# Characterization of laminar ultra-lean H₂-air annular flames using spectral proper orthogonal decomposition

Journal Title
XX(X):1-12
©The Author(s) 2016
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DOI: 10.1177/ToBeAssigned
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SAGE

Nicole Lopes Junqueira<fnref n="1" />,<fnref n="2" />, Luís Fernando Figueira da Silva<fnref n="2" /> and Valéry Morgenthaler<fnref n="1" />

## Abstract

Lean hydrogen-air flames have the benefit of consuming less fuel and reaching lower temperatures than stoichiometric flames, but they exhibit intrinsic instabilities that affect flame dynamics and propagation. The goal of this study is to analyze the spatio-frequential structure of ultra-lean, burner stabilized, hydrogen-air flames in two distinct situations. The first corresponds to the presence of intrinsic instabilities only, whereas, in the second, longitudinal pulsations related to the burner acoustic arise. The analysis uses measured time resolved schlieren data and the spectral proper orthogonal decomposition technique (SPOD). The SPOD is a reduction technique for isolating patterns and structures in complex flows, commonly applied in aeroacoustics and jet studies to represent flow structures. The results for the intrinsically unstable flame show a monotonic energy decay with frequency, with an exponential decay of around 19 modes per energy decade. The spatial distribution remains similar across frequencies, depicting intercalated wave packets across the burner, resembling the intrinsic instability's transversal movement, as observed in schlieren images. For the longitudinal pulsation case, the energy spectrum reveals multiple peaks, with the first mode at the harmonic frequencies exhibiting significantly higher energy. From the second mode on, an exponential decrease of about 18 modes per energy decade is observed. The spatial distribution varies with frequency, where the first mode at harmonics displays longitudinally intercalated wave packets, whereas the second mode appears to capture a superposition from both instabilities. The SPOD analysis highlights the distinct interactions between intrinsic instabilities and longitudinal pulsations due to burner acoustic modes across different frequencies and spatial modes.

## Keywords

Lean hydrogen-air flames, Intrinsic flame instability, Annular burner, Schlieren imaging, Spectral proper orthogonal decomposition

## Introduction

Hydrogen, a source of potentially sustainable alternative fuel, is being increasingly studied<fnref n="1" />. The behavior of the stable premixed laminar H₂-air flames when the Lewis number controlling the combustion is close to unity, i.e., mixtures close to stoichiometry, is well known<fnref n="2" />. However, burner stabilized, ultra-lean H₂-air mixtures are still to be thoroughly examined. In this work, the term “ultra-lean” is defined as mixtures with equivalence ratios in the range 0.28 to 0.40, close to the lean blow-off limit, and characterized by a Lewis number significantly less than one (Le≈ 0.28–0.55). Those mixtures are interesting for a few aspects: they consume less fuel and generate lower NOx emissions than stoichiometric ones, and temperature of the burned gases is smaller, i.e., they impose less thermal stresses on the combustion chambers.

However, lean premixed hydrogen/air flames exhibit intrinsic flame instabilities that influence flame dynamics and propagation³,⁴. One of these instabilities, present in all premixed flames, is the Darrieus-Landau instability, that arises from density variations across the flame sheet⁴. Another intrinsic instability is the thermodiffusive one, controlled by the hydrogen's low Lewis number ⁵,⁶, that is associated to a substantial difference between the diffusivities of temperature and fuel, thus leading to differential diffusion effects within the flame front. These

instabilities play a crucial role in shaping the behavior and characteristics of lean premixed hydrogen-air flames⁷.

Several recent works, based on detailed numerical simulations, have analyzed the of flame front structure, flame speed enhancements, and the corresponding influence of parameters such as temperature, pressure, and equivalence ratio⁶⁻¹¹. These studies also showed that thermodiffusive instabilities not only change the flame speed but also introduce intrinsic length scales in the flame¹²,¹³, which could impact the design and optimization of hydrogen combustion practical systems.

Recent application-oriented experimental studies also explore lean hydrogen-air flames, focusing on flame structure, thermo-acoustic instabilities, and propagation¹⁴⁻¹⁶, thus contributing to the development of hydrogen-based combustion systems. In particular, the combustion instabilities have been studied to reveal the oscillatory modes and frequencies that influence flame behavior and efficiency in

<footnote n="1">¹Ansys, France</footnote>

<footnote n="2">²Institut Pprime CNRS - ISAE-ENSMA and Université de Poitiers, France</footnote>

<footnote>Corresponding author:</footnote>

<footnote>Nicole Lopes Junqueira, Institut Pprime CNRS - ISAE-ENSMA and Université de Poitiers, France.</footnote>

<footnote>Email: nicole.lopes-monteiro-de-barros-junqueira@ensma.fr</footnote>
lean hydrogen-air combustion¹⁶. However, the dynamics of burner stabilized, unstable ultra-lean laminar hydrogen-air flames has not yet been thoroughly experimentally examined. While existing experimental studies have advanced understanding of these instabilities, further insights can be gained by examining the dominant energy modes and spatial-temporal structures within these flames, which is the goal of this study.

To isolate patterns and structures within complex, turbulent flows, in the field of aero-acoustics and jets, the spectral proper orthogonal decomposition (SPOD) has been applied to a wide range of purposes¹⁷⁻²⁰. This technique empirically obtains hidden structures of the data and it is particularly effective for analyzing data where coherent, energy-dominant structures evolve in both time and space²¹,²². It finds an optimal orthogonal basis, with modes that vary in both space and time²³. These modes are orthogonal under a space-time inner product, meaning that they can be ranked in terms of energy. The SPOD thus combines temporal and spatial information, delivering a set of orthogonal modes ranked by energy at discrete frequencies. Accordingly, this study aims to characterize the spectral proper orthogonal decomposition structures of the instantaneous schlieren of the unstable lean premixed hydrogen-air laminar flames stabilized on an annular burner for the equivalence ratios between the lean blowoff and the flashback limits. This will enable us to unveil the spectral and modal behavior of these ultra-lean flames, in cases where thermodiffusive instability arises only, and, where flame pulsations associated to the burner acoustics modes are also observed.

## Methodology

In this section it is presented the developed burner, the experimental method and the spectral proper orthogonal decomposition.

### Experimental methodology

The burner developed for this study, shown in Fig. 1 is based on the “ANÔ” burner, that was used to perform a stochastic characterization of unstable lean hydrogen-air flames⁵. The new slit burner was designed to provide a smaller pressure drop throughout the inlet system, and to enable the stabilization of lower Reynolds number flames. The central part of the new annular-slit burner is open, thus allowing ambient air to flow freely. As it is shown in Fig. 1a, the mixture of hydrogen and air occurs in a ellipsoidal-shaped settling chamber connected by ten 26 cm length tubes of 6 mm inner diameter to the bottom of the burner. The burner cross section is illustrated in Fig. 1b: a porous plate, which homogenizes the mixture and stabilizes the flow is located upstream to the burner slit at a distance of 80 mm. The flame is located outside the burner, attached to two discs that form a 2 mm annular slit, with a thickness of 10 mm. The inner radius of the slit is 39 mm and the outer radius is 41 mm. The resulting perimeter to slit ratio is around 120, which allows us to observe the nearly two-dimensional flame instabilities. The material of the burner is stainless steel and the outer disk surface temperature is monitored by a thermocouple.

To explore the ultra-lean laminar hydrogen-air flames, three mass flow controllers (Brooks models SLA5850 and SLA5851) were used, with flow meter ranges of 20, 50 and 100 nlmph. In this burner, the blow-off limit is around the equivalence ratio of $φ =0.26$, whereas the first point at which the flame is anchored on both sides of the slit is $φ =0.28$, and the flashback limit is around the equivalence ratio of 0.40. The obtained Reynolds number, defined using the fuel-air mixture flow rate and the burner hydraulic diameter, ranges from 480 to 860, and the equivalence ratio ranges from 0.28 to 0.40.

The experimental technique that is the focus of this work is the dual-lens schlieren imaging⁵,⁹, that is an optical method that visualizes the refractive index gradients, which are directly linked to density variations in the flow²⁴. In gas-phase subsonic combustion, these density variations are related to temperature gradients across the flame front, allowing us to observe thermal structures from the schlieren images. In this setup, a collimated LED light source (LTCLHP080-W, 100 mm diameter) is located 58 mm from the burner center. The light path on this schlieren setup is: the light passes *though the flame*, it is focused by a 100 mm diameter, 750 mm focal length plano-convex lens to a knife edge of 45° that blocks the non-deviated light. Then, the deviated-light is focused by a bi-convex lens of 95 mm diameter and 150 mm focal distance to the camera, Photron FASTCAM Mini AX200 CMOS detector. The schlieren system's depth of field was estimated to be approximately 0.75 mm, following the approach described by Settles (2001)²⁴, indicating that only a thin region of the flame is sharply focused. Note that as this is an in-line schlieren configuration, the recorded images represent an integration of refractive index gradients along the optical axis. Thus, it is not possible to distinguish whether observed structures originate from the front or back side of the flame.

The camera operated with two temporal resolutions, depending on the considered flame regime: 2,000 and 13,500 frames per second, with a shutter speed of 4 µs, and a spatial resolution of 10 px/mm. For each flow condition a set of 10,000 images was obtained. The raw images, sized 896 × 560 px in the (x, y) directions, were processed using a pixel-wise adaptive low-pass filter with a 5 × 5 pixel neighborhood to estimate local image statistics, thus reducing Gaussian white noise while preserving image features across each frame of the sequence. Then, a set of instantaneous frames is used to characterize the flame through the spectral proper orthogonal decomposition.

Two successive sets of experiments were carried out: the first aimed to determine the flashback and blow-off limits, and the second one to characterize the flame regimes as a function of the equivalence ratio and Reynolds number.

### Spectral proper orthogonal decomposition

Figure 2 shows a scheme of the SPOD algorithm²³. It begins by dividing the temporal dataset, denoted by $Sc(t)$, into overlapping time-domain blocks ($Sc^{(B_i)}$). For each of these blocks, the Discrete Fourier Transform (DFT) is applied to convert the data to the frequency domain ($\hat{Sc}^{(B_i)}$). After obtaining the frequency representation of each block, the data is reorganized by frequency. For each frequency, Proper
<figure><img src="image_1.png" /><figcaption>**Figure 1.** Annular slit burner scheme. (a) the overall installation; (b) the burner cross section with a mean schlieren and the representation of the two regions of interest (ROI): the complete annular flame (ROI 1); central of the burner (ROI 2).</figcaption></figure>

Orthogonal Decomposition (POD) is then applied to the frequency domain data, obtaining the SPOD modes $\hat{Sc}_{f_k}$. This procedure extracts the dominant coherent structures for each frequency by identifying the modes that carry the most energy (i.e., those associated with the largest eigenvalues). Although SPOD is formally defined as an eigenvalue decomposition of the cross-spectral density matrix at each frequency, it is often computed using a Singular Value Decomposition (SVD) of a matrix constructed from the frequency-domain blocks. In this case, the squared singular values correspond to the mode energies. Such an algorithm has already been presented and used for different applications, for additional details refer to$^{23,25,26}$.

$$ Sc(t) = \left\{ \left( \begin{array}{ccccccccc} B_1 & & & & & & & & B_3 \\ & \ddots & & & & & & & \\ & & \ddots & & & & & & \\ & & & B_M \end{array} \right) \right\} $$

<figure><img src="image_2.png" /><figcaption>**Figure 2.** Spectral proper orthogonal decomposition (SPOD) scheme, adapted from$^{26}$.</figcaption></figure>

From the SPOD, the energy spectrum that describes how the energy content of each mode varies as a function of frequency is obtained. Here, the term “energy” refers to the eigenvalue (or squared singular value) associated with each mode, which quantifies its contribution to the total variance of the dataset at a given frequency. In other words, it reflects how much of the data’s coherent structure is captured by that mode. Overall, in a POD decomposition, the first few

modes have a higher energy, and as the number of modes increases, this energy decreases. This characteristic comes from the singular value decomposition used in the POD step, where the singular values are sorted in descending order, and the leading modes capture the dominant coherent structures in the dataset. As a result, it is possible to evaluate whether a low-rank approximation to the problem is feasible, since the most relevant information in the system (in this sense) has a higher energy. However, because SPOD requires the convergence of second-order space-time statistics, large datasets and long time series are necessary for accurate results$^{23}$.

In this work, the SPOD algorithm is applied to different regions of the instantaneous images: the complete annular flame and the central region of the burner. Those regions of interest have an area of 896×210 px (ROI 1) and 200×210 px (ROI 2), respectively. Those regions are shown overlaid on the schlieren image in Fig 1b, where the blue rectangle shows the ROI 1, and, the orange rectangle is the ROI 2. In both cases, the vertical height (y-direction in Fig. 1b) of the image is the same and spans from the burner surface to around 10 mm above the tip of the flame, for the highest flame height obtained in the set of experiments. Each processed temporal dataset, that represents a Reynolds number and equivalence ratio ($Re, \phi$) pair, is formed by 8,200 snapshots of the instantaneous schlieren images for the ROI 2 and 6,000 snapshots for the ROI 1. Regarding the SPOD parameters, the default spectral estimation parameters are used: the Hamming temporal window; 50% of the length of the temporal window as the number of segments overlapped; uniform spatial inner product weight. Therefore, for each ($Re, \phi$) point at the ROI 2, there are a total of 31 temporal blocks, where each block has a total of 512 snapshots, and 256 of which are overlapping for the SPOD applied to the central region of the burner. For the full burner region, ROI 1, there are 22 temporal blocks with 528 snapshots, where 264 of those are overlapping. Note that the number of time blocks limits the number of statistically independent realizations available per frequency, which in turn limits the number of resolvable SPOD modes. The
SPOD modes are complex-valued fields that contain both
amplitude and phase information. In this study, the real part
of the mode is shown only. As such, the spatial distribution
discussed corresponds to a snapshot at an arbitrary phase
angle.

Results and Discussion

The first part of this section presents the particularities of
the newly developed burner regarding the flame regimes and
its flame structures, that were using the classical schlieren
imaging technique. This is followed by the spectral proper
orthogonal decomposition analysis, where is presented and
discussed the energy and the wave packet of the different
flame structures found.

Flame regimes and structures

In this burner two characteristic flame structures are
observed: the first is when only the intrinsic instabilities
of the lean flames arise, i.e., the thermo-diffusive and the
Darieus-Landau. The second flame structure is when there is
a longitudinal pulsation caused by the coupling of the natural
instability of the flame heat release rate with the acoustic
modes of the burner. For the sake of simplicity, here, we
refer to these cases as the intrinsic instability and longitudinal
pulsation cases, respectively. In Fig. 3, a regime diagram
is showing, as a function of the Reynolds number and the
equivalence ratio.

<figure><img src="image_3.png" /><figcaption>**Figure 3.** Flame regime diagram. The Filled symbols indicate experiments for which high-speed imaging was recorded. The red and blue lines denote the flashback and blow-off limits of the burner, respectively. The green region highlights flame regimes where the coupling between intrinsic instability and longitudinal pulsation is observed. The two black-filled symbols correspond to the experimental cases selected for detailed analysis in this work.</figcaption></figure>

The diagram shows the experimentally identified insta-
bility boundaries, along with the blow-off and flashback
limits, indicated by blue and red lines, respectively. These
boundaries and limits were determined using the imple-
mented schlieren technique. The symbols represent different
observed flame behaviors: thermodiffusive instability (◊),
acoustic coupling (⊙), blow-off (▽), and flashback (△),

where the filled symbols denote the cases for which high-
speed schlieren recordings were acquired. The green region
highlights the range of operating conditions where the longi-
tudinal acoustic pulsation is observed.

It is possible to notice that the region where the
longitudinal pulsations occurs lies between the equivalence
ratios of 0.30 and 0.35. Moreover, it was observed during the
experiments that, within this equivalence ratio (φ ≈ 0.30–
0.35), a characteristic acoustic signature associated with the
instability coupling is clearly audible at lower Reynolds
numbers (Re < 800). At higher Reynolds numbers (Re >
800), the audible sound intensity decreases and becomes
difficult to detect without dedicated acoustic instrumenta-
tion, although the longitudinal pulsation remains evident in
the schlieren visualizations albeit with a lesser intensity.
Although the presence of an acoustic feedback loop is
suggested by the data, detailed acoustic measurements are
not the subject of current work. Therefore, only the analysis
of the different flame structures obtained is presented here.

Figures 4 and 5 shows two examples of representative schlieren images, one for the intrinsic instability and another when there is a longitudinal pulsation. A typical mean schlieren image of the annular flame is shown in Figs. 4a and 5a. For both these regimes, it is possible to observe the boundaries between the fresh and burned gases, and between those the ambient air, as well as the open flame tip, characteristic of these low Lewis number, around 0.38 and 0.61, flames. These mean images are similar to those analyzed in the previous burner version⁵.

<figure><img src="image_4.png" /><figcaption>**Figure 4.** Schlieren images for the longitudinal pulsation ($Re, \phi$) = (645, 0.29): mean (top) and instantaneous (bottom) images.</figcaption></figure>

Regarding the instantaneous images, it is possible to
differentiate the two regimes. Note that these instantaneous
images are obtained by dividing each frame by the schlieren
mean, thus corresponding to the fluctuating part of the
temperature gradient. For the case where only the intrinsic
instabilities appear, Fig. 4b shows the flame surface wrinkles
along the transversal direction of the burner. These wrinkles
move rather slowly along the transversal direction. Note that
the flame is geometrically constrained in the axial direction:
its observed height in schlieren is a few thermal flame
thicknesses, which suppresses longitudinal thermodiffusive
modes and leaves predominantly transverse instabilities.
A rather different instantaneous structure is observed for
the case where there is a longitudinal pulsation, Fig. 5b.
<figure><img src="image_5.png" /><figcaption>(a) Mean schlieren</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) Instantaneous schlieren</figcaption></figure>

<caption>**Figure 5.** Schlieren images for the longitudinal pulsation ($Re, \phi$) = (645, 0.31): mean (top) and instantaneous (bottom) images.</caption>

Indeed, the flame tip opens and closes at a certain frequency, causing a pocket of fresh gases to detach and, perhaps, burn downstream of the average flame tip. Since these flame structures are typical of those found throughout the (Re, φ) ranges given in Fig. 3, for the sake of brevity, only these two cases will be discussed here: the intrinsic instability (Re = 645, φ = 0.29) and the longitudinal pulsation regime (Re = 645, φ = 0.31). These cases were chosen to illustrate that a small variations on the equivalence ratio may lead to significantly different flow fields. Note that this Reynolds number value was chosen to lie in the middle of the experimental range given in Fig 3.

## Intrinsic instability case SPOD

In order to examine the measured flame structures via SPOD, Fig. 6a shows the SPOD mode energy (SPOD eigenvalue) spectrum for the intrinsic instability case considering the full image, i.e., ROI 1. It can be seen, for each frequency range obtained through the direct Fourier transform, the energy contained in each SPOD mode. The first mode, highlighted in red, always has the highest energy, which decreases monotonically with increasing frequency. The subsequent modes are colored from black to light gray, thus emphasizing the importance of their energy contribution. For the lowest frequency in the spectra, the energy is around $10^1$ and for the highest frequency, the energy drops to the order of $10^{-4}$. This figure suggests that the energy decay rate beyond $f \approx 200$ Hz is constant ($E_{SPOD} \propto f^{-n}$), with $6 \le 2n \le 7$. Beyond approximately 800 Hz the flat spectrum indicates no information remains in the dataset.

To analyze the percentage of the modal energy distribution at a given frequency, Fig. 6b shows the SPOD energy normalized by the sum as a function of each mode for the arbitrary frequency of 367 Hz. The obtained 22 modes are depicted, where the first three modes have an energy higher than $10^1$ and the last two modes have an energy around $10^0$, thus the majority of these modes are within an energy range of one order of magnitude. Furthermore, it can be seen that the energy difference between the first and second modes is smaller than 2%. Overall, the energy decays by one decade approximately every 19 modes. This slow mode decay rate suggests that, in order to build a reduced order model of

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

(a)

<figure><img src="image_8.png" /></figure>

(b)

<caption>**Figure 6.** SPOD of the intrinsic instability case (Re = 645, φ = 0.29), ROI 1: (a) Energy (SPOD eigenvalue) spectrum, the first mode is the red line and the following modes go from black to light gray; (b) Energy content (%), normalized by the sum, as a function of the SPOD modes for the arbitrary frequency (70 Hz).</caption>

intrinsically unstable flames, a large number of modes would have to be accounted for.

The spatial distribution of the modes, also known as wave packets<sup>17,18</sup>, is illustrated in Figs. 7 and 8, showing the real part of the SPOD modes, which captures wave-like structures at an arbitrary phase angle. Even though the spatial SPOD modes are complex-valued, here it is discussed the real part of each mode at a fixed, arbitrary, phase angle. This is a common practice when analyzing wave packet structures (e.g., Schmidt & Colonius, 2020<sup>23</sup>). Accordingly, spatial features such as extrema or color intensity are phase-dependent and do not directly represent the mode's magnitude. To understand how this distribution behaves along the burner, Fig. 7a shows the wave packet of the first mode for different frequencies obtained from 6,000 frames of instantaneous schlieren. Here, colored regions represent the
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Spatial distribution of the first SPOD mode (wave packet), obtained from the full burner (ROI 1). (a) Intrinsic instability case, for the frequency 70 Hz. (b) Longitudinal pulsation case, for the fundamental frequency (366 Hz). The contours show the real part of the complex SPOD mode, the phase is arbitrary and does not correspond to a specific physical time. The colorbar goes from [-10<sup>-2</sup>, 10<sup>-2</sup>].</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Spatial distribution of the first and second SPOD modes ($m_1, m_2$) for three arbitrary frequencies ($f_1, f_2, f_3$) = (70, 207 and 367 Hz) in ROI 2. The frequency increases from left to right and the modes increases from top to bottom. The colorbar is identical to Fig. 7.</figcaption></figure>

real part of the SPOD mode, with red and blue indicating positive and negative values, respectively. The colorbar range ([-10<sup>-2</sup>, 10<sup>-2</sup>]) was selected to emphasize the subtle features of the wavepackets.

In Fig. 7a, we can see that the wave packet has a distinctive pattern across the burner. The blue and red zones are intercalated transversely and are regularly arranged, being closely spaced farther away from the center due to the burner curvature effect. This wave packet structure is a reminiscent of the transversal structure of the instability across the burner. Indeed, compared to Fig. 4b, this wave packet resembles the thermodiffusive instability observed in the schlieren. Another aspect to note is the intensity of the colors on the SPOD spatial distribution: the more intense colors indicate higher absolute values of the real part of the SPOD mode at the chosen phase. Accordingly, the absolute value of the real part is higher at the sides of the burner than in the center, which indicates that there is a stronger signal at the sides of

the burner, as in the schlieren. This is related to the larger light deflection angle at the sides, compared to the center.

Since the SPOD behavior is similar across the burner, the central region (ROI 2) is analyzed in further detail. Figure 8 shows the wave packet of the first and second modes for three arbitrary frequencies chosen to represent the lower, intermediate and higher frequency ranges of Fig. 6a. Regarding mode 1 behavior, note that a similar pattern to Fig. 7a is seen for all the frequencies, Figs. 8a - 8c. Indeed, departing from the burner surface, the highest absolute real part of the mode are near the center of the ROI 2, and decrease further downstream. Moreover, the wave packet structures exhibit a transverse pattern of alternating red and blue zones that seem similar across these frequencies. This pattern becomes more closely spaced away from the center, consistent with the burner's curvature and the expected transverse structure of the instability.
At frequencies of 70 Hz and 207 Hz, the wave-packet patterns exhibit a more regular and coherent spatial structure throughout the burner region. In contrast, the pattern at 367 Hz becomes irregular away from the burner. This discrepancy correlates with the energy content and modal convergence: the first mode energy at 367 Hz is significantly lower (below $10^{-1}$), suggesting that the corresponding eigenvalues contain limited physical information and that the mode may not be fully converged.

To further evaluate the reliability of these modes, a convergence analysis was performed by comparing modes computed from the full dataset with those obtained from subsets containing 80% of the data¹⁸. The convergence metric, defined as the he squared normalized inner products between modes from different subsets, indicates satisfactory convergence ($\beta^2 > 0.80$) at frequencies near 70 Hz and 207 Hz, confirming the robustness of the mode shapes at these frequencies. Conversely, the first mode at 367 Hz exhibits a substantially lower convergence metric, $\beta^2$ near 0.40, consistent with its low energy content and irregular spatial pattern. This suggests that the mode at 367 Hz is less statistically robust, likely due to insufficient data sampling rather than intrinsic physical instability. Accordingly, the irregular wave-packet pattern observed at 367 Hz is primarily attributed to convergence limitations, whereas the more regular structures at 70 Hz and 207 Hz reflect well-converged modes that capture the intrinsic flame dynamics.

Concerning now mode 2, the overall wave packet patterns are very similar for the each given frequencies, Figs. 8d - 8f, when compared to those of the first mode. The main difference regarding the maximum real part of the mode occurs in the transverse direction also.

## Longitudinal pulsation case

Figure 9a shows the SPOD energy spectrum for the case where the longitudinal pulsation occurs. As in the previous case, the energy spectrum of the SPOD modes has a higher energy in the first mode and decreases with the number of modes. However, one remarkable difference between the spectra is that, in Fig. 9a, several peaks are clearly seen. The frequencies of these peaks are the system's harmonics, with the fundamental frequency at 366 Hz, the second harmonic at 732 Hz, the third at 1098 Hz, and so on. The energy difference between the first and the second modes is higher, thus ensuring that these frequencies are significant for the system studied. Note that, for the frequencies between the harmonics, the differences between the energy of the modes are lower and, similar to that of Fig. 6a. This is also the case for the second to the last modes in the harmonics frequencies. Again, the region of high-frequency beyond 4,000 Hz has very low constant energy, bearing no information, which is corroborated by the spatial distribution of the modes on these frequencies, not shown here for the sake of brevity.

To further examine the energy distribution in the harmonics, Fig. 9b shows the SPOD energy as a function of each mode for the fundamental frequency, 366 Hz. The first remarkable difference between this case and the previous, shown in Fig. 6b, is the energy content in the first mode, that is much higher than the second mode, with energies around 49% and 8% respectively. In total, 22 modes are shown and, besides the first mode, the energy of the modes

<figure><img src="image_11.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b)</figcaption></figure>

**Figure 9.** SPOD of the longitudinal pulsation case ($Re = 645$, $\phi = 0.31$), ROI 1: (a) Energy spectrum, the first mode is the red line and the following modes goes from black to light gray. The dots mark the frequencies that are analyzed, the green dots are the harmonics and the pink dots are the frequencies between the harmonics that have a local minimum of energy. (b) Energy content (%), normalized by the sum, as a function of the SPOD modes for the fundamental frequency (366 Hz).

is exponentially decreasing. Furthermore, beyond the second mode, the energy decays by one decade approximately every 18 modes, which is similar to what was observed in the previous case.

Regarding the spatial distribution of the SPOD modes, Fig. 7b shows the wave packet of the first mode for a frequency of 366 Hz in ROI 1. It is possible to observe that, in most of the domain, the **real part of the mode** is nearly zero across most of the domain, having the local maxima and local minimum concentrated on both sides of the image, the lateral part of the burner. These lateral structures correspond to the frontal view of the flame, which exhibit
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Spatial distribution of the first and second SPOD modes ($m_1, m_2$) for the fundamental frequency ($f_1$ = 366 Hz) and the second and third harmonics ($f_2, f_3$) = (732 Hz and 1098 Hz) in ROI 2. The frequency increases from left to right and the modes increases from top to bottom. The contours show the real part of the complex SPOD mode. The phase is arbitrary and does not correspond to a specific physical time. The colorbar is identical to Fig. 7.</figcaption></figure>

coherent motion consistent with the periodic opening and closing of the tip driven by the longitudinal pulsations.

The alternating positive (red) and negative (blue) regions in the real part of the mode are associated to the presence of flame fronts and unburned gas pockets oscillating in a symmetrical pattern. However, because the strongest features in the real part are confined to the flame tips at the lateral edges in Fig. 7b, the burner central region—where the longitudinal dynamics are seen from a different angle—shows relatively weak signal. Due to this dominance of the flame's lateral region, in contrast with the intrinsic instability case previously presented, Fig. 7b does not show a transversal pattern across the burner for the fundamental frequency, 366 Hz. As a result, to better isolate and analyze the axial (longitudinal) pulsation pattern near the burner centerline, the following analysis considers the center of the burner (ROI 2) only.

In order to evidence the effect of the longitudinal pulsation on the wave packets, these will be examined for the frequencies marked by dots in Fig. 9a. Accordingly, six frequencies will be analyzed for the case with longitudinal pulsation: the three harmonics; a low frequency (220 Hz) and the two frequencies between the harmonics that have the lowest energy (586 and 952 Hz). Accordingly, Figs. 10 and 11 show the spatial distribution of the three first SPOD modes for these different frequencies.

For the first mode (Figs. 10a - 10c) the real part of the SPOD mode displays a spatial pattern along the y-direction, with alternating red (positive) and blue (negative) regions shown at a fixed, arbitrary phase angle. This can be seen for the first mode of all harmonic frequencies, in Fig. 9a. The main difference between these wave packets is the longitudinal spacing of the wave-like features, which

we refer to as an apparent wavelength, estimated here as the distance between repeating positive extrema in the real part of the mode. Indeed, higher frequencies correspond to smaller wavelengths, i.e., around 18 mm, 9.7 mm, and, 6.5 mm for the fundamental, first and second harmonic, respectively. Another difference is the position of the most intense color that corresponds to the maximum of the real part at the chosen phase: for the fundamental frequency, it is found close to the burner surface, between 0 and 5 mm in the y-direction. However, as the frequency increases, this local maximum position is now located progressively further downstream: for the second and third harmonics it is located around 13 and 15 mm in the y-direction, respectively.

The first mode of the fundamental frequency is related to the instantaneous schlieren shown in Fig. 5b. Comparing both images, along the y-direction, the wave packet structure is a consequence of the pulsating flame behavior observed in the schlieren, in which pockets of fresh gases periodically separate from the flame. The acoustic of the burner system seems to dominate the flame movement, since the first mode is the most energetic.

Even if the second SPOD mode has less energy than the first, it is interesting to analyze the corresponding spatial distribution. Accordingly, Figs. 10d - 10f show the spatial distribution of the wave packets for the first three harmonics of the second mode. These figures show that the wave packet has distinctive structure which seems to be a superposition of longitudinal and transversal patterns. Moreover, it is possible to notice that the regions connecting the positive extreme has an inclination with respect to the x-axis of the burner. This inclination has a tendency to decrease as the frequency increases: for to the fundamental frequency, 366 Hz, this inclination is approximately 72°, and it decreases to 64°
<figure><img src="image_14.png" /><figcaption>**Figure 11.** Spatial distribution of the first and second SPOD modes ($m_1, m_2$) for the three intermediate frequencies ($f_1, f_2, f_3$) = (220, 586 and 952 Hz) in ROI 2. The frequency increases from left to right and the modes increases from top to bottom. The contours show the real part of the complex SPOD mode. The phase is arbitrary and does not correspond to a specific physical time. The colorbar is identical to Fig. 7.</figcaption></figure>

and 52° for the second and third harmonics, respectively. Therefore, this wave packet structure of the second mode seems to be influenced both by the longitudinal pulsations, which wavelength decreases with frequency, and by the intrinsic instabilities, which wavelength is independent of the frequency.

Regarding the frequencies that are not the harmonics, Fig. 11 shows the wave packet for first and second modes of the frequencies 220, 586 and 952 Hz. For the frequency 220 Hz, Fig. 11a and 11d show that the wave packet of the first and second modes resembles that of the intrinsic instability case for the frequencies 70, 207 and 367 Hz, which has a transversal pattern. This suggest that the longitudinal pulsations do not have an impact on the wave packets on the low frequencies. Concerning now the frequencies between the harmonics, the first mode, Figs. 11b and 11c, resembles the second mode of the wave packets of the harmonics, i.e., an inclination is seen, which means that, even though the frequency is not an harmonic, the pulsations of the system still influence the wave packet. Concerning the second mode, Figs. 11d - 11f, the wave packets resemble their respective first mode wave packet.

Regarding the convergence analysis for the longitudinal pulsation case, the focus is placed on the first two harmonic frequencies observed in the energy spectrum, at 366 Hz and 732 Hz. For the full dataset, the SPOD energy of the first mode is 13.27 and 2.61 for the fundamental and first harmonic, respectively, while for the 80% subset the corresponding values are 13.18 and 2.50. The squared normalized inner products between the modes from the full and reduced datasets are $\beta^2 = 0.98$ at 366 Hz and $\beta^2 = 0.97$ at 732 Hz, indicating excellent agreement in mode shapes. Moreover, $\beta^2$ remains above 0.70 for most frequencies,

suggesting that the first mode is generally well converged across the analyzed spectral range. These results confirm that the modal structures associated with the fundamental and its first harmonic are statistically robust and insensitive to a moderate reduction in the number of realizations.

## Parametric analysis

A parametric analysis of the longitudinal pulsation signature is performed by computing the SPOD mode energy as a function of frequency for two different series of experiments: (i) constant Reynolds number and varying equivalence ratio, and (ii) constant equivalence ratio ($\phi = 0.31$) and varying Reynolds number. The SPOD is computed using ROI 1 and the same spatial windowing throughout the study, ensuring consistent comparisons across cases, even if the SPOD energy normalization precludes a direct comparison between peak values. The spectra, Fig. 12, are plotted on identical frequency axes to permit a direct identification of shifts in harmonic peaks and changes in spectral eigenvalues for the first mode.

Accordingly, to characterize the influence of the equivalence ratio, Fig. 12a, shows the first mode for three equivalence ratios for a constant Reynolds number of 645. The first harmonic shifts to increasing frequencies with increasing equivalence ratio. Indeed, for the leanest mixture, $\phi = 0.31$, the first harmonic is found to occur at 366 Hz, whereas for $\phi = 0.33$ the first peak lies between 366 Hz and 390 Hz. At $\phi = 0.35$, the first harmonic is observed at 390 Hz. A similar increasing trend is observed for the other harmonics also.

Concerning now the constant equivalence ratio analysis, which first mode results are shown in Fig. 12b, the first harmonic is essentially insensitive to Reynolds number within the studied range. For Re = 486, 645 and 729 the spectral
<figure><img src="image_15.png" /><figcaption>**Figure 12.** SPOD energy spectra of the first mode, ROI 1: (a) constant Reynolds number and different equivalence ratio; (b) constant equivalence ratio and different Reynolds number.</figcaption></figure>

maxima lie within the band 350–375 Hz. It is important to highlight that no significant shift of the harmonic frequency is seen to arise with Reynolds number. However, the amplitude of the harmonic peaks decreases as Reynolds number increases, indicating that higher Re reduces the coherent energy contained in the first longitudinal mode, thus suggesting that these harmonic perturbation are damped at higher Reynolds number.

Overall, the results show that the first longitudinal harmonic changes more noticeably with equivalence ratio than with Reynolds number within the studied range. The reduction in harmonic peak amplitude observed at higher Reynolds numbers may be related to the approach toward the laminar–turbulent transition regime, which enhances small-scale fluctuations and mixing²⁷. This disrupts the coherent hydrodynamic structures that contribute to acoustic coupling²⁸. Such a loss of coherence lowers the efficiency of energy transfer between the flame and the longitudinal acoustic mode, thereby reducing the harmonic peak amplitude and, at sufficiently high Reynolds numbers, suppressing thermoacoustic coupling. In contrast, increasing the equivalence ratio primarily shifts the first harmonic to higher frequencies, indicating that flame composition has a stronger influence on the oscillation frequency than on its amplitude.

## Conclusions

The SPOD characterization analysis of ultra-lean H₂-air annular flames provides observations into the energy distribution and spatial structures of intrinsic instabilities and longitudinal pulsations in the burner system. In particular, when intrinsic instabilities are present only, the schlieren fluctuations energy spectrum decays monotonically with frequency. The SPOD mode energy was found to decrease rather slowly; i.e., an exponential decrease over 19 modes per energy decade. Both the frequency and mode energy behaviors suggest that it would be hard to propose a low rank representation to this intrinsically unstable flame regime. The spatial distribution analysis reveals intercalated wave packet structures that remain similar across frequencies, resembling the transverse movement of instability along

the burner. These findings underline the persistence and spatial organization of intrinsic instabilities across modes and frequencies.

In contrast, the longitudinal pulsation case analysis shows a distinct energy spectrum with pronounced peaks, i.e., harmonic frequencies, with significant energy content. The first mode in the fundamental frequency carries substantially higher energy than subsequent modes, whereas the decay rate from the second mode on resembles the decay observed in intrinsic instability case, with a 18 mode-per-decade energy decay. Regarding the spatial distribution of the SPOD modes at the system's harmonics, the first mode exhibits a wave packet that is intercalated longitudinally, with a wavelength that decreases with the frequency. On the other hand, the second mode appears to be a superposition of longitudinal and transverse patterns. This spatial arrangement emphasizes the complex coupling between flame heat release and burner acoustics, highlighting how both intrinsic instabilities and longitudinal pulsation due to the acoustic modes of the burner interact in distinctive behavior across different frequencies and spatial modes. The parametric analysis shows that the first longitudinal harmonic varies with equivalence ratio, increasing from 366Hz to 390Hz in the range studied, whereas it remains unchanged with Reynolds number. The amplitude of the harmonic peaks decreases with increasing Reynolds number.

Future work will focus on quantifying the transverse wavelength of the intrinsic instability by performing a discrete Fourier transform in the streamwise direction at fixed transverse positions. Moreover, applying SPOD to subdomains — such as ROI 2 — and re-projecting the results onto the full spatial domain using extended (S)POD techniques²⁹ could help isolate and interpret specific coherent structures. Finally, a minimum model is under development to characterize the observed mode interactions and thus the dynamics of the studied ultra-lean flame behavior.

## Acknowledgements

For the purpose of Open Access, a CC-BY public copyright license has been applied by the authors to the present
document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission. The authors are grateful for the experimental support provided by V. Montassier, A. Claverie and N. Papin.

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