# Azimuthal Instabilities Arising in an Annular Combustor Equipped With Pure Hydrogen Injection Units

Nicolas Vaysse, Daniel Durox, Ronan Vicquelin, Sébastien Candel, Antoine Renaud

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Nicolas Vaysse, Daniel Durox, Ronan Vicquelin, Sébastien Candel, Antoine Renaud. Azimuthal Instabilities Arising in an Annular Combustor Equipped With Pure Hydrogen Injection Units. Journal of Engineering for Gas Turbines and Power, 2025, 148 (1), pp.011008. 10.1115/1.4069546. hal-05340944

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Submitted on 31 Oct 2025

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# Azimuthal instabilities arising in an annular combustor equipped with pure hydrogen injection units

Nicolas Vaysse¹, Daniel Durox¹, Ronan Vicquelin¹, Sébastien Candel¹, and Antoine Renaud¹

¹Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3, rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France

## Abstract

At a stage where hydrogen is being considered as a possible fuel for aeroengines or industrial gas turbines, it is important to examine operability issues and in particular those induced by the coupling between acoustic modes of the system and combustion. The aim of this investigation is to analyze instabilities coupled by azimuthal modes which are known to be less well damped and also feature eigenfrequencies that fall in the range where flames are most susceptible to disturbances. The present study is carried out in a laboratory scale annular combustor equipped with 16 burners, each including cross-flow hydrogen injection in a swirled air flow. A mapping of the operational domain reveals many different regimes of instability, some of them arising at high frequencies. An examination of the pressure signals detected by the eight microphones indicate that the levels of oscillations may be quite high reaching peak values of more than 2000 Pa. It is found that instabilities are most often coupled by standing modes with well defined nodal lines. The pressure signals are used to infer the nature of the modes and determine their azimuthal structure. Various types of instabilities are identified involving the 1A1L, 2A1L, 2A2L, 3A1L and 4A1L modes at frequencies ranging from 800 to 2800 Hz in agreement with theoretical estimates of the resonant frequencies. The pressure and OH* light intensity data are used in a second stage to characterize

the growth rates of the various unstable regimes and deduce the damping rates pertaining to the various modes. It is shown that flames established by injection of pure hydrogen are highly sensitive to disturbances and that they easily couple with azimuthal modes giving rise to higher order modes at elevated frequencies.

## nomenclature

### Physical parameters

<table><tr><td>A<sub>m</sub><sup>+</sup></td><td>Counterclockwise wave (CCW) amplitude [kg m<sup>-1</sup> s<sup>-2</sup>]</td></tr><tr><td>A<sub>m</sub><sup>-</sup></td><td>Clockwise wave (CW) amplitude [kg m<sup>-1</sup> s<sup>-2</sup>]</td></tr><tr><td>&phi;&#x0307;<sub>m</sub><sup>+</sup></td><td>Counterclockwise wave (CCW) phase [-]</td></tr><tr><td>&phi;&#x0307;<sub>m</sub><sup>-</sup></td><td>Clockwise wave (CW) phase [-]</td></tr><tr><td>c</td><td>Mean speed of sound</td></tr><tr><td>P<sub>th</sub></td><td>Flame thermal power [kW]</td></tr><tr><td>&phi;</td><td>Global equivalence ratio [-]</td></tr><tr><td>&Pi;</td><td>Dimensionless pressure amplitude [-]</td></tr><tr><td>&omega;<sub>mn</sub></td><td>Angular frequency of the m, n-th mode [rad s<sup>-1</sup>]</td></tr><tr><td>f<sub>mn</sub></td><td>Eigenfrequency of the m, n-th mode [s<sup>-1</sup>]</td></tr><tr><td>&rho;<sub>0</sub></td><td>Ambient density [kg m<sup>-3</sup>]</td></tr><tr><td>p<sub>0</sub></td><td>Ambient pressure [kg m<sup>-1</sup> s<sup>-2</sup>]</td></tr><tr><td>S</td><td>Swirl number [-]</td></tr><tr><td>s<sub>r</sub></td><td>Spin ratio [-]</td></tr><tr><td>U<sub>b</sub></td><td>Bulk velocity [m s<sup>-1</sup>]</td></tr></table>

### Geometrical parameters
$\theta$ Azimuthal angle [-]

$\Theta_+, \Theta_-$ Angular ranges [-]

$l_c$ Combustor length [m]

$l'_c$ Length including end correction
$l'_c = l_c + \delta_a$ [m]

$\delta_a$ End correction [m]

$P_a$ Mean annular perimeter [m]

$h_r$ Recess length between H₂ injection plane and injector outlet [mm]

## Superscripts and subscripts

$b$ bulk value

$m$ Azimuthal mode number [-]

$n$ Axial mode number [-]

# 1 Introduction

Much current research in combustion is concerned with the replacement of fossil fuels by hydrogen or hydrogen enriched blends. One practical goal is to ensure that systems fed in this way will be operable, enable a stable combustion, and ensure low NOx emissions. It is known however that the use of hydrogen raises a wide variety of issues which need to be considered. In the case of gas turbine combustors and more specifically those that could be considered for aeroengine applications, one important question is that of thermo-acoustic instabilities. This needs to be analyzed not only in the case of single flames but in annular configurations comprising multiple injection units, geometries that are of the kind used in practical devices and feature a rotational symmetry. In such configurations, the flames may couple with azimuthal modes which are generally considered as the most dangerous because they are less well damped and correspond to resonance frequencies that fall in the range where the flames are most susceptible to incident disturbances.

This type of analysis has been carried out in different annular combustors operating with gaseous or liquid hydrocarbon fuels. A laboratory scale annular system developed in Cambridge, initially used to examine combustion instabilities when operated with premixed ethylene-air, [1], was used in many further investigations at Norwegian University of Science and

Technology (NTNU) [2]. The annular system called MICCA designed at EM2C Lab was initially used to examine instabilities coupled by azimuthal modes [3, 4, 5]. Other annular systems were developed more recently at Zhejiang University [6] and at Indian Institute of Technology Madras [7] to study dynamics of annular combustion systems, respectively operated with propane/air and liquid petroleum gas. Efforts were also made to operate annular combustors at higher pressures. The annular facility at Technical University of Munich (TUM) [8] was used recently to study effects of quarter wave dampers [9] while a NTNU test rig was run in the recent period to characterize the thermo-acoustic coupling with azimuthal modes at elevated pressures [10, 11]. The unstable oscillations investigated in the literature predominantly involve the first-order azimuthal mode, but some observations indicate that harmonic components are present [10] and a coupling with the second azimuthal-first longitudinal (2A1L) mode is reported in [6]. These investigations have also revealed a strong dependency of the triggering of thermo-acoustic instabilities with respect to the hydrogen content. Azimuthal mode shifting is observed in the NTNU annular chamber [11], where the inhibitor effect of high hydrogen enrichment is explained to be a result of the cancellation taking place between distinct out-of-phase regions in the flame [12]. Experiments on can-annular configurations and micromix simulations [13, 14] indicate that the addition of hydrogen promotes thermoacoustic coupling in the high frequency range.

Most of these studies have concerned annular combustors fed with premixed gases, liquid hydrocarbon fuels injected in the form of droplet sprays and in the more recent period gaseous blends of methane and hydrogen. The case of pure hydrogen injection in annular combustors has apparently not been covered although a number of studies are dealing with thermo-acoustic instabilities of single H₂ flames. The present effort is aimed at filling this gap by providing experimental data on combustion instabilities in annular systems equipped with multiple injection units fed with pure hydrogen, a situation that is of much practical interest but is less well documented. The objective is to analyze a system in which pure hy-
<figure><img src="image_2.png" /><figcaption>Figure 1: Experimental setup: (a) Photograph of MICCA-H2 without the central mask, (b) Photomultiplier locations for the OH* measurements and position of the removable cylindrical mask (the dotted circle) located inside the annulus, (c) Details of CFI-X1. The height $h_r$ designates the recess between the H₂ injection holes and the outlet, (d) MICCA-H2 under hot fire operation.</figcaption></figure>

drogen is injected through swirling units placed in an annular chamber, explore the domain of operation, define regions of instability, identify the corresponding modes of oscillation, analyze the nature of these modes and examine the heat release rate fluctuations induced by these modes.

At this point, it is natural to review the literature dealing with the combustion dynamics of hydrogen flames in the case of thermo-acoustic instabilities. Much of the research effort has concerned flames formed by adding hydrogen to a hydrocarbon fuel, in most cases methane. Such blends are considered for gas turbine applications. These studies are generally carried out with single burners. Several phenomena have been examined: transitions in flame shapes and sensitivity to incident disturbances [15, 16, 17, 18, 19], reduction in time delay between the flow perturbation and the flame response and its effect on the flame transfer function [19, 20, 21, 22], and the consequences of this phenomenon in terms of thermoacoustic coupling [23, 24, 25]. It is interesting to note that some studies have indicated a gain reduction in the FTF when the hydrogen content is augmented, a feature that was found in both experimental and numerical investigations of swirled burners [26, 27, 19]. It is also observed in some recent

studies that an increase in hydrogen content could lead to enhanced heat release rate fluctuations or to an augmented acoustic damping [11, 28]. Recent works have emphasized the dynamics of pure hydrogen flames [29, 30, 31, 32, 33]. Hydrogen flames are shown to be sensitive to high-frequency modes [34, 29, 30] in multi-element configurations and in [35] with a single burner where high frequency transverse self-excited modes are observed. Numerical simulations [36, 31] conducted on premixed conical flames of hydrocarbons and air show that the dynamics of these flames exhibit a greater sensitivity to high frequencies when they are enriched with hydrogen. Flame transfer functions show that the cut-off frequencies are pushed towards high frequencies, and that this leads to a greater thermo-acoustic sensitivity at high frequency modes. The unstable frequency range of swirling hydrogen flames has been measured, as well as the energy source term, growth rates and limit-cycle damping terms through experiments and low-order modeling [32]. The flame response of hydrogen swirling flames has been investigated using the flame describing function framework [32] and more recently through acoustic transfer matrices measurements [33]. A more complete list of works on the dynamics of flames operating with pure hydrogen or
in mixture with a hydrocarbon can be found in the two review articles [37, 38].

In distinction with these studies, the present investigation is carried out in a laboratory scale annular combustor equipped with injection units in which hydrogen is delivered in cross-flow in a swirling stream of air. The experimental setup is briefly described in section 2. Modal structures of the chamber are examined in section 3. This section also includes estimates of the modal eigenfrequencies that will be used in a first identification of the modes coupling the unstable oscillations observed experimentally. The domain of instability is explored in section 4 with respect to the global equivalence ratio and to the thermal power and the coupling modes are reconstructed in section 5. Oscillations are then characterized through a combination of acoustic pressure measurements and OH* emission detection in the UV range. These measurements are used to determine acoustic energy source terms and calculate the corresponding growth rates (section 6). Images of individual flame motion are then examined in section 7.

## 2 Experimental setup

The experimental facility is derived from the MICCA annular combustor at EM2C Lab (Fig. 1a), that was previously used to examine combustion dynamics of hydrocarbon fuels. The combustion chamber is formed by two concentric quartz cylinders of diameters 300 mm and 400 mm providing optical access to the combustion regions. The two quartz cylinders are placed on a rigid backplane that comprises sixteen injection units. In each of these units gaseous hydrogen is delivered in cross-flow in a swirling flow of air (Fig. 1c). It is known from cross-flow injection in supersonic combustion that the penetration of hydrogen is limited by its low density. In the present arrangement, mixing is enhanced by the combination of the swirling air stream and the converging injector outlet [39]. It is worth noting that a similar arrangement was used in [40] to deliver a blend of methane and hydrogen.

The fuel injector, denoted as CFI-X1, comprises a 5.3 mm slender cone that is slightly longer than

the one used in [41, 39]. Six radial channels having a diameter of 1.5 mm are drilled in its base and convey gaseous hydrogen. The injection plane is located at a distance $h_r = 11.1$ mm with respect to the injection unit outlet (Fig. 1c). The air passes through a radial swirler and flows in the axial direction. The swirler comprises 6 non-radial channels characterized by their diameter $d_{sc} = 4$ mm, and the distance between the axis of the channels and the central axis of the swirler $R_0 = 4.6$ mm. When this swirler is placed in the injection unit it generates a rotating flow featuring a measured swirl number $S = 0.60$ and a pressure drop coefficient on the air stream $\sigma = \Delta p / [(1/2)\rho U_{b,air}^2] = 3.25$, where $U_{b,air}$ is the axial bulk velocity at the injector outlet.

Air and hydrogen are delivered at ambient temperature through two Bronkhorst EL-FLOW controllers with full-scale capacities of 2500 NL/min for air and 505 NL/min for hydrogen and with a relative accuracy of 0.5%. The mass flow rates of air and hydrogen corresponding to the different test cases are gathered in Table 1. The system comprises 8 mi-

<table><caption>Table 1: Mass flow rates of air and hydrogen and bulk velocity for the different test cases.</caption><thead><tr><th>Test case</th><th>ṁ<sub>air</sub> (g/s)</th><th>ṁ<sub>H2</sub> (g/s)</th><th>φ</th><th>U<sub>b</sub> (m/s)</th></tr></thead><tbody><tr><td>A</td><td>17.8</td><td>0.36</td><td>0.69</td><td>23.7</td></tr><tr><td>B</td><td>24.5</td><td>0.36</td><td>0.50</td><td>30.7</td></tr><tr><td>C</td><td>20.1</td><td>0.36</td><td>0.61</td><td>26.1</td></tr><tr><td>D</td><td>21.9</td><td>0.36</td><td>0.56</td><td>28.0</td></tr><tr><td>E</td><td>28.0</td><td>0.48</td><td>0.58</td><td>36.0</td></tr><tr><td>F</td><td>28.1</td><td>0.56</td><td>0.68</td><td>37.3</td></tr><tr><td>G</td><td>30.8</td><td>0.54</td><td>0.60</td><td>39.9</td></tr></tbody></table>

crophones (P1 to P8, see Fig. 1b) arranged periodically. They are side-mounted on waveguides, and the waveguide inlets (shown in red in Fig. 1b) are on the chamber backplane. The system also comprises an array of eight photomultipliers fitted with bandpass filters centered on OH* emission in combination with spatial filters such that the PMs only detect the light emitted by the flame they are facing (Figs. 1a and 1b). Each waveguide is terminated by a 25 m pipe that eliminates wave reflections. Flame
images shown in 7 are captured using a Princeton Instruments PI-MAX4 intensified camera, equipped with an Asahi Spectra 310 nm filter for OH* chemiluminescence. For measurements recorded with photomultipliers and for flame images taken with the intensified camera, a dark metal tube 0.20 m in diameter and 0.10 m long is centered on the axis of the annular chamber, inside the inner quartz tube, to stop the light emitted by flames located opposite the flames examined. This circular mask is not shown in Fig. 1a, but is drawn in Fig. 1b. It can be seen in Fig. 1 of reference [5].

## 3 Modal structures and eigen-frequencies

It is interesting to examine the acoustic modal structures of the MICCA annular combustor. This will be used to identify the modes involved in the coupling giving rise to the unstable oscillations observed when this facility is operated with pure hydrogen. As indicated in [42], the MICCA chamber is decoupled from the upstream plenum by injectors that are weakly transparent to acoustic waves. This decoupling also results from the important change in area that takes place in these injection units. It is then reasonable to assume that the coupling modes are essentially those of the chamber whose boundaries are a back-plane which can be assimilated to a rigid wall and an outlet which is open to the atmosphere. One has to specifically consider azimuthal-longitudinal modes of order m and n respectively that will be designated as mAnL. At the frequencies of operation, one may neglect any radial dependence, so that the pressure modal distribution can be cast in the form:

$$ \Psi_{mn}(x, \theta, t) = [A_m^+ e^{im\theta+i\phi_m^+} + A_m^- e^{-im\theta+i\phi_m^-}] e^{-i\omega_{mn}t} \psi_n(x) \quad (1) $$

In this expression $A_m^+ \exp(i\phi_m^+)$ and $A_m^- \exp(i\phi_m^-)$ represent the complex amplitudes of the counterclockwise (CCW) and clockwise (CW) spinning waves respectively, $\theta$ is the azimuthal coordinate considered positive in the CCW direction and $\omega$ is the angular frequency. In the previous expression, $\psi_n(x) = \cos[(2n - 1)\pi x/(2l')]$ is the axial wavefunc-

tion satisfying the boundary conditions on the chamber backplane and at its exhaust, with $l'_c = l_c + \delta_a$, $l_c$ being the chamber length and $\delta_a$, the end correction. The determination of the end correction for the MICCA annular combustor was carried out under cold flow conditions by exciting the first resonant acoustic mode with an external driver unit. A microphone was moved axially from inside the chamber to the near field adjacent to the outlet. This measurement is described in [43]. It has been found that $\delta_a \simeq 0.04$ m so that $l'_c \simeq 0.44$ m or $l'_c = 0.24$ m for the longer and shorter chambers respectively. The eigenfrequencies corresponding to the mAnL modes are given by:

$$ f_{mn} = c \left[ \left( \frac{m}{P_a} \right)^2 + \left( \frac{2n - 1}{4l'_c} \right)^2 \right]^{1/2} \quad (2) $$

where $P_a = 2\pi R$ is the mean perimeter of the system and c, the speed of sound, assumed constant in the chamber (uniform temperature). In the MICCA experiments, the perimeter is $P_a \simeq 1.1$ m. The speed of sound c is of the order of 750 m/s but the exact value is not well known because the temperature distribution inside the combustion chamber cannot be measured. This temperature is a function of the equivalence ratio but also depends on the amount of air entrained into the chamber through the open outlet and on the heat fluxes that leave the chamber due to radiation from the hot gases and transfer by convection at the walls of the chamber. It is then convenient to define a modal coefficient $\Gamma_{mn}$ which only depends on the geometry of the combustor and on the modal indices m and n and calculate the eigenfrequencies as a product of these coefficients by the mean speed of sound in the combustor:

$$ \Gamma_{mn} = \left[ \left( \frac{m}{P_a} \right)^2 + \left( \frac{2n - 1}{4l'_c} \right)^2 \right]^{1/2} \quad \text{so that} \quad f_{mn} = \Gamma_{mn}c \quad (3) $$

The coefficients $\Gamma_{mn}$ calculated for the two values of $l_c$ are gathered in Table 2 for the first four azimuthal modes of the MICCA chamber. One also finds in this table, for chamber length $l_c$, a first column corresponding to the eigenfrequencies estimated by taking a mean speed of sound c = 730 m/s which is apparently the mean speed of sound when the facility is operated at an equivalence ratio $\phi \simeq 0.57$, while
a second column pertains to $c = 770$ m/s, a value which should be used when $\phi \simeq 0.69$. The frequency estimates obtained theoretically will be used to identify the modes that couple unstable oscillations in the MICCA combustor. This will be achieved by comparing the theoretical values to the frequency of the instability $f$ determined from the power spectral density (PSD) of the pressure signals detected by the microphones plugged on the chamber backplane. These signals are sampled at a frequency $f_s = 16384$ Hz. The PSDs are calculated using Welch's method [44] applied to 63 blocks of 4096 samples using a 50% overlap and each block of samples is weighted by a Hamming window. The frequency resolution for these conditions is thus $\Delta f = 4$ Hz. As will be seen in the experimental results, the oscillation frequencies differ from the eigenfrequencies of the chamber modes, since the flame response can induce a frequency shift ([45]). However, this shift is limited to less than 5% in relative difference with respect to the eigenfrequencies associated to the mAnL modes. The complex wave amplitudes $A_m^+ \exp(i\phi_m^+)$ and $A_m^- \exp(i\phi_m^-)$ may be extracted from the eight microphone signals. The pressure field is represented as a sum of the modes as in Eq. 1. The eigenfrequencies are deduced from the signals themselves. A least squares algorithm is used because the system is overdetermined (there are more chamber microphones than unknowns). One may also deduce the instability amplitude from the travelling waves amplitudes:

$$A_m = [(A_m^+)^2 + (A_m^-)^2]^{1/2} \quad (4)$$

which corresponds to a value averaged in time and space. It is also instructive to determine the spin ratio

$$s_r = \frac{A_m^+ - A_m^-}{A_m^+ + A_m^-} \quad (5)$$

This ratio defines the modal nature. When $s_r = 0$ the mode is standing. When $s_r = +1$ or $-1$ the mode is purely spinning in the CCW or CW direction, respectively. Other values of $s_r$ correspond to mixed modes formed by combinations of counterclockwise and clockwise travelling azimuthal waves. A more practical classification consists in defining the standing modes as those corresponding to $|s_r| < 1/3$, the mixed modes being such that $1/3 < |s_r| < 2/3$ and the spinning modes as having $|s_r| > 2/3$ [2].

# 4 Operating conditions and combustion instabilities

Systematic recordings of pressure fluctuations at the flame location and volume-averaged OH* light intensity measurements for various thermal powers and equivalence ratio are used to obtain a first indication on the regimes of instability of the system. In the present study, the combustor is equipped with the 400 or 200 mm-long quartz tubes. The maximum in the power spectral densities defines the sound pressure level and frequency for unstable points of operation. The operating conditions gathered in Table 3 correspond to 7 cases. The first four (A, B, C, D) were carried out with 400 mm quartz tubes. The other recordings correspond to experiments performed with the 200 mm quartz tubes. In test E, the oscillations comprise periodic bursts. The signals E-1, E-2, E-3 correspond to the analysis of three different bursts. When MICCA-H2 is equipped with the $l_c = 400$ mm chamber, the thermal power is kept constant at 43.2 kW. The global equivalence ratio is varied between $\phi = 0.50$ and $\phi = 0.69$. The $l_c = 200$ mm geometry is investigated with higher thermal powers spanning from 57.3 to 67.2 kW. The frequencies of the various instabilities quoted in Table 3 fall between 817 and 2834 Hz. It is thus possible to look for near matches between these frequencies and those estimated in the previous section by considering the four first azimuthal modes of the system. One finds that the frequency corresponding to A is close to that calculated for the mode 1A1L (825 Hz) for $l_c = 400$ mm and a mean speed of sound of 770 m/s (last column in Table 2), which is consistent with the high equivalence ratio $\phi = 0.69$, whereas the lowest equivalence ratio used with this geometry ($\phi = 0.50$ in B) leads to an oscillation frequency of 1328 Hz which is slightly smaller than the eigenfrequency (1391 Hz) calculated for the 2A1L mode for a mean speed of sound $c = 730$ m/s. The frequencies appearing in the C and D cases (1833 Hz and 2027 Hz) are close to the eigenfrequencies (respectively 1819 Hz and 2034 Hz) calculated for the 2A2L and 3A1L modes for $l_c = 400$ mm and a mean speed of sound $c = 730$ m/s. For the short com-
<table><caption>Table 2: Calculated coefficients Γ<sub>mn</sub> and estimated eigenfrequencies of the first modes of the chamber. The eigenfrequencies are given for two different values of the mean speed of sound: c = 730 m/s and c = 770 m/s, depending of the equivalence ratio of the operating points. These frequency estimates are used to obtain a first identification of the coupling modes.</caption><thead><tr><th rowspan="2">Mode</th><th colspan="3">l<sub>c</sub> = 0.2 m</th><th colspan="3">l<sub>c</sub> = 0.4 m</th></tr><tr><th>Γ<sub>mn</sub></th><th>f<sub>mn</sub> (Hz)</th><th>f<sub>mn</sub> (Hz)</th><th>Γ<sub>mn</sub></th><th>f<sub>mn</sub> (Hz)</th><th>f<sub>mn</sub> (Hz)</th></tr></thead><tbody><tr><td>1A1L</td><td>1.380</td><td>1009</td><td>1065</td><td>1.072</td><td>783</td><td>825</td></tr><tr><td>2A1L</td><td>2.095</td><td>1530</td><td>1613</td><td>1.905</td><td>1391</td><td>1467</td></tr><tr><td>2A2L</td><td>3.615</td><td>2639</td><td>2784</td><td>2.492</td><td>1819</td><td>1919</td></tr><tr><td>3A1L</td><td>2.919</td><td>2131</td><td>2248</td><td>2.786</td><td>2034</td><td>2145</td></tr><tr><td>4A1L</td><td>3.783</td><td>2761</td><td>2913</td><td>3.680</td><td>2687</td><td>2834</td></tr></tbody></table>

bustion chamber $l_c = 200$ mm, the frequencies of the E and F oscillations (2209 Hz and 2248 Hz) are close to the eigenfrequency of the 3A1L mode (2248 Hz) calculated by assuming a mean speed of sound $c = 770$ m/s. The oscillation arising in the G test of 2834 Hz lies between the previously calculated eigenfrequencies of the 4A1L mode (2761 Hz and 2913 Hz) indicating an intermediate mean speed of sound between 730 and 770 m/s. This comparison already allows an identification of the modes involved in the couplings observed experimentally. It is therefore possible to fill the seventh column in Table 3.

The nature of the modes may be confirmed by examining the signals detected by the eight microphones. Short records of these signals are shown in Figs. 2, 3 and 4. Figure 2 gathers test cases A, B and C which were obtained for a low thermal power of around 43.2 kW. Figure 3 corresponds to test case E and shows microphone signals for three different bursts. Tests performed at the highest thermal powers (F and G) are shown in Fig. 4.

The diagrams appearing in these figures show the annular combustor viewed from above together with short records of the microphone signals. These records can be used to determine the frequency of oscillation and compare this value with estimates of the eigenfrequencies of the acoustic eigenmodes. One may then presume the modal nature and use the various signals to determine the modal type simply by

looking at the oscillation periods and amplitudes. Results of this modal identification are gathered in the last two columns in Table 3. One finds that the modes that couple the unstable oscillation feature an azimuthal mode number $m = 1$ to 4 and axial wavenumbers $n = 1$ or 2. The amplitudes of the signals gathered by the microphones seem to depend directly on the microphone position, indicating that most of these modes are of the standing type. It is also possible to use the signals to determine the location of the nodal and antinodal lines. These lines are represented respectively in blue and in red on the diagrams. Maps can be drawn showing the dependency of the frequency and the oscillation amplitude with the operating conditions, namely thermal power and equivalence ratio, for the two chamber lengths tested. Results are shown in Fig. 5.

It is interesting to note that the flames are coupled with higher order modes when the MICCA combustor is fed with pure hydrogen and that the azimuthal mode number $m$ increases as the thermal power is augmented (at a nearly constant global equivalence ratio). Indeed, one may recall that the only coupling observed when the MICCA system is fed with premixed propane and air [46] or with heptane or decane [4] or blends of these two fuels [42] is through the 1A1L mode and that the oscillation is generally of the mixed type. The dynamical behavior of the MICCA system when it is fed with hydrogen thus notably differs from that observed when it is fed with
<table><caption>Table 3: Characteristics of test cases, modal features, modal nature and signal type (C: continuous record, B:bursts).</caption><thead><tr><th>Case</th><th>l<sub>c</sub> (mm)</th><th>P<sub>th</sub> (kW)</th><th>φ</th><th>Type</th><th>f (Hz)</th><th>Mode</th><th>Nature</th></tr></thead><tbody><tr><td>A</td><td>400</td><td>43.2</td><td>0.69</td><td>C</td><td>817</td><td>1A1L</td><td>Standing</td></tr><tr><td>B</td><td>400</td><td>43.2</td><td>0.5</td><td>C</td><td>1328</td><td>2A1L</td><td>Standing</td></tr><tr><td>C</td><td>400</td><td>43.2</td><td>0.61</td><td>C</td><td>1833</td><td>2A2L</td><td>Standing</td></tr><tr><td>D</td><td>400</td><td>43.2</td><td>0.56</td><td>C</td><td>2027</td><td>3A1L</td><td>Spinning</td></tr><tr><td>E-(1)</td><td>200</td><td>57.3</td><td>0.58</td><td>B</td><td>2209</td><td>3A1L</td><td>Standing</td></tr><tr><td>E-(2)</td><td>200</td><td>57.3</td><td>0.58</td><td>B</td><td>2209</td><td>3A1L</td><td>Standing</td></tr><tr><td>E-(3)</td><td>200</td><td>57.3</td><td>0.58</td><td>B</td><td>2209</td><td>3A1L</td><td>Standing</td></tr><tr><td>F</td><td>200</td><td>67.2</td><td>0.68</td><td>C</td><td>2248</td><td>3A1L</td><td>Standing</td></tr><tr><td>G</td><td>200</td><td>65.1</td><td>0.60</td><td>C</td><td>2834</td><td>4A1L</td><td>Standing</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Figure 2: Short microphone signal records corresponding to test cases A, B and C. The nodal and antinodal lines are plotted in blue and red respectively. $\Theta_{j+}$ and $\Theta_{j+}$ designate the azimuthal ranges where the azimuthal wavefunction takes positive or negative values respectively.</figcaption></figure>

hydrocarbon fuels.

# 5 Reconstruction of the modes and determination of the modal nature

The eight pressure recordings can be used to retrieve the time evolution of the complex amplitudes $A_m^+ \exp(i\phi_m^+)$ and $A_m^- \exp(i\phi_m^-)$ presented in equation 1. Indeed, one needs, for each azimuthal order $m$, two

equations to determine the clockwise and counterclockwise complex amplitudes, and one equation to determine the complex amplitude corresponding to the zero-azimuthal-order mode. Hence, seven equations are required to solve the system corresponding to azimuthal modes up to the third order. Since 8 microphone signals are available, it is thus possible to reconstruct the 1A1L, 2A1L, 2A2L and 3A1L modes. For the 4A1L mode, since there are only 8 microphones, this method leads to an underdetermined system, for which a bypass technique is presented in appendix A. The temporal evolution of the 7 modal
<figure><img src="image_4.png" /><figcaption>Figure 3: Short microphone signal records corresponding to test cases E for three different bursts. The nodal and antinodal lines are plotted in blue and red respectively. $\Theta_{j+}$ and $\Theta_{j+}$ designate the azimuthal ranges where the azimuthal wavefunction takes positive or negative values respectively.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Short microphone signal records corresponding to test cases F and G. The nodal and antinodal lines are plotted in blue and red respectively. $\Theta_{j+}$ and $\Theta_{j+}$ designate the azimuthal ranges where the azimuthal wavefunction takes positive or negative values respectively.</figcaption></figure>

amplitudes are calculated for all the recordings and plotted in Fig. 6a for the recording F as an example. It can be seen that in this case, the amplitudes $A_3^+$ and $A_3^-$ strongly dominate all the others, and that they remain at similar levels. This confirms that the

azimuthal order is $m = 3$ for the F recording, while the latter indicates a standing mode since the CW and CCW components are of equivalent amplitudes. This is completed by a joint probability density function $P(A_3^+, A_3^-)$ plotted in Fig. 6b together with lines
<figure><img src="image_6.png" /><figcaption>Figure 5: Frequency (left) and amplitude (right) of oscillations measured in the MICCA-H2 combustor, for various thermal powers $P_{th}$ and global equivalence ratios.</figcaption></figure>

corresponding to a standing mode and the limits of CW and CCW spinning modes. Finally, the angular locations of the nodal lines can be determined using the phases of the CW and CCW complex amplitudes:

$$ \theta_N^k = \frac{1}{2m} [\phi_m^{-} - \phi_m^{+} + (2k+1)\pi], \quad \text{for } k = 0, \dots, m-1 \qquad (6) $$

The temporal evolution of the first nodal line is represented in a polar plot in Fig. 6c where the radius corresponds to the time of the recording and the angle to the angular position of the nodal line. It shows that the nodal line slightly fluctuates around its mean position, but by less than 2°.

The modal nature, i.e. spinning, standing or mixed, that was only inferred in the previous section by examining the raw pressure signals, may now be confirmed by calculating the spin ratio $s_r$ as defined in Eq. 5. The values of the spin ratio integrated in time are displayed in Fig. 7. The shorter quartz cylinders leads to the observation of standing modes, all of the third and fourth azimuthal order, whereas using the $l_c = 400$ mm cylinders brings a diversity of modes including 1A1L, 2A1L, 2A2L and 4A1L standing modes and 3A1L spinning modes.

## 6 Acoustic energy source terms and growth rates

It is first interesting to study the linearity of the pressure signals and OH* intensity signals emitted by the flame. For each of the 2A1L and 3A1L modes, the pressure and OH* intensity fluctuations are analyzed using Welch's periodogram method. The Power Spectral Density (PSD) of one pressure signal and the closest OH* intensity signal are represented in Fig. 8 with a spectral resolution of $\Delta f = 2$ Hz. For the 2A1L mode (recording B represented in Fig. 8a), the pressure spectrum features 3 peaks, corresponding to the 2A1L fundamental mode at 1328 Hz and its first two harmonics at 2656 Hz and 3984 Hz. In the OH* intensity spectrum, the same peaks are observed together with a fourth peak corresponding to the third harmonic at 5312 Hz. The difference between the pressure level of the fundamental and the first harmonic is 28 dB, which is much higher than the difference observed in the OH* spectrum (14 dB). It means that whereas the pressure signal can be considered as linear, the OH* signal features much higher harmonic content and should be treated non-linearly. The same conclusions can be drawn for the pressure spectrum of the 3A1L mode 8b, where only two peaks are visible within the [0-6000] Hz range: the 2249 Hz peak corresponding to the fundamental frequency, and the 4498 Hz pertaining to the first harmonic. Once again, the pressure level of the first harmonic is significantly lower than the fundamental level (21 dB), indicating that the pressure signal is nearly linear. Here, the harmonic content of the OH* intensity signal is of the same order (20 dB below the fundamental level) and in that particular case, both signals are nearly linear. However, the overall observations indicate that the acoustics can be treated linearly while the representation of combustion using the Flame Describing Function (FDF) framework suitably represents the dynamics of high-order azimuthal modes in the linear or weakly non-linear ranges.

The reconstructed pressure field can then be used to calculate the pressure fluctuations at the burner locations, which may then be compared to the volume-integrated OH* intensity fluctuations over each flame.
<figure><img src="image_7.png" /><figcaption>Figure 6: (a) Time series of the complex amplitude modules for the F recording. (b) Joint probability density function of the two main components of the pressure mode for the F recording. The continuous grey line represents the standing mode case, whereas the grey dashed lines are the limits for the ACW spinning mode (bottom sector) and the CW spinning mode (left sector). (c) Radial plot showing the evolution of the nodal line, calculated from the two main complex amplitude phases, for the F recording. The radius represents the time and the angular position corresponds to the true nodal line location. The time-averaged positions of the three nodal lines are plotted in blue.</figcaption></figure>

Under the strong assumption that this light intensity represents to the volume integrated heat release rate $\dot{Q}'$ and that $\dot{Q}'/\dot{Q}_0 \simeq I'_{OH*}/\bar{I}_{OH*}$, an hypothesis that was partially verified for one given operating point in [32], one may obtain an estimate of the Rayleigh source term corresponding to the j-th flame

$$ \langle S_j \rangle = \frac{\gamma - 1}{\rho_0 c^2} \int_{T,V} p'_j \dot{q}'_j dV \quad (7) $$

Assuming that the flames are compact with respect to the acoustic wavelength and that they act like point sources at $x_{fj}$ located at an angle $\theta_j$ and a distance $x_f$ from the chamber backplane, one may write $\dot{q}'_j = \dot{Q}'_j \delta(x - x_{fj})$ and the Rayleigh source term of the j-th flame becomes

$$ \langle S_j \rangle = \frac{\gamma - 1}{\gamma} \dot{Q}_0 \frac{1}{2} \text{Re} \left\{ \frac{p'(\theta_j, x_f)}{p_0} \left[ \frac{I'_{OH*}(\theta_j)}{\bar{I}_{OH*}} \right]^* \right\} \quad (8) $$

where $\rho_0 c^2$ is replaced by $\gamma p_0$. This acoustic energy source term normalized by the thermal power $\dot{Q}_0$ released by each flame is plotted in Fig. 9 as a function of the reduced pressure amplitude $\Pi = p_j/(\rho_0 U_b^2)$ at

the eight burners equipped with an OH* photomultiplier, in the case of a 3A1L standing mode (recording F). Calculations carried out along the lines of [42] indicate that this acoustic energy source term should evolve as the square of the pressure fluctuations amplitude at the j-th burner $|p_j/p_0|$, a law that is indeed retrieved with this experimental dataset. As derived in [42], the growth rate can be evaluated from the sum of the 16 time- and volume-averaged Rayleigh source terms $\langle S_j \rangle$ and from the time average volume integrated acoustic energy density of the mode $\langle E \rangle$ using :

$$ \omega_i = \frac{\langle S \rangle}{2\langle E \rangle} \quad (9) $$

This quantity represents the rate of growth induced by the Rayleigh source term. By subtracting the damping rate one obtains the effective growth rate $\omega'_i = \omega_i - \alpha$. At the limit cycle $\omega'_i = 0$ so that the growth rate is equal to the damping rate $\omega_i^{lc} = \alpha$. This method enables to compute a damping rate $\alpha = \omega_i^{lc} = 193 \text{ s}^{-1}$ for the recording F, a 3A1L standing mode with a frequency of 2248 Hz and a reduced
<figure><img src="image_8.png" /><figcaption>Figure 7: Values of the time-averaged spin ratio obtained after modal reconstruction, for various thermal powers $P_{th}$ and equivalence ratios $\phi$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Figure 8: Power Spectral Density of (left) the pressure signal recorded by microphones (a) M4 and (b) M6 and (right) the OH* intensity signal recorded using photomultipliers (a) PM4 and (b) PM8. The chosen microphones are close to a pressure antinode, and the associated PM is one of the two closest, in order to maximize the signal-to-noise ratio. The recordings correspond to (a) a 2A1L mode in recording B and (b) a 3A1L mode in recording F. A lowpass filter with a 6000 Hz cut-off frequency is first applied to avoid spectral aliasing. The spectral resolution is $\Delta f = 2$ Hz.</figcaption></figure>

$$ \text{amplitude } A/(\rho_0 U_b^2) = 1.39. $$

## 7 Observation of unsteady flame motion

In addition to the global analysis carried out in the previous sections, it is also interesting to obtain a more local information on the dynamics of the flames within an oscillation cycle, by making use of phase-averaged images formed from data originating from the ICCD camera equipped with a 310 nm band-pass filter selecting the OH* emission signal from the flame. The camera is installed in front of the injector 16, which is located between microphones M6 and M7, near microphone M7, as shown in Fig. 1b. In the 1A1L case (recording A), the nodal line is located close to the diameter connecting injectors 5 and 13. The injector 16 is thus located next to the antin-

<figure><img src="image_10.png" /><figcaption>Figure 9: Rayleigh acoustic source term for 8 burners as a function of the dimensionless pressure fluctuations amplitude. The recording (F) corresponds to a 3A1L standing mode. The dashed line corresponds to a theoretical estimate calculated along the lines of [42] $S_j = C\Pi^2\dot{Q}_0$ with $C = 6 \times 10^{-3}$.</figcaption></figure>
odal line, characterized by high pressure amplitudes. Phase-averaged OH* chemiluminescence images presented in Fig. 10a indicate that the flame periodically attaches to the injector, and detaches. When the flame is lifted above the burner, it widens and forms a pocket of burnt gases (first phase image in Fig. 10a). This structure is then convected downstream as can be seen in the phase images 2, 3 and 4 in the wake of the flame. Using these three images, one obtains a convection velocity equal to 25 m/s. The images are line-of-sight integrated images, so a slight asymmetry can be seen in the flame motion. It is however too weak to determine whether it is an effect of the azimuthal component of the acoustic velocity field. It is then interesting to analyze the link between the strong axial motion and the axial velocity fluctuations $u'$. These fluctuations are correlated with the OH* signal emitted from the flame, a relationship that can be measured in a single-sector configurations present at lab EM2C called SICCA-H2 [32]. The single flame is acoustically forced at the frequency of the 1A1L mode observed in MICCA (817 Hz) and the response of OH* is measured by a photomultiplier equipped with a 310 nm bandpass filter. The axial velocity fluctuation $u'$ are measured under cold-flow conditions using a laser Doppler anemometer, a method already described in [32]. The time evolution of $u'$ over a cycle is plotted in Fig. 10a. It indicates that the attachment of the flame occurs when $u'$ is negative, and that the flame remains attached to the injector until $u'$ is positive again. At this stage, the flame lifts-off coherently with $u'$ reaching its maximal value. When the flame is located near a nodal line (as in the 4A1L case) its dynamics notably differs from that observed previously (near the antinodal line). It exhibits a lateral motion that is apparent in Fig. 10b. The axial motion described previously is still present, but the lateral motion dominates. This can be explained by the fact that next to a pressure node, the azimuthal velocity reaches its maximum amplitude. The azimuthal velocity can be deduced from the pressure fluctuation signal reconstructed at the burner 16 location, and the linearized momentum equation projected in the transverse direction:

$$ \frac{\partial u_{\theta}}{\partial t} = -\frac{1}{\rho_0} \frac{1}{r} \frac{\partial p}{\partial \theta} \qquad (10) $$

The comparison of the OH* images with the time evolution of $u_{\theta}$ indicates that the lateral motion of the flame is correlated with the azimuthal component of the acoustic velocity field. When this velocity component is negative the flame is displaced leftwards. When this velocity becomes positive the flame moves to the right. It is important to note that when the azimuthal order increases, the angular sectors between two nodal lines become narrower, making it more difficult for a given flame to be exactly on a pressure node or antinode. At high-order modes, it will thus be necessary to take into account these lateral motions.

## 8 Conclusion

Experiments carried out in an annular combustor fed with hydrogen and air through a set of swirling injection units indicate that combustion instabilities appear in a wide range of operating conditions. It is found in particular that the oscillation frequency and azimuthal mode numbers increase with the thermal power. The modes observed in these experiments are generally of the standing type as inferred from the signals detected by microphones plugged on the chamber backplane. This is confirmed by a reconstruction of the modes using the eight pressure recordings, which allows to determine the time evolution of the spin ratio and the nodal line locations. From these observations one may infer that hydrogen-fed annular combustors may raise dynamical issues and be prone to combustion instability involving high-order azimuthal modes. This is consistent with some recent single-burner studies indicating that hydrogen flames are sensitive to disturbances over a broad range of frequencies. The study of the power spectral density of both pressure and OH* intensity signals indicate that the former can be considered as linear whereas the latter has a non-linear behavior. The observation of the motion through phase average images indicates that when the flame is located close to an antinodal line, its movement is mainly axial and consists in a cyclic attachment to the injector lip and detachment from that lip. As shown by measuring the response of OH* intensity fluctuations to
<figure><img src="image_11.png" /><figcaption>Figure 10: Top: Phase-averaged OH* images of the flame at burner 16 taken during recordings (a) A corresponding to a 1A1L mode and (b) G pertaining to a 4A1L mode. The phase reference is taken when the pressure recorded with microphone M7 reaches zero with a falling slope. Bottom: the left axis shows pressure fluctuations reconstructed at the flame location and volume-integrated OH* intensity fluctuations, whereas the right axis is associated with relative axial velocity fluctuations (a) and azimuthal acoustic velocity fluctuations in (b).</figcaption></figure>

axial velocity fluctuations, this axial motion is driven by the axial velocity fluctuations, whose amplitude is higher near the pressure antinode. As the flame gets closer to the nodal line, the intensity of this axial motion diminishes whereas a strong lateral motion appears due to the high transverse velocities induced by the coupling mode. This study of hydrogen-fueled annular combustor is probably the first to point out that in such configurations, oscillations occur on a much broader frequency range than those found with hydrocarbon fuels and that in general, the frequency and azimuthal modal number of oscillations increase with the thermal power of the system.

## Acknowledgments

This work was partially supported by the FlyHy project of the Agence Nationale de la Recherche (ANR-21-CE05-0008). For open access purposes, a

CC-BY license has been applied by the authors to this document and will be applied to any subsequent version up to the author's manuscript accepted for publication resulting from this submission.

## A Reconstruction of the 4A1L mode

The identification of the 4A1L mode raises an indeterminacy problem. Since only 8 microphones are available to resolve a fourth-azimuthal order system that now counts 9 unknown parameters the system of equations defining the modal amplitudes is underdetermined and it is not possible to obtain $A_4^+$ and $A_4^-$ and define the nature of the coupling mode (spinning, standing or mixed). It is also not possible to locate the nodal lines. Thus, another reconstruction method is designed, which relies on the OH* light intensity
data. One considers again that $\dot{Q}'/\dot{Q}_0 \simeq I'_{OH^*}/\bar{I}_{OH^*}$ and one assumes that the signals recorded by the photomultipliers are representative of waves travelling in the annulus. Now, a close examination of the microphone signals indicates that M1 and M6 are similar and this is the case also for M5 and M7, and for M4 and M8 (Fig. 4b). One may then consider that the diameter connecting M2 and M6 is an axis of symmetry and as a consequence that this diameter is also an axis of symmetry for the signals detected by the 8 photomultipliers. It is thus possible to assign signals to virtual photomultipliers that are symmetrically located with respect to the real photomultipliers. One may then obtain 16 equations to resolve the 9-unknown system. This technique leads to the determination of 9 complex amplitudes corresponding to the OH* fluctuations mode. In essence, this provides a decomposition of $I'_{OH^*}/\bar{I}_{OH^*}$ or of the relative heat release rate fluctuations in terms of two waves travelling in the positive and negative directions.

$$ \frac{\dot{Q}'}{\dot{Q}_0} = [B_m^+ e^{(im\theta+i\beta_m^+)} + B_m^- e^{(-im\theta+i\beta_m^-)}]e^{-i\omega t} \quad (11) $$

The coefficients $A_{mn}^{\pm}$ may be deduced from $B_m^{\pm}$ by making use of the describing function $F_p$ which links the relative heat release rate fluctuation to the dimensionless pressure fluctuations:

$$ F_p = \dot{Q}'/\dot{Q}_0/[p'/(\rho_0 U_b^2)] = G_p \exp(i\phi_p) \quad (12) $$

Now, the reduced pressure amplitude $\Pi$ at the eight microphones is nearly the same so that the gain and phase of this describing function also have constant values. Combining equations (1), (11) and (12) one obtains

$$ A_{mn}^{\pm} = (\rho_0 U_b^2 / G_p) B_m^{\pm} \quad \text{and} \quad \phi^{\pm} = \beta_m^{\pm} - \phi_p \quad (13) $$

so that the spin ratio may be calculated from $s_r = (B_m^+ - B_m^-)/(B_m^+ + B_m^-)$. One also notes that the nodal lines locations may be obtained directly from $\beta_m^{\pm}$.

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