# Analysis of Thermo-Acoustic Instabilities Induced by Hydrogen Swirling Flames

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

► To cite this version:

Nicolas Vaysse, Daniel Durox, Ronan Vicquelin, Sébastien Candel, Antoine Renaud. Analysis of Thermo-Acoustic Instabilities Induced by Hydrogen Swirling Flames. Journal of Engineering for Gas Turbines and Power, 2024, 147 (1), 10.1115/1.4066242. [hal-04955008](https://HAL Id: hal-04955008)

https://centralesupelec.hal.science/hal-04955008v1

Submitted on 6 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Analysis of Thermo-Acoustic Instabilities Induced by Hydrogen Swirling Flames

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

A considerable research effort has concerned combustion dynamics of systems fed with hydrocarbon fuels. The case of pure hydrogen/air flames deserves to be specifically considered because hydrogen is highly reactive, has a tendency to develop thermo-diffusive instabilities, is envisaged in many future applications, most notably in gas turbines and is less well documented. Thermo-acoustic instabilities of pure hydrogen flames are here investigated in a configuration where hydrogen is injected in-crossflow in a swirling stream of air. The study is focused on operating conditions that lead to oscillatory regimes. Using Abel-transformed phase-averaged images of OH* emission and visible light emission in burnt gases, it is shown that the OH* signal evolves approximately in phase with the heat release rate. This signal is then used to determine the local Rayleigh source term that feeds acoustic energy in the oscillation. The contributions of this term are examined using a space-time analysis based on an integration of the source term in the transverse direction. This procedure allows a detailed analysis of the processes that contribute to the acoustic energy in the system, showing, in particular, that a strong positive addition of acoustic energy results from a roll-up of the flame tip and from the quick cyclic back propagation of the flame to the injector tip. A global integration of the Rayleigh source term is then used together with a volume-integrated

acoustic energy to estimate the growth rate associated to these driving processes and estimate the damping rate. A special experimental method is then exploited to determine the effective growth rate of the instability. The system allowing a sweep in frequency, self-sustained instabilities obtained at different frequencies are used to extract the specific instability frequency band of the burner. Finally, the flame is externally forced in order to measure its flame describing function.

**Keywords:** Hydrogen combustion, Thermo-acoustic instabilities, Flame Describing Functions, Growth rate, Acoustic damping

## Nomenclature

### Physical parameters

$P_{th}$ : Flame thermal power [kW]

$\phi$ : Global equivalence ratio [-]

$\dot{Q}$ : Heat release rate [kW]

$u$ : acoustic velocity [m.s$^{-1}$]

$q_v$ : volumetric flow rate [m$^3$.s$^{-1}$]

$\omega_i$ : Growth rate [s$^{-1}$]

$\omega'_i$ : Effective growth rate [s$^{-1}$]

$\alpha$ : Damping rate [s$^{-1}$]

### Geometrical parameters

$l_c$ : Combustor length [mm]

$S$ : Swirl number [-]

$h_r$ : Recess length between injection holes and
injector outlet [mm]

Describing functions

$F_I$: Injector Describing Function (IDF)

$F_{IF}^o$: Total Flame Describing Function (TDF)

$F_F^o$: Flame Describing Function (FDF)

$G$: Gain of Describing Function [-]

$\phi$: Phase of Describing Function [rad]

Superscripts and subscripts

b : bulk value

# 1 Introduction

Current industrial objectives of replacing traditional hydrocarbon fuels by hydrogen-enriched blends or pure hydrogen require novel injection schemes and combustor architectures, that will enable a stable combustion and ensure low NOx emissions. The present work is aimed at examining combustion dynamics issues leading to instabilities of an injection unit that relies on cross-flow injection of hydrogen in a swirled airflow. This configuration is investigated for its capacity to induce a sufficient level of premixing and to establish stable flames anchored at a distance from the injection unit. It was found in a previous study that this system features unstable ranges of operation depending on the global equivalence ratio and thermal power values [1]. The present investigation is aimed at (1) characterizing these unstable regimes in terms of flame shape, pressure, heat release rate fluctuations and show how the flame dynamics contributes to the acoustic energy source term using a space-time analysis of this term, (2) defining for one operating point, a flame describing function by two methods, an indirect method to determine the specific instability frequency band of the burner, and a more classical direct method, (3) obtaining an estimate of the damping rate and through a special technique determining the effective growth rate and seeing if it can be retrieved from the measured flame describing function.

At this point, it is natural to review the literature dealing with the combustion dynamics of flames formed by hydrogen added to other fuels, but also by considering the cases of injection of pure hydrogen. Most of these studies indicate that the addition of hydrogen to gaseous methane that is considered for gas turbines and the use of hydrogen as a fuel in aeroengines enhance dynamical problems. For example, the increase of burning velocity and the broader flammability limits corresponding to hydrogen enriched natural gas tend to promote flashback which may occur for example at ignition as a result of the overpressure induced by the rapid reaction of fresh mixture in the chamber [2, 3, 4, 5, 6]. The change in fuel composition with the addition of hydrogen with higher exothermic power may also enhance the flame sensitivity to incoming disturbances and modify the thermo-acoustic coupling between the flame and the combustion chamber. A first mechanism is the reduction of flame size due to the higher burning velocity, and greater resistance to strain rate [7]. Hydrogen enrichment may also induce transitions from “V” to “M” flame shapes observed at lower equivalence ratios [8], with more compact flames [9, 10, 11]. Flames enriched with hydrogen are also characterized by a reduced time delay between upstream flow perturbations and the burning region, a feature that is reflected in the Flame Transfer Functions (FTF) which exhibit a reduction in the phase slope [12, 13, 11, 14]. The change in thermo-acoustic characteristics is also a consequence of flame properties such as burning velocity and adiabatic temperature [15, 16]. Gain reductions in the FTF have been reported in both experimental and numerical swirled configurations when the hydrogen content is augmented [17, 18, 19, 14, 11]. This might eventually reduce the sensitivity of such flames to disturbances. The possibility of using hydrogen pilot injection in active control of instabilities was investigated on this basis [20, 21, 18, 19]. For V-flame burners, generally a constructive coupling of hydrogen combustion with vortex shedding was observed both experimentally and numerically [22, 14, 11, 23]. Experiments have further underlined the complex dependency of hydrogen content on the control itself, since an increase in hydrogen concentration could lead to an in-
crease in heat release rate fluctuations or to enhanced acoustic damping [24, 25].

The previous studies were mainly carried out in single injector configurations. Multiple injection systems 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 [24], 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 [26]. Both experiments on can-annular configurations [27] and micromix simulations [28] have shown an enhancement of the thermo-acoustic coupling in the high frequency range when hydrogen is added. In [28] a low-order model is used to predict growth rate in the low-frequency range implying that combustors using hydrogen micromix injection units are less unstable than lean premixed prevaporized kerosene configurations.

The combustion dynamics of systems using pure hydrogen injection is less well documented. There are however indications that multi-slit configurations designed to avoid flashback may also give rise to high-frequency transverse modes [29, 30] and to low frequency axial modes.

This brief literature review indicates that it is worth examining pure hydrogen injection, specifically considering combustion instabilities as a function of injection unit parameters and studying the nature of the coupling observed between such flames and acoustic modes of the system. This is done here by focusing on the special case of cross-flow injection of hydrogen in a swirling stream of air. The experimental setup is briefly described in section 2. The domain of instability is explored with respect to the global equivalence ratio and to the thermal power. Self-sustained oscillations are then characterized for one operating point through optical diagnostics and acoustic pressure measurements. The growth rates and damping terms are evaluated using limit-cycle and instability growth recordings. The effect of the chamber length

on the flame stability is then investigated. An optical method is then used to determine the Flame Describing Function and see if it can be used to retrieve the growth rate determined experimentally.

## 2 Experimental setup

The experimental facility is derived from the SICCA burner at EM2C Lab, previously used for premixed and liquid spray hydrocarbon combustion. In this system shown in Fig. 1, gaseous hydrogen is injected in cross-flow in a swirling air flow. Cross-flow injection is commonly used in supersonic combustion engines. The penetration of hydrogen is however limited by its low density. In the present arrangement, mixing is enhanced by the swirling air stream [31]. A similar scheme was used for an injection of a blend of CH₄ with H₂ [32]. The injector, denoted as CFI-X1, comprises an 5.3 mm slender cone that is slightly longer than the one used in [1, 31]. Six radial channels of diameter 1.5 mm are drilled in its base and convey gaseous hydrogen. In this work, the injection unit is set up with a recess $h_r = 11.1$ mm and a swirler, denoted as 707, made of 6 non-radial channels characterized by their holes diameter $d_{sc} = 4$ mm, and the distance between the axis of the holes and the central axis of the swirler $R_0 = 4.6$ mm. The swirler is characterized in [33] in terms of swirl number $S = 0.60$ and pressure drop coefficient $\sigma = \Delta p/[(1/2)\rho U_b^2] = 3.25$, with $U_b$ the axial bulk velocity at the injector outlet. Air and hydrogen are delivered at ambient temperature through two *Bronkhorst* EL-FLOW meters with full-scale capacities of 209 NL/min for air and 30 NL/min for hydrogen and with a relative accuracy of 0.5%. The mass flow rates of air and hydrogen are $\dot{m}_{air} = 1.39 \text{ g s}^{-1}$ and $\dot{m}_{H_2} = 22.5 \text{ mg s}^{-1}$, leading to a thermal power $P_{th} = 2.7 \text{ kW}$ and a global equivalence ratio $\phi = 0.55$. These values are varied in section 3. The combustion chamber comprises a 150 mm long quartz tube open to the atmosphere with a 69 mm inner diameter. The system is not adiabatic: there is significant heat transfer at the level of the cylindrical combustion chamber which is made of quartz. To have reproducible operating conditions, the metal part of the injector is cooled from
the outside and heat transfer takes place with the metal plate at the bottom of the chamber. To allow pressure measurements in the vicinity of the chamber backplane a 15 mm high metal piece is placed under the quartz tube, as shown in Fig. 1b. A waveguide is plugged on this piece and is equipped with a microphone. The waveguide is terminated by a 25 m pipe that eliminates wave reflections. This metal piece is removed for flame visualisations and for velocity measurements close to the burner outlet. To obtain the same chamber length, a metal cylinder with the same height of 15 mm is placed at the end of the quartz tube. Several quartz tube lengths are available, so that this dimension can be varied from 55 mm to 430 mm.

Flame images are captured using a Princeton Instruments PI-MAX4 intensified camera, equipped with an Asahi Spectra 310 nm filter for OH* chemiluminescence or a glass plate to cut UV light originating from OH* and only record visible light emission from hot water vapor or excited hydrogen peroxide [34, 35]. The sensor of this camera cuts the light over 700 nm.

For velocity measurements, in cold conditions, close to the burner outlet, a LDA system is used. The air is then seeded with fine oil droplets.

## 3 Effect of operating conditions on flame stability

Systematic recordings of pressure fluctuations at the flame location and spatially-averaged OH* measurements for various thermal powers and equivalence ratio can be used to draw a stability map of the system equipped with the CFI-X1 injector fed with pure hydrogen. The burner is equipped with the 150 mm-long quartz tube placed on the annular bottom piece that supports the microphone. The total chamber length is thus $l_c = 165$ mm. The power spectral density of the pressure signal is determined using Welch's method. The maximum in this spectrum defines the sound pressure level and frequency for unsta-

<figure><img src="image_2.png" /><figcaption>Figure 1: Experimental setup: full burner for (a) flame visualization, (b) pressure measurements, and (c) details of CFI-X1. The height $h_r$ is the recess between the injection holes and the chamber backplane.</figcaption></figure>
ble points of operation, with a frequency resolution of $\Delta f = 8$ Hz. Figure 2 includes two maps, the first one showing the peak sound pressure level ($SPL_{\Delta f}$) of the pressure oscillations, and the second one displaying the frequency of the thermo-acoustic oscillations. The pressure RMS value at the backplane of the chamber reaches 1450 Pa in the strongest instability conditions, corresponding to 137 dB in this configuration.

The black line in the frequency map delimits the unstable region based on two criteria: an operating point is considered as unstable if the root-mean-square value of pressure fluctuations is greater than twice its value in a stable baseline case $p_{RMS}^{bl} = 78$ Pa. Additionally, the maximum sound pressure level in the spectrum must exceed the baseline value $SPL_{\Delta f} = 95$ dB by more than $\Delta SPL = 20$ dB. These threshold values are determined from experiments that indicate a steep change in noise level when the system evolves from a stable operation to an unstable oscillatory regime. The SPL peak threshold value has only little influence on the following results. It only defines a threshold for the frequency map by avoiding computing frequencies for stable points. In Fig. 2b, one can see that the unstable range is composed of two connected regions corresponding to relatively high equivalence ratios $\phi > 0.4$ and moderate thermal power levels $P_{th} > 1.5$ kW. The unstable points are characterized by intermediate values of the bulk velocity $U_b \in [17; 35]$ m.s$^{-1}$ (see Fig. 2a). Outside of this range, the combustion is much more stable. Below $20$ m.s$^{-1}$, the flame edge is close to the injector tip and the flame adopts a nearly cylindrical shape, whereas for $U_b > 35$ m.s$^{-1}$ they are stabilized at a distance from the injector (lifted flames. See [1] to have more details on the flame morphology related to the operating points). Figure 2b indicates that the frequency of these self-sustained oscillations lies between 770 Hz and 910 Hz. This is consistent with a coupling by the first longitudinal mode of the combustion chamber :

$$f = \frac{c}{4(l_c + \delta_c)} \quad (1)$$

where $\delta_c = 0.4d_c$ is the open-end correction for a cylindrical tube outlet, $l_c$ being the chamber length and $c$ the mean speed of sound in the chamber. At a

<figure><img src="image_3.png" /><figcaption>(a) Sound Pressure Level SPL$_{\Delta f}$</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) Frequency f</figcaption></figure>

Figure 2: Stability maps for a chamber length $l_c = 165$ mm. The pressure signals are recorded at a total number of 28 points defining a structured grid ("+" symbols), and then interpolated to a 500x500 grid using Akima interpolation method. (a) maximum Sound Pressure Level in the spectrum and isolines of bulk velocity (b) frequency of the SPL peak with 115 dB-isoline. The stable domain is colored in gray.
thermal power $P_{th} = 2.7$ kW and global equivalence ratio $\phi = 0.55$, the frequency is equal to 848 Hz. Using (1) and without taking into account the possible slight frequency shift linked to the flame, one may determine a mean speed of sound $c = 653 \text{ m.s}^{-1}$ corresponding to a mean temperature $T = 1060 \text{ K}$.

# 4 Investigation of a self-sustained oscillation

An example of flame dynamics for an unstable case is analyzed in this section for a 150 mm-long tube. The thermal power is $P_{th} = 2.7$ kW and the global equivalence ratio is $\phi = 0.55$. Self-sustained oscillations occurring at a frequency of $890 \pm 4$ Hz correspond to a quarter-wave longitudinal mode of the combustion chamber. This frequency is slightly higher here than the frequency 848 Hz derived from the map in Fig. 2b because of the removal of the additional 15 mm length brought by the chamber microphone port. Phase-averaged OH* emission images of this instability are shown in Fig. 3. The phase averaging is done by triggering the camera with a predefined set of time delays corresponding to the 12 phase shifts, starting from the moment where the pressure oscillation measured by the plenum microphone (MP in Fig. 1a) crosses zero with a falling slope, corresponding to $\theta = 0^\circ$ in Fig. 3. For each phase, 30 images (100 µs exposure time) are averaged. These phase-averaged images reveal a longitudinal motion of the flame characterized by a cyclic detachment and rapid back propagation towards the burner outlet. A toroidal vortex is formed at the top of the flame. The vortex in which the flame is rolled up is convected upwards together with the stream of fresh reactants.

Another view of this thermo-acoustic instability can be obtained by replacing the OH* filter with a glass plate that cuts off UV radiation. It should be noted that the intensified camera sensor filters light beyond 700 nm. Since water vapor radiates in near infrared and infrared regions, the light emission, mainly dark blue, is attributed to burnt gases radiating in the visible range [35] and this filter gives access

<figure><img src="image_5.png" /><figcaption>Figure 3: Phase-averaged OH* chemiluminescence images of the unstable flame at $P_{th} = 2.7$ kW and $\phi = 0.55$, obtained with the 150 mm-long tube. The phase angle is denoted by $\theta$. Injector parts are represented in white, and the invisible region in black. The images are obtained after an Abel deconvolution.</figcaption></figure>

to the burnt gas location in the visible domain. The corresponding phase-averaged images are presented in Fig. 4, where the contour of OH* images obtained with the Otsu thresholding method has been superimposed to the visible blue emission.

At first, burnt gases radiating in the visible range are located inside the reactive front materialized by OH* emission and are always in contact with the burner outlet. Burnt gases might also be present in the outer recirculation zone, close to the quartz wall, but the signal level recorded in Fig. 4 in that zone remains low probably because of the low ambient temperature in that region. Although the flame front, visualized by OH*, detaches from the injector between $\theta = 330^\circ$ and $\theta = 60^\circ$ (Fig. 3), the burnt gas region radiating visible light remains attached during the whole period (Fig. 4).

Second, the intensity of the visible emission is fairly homogeneous over the flame length, except for regions that are in the flame front vicinity where the signal is slightly stronger. This could be explained either by the presence of visible burnt gases directly next to the flame front, or by molecular OH radicals
<figure><img src="image_6.png" /></figure>

Figure 4: Contours of the OH* emission superimposed to the phase averaged-visible emission images of the unstable flame at $P_{th} = 2.7$ kW and $\phi = 0.55$, obtained with the 150 mm-long tube. The phase angle is denoted by $\theta$. Injector parts are represented in gray, and the invisible region in black. The images are obtained after an Abel deconvolution.

involved in the suspected reactions for the blue continuum, as proposed by Fiala and Sattelmayer [35]. In this work, the authors show that the OH* and blue emissions are located in the same zones, and that the role of OH radicals in both mechanisms (de-excitation of OH* and hydrogen peroxide formation) can explain this co-location. They also suggest that both radiations can be used to describe the dynamic behavior. In the present study, the advection of hot burnt gases downstream the flame front leads to a wider blue emission zone than the OH* zone and one can examine more accurately the dynamics of the two emissions.

The phase-averaged images can be used to compare the time evolution of the spatially-averaged OH* intensity fluctuations with those of the visible light intensity. Figure 5 shows the temporal trace of those two quantities, showing that they are almost in phase. A Cross Power Spectral Density (CPSD) computation confirms that the phase shift between these two emissions is small ($0.17\pi$). On the other hand, the

<figure><img src="image_7.png" /></figure>

Figure 5: Comparison of the relative fluctuations of OH* and visible light intensity, both computed using a radius-weighted spatial averaging of the phase-averaged images. The gain and phase shift between the two signals are respectively 0.52 and $0.17\pi$. Traces of the pressure fluctuations in the plenum (MP, black) and the chamber (MPC, red) are plotted in dashed lines.

fluctuations of OH* intensity are twice as large as those of the visible emission. Since visible radiation is associated with the production of hot gases and the OH* emission appears to oscillate with a very small phase shift with visible radiation, one can assume that the OH* signal should be, approximately, in phase with the variation of the heat release rate (HRR). This hypothesis will be generally validated in section 6.

In Fig. 4, the visible emission clearly shows the trace of the toroidal vortex shedding due to the instability as already seen in [13, 14, 22, 23]. The phase-averaged images allow to compute the phase velocity of these vorticity waves : at $\theta = 300^\circ$, one can clearly see both vortices, separated by 18.1 mm, which corresponds to a phase velocity $V_\phi = 13.4 \text{ m s}^{-1} = 0.47 U_b$, i.e. half the bulk velocity $U_b = 28.4 \text{ m s}^{-1}$.

**Space-time analysis of the Rayleigh source term** It is next interesting to use the phase-averaged images in a space-time analysis of the Rayleigh source term. This term appears in the acoustic energy balance equation. If its sign is positive acoustic energy is being added to the oscillation by the unsteady combustion process. If its sign is
negative, energy is being extracted from the oscillation.

Under some fairly general assumptions, the acoustic energy balance takes the form:

$$ \frac{\partial \mathcal{E}}{\partial t} + \nabla \cdot \mathcal{F}_a = \mathcal{S} - \mathcal{D} \quad (2) $$

where $\mathcal{E}$, $\mathcal{F}_a$, $\mathcal{S}$ and $\mathcal{D}$ are respectively, the acoustic energy density, acoustic energy flux, Rayleigh source term and volumetric damping term representing dissipation processes:

$$ \mathcal{E} = \frac{1}{2} \rho_0 c^2 + \frac{1}{2} \rho_0 v'^2 \quad (3) $$

$$ \mathcal{F}_a = p'v' \quad (4) $$

$$ \mathcal{S} = \frac{\gamma - 1}{\rho_0 c^2} p' \dot{q}' \quad (5) $$

The previous expressions correspond to instantaneous quantities. The following development is focused on the source term $\mathcal{S}$.

From what was seen in the previous section, it is possible to assume that the OH\* emission is a tracer of the heat release rate, at least that the two quantities evolve in phase. One may then deduce the spatial distribution of this last quantity at different phases during the cycle of oscillation. This does not imply that the relative oscillation of the OH\* signal is equivalent to the relative oscillation of the heat release rate (HRR), but this does not call into question the analysis of the Rayleigh source term, since the important element is the phase between the pressure and the HRR to know if the contribution is additive or subtractive. In the problem at hand the period averages and time average of light intensity are axisymmetric because the mean flow and its modulation by the longitudinal pressure mode are rotationally symmetric. We now consider a cycle of oscillation with a period $T$ and $N$ instants during this cycle, $t_n = nT/N$, where the light emission distribution is determined by taking period averages. Let $I(r, z, t_n)$ designate the period average light distribution at $t_n$. The mean light intensity distribution may be obtained by summing the period average images over all instants during the cycle

$$ \bar{I}(r, z) = \frac{1}{N} \sum_{n=0}^{N-1} I(r, z, t_n) \quad (6) $$

The period average fluctuation distribution may then be obtained by subtracting the mean distribution from the period average distributions corresponding to the various instants $t_n$:

$$ \tilde{I}(r, z, t_n) = I(r, z, t_n) - \bar{I}(r, z) \quad (7) $$

It is also useful to define the total light intensity level

$$ I_0 = \int \bar{I}(r, z) 2\pi r dr dz \quad (8) $$

Using the assumption concerning the relation between volumetric heat release rate fluctuations and light intensity fluctuations, one may write:

$$ \frac{\tilde{q}(r, z, t_n)}{\dot{Q}_0} \approx \frac{\tilde{I}(r, z, t_n)}{I_0} \quad (9) $$

For the space-time analysis the key idea is to keep the axial coordinate and the time instants $t_n$. This is achieved by integrating the period average distributions with respect to the radial coordinate. It is also convenient to sum the light intensity in thin axial slices $\delta z$. One obtains in this way:

$$ \frac{\langle I(z, t_n) \rangle}{I_0} = \int_z^{z+\delta z} \int_0^R \frac{\tilde{I}(r, z, t_n)}{I_0} 2\pi r dr dz \quad (10) $$

Since the pressure fluctuation only depends on $z$ and $t$ and remembering that $\rho_0 c^2 = \gamma p_0$ where $p_0$ is the ambient pressure, one may express the Rayleigh source term integrated in the radial direction and in a slice $\delta z$ as

$$ \frac{\gamma - 1}{\rho_0 c^2} \int_0^R p' \dot{q}' 2\pi r dr = \dot{Q}_0 \frac{\gamma - 1}{\gamma} \frac{\tilde{p}(z, t_n) \langle I(z, t_n) \rangle \delta z}{p_0 I_0} \quad (11) $$

Using the previous expression, one can then plot a Rayleigh index based on the OH\* light emission and on the pressure recorded at the chamber backplane (Fig. 6). This index defines the source term integrated over a cylindrical volume of radius $R$ and thickness $\delta z$. Constructive coupling between heat release rate and pressure occurs in the vortex region (between 13 and 22 mm) and in the vicinity of the flame anchorage, close to $z=0$ mm, during the flame back propagation step that follows. During the short duration where detachment of the flame takes place, the Rayleigh index becomes negative and tends to damp the instability. The zone of positive Rayleigh index is then rapidly shifted downwards, and the flame lift-
<figure><img src="image_8.png" /><figcaption>Figure 6: Temporal height-averaged maps of the estimated Rayleigh source term, computed using the product of OH* intensity fluctuations and pressure fluctuations. The results have been interpolated in time from the phase-averaged images. Axial variations of the fluctuations in OH* intensity are based on radius-weighted average of the Abel-inverted OH* signal at each height.</figcaption></figure>

off sequence, between $3\pi/2$ and $5\pi/2$, is characterized at the flame foot by rising OH* fluctuations coupled with negative pressure values. However, the level of negative Rayleigh index is much smaller than the level of positive Rayleigh index values. Furthermore, this negative zone sequence occurs at a time where a strong amplification is taking place further downstream at a distance of about 7 mm, corresponding to the flame top extinguishing (from $\theta = 150^\circ$ and $\theta = 270^\circ$ in Fig. 3). The motion that appears in red in Fig. 6 between $3\pi/2$ and $5\pi/2$, is slower than the flame foot back propagation (between $\pi/2$ and $\pi$) and is delayed in time with respect to it. This zone corresponds to lower OH* intensity than the mean intensity and negative pressure, which leads to a positive value of the Rayleigh index. When the Rayleigh index is integrated over the whole flame and over a complete cycle, it is found to be notably positive, corresponding to a strong energy input to the oscillation.

# 5 Growth rate and damping

The previous data gathered during limit cycle operation of the system is now used to determine characteristic parameters of this instability process. It is convenient to begin with the acoustic energy balance averaged over one period of oscillation, integrated over

the whole combustor volume $V$:

$$ \frac{d\langle E \rangle}{dt} = \langle S \rangle - \langle D \rangle - \langle F \rangle \quad (12) $$

where $\langle E \rangle$ designates the acoustic energy integrated over a period and over the volume $V$. $\langle S \rangle$ and $\langle D \rangle$ are the period average volume integrated Rayleigh source term and acoustic damping respectively and $\langle F \rangle$ represents the normal acoustic energy fluxes integrated over a period and over the control surface of the volume $V$ (see [12] section 2.6).

One recognizes in this expression a growth rate $\omega_i$ (which may be positive or negative depending on the sign of $\langle S \rangle$ and a damping rate $\alpha$:

$$ \omega_i = \frac{\langle S \rangle}{2\langle E \rangle}, \quad \alpha = \frac{\langle D \rangle + \langle F \rangle}{2\langle E \rangle} \quad (13) $$

The volume-averaged Rayleigh source term $\langle S \rangle$ is directly computed using pressure measurements at the flame location and OH* intensity fluctuations:

$$ \langle S \rangle = \dot{Q}_0 \frac{\gamma - 1}{\gamma} \frac{1}{T} \int_T \frac{p'(t)}{p_0} \frac{I'(t)}{I_0} dt \quad (14) $$

In the present case one finds $\langle S \rangle = 3 \pm 0.7$ W. The period average volume integrated acoustic energy for the quarter wave mode is written as [12]:

$$ \langle E \rangle = \frac{|p'|^2}{\rho_0 c^2} \frac{1}{4} V \quad (15) $$

where $V$ is the volume of the chamber, and in this case $\langle E \rangle = 4.6 \pm 0.5$ mJ. The corresponding growth rate is then $\omega_i = 342 \pm 31$ s$^{-1}$.

This value of the growth rate $\omega_i$ is obtained at the limit cycle, ie when $d\langle E \rangle/dt = 0$. This is so if $(\omega_i)_{lc} = \alpha$ and one deduces that $\alpha = 342 \pm 31$ s$^{-1}$. It is however interesting to evaluate the growth rate during the transient phase that leads to the limit cycle and in particular obtain an estimate of the growth rate at low oscillation amplitudes when the system operates in the linear regime. To this purpose, one may first try to determine the effective growth rate $\omega'_i$, that can be deduced by recording the pressure signal during the transient preceding the limit cycle. For this purpose, the 165 mm tube is slightly tilted with a small angle using a removable wedge, during a self-sustained instability, to create an open-
<figure><img src="image_9.png" /></figure>

Figure 7: Typical temporal trace of the pressure fluctuations measured at the flame location during the growth of the thermo-acoustic instability. The tube length is $l_c = 165$ mm and the instability frequency is 848 Hz. The orange line represents the envelope computed using a Hilbert transform. The domain of interest is located between the two green lines.

open boundary condition in the tube. This 1-mm gap damps the instability and essentially eliminates the oscillation. When the tube is put straight back to its original position, the condition switches back to closed-open and a quarter-wave acoustic mode couples with the flame: the instability starts to grow. The tube takes roughly 14 ms to retrieve its original position and the recording begins after this time delay. A typical record of this growth is visible in Fig. 7, where four phases can be observed: a first stage with the start of growth, which stops at around 2358 ms, as the tube tilts slightly to the opposite position, before falling back onto its base at around 2360 ms. Exponential growth can then be observed. Saturatio n is reached at around 2372 ms, lasting around 5 cycles, before decreasing slightly and stabilizing at a nearly constant amplitude of oscillation.

A Hilbert transform is used to obtain the envelope of the pressure signal $A$. Its logarithmic temporal derivative corresponds to the growth rate during this phase. It is also possible to deduce the phase drift of this signal with respect to the initial rate of oscil lation. This phase $\varphi(t)$ may be unwrapped and an instantaneous frequency may be deduced from this

<figure><img src="image_10.png" /></figure>

Figure 8: Time evolution of the effective growth rate $\omega'_i$ and the growth rate $\omega_i$.

<figure><img src="image_11.png" /></figure>

Figure 9: Time evolution of the instantaneous frequency $f_t$.
<figure><img src="image_12.png" /><figcaption>Figure 10: Frequency-growth rate trajectory plotted as a function of the pressure amplitude A.</figcaption></figure>

quantity

$$ \omega'_{i}(A) = \frac{1}{A} \frac{dA}{dt}, \quad \omega_{r}(A) = \frac{d\varphi}{dt} \qquad (16) $$

In general and assuming that the damping rate $\alpha$ remains constant with respect to the amplitude:

$$ \omega_{i}(A) = \omega'_{i}(A) + \alpha \qquad (17) $$

The effective growth rate vanishes at the limit cycle $\omega'_{i}(A_{lc}) = 0$ so that, as already indicated, $\omega_{i}(A_{lc}) = \alpha$. During the transient phase one has: $\omega_{i}(A) = \omega'_{i}(A) + \omega_{i}(A_{lc})$, which becomes in the linear regime: $\omega_{i}(A \approx 0) = \omega'_{i}(A \approx 0) + \omega_{i}(A_{lc})$.

The time-evolution of the growth rate $\omega_{i}$ which corresponds to changes in the amplitude of oscillation displayed in Fig. 8 indicates that $\omega_{i}$ diminishes from its linear value and tends to zero when the oscillation reaches the limit cycle. The decrease is however not monotonic. Fig. 9 shows the temporal evolution of the instantaneous frequency $f_{r} = \omega_{r}/2\pi$. The effective growth rate $\omega'_{i}$ can be plotted together with the instantaneous frequency $f_{r}$ on a trajectory as in Fig. 10. This representation clearly shows that the growth rate, initially positive, leads to an increase in amplitude together with an increase in the growth rate. This causes the amplitude to grow exponentially, until the growth rate starts to drop, goes to zero and becomes negative. During the whole growth, the instantaneous frequency decreases from 878 Hz to 842 Hz. Then, the saturation phase is characterized simultaneously by a slight decrease in amplitude and a novel increase of the growth rate back to zero.

Repeated experiments lead to consistent effective

<figure><img src="image_13.png" /><figcaption>Figure 11: Mapping of the results obtained from different recordings of instability growth: linear growth rate $\omega'_{i}$, damping $\alpha$ measured at the limit cycle and amplitude of oscillation $A_{lc}$.</figcaption></figure>

growth rate values in the linear regime $\omega'_{i}(A \approx 0) = 108 \pm 17 \text{ s}^{-1}$. For each experiment, the value of $\omega'_{i}(A \approx 0)$ is computed by averaging the part of the signal $\omega'_{i}(A)$ that remains greater than 30 s$^{-1}$, in order to filter out the very beginning of the growth, where the signal is noisy, and the transition towards the limit cycle regime, at the end of the growth. According to Eq. 17, this results in a value for the total growth rate equal to $\omega_{i}(A \approx 0) = 450 \pm 47 \text{ s}^{-1}$. In Fig. 11, each point corresponds to a couple $\{\alpha, \omega'_{i}(A)\}$, colored according to the amplitude of oscillation.

# 6 Specific instability band of the burner

It is next interesting to determine the instability band of the system by varying the chamber length. It is thus possible to sweep a frequency range and examine its influence on the thermo-acoustic stability of the operating point ($P_{th} = 2.7 \text{ kW} ; \phi = 0.55$). The volume-integrated OH* emission is recorded using the photomultiplier used to draw the maps of Fig. 2. A power spectral density of the pressure at the flame location enables to measure the peak sound pressure level SPL$_{\Delta f}$ and the corresponding frequency f as a function of the combustor length $l_{c}$. Results shown in Fig. 12 indicate the existence of three zones: small combustion chambers that lead to thermo-acoustic
<figure><img src="image_14.png" /><figcaption>Figure 12: Evolution of the SPL peak frequency as a function of the combustor length $l_c$. Colors correspond to the SPL value of the peak in the spectrum. The black dashed line indicates the frequency expected from an isothermal quarter-wave mode model taking the $l_c = 165$ mm as reference. The red dashed line is the three quarter-wave mode obtained with this model.</figcaption></figure>

instabilities coupled by the quarter-wave mode, intermediate lengths ($l_c \in [280; 400]$ mm) where the regime is stable and longer chambers, for which the expected quarter-wave mode frequency is even lower but the system becomes unstable again but with a coupling with the three quarter-wave mode, leading to frequencies of the same order of magnitude as those of the first unstable range. The frequencies for which the flame is unstable are located between 600 Hz and 1100 Hz, which is consistent with the frequency map displayed in Fig. 2b. The black dashed line in Fig. 12 shows the quarter-wave mode frequency computed from Eq. 1 assuming a constant speed of sound, equal to the one computed on the point ($P_{th} = 2.7$ kW; $\phi = 0.55$). The red dashed line represents the three quarter-wave mode, calculated with the same model. The frequencies of the self-sustained oscillations are located close to these dotted curves. It can be seen that they do not perfectly follow the quarter-wave curve as the response of the flame slightly shifts the frequency relative to the quarter-wave modes, the more so as the phase shift between the heat release rate oscillation and the pressure oscillation increases [36].

A cross power spectral density between the pressure and the OH* intensity fluctuations can be used to compute a gain and a phase shift between these

two signals. Results are gathered only for the points with a coherence value greater than 0.95. The evolution of the phase is plotted in Fig. 13 as a function of the frequency with each point colored by the SPL amplitude of the instability. It can be seen that the phase increases more or less steadily with frequency. The first unstable points appear for a phase shift of $-3\pi/8$ and the last points for a phase shift of $+\pi/4$. This phase shift band is not perfectly centered around 0, but it is close. If this were the phase shift between heat release rate and pressure, theory states that this band should be centered on zero [12, 36]. Additionally, points with similar phases can be either stable or unstable, depending on the gain of their transfer functions. From the experimental results, one can conclude that the OH* signal does not evolve perfectly in phase with the heat release rate, but that it is not far from it. With a phase shift range $[-3\pi/8; +\pi/4]$, the centering difference can be estimated to be about $\pi/16$. One may therefore assume that OH* evolves nearly in phase with the heat release rate, as suggested in section 4. The instability band, specific to the burner considered here, operating with the conditions $P_{th} = 2.7$ kW and $\phi = 0.55$, means that this type of burner, installed in a combustion chamber might trigger a thermo-acoustic instability if one of the acoustic modes in the chamber falls within this frequency band, provided of course that the damping rates are the same. This method is not equivalent to measuring a transfer function, but it does indicate the frequency band that must be avoided to prevent the occurrence of such dynamical phenomena.

# 7 Flame Describing Function measurements

The flame transfer function (FTF) links the volumetric flowrate fluctuations at the outlet of the injector and the resulting heat release rate fluctuations in the flame. It is known that this FTF depends on the incoming acoustic disturbance amplitude $A$ and takes the form of a describing function (FDF):

$$ \mathcal{F}_F(\omega, A) = \frac{\dot{Q}'(\omega, A)/\bar{Q}}{q'_v(\omega, A)/\bar{q}_v} = G_F(\omega, A)e^{i\varphi_F(\omega, A)} \quad (18) $$
<figure><img src="image_15.png" /><figcaption>Figure 13: Evolution of the phase difference between chamber pressure and OH* intensity fluctuations, as a function of the peak frequency of the cross power spectral density. Colors correspond to the SPL value of the peak in the spectrum.</figcaption></figure>

with $\omega$ the angular frequency, $\dot{Q}'$ and $\bar{Q}$ respectively the fluctuating and mean parts of the heat release rate, and $q'_v$ and $\bar{q}_v$ respectively the fluctuating and mean parts of the volumetric flowrate. The volumetric flowrate fluctuations are here derived from the fresh gases velocity fluctuations using a Laser Doppler Anemometry (LDA) system from Dantec Dynamics. The experimental setup used for this determination is shown in Fig. 14. Previous work performed on the SICCA-Spray setup [37] have shown that in cold conditions, it is possible to approach the volumetric flowrate fluctuations using the velocity fluctuations by choosing a specific measurement location for the LDA system such that:

$$q'_{v}/\bar{q}_{v} = u'/\bar{u} \qquad (19)$$

However, this method cannot be used here in hot conditions, because there is a notable difference between the velocity profiles corresponding to reactive and cold conditions. This complicates the direct measurement of FDFs of hydrogen flames. One may then use a two step method in which one first determines the injector describing function $\mathcal{F}_I$ linking the incoming acoustic velocity fluctuations in the plenum $u_a$ to the velocity fluctuations at the injector outlet $u'$. This describing function is measured under cold flow conditions, where Eq. 19 is still valid.

$$\mathcal{F}_I(\omega, A) = \frac{u'(\omega, A)/\bar{u}}{u'_a(\omega, A)/\bar{u}_p} = G_I(\omega, A)e^{i\varphi_I(\omega, A)} \quad (20)$$

<figure><img src="image_16.png" /><figcaption>Figure 14: Experimental setup for FDF measurement. The chamber length is $l_c = 300$ mm to ensure stable operation of all the tested operating points. Adapted from SICCA-Spray [37]. The LDA system is only used in cold flow conditions whereas the photomultiplier is used for OH* intensity measurement in reactive conditions.</figcaption></figure>

where $\bar{u}_p$ is the bulk velocity in the plenum. The sampling point is always located at radius $r = 3.5$ mm and $z = 2.5$ mm. The air flow is seeded with small oil droplets (mean diameter 2.6 µm) using an air nebulizer [38]. The LDA datarate obtained for this operating point is about 32,000 samples per second. The signals acquired by the LDA processor are then interpolated at a frequency of 32,768 Hz. The acoustic velocity $u'_a$ in the plenum is measured using two Brüel & Kjær type 4938 microphones (MP1 and MP3) with a relative precision of 1%. They are placed at a vertical distance $\Delta x_p = 20$ mm to each other, which allows to access the acoustic velocity in the plenum using:

$$u'_{a} = (p'_{3} - p'_{1})/(\rho_{p}i\omega\Delta x) \qquad (21)$$

with $\rho_p = 1.204 \text{ kg m}^{-3}$ and $\omega$ the angular frequency of modulation. The analytical signals $p'_1$ and $p'_3$ are deduced from the real pressure signals recorded by microphones MP1 and MP3 using Hilbert transforms. The dependency of the transfer function with the modulation frequency is obtained by exciting the driver units with a set of input voltages ranging from 1.700 Vpp to 2.900 Vpp using a sine wave generator.
<figure><img src="image_17.png" /><figcaption>Figure 15: Evolution of gain and phase of the injector describing function (Eq. 20) measured in cold conditions. The air flow rate is taken as the total fresh gases volumetric flow rate of operating conditions $P_{th} = 2.7$ kW and $\phi = 0.55$. Color axis shows the amplitude of the velocity signal measured in the plenum, denoted as $u'_{rms,pl}$, normalized by the air bulk velocity in the plenum.</figcaption></figure>

<caption>Figure 16: Evolution of gain and phase of the total injector and flame describing function (Eq. 22) measured in reactive conditions. The operating point is $P_{th} = 2.7$ kW and $\phi = 0.55$. Color axis shows the amplitude of the input acoustic velocity signal measured in the plenum, denoted as $u'_{rms,pl}$, normalized by the air bulk velocity in the plenum.</caption>

Given the large amount of data, two linear frequency sweeps with a duration of 111s each are performed. The first one ranges from 300 Hz to 800 Hz while the second one extends from 700 Hz to 1200 Hz. In hot conditions, the MP1, MP3 and PM signals are acquired by a program running under National Instruments LabVIEW software. The sampling frequency is 16,384 Hz and successive blocks of 32,768 points are recorded during the frequency sweep, resulting in 2s long blocks of acquisition. Welch's method is then used to determine the cross power spectral density between $u'_a$ and $u'$. Each 2s block is divided into 8 Hamming windows that overlap by 50%, providing the gain and phase of the transfer function for the considered frequency $f$ in each sub-block. Resulting describing functions for the injector under cold conditions are displayed in Fig. 15.

In the second step of this procedure one determines the flame describing function $\mathcal{F}_{IF}^o$ between the relative heat release rate fluctuations and the relative acoustic velocity fluctuations in the plenum calculated under hot fire conditions:

$$ \mathcal{F}_{IF}^o(\omega, A) = \frac{\dot{Q}'(\omega, A)/\dot{\bar{Q}}}{u'_a(\omega, A)/\bar{u}_p} = G_{IF}^o(\omega, A)e^{i\varphi_{IF}^o(\omega, A)} \quad (22) $$

To evaluate the relative fluctuations of heat release rate, it is assumed that in this partially premixed system the relative intensity fluctuations of OH* emission is approximately proportional to the relative heat release rate fluctuations in the flame. This is

admittedly a strong assumption because the flame is not fully premixed but it was shown in sections 4 and 6 that the OH* signal was an acceptable tracer of the flame front and that it represented to some extent the heat release rate in the flame. It was verified for phase, but it could not be verified for amplitude, but the first use of the FDF is to give the phase delay between the acoustic perturbation and the variation of the HRR. Secondly, the gain of the FDF will be useful to characterize the growth of the instability and the limit cycle as a function of the acoustic damping of the combustor. The OH* signal radiated by the flame is captured using a 308 nm band-pass filter placed on a photomultiplier. The air flow is forced identically as for the injector describing function measurements. The resulting total describing function is shown in Fig. 16.

The flame describing function (FDF) $\mathcal{F}_F^o$ can then be derived from the first two describing functions:

$$ \mathcal{F}_F^o(\omega, A) = \frac{\mathcal{F}_{IF}^o(\omega, A)}{\mathcal{F}_I(\omega, A)} = \frac{G_{IF}^o(\omega, A)}{G_I(\omega, A)} e^{i(\varphi_{IF}^o(\omega, A) - \varphi_I(\omega, A))} \quad (23) $$

The FDF extracted from this two step procedure is displayed in Fig. 17. It features a gain that takes values that are greater than one in a range of frequencies between 700 and 900 Hz. The phase varies linearly between 600 and 860 Hz with a slope of $\pi/400$ rad.Hz$^{-1}$. Using a simple time lag model, one shows that this slope is equal to $2\pi\tau$, where $\tau \approx 1.3$ ms is the time lag of the flame. This value is
close to the convective time obtained by dividing the height of the flame $l_f \approx 30$ mm by the bulk velocity $U_b = 28.4 \text{ m s}^{-1}$. The dependence on the amplitude of the incoming disturbances appears most clearly in the range of frequencies from 600 to 800 Hz where the changes in excitation voltages are reflected in variations in relative velocity fluctuations. Beyond 800 Hz, the relative velocity fluctuations at the injector outlet cannot exceed about 0.1 and it is not possible to examine the effect of amplitude on the FDF. It is possible to compute the linear growth rate $\omega_i(A \simeq 0)$ using an expression derived along the lines of [12]:

$$ \omega_i = \frac{\gamma - 1}{\gamma} \frac{\dot{Q}_0}{p_0 V} \frac{G_F}{M G \zeta} \cos(\varphi_F - \varphi_\zeta) \quad (24) $$

In this expression $\zeta$ represents the effective impedance at the flame, $G_\zeta$ and $\varphi_\zeta$ are the modulus and phase of this impedance and $M = 0.075$ is the Mach number at the flame. One has $\dot{Q}_0 = P_{th} = 2.7 \text{ kW}, p_0 = 10^5 \text{ Pa}$ and $V = 0.617 \text{ m}^3 \text{ s}^{-1}$ is the volume of the chamber. The FDF at 850 Hz features a gain $G_F \simeq 2$ and its phase is $\varphi_F \simeq 0.76\pi$. The impedance term $\zeta$ has been measured in this burner for low velocity fluctuations in [39]: for $u'/\bar{u} = 0.1$, one obtains $G_\zeta \simeq 0.49$. The impedance phase at the injector is controlled by the jet submitted to pressure disturbances at the outlet and considering that the velocity is augmented when the pressure is diminished in a quasi-steady fashion $p' \simeq -\rho\bar{u}u'$ so that $\varphi_\zeta = \pi$. This value is also retrieved when using the phase between pressure and OH* intensity measured in section 6: $\varphi_\zeta = \varphi_F - \varphi_p \simeq 0.76\pi - (-0.24\pi) \simeq \pi$. One obtains the growth rate in the low amplitude regime $\omega_i = 495 \text{ s}^{-1}$, which is of the same order as what has been determined in section 5. This indicates that the FDF can be used to estimate growth rates, a possibility that can be exploited in the prediction of combustion instabilities.

# 8 Conclusion

Thermo-acoustic instabilities are examined in a system where pure hydrogen is injected in cross-flow in a swirling stream of air, in a situation that shares some common features with that found in the “mi-

<figure><img src="image_18.png" /><figcaption>Figure 17: Evolution of gain and phase of the flame describing function (FDF, Eq. 23) measured optically with operating conditions $P_{th} = 2.7 \text{ kW}$ and $\phi = 0.55$. Color axis shows the amplitude of the velocity signal measured at the outlet, denoted as $u'_{c,r}$, normalized by its mean value at each given frequency.</figcaption></figure>

cromix” geometry. This study is motivated by the possible use of hydrogen as a fuel in gas turbines and aeroengines.

* The unstable oscillations are coupled by axial modes of the chamber. The flame motion characterized by phase-averaged images indicates that the unsteady motion comprises the formation of a toroidal vortex at the flame tip during the first part of the cycle and a subsequent backpropagation of the flame towards the injector exhaust during a second part of the cycle. A space-time analysis of the Rayleigh source term indicates that these two processes feed energy in the coupling mode.

* The growth rate associated with this mechanism is estimated by integrating the Rayleigh source term and the acoustic energy density over the combustor volume. These two quantities are determined at the limit cycle giving access to the damping rate which is of the order of 300 s$^{-1}$.

* A special experimental procedure is then used to determine the effective growth rate (the growth rate associated with the flame response minus the damping rate) during the transient phase leading to the limit cycle. At low amplitudes of oscillation, in the linear regime, this effective growth rate is of the order of 100 s$^{-1}$. This yields a growth rate associated with the flame which takes a value of about 400 s$^{-1}$.
* An approximate determination of the flame describing function is obtained by measuring the flame response to acoustic modulations of the upstream flow and accounting for the transfer function between the plenum and the injector exhaust. The gain of the optical FDF is high between 600 Hz and 900 Hz and the phase increases linearly in this range corresponding to a delay $\tau \simeq 1.3$ ms, which is coherent with a collective time lag model.

* Experiments carried out by changing the length of the combustion chamber allow to examine the phase relation between the pressure oscillation and the heat release rate in the flame. This is used to identify three ranges of operation: an unstable domain for short combustor chambers corresponding to higher frequencies in the quarter-wave mode, a stable domain for intermediate lengths and a domain of thermo-acoustic instabilities coupled by the three quarter-wave mode for longer combustors. This method defines an instability range of frequencies (in the present study, between 600 Hz and 1100 Hz). A combustor equipped with this injection unit and operating at the power and equivalence ratio conditions used in the present study will be stable if there are no eigenmodes in this range, provided that the flame transfer function and the injector impedance do not change.

## 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. The authors wish to thank the three reviewers of this article for their helpful suggestions.

## References

[1] Vaysse, Nicolas, Durox, Daniel, Vicquelin, Ronan, Candel, Sébastien and Renaud, Antoine. "Stabilization and Dynamics of Pure Hydrogen Swirling Flames Using Cross-Flow Injection." *ASME Turbo Expo 2023: Turbomachinery Technical Conference and Exposition*. 2023. American Society of Mechanical Engineers Digital Collection.

[2] Yahou, Tarik, Dawson, James R and Schuller, Thierry. "Impact of Chamber Back Pressure on the Ignition Dynamics of Hydrogen Enriched Premixed Flames." *Symp. Int. Combust.* (2022): p. 9.

[3] Aniello, A., Poinsot, T., Selle, L. and Schuller, T. "Hydrogen Substitution of Natural-Gas in Premixed Burners and Implications for Blow-off and Flashback Limits." *International Journal of Hydrogen Energy* Vol. 47 No. 77 (2022): pp. 33067-33081.

[4] Pers, H., Aniello, A., Morisseau, F. and Schuller, T. "Autoignition-Induced Flashback in Hydrogen-Enriched Laminar Premixed Burners." *International Journal of Hydrogen Energy* (2022).

[5] Prieur, Kevin, Vignat, Guillaume, Durox, Daniel, Schuller, Thierry and Candel, Sébastien. "Flame and Spray Dynamics During the Light-Round Process in an Annular System Equipped With Multiple Swirl Spray Injectors." *Journal of Engineering for Gas Turbines and Power* Vol. 141 No. 061007 (2019).

[6] Töpperwien, Karl, Collin-Bastiani, Félix, Riber, Eleonore, Cuenot, Bénédicte, Vignat, Guillaume, Prieur, Kevin, Durox, Daniel, Candel, Sébastien and Vicquelin, Ronan. "Large-Eddy Simulation of Flame Dynamics During the Ignition of a Swirling Injector Unit and Comparison With Experiments." *Journal of Engineering for Gas Turbines and Power* Vol. 143 No. 021015 (2021).
[7] Schimek, Sebastian, Göke, Sebastian, Schrödinger, Christina and Paschereit, Christian Oliver. "Flame Transfer Function Measurements With CH4 and H2 Fuel Mixtures at Ultra Wet Conditions in a Swirl Stabilized Premixed Combustor." *Vol. 2 Combust. Fuels Emiss. Parts B*: pp. 1335-1344. 2012. American Society of Mechanical Engineers, Copenhagen, Denmark.

[8] Shanbhogue, S. J., Sanusi, Y. S., Taamallah, S., Habib, M. A., Mokheimer, E. M. A. and Ghoniem, A. F. "Flame Macrostructures, Combustion Instability and Extinction Strain Scaling in Swirl-Stabilized Premixed CH4/H2 Combustion." *Combustion and Flame* Vol. 163 (2016): pp. 494-507.

[9] Chterev, Ianko and Boxx, Isaac. "Effect of Hydrogen Enrichment on the Dynamics of a Lean Technically Premixed Elevated Pressure Flame." *Combustion and Flame* Vol. 225 (2021): pp. 149-159.

[10] Aguilar, José G., Æsøy, Eirik and Dawson, James R. "The Influence of Hydrogen on the Stability of a Perfectly Premixed Combustor." *Combustion and Flame* Vol. 245 No. 112323 (2022).

[11] Kwak, Sanghyeok, Choi, Jaehong, Ahn, Myunggeun and Yoon, Youngbin. "Effects of Hydrogen Addition on the Forced Response of H2/CH4 Flames in a Dual-Nozzle Swirl-Stabilized Combustor." *International Journal of Hydrogen Energy* Vol. 47 No. 65 (2022): pp. 28139-28151.

[12] Schuller, Thierry, Poinsot, Thierry and Candel, Sébastien. "Dynamics and Control of Premixed Combustion Systems Based on Flame Transfer and Describing Functions." *J. Fluid Mech*. Vol. 894 (2020): p. P1.

[13] Joo, Seongpil, Kwak, Sanghyeok, Lee, Jongguen and Yoon, Youngbin. "Thermoacoustic Instability and Flame Transfer Function in a Lean

Direct Injection Model Gas Turbine Combustor." *Aerospace Science and Technology* Vol. 116 (2021): p. 106872.

[14] Æsøy, Eirik, Aguilar, José G., Wiseman, Samuel, Bothien, Mirko R., Worth, Nicholas A. and Dawson, James R. "Scaling and Prediction of Transfer Functions in Lean Premixed H2/CH4-flames." *Combustion and Flame* Vol. 215 (2020): pp. 269-282.

[15] Park, Jaehyun and Lee, Min Chul. "Combustion Instability Characteristics of H2/CO/CH4 Syn-gases and Synthetic Natural Gases in a Partially-Premixed Gas Turbine Combustor: Part I—Frequency and Mode Analysis." *International Journal of Hydrogen Energy* Vol. 41 No. 18 (2016): pp. 7484-7493.

[16] Yoon, Jisu, Joo, Seongpil, Kim, Jeongjin, Lee, Min Chul, Lee, Jong Guen and Yoon, Young-bin. "Effects of Convection Time on the High Harmonic Combustion Instability in a Partially Premixed Combustor." *Proceedings of the Combustion Institute* Vol. 36 No. 3 (2017): pp. 3753-3761.

[17] Lim, Zhengli, Li, Jingxuan and Morgans, Aimee S. "The Effect of Hydrogen Enrichment on the Forced Response of CH4/H2/Air Laminar Flames." *International Journal of Hydrogen Energy* Vol. 46 No. 46 (2021): pp. 23943-23953.

[18] Oztarlik, Gorkem, Selle, Laurent, Poinsot, Thierry and Schuller, Thierry. "Suppression of Instabilities of Swirled Premixed Flames with Minimal Secondary Hydrogen Injection." *Combustion and Flame* Vol. 214 (2020): pp. 266-276.

[19] Ghani, Abdulla and Polifke, Wolfgang. "Control of Intrinsic Thermoacoustic Instabilities Using Hydrogen Fuel." *Proceedings of the Combustion Institute* Vol. 38 No. 4 (2021): pp. 6077-6084.

[20] Barbosa, Séverine, de La Cruz Garcia, Marta, Ducruix, Sébastien, Labegorre, Bernard and Lacas, François. "Control of Combustion Instabilities by Local Injection of Hydrogen." *Proceed-
*ings of the Combustion Institute* Vol. 31 No. 2
(2007): pp. 3207–3214.

[21] Nam, Jaehyun and Yoh, Jack J. "A Numerical Investigation of the Effects of Hydrogen Addition on Combustion Instability inside a Partially-Premixed Swirl Combustor." *Applied Thermal Engineering* Vol. 176 (2020): p. 115478.

[22] Poinsot, Thierry J., Trouve, Arnaud C., Veynante, Denis P., Candel, Sebastien M. and Esposito, Emile J. "Vortex-Driven Acoustically Coupled Combustion Instabilities." *J. Fluid Mech*. Vol. 177 (1987): pp. 265–292.

[23] Kangkang, Guo, Boqi, Xu, Yongjie, Ren, Yiheng, Tong and Wansheng, Nie. "Three-Dimensional Numerical Analysis of Longitudinal Thermoacoustic Instability in a Single-Element Rocket Combustor." *Front. Energy Res*. Vol. 10 (2022).

[24] Indlekofer, Thomas, Ahn, Byeonguk, Kwah, Yi Hao, Wiseman, Samuel, Mazur, Marek, Dawson, James R. and Worth, Nicholas A. "The Effect of Hydrogen Addition on the Amplitude and Harmonic Response of Azimuthal Instabilities in a Pressurized Annular Combustor." *Combustion and Flame* Vol. 228 (2021): pp. 375–387.

[25] Zhang, Jianan and Ratner, Albert. "Experimental Study of the Effects of Hydrogen Addition on the Thermoacoustic Instability in a Variable-Length Combustor." *International Journal of Hydrogen Energy* Vol. 46 No. 29 (2021): pp. 16086–16100.

[26] Ahn, Byeonguk, Indlekofer, Thomas, Dawson, James R. and Worth, Nicholas A. "Heat Release Rate Response of Azimuthal Thermoacoustic Instabilities in a Pressurized Annular Combustor with Methane/Hydrogen Flames." *Combustion and Flame* Vol. 244 (2022): p. 112274.

[27] Moon, Kihun, Choi, Yongseok and Kim, Kyu Tae. "Experimental Investigation of Lean-Premixed Hydrogen Combustion Instabilities in a Can-Annular Combustion System." *Combustion and Flame* Vol. 235 (2022): p. 111697.

[28] Abbot, David, Giannotta, Alessandro, Sun, Xiaoxiao, Gauthier, Pierre and Sethi, Vishal. "Thermoacoustic Behaviour of a Hydrogen Micromix Aviation Gas Turbine Combustor Under Typical Flight Conditions." *ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition*: p. 12. 2021. American Society of Mechanical Engineers Digital Collection.

[29] Kang, Hyebin and Kim, Kyu Tae. "Combustion Dynamics of Multi-Element Lean-Premixed Hydrogen-Air Flame Ensemble." *Combust. Flame* Vol. 233 (2021): p. 111585.

[30] Lee, Taesong and Kim, Kyu Tae. "High-Frequency Transverse Combustion Instabilities of Lean-Premixed Multislit Hydrogen-Air Flames." *Combustion and Flame* Vol. 238 (2022): p. 111899.

[31] Vaysse, N, Durox, D, Soundararajan, P Rajendram, Vicquelin, R, Candel, S and Renaud, Antoine. "Structure and Light Emission of Swirling Flames Produced by Pure Hydrogen Injection in Cross-Flow." *Proc. Eur. Combust. Meet.*: p. 6. 2023. Rouen, France.

[32] Karyeyen, Serhat, Feser, Joseph S. and Gupta, Ashwani K. "Hydrogen Concentration Effects on Swirl-Stabilized Oxy-Colorless Distributed Combustion." *Fuel* Vol. 253 (2019): pp. 772–780.

[33] Vignat, Guillaume. "Injection and combustion dynamics in swirled spray flames and azimuthal coupling in annular combustors." Ph.D. Thesis, Université Paris-Saclay. 2021.

[34] Padley, P. J. "The Origin of the Blue Continuum in the Hydrogen Flame." *Trans. Faraday Soc.* Vol. 56 (1960): p. 449.

[35] Fiala, Thomas and Sattelmayer, Thomas. "Modeling of the Continuous (Blue) Radiation in Hydrogen Flames." *International Journal of Hydrogen Energy* Vol. 41 No. 2 (2016): pp. 1293–1303.
[36] Durox, Daniel, Moeck, Jonas P., Bourgouin, Jean-François, Morenton, Pascal, Viallon, Marc, Schuller, Thierry and Candel, Sébastien. "Flame Dynamics of a Variable Swirl Number System and Instability Control." *Combustion and Flame* Vol. 160 No. 9 (2013): pp. 1729-1742.

[37] Rajendram Soundararajan, Preethi, Durox, Daniel, Renaud, Antoine, Vignat, Guillaume and Candel, Sébastien. "Swirler Effects on Combustion Instabilities Analyzed with Measured FDFs, Injector Impedances and Damping Rates." *Combustion and Flame* Vol. 238 (2022): p. 111947.

[38] Durox, D., Ducruix, S. and Lacas, F. "Flow Seeding with an Air Nebulizer." *Experiments in Fluids* Vol. 27 No. 5 (1999): pp. 408-413.

[39] Latour, V., Durox, D., Renaud, A. and Candel, S. "Experiments on Symmetry Breaking of Azimuthal Combustion Instabilities and Their Analysis Combining Acoustic Energy Balance and Flame Describing Functions." *J. Fluid Mech.* Vol. 985 (2024): p. A31.