few operating points and do not cover the entire domain of operation, limiting the scope of the analysis. The logical step taken in the present investigation is to collect FDFs for a large number of flow conditions of the laboratory-scale annular combustor MICCA-Spray. This is achieved using a single-injector system, SICCA-Spray, representing one sector of MICCA-Spray and that allows external flame modulation. The collected FDF data correspond to injectors of two types, characterized by different combustion dynamics in MICCA-Spray. This FDF database, in combination with an analytical framework derived from acoustic energy balance equations, serves to determine growth rates and define a theoretical instability domain. A comparison with the stability maps obtained in the annular combustor indicates that the general layout of these maps can be retrieved for the two injector types, validating the relevance of this data-driven model-based analysis of thermo-acoustic instabilities.

**Keywords:** Flame describing functions, Annular combustors, Combustion instabilities, Reduced-order modeling

## Novelty and significance statement

The novelty of this work lies in the reported flame describing function (FDF) database, measured in the single-injector setup SICCA-Spray, for a wide range of operating conditions corresponding to the operation domain of the MICCA-Spray annular combustor, and for two types of injectors leading to different flame dynamics (stable and unstable). An analytical framework is then used to determine growth rates of oscillation based on the FDF data, enabling to perform a stability analysis and interpret the observations in MICCA-Spray: the differences in flame dynamics observed between the two injectors are successfully retrieved, and for the unstable injector, stable and unstable regions of the operating domain can also be distinguished.

This work is significant because it provides an analytical framework of interest from a theoretical standpoint and for practical applications that is validated against a broad experimental dataset.

## Authors' contributions

V.L.: Performed experiments, processed data, developed theoretical framework, wrote parts of the paper

D.D.: Designed the SICCA-Spray and MICCA-Spray facilities, performed experiments, wrote parts of the paper

A.R.: Participated in data processing and interpretation, wrote parts of the paper

S.C.: Designed research effort, developed theoretical framework, interpreted data, wrote parts of the paper

<footnote n="1">¹Corresponding author: Véranika Latour. Email address: veranika.latour@centralesupelec.fr</footnote>
Nomenclature

Roman symbols

<table><tr><td>A<sub>+</sub>, A<sub>-</sub></td><td>Amplitudes of the counterclockwise and clockwise spinning waves</td></tr><tr><td>A</td><td>Instability amplitude</td></tr><tr><td>c</td><td>Speed of sound</td></tr><tr><td>c<sub>p</sub></td><td>Specific heat at constant pressure</td></tr><tr><td>c<sub>v</sub></td><td>Specific heat at constant volume</td></tr><tr><td>&lang;D&rang;</td><td>Volume integrated period average acoustic losses</td></tr><tr><td>&lang;E&rang;</td><td>Volume integrated period average acoustic energy</td></tr><tr><td>&lang;F&rang;</td><td>Volume integrated period average acoustic fluxes</td></tr><tr><td>f</td><td>Frequency</td></tr><tr><td>F</td><td>Flame describing function</td></tr><tr><td>G<sub>F</sub>, &phi;<sub>F</sub></td><td>Flame describing function gain and phase</td></tr><tr><td>G<sub>&zeta;</sub>, &phi;<sub>&zeta;</sub></td><td>Impedance modulus and phase</td></tr><tr><td>M</td><td>Mach number</td></tr><tr><td>N</td><td>Number of injectors</td></tr><tr><td>p<sub>0</sub>, p'</td><td>Reference pressure and pressure fluctuations</td></tr><tr><td>&Delta;p</td><td>Pressure drop</td></tr><tr><td>P</td><td>Thermal power</td></tr><tr><td>q&#773;<sub>v</sub>, q'<sub>v</sub></td><td>Mean volume flow rate and volume flow rate fluctuations</td></tr><tr><td>Q&#775;<sub>0</sub>, Q'<sub>0</sub></td><td>Mean heat release rate and heat release rate fluctuations</td></tr><tr><td>r</td><td>Specific gas constant</td></tr><tr><td>&lang;S&rang;</td><td>Volume integrated period average source term</td></tr><tr><td>S<sub>N</sub></td><td>Swirl number</td></tr><tr><td>s'</td><td>Entropy fluctuations</td></tr><tr><td>T</td><td>Period of oscillation</td></tr><tr><td>T<sub>0</sub>, T'</td><td>Reference temperature and temperature fluctuations</td></tr><tr><td>V</td><td>Combustion chamber volume</td></tr><tr><td>v&#773;, v'</td><td>Mean velocity and velocity fluctuations</td></tr><tr><td>x<sub>f</sub></td><td>Flame barycenter position</td></tr></table>

Greek symbols

<table><tr><td>&alpha;</td><td>Damping rate</td></tr><tr><td>&gamma;</td><td>Specific heat ratio</td></tr><tr><td>&phi;</td><td>Equivalence ratio</td></tr><tr><td>&omega;<sup>R</sup><sub>i</sub></td><td>Growth rate corresponding to the Rayleigh source term</td></tr><tr><td>&omega;<sub>i</sub>'</td><td>Effective growth rate: &omega;<sub>i</sub>' = &omega;<sub>i</sub><sup>R</sup> &minus; &alpha;</td></tr><tr><td>&omega;</td><td>Angular frequency, &omega; = 2&pi;f</td></tr><tr><td>&rho;</td><td>Density</td></tr><tr><td>&sigma;</td><td>Pressure drop coefficient</td></tr><tr><td>&theta;</td><td>Azimuthal position</td></tr><tr><td>&theta;<sub>0</sub></td><td>Antinodal line azimuthal position</td></tr><tr><td>&tau;<sub>u'<sub>-</sub>&#775;&#775;&#775;v'</sub>, &phi;<sub>u'<sub>-</sub>&#775;&#775;&#775;v'</sub></td><td>Time delay and phase between heat release rate and velocity fluctuations</td></tr><tr><td>&psi;(x)</td><td>Axial wavefunction</td></tr><tr><td>&psi;<sub>f</sub></td><td>Axial wavefunction evaluated at the flame barycenter position: &psi;<sub>f</sub> = &psi;(x<sub>f</sub>)</td></tr></table>
# 1. Introduction

Thermo-acoustic instabilities, induced by a resonant coupling between combustion and the acoustics of the system, give rise to operational issues of combustion devices. They often cause severe damage to the system and may lead in extreme cases to failure [1, 2]. The geometry of the system and more specifically that of the chamber plays a crucial role as it determines the nature and frequency of the acoustic modes involved in the combustion/acoustics coupling. In aero-engines or ground-based gas turbines, the commonly used annular geometries give rise to modes featuring an azimuthal structure. These modes are generally considered the most dangerous as they are less well-damped and because the associated eigenfrequencies often fall in the range where the flames are most susceptible to disturbances.

Predicting combustion instabilities induced by these modes at an early stage of the design process raises many scientific issues because of the variety of physical phenomena involved, including swirling flows, turbulence, fuel injection as a spray of droplets, exothermic reactions and acoustics, to enumerate a few, that need to be represented in a reduced-order modeling framework. A common approach that circumvents the associated difficulties consists in using flame transfer functions (FTFs) or their nonlinear extension, flame describing functions (FDFs) [3, 4], to model the response of flames to acoustic disturbances. These FTFs (or FDFs) in combination with acoustic representations of combustion systems provide reduced-order models that constitute useful tools in thermo-acoustic analysis and are the subject of ongoing research.

The present article aims at advancing such reduced-order descriptions of combustion instabilities in two directions. The first consists in building a large database of FDFs, mapping the operating domain of an annular system for two types of injection units leading to different flame dynamics in a laboratory-scale annular combustor (MICCA-Spray). The second direction is to use this database to determine the growth rates of perturbations using analytical expressions derived from the acoustic energy balance equations in combination with the FDF framework. In turn, the growth rates obtained in this way can be used to determine a domain of instability that may be compared to the one observed experimentally.

At this stage, it is worth reviewing the state of the art in the field of combustion dynamics and more specifically consider research efforts dealing with instabilities coupled by azimuthal modes in annular systems. Such instabilities are often found in practice but are difficult to investigate on full-scale systems. However, recent experiments in laboratory-scale annular geometries featuring self-sustained oscillations and in transversely forced linear combustors have provided valuable insights essentially through pressure and heat release rate recordings [5–10]. On the numerical level, instabilities coupled by azimuthal modes are not easy to calculate either because their large eddy simulations require a full representation of the annular geometry that implies a large grid, typically of the order of a few hundred million nodes [11], and millions of timesteps to cover a sufficient number of cycles of oscillation. In this general context, theoretical analysis remains particularly valuable in the interpretation of data gathered from experiments and simulations. Much of the recent efforts on that topic have been aimed at building more reliable tools and improving reduced-order models. Most of these models use acoustic transfer matrix networks of the combustion system [12–19] and, as already indicated, represent the flame response to disturbances in terms of FTFs or FDFs. Another approach consisting in using FTFs or FDFs in conjunction with Helmholtz solvers [20–23] is also possible. Mitigation and control strategies, such as azimuthal staging, can also be investigated using analytical descriptions of annular combustors in which the differences in flame dynamics between the burners are represented by different FDFs [24–26]. A broad set of staging experiments together with analytical interpretation is also provided in [27].

FTFs or FDFs are often determined experimentally in single-sector configurations equipped with driver units to modulate the flow. Effects of the presence of side flames on FDF measurements have been investigated using a linear array of three injectors [28], and differences were observed with measurements in a single-injector configuration when the flames are close to each other and give rise to strong interactions. Azimuthal forcing of an annular combustor was also performed [29] to examine the FDF dependence on the amplitude, nature, and orientation angles of the incoming disturbances. In their test conditions, the authors observed differences in the FDFs depending on the acoustic field direction of rotation (clockwise or anti-clockwise), and an “azimuthal FDF” was proposed to describe and interpret these observations. Theoretical investigations involving FDFs can be found in the literature [30] as well as attempts to extract these functions from numerical calculations [16, 31].

Recent experiments in single-injector setups indicate that FTFs and FDFs depend on the fuel type [32], fuel composition that is injected as a spray [33], reactant mixture composition for blends of CH₄/H₂ [34] and NH₃/H₂/N₂ [35], hydrogen injection method for CH₄/H₂ mixtures (premixed or pilot tube) [36], and atomizer recess (i.e. position of

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the atomizer with respect to the outlet of the injector) for swirled spray injection units [37]. In comparison, the FDF
dependence with respect to operating conditions needs to be documented in a more extensive fashion. Measurements
at three different operating points can be found [32], and an instability index is derived from FDF data to interpret the
observations in the annular system MICCA-Spray. Nevertheless, to the authors' knowledge, the systematic determina-
tion of FTFs or FDFs for a wide range of thermal powers and equivalence ratios has not been attempted yet. The FDF
mapping of the operating domain of an annular system, reported in the present article, is aimed at providing such a
database. These data are then exploited in combination with a stability analysis framework derived in a previous study
[27] to interpret stability maps of the annular system operating with liquid heptane atomized in swirling injectors and
delivered to the combustor as a spray. Due to experimental limitations, FDF data do not cover the range of oscillation
amplitudes observed at limit cycle (expected to lie, in terms of relative velocity fluctuations, between 20% and 30%).
However, the different modulation amplitudes tested reveal trends in the flames' responses with increasing oscillation
level, enabling to gain insights into the onset of instabilities and the trajectories towards limit cycle.

This work comprises four main sections. Section 2 presents the MICCA-Spray annular combustor, the SICCA-Spray experimental setup used for FDF measurements, and the flow characteristics of the two types of injection units investigated in this study. Section 3 provides the stability maps of the annular combustor MICCA-Spray. FDF data are reported in section 4 for the two injection units investigated, and the FDFs dependence on equivalence ratio, thermal power, and modulation amplitude is discussed. A stability analysis framework is introduced in section 5 and exploited in combination with the FDF database to obtain “growth rate maps” of the MICCA-Spray combustor calculated for the two types of injectors. The maps obtained in this way are then compared to the stability maps of MICCA-Spray, to see if the stable and unstable regions of the annular combustor are correctly retrieved.

## 2. Experimental setups

Two experimental setups are used in this study: the annular combustor MICCA-Spray (described in subsection 2.1), and the single-sector setup, SICCA-Spray, used for FDF measurements (subsection 2.2). These two configurations are equipped with modular injection units. Injectors equipped with two different swirlers, namely “707” and “727”, leading to different flame dynamics in MICCA-Spray, are investigated in the present study and characterized in the last subsection (2.3).

<figure><img src="image_4.png" /><figcaption>Figure 1: The MICCA-Spray annular combustion chamber (left) and the SICCA-Spray setup for FDF measurements (right).</figcaption></figure>

### 2.1. *The MICCA-Spray combustor*

In the MICCA-Spray annular combustor, shown in Fig. 1 (left), the chamber, formed by two cylindrical quartz walls of height $l = 400$ mm, is equipped with sixteen swirling spray injectors. The inner diameter of the outer quartz tube is 400 mm and the outer diameter of the inner tube is 300 mm. The liquid heptane fuel mass flow rate is controlled through a Bronkhorst CORI-flow controller with a relative accuracy of 0.2% and the air mass flow rate through two Bronkhorst EL-FLOW mass flow controllers with a relative accuracy of 0.5% (full scale). Eight Brüel & Kjær microphones are plugged on waveguides connected to the chamber backplane in order to protect these sensors
from the chamber's hot temperature environment. The waveguide ports are located on the mean circumference, at equal distance from two adjacent injectors. The microphones provide the instability amplitude and frequency using a wave decomposition procedure described in section 3.

## 2.2. The SICCA-Spray configuration

The SICCA-Spray setup displayed in Fig. 1 (right) represents a sector of the annular chamber MICCA-Spray. It is equipped with an injector identical to those used in MICCA-Spray, and the fuel and air mass flow rates are also controlled through a Bronkhorst CORI-flow and two Bronkhorst EL-FLOW mass flow meters. Two driver units plugged at the bottom of the experimental setup serve to modulate the flow and induce fluctuations in the air volume flow rate. A cylindrical quartz tube, of diameter equal to the spacing between two adjacent injectors in MICCA-Spray ($d_c = 69$ mm), delimits the combustion chamber, and the backplane surface area in SICCA-Spray ($A_{SICCA} = 37.4$ cm²) is close to that of a sector in MICCA-Spray ($A_{MICCA} = 35.7$ cm²). The system operates in a stable fashion or exhibits self-sustained instabilities, depending on the chamber's length. A small chamber length (100 mm) is here used to avoid self-sustained oscillations, hence allowing measurements under external modulation by the driver units.

The determination of the flame response to incident disturbances in the form of relative heat release rate fluctuations $\dot{Q}'/\overline{Q}$ requires in principle a measurement of relative fluctuations in volume flow rate $q'_v/\overline{q}_v$ and in equivalence ratio $\phi'/\overline{\phi}$. It is shown however in [38] that relative equivalence ratio fluctuations $\phi'/\overline{\phi}$ are in the present experiments small compared to relative volume flow rate fluctuations $q'_v/\overline{q}_v$ so that the flame behaves in a quasi-premixed fashion. It is then admissible to deduce relative heat release rate fluctuations from relative light intensity fluctuations originating from OH* radicals, using $\dot{Q}'/\overline{Q} \simeq I'_{OH*}/\overline{I}_{OH*}$. A photomultiplier, equipped with a narrow band filter centered at 310 nm, is used to record the OH* light intensity from the flame.

Relative volume flow rate fluctuations needed to estimate the FDFs cannot be measured in the plenum because the injection unit is weakly transparent to acoustic waves [39]. Hence the relative flow rate fluctuations seen by the flame do not correspond to those measured on the upstream side of this unit. Direct measurements of flow rate fluctuations are not easy in situations where the flow at the injector outlet is highly non-uniform. It is shown however in [38] that flow rate fluctuations at the outlet of the injector may be obtained by measuring relative velocity fluctuations at a reference point in the flow where these two quantities closely match so that $q'_v/\overline{q}_v \simeq v'/\overline{v}$. The velocity fluctuations are measured using a Dantec Dynamics laser Doppler anemometry (LDA) system at a reference point, shown to depend on the swirler characteristics of the injector unit [38]. For the two swirling injectors investigated in this study (which will be presented in detail in sec 2.3), the measurement point is located at a distance $r = 4.0$ mm from the center of the burner for an injector equipped with swirler 727, and at $r = 3.5$ mm for an injector equipped with swirler 707, and a height $z = 2.5$ mm above the backplane of the chamber [38]. Phase Doppler anemometry indicates that the fuel spray comprises a population of droplets with a mean diameter of less than 4 µm at the measurement point ($z = 2.5$ mm and $r = 3.5$ mm or $r = 4.0$ mm) [38]. These droplets are hence small enough to be used as flow tracers in the frequency range of interest. Heat release rate and volume flow rate fluctuations may then be used to determine the FDF defined by the following expression

$$ \mathcal{F}(f, q'_v) = \frac{\dot{Q}'/\overline{Q}}{q'_v/\overline{q}_v} = G(f, q'_v) e^{i\varphi(f, q'_v)} \quad (1) $$

## 2.3. Injector units characteristics

The exploded view of the injectors presented in Fig. 2 (left) shows four main elements: an air distributor, a liquid fuel atomizer, a swirler, and a terminal plate with a convergent conical section. Changing the swirler (swirler's channel orientation and radius) changes the pressure drop and the swirl number of the injection unit. Two injectors equipped with two different swirlers, named "707" and "727", are investigated in the present study. In what follows, when an injector unit is equipped with swirler 707 (727 respectively), it will be designated as "injector 707" ("injector 727" respectively).

The pressure drop, pressure drop coefficient, and conventional swirl number, measured under cold flow conditions for an air mass flow rate of 2.6 g/s [40], are gathered in Tab. 1. Injector 727 features a higher swirl number and a higher pressure drop value. Injector 707 leads to stable operation in MICCA-Spray [41] and injector 727 gives rise to self-sustained oscillations in the system for a wide range of operating conditions. Abel-inversed OH* chemiluminescence
images of the flames formed by injectors 707 and 727 under stable operation in the SICCA-Spray burner (set at an operating point of thermal power $P= 7.4$ kW and equivalence ratio $\phi = 0.9$) are presented in Fig. 2, right. The flame formed by injector 707 is longer than that established by injector 727. The latter is more compact with a hollow M-shape due to the higher swirl number of this unit.

<table><caption>Table 1: Injectors conventional swirl number ($S_N$), pressure drop ($\Delta p$), and pressure drop coefficient ($\sigma$), measured under cold flow conditions for an air mass flow rate of 2.6 g/s [40]</caption><thead><tr><th></th><th>707</th><th>727</th></tr></thead><tbody><tr><td>Swirl number ($S_N$)</td><td>0.64</td><td>0.74</td></tr><tr><td>Pressure drop ($\Delta p$, kPa)</td><td>3.65</td><td>5.70</td></tr><tr><td>Pressure drop coefficient ($\sigma$)</td><td>3.33</td><td>5.20</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>Figure 2: Left: Exploded view of the injector and top view of the swirler. Right: OH* light intensity images, obtained after an Abel inversion of the accumulated images on a CCD camera, of the flames formed by injectors 707 and 727 under stable operation in the SICCA-Spray burner (driver units deactivated), operating at $P= 7.4$ kW and $\phi = 0.9$.</figcaption></figure>

### 3. Annular combustor stability maps

The stability maps of the annular combustor MICCA-Spray described in this section will be interpreted using the FDF data collected in the single-sector experiment, SICCA-Spray (section 4), and the analysis will be pursued in section 5.

The annular combustor MICCA-Spray is here operating with liquid heptane. When MICCA-Spray is equipped with injectors 707, combustion takes place in a stable fashion [41] and the system is stable over the whole range of operating conditions. In contrast, large amplitude pressure oscillations are observed for a wide range of operating conditions when the annular chamber is equipped with injectors of the 727 type [33, 41].

Stability maps of MICCA-Spray equipped with injector 727, displayed in Fig. 3, are obtained from observations at thirty operating points of thermal power $P= 92, 99, 103, 110$ and $118$ kW, and global equivalence ratio $\phi = 0.75, 0.85, 0.9, 0.95, 1.0$ and $1.05$. The instability amplitude and frequency are determined from pressure recordings at each operating point.

The pressure field in the annular chamber can be cast in the following form:

$$p'(\theta, t) = A^{+}e^{(i\theta - i\omega t)} + A^{-}e^{(-i\theta - i\omega t)} \quad (2)$$

with $\theta$, the azimuthal angle considered positive in the clockwise direction, $\omega$, the angular frequency, and $A^+$ and $A^-$, the amplitudes of the counterclockwise and clockwise spinning waves respectively. $A^+$ and $A^-$ are determined
from the pressure signals recorded by the eight microphones placed around the annulus, by solving an overdetermined system of equations using the least square method.

The instability amplitude is characterized by the parameter A, defined as

$$A = [|A^+|^2 + |A^-|^2]^{1/2} \quad (3)$$

One can easily show that, for a standing mode, A represents the root mean square (rms) value of the pressure field at the pressure antinode.

The instability characteristics are deduced from 8-second-long microphone recordings at a sampling rate of 32 768 Hz for the thirty operating conditions. The instability frequency, f, is determined from the power spectral density (PSD) of the microphone signals, calculated using Welch's method applied to 63 blocks of 8192 samples with a 50% overlap and a Hamming window weighting. These parameters yield a frequency resolution Δf = 4 Hz.

The instability amplitudes and frequencies thus determined for the thirty operating conditions are reported in the (P,φ) plane. The “stability maps”, presented in Fig. 3, result from a 2D linear interpolation between the thirty operating points investigated, in order to better visualize the shapes of the stable and unstable regions. The amplitude map (Fig. 3, top) defines the annular system’s stable and unstable regions and indicates the level of oscillation. In the region of instability bounded by the black contour, the following two criteria are satisfied:

* The root mean square (rms) pressure is a few times (in the present case three times) higher than a baseline level corresponding to the rms pressure level of noise in the presence of combustion, measured for a stable operating point. In the baseline configuration, this level is typically 80 Pa.

* The power spectral density exhibits a peak, SPL<sub>Δf</sub><sup>m</sup>, that exceeds the maximum level measured in the baseline configuration, SPL<sub>Δf</sub><sup>bl,m</sup>, by a large amount: SPL<sub>Δf</sub><sup>m</sup> ≥ SPL<sub>Δf</sub><sup>bl,m</sup> + ΔS (here ΔS = 25 dB). This condition ensures that the signal is dominated by a well-defined frequency tone.

<figure><img src="image_6.png" /><figcaption>Figure 3: Amplitude (left) and frequency (right) stability maps obtained in the annular combustor MICCA-Spray equipped with injector 727 [33]. Hashed lines intersections correspond to the 30 experimental operating points, and the black contours delimit stable and unstable regions. The annular combustor is stable when it is equipped with injector 707.</figcaption></figure>

The frequency map displayed in Fig. 3 shows how the frequency of oscillation evolves as a function of thermal power and equivalence ratio. The frequency is close to the estimated value of the 1A1L mode (f<sub>11</sub> = 808 Hz), identified as the mode involved in the combustion/acoustics coupling, as explained in [33]. The oscillation frequency is seen to depend on the operating point. This is due in part to variations in temperature that are mainly induced by changes in equivalence ratio producing changes in the speed of sound. The frequency variations with respect to the 1A1L value are also linked to the fact that flame dynamics intervene in the resonant coupling so that the frequency shift with respect to the acoustic 1A1L eigenfrequency can take positive or negative values [42].
4. FDF measurements in SICCA-Spray

The single-sector setup (SICCA-Spray), described in section 2.2, is now used to explore the domain of operation of the annular combustor MICCA-Spray to determine the FDFs corresponding to thirty operating points of thermal powers P= 5.8, 6.2, 6.4, 6.9 and 7.4 kW, and equivalence ratios φ = 0.75, 0.85, 0.9, 0.95, 1.0 and 1.05. These operating points match those used to obtain the stability maps of the annular combustor MICCA-Spray plotted in section 3. The thermal power values chosen for FDF measurements are equal to those corresponding to a single injector in MICCA-Spray, that is to say, 1/16th of the total thermal power in that rig.

The FDFs are determined for three levels of modulation amplitude of the driver units. The corresponding relative velocity fluctuations as a function of frequency, measured in SICCA-Spray for an operating point of thermal power P= 6.9 kW and equivalence ratio φ = 0.85, are presented in Fig. 4, for injectors 707 (Fig. 4, left) and 727 (Fig. 4, right). Two peaks corresponding to the resonance of the system are observed around 500 Hz and 750 Hz and higher relative velocity fluctuation levels are measured for frequencies below 650 Hz for injector 707 than 727. In the frequency range of interest for the interpretation of the stability maps of MICCA-Spray (around 800 Hz, corresponding to the 1A1L mode of the MICCA-Spray chamber), one can see that the values of relative velocity fluctuations reached lie between 5 % and 10% for both injectors 707 and 727.

<figure><img src="image_7.png" /><figcaption>Figure 4: Relative velocity fluctuations, $u'/\bar{u}$, as a function of frequency for three modulation amplitude levels ('low', 'intermediate', and 'high') measured in SICCA-Spray (thermal power P= 6.9 kW and equivalence ratio φ = 0.85) for the two injection units: 707 (left) and 727 (right). Measurements are carried out at a point where $q_v'/\bar{q}_v \simeq v'/\bar{v}$, located z = 2.5 mm above the backplane and r = 3.5 mm from the center of the injection unit for injector 707 and r = 4.0 mm for injector 727 [38].</figcaption></figure>

One can first explore the collected FDF database by plotting the FDF gain and phase as a function of frequency for a fixed equivalence ratio value. FDF measurements, obtained for injectors 707 and 727 for an intermediate modulation amplitude, are displayed in Fig. 5, for two equivalence ratios ($\phi$ = 0.85 and $\phi$ = 0.95). For frequencies around 800 Hz, injector 707 is characterized by lower FDF gains and higher FDF phase values than injector 727. One can also observe that, for both injectors 707 and 727, at a given equivalence ratio $\phi$, the FDF phase takes higher values when the thermal power decreases. This trend can be explained by considering the time delay, $\tau_{u'-\dot{Q}'}$, between velocity oscillations and heat release rate fluctuations. Introduced by Crocco [43] in the early days of combustion instabilities investigations in rocket engines, delay effects have been the subject of numerous models and investigations [32, 44-47]. For spray flames, the time delay results from characteristic times associated to various processes such as liquid fuel atomization, vaporization, mixing, convection, and chemical processes involved in combustion [32]. For the case at hand, a reduction in thermal power corresponds to a reduction in fuel flow rate. If the equivalence ratio value remains unchanged, this reduction in fuel flow rate has to be accompanied by a reduction in the air flow rate and consequently of the bulk velocity $U_b$ at the injector exit. The flames investigated here are formed by high-swirl number injectors. At a fixed equivalence ratio and for a given injector, their lengths only weakly depend on the thermal power P, as confirmed by a systematic study of flame structures in [48]. The characteristic delay $\tau_{u'-\dot{Q}'}$ then evolves like the inverse of the bulk velocity $U_b$, and one may write $\Delta\tau_{u'-\dot{Q}'}/\tau_{u'-\dot{Q}'} \simeq -\Delta P/P$. One then expects to find that a reduction in thermal power induces an increase in the phase curve slope. To verify this hypothesis, one can extract

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<figure><img src="image_8.png" /><figcaption>Figure 5: FDF gain and phase curves for injectors 707 (left) and 727 (right) obtained at the intermediate modulation amplitude, for $\phi = 0.85$ and $\phi = 0.95$. For each equivalence ratio value, FDFs are plotted for the five thermal powers tested ($P = 5.8, 6.2, 6.4, 6.9$ and $7.4$ kW).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 6: Time delay $\tau_{u'-\dot{Q}'} = d\varphi_{u'-\dot{Q}'}/d\omega$ as a function of the thermal power $P$ for injectors 707 (left) and 727 (right), grouped by equivalence ratio values ($\phi = 0.75, 0.85, 0.9, 0.95, 1.0$, and $1.05$), and determined from the FDF data collected for an intermediate modulation amplitude level.</figcaption></figure>

the time delay from the FDF phase curves, by taking the derivative of the phase with respect to the angular frequency: $\tau_{u'-\dot{Q}'} = d\varphi_{u'-\dot{Q}'}/d\omega$. Calculations are only carried out when the phase curves are nearly linear in the frequency range between 300 and 800 Hz. One may then determine the delay from the slope of a linear fit of the phase curve. The time delays, grouped by equivalence ratio values, are reported as a function of the thermal power $P$ in Fig. 6 for injectors 707 and 727. One can indeed observe a decrease in the time delay $\tau_{u'-\dot{Q}'}$ as the thermal power increases. The change in time delay with the equivalence ratio at a fixed thermal power value is more difficult to interpret. If flame images reported in [48] show close flame structures for a fixed equivalence ratio but different thermal powers, more pronounced differences are found between flames of same thermal power but different equivalence ratio values.

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The effect of the modulation amplitude level on the relative position of FDF phase curves for a fixed equivalence ratio are investigated in Appendix A, to verify that the trends observed in Fig. 5 are essentially due to a change in thermal power, and not to differences in relative velocity fluctuations between the different operating points.

<figure><img src="image_10.png" /><figcaption>Figure 7: Injector 727 FDF gain (left) and phase (right) measurements for a thermal power value $P= 6.4$ kW and 6 equivalence ratio values ($\phi = 0.75, 0.85, 0.9, 0.95, 1$, and $1.05$) for the lowest and highest modulation amplitudes.</figcaption></figure>

It is next worth examining how FDFs change with the equivalence ratio, for a fixed thermal power value. FDFs corresponding to six equivalence ratios and the lowest and highest modulation amplitudes are shown in Fig. 7 for injector 727, for a thermal power $P= 6.4$ kW. For the modulation amplitudes tested, the phase curves associated with equivalence ratios $\phi = 0.9$ and $\phi = 0.95$ constitute the lower limit, and the phase curve associated with $\phi = 0.75$, the upper limit. Examining the phase curves associated with equivalence ratios above $\phi = 0.85$ and the range of frequencies between 750 and 830 Hz, one finds that the plots corresponding to increasing modulation levels move towards lower phase values and the gains decrease.

# 5. Stability analysis and interpretation using the FDF database

One now seeks to retrieve the general layout of the stability maps described in section 3 using the FDF database measured in the single-injector configuration, SICCA-Spray. To that end, a stability analysis framework has to be described first, which is the subject of the next subsection (subsection 5.1). This framework is then used together with the FDF data to calculate the growth rates corresponding to the different operating points (subsection 5.2). The values obtained in this way are then compared with the stability maps of MICCA-Spray.

## 5.1. Stability analysis framework

The stability analysis of combustion systems is often carried out by expanding the pressure on the normal modes of the system and examining the dispersion relation derived from the dynamical equation governing the modal amplitudes (see for example [24, 49]). It is generally assumed in these models that the flames are compact with respect to the acoustic mode so that the location of the j-th flame is defined by its azimuthal position, $\theta_j$, and the axial position of its barycenter, $x_f$ [42]. This assumption will also be made in what follows. The approach chosen in this study uses,
as a starting point, a balance equation for acoustic energy, expressed in terms of period averages and integrated over the combustor volume V

$$ \frac{d\langle E \rangle}{dt} = \langle S \rangle - \langle D \rangle - \langle F \rangle \quad (4) $$

where $\langle E \rangle$, $\langle S \rangle$, and $\langle D \rangle$ are the integrals over the volume of the period averages of the acoustic energy density, Rayleigh source term and losses, with

$$ \langle E \rangle = \frac{1}{T} \int_{[T],V} \left[ \frac{1}{2} \frac{p'^2}{\rho_0 c^2} + \frac{1}{2} \rho_0 v'^2 \right] dV dt \quad (5) $$

$$ \langle S \rangle = \frac{(\gamma - 1)}{\rho_0 c^2} \frac{1}{T} \int_{[T],V} p' \dot{q}' dV dt \quad (6) $$

and $\langle F \rangle$ is the surface integral of the normal acoustic energy flux

$$ \langle F \rangle = \int_{\Sigma} \mathbf{F} \cdot \mathbf{n} \, dA \quad (7) $$

with

$$ \mathbf{F} = \frac{1}{T} \int_{T} p'v'dt \quad (8) $$

In the previous expressions, $\gamma$ is the specific heat ratio, $p'$, $v'$ and $\dot{q}'$, the pressure, velocity and local heat release rate fluctuations, $\rho_0$, the ambient density, $c$, the speed of sound, and $T = 2\pi/\omega$ designates the period of the oscillation. If the boundaries are passive, the normal acoustic fluxes are positive or null and the integrated term $\langle F \rangle$ generally contributes to a net reduction in the acoustic energy $\langle E \rangle$. Losses represented by $\langle D \rangle$ are due to a variety of processes including viscous dissipation, friction at the walls, relaxation phenomena when the flow contains particles or mode conversion. This term is always positive ($\langle D \rangle > 0$) and induces a net reduction in the acoustic energy.

As discussed by Lieuwen in [50], the expressions of the source term and energy flux depend on the definition chosen for the energy density. An example is given in [51, 52], where a “fluctuation energy”, $e_{FE}$, taking into account entropy perturbations, $s'$, is considered:

$$ e_{FE} = \left[ \frac{1}{2} \frac{p'^2}{\rho_0 c^2} + \frac{1}{2} \rho_0 v'^2 + \frac{p_0}{rc_p} \frac{s'^2}{2} \right] \quad (9) $$

where $r$, $c_p$ and $p_0$ respectively designate the specific gas constant, the specific heat at constant pressure and the ambient pressure.

When the balance of fluctuation energy derived in [51, 52] is integrated over the volume of the combustor, one obtains an alternative expression for the source term with various contributions, among which, “Chu’s term”, which is sometimes quoted as an alternative to the Rayleigh source term $\langle S \rangle$:

$$ \langle S \rangle_{Chu} = \frac{1}{T} \int_{[T],V} \frac{T'}{T_0} \dot{q}' dV dt \quad (10) $$

This expression exhibits a product of the local temperature fluctuations, $T'/T_0$, by the heat release rate fluctuations which differs from the product between pressure and heat release rate fluctuations appearing in the Rayleigh source term, $\langle S \rangle$. One may then be led to think that this new expression should replace the more standard one. On the practical side, this would be difficult because it is not easy to measure temperature fluctuations in flames, and even less so when these fluctuations take place at a few hundred Hz. In addition, extracting temperature fluctuations associated with the oscillatory motion from the background of fluctuations associated to turbulence poses another challenge. It is however interesting to push the analysis one step further by expressing the temperature fluctuations in terms of entropy and pressure fluctuations

$$ \frac{T'}{T_0} = \frac{1}{\gamma} \frac{s'}{c_v} + \frac{(\gamma - 1) p'}{\gamma p_0} \quad (11) $$

with $c_v$ the specific heat at constant volume. Inserting this expression in Eq. 10, one obtains:
$$ \langle S \rangle_{Chu} = \frac{(\gamma - 1)}{\gamma} \frac{1}{T} \int_{[T],V} \frac{p'}{p_0} \dot{q}' dV dt + \frac{1}{\gamma} \frac{1}{T} \int_{[T],V} \frac{s'}{c_v} \dot{q}' dV dt \quad (12) $$

The first term in this expression is easily identified as the Rayleigh source term $\langle S \rangle$, and the second is an additional term associated to entropy fluctuations. Entropy waves and their effects on thermoacoustic instabilities are discussed in [51–53]. It is however shown in Appendix B that, for the situation at hand, the entropy fluctuations contribution to the source term vanishes identically at zero Mach number and is generally small compared to that of the Rayleigh integral. Terms associated to unsteady vortical motions can also intervene [50, 54], but it is generally admitted that their order of magnitude is lower than that of terms induced by heat release rate fluctuations. Hence the main contribution to the source term may be assumed, with some confidence, to originate from the Rayleigh integral $\langle S \rangle$ defined by Eq. 6.

One may then use the balance of acoustic energy (Eq. 4) obtained by considering the period average acoustic energy density expressed in Eq. 5, in which the Rayleigh integral $\langle S \rangle$ may be positive and augment the acoustic energy $\langle E \rangle$, while $\langle D \rangle + \langle F \rangle$ are essentially positive and contribute to its net reduction. It is worth noting that a recent analysis [55] indicates that, under some rather general conditions, $\langle S \rangle$ is always positive. It is then a simple matter to obtain the rate of change in energy density by dividing Eq. 4 by $\langle E \rangle$. Since pressure fluctuations are directly accessible experimentally from microphone signals, it is preferable to divide this equation by $2\langle E \rangle$ to get the rate of change of this variable:

$$ \frac{1}{2\langle E \rangle} \frac{d\langle E \rangle}{dt} = \frac{\langle S \rangle}{2\langle E \rangle} - \frac{\langle D \rangle + \langle F \rangle}{2\langle E \rangle} \quad (13) $$

One may then write this expression in the form:

$$ \omega'_{i} = \omega_{i}^{R} - \alpha, \quad \text{where } \omega_{i}^{R} = \frac{\langle S \rangle}{2\langle E \rangle}, \quad \text{and } \alpha = \frac{[\langle D \rangle + \langle F \rangle]}{2\langle E \rangle} \quad (14) $$

with $\omega'_i$, the “effective growth rate” of pressure and $\omega_i^R$, the rate of energy input in the system resulting from a source term associated to unsteady heat release rate fluctuations, which might be called “Rayleigh growth rate”. For simplicity, this term will be designated in what follows as “growth rate”, but to comply with a reviewer’s request, the superscript “R” is used as a reminder that the source term considered here is that associated to the Rayleigh integral (Eq. 6). As discussed earlier and underlined by the reviewer, this term may not be the only source of energy input in the system, but as shown by an order of magnitude analysis, it constitutes the dominant source of acoustic energy in the situation considered in this work. In Eq. 14, $\alpha$ designates a global damping rate, representing the acoustic losses in the system resulting from dissipation mechanisms, mode conversion processes and outgoing acoustic energy fluxes. The damping rate $\alpha$ was, for example, estimated in MICCA-Spray from acoustic energy balance considerations in [56], and calculated from simultaneous experimental recordings of heat release rate and pressure fluctuations at limit cycle, confirming another estimate obtained in [27].

The acoustic energy will hence increase in the system if the right hand side of Eq. 13 is positive. According to Eq. 14, the system will be linearly unstable if the growth rate of small perturbations $\epsilon$ exceeds the damping rate: $\omega_i^R(\epsilon) > \alpha$. The effective growth rate will be in that case $\omega'_i(\epsilon) = \omega_i^R(\epsilon) - \alpha$, where $\omega_i^R(\epsilon)$ designates the growth rate of small perturbations. There are also cases where small perturbations are stable, i.e. $\omega_i^R(\epsilon) < \alpha$, but finite amplitude perturbations are unstable and such that $\omega_i^R(A)$ exceeds $\alpha$. In these two cases, respectively corresponding to supercritical and subcritical instability, the system will finally reach a limit cycle (see [4, 57]).

One now seeks to evaluate the period average source term $\langle S \rangle$ and acoustic energy term $\langle E \rangle$ for an annular combustor featuring N injectors. The reader will find the detailed derivation in [27] and only the main steps are recalled here.

Considering a coupling by the 1A1L mode, pressure fluctuations can be cast in the following form:

$$ \hat{p}(x, \theta, t) = [A^{+}e^{(i\theta - i\omega t)} + A^{-}e^{(-i\theta - i\omega t)}]\psi(x) \quad (15) $$

with $\psi(x)$, the axial modal function corresponding to the 1L mode and $\theta$, the angular position in the annulus. In this case, one can express the period average acoustic energy density as:

$$ \langle E \rangle = \frac{1}{4} \frac{1}{\rho_0 c^2} [|A^+|^2 + |A^-|^2] V \quad (16) $$
The period-average source term is then decomposed as the sum of the contribution of the N flames, $\langle S \rangle = \sum_{j=1}^{N} \langle S_j \rangle$, and one obtains for the period-average source term $\langle S_j \rangle$ at flame j:

$$ \langle S_j \rangle = \frac{1}{2} \dot{Q}_0 \frac{(\gamma - 1)}{\gamma} \frac{1}{p_0^2} \frac{1}{M} \frac{G_{Fj}}{G_{\zeta j}} \psi_f^2 \cos(\varphi_{Fj} - \varphi_{\zeta j}) [|A^+|^2 + |A^-|^2 + 2|A^+||A^-| \cos(2(\theta_j - \theta_0))] \quad (17) $$

where M is the Mach number of the flow at the flame and $\psi_f = \psi(x_f)$, the axial wavefunction evaluated at the flame barycenter position $x_f$. $G_{Fj}$ and $\varphi_{Fj}$ correspond to the gain and phase of the flame describing function $\mathcal{F}$ linking heat release rate and velocity fluctuations at the flame

$$ \mathcal{F} = \frac{\dot{Q}/\dot{Q}_0}{\hat{v}/v} = G_{Fj} \exp(i\varphi_{Fj}), \quad (18) $$

$G_{\zeta j}$ and $\varphi_{\zeta j}$ represent the gain and phase of the effective impedance $\zeta$, connecting velocity and pressure fluctuations at the flame

$$ \zeta = \frac{\hat{p}}{\rho_0 c \hat{v}} = G_{\zeta j} \exp(i\varphi_{\zeta j}) \quad (19) $$

Assuming, in addition, that the gains and phases of the FDF and impedance take the same values for all flames, $G_{Fj} = G_F$, $\varphi_{Fj} = \varphi_F$, $G_{\zeta j} = G_\zeta$, and $\varphi_{\zeta j} = \varphi_\zeta$, that the angles $\theta_j$ are regularly distributed, $\theta_j = (j - 1)2\pi/N$, one gets for an annular combustor equipped with N identical injectors

$$ \omega_i^R = \frac{(\gamma - 1) N \dot{Q}_0}{\gamma p_0 V} \psi_f^2 \frac{G_F}{M G_\zeta} \cos(\varphi_F - \varphi_\zeta) \quad (20) $$

Equation (20) shows that the growth rate depends linearly on the FDF gain, $G_F$. As the gain values are substantially lower for injector 707 than injector 727 in the frequency range of interest (around 800 Hz), one expects to find higher growth rate values associated to injector 727 than to injector 707. It also appears that the sign of the growth rate is determined by that of $\cos(\varphi_F - \varphi_\zeta)$, which depends on the relative value of the FDF phase with respect to the effective impedance phase $\varphi_\zeta$. Injector 707 presents a higher phase than injector 727 (see sec. 4), and these values have to be compared to the phase of the effective impedance to obtain the sign of the growth rate $\omega_i^R$, as will be discussed in section 5.2.

For the theoretical estimation of the growth rate, the FDF gain and phase, $G_F$ and $\varphi_F$, the product $MG_\zeta$, and the phase of the effective impedance $\varphi_\zeta$, need to be evaluated at the frequency of oscillation. This is not known exactly because it is shifted with respect to the coupling mode eigenfrequency, as highlighted by the frequency stability map displayed in Fig. 3 (bottom). However, the frequency shift does not exceed 4% of the acoustic eigenfrequency, and it is considered, as a first approximation, that the frequency of oscillation is given by the estimated resonant frequency of the 1A1L mode, $f_{1A1L}$.

Finally, the determination of the impedance phase $\varphi_\zeta$ and of the product $MG_\zeta$ is discussed in Appendix C. It is shown that the impedance $\zeta$ can be assumed the same for the two types of injectors, and the values finally retained for this study are $MG_\zeta = 0.09$ (also see Appendix C for how this value was adjusted for the different operating points) and $\varphi_\zeta = 1.2\pi$.

The FDF database may now be used in combination with Eq. (20) to interpret the differences in combustion dynamics observed in MICCA-Spray for the two types of injector investigated. One wishes to see whether one can retrieve the stable behaviour found for injector 707 and the unstable domain corresponding to injector 727 observed in MICCA-Spray when it is equipped with these injectors.

## 5.2. Growth rate maps

The growth rates $\omega_i^R$ are here evaluated for the thirty operating points, at three levels of modulation amplitude, using Eq. (20) and the FDF data collected in SICCA-Spray. The results are reported in the form of “growth rate maps” displayed in Fig. 8 for injectors 707 and 727. The interpolated growth rate maps, obtained from 2D linear interpolation between the thirty operating points, are also provided for injector 727 in Appendix D.

The maps obtained for injector 707 (Fig. 8, top) indicate that the calculated growth rates take negative or low positive values (between -200 and 80 s⁻¹) for the range of operating conditions explored. This can be explained by
<figure><img src="image_11.png" /><figcaption>Figure 8: Growth rate ($\omega_i^R$) maps obtained from FDF data for the three modulation amplitudes tested: injector 707 (top row) and injector 727 (bottom row). The thermal power values correspond to the thermal power in MICCA-Spray to enable direct comparison with the MICCA-Spray stability maps. The contour separating stable and unstable regions in MICCA-Spray, obtained in Fig. 3, is superimposed to the growth rate maps for injector 727 (bottom row) to facilitate the interpretation.</figcaption></figure>

the combination of the low values taken by the FDF gain around the frequency of interest ($f_{1A1L}$= 808 Hz) and the position of the phase curves, leading to an important phase difference with the impedance $\varphi_\zeta$ (higher than $\pi/4$), hence the $\cos(\varphi_F - \varphi_\zeta)$ term in Eq. (20) taking negative or low positive values. A previous study [27] enabled to obtain an estimate of the damping term value in MICCA-Spray between $\alpha = 310 \text{ s}^{-1}$ and $\alpha = 390 \text{ s}^{-1}$. These calculated values for the growth rate are below the damping rate estimate, and these findings are consistent with the fact that MICCA-Spray operates in a stable fashion when it is equipped with injectors 707.

The maps obtained for injector 727 (Fig. 8, bottom) reveal regions where the growth rate $\omega_i^R$ takes negative or low positive values and regions where it takes high positive values. The three maps present a layout close to that found in the amplitude stability map (Fig. 3, top). The contour plot separating the stable and unstable regions in MICCA-Spray is superimposed to the growth rate map of injector 727 to facilitate its interpretation. One can see that stable regions in MICCA-Spray correspond to points where the growth rate features negative or low positive values (from -446 to $22 \text{ s}^{-1}$), and unstable regions to points where $\omega_i^R$ features positive values (from 60 to $565 \text{ s}^{-1}$).

At this stage, one can refine the analysis by examining the link between the analytically calculated growth rate values and the experimentally measured pressure oscillation amplitude at limit cycle, $\mathcal{A}$. Figure 9 shows the thirty operating points in the ($\mathcal{A}, \omega_i^R$) plane for injector 727, for the highest and lowest levels of modulation amplitude. The estimate of the damping rate, obtained in [27], corresponds to the vertical grey band superimposed to the plot. Two groups of unstable points, corresponding to different trajectories in the amplitude-frequency state space (see for example the discussion in [4]) can be distinguished for injector 727:

* High power unstable operating points ($\mathcal{P}$ = 110 kW and $\mathcal{P}$ = 118 kW, corresponding to orange and blue points in Fig. 9) present linear growth rates values $\omega_i^R(\epsilon)$ higher than the estimate of the damping rate $\alpha$. For these points, the growth rate is expected to decrease with the level of relative velocity fluctuations, until it eventually reaches the damping rate value, establishing limit cycle oscillations. These operating conditions are associated
<figure><img src="image_12.png" /><figcaption>Figure 9: Limit cycle pressure oscillation amplitude determined experimentally, $\mathcal{A}$, as a function of the analytically calculated growth rate, $\omega_i^R$, from FDF data for the lowest (left) and highest (right) levels of modulation amplitude for injector 727. The different colors correspond to the thermal power in MICCA-Spray to enable a direct comparison with the MICCA-Spray stability map. The rectangular grey band superimposed to the plot correspond to an estimate of the damping rate $\alpha$, obtained in [27].</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Figure 10: Growth rate trajectories for injector 707 (a) and 727 (b). Stable points in MICCA-Spray are marked with black diamonds in subfigure (b), and the dashed lines represent three linearly extrapolated trajectories, corresponding to a stable (purple), a linearly unstable (blue), and a nonlinearly unstable (yellow) case. The different colors correspond to the power values in MICCA-Spray and the grey horizontal band to an estimate of the damping rate value in MICCA-Spray, $\alpha$, obtained in [27]. (c): Idealized representation of the possible growth rate trajectories as a function of the level of relative velocity fluctuations for the two identified categories of unstable points, and stable situations.</figcaption></figure>

with the highest limit cycle pressure oscillation levels.

* For intermediate power operating points ($\mathcal{P} = 99$ kW and $\mathcal{P} = 103$ kW, purple and yellow points in Fig. 9), $\omega_i^R$ takes low negative and then low positive values (below $200 \text{ s}^{-1}$) as the level of modulation amplitude increases. These operating conditions probably correspond to a subcritical situation where $\omega_i^R(\epsilon)$ is small and the system is linearly stable. However, as the level of oscillation increases, the growth rate is expected to take higher values until it overcomes the damping rate: the system becomes unstable and a limit cycle can finally be reached. For these points, a notable variation in the phase value is observed when the modulation amplitude increases, hinting nonlinear effects. These points are generally associated with lower levels of limit cycle amplitudes than those of the former category. But to confirm this general interpretation, FDFs are required for a range of relative velocity fluctuations of the order of that observed during self-sustained oscillations in MICCA-Spray, which are expected to be around 20% and 30%.
The FDF data only covering relative velocity fluctuations levels up to 10%, the complete growth rate ($\omega_i^R$) trajectories leading to the limit cycle cannot be reconstructed. It is however possible to represent the initial trajectories of $\omega_i^R$, associated to low levels of oscillation, as a function of the level of relative velocity fluctuations $v'/\bar{v}$ using the available FDF data. This is done in Fig. 10, for all the operating points (colored by the thermal power value in MICCA-Spray), for injectors 707 (Fig. 10, (a)) and 727 (Fig. 10, (b)). One can see that, for injector 707 (Fig. 10, (a)), the calculated growth rates all lie below the grey band representing the estimate of the damping rate in MICCA-Spray (obtained in [27]). The corresponding trajectories are essentially horizontal, indicating that they will not cross the damping rate level and confirming that the annular combustor equipped with this type of injectors will not exhibit oscillations. For injector 727 (Fig. 10, (b)), differences in the trajectories appear between linearly and nonlinearly unstable points, and stable points. An attempt can then be made to extend the truncated trajectories by linear extrapolation. This shows that some of the trajectories cross the damping rate threshold band at a finite value indicating that some operating points are indeed linearly stable but nonlinearly unstable. One may portray this situation with an “idealized” representation of the possible growth rate trajectories as a function of the oscillation amplitude level plotted in Fig. 10 (c). Starting from a small initial perturbation, the growth rate increases or decreases depending on the supercritical or subcritical nature of the unstable point. In the subcritical case, some of the extrapolated trajectories cross the damping rate level giving rise to an unstable limit cycle. Beyond that point, $\omega_i^R$ increases, reaching a maximum value before decreasing and finally crossing the damping rate band establishing a stable limit cycle. There are also points that are not unstable for which the growth rate remains below the damping rate band whatever the amplitude level.

## 6. Conclusion

A flame describing function (FDF) database is established using systematic measurements in a single-injector combustor (SICCA-Spray) which represents a sector of an annular configuration (MICCA-Spray). The FDFs are determined for thirty operating points corresponding to the operating domain of the annular system, and for two types of injectors (707 and 727) leading to different flame dynamics. Injector 707 is characterized by lower FDF gain and higher FDF phase values than injector 727. A stability analysis framework exploiting the FDF database in combination with values of the effective impedance, connecting the pressure and velocity fluctuations acting on the flame, is then used to calculate the growth rates associated with the different operating points of the domain. These growth rates, plotted in the form of “growth rate maps”, provide insights into the behavior of the annular system. The differences in flame dynamics experimentally observed in MICCA-Spray for the two types of injectors are well retrieved: the FDF data collected for injector 707 combined with the analytical framework yields negative or low positive growth rates for all the operating points tested, which is consistent with the fact that MICCA-Spray operates in a stable fashion when it is equipped with these injection units. For injector 727, the general layout and location of the unstable region predicted with the analytical framework is close to that determined experimentally in the annular combustor. In addition, the model enables to distinguish two categories of unstable points for this injector: linearly unstable points, observed for high power operating conditions, featuring growth rate values higher than the damping rate for low levels of modulation amplitude, and nonlinearly unstable points, corresponding to intermediate power values and associated to growth rates that are initially lower than the damping rate but that are expected to increase with the level of oscillation and may eventually exceed this threshold, giving rise to limit cycles. The flame dynamics associated to different injectors and a broad range of operating conditions are hence successfully retrieved, showing the relevance of a data-driven model-based approach to sort out stable and unstable cases in an annular configuration.

## Acknowledgements

This work was partially supported by the FlySAFe project of the Agence Nationale de la Recherche ANR-22-CE05-0022-02.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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