# Coupled thermoacoustic interactions in hydrogen enriched lean combustion

Abhishek Kushwaha1,2<EMAIL_ADDRESS>Amitesh Roy3  Ianko Chterev4
Isaac Boxx5  R. I. Sujith1,2  1Department of Aerospace Engineering, IIT
Madras, Chennai, Tamil Nadu, 600036, India 2 Centre of Excellence for Studying
Critical Transition in Complex Systems, IIT Madras, Chennai, Tamil Nadu,
600036, India 3Institute for Aerospace Studies, University of Toronto, Ontario
M3H5T6, Canada 4Institute for Combustion Technology, German Aerospace Centre
(DLR), Stuttgart 70569, Germany 5Chair of Optical Diagnostics for Energy,
Process and Chemical Engineering. RWTH Aachen University, Aachen 52062,
Germany

###### Abstract

In this paper, we present a framework to study the synchronization of flow
velocity with acoustic pressure and heat-release rate in technically-premixed
swirl flames. The framework uses the extended proper orthogonal decomposition
to identify regions of the velocity field where velocity and heat release
fluctuations are highly correlated. We apply this framework to study coupled
interactions associated with period-1 and period-2 type thermoacoustic
instability in a technically-premixed, swirl stabilized gas turbine model
combustor operated with hydrogen-enriched natural gas. We find the structures
in flame surface and heat release rate that are correlated to the dominant
coherent structures of the flow field using extended POD. We observe that the
correlated structures in the flow velocity, flame surface and heat release
rate fields share the same spatial regions during thermoacoustic instability
with period-1 oscillations. In the case of period-2 oscillations, the
structures from flame surface and heat release rate field are strongly
correlated. However, these structures contribute less to the coherent
structures of the flow field. Using the temporal coefficients of the dominant
POD modes of the flow velocity field, we also observed 1:1 and 2:1 frequency
locking behaviour among the time series of acoustic pressure, heat release
rate and the temporal coefficients of the first two dominating POD modes of
velocity field during the state of period-1 and period-2 oscillations,
respectively. These frequency-locked states, which indicate the underlying
phase-synchronization states, are then correlated with coherent structures in
the flow velocity field.

Abstract

##  1 Introduction

Reducing global greenhouse emissions is essential to ensure the habitability
of our planet. One of the major sources of these emissions comes from the
combustion of hydrocarbon fuels which continues to be the primary source of
power generation in modern gas turbines utilized in aviation and thermal power
plants. Lean combustion with the addition of hydrogen has become a significant
alternative for enhancing lean combustion limits, providing high energy, and
lowering carbon emissions. (Janus et al., 1997; Schefer, 2003; Birol, 2019;
Hord, 1978).

Despite evident advantages, the effect of hydrogen enrichment on the
thermoacoustic stability of turbulent combustors is not well-understood and
remains an active topic of research. Turbulent lean premixed flames are
particularly sensitive to acoustic fluctuations. Under suitable conditions,
the positive feedback among the acoustic field, turbulence and the flame leads
to thermoacoustic instability which results in large-amplitude pressure
fluctuations (Lieuwen, 2021; Sujith & Pawar, 2021). Thermoacoustic instability
(TAI) develops when heat release rate and pressure oscillations occur in-phase
and the acoustic driving is greater than acoustic damping, leading to the
energy increment to the acoustic field of the combustor (Rayleigh, 1878; Chu,
1965; Putnam, 1971). The amplitude of pressure oscillations increases
nonlinearities in the acoustic damping and the acoustic driving (Lieuwen &
Yang, 2005; Sujith & Pawar, 2021). The balance between the acoustic losses and
growth leads to limit cycle oscillations (Dowling, 1997). The state of TAI can
be detrimental to components of the engine, imparting structural damages,
resulting in unscheduled shutdowns and mission failures (Juniper & Sujith,
2018).

Large-scale coherent structures emerge during the onset of combustion
instability in combustion systems (Poinsot et al., 1987; Sterling & Zukoski,
1987; George et al., 2018). The mutual synchronization of the reactive flow
field with pressure oscillations is associated with the shift from stable
combustion operation to the state of TAI (Mondal et al., 2017; Pawar et al.,
2017; Singh et al., 2022). The goal of this study is to present a new
framework based on the extended POD for quantifying the synchronization of
coherence structures in the flow with heat release rate and pressure
oscillations in a swirl-stabilized combustor. To accomplish this, we quantify
the mutual synchronization of these coherent flow structures with pressure
oscillations during various dynamical states attained at various levels of
hydrogen enrichment.

###  1.1 Dynamics of swirling flows and flames

Swirling flames are used extensively in aircraft and gas turbine engines as
they are one of the simplest and robust methods of stabilizing turbulent flame
(Gupta et al., 1984; Hoffmann et al., 1994; Reddy et al., 2006; Freitag &
Janicka, 2007; Candel et al., 2014). The use of swirl flow leads to better
air-fuel mixing (Syred et al., 2014), reduction in fuel consumption and
decrement in exhaust pollutant emission (Rashwan et al., 2016).

Intense swirling flow gives rise to a central recirculation zone or a ‘vortex
breakdown bubble’ (VBB) (Harvey, 1962), which provides hot gases to the flame
at the base, improving the static stability of the flame. The flow returns
along the flow center-line when the swirl number exceeds a critical threshold,
forming an inner recirculation zone (IRZ) through helical instability (Gupta
et al., 1984; Choi et al., 2007). The axial flux of azimuthal momentum divided
by the axial flux of axial momentum is known as the swirl number. At
sufficiently intense swirling conditions, precessing vortex cores (PVCs) form
through helicoidal instabilities (Liang & Maxworthy, 2005; Gallaire et al.,
2006; Qadri et al., 2013). On the other hand, the outer recirculation zone
(ORZ) is formed when swirling flows encounter sudden expansion, leading to the
flow recirculation along the dump plane. Turbulent flames can stabilize either
in the central or IRZ or in the ORZ or in both zones depending upon the
overall geometry of the combustor (Gupta et al., 1984). Further, the
hydrodynamic stability is altered significantly due to the presence of density
stratification in non-isothermal conditions of turbulent flames. Indeed,
density stratification has been shown to suppress the formation of PVCs
(Oberleithner et al., 2013). Thus, the overall flow behavior is strongly
affected by the global stability of swirling flows and flames (Oberleithner et
al., 2011; Stöhr et al., 2011).

###  1.2 Effect of hydrogen enrichment on swirling turbulent flames

Hydrogen enrichment can significantly alter the characteristics of turbulent
flames. Comparing the size of the reaction zone to a pure methane-air flame
stabilized by swirling, Wicksall et al. (2005) noticed that the addition of
hydrogen to methane makes the flame more resilient. The increased
concentration of OH, H, and O radicals in the ensuing
$\textrm{H}_{2}$-enriched flame directly contributes to the increase of flame
stability limits (Schefer, 2003) in premixed turbulent combustors (Cozzi &
Coghe, 2006). One key feature of $\textrm{H}_{2}$-enrichment is the much
higher burning velocity of the enriched methane mixture relative to pure
methane due to the fast reaction rate of hydrogen observed in both numerical
simulations (Hawkes & Chen, 2004; Nam et al., 2019; Guo et al., 2020; Xia et
al., 2022) and experiments (Mandilas et al., 2007; Strakey et al., 2007;
Nakahara & Kido, 2008; Zhang et al., 2020; Pignatelli et al., 2022).

Higher reaction rates also manifest in much larger increment in flame surface
density upon $\textrm{H}_{2}$-enrichment (Halter et al., 2007; Guo et al.,
2010) and enhancement of flame wrinkling due to interactions of small-scales
of turbulence with the flame front (Emadi et al., 2012). Further, addition of
hydrogen alters the manner in which flame interacts with turbulence and
pressure fluctuations, thereby leading to increased rate of reaction of
hydrogen blended methane (Kim et al., 2009). The addition of $textrmH_{2}$
enhances flame anchoring and lean combustion limits by strengthening the
flame’s resistance to strain brought on by turbulence (Schefer et al., 2002;
Zhang et al., 2011).

As such, H2 enrichment in methane increases the flame surface area and the
flame wrinkling and also contributes to the state of TAI by increasing the
heat release rate fluctuations (Zhang & Ratner, 2019). Moreover, the overall
change in the flame shape at various levels of $\textrm{H}_{2}$-enrichment is
also a strong contributor to TAI while affecting pollutant emissions (Schmitt
et al., 2007). Tuncer et al. (2009) reported that a shift in the flame center
of mass towards the inlet due to an increase in the burning velocity with
hydrogen enrichment, controls the shape of the flame. Similarly, swirling
methane-air flame under different levels of H2 enrichment undergoes drastic
change from columnar, V-shape to M-shape flame (Davis et al., 2013; Taamallah
et al., 2015; Shanbhogue et al., 2016; Chterev & Boxx, 2019). In fact, these
effects persist at elevated pressures at various levels of H2 enrichment
(Chterev & Boxx, 2021) which determine the exact flame shape depending upon
the stability of the swirling flow. García-Armingol et al. (2014) found that a
V-flame transitions to an M-flame whenever the swirling flow features an outer
reaction zone.

Using linear stability analysis, Oberleithner et al. (2015) showed that
V-shape flames possess strong radial gradients, thus suppressing the formation
of precessing vortex cores (PVC) and the flame predominantly anchors along the
shear layer. In contrast, M-flames anchor along the outer recirculation zones
and possess smooth density gradients. These flames also exhibit PVCs. However,
the role of PVCs in driving thermoacoustic instabilities have not yet been
resolved satisfactorily (Candel et al., 2014). While initially thought to be a
contributor of thermoacoustic instability (Gupta et al., 1984; Huang & Yang,
2009), in many cases PVCs have been found to have limited impact on the
thermoacoustic characteristics of axisymmetric swirl combustors (Boxx et al.,
2010; Moeck & Paschereit, 2012; Oberleithner et al., 2013). This is due to
PVC-induced anti-symmetric flame perturbations, which have no impact on
oscillations in the global heat release rate (Candel et al., 2014).

In the specific context of thermoacoustic instability in swirl combustors,
addition of $\textrm{H}_{2}$ can shift the state to TAI from combustion noise
(Janus et al., 1997). Here, the term ”combustion noise” refers to a steady
operating condition characterized by low amplitude aperiodic acoustic
oscillations with broadband spectra (Candel et al., 2009) and fractal
characteristics (Nair & Sujith, 2014). The addition of hydrogen shifts the
occurrence of the state of TAI (Figura et al., 2007) to lower equivalence
ratios with the reduction in dynamic pressure amplitude. These characteristics
were ascribed to the hydrogen present in the combination of $textrmCH_{4}$ and
$textrmH_{2}$, which caused a higher rate of reaction and a shorter convective
time scale. Hong et al. (2013) reported a varying frequency and flame response
when hydrogen was varied as a result of change in the relative phase between
heat release rate and pressure fluctuations. In addition to seeing several
developments related to flame transfer function, Æsøy et al. (2020) showed a
reduction in the phase difference between heat release rate and pressure for
hydrogen-enriched fuel. Lee & Kim (2020) separately investigated the dynamics
of a mesoscale burner for the combustion of methane and hydrogen and reported
that the dynamics is connected with higher eigenmodes for pure hydrogen
operation. Similarly, many research studies show different effects of
hydrogen-enrichment on the combustion dynamics (Taamallah et al., 2015;
Shanbhogue et al., 2016; Zhang & Ratner, 2019; Chterev & Boxx, 2019).

###  1.3 Modal decomposition of thermoacoustic systems

Coherent structures are intimately related to the overall thermoacoustic
behaviour of turbulent combustors (Poinsot et al., 1987; Sterling & Zukoski,
1987; Schadow & Gutmark, 1992) as these structures dictate the behaviour of
the flow. Using techniques such as spark-Schlieren imaging and C2 radiation
mapping, these studies tracked the formation, growth and decay of coherent
structures. These structures affect the mixing process of the unburnt fuel-air
mixture with burnt hot gases. The mutual synchronization of the pressure
fluctuations with the periodic shedding of coherent structures are one of the
key mechanisms underlying the emergence of the state of TAI (Poinsot et al.,
1987; Pawar et al., 2017; George et al., 2018).

One of the most significant techniques for analysing coherent patterns in
turbulent flow is the proper orthogonal decomposition (POD) approach (Lumley,
1967; Sirovich, 1987). POD method reduces turbulent flow fields in terms of
orthogonal modes which are obtained by optimizing the
$\boldsymbol{\mathit{L}}^{2}$-norm or the kinetic energy of the velocity
field. The POD modes are orthogonal and characterize the most energy
containing coherent structures. Fluid dynamical systems have strongly coupled
subsystems and to understand the effect of coherent structures in the flow on
other correlated variables, Borée (2003) introduced the extended proper
orthogonal decomposition (EPOD). The method is used to project other
correlated variables such as concentration, temperature, etc., on the POD
modes of the velocity field and obtain analogous flow structures in such
correlated variables.

In the context of thermoacoustics, the relationship between coherent
structures and pressure variations has been investigated with POD. For
instance, Sui et al. (2017) used the POD method to understand the behavior of
acoustic fluctuations of different eigenmodes in space using the reconstructed
2-D pressure field of a Rijke tube. They reported that the heating source
mitigates the pressure fluctuations. They also showed 1:1 and 2:1 coupling
among different filtered modal coefficients using the traditional Lissajous
patterns (axisymmetric patterns). The Lissajous patterns are the
representation method for a pair of signals by which the frequency locking
behaviour can be provided in terms of the ratio between the dominating
frequencies of the pair of the signals (Greenslade Jr, 1993). Boxx et al.
(2010) used POD to study PVC and showed that the dominant vortices in the most
energetic mode are rotating in the counter-clockwise direction. However, the
second mode has the dominating vortices rotating clockwise. With the help of
POD modes, they reported that the first two dominating modes represent a
helical PVC and the third modes is associated with the thermoacoustic
pulsation in the axial direction. Duwig & Iudiciani (2010) used EPOD to study
the correlation among the modes of the reacting flow field and the unsteady
flame using PLIF data for the acoustically excited flame. Similarly, EPOD has
also been used to determine the effect of coherent structures on correlated
modes of related variables such as pressure, temperature, scalar
concentrations (OH* and CH* intensity fields) in various thermoacoustic
systems (Sieber et al., 2017; Wang et al., 2019; Lohrasbi et al., 2021).

###  1.4 Contributions of the present study

In this study, we aim to provide a framework to understand the synchronization
of the flow velocity field with heat release rate and acoustic pressure and to
clarify the manner in which the addition of $\textrm{H}_{2}$ affects coherent
flow structures of a highly turbulent swirling flame, influence heat release
rate fluctuations and modify the thermoacoustic behavior in a partially
premixed, swirl-stabilized flame. To characterize the coupled interaction, we
use POD to obtain the dominant modes of the measured velocity field. With the
help of dominating POD modes, we unravel their effect on the heat release rate
fluctuations through the use of EPOD modes. In this study, we are referring
the $x$-component of the velocity ($u_{x}$), acquired in the Cartesian frame
of reference as transverse component and the $y$-component of the velocity
($u_{y}$) as the axial component of the flow field. This also enables us to
correlate how the thermoacoustic behaviour is affected by the addition of
$\textrm{H}_{2}$. We compare the behavior of such extended POD modes during
the state of period-1 and period-2 thermoacoustic instability (Kushwaha et
al., 2021) and contrast their characteristics with the baseline case of
combustion noise. By analyzing the spatial EPOD modes simultaneously, we
relate the behavior of the dominant modes of the swirling flow, the flame
structure and the heat release rate field with the self-excited acoustic mode
of the combustor. In this manner, we delineate the overall synchronization
behaviour of the swirling flames during various dynamical states and relate
them to the dominant modes or the coherent structure of the swirling flow.

The rest of the paper is organized as follows. In §2, the details of the
experimental set-up and the imaging techniques used are provided. In §3, we
discuss the methodology used for characterizing the coupled interactions and
check the existence of the chaos. This is followed by results and discussion
in §4. We summarize our major findings in §5.

##  2 Experimental setup

Figure 1: (a) Schematic of the swirl-stabilized PRECCINSTA combustor. Various
features of the swirling flow are identified: OSL - outer shear layer, ISL -
inner shear layer, ORZ - outer recirculation zone and IRZ - inner
recirculation zone. The domain of particle image velocimetry (PIV),
OH*-chemiluminescence and planar laser-induced fluorescence (PLIF) imaging are
also indicated (Kushwaha et al., 2021). Instantaneous (b) vorticity field, (c)
OH-PLIF intensity showing the flame area, and (d) OH*-chemiluminescence field
showing the heat release rate distribution are also shown for reference.

The experiments were performed in a technically premixed gas turbine model
combustor (PRECCINSTA configuration) shown in figure 1. After passing through
a swirl generator embedded with 12 radial swirl channels, air enters the
cylindrical plenum. Fuel (methane with variable quantities of hydrogen
addition) is injected into each swirl channel through a 1 mm orifice. The
reactant mixture then passes through a conical nozzle with an exit diameter
($D=27.85$ mm) and reaches the combustion chamber. Within the nozzle is a
conical centerbody with a rounded tip. The chamber has a height of 114 mm and
a square cross-section area of 85 $\times$ 85 mm2. The combustor has quartz
windows and provide optical access for performing various flow diagnostics.
For all the experiments reported here, the combustor is maintained at the same
initial conditions.

Experiments involving the dynamics of the state of TAI of
$\textrm{CH}_{4}$-air flame at various levels of $\textrm{H}_{2}$ enrichment
are performed while keeping the global equivalence ratio constant. This is
ensured by simultaneously decreasing $\dot{m}_{\rm CH_{4}}$ and increasing
$\dot{m}_{\rm H_{2}}$. The dynamical states, analysed in this paper, have the
operating conditions, indicated in table 1. There is also a slight variation
from 22.17 kW to 25.31 kW in the thermal power $P_{\textrm{th}}$ when the
level of $\textrm{H}_{2}$ is varied. Based on the nozzle’s exit diameter, the
nominal Reynolds number is in the order of $2x10^{4}$. A more comprehensive
explanation of the experimental setup, instrumentation, equipment sensitivity
and uncertainty, and optical diagnostics are presented in § S1 of supplemental
materials.

Sr. No. | $\textrm{H}_{2}(\%)$ | $Re(\times 10^{4})$ | $\phi_{g}$ |  $P_{\textrm{th}}$ (kW) |  $p^{\prime}_{\textrm{rms}}$ (kPa) | Dynamical state
---|---|---|---|---|---|---
1 | 0 | 2.41 | 0.65 | 25.31 | 0.09 | Chaotic
2 | 20 | 2.74 | 0.65 | 22.26 | 0.91 | P2 LCO
3 | 50 | 2.64 | 0.65 | 22.50 | 0.71 | P1 LCO
Table 1: The values of different operating parameters during the experiments.
Keys: $\textrm{H}_{2}(\%)$ \- Volumetric percentage of $\textrm{H}_{2}$ in the
fuel; $Re$ \- Nominal Reynolds number; $\phi_{g}$ \- Global equivalence ratio;
$P_{\textrm{th}}$ \- Thermal power rating (kW) and dynamical states. The
experimental cases analysed here is highlighted.

##  3 Method of analysis

###  3.1 Extended proper orthogonal decomposition for obtaining correlated
flow structures

Proper orthogonal decomposition (POD) is a data-driven technique to reduce
large-scale, high-dimensional processes or data-sets into low-dimensional
deterministic modes (Berkooz et al., 1993; Oberleithner et al., 2015; Taira et
al., 2017). POD decomposes snapshots of space and time-dependent observables
into orthogonal modes, ranks them according to their energy and obtains the
spatial structure of the corresponding modes. Here, we use the snapshot POD
method suggested by Sirovich (1987). It has a spatial average operator to
extract coherent structures from the time-resolved snapshots of fields such as
velocity, pressure, temperature and concentration. The method uses
$\boldsymbol{\mathit{L}}^{2}$-norm for classifying the modes, which is
equivalent to the fluctuating kinetic energy for the velocity field and
variance for other fields.

The velocity fluctuations $(\mathbf{u}^{\prime})$ can be decomposed in terms
of the spatial modes $(\boldsymbol{\psi}_{i})$ and temporal modes
$(\boldsymbol{a}_{i})$ in the following manner:

$\mathbf{u}^{\prime}(x,y,t):=\mathbf{u}(x,y,t)-\bar{\mathbf{u}}(x,y):=\sum^{N}_{i=1}a_{i}(t)\
\boldsymbol{\Psi_{i}}(x,y),$ (1)

where, $\bar{\mathbf{u}}(x,y)$ refers to time average of the instantaneous
velocity field $\mathbf{u}(x,y,t)=u_{x}\hat{i}+u_{y}\hat{j}+u_{z}\hat{k}$ with
a resolution of $m\times n$. Depending upon the component of velocity of
interest, we reshape each image as a row and form a two-dimensional matrix
that has a size of $N\times(m\times n)$ (Taira et al., 2017), where $N$ is the
total number of images. Each snapshot is considered to be an element of the
square-integrable vector field $\boldsymbol{\mathit{L}}^{2}(\xi)$ specified in
the Hilbert space. The class of square-integrable functions is unique for
compatibility with an inner product, which allows us to define conditions of
orthogonality. The inner product of two vector $\boldsymbol{\alpha}$ and
$\boldsymbol{\beta}$ fields in the Hilbert space is defined as

$(\boldsymbol{\alpha,\beta})_{\xi}\coloneqq\int_{\xi}\boldsymbol{\alpha}\cdot\boldsymbol{\beta}\
d\boldsymbol{x},$ (2)

where, in a spatial domain $\xi\subset\rm I\\!R^{3}$, $\boldsymbol{x}$ is a
point and the corresponding norm $||\boldsymbol{\alpha}||_{\xi}$ is given by

$||\boldsymbol{\alpha}||_{\xi}\coloneqq\sqrt{(\boldsymbol{\alpha},\boldsymbol{\alpha})_{\xi}}.$
(3)

Usually, Compared to the number of snapshots, the number of spatial points is
significantly bigger. So, we quantify the relation between individual
snapshots by formulating the correlation matrix R(N×N)

$R_{i,j}\coloneqq\frac{1}{N}\left[\mathbf{u}^{\prime}\left(\mathbf{x},t_{i}\right),\mathbf{u}^{\prime}\left(\mathbf{x},t_{j}\right)^{T}\right].$
(4)

Here, each snapshot is stored in $\mathbf{x}=x_{i}$ after reshaping from size
$(m$ × $n)$, and $T$ indicates the transpose of the matrix.

Using eigenvalue decomposition, we calculate the eigenvectors and eigenvalues
of the correlation matrix R,

$\centering\textbf{R}a_{i}=\lambda_{i}a_{i}.\@add@centering$ (5)

R has real and non-negative eigenvalues $\lambda_{i}\geq 0$. We assume the
eigenvalues are arranged in decreasing order without losing generality, i.e.
$\lambda_{1}\geq\lambda_{2}\geq.....\geq\lambda_{N}.$ The eigenvectors
$\textbf{a}_{i}$ are called the temporal coefficients of the velocity field
and follow the orthogonality condition

$\frac{1}{N}\sum^{N}_{k=1}a_{i}(t_{k})a_{j}(t_{k})=\lambda_{i}\delta_{ij},$
(6)

where, $\delta_{ij}$ is the Kronecker Delta function.

The spatial modes are determined as follows using the projection of the
snapshots on the temporal coefficients,

$\boldsymbol{\Psi_{i}}(\mathbf{x}):=\frac{1}{N\lambda_{i}}\sum^{N}_{j=1}a_{i}(t_{j})\mathbf{u}^{\prime}(\mathbf{x},t_{j}).$
(7)

The spatial modes are orthogonal by construction, viz,
$(\mathbf{\Psi}_{p},\mathbf{\Psi}_{q})_{\xi}=\delta_{pq}$. After reshaping
$\boldsymbol{\Psi_{i}}(\mathbf{x})$, we can get spatial modes
$\boldsymbol{\Psi_{i}}(x,y)$ having $m$ and $n$ as rows and columns,
respectively.

Additionally, The correlation between the flow structures and structures in
other quantities such as concentration, temperature, or pressure, can be
identified using the POD method. This can be achieved by the extended POD
analysis (Borée, 2003). The velocity field $u(\mathbf{x},t)$ is measured
concurrently with a collection of intensity fields $I(\mathbf{x},t)$.
$I(\mathbf{x},t)$ is first decomposed into the average $\bar{I}(\mathbf{x},t)$
and a fluctuating part $I^{\prime}(\mathbf{x},t)$

$I(\mathbf{x},t)=\bar{I}(\mathbf{x})+I^{\prime}(\mathbf{x},t).$ (8)

Similar to the spatial modes $\Psi(\mathbf{x})$ of velocity, a set of extended
POD modes $\boldsymbol{\Omega}_{i}$(x) can be defined as

$\boldsymbol{\Omega_{i}}(\mathbf{x})=\frac{1}{N\lambda_{i}}\sum^{N}_{j=1}\textbf{a}_{i}(t_{j})I^{\prime}(\mathbf{x},t_{j}),$
(9)

where $\boldsymbol{a_{i}}(t_{j})$ are the temporal coefficients of the POD
modes of velocity fields. In this manner, we correlate coherent structures
observed in the swirling flow with the extended modes in the OH-PLIF ($I_{P}$)
and OH*-chemiluminescence fields ($I_{C}$).

###  3.2 Temporal synchronization of coherent structures

The POD and extended POD analysis quantifies the spatial extent of flow and
flame structures. The temporal interactions of these spatially extended modes
are equally important, and we quantify these interactions based on their
synchronization characteristics. This is achieved by time series analysis of
the POD temporal coefficients along with pressure and global heat release rate
fluctuations.

We compute the instantaneous phase using the concept of analytic signals and
by utilizing the Hilbert transform (Rosenblum et al., 1996). For a given
signal $x(t)$ normalized by its global maxima, the instantaneous amplitude
$A(t)$ and phase $\phi(t)$ can be obtained from the complex analytic signal

$\xi(t)=x(t)+i\mathcal{H}\left(x(t)\right)=A(t)\exp\left(i\phi(t)\right)$ (10)

where $A(t)$ and $\phi(t)$ denote the instantaneous amplitude and phase of the
analytic signal. The Hilbert transform is
$\mathcal{H}(x(t))=1/\pi\int^{\infty}_{-\infty}x(\tau)/(t-\tau)d\tau$, where
the integral is evaluated at the Cauchy principle value. The analytic nature
of temporal coefficients is verified for different states by plotting the
phase space trajectory of $\xi(t)$ and adjudging its center of rotation. For
periodic signals, there is a unique center of rotation, allowing us to
uniquely specify the phase of the signal. As instantaneous phase is a
monotonically increasing or decreasing quantity, it is first wrapped to the
interval $[-\pi,\pi]$. Finally, synchronization among pairs of signals are
evaluated by measuring the instantaneous phase difference:
$\Delta\phi_{x_{1},x_{2}}=\phi_{x_{1}}(t)-\phi_{x_{2}}(t)$. Signals are
considered to be phase locked when the difference $\Delta\phi_{x_{1},x_{2}}$
becomes bounded to a small interval $\epsilon$ ($\leq 2\pi$) around the mean
phase $C$, to wit,
$|\Delta\phi_{x_{1},x_{2}}(t)-C|=|\phi_{x_{1}}-\phi_{x_{2}}-C|\leq\epsilon$
(Pikovsky et al., 2001).

The temporal interactions of the identified structures are further visualised
using the phase portraits or Lissajous plots for pair of signals. The exact
relation between the signals are inferred based on the dominant frequencies
($f_{1},f_{2}$) of the signals and the mean phase difference
($\Delta\phi_{m}$). This is done by constructing the pair of analytical
signals based on superposition of the dominant frequencies,

$\begin{split}x_{1}&=A_{1}\sin(2\pi f_{1}t+\Delta\phi_{m})+A_{2}\sin(2\pi
f_{2}t+\Delta\phi_{m}),\\\ x_{2}&=B_{1}\sin(2\pi f_{1}t)+B_{2}\sin(2\pi
f_{2}t),\end{split}$ (11)

where $f_{1}$ is the dominating frequency of the signal and $f_{2}$ is the
multiple of $f_{1}$, i.e., $f_{2}$ = 2$f_{1}$. $A_{1},A_{2},B_{1}$ and $B_{2}$
are the amplitudes associated with the dominant modes $f_{1}$ and $f_{2}$ for
different pair of signals.

###  3.3 Quantifying chaotic evolution of coherent structures

In order to quantify chaotic time evolution of coherent structures, we perform
$0-1$ test which quantifies the unbounded growth of phase space trajectories
typical of chaos (Gottwald & Melbourne, 2004). For any input time series
$\gamma(t)$, the first step is to compute the translation variables $x(n)$ and
$y(n)$ such that

$x(n)=\sum_{j=1}^{n}\gamma(t)\cos(tc),\enspace
y(n)=\sum_{j=1}^{n}\gamma(t)\sin(tc)$ (12)

where, $n=1,2,...,N$, where $N$ denotes the number of data points present in
time signal. For our analysis, we have selected the value of $c$ in the
interval $(\pi/5,4\pi/5)$ (Nair et al., 2013). The nature of these two
translation variables provide indications to determine if chaos exists in the
system. For periodic or quasi-periodic oscillations, the variables show
bounded behavior. However, for chaotic dynamics, their behavior is unbounded
and irregular. The evolution of the trajectory in the $x-y$ plane for
increasing $n$ can be computed with the help of modified mean square
displacement $M(n)$ as follows (Ashwin et al., 2001):

$\displaystyle
M(n)=\frac{1}{N}\sum_{j=1}^{n}\left[(x(j+n)-x(j))^{2}+(y(j+n)-y(j))^{2}\right]-\frac{1}{N}\sum_{j=1}^{n}\gamma(t)\frac{1-\cos(jc)}{1-\cos(c)}$
(13)

For a chaotic state, the displacement $M(n)$ between the translational
variables grows monotonically with $n$, while it becomes nearly constant for
regular states. With the use of linear regression, the asymptotic growth rate
($K$) of the mean displacement is computed as follows:

$\centering K=\lim_{n\to\infty}\frac{\log M(n)}{\log n}.\@add@centering$ (14)

Gottwald & Melbourne (2009) pointed that for the small values of n, the linear
regression methods results in altering the $K$ values. To estimate the $K$
value, we use a correlation method which performs better than the linear
regression method. In this method, $K$ is the correlation coefficients for the
vectors $\zeta=(1,2,3,...,n)^{T}$ and $\Delta=(M(1),M(2),M(3),...,M(n))^{T}$.
Subsequently, the correlation coefficient can be defined as

$K=\textrm{corr}(\zeta,\Delta)=\frac{\textrm{cov}(\zeta,\Delta)}{\sqrt{\textrm{var}(\zeta)\textrm{var}(\Delta)}}\in[-1,1]$
(15)

where,

$\textrm{cov}(\zeta,\Delta)=\frac{1}{n}\sum_{j=1}^{n}\left[\zeta(j)-\bar{\zeta}\right]\left[\Delta(j)-\bar{\Delta}\right],\enspace\textrm{var}(\zeta)=\textrm{cov}(\zeta,\zeta),\enspace\textrm{and}\enspace\bar{\zeta}=\frac{1}{n}\sum_{j=1}^{n}\zeta(j).$

The value of $K$ lies between 0 and 1. For a chaotic signal, $K$ takes a value
close to 1, and for a regular signal, it approaches a value close to 0.

##  4 Results and discussion

We begin by examining three distinct dynamical states - chaotic, period-1 and
period-2 oscillations exhibited by the combustor at different levels of
$\textrm{H}_{2}$ enrichment. The operating conditions corresponding to these
states are highlighted in table 1. The combustor is initially in a
thermoacoustically stable state, operating only on $\textrm{CH}_{4}$ at a
premixed equivalence ratio of $\phi=0.65$ and a thermal power of 25.31 kW.
During these conditions, thermoacoustic system is characterized with low-
amplitude, chaotic pressure oscillations. We refer to this baseline state as
chaotic state. When $\textrm{CH}_{4}$-air flame is enriched with
$\textrm{H}_{2}$, the pressure fluctuations inside the combustor exhibits high
amplitude period-1 and period-2 limit cycle oscillations (LCO) at 50% and 20%
hydrogen enrichment. We refer to these states as period-1 and period-2 states,
respectively.

We disentangle large-scale coherent structures in the flow velocity field and
their impact on the flame surface and heat release rate field by performing
the POD and extended POD analysis. Large-scale structures in turbulent flows
contain majority of the turbulent kinetic energy which commences the cascade
process and turbulence phenomenology. Thus, POD modes with the largest kinetic
energy content along with correlated flame structures would dominate the flow
and flame dynamics. This can be observed simply by noting that the turbulent
kinetic energy $E$ and fluctuating kinetic energy $E_{i}$ contained in each
POD mode are connected via the relations between the eigenvalues of the
correlation matrix (4) and the flow field fluctuations:

$\displaystyle
E_{i}\coloneqq\frac{1}{2}\overline{(\textbf{u}^{\prime},\boldsymbol{\Psi}_{i})^{2}}=\frac{1}{2}\lambda_{i},\qquad
E=\sum_{i=1}^{N}E_{i}=\frac{1}{2}\sum_{i=1}^{N}\lambda_{i}.$ (16)

Figure 2: Fraction of kinetic energy contained in the spectrum of POD modes
(16) obtained from the flow velocity field ($\textbf{u}^{\prime}$) for the
three dynamical states highlighted in Table 1. The key role played by large-
scale coherent structures during the states of thermoacoustic instabilities
(period-1 and period-2) is highlighted by the first two principal modes
containing an order of magnitude higher kinetic energy than observed during
stable operation.

Figure 2 shows the percentage of the turbulent kinetic energy ($E_{i}/E$) that
the first 100 modes of the velocity field contain for different dynamical
states. For all the cases, the first 100 modes contain over 90% of the
turbulent kinetic energy, with the first few principal modes accounting for a
major fraction of the total kinetic energy. During chaotic oscillations, the
first three POD modes contain $14.74\%$ of the total kinetic energy. Further,
for period-1 LCO, the first two modes contain $19.87\%$ of the turbulent
kinetic energy. The third mode contains an order of magnitude less fraction of
the total energy. The energy content decreases drastically for higher POD
modes. For period-2 LCO, the first three modes contain comparable proportion
of the total kinetic energy, with the total amounting to around $29.86\%$ of
the total energy.

In keeping with the definition, the spatial structures associated with the
first few principal modes are expected to dominate flow field dynamics. For
subsequent analysis and visualization, we consider the transverse component of
the velocity field ($u_{x}$) and its effect on the turbulent flame. Our
conclusions remain the same when the axial component ($u_{y}$) is considered
instead (see supplementary information).

###  4.1 Chaotic oscillations in the absence of H2

Figure 3: Flow and flame dynamics for the purely $\textrm{CH}_{4}$-air flame
for the baseline condition. Mean of the (a) transverse velocity component
$u_{x}$, (b) flame brush obtained from OH-PLIF images, and (c) heat release
rate fluctuations obtained from OH*-chemiluminescence. Panels (d-f) shows the
first three POD modes of $u_{x}^{\prime}$ and the extended modes associated
with $I_{P}^{\prime}$ and $I_{C}^{\prime}$.

The flame and flow dynamics of the swirl combustor operating only on
$\textrm{CH}_{4}$ at $\phi=0.65$ are shown in figure 3. The first row shows
the mean of the transverse velocity component ($\bar{u}_{x}$), the flame brush
obtained from time-averaged OH-PLIF imaging ($\bar{I}_{P}$) and the mean heat
release rate obtained from OH*-chemiluminescence imaging ($\bar{I}_{C}$). The
time averaging is performed over 3000 instantaneous images (or 0.3 seconds).
Figure 3a shows high transverse velocity along the shear layers, with the
opposite signs indicating the clockwise direction of the swirling flow. In
addition, the flow shows very weak inner and outer recirculation (also seen
from figure 2a showing $\bar{u}_{y}$ in supplementary materials). Hence, the
flame can be seen to stabilize along the inner shear layer, resulting in a
characteristic V-shape of the turbulent flame brush (figure 3b). The thickness
of the flame brush can also be seen to increase downstream in comparison to
the attachment point at the exit of the dump plane. As chemiluminescence
images are line-of-sight integrated in the out-of-plane direction, increased
flame brush thickness can be seen to result in sustained, continuous band of
heat release rate profile downstream of the combustor (figure 3c).

Figure 3d shows the first three POD modes of the transverse ($u_{x}$)
velocity, together which account for $14.74\%$ of TKE energy (figure 2). The
first POD mode of the transverse velocity shows a coherent structure that
grows in magnitude in the streamwise direction. The structure indicates a
vortex bubble with positive velocity magnitude, depicting the overall
direction of its motion. In contrast, mode 2 and mode 3 depict modes with same
wavenumber and frequency, whose magnitude increases and then decays
downstream. These structures are shifted in the streamwise direction by
approximately quarter wavelength. Mode 2 and 3 are part of an oscillating
process, linked to a helical mode instability around the recirculation bubble
observed in POD mode 1. This can be observed in the Lissajous plot between
mode coefficients $a_{2}-a_{3}$ (figure 4). We further note here that the
first three POD modes associated with $u_{y}$ along the shear layer (in figure
2d in supplemental materials) also indicate the presence of the same coherent
structures with the same axial wavenumber with each mode shifted axially.
Additionally, all the three modes exhibit advection of the central
recirculation zone (in figure 2d in supplemental materials).

Figure 4: (a-c) Phase portrait of the pairs of temporal coefficients ($a_{1}$
\- $a_{3}$) of the velocity field for the state of chaotic oscillations. The
red lines indicate a few periodic orbits.

The extended POD modes of the flame profile ($I_{p}$) associated with the POD
modes of $u_{x}$ are also shown in figure3e. We notice a number of salient
features of the flame dynamics from the extended POD modes of the flame
surface $I_{p}$ in figure 3e. The dominant POD mode of $u_{x}$ associated with
the central recirculation bubble, does not correlate with extended POD modes
of the flame surface. Only the weaker structures along the shear layer are
reflected in the extended POD modes of the flame surface. This is due to the
V-shaped flame (figure 3b) which stabilizes along the inner shear layer and
does not extend into the inner recirculation zone. The increased activity in
the flame surface is further emphasized in the second and third extended POD
modes of the flame surface. The coherent structures present along the shear
layer in the second and third POD modes of $u_{y}$ are clearly correlated to
the extended POD mode structures. The extended POD mode structure is also
shifted by a quarter wavelength. The amplitude of the structure can be see to
increase downstream of the flame attachment region, identifying regions
contributing to the most of the changes in the flame dynamic. Interestingly,
while the coherent structure is anti-symmetric (mode 2 & 3 in figure 3d) due
to helicoidal instability in the flow field, the extended mode structure of
the flame surface clearly retains axisymmetry.

Figure 5: Characterization of the thermoacoustic response during the state of
chaotic oscillations. (a) Time series and (b) power spectra of fluctuations in
acoustic pressure, heat release rate and the temporal coefficients of the
first three POD modes. (c) Phase space trajectory in the complex analytical
plane for the part of the signals indicated in panel (a).

The effect of these local flow and flame dynamics on the global heat release
rate profile can be understood from extended mode structure of heat release
rate distribution shown in figure 3f. As with the extended structure of flame
surface, the first extended mode of heat release rate does not correlate with
the central recirculating zone, corroborating the fact that the central vortex
bubble arising due to the swirl does not affect the fluctuations in the heat
release rate. The second and third extended modes on the other hand correlate
well with the coherent structure present in the velocity field. Indeed, we
notice that structure of extended POD of heat release exists along the shear
layers along with the extended mode structure of the flame surface. The second
and third mode are again related to the same structure, only displaced by a
quarter wavelength. The banded nature of these structures can be ascribed to
the fact that the chemiluminescence imaging is line-of-sight integrated,
revealing the annular region over which heat is release from the three
dimensional flame structure. We note here that our observations remain the
same when the streamwise velocity $u_{y}$ is used for obtaining extended POD
modes from flame surface and heat release rate (see figure 2e,f in
supplemental materials). These observations show that the structures from the
heat release rate distribution and flame surface corresponding to the most
dominant POD mode do not correlate. The coherent structures from other modes
show some correlation with the structures from other fields. However, due to
low turbulent kinetic energy of mode 2 and 3, their overall effect remains
minimal.

Signals $\downarrow$ Dynamical states $\rightarrow$ | Chaotic oscillations | Period-1 LCO | Period-2 LCO
---|---|---|---
$p^{\prime}$ | 0.99 | 0.08 | 0.01
$\dot{q}^{\prime}$ | 0.98 | 0.11 | 0.05
$a_{1}$ | 0.62 | 0.15 | 0.27
$a_{2}$ | 0.99 | 0.12 | 0.32
$a_{3}$ | 0.99 | 0.61 | 0.79
Table 2: Results of the 0-1 test for different signal during all the dynamical
states discussed in this paper. The values close to 1 indicate the existence
of chaos in the signal. Moreover, the signal with regular dynamics show the
values close to 0.

We next relate the manner in which these spatial modes are related to the
thermoacoustic response in the system. Figure 5 presents the normalized time
series of the acoustic pressure ($p^{\prime}$) and heat release rate
fluctuations ($\dot{q}^{\prime}$) along with the time evolution of the
temporal coefficients of the first three POD modes. Here,
$\dot{q}^{\prime}(t)$ is obtained by a global summation from individual
chemiluminescence images. Each of the time series are normalized by their
respective maxima to aid comparison. The aim is to relate the manner in which
the spatial modes evolve in time and relate to fluctuations in $p^{\prime}$
and $\dot{q}^{\prime}$. In each case, the time evolution is characterized by
the power spectral density and the phase portrait of the analytic signal
defined according to (10).

The present baseline case corresponds to the low-amplitude stable operation of
the combustor. The pressure and heat release rate oscillations during this
baseline case remains chaotic, as reported earlier in (Kushwaha et al., 2021).
This can be observed in figure 5b where the power spectral density for
$p^{\prime}$, $\dot{q}^{\prime}$ and the temporal coefficients
($a_{1},a_{2},a_{3}$) depict a broadband behavior. In addition, the phase
portrait featuring the analytic representation of these signals depict the
absence of a unique center of rotation. Hence, the signals are non-analytic,
and a phase cannot be ascribed to these signals.

We perform the 0-1 test (described in §3.3) for these signals to ascertain
whether their dynamics is chaotic or not. The value of the asymptotic growth
rate $K$ of the displacement $M$ between translational variables is tabulated
in Table 2. We note here that all the signals display values which are close
to unity, implying chaotic evolution of each of these quantities. Thus, the
heat release rate and pressure oscillations display chaos during stable
combustor operation, corroborating past results (Nair & Sujith, 2014; Kushwaha
et al., 2021).

###  4.2 Period-1 limit cycle oscillations at 50% $\textrm{H}_{2}$ enrichment

Figure 6: Flow and flame dynamics during Period-1 TAI with 50% hydrogen
enrichment. Mean of the (a) transverse velocity component $\bar{u}_{x}$, (b)
flame brush obtained from OH-PLIF images, and (c) heat release rate
fluctuations obtained from OH*-chemiluminescence. Panels (d-f) shows the first
three POD modes of $u_{x}^{\prime}$ and the extended modes associated with
$I_{P}^{\prime}$ and $I_{C}^{\prime}$.

The combustor shows the state of TAI with the addition of hydrogen (50%) at a
constant equivalence ratio of $\phi=0.65$ and a thermal power of $P=20$ kW. We
observe sustained, period-1 LCO with an amplitude of
$p^{\prime}_{\textrm{rms}}=0.91$ kPa. Figure 6 shows the time-averaged
velocity field, flame brush and heat released rate. Figure 6a shows high
transverse velocity along the shear layers having opposite sign which
indicates swirling flow. The shear layer is much broader in comparison to the
baseline case. The flame also stabilizes along the shear layers and extends
towards the outer recirculation zone, making it an M-shaped flame (figure 6b).
The flame can be seen to be anchored very close to the dump plane. The time-
averaged image of the heat release distribution field shows a concentrated
region of high heat release rate. The location of the peak of $\bar{I}_{C}$ is
located much closer to the flame anchor point, in contrast to the baseline
chaotic case where the peak occurred much further downstream.

Figure 6d shows the first three dominant POD modes associated with the
transverse velocity fluctuations $u_{x}^{\prime}$. These three modes
collectively hold $30\%$ of the total kinetic energy. The first and second POD
modes indicate the presence of the same coherent structure in the form of a
toroidal vortex of the same frequency and wavenumber. The modes can be seen to
be shifted by a quarter of a wavelength and $\pi/2$ radians (see figure 9a).
In contrast to the baseline case, the dominant coherent structure during
period-1 LCO comprises the helical toroidal vortex. The amplitude of the
toroidal vortex can be observed to be much stronger than it was for the
baseline case (cf. figure 3). The toroidal vortex can also be seen to extend
to a much larger radial domain, with the anti-phase features alternating
asymmetrically on each side of centre flow axis ($x=0$). In addition to the
toroidal vortex, the first two POD modes also show the presence of an anti-
symmetric mode structure along the IRZ which convects in time. The third mode
shows the presence of an axisymmetric coherent structure extending along the
length of the combustor.

Figure 7: Characterization of the dynamic response during period-1 TAI. (a)
Time series and (b) power spectra of acoustic pressure, heat release rate and
temporal coefficients of first three POD modes. The time series are normalised
by their respective maxima values. (c) Phase space trajectory in the complex
analytical plane for the part of the signal indicated in panel (a).

The extended POD modes of the flame surface is shown in figure 6e. For the
first two modes, we notice that the extended structure is distinctly M-shaped
with branches of the flame surface extending along the nodal line of the
toroidal as well as the inner coherent structure. The two extended modes,
which are shifted by $\pi/2$ radians, are associated with positive and
negative intensities, indicating the correlation between POD mode and the
extended POD mode structure of the flame surface. In addition, the first and
the second EPOD modes have the maximum intensity along the shear layers.
However, we observe small scale structures in the third EPOD mode structure of
the flame. Figure 6f present the EPOD modes of the heat release rate
distribution. The first two EPOD modes show the structures in the IRZ and
along the shear layer. The structures along shear layer confirms the existence
of a toroidal vortex in the flow field (figure 8d). However, we do not observe
the high intensity structure in the third EPOD modes of the OH-PLIF images and
OH*-chemiluminescence. With the random arrangement of structures in the third
extended modes of the flame surface and heat release rate, the coherent
structures from the flow field do not correlate with the inner recirculating
zone.

Figure 8: (a-f) Phase portrait of the temporal coefficients of the flow
velocity field with respect to acoustic pressure and heat release rate along
with analytical curves having the same dominating frequencies and phase
differences as the original data and the corresponding phase difference with
the red line indicating the mean value of phase difference, for the state of
TAI with period-1 LCO.

To probe the dynamics further, the time series, power spectra and analytical
signal are shown in figure 7. The time series of $p^{\prime}$ and
$\dot{q}^{\prime}$ signals, normalised with their respective global maxima
values, show high amplitude fluctuations compared to that during the chaotic
oscillations (figure 7a). We also observe a distinct peak at 415 Hz and its
higher harmonic at 827 Hz for the $p^{\prime}$ and $\dot{q}^{\prime}$ signals
(figure 7b). The temporal coefficients $a_{1}$ and $a_{2}$ are periodic in
nature (figure 7a) and exhibit distinct peak at 415 Hz (figure 7b). However,
the third mode does not show periodic behaviour (figure 7a) and we observe a
broadband power spectrum without any distinct peak (figure 7b). However, the
temporal coefficient of the third mode shows aperiodic behaviour (figure 7a)
and exhibit a broadband power spectrum without any distinct frequency peak
(figure 7b). The trajectory on the complex plane also does not have a unique
center of rotation, indicating that the signal is not analytic. Further, from
Table 2, we notice that the $0-1$ test of $a_{1}$ indicates a value of
$K=0.61$, indicating a possible chaotic evolution of mode 3.

In order to understand the coupled behaviour between the variables, we show
the Lissajous plots and determine the phase difference between various pairs
of signals in figure 8. The phase space trajectory plotted in the complex
analytic plane, shown in figure 7c, clearly depicts a unique center of
rotation for all the pair of signals except for $a_{3}$. This indicates that
all signals other than $a_{3}$ are analytic in nature and their phases are
well-defined. We compute the mean of the instantaneous phase difference and
show it with a red line. With the help of dominant frequencies and the mean
phase difference for each pair of signal, we generate the analytical curve
according to (11) and depict it in the Lissajous plot. These analytical curves
identify the phase and frequency relationship among the dominant POD modes and
the pressure and heat release rate fluctuations. We further note that the
pressure fluctuations correspond to the acoustic standing wave of the
combustor, ensuring that the phase jumps by 2$\pi$ when the nodal line is
crossed. Thus, the phase relationships discussed next are expected to remain
unchanged close to the dump plane of the combustor where the POD are shown in
figure 7.

In figure 8a, the phase plot between $a_{1}$ and $a_{2}$ shows that the first
two POD modes are phase-locked around $\pi/2$ and are part of the same
oscillating coherent structure. We also notice that $p^{\prime}$ and
$\dot{q}^{\prime}$ are phase-locked close to a mean value of
$\langle\Delta\phi\rangle=\pi/3$ (figure 8d), implying that the Rayleigh
criteria is fulfilled during the state of TAI. $\langle\cdot\rangle$ indicates
a time-averaged value. The relationship between the dominant POD modes and
$p^{\prime}$ and $\dot{q}^{\prime}$ are quite interesting. We note that
$a_{1}$ is phase-locked to $p^{\prime}$ at $\Delta\phi=3\pi/4$ denoted by the
elliptical Lissajous plot (figure 8b) and to $\dot{q}^{\prime}$ at
$\Delta\phi=\pi/2$ as denoted by the circular plot (figure 8c). Next, we
notice that the second mode $a_{2}$, which is associated with the same
dominant coherent structure, is phase-locked to $p^{\prime}$ at $-\pi/2$
(figure 8e) and with $\dot{q}^{\prime}$ with a phase difference of
$\approx-5\pi/8$.

One of the mechanisms through which the state of TAI occurs is often related
to the presence of coherent structures. Previous researches have shown that
the heat release rate fluctuations induced by the coherent structures are in-
phase with the pressure fluctuations (Poinsot et al., 1987; Chakravarthy et
al., 2007; Pawar et al., 2017). However, the phase relationships presented
above suggest a non-trivial manner in which the coherent modes interact with
$p^{\prime}$ and $\dot{q}^{\prime}$ fluctuations. The present relationship
suggests that most energetic coherent structures evinces heat release rate
fluctuations delayed by a phase-lag of $\pi/2$, which are subsequently related
to $p^{\prime}$ fluctuations after a phase delay of $\pi/3$. However, the
mismatch between the mean-phase relationship obtained from
$\langle\Delta\phi_{a_{1},p^{\prime}}\rangle$ and
$\langle\Delta\phi_{a_{1},\dot{q}^{\prime}}\rangle$ with
$\langle\Delta\phi_{p^{\prime},\dot{q}^{\prime}}\rangle$ indicates additional
mechanism through which the coherent structure is coupled to the pressure
fluctuations.

Thus, we have identified the phase-relationships between the coherent
structures containing the most of the energy, flame fluctuations, global heat
release rate fluctuations and the pressure field. The structures from the
flame surface at the central plane and heat release rate distribution field
are highly correlated with the coherent structures from the velocity field and
temporally shows the 1:1 frequency locking behaviour among the subsystems in
the combustor.

###  4.3 Period-2 limit cycle oscillations (P2 LCO)

Let us now turn our attention to the coupled dynamics observed during the
state of TAI with period-2 LCO when the combustor is operated at a constant
equivalence ratio $\phi=0.65$ and thermal power $P=20$ kW with only $20\%$
hydrogen. Like earlier, we plot the mean, POD modes and extended POD modes in
figure 9 and the temporal dynamics in figure 10.

Figure 9: Mean of the (a) transverse velocity component $u_{x}$, (b) flame
brush obtained from OH-PLIF images, and (c) heat release rate fluctuations
obtained from OH*-chemiluminescence. Panels (d-f) shows the first three POD
modes of $u_{x}^{\prime}$ and the extended modes associated with
$I_{P}^{\prime}$ and $I_{C}^{\prime}$. for the state of period-2 LCO with 20%
hydrogen by volume of the fuel.

In the mean transverse velocity field (figure 9a), we notice high velocity
fluctuations along the shear layers and the outer recirculation region (ORZ)
with the opposite signs. The strength of fluctuations are lower than that
observed during period-1 LCO (figure 6a). Moreover, the mean of the OH-PLIF
image shows the presence of a columnar shaped flame having very high intensity
stabilized along the inner recirculation zone of the combustor (figure 9b).
Most of the heat release rate occurs along the inner recirculation zone and
extends downstream of the combustor as shown in figure 9c. This is in contrast
to the mean profile observed for the M-flame during period-1 oscillations
(figure 6a). These results indicate that the flame surface shape and the heat
release rate distribution are entirely changed from the state of period-1 LCO
with small variation in the volume of hydrogen to the fuel.

To study the state of period-2 LCO thoroughly, the first three dominating POD
modes of the transverse component are presented in figure 9d which
cumulatively contains $23\%$ of the total kinetic energy. The first two modes
show that the coherent structures are symmetrically positioned on each side of
the flow axis and have anti-phase behaviour. These coherent structures confirm
the existence of the helical nature of the flow. Moreover, very weak small-
scale coherent structures with low velocity fluctuations can also be seen in
IRZ. Unlike during period-1 LCO (figure 6d), the coherent structure can be
seen to be limited only along the periphery and do extend inwards along the
shear layer. However, the third mode shows the presence of a convecting
structure along the axial direction (figure 9d). We also see similar elongated
coherent structures in the POD modes of the axial velocity component (see
figure 4 in supplementary material). These observations shows that the flow
structures from the dominant modes confirm the existence of helical
instability. However, the heat release rate field indicate that the maximum
heat release occurs in the IRZ and does not follow the path of the helical
structures in the flow.

Figure 10: (a) time series of acoustic pressure, heat release rate and
temporal coefficients of first three POD modes normalised by their respective
maxima values; (b) the corresponding power spectra and (c) the analytical
signals in the complex plane, corresponding to the time series enclosed in the
red dashed box during the state of period-2 LCO.

The EPOD modes of the flame surface corresponding to the first and second POD
modes of velocity field exhibit structures with high flame surface
fluctuations along the shear layer near to the inlet of the combustor (figure
9e). That indicates a longer flame surface than that of in case of period-1
LCO due to low volume of hydrogen as combustion with higher volume has smaller
flame (Zhen et al., 2012). The EPOD modes of the flame surface correlate well
with the convecting coherent structure. Moreover, the third EPOD mode shows an
asymmetrically distributed flame surface, convecting downstream with the flow
(figure 9e). We also see similar structures near the IRZ in the extended modes
of the flame surface using the axial velocity component of the flow (figure 4
in Supplementary material). The first two EPOD modes of heat release rate
distribution confirm that the maximum heat release rate occurs downstream of
the combustor (figure 9f). The first two EPOD modes of heat release rate
distribution are symmetrical about the axis at $x=0$. Similar to the third
EPOD mode of the OH-PLIF images, the structures occur without any pattern in
the third EPOD mode of heat release rate distribution field. The results from
the first two modes clarify that the spatial locations of the coherent
structures do not match with the locations of structures from EPOD modes. The
coherent structures in the POD modes of axial velocity component shares the
spatial positions with the structures from the corresponding EPOD of flame
surface and heat release rate distribution fields (figure 4 in Supplementary
material). Moreover, the structures from EPOD modes of flame surface and heat
release rate distribution fields match.

To understand the time dynamics of the state of period-2 LCO, we show the time
series, power spectra and analytical signal in figure 10. The normalised time
series of $p^{\prime}$ and$\dot{q}^{\prime}$ signals show high amplitude
fluctuations. The acoustic pressure shows period-2 LCO, characterised by two
dominant timescales. We observe that the signal repeats itself in every two
oscillations (Kushwaha et al., 2021). The heat release rate signal shows
period-1 LCO (figure 10a). The power spectra of $p^{\prime}$ and
$\dot{q}^{\prime}$ show a distinct peak at 283 Hz with its higher harmonics at
566 Hz and 849 Hz. The fundamental frequency and first harmonic in the
$p^{\prime}$ power spectrum have the same order of power (figure 10b). On the
complex plane, the trajectory of the analytical signal forms a double-looped
pattern. We observe a unique center of rotation for the trajectory for
$\dot{q}^{\prime}$ (figure 10c) indicating that the signals are analytic. The
temporal coefficients $a_{1}$ and $a_{2}$ have period-2 behaviour (figure 10a)
and exhibit distinct peaks at 283 Hz with higher harmonics at 566 Hz and 849
Hz (figure 10b). The signals $a_{1}$ and $a_{2}$, exhibiting the centers of
rotation on the complex plane, are analytic (figure 10c). In case of the third
mode, we do not see a periodic behaviour in the time series, as can be seen in
figure 10a. Further, we notice a broadband spectrum without any distinct peak
in the power spectrum (figure 10b). The trajectory of the signal does not have
a unique center of rotation (figure 10c) which suggests that the signal is not
analytic.

Figure 11: (a-f) Phase portrait of the temporal coefficients of the flow
velocity field with respect to acoustic pressure and heat release rate along
with analytical curves having the same dominating frequencies and phase
differences as the original data and (i - vi) the corresponding phase
difference with the red line indicating the mean value of phase difference,
for the state of TAI with P2 LCO.

In figure 11, we show the phase portraits for different pairs of signals and
the phase difference between the signals for the corresponding pairs for the
state of TAI with period-2 LCO. Unlike the state of period-1 LCO, we observe
different patterns similar to the Lissajous figures. The phase portrait for
the pair of the signals $a_{1}$ and $a_{2}$ shows that the coherent structures
are phase-locked with the phase difference around $\pi/2$ (figure 11a). The
reconstructed curve exhibit the circular pattern which confirms the 1:1
frequency locking between $a_{1}$ and $a_{2}$. The pair of the signals $a_{1}$
and $p^{\prime}$ show the phase locking behaviour with the mean phase
difference between 0 and $\pi/4$ (figure 11a). However, the instantaneous
phase difference between $a_{1}$ and $p^{\prime}$ varies a lot that could be
due to the presence of high turbulence in the flow. The reconstructed curve in
the phase portrait show a double-lobbed pattern (figure 11a) which is same as
the Lissajous patterns indicating 2:1 frequency locking behaviour. Similarly,
other pairs of signals also exhibit the patterns with the two lobes (figures
11c-f) which are traditionally observed for 2:1 frequency locking between the
signals. The traditional 2:1 Lissajous patterns has lobes with equal shape and
size (at least symmetric about one axis). However, we observe patterns with
unequal shape and size due to the presence of higher harmonics in the
experimental data. These results confirm that the temporal dynamics has both
1:1 and 2:1 frequency locking behaviour during the state of TAI with period-2
LCO.

Consequently, we have found that there is a phase-relationship among the
dominant coherent structures of the flow velocity field, the acoustic pressure
field and the heat release rate field. The flame surface structures are high
correlated with the structures in heat release rate. However, these structures
contribute less to the coherent structures of flow velocity field. Temporal
analysis exhibits that the subsystems of the combustor demonstrate the 1:1 and
2:1 frequency locking behaviour.

##  5 Conclusion

In this paper, we present a framework for analyzing the synchronization
characteristics of multiple flow parameters that are acquired simultaneously.
This framework utilizes the extended proper orthogonal decomposition to look
for correlations between coherent structures measured observed in planar
velocity field measurements, and parameters such such heat-release rate,
acoustic pressure fluctuations and OH-distribution. We characterize the
coherent structures in the flow field using the proper orthogonal
decomposition (POD). We also describe the structures in the OH-PLIF images and
heat release rate distribution field using extended POD during the occurrence
of various dynamical states of chaotic oscillations, thermoacoustic
instability with period-1 and period-2 limit cycle oscillations. These
dynamical states are observed with the variation of the amount of hydrogen in
the fuel by keeping the thermal power and equivalence ratio constant.

During the state of chaotic oscillations, we hardly find correlations among
the structures in the flow, the flame and heat release rate distribution
fields. However, during the state of P1 LCO, we observe that the structures
with high energy coherent structures correlate with the high energy structures
of the flame surface and heat release rate distribution fields as the
structures occur at the same spatial regions. During the P2 LCO, the
structures from the flame surface are strongly correlated with the structures
of the heat release rate field. However, the structures from these fields
contribute less to the coherent structures of the velocity field.

The time series analysis shows that the temporal coefficients of the first two
modes dominate and decides the nature of the flow field during thermoacoustic
instability with period-1 and period-2 limit cycle oscillations. Moreover, the
patterns in the phase portrait of different pairs of the signals demonstrates
1:1 frequency locking behaviour among the acoustic field, heat release rate
and the flow field during period-1 limit cycle oscillations. In case of
period-2 limit cycle oscillations, we see both 1:1 and 2:1 frequency locking
behaviour.

Thus, with the help of temporal coefficients of dominating POD modes of the
flow field, we provide a framework to understand the synchronization of the
flow velocity field with the acoustic pressure and the heat release rate.
Apart from this, we also find the correlations of the coherent structures in a
flow field with the flame and heat release rate distribution field using the
EPOD modes of OH-PLIF images and OH*-chemiluminescence fields. This study
enhances the fundamental understanding of the coherence in the flow velocity
and heat release rate distributions leading to different types of
thermoacoustic instabilities which can occur in practical gas turbines.

Acknowledgements: Abhishek Kushwaha would like to acknowledge the Ministry of
Human Resource Development, India for providing Ph.D scholarship. The authors
would like to acknowledge Dr. Samadhan A. Pawar (UTIAS, Canada) for helpful
discussions on this topic.

Funding: This project has received funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(Grant Agreement No. 682383). This research work was also supported by the IoE
initiative (SB/2021/0845/AE/MHRD/002696), IIT Madras, India.

Declaration of interests: The authors report no conflict of interest.

Data availability statement: The data that support the findings of this study
are available upon reasonable request from the corresponding authors.

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