# Comparison of pressure-based flame describing functions measured in an annular combustor under self-sustained oscillations and in an externally modulated linear combustor

Abdallah Alhaffar, Véranika Latour, Clément Patat, Daniel Durox, Antoine Renaud, Jean-Bernard Blaisot, Sébastien Candel, Françoise Baillot

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## ► To cite this version:

Abdallah Alhaffar, Véranika Latour, Clément Patat, Daniel Durox, Antoine Renaud, et al. Comparison of pressure-based flame describing functions measured in an annular combustor under self-sustained oscillations and in an externally modulated linear combustor. Proceedings of the Combustion Institute, 2024, 40 (1-4), pp.105249. 10.1016/j.proci.2024.105249. hal-05235187

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# Comparison of pressure-based flame describing functions measured in an annular combustor under self-sustained oscillations and in an externally modulated linear combustor

Abdallah Alhaffar<sup>a,*</sup>, Véranika Latour<sup>b</sup>, Clément Patat<sup>a</sup>, Daniel Durox<sup>b</sup>, Antoine Renaud<sup>b</sup>, Jean-Bernard Blaisot<sup>a</sup>, Sébastien Candel<sup>b</sup>, Françoise Baillot<sup>a</sup>

## ARTICLE INFO

**Keywords:**

Combustion instabilities

Annular combustor

Transverse acoustic mode

Acoustic forcing

Flame describing function

## ABSTRACT

One important aspect in the analysis of combustion instabilities in multiple-injector annular systems is that of representing such oscillations with a linear array of injectors. This issue is considered in the present work by comparing flame dynamics in a self-sustained oscillations situation in the annular configuration MICCA-Spray, with experiments carried out in TACC-Spray, a rectangular combustor equipped with five injectors externally modulated by driver units. In the annular system, the coupling mode is azimuthal, and in the linear array, the flames are submitted to a transverse acoustic field. An original method is used to change the limit-cycle pressure oscillations amplitude in the annular system and favor a standing mode with a well-defined nodal line position. It consists in using different arrangements of two types of injectors, leading to various azimuthal staging patterns. The range of pressure oscillation amplitudes is swept in TACC-Spray by changing the external acoustic forcing level. It is thus possible to compare the flame dynamics in forced and self-sustained situations using these two experimental facilities. This is done by examining flame describing functions (FDFs) based on downstream pressure fluctuations measured in the two configurations for different pressure oscillation amplitudes. The trends observed in the two systems are in good agreement, and there is a reasonably good match between the values of the gains and phases determined in the two configurations. The pressure-based FDFs are determined for flames located at different positions with respect to the acoustic mode. It is found that the transverse or azimuthal velocity component has only a weak influence on the pressure-based FDF, the strongest impact being observed in the neighborhood of the velocity antinode. This study demonstrates the ability of a linear-array setup modulated by a transverse mode to reproduce conditions of azimuthal coupling leading to self-sustained limit cycles in an annular combustor.

## 1. Introduction

Thermoacoustic instabilities continue to be an issue in the development of combustion chambers of various high-performance devices, such as rocket engines, aero-engines and gas turbines. These instabilities result from the positive interaction between acoustic disturbances and the unsteady heat release rate (HRR) produced by the flames. In gas turbines and aeroengines, the combustor features multiple gaseous or spray flames, stabilized by swirling flows, and most often confined in an annular geometry. However, annular combustors can be very sensitive to thermoacoustic instabilities due to their geometry. It is therefore important to develop tools and models to be able to predict the occurrence of these undesired phenomena in the combustor design process.

A key factor in this direction is to gain an understanding of the physical mechanisms driving these instabilities. In this context, experimental studies have been carried out in laboratory-scale self-unstable annular combustors coupled by azimuthal modes [1–6], in addition to theoretical and numerical studies [7–13]. Transverse acoustic forcing experiments have also been performed in single-injector combustors [14–18], and multiple-injector linear setups corresponding to unfolded sectors of annular combustors [19,20] to characterize the flames’ response to acoustic disturbances.

In annular combustors, pressure oscillations mainly couple with azimuthal modes of the cavity [2,3,21], inducing axial flow fluctuations downstream of the injectors and generating HRR oscillations. One may

<footnote><sup>a</sup> Univ Rouen Normandie, INSA Rouen Normandie, CNRS, CORIA UMR 6614, F-76000 Rouen, France</footnote>

<footnote><sup>b</sup> Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire EM2C, 91192 Gif-sur-Yvette, France</footnote>

<footnote>* Corresponding author.
E-mail address: abdallah.alhaffar@coria.fr (A. Alhaffar).</footnote>
recall at this point that the Rayleigh source term, $S_{Ra}$, also designated as the Rayleigh index, expresses the link between pressure and HRR oscillations. For an instability to grow, $S_{Ra}$ must be positive, and this necessary condition may be written as:

$$S_{Ra} = \frac{1}{T} \int_{T} \int_{V} \frac{\gamma - 1}{\bar{\rho}c_{0}^{2}} p'q'dVdt > 0 \quad (1)$$

where V is a volume comprising the whole heat release region, T a sufficiently long time interval, γ the specific heat ratio, $\bar{\rho}$ the mean gas density, $c_0$ the speed of sound, $p'$ and $q'$ the fluctuations of the local acoustic pressure and volumetric HRR respectively. In addition to the positivity condition, an oscillation will occur only if $S_{Ra}$ is sufficiently large compared to the acoustic energy losses in the system.

The flame response to downstream pressure oscillations can be characterized by the pressure-based FDF, $F_p$ [19,22]. A first advantage of this approach relies on the fact, for compact flames, pressure oscillations measurements are less sensitive to the location where they are made, in contrast with velocity fluctuation measurements at the injector exit, that are strongly dependent on the location especially for swirling flows which present large spatial velocity gradients [23, 24]. Besides, in practical applications, pressure oscillations are easier to measure. A second advantage is that $F_p$ implicitly contains the information about the injector impedance, i.e., the conversion of downstream pressure oscillations to velocity oscillations through the injector. Finally, it is noteworthy that in many theoretical investigations dealing with the nonlinear modeling of combustion instabilities, it has been found convenient to express the HRR as a function of pressure fluctuations [25,26], in the form of a third order polynomial [11], or as a combination between acoustic pressure and transverse velocity fluctuations [10]. It is hence worth quantifying the flame response in terms of HRR to downstream pressure disturbances to better understand the relationship between these two key quantities and provide experimental data for modeling purposes.

For non-sooty flames with weak equivalence ratio fluctuations, the oscillations of the light emitted by OH* radicals, $I'$, can be used to quantify HRR oscillations [24,27]. The following pressure-based FDF was introduced for spray flames in [19]:

$$F_p(f, \Pi) = \frac{\hat{I}/\bar{I}}{\hat{\rho}/\rho_a U_b^2} = G(f, \Pi) e^{i\phi(f, \Pi)} \quad (2)$$

where $\hat{\rho}$ and $\hat{I}$ are the Fourier transforms of $p'(t)$ and $I'(t)$ at the resonance frequency $f$. $\bar{I}$ is the mean light intensity emitted by OH*. $\Pi$ is the reduced acoustic pressure amplitude defined as $\Pi = p_{rms}/\rho_a U_b^2$, where $p_{rms}$ is the root mean square (rms) acoustic pressure amplitude obtained at the resonance frequency $f$, here measured at the position of the studied flame, $\rho_a$ the air density and $U_b$ the air bulk velocity at the injector exit, both calculated at 20 °C. G is the gain of the FDF, corresponding to the ratio between the reduced rms amplitude of the flame intensity oscillation $I_{rms}/\bar{I}$, and $\Pi$; $\phi$ corresponds to the phase delay between $p'(t)$ and $I'(t)$.

This work aims to measure pressure-based FDFs in forced and self-sustained pressure oscillation conditions, and compare the flame dynamics in both situations. To this end, two experimental setups are used: the TACC-Spray linear combustor (designated as “TACC” in what follows), operated by CORIA, and the annular combustor MICCA-Spray (designated as “MICCA”) at EM2C Laboratory. TACC corresponds to an unfolded sector of three burners of MICCA, and previous investigations [19,28] showed that, when submitted to a transverse acoustic forcing exciting the 2T1L mode of the rectangular chamber, TACC enables to qualitatively reproduce flame phenomena observed during self-sustained thermoacoustic instabilities coupled by the 1A1L mode in MICCA.

The objective of the present investigation is to demonstrate the ability of TACC to quantitatively reproduce the flame dynamics observed in MICCA. This is done by comparing pressure-based FDFs at several flame locations in a standing transverse acoustic field for different levels

of pressure oscillations. The range of pressure oscillation amplitudes is swept in TACC by changing the external acoustic forcing level. In MICCA the level of pressure fluctuations at limit cycle is varied by mixing two types of injection units, designated as “707” and “727”, in different arrangements. When MICCA is equipped with injectors 707, the system operates in a stable fashion, and when injectors 727 are mounted, the system presents self-sustained oscillations for a broad range of operating conditions [6]. Staging these two types of injectors enables to favor a well-defined standing mode, with a controlled nodal line position, as discussed in [29].

This article comprises three sections. The two experimental setups are presented in Section 2. Pressure-based FDF measurements in forced (TACC) and self-sustained (MICCA) situations are then described in Section 3, and finally, the results are compared for the two situations in Section 4.

## 2. Experimental setups

The two experimental setups MICCA and TACC are equipped with the same swirling spray injectors, described in Section 2.3, fed by liquid n-heptane and air. The thermal power is set to 7.4 kW/flame and the global equivalence ratio to 0.9. This leads to a fuel mass flow rate of 0.166 g/s, an air mass flow rate of 2.78 g/s and a bulk velocity at the injector outlet of $U_b$ = 46 m/s. Previous studies on TACC and MICCA [23,24] showed that, with the injection units used (described later in Section 2.3), equivalence ratio fluctuations are negligible compared to volume flow rate fluctuations. Hence, relative fluctuations of the global HRR can be approximated by the relative fluctuations of the light intensity emitted by OH* radicals: $\hat{Q}'/\bar{Q} \approx I'/\bar{I}$. This normalization is based on the OH* intensity signal measured from each burner. The mean HRR can be considered to be the same for all flames since the fuel flow rates delivered by the various injectors are identical and these systems operate on the lean side providing an excess of air.

### 2.1. The MICCA annular combustor

The laboratory-scale annular combustor MICCA, shown in Fig. 1(a), is used to investigate flame dynamics under self-sustained oscillations coupled by azimuthal modes. Two cylindrical quartz walls of length $l$ = 400 mm, of outer diameter $d_{out}$ = 300 mm for the inner wall and inner diameter $d_{in}$ = 400 mm for the outer wall, delimit the combustion chamber while providing optical access to the flames. Sixteen swirling spray injectors are mounted on the chamber backplane, and deliver liquid n-heptane as a hollow cone spray of droplets. The fuel and air mass flow rates are controlled through a Bronkhorst CORI-FLOW controller with a relative accuracy of 0.2%, and two Bronkhorst EL-FLOW mass flow meters, with a relative accuracy of 0.6%.

Eight Brüel & Kjær microphones are flush-mounted on waveguides and plugged onto the chamber backplane to record pressure fluctuations. An array of 8 photomultipliers (PM), equipped with a filter centered at 310 nm, is used to record the UV light signal, $I$, emitted by OH* radicals of eight adjacent flames. A spatial mask is placed in front of each PM to make sure that the sensors only record the light emitted by one flame. The microphones and PMs positions are displayed in Fig. 1(b).

### 2.2. The TACC linear combustor

TACC is composed of a fixed base and movable lateral and front walls. It has an adjustable length $L_c$ such that the chamber can be used as an acoustic resonant cavity at various frequencies $f$. To prevent outer air entrainment into the combustor through the top, which might disturb the flames environment, a convergent unit with a reduced exhaust section is installed on the top of the chamber. The width of this section is five times smaller than the combustor width ($\lesssim e_c/5$) (see Fig. 2(a)).
<figure><img src="image_5.png" /><figcaption>Fig. 1. (a) The MICCA-Spray annular combustor with an array of eight photomultipliers. (b) Microphones (labelled “MX”) and photomultipliers positions (labelled “PMX”). (c) Exploded view of the injector, and top view of the swirler.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. TACC-Spray combustor ($L_c$ = 780 mm, $h_c$ = 200 mm, $e_c$ = 55 mm) with its convergent part ($h_c$ = 60 mm, exit width = 10 mm): (a) Three swirling spray flames of n-heptane without acoustics; (b) Top view showing the 3 burners (red). The two lateral injectors (blue) supply swirling air only. M1, M2: microphones, PM: photomultiplier and AD: Acoustic Driver unit.</figcaption></figure>

The injection system features five swirling injectors (presented in Section 2.3) mounted on the fixed base: three central spray injectors and two lateral air injectors without a fuel atomizer. The lateral injectors allow to set the central flame in local aerodynamical and thermal conditions similar to those prevailing in the annular combustor MICCA. This is achieved by placing the central flame between two lateral swirling flames and this flame array is stabilized, without any wall, by means of two additional lateral streams of swirling air that are transparent to the transverse acoustic field. The ratio of the injector diameter to the spacing between adjacent injectors is quite small in MICCA and TACC (~ 0.11). This leads to a weak interaction between the neighboring flames in the two setups (see Fig. 2(a) and [30,31]).

An acoustic forcing system is implemented in TACC. A signal of chosen amplitude and frequency is delivered to two Beyma CP850ND acoustic drivers units (AD) fixed on the lateral walls of the cavity, to generate a standing transverse acoustic field inside the cavity that has an adjustable length (see Section 3.2).

TACC is equipped with various diagnostics (Fig. 2(b)). Temperature measurements are performed using type K thermocouples flush-mounted on the front walls. The acoustic pressure, $p'(t)$, is measured with type 4182 B&K microphones also flush-mounted on the front wall. A PM equipped with an OH\* interference filter (305 ± 12 nm) is used to measure the intensity $I$ of the central flame. The pressure and flame intensity measurements are synchronized with a National Instruments DAQ system at a sampling frequency of about 82 kHz.

## 2.3. Injectors' characteristics

MICCA and TACC are equipped with the same injection units, presented in Fig. 1(c). The injectors used are composed of four main elements: an air distributor, a liquid fuel atomizer, a swirler imparting

a swirling motion in the clockwise direction to the mixture of air and liquid fuel droplets, and a terminal plate. These units are modular: changing the swirler (channel radius and orientation) changes the pressure drop and swirl number of the injection unit.

Two injectors, named “707” and “727”, are used in this study. Their characteristics (swirl number, pressure drop, and pressure drop coefficients), measured under cold flow conditions, are gathered in Table 1. Injector 727 has a higher pressure drop and swirl number values than injector 707, which is reflected in a shorter and more compact flame established by injector 727 [6]. The flames formed by the two injectors have the same general appearance and geometrical shape in TACC and in MICCA, and the typical flame shape is depicted in Fig. 2(a). The two kinds of injectors lead to different flame dynamics in MICCA. The annular combustor operates in a stable fashion when it is equipped with sixteen 707 injectors, whereas self-sustained oscillations coupled with the 1A1L mode of the chamber are observed for a broad range of operating conditions when all the injectors are of the 727 type [6].

The air manifold feeding the injectors are different in the two facilities. However, this does not have a major effect on the dynamics of the system. First, the area ratio between the injector outlet and the sector surface area is quite large inducing a decoupling between the chamber and the upstream manifold. Then, the injectors are weakly transparent to acoustic waves [30] and they are placed in a rigid backplane. Therefore, the impedance at the flame is notably dominated by the presence of the rigid wall.

## 3. Pressure-based FDF measurements

Pressure-based FDFs, as defined in Eq. (2), are obtained from simultaneous pressure and OH\* intensity recordings in a forced situation
<caption>**Table 1**
Injection units characteristics. $\Delta p = (1/2)\sigma\rho_a U_b^2$ obtained from measurements under cold flow conditions with an air flow rate of 2.6 g s$^{-1}$.</caption>

Source: Adapted from [6].

<table><thead><tr><th>Injector</th><th>707</th><th>727</th></tr></thead><tbody><tr><td>Swirl number (S<sub>N</sub>)</td><td>0.6</td><td>0.74</td></tr><tr><td>Pressure drop (kPa) (Δp)</td><td>3.65</td><td>5.70</td></tr><tr><td>Pressure drop coefficient (σ)</td><td>3.33</td><td>5.20</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Fig. 3.** Self-sustained instability amplitude $A$ and frequency $f$ for seven injector arrangements (A-G) in MICCA. Red: injectors 707; blue: injectors 727.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Classifications of the injectors as “PAN”, “IAN”, and “VAN” in MICCA depending on their location with respect to the nodal line position for a standing mode.</figcaption></figure>

(TACC) and under self-sustained situation (MICCA). Measurements are carried out for various levels of pressure oscillation amplitudes and for flames located at different positions with respect to the pressure nodal line: in the neighborhood of the pressure antinode (PAN), the intensity antinode (IAN) - referring to the position where the acoustic flux is maximum - and the velocity antinode (VAN). The coherence level at the frequency of oscillations between signals $p'(t)$ and $I'(t)$ has been checked to be higher than 0.9 and the data points not fulfilling this condition were removed from the data set.

### 3.1. Self-sustained situation (MICCA)

It is shown in a previous investigation [29] that mixing two categories of injection units enables to modulate the amplitude of limit cycle oscillations in MICCA and, for certain arrangements, favor a standing mode and control the nodal line position. This possibility is exploited in this work to gather pressure-based FDF data under self-sustained oscillation conditions, as defined in Eq. (2), for flames located at different positions with respect to the nodal line and for various pressure oscillation amplitudes.

The pressure field at each injector position is reconstructed as a sum of two waves propagating in the counterclockwise (CCW) and clockwise

(CW) directions corresponding to the 1A azimuthal mode:

$$p' = A_{+}e^{i\theta-i\omega t} + A_{-}e^{-i\theta-i\omega t} \quad (3)$$

where $\omega$ is the angular frequency and $\theta$ the azimuthal angle considered positive along the counterclockwise direction. $A_+$ and $A_-$ represent complex amplitudes of the two CCW and CW waves determined from microphone measurements. The instability amplitude may then be defined as follows:

$$A = (|A_{+}|^2 + |A_{-}|^2)^{1/2} \quad (4)$$

Fig. 3 reports the instability amplitudes and frequencies observed experimentally for the seven arrangements investigated. The different configurations lead to self-sustained oscillations, characterized by frequencies in the [775–805] Hz range, and instability amplitudes between 300 and 1400 Pa. Placing two groups of injectors 707 in diametrically opposed positions enables to fix a well-defined standing mode with a nodal line defined by the diameter connecting the middle points of these two groups of injectors.

In what follows, the pressure-based FDF data points will be labeled as “PAN”, “IAN”, or “VAN”, depending on the flame’s position with respect to the nodal line. As shown in Fig. 4, injectors that are located between ±30° around the nodal line will be denoted as “VAN”, those between ±30° around the antinodal line as “PAN”, and the others as “IAN”.

Fig. 5 shows rms pressure values and pressure-based FDF gains $G_M$ and phases $\varphi_M$ obtained from simultaneous microphone and PM recordings at eight different injector locations for five configurations in MICCA (A to E). Gain and phase values associated to injectors 727 appear in blue and those corresponding to injectors 707 in red. The phase values $\varphi_M$ are found to lie between [-$\pi$/2, $\pi$/2] for the two types of injector (see Fig. 5, bottom row). Injectors 707 and 727 mainly differ by their gain, which is substantially lower for injector 707 (see Fig. 5, second row).

### 3.2. Forced situation (TACC)

The 2T1L mode is excited inside the TACC chamber at a resonance frequency of 820 ± 5 Hz, close to the self-sustained oscillation frequencies observed in MICCA. This is done by fixing the chamber length at 780 mm. There are some differences in frequency between experiments in TACC and in MICCA but the corresponding relative variations $\Delta f/f$ of about 2–4% will only have a minor impact on the data. This conclusion can be inferred from tests carried out previously (see [6]) which indicate that frequency variations of a few percent only weakly change the measured velocity-based FDFs and that this is the case for the two injectors (707 and 727) used in the present experiments.

The position of the central flame can be changed by moving the lateral walls of the chamber. The microphone M1 (see Fig. 2(b)) is always placed at the central flame location, and its signal is used in the calculation of the pressure-based FDF. Fig. 6 reports the transverse acoustic field inside the TACC cavity and the domains “PAN”, “IAN” and “VAN” defined basing on the method used in MICCA. These domains are similar to those in MICCA. “VAN” is delimited between ±8% of the chamber length around the pressure node, “PAN” between by ±8% of the chamber length around the pressure antinode, and the rest as “IAN”. The difference between the resonance modes, 1A1L in MICCA and 2T1L in TACC, leads to a steeper pressure gradient at IAN in TACC.

## 4. Comparison between self-sustained and forced situations

The pressure-based FDF gain and phase values obtained at PAN, IAN and VAN, for injectors 727 and 707 are presented in Fig. 7. The data gathered in the forced and self-sustained situations have many features in common.
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Top: root mean square of the pressure field at eight injector locations in MICCA for configurations A, B, C, D, and E. Center and bottom: pressure-based FDF gain and phase values, $G_M$ and $\phi_u$, at different photomultipliers (PM) positions obtained under self-sustained oscillations in MICCA. Red points correspond to injectors 707 and blue points to injectors 727. The oscillations frequencies are the same as those highlighted in Fig. 3. All the points appearing in these graphs have a coherence between the pressure and photomultiplier signals higher than 0.9.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Standing transverse acoustic field in TACC: “PAN”, “IAN” and “VAN” domains are defined from the method used in MICCA. Black solid symbols: with combustion ($p_{rms}^{max}$ = 425 Pa, $f$ = 784 Hz); open symbols are without combustion: blue ($p_{rms}^{max}$ = 66 Pa, $f$ = 773 Hz), red ($p_{rms}^{max}$ = 355 Pa, $f$ = 718 Hz), green ($p_{rms}^{max}$ = 498 Pa, $f$ = 718 Hz); dash-dotted line: simulation ($f$ = 722 Hz).

Source: Adapted from [28].</figcaption></figure>

Starting with injector 727 (Fig. 7, left), one observes a decrease in the gain with the reduced pressure $\Pi$ in both forced and self-sustained situations. The same slope is noticed at PAN and IAN, indicating a strong nonlinear response of the flame. The gain values match well between the two experiments at PAN and VAN. In the forced situation, the gain at IAN is lower by 20% than at PAN, while in the self-sustained situation no significant difference is observed except for one point. The phase angle takes values in the [-$\pi$/2, $\pi$/2] range corresponding to the potential instability range according to the Rayleigh criterion.

For injector 707 (Fig. 7, right), the results are mostly in agreement even though discrepancies are more apparent. The gain remains practically constant indicating a quasi-linear flame response for forced and self-sustained situations whatever the flame position. Results in the forced situation show gain values ($G_T$) around 0.3, and those obtained under self-sustained oscillations lie around 0.25. The phase angles are in the [-$\pi$/2, $\pi$/2] range for the forced and the self-sustained situations. In the latter case, the gain and phase values are dispersed over a

broader interval, even though the coherence level is higher than 0.9, which guarantees that the measurements are reliable. The dispersion is probably due to the fact that injectors 707 are used to fix the nodal line position, and as a result, these injectors operate mainly near VAN or IAN locations, where the pressure fluctuation level is relatively low and the uncertainty is correspondingly larger. It is also probable that part of the dispersion is due to the variability in the response of the different injectors, as also discussed in [32], where variations in the response of flames, submitted to spinning waves in a forced annular combustor, were observed.

For both injectors, it is not surprising to observe at VAN significant values for the gain of the pressure-based FDF. In fact, in the vicinity of the nodal line, the pressure fluctuations are low, but the flame intensity oscillations are also weak. Their ratio is less well defined and it may take some large values.

The linear evolution of the gain with the pressure oscillation amplitude for injector 707 as well as the nonlinear one for injector 727, which have been highlighted above, differ from that assumed in some current models where the HRR is expressed as a third degree polynomial of the pressure oscillations [10,25]. In these models the changes in gain as a function of the amplitude of oscillation is probably too fast at least with respect to the present experimental results.

This comparative analysis hence demonstrates the ability of TACC to reproduce, under acoustic forcing, the unstable flame dynamics of MICCA for different levels of pressure oscillations. The collected experimental data also enable to gain insights into the dynamics of the flames formed by injectors 707 and 727. The flames formed by the two injectors respond to pressure perturbations with similar phase values and the responses mainly differ in the gain, with a lower gain measured for injector 707.

It is now interesting to use the collected data to determine the Rayleigh index $\bar{R}_a$ defined by $\bar{R}_a = (1/T) \int_T p' / \rho_a U_b^2 \times I' / \bar{I} dt$ in TACC and compare the values obtained for the flames formed by the two
<figure><img src="image_11.png" /><figcaption>Fig. 7. Pressure-based FDFs at PAN, IAN and VAN, obtained under forced oscillations in TACC ($G_y$ and $\phi_y$) and self-sustained oscillations in MICCA ($G_m$ and $\phi_m$) operated for seven injectors configurations (A to G), each of them leading to flame responses at PAN, IAN, and VAN. (a) Injector 727, (b) Injector 707.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 8. Rayleigh index in TACC. Circles for 727, triangles for 707. Purple for PAN, yellow for IAN, green for VAN.</figcaption></figure>

injectors. The results are reported in Fig. 8. One can notice that the calculated Rayleigh index is higher for injector 727 than for injector 707 for the different positions of the flame in the acoustic field (PAN, IAN or VAN). The calculated Rayleigh index values hence enable to explain why, when all the injectors are of the 707 type in MICCA, the annular system operates in a stable fashion, and when injectors 727 are mounted, unstable operation is observed. One can expect that the total source term $S_{Ra}$ exceeds the damping term in the second case, but not in the first one, hence leading to unstable and stable operation respectively.

## 5. Conclusion

A comparison of the flame dynamics in an annular combustor featuring self-sustained oscillations coupled by an azimuthal mode and an externally modulated linear array of injectors is made. This is achieved by measuring pressure-based flame describing functions for different levels of pressure oscillation amplitude and different locations with respect to the pressure nodal line position in the forced and self-sustained situations. The changes in pressure oscillation amplitude are obtained in the linear array by changing the level of excitation of the driver units. The amplitudes are swept in the annular system by mixing two types of injectors in various arrangements, enabling to operate with a well-defined standing mode and to fix the nodal line position. It is found that the pressure-based FDFs feature similar trends in the forced and self-sustained situations, and that the agreement is good at the pressure antinode and in the vicinity of the pressure

node. It is also achieved at the intensity antinode. At this point, the acoustic velocity has a transverse or azimuthal component which does not seem to influence the pressure-based FDF in a significant manner. This study hence demonstrates that a linear array setup modulated by a transverse mode (TACC) can be used to reproduce conditions of azimuthal coupling leading to self-sustained limit cycle oscillations in an annular combustor (MICCA).

## Novelty and significance statement

The novelty of this research lies in: (i) the original method used to obtain limit cycle thermo-acoustic oscillations featuring a range of amplitudes in a laboratory scale annular combustor, MICCA-Spray; (ii) the demonstration that a linear array of injectors (TACC-Spray) modulated in the transverse direction can be used to reliably reproduce conditions prevailing in the annular combustor MICCA-Spray for a wide range of instability amplitudes at the frequency of oscillation corresponding to the coupling mode in that system; (iii) flame describing functions linking the heat release rate fluctuations to the downstream pressure fluctuations obtained from data gathered in the two facilities closely match. The data in the form of FDFs based on pressure may be useful to theoretical investigations of nonlinear thermo-acoustic instabilities and open perspectives to the instability analysis of self-sustained oscillations that may be easier to apply to industrial configurations.

## CRediT authorship contribution statement

**Abdallah Alhaffar:** Performed experiments in TACC, Processed and analyzed data, Writing – original draft. **Véranika Latour:** Performed experiments in MICCA, Processed and analyzed data, Writing – original draft. **Clément Patat:** Performed experiments in TACC, Processed and analyzed data, Writing – original draft. **Daniel Durox:** Designed experimental facility MICCA, Operated MICCA, Conceived novel method of mixing injector families, Performed experiments in MICCA. **Antoine Renaud:** Participated in data processing and interpretation, Writing – original draft. **Jean-Bernard Blaisot:** Participated in data interpretation, Writing – original draft. **Sébastien Candel:** Designed research effort, participated in data interpretation, Writing – original draft. **Françoise Baillot:** Designed research effort, participated in data interpretation, Writing – original draft.
Declaration of competing interest

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

Acknowledgments

This work was supported in part by project FlySAFe (ANR-22-CE05-
0022) of the French National Research Agency (ANR) and Région
Normandie. The authors thank RENADIAG for the technical support
provided to this study and the reviewers of this article for their helpful
comments.

References

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