3.3.1. Convective scaling of the dominant thermoacoustic modes

We have shown that the thermoacoustic system exhibits distinct modal behaviours that depend critically on fuel composition. With pure ${\rm CH}_4$ fuel, the system exhibits periodic limit cycle oscillations at $f_1 = 179$ Hz. Progressive hydrogen enrichment triggers the emergence of a different thermoacoustic mode, characterised by a spectral peak at $f_2=133$ Hz that gradually becomes dominant with increasing $P_{\textrm{H}_2}$ . The observed frequency reduction $f_2\lt f_1$ provides crucial insight into the nature of this mode transition. While the addition of ${\rm H}_2$ to the fuel blend raises the hot gas temperature in the combustor, which would typically be expected to raise the acoustic mode frequency, the reduction in frequency from $f_1$ to $f_2$ indicates that the $f_2$ mode does not originate from the cavity acoustic mechanism. Instead, this behaviour suggests a fundamentally different instability mechanism. To investigate this mode switching, we compare the mean flame structures of two representative cases: (i) instability at $f_1$ for $P_{\textrm{CH}_4}$ = 1 and $P_{\textrm{H}_2}$ = 0; and (b) at $f_2$ for $P_{\textrm{CH}_4}$ = 0.3, $P_{\textrm{H}_2}$ = 0.5 and $P_{\textrm{NH}_3}$ = 0.2. Figure 8 reveals that mode switching is accompanied by dramatic flame morphological changes. The system unstable to the $f_1$ mode produces a long V-shaped flame structure with its root anchored to the dump plane (figure 8 a). In contrast, the $f_2$ mode results in a substantially shorter, flatter and more compact flame at the same thermal power (figure 8 b). This transformation is attributed to the higher diffusivity of ${\rm H}_2$ , which enhances the flame speed and modifies the Lewis number, thereby altering the flame propagation characteristics and local strain rates. The integrated heat release rate profiles along the length of the combustor (figure 8 c–d) reveal the upstream migration of peak intensity from $x=30{-}60$ mm for the pure methane flame to $x=15{-}45$ mm for the hydrogen-enriched flame ( $P_{\textrm{H}_2}$ = 0.5). This mode-dependent spatial restructuring of flame directly affects the underlying flame–acoustic coupling, modifying the thermoacoustic feedback mechanism.

Figure 8.

Mean flame shape with respective integrated heat release rate in $z$ -direction along the length of the combustor, $x$ (mm): ( $a$ ) $P_{\textrm{H}_2} =$ 0 ( $P_{\textrm{NH}_3} =$ 0, $P_{\textrm{CH}_4} =$ 1.0) and ( $b$ ) $P_{\textrm{H}_2} =$ 0.50 ( $P_{\textrm{NH}_3} =$ 0.20, $P_{\textrm{CH}_4} =$ 0.30). (e) Variation in frequency of thermoacoustic oscillations at $P_{\textrm{H}_2} =$ 0 ( $P_{\textrm{NH}_3} =$ 0, $P_{\textrm{CH}_4} =$ 1.0) and $P_{\textrm{H}_2} =$ 0.50 ( $P_{\textrm{NH}_3} =$ 0.20, $P_{\textrm{CH}_4} =$ 0.30), as functions of mean flow velocity in the burner section.

To further elucidate the characteristics of the $f_1$ and $f_2$ modes, and gain insight into the underlying mode-switching mechanism, we examine their frequency scaling behaviour as a function of mean flow velocity. This approach allows an assessment of the sensitivity of each mode to convective time scales, thereby distinguishing whether the instability is governed by cavity acoustic resonance or by the flame-intrinsic feedback processes. Figure 8(e–f) shows the frequency scaling with mean flow velocity for two representative fuel compositions: (i) pure methane ( $P_{\textrm{CH}_4}$ = 1) and (ii) hydrogen-enriched fuel with $P_{\textrm{H}_2}$ = 0.5, $P_{\textrm{CH}_4}$ = 0.30 and $P_{\textrm{NH}_3}$ = 0.2. It should be noted that an increase in the mean flow velocity at constant equivalence ratio occurs when the thermal power increases. For a combustor operating with pure ${\rm CH}_4$ , increasing thermal power from 2.4 kW to 4.3 kW causes the mean velocity to increase from 1.56 m s $^{-1}$ to 2.86 m s $^{-1}$ . Figure 8(e) shows that the dominant frequency increases linearly from 142 to 185 Hz as the mean velocity increases from 1.56 to 2.86 m s $^{-1}$ . This relationship holds throughout the regime, indicating stable flame anchoring without blowout. Similarly, hydrogen-enriched operation with the fuel blend comprising $P_{\textrm{CH}_4}$ = 0.30, $P_{\textrm{H}_2}$ = 0.50 and $P_{\textrm{NH}_3}$ = 0.30 reveals a consistent scaling trend. When thermal power increases from 3.8 to 4.9 kW, the mean velocity increases from 2.63 to 3.41 m s $^{-1}$ . Figure 8(f) demonstrates that the dominant frequency varies linearly from 130 to 159 Hz, as the mean velocity progressively increases from 2.63 to 3.41 m s $^{-1}$ , albeit with a different slope compared with the pure methane case. Regardless of whether the combustor operates with pure ${\rm CH}_4$ or hydrogen-enriched fuel blends, the linear scaling of frequency with the mean flow velocity observed in the two fuel configurations (figure 8 e–f) strongly suggests that the thermoacoustic mode is governed by characteristic time delays intrinsically associated with the flame response. This trend represents a hallmark of the ITA mode of instability. Unlike the acoustic cavity modes, which are eigenfrequencies determined by the combustor geometry, ITA modes arise from flame-intrinsic feedback mechanisms driven by convective time scales and upstream velocity perturbations (Bomberg et al. Reference Bomberg, Emmert and Polifke2015; Emmert et al. Reference Emmert, Bomberg and Polifke2015; Hoeijmakers et al. Reference Hoeijmakers, Kornilov, Arteaga, de Goey and Nijmeijer2016).

To provide a quantitative validation of the convective scaling behaviour, we employ the theoretical framework established by Albayrak et al. (Reference Albayrak, Steinbacher, Komarek and Polifke2018), which relates the ITA mode frequency to $u_{\kern-2pt B}/L_{\kern-1pt F}$ , the inverse of the characteristic flame time scale, where $L_{\kern-1pt F}$ is the flame length and $u_{\kern-2pt B}$ is the mean flow velocity. This scaling is described by (3.1), which enables direct comparison between theoretically predicted and experimentally observed ITA mode frequencies across different operating conditions:

(3.1) \begin{align} \frac {f_{\textit{ITA},2}}{f_{\textit{ITA},1}} = \frac {L_{F,1}}{L_{F,2}}\Big (\frac {u_{\kern-1pt B,2}}{u_{\kern-1pt B,1}}\Big )^{0.8}, \end{align}

where indices 1 and 2 represent two different thermal power ratings of a combustor.

Figure 9(a–c) shows the mean flame structure at the $f_2$ mode of instability at three mean flow velocities: (a) 3.1 m s $^{-1}$ ; (b) 3.52 m s $^{-1}$ and (c) 3.93 m s $^{-1}$ , corresponding to thermal power levels of 4500, 5100 and 5700 W, respectively. These conditions are associated with dominant spectral peaks in the PSD at 139.5, 148.5 and 159 Hz, respectively. As thermal power increases, the flame exhibits a marked increase in luminosity, indicating an intensified heat release rate, while the overall flame macrostructures remain largely unchanged. This trend is confirmed by the integrated heat release rate profiles shown in figure 9(d), which presents the streamwise distribution of this quantity, showing a clear intensity increase with increasing power input. Figure 9(d) also reveals that flame length, which is defined as the distance from the dump plane to the peak heat release location, increases from 28 to 32 mm as the thermal power rises from 4500 to 5700 W. Using these measured flame lengths $L_{\kern-1pt F}$ and corresponding mean flow velocities $u_{\kern-2pt B}$ , we compute the convective time scales ( $\tau _c = L_f/u_{\kern-2pt B}^{0.8}$ ) to be approximately 18.8, 17.4, and 15.8 ms for the three operating conditions, respectively. These time scales characterise the intrinsic flame response and will be instrumental in interpreting the network model analysis presented in § 3.3.2. To evaluate whether the observed frequency evolution of the $f_2$ mode adheres to convective scaling, we apply the relation outlined in (3.1), which links the dominant frequency to the inverse of the flame time scale. Figure 9(e) demonstrates a strong agreement between the theoretically predicted and experimentally observed frequency trends, demonstrating that ITA modes are fundamentally governed by convective transport processes within the flame, thereby distinguishing ITA modes from the acoustic modes.

Figure 9.

( $a$ ) Heat release rate integrated in z-direction along the length of the combustor at three different thermal power (i.e. mean flow velocity) conditions ( $P_{\textrm{H}_2}$ = 0.50, $P_{\textrm{NH}_3}$ = 0.20, $P_{\textrm{CH}_4}$ = 0.30) and the respective mean flame shapes: ( $b$ ) $k=$ 4500 W; ( $c$ ) $k=$ 5100 W and ( $d$ ) $k=$ 5700 W. (e) Normalised change in the dominant ITA frequency as a function of the bulk flow velocity within the burner for the three thermal powers, the red solid line shows the trend for the model and dotted is from experimental.

Figure 9. Long description

The image contains multiple graphs depicting heat release rates and flame shapes at different thermal power conditions and the normalized change in dominant frequency as a function of bulk flow velocity. Panel A: A heat map showing the heat release rate integrated in the z-direction along the length of the combustor at 4500 W. The x-axis represents the distance in millimeters, and the y-axis represents the height in millimeters. The color scale indicates the intensity of the heat release rate, ranging from 0 to 1. Panel B: A heat map showing the heat release rate at 5100 W. The axes and color scale are the same as in Panel A. Panel C: A heat map showing the heat release rate at 5700 W. The axes and color scale are the same as in Panels A and B. Panel D: A line graph showing the heat release rate as a function of distance along the combustor for three different thermal power conditions (4500 W, 5100 W, and 5700 W). The x-axis represents the distance in millimeters, and the y-axis represents the heat release rate. The graph includes three lines, each representing a different thermal power condition, with corresponding frequencies labeled. Panel E: A scatter plot showing the normalized change in the dominant intrinsic thermoacoustic (ITA) frequency as a function of the bulk flow velocity within the burner for the three thermal powers. The x-axis represents the bulk flow velocity in meters per second, and the y-axis represents the normalized change in the dominant ITA frequency. The plot includes data points and trend lines for both the model and experimental results.

3.3.2. Acoustic network approach for thermoacoustic mode prediction

Next, we use an acoustic network model to identify unstable eigenmodes and characterise flame-acoustic interactions within the combustor (detailed methodology provided in Appendix A). This approach enables systematic differentiation between acoustic cavity modes and intrinsic thermoacoustic (ITA) modes through eigenfrequency analysis in the complex plane. We represent the thermoacoustic system as a network of three interconnected acoustic elements: a plenum, a burner and a combustion chamber, within which one-dimensional acoustic wave propagation is assumed. The model assumes uniform cold flow properties ( $T_u$ = 293 K) in the plenum and burner sections, while the combustion chamber temperature is set to $T_d$ = 1600 K to reflect realistic operating conditions. The geometric parameters corresponding to the experimental configuration are summarised in table 1. We model the flame as a compact heat source located at the combustion chamber inlet, characterising its unsteady dynamics with the $n{-}\tau$ model: $\digamma (s) = ne^{-s\tau }$ , where $n \geqslant 0$ represents the interaction index that quantifies flame–acoustic coupling strength, and $\tau \geqslant 0$ denotes the time delay between acoustic perturbations and heat-release-rate oscillations (Silva, Yong & Magri Reference Silva, Yong and Magri2019). To assess the stability characteristics of the system, we solve the eigenvalue problem numerically using the taX acoustic simulation software (Emmert Reference Emmert2016), computing eigenfrequencies across a wide range of flame interaction indices ( $n$ ) and time delays ( $\tau$ ). This parametric sweep enables the identification of different thermoacoustic modes and their dependence on flame response parameters (Silva Reference Silva2023).

Table 1.

Parameters for the acoustic network model. Thermodynamic properties are $\bar {p} = 101325$ Pa and $\gamma = 1.4$ .

Table 1. Long description

A table with three rows and three columns. The columns are labeled Section and Parameters. The rows list the sections as Plenum, Burner, and Combustor. For each section, the table provides the length, diameter, and temperature. Row 1: Section, Plenum; Parameters, Length, l 0 equals 0.2 meters; Diameter, d 0 equals 0.2 meters; T u equals 293 K. Row 2: Section, Burner; Parameters, Length, l 1 equals 0.3 meters; Diameter, d 1 equals 0.028 meters; T u equals 293 K. Row 3: Section, Combustor; Parameters, Length, l 2 equals 0.8 meters; Diameter, d 2 equals 0.075 meters; T d equals 1600 K.

Figure 10(a) presents a thermoacoustic stability map for ${\rm CH}_4$ combustion, displaying eigenfrequency loci in the complex plane defined by frequency ( $f$ ) and growth rate ( $\sigma$ ) coordinates. The analysis encompasses interaction indices $n \in$ [0.05, 4] and time delays $\tau \in$ [25, 27.8] ms, where $\tau$ values are estimated from experimental measurements of pure ${\rm CH}_4$ flames. The thermoacoustic map reveals distinct eigenfrequency trajectories originating from three fundamental instability mechanisms: (i) the plenum Helmholtz mode at frequency $f_H$ ; (ii) the cavity acoustic mode at frequency $f_{acoustic}$ and (iii) the intrinsic thermoacoustic (ITA) mode at frequency $f_{\textit{ITA}}$ . Each mechanism exhibits characteristic behaviour that enables definitive mode identification. Cavity acoustic modes typically manifest as star-shaped spiral trajectories centred around specific frequencies, converging to natural eigenfrequencies of passive acoustic modes in the limit $n\rightarrow 0$ (Silva & Polifke Reference Silva and Polifke2019; Orchini et al. Reference Orchini, Silva, Mensah and Moeck2020; Silva Reference Silva2023). Figure 10(a) shows that trajectories around $f_{acoustic}$ approach 244 Hz, with large negative growth rates as $n\rightarrow 0$ , regardless of variations in $\tau$ . This behaviour indicates strongly damped cavity acoustic modes across the investigated parameter ranges. The stability map also reveals groups of clustered quasi-vertical trajectories characteristic of ITA modes. In the limit of $n\rightarrow 0$ , the eigenfrequencies associated with these quasi-vertical trajectories exhibit highly negative growth rates, reflecting strong damping under weak flame–acoustic coupling conditions. This behaviour constitutes a defining characteristic of intrinsic thermoacoustic modes (Silva et al. Reference Silva, Yong and Magri2019; Silva Reference Silva2023). For non-zero values of $n$ and $\tau$ , the network model predicts an infinite number of intrinsic modes (Mukherjee & Shrira Reference Mukherjee and Shrira2017; Albayrak et al. Reference Albayrak, Steinbacher, Komarek and Polifke2018). In the present study, we show only the first nine intrinsic modes in the stability map for visual clarity. Each trajectory cluster represents a different harmonic of $f_{\textit{ITA}}$ , with the flame-intrinsic frequencies given by $f = (2n_m+1)/2\tau$ , where $n_m$ denotes the mode number. Along a given trajectory, $n$ varies while $\tau$ remains constant. Figure 10(a) demonstrates that these ITA modes become unstable when $n$ exceeds a critical threshold within the time delay range $\tau \in$ [25, 27.8] ms. Variation in $\tau$ causes progressive horizontal shifts in ITA trajectories. Specifically, increasing $\tau$ decreases the eigenfrequency of ITA modes, consistent with the inverse relationship prescribed by the intrinsic frequency formula. When convective time delays derived from pure ${\rm CH}_4$ combustion are used, the predicted frequencies spanning 155 to 185 Hz (see figure 8 e), corresponding to the fourth harmonic of the ITA mode, demonstrate remarkable agreement with experimental observations. To validate this mode identification, figure 10(b) presents the experimentally observed dominant frequency $f_1$ (open circular markers) as a function of $\tau$ estimated from the pure ${\rm CH}_4$ flame. Theoretical modal lines (grey solid lines) represent the relationship $f = (2n_m+1)/2\tau$ , linking ITA eigenfrequencies with mode number $n_m$ and time delay $\tau$ . The experimental dominant frequency closely follows the modal grid line corresponding to the fourth harmonic, providing compelling evidence for the intrinsic nature of the observed modes.

Figure 10.

(a) Eigenfrequencies of the system with a wide range of flame interaction index ( $n=[0.05 {-} 4]$ ) and the time delay ( $\tau = [25{-}27.8]$ ms) computed using taX software. A closed inlet (R $_{in}$ = 1) and an open outlet (R $_{out}$ = −1) are considered. (b) Theoretical flame instrinsic thermoacoustic frequency $f_{\textit{ITA}}$ as functions of $\tau$ for different mode number, with $f_{\textit{ITA}}$ given by $ (2n_m+1)/2\tau$ . The time delay values are determined from $\tau = L/u_{\kern-2pt B}^{0.8}$ , accounting for variations in flame length $L$ and bulk velocity $u_{\kern-2pt B}$ for pure ${\rm CH}_4$ flame, as demonstrated in figure 8(e).

Next, we perform the network analysis to predict the thermoacoustic eigenfrequencies and identify mode-switching behaviour during hydrogen-enriched combustion. As shown previously (figure 8), hydrogen enrichment substantially reduces the flame length from 51 mm (for pure ${\rm CH}_4$ ) to 28 mm (for $P_{\textrm{H}_2}$ = 0.5), causing time delays to decrease from approximately $\tau$ = 25 ms to 11 ms. This reduction in characteristic time scales fundamentally alters the thermoacoustic behaviour. Figure 11(a) shows a thermoacoustic stability map obtained from the network analysis. The same interaction indices $n \in$ [0.05, 4] are used as before, but with updated time delays $\tau \in$ [10.7, 12.8] ms, estimated from experimental measurements of ${\rm H}_2$ -enriched fuel combustion. When this physically appropriate convective time delay is used in the network analysis, the model reliably identifies the ITA mode characteristics under these modified conditions. As before, the stability map shows quasi-vertical trajectories in the complex plane with highly negative growth rates in the limit $n\rightarrow 0$ , reflecting strong damping under weak flame–acoustic coupling conditions. When $n$ exceeds a threshold value, the eigenfrequency trajectory intersects the neutral stability axis at specific frequencies given by $f = (2n_m+1)/2\tau$ . A comparison between the thermoacoustic map (figure 11 a) and the experimentally observed frequencies (figure 11 b) reveals remarkable agreement. The eigenfrequencies of the first harmonic of the ITA mode demonstrate excellent correspondence with the experimentally observed dominant frequencies. Specifically, ITA mode trajectories approach neutral stability at discrete frequencies $f_2$ = 130–160 Hz (see figure 8 f) in the thermoacoustic map for convective time delays $\tau _c \in$ [10.7–12.8] ms, which are the first harmonics of the ITA mode as seen in figure 11(b). This systematic frequency–time-delay relationship is consistent with the flame-intrinsic feedback mechanism that endows ITA modes with four distinctive characteristics that differentiate them from acoustic cavity modes: (i) their frequencies do not depend on the acoustic eigenfrequencies of the combustor (Silva Reference Silva2023); (ii) they become more unstable as acoustic losses increase (Silva et al. Reference Silva, Merk, Komarek and Polifke2017; Wildemans et al. Reference Wildemans, Kornilov, de Goey and Lopez Arteaga2023); (iii) the phase of the flame transfer function at ITA frequencies is close to $-\pi$ (Xu et al. Reference Xu, Zheng, Wang, Feng, Tian, Li and Qi2021); (iv) and their frequencies scale with the convective time delay upstream of the flame, $\tau$ (Albayrak et al. Reference Albayrak, Steinbacher, Komarek and Polifke2018). These characteristics have been confirmed across a wide range of configurations, from theoretical studies of simple configurations (Bomberg et al. Reference Bomberg, Emmert and Polifke2015; Emmert et al. Reference Emmert, Bomberg and Polifke2015; Hoeijmakers et al. Reference Hoeijmakers, Kornilov, Arteaga, de Goey and Nijmeijer2014) to experimental studies of laminar premixed flames (Hoeijmakers et al. Reference Hoeijmakers, Kornilov, Arteaga, de Goey and Nijmeijer2014; Wildemans et al. Reference Wildemans, Kornilov, de Goey and Lopez Arteaga2023), turbulent premixed swirl-stabilised combustors (Albayrak et al. Reference Albayrak, Steinbacher, Komarek and Polifke2018; Xu et al. Reference Xu, Zheng, Wang, Feng, Tian, Li and Qi2021) and partially premixed configurations (Murugesan et al. Reference Murugesan, Singaravelu, Kushwaha and Mariappan2018). The presence of the same essential ITA characteristics across both premixed and partially premixed configurations indicates that the underlying physics governing ITA modes is not tied to a specific premixing arrangement. The agreement with eigenfrequency, along with the convective scaling behaviour confirmed here, provides compelling evidence of ${\rm H}_2$ -driven mode switching. Collectively, these results indicate that hydrogen enrichment shifts the ITA eigenfrequency from the fourth harmonic (under pure ${\rm CH}_4$ conditions) to the first harmonic under ${\rm H}_2$ -enriched conditions with $P_{\textrm{H}_2}$ = 0.5). This harmonic shift represents a fundamental change in the dominant instability mechanism rather than a simple frequency adjustment. The mode switching coincides with substantial changes in the flame macrostructure, which reduce the characteristic time scale and alter the thermoacoustic feedback mechanism. This correspondence between predicted eigenfrequency shifts and observed flame structure modifications provides corroborative evidence that flame time scales govern the intrinsic thermoacoustic feedback mechanism (Albayrak et al. Reference Albayrak, Steinbacher, Komarek and Polifke2018).

Figure 11.

(a) Network analysis results for ${\rm H}_2$ -enriched conditions, analogous to figure 10(a) but for ${\rm H}_2$ enriched conditions, with time delay varied over the range $\tau = [10.7{-}12.8]$ ms. ( $b$ ) Theoretical flame instrinsic thermoacoustic frequency $f_{\textit{ITA}}$ as a function of time delay $\tau$ for different mode numbers $n_m$ , where $f_{\textit{ITA}}$ = $ (2n_m+1)/2\tau$ . The time delay values are determined from $\tau = L/u_{\kern-2pt B}^{0.8}$ , accounting for variations in flame length $L$ and bulk velocity $u_{\kern-2pt B}$ with ${\rm H}_2$ enrichment, as demonstrated in figure 8(f).

Figure 11. Long description

Panel A: A scatter plot shows the network analysis results for enriched conditions. The horizontal axis represents frequency (f) in Hertz, ranging from 0 to 350. The vertical axis represents sigma (σ), ranging from -200 to 50. The plot includes multiple data points colored from red to yellow, indicating different time delays (τ) in milliseconds. Key frequencies such as f_H, f_2, and f_acoustic are marked. Panel B: A line graph displays the theoretical flame intrinsic thermoacoustic frequency as a function of time delay (τ) in seconds. The horizontal axis ranges from 0 to 50 seconds, and the vertical axis ranges from 120 to 200 Hertz. The graph includes multiple lines representing different mode numbers (n_m) and red circles indicating experimental data points.

The network analysis thus successfully captures the physical basis for hydrogen-induced mode transitions through the systematic relationship between fuel composition, flame characteristics and eigenfrequency behaviour. The identification of harmonic shifts as the mechanism for mode switching provides crucial insights into the control of thermoacoustic instability. Unlike cavity acoustic modes that depend on combustor dimensions, ITA modes scale with flame properties and can be controlled through fuel composition modifications, offering new strategies for control of thermoacoustic instability in alternative fuel applications. In this subsection, we have shown that variation in the hydrogen content causes the thermoacoustic system to undergo nonlinear dynamical transitions accompanied by mode switching. Next, we examine the effects of ammonia addition on self-excited thermoacoustic instability.

3.3.3. Impact of $\textit{NH}_{\it 3}$ enrichment in $\textit{CH}_{\it 4}{-}\textit{H}_{\it 2}$ blends on thermoacoustic dynamics

As previously established, the thermoacoustic system operating with pure ${\rm CH}_4$ exhibits limit cycle oscillations with a dominant frequency of $f_1$ = 179 Hz and peak amplitudes of approximately 500 Pa. When ${\rm CH}_4$ is enriched with ${\rm NH}_3$ by incrementally increasing the ammonia fraction from $P_{\textrm{NH}_3}$ = 0–0.4 while simultaneously decreasing the methane fraction from $P_{\textrm{CH}_4}$ = 1 to 0.6, while $P_{\textrm{H}_2}$ is held constant at 0, the system continues to exhibit periodic oscillations, but with reduced oscillation amplitudes. This behaviour is evident from the time trace of $p'$ (figure 12 a), which shows no significant amplitude modulation, and from the PSD (figure 12 d) where the dominant peaks remain at approximately 179 Hz. However, when $P_{\textrm{NH}_3}$ is increased further to 0.6 with proportionally decreasing $P_{\textrm{CH}_4}$ while keeping $P_{\textrm{H}_2}$ fixed at 0, the oscillation amplitude decreases substantially to approximately 40 Pa and the signal becomes aperiodic, as evidenced by the time trace in figure 12(a). The corresponding PSD (figure 12 d) shows a shallow spectral peak, indicating the suppression of thermoacoustic instability. These findings demonstrate that while enrichment of ${\rm CH}_4$ with ${\rm NH}_3$ does not alter the nonlinear dynamical state, it can reduce the oscillation amplitudes. For $P_{\textrm{NH}_3} \gt 0.70$ , the combustor fails to ignite, marking the combustion stability limit associated with high ${\rm NH}_3$ content.

Figure 12.

( $a{,}b{,}c$ ) Time series and ( $d{,}e{,}f$ ) power spectral density of $p^\prime$ in the combustor at ( $a{,}d$ ) limit cycle ( $P_{\textrm{H}_2} =$ 0), ( $b{,}e$ ) quasiperiodic ( $P_{\textrm{H}_2} =$ 0.30) and ( $c{,}f$ ) period-3 states ( $P_{\textrm{H}_2} =$ 0.50) with increasing $P_{\textrm{NH}_3}$ (and decreasing $P_{\textrm{CH}_4}$ ).

Figure 12. Long description

Panel A: Three line graphs show the time series of pressure oscillations in Pascals over time in seconds at different ammonia concentrations. The concentrations are 0.6, 0.4, 0.3, and 0 percent. Panel B: Three line graphs display the time series of pressure oscillations in Pascals over time in seconds at different ammonia concentrations. The concentrations are 0.6, 0.4, 0.3, and 0 percent. Panel C: Three line graphs illustrate the time series of pressure oscillations in Pascals over time in seconds at different ammonia concentrations. The concentrations are 0.5, 0.4, 0.3, and 0 percent. Panel D: Four line graphs present the power spectral density in decibels per Hertz against frequency in Hertz at different ammonia concentrations. The concentrations are 0.6, 0.4, 0.3, and 0 percent. Panel E: Four line graphs show the power spectral density in decibels per Hertz against frequency in Hertz at different ammonia concentrations. The concentrations are 0.6, 0.4, 0.3, and 0 percent. Panel F: Four line graphs depict the power spectral density in decibels per Hertz against frequency in Hertz at different ammonia concentrations. The concentrations are 0.5, 0.4, 0.3, and 0 percent.

Next, we examine the thermoacoustic behaviour when ${\rm NH}_3$ is introduced into the fuel blends containing a relatively high hydrogen fraction, specifically $P_{\textrm{H}_2}$ = 0.3. Figure 12(b) shows that when the ${\rm CH}_4{-}{\rm H}_2$ fuel blend is enriched with ${\rm NH}_3$ at $P_{\textrm{NH}_3}$ = 0.3, the system exhibits quasiperiodic oscillations $-$ a continuation of the dynamical trend observed at lower ${\rm NH}_3$ fraction ( $P_{\textrm{NH}_3}$ = 0.1 and 0.2, as seen in figures 6 and 7). The time trace of $p'$ (figure 12 b) shows more pronounced amplitude modulation than in the ${\rm NH}_3$ free fuel blend of ${\rm CH}_4{-}{\rm H}_2$ . This behaviour is corroborated by the PSD (figure 12 e), which reveals sharp peaks at $f_1$ and $f_2$ , and several additional peaks at their linear combinations, all of which are characteristic of quasiperiodic dynamics. As the ${\rm NH}_3$ fraction is further increased in the ${\rm CH}_4{-}{\rm H}_2$ fuel blend up to $P_{\textrm{NH}_3}$ = 0.4, while $P_{\textrm{H}_2}$ is held at 0.3, the system continues to exhibit quasiperiodic oscillations. While the oscillation amplitudes decrease slightly, the spectral amplitude at $f_2$ remains strong, whereas that at $f_1$ weakens considerably, as evidenced by the PSD in figure 12(e). This selective suppression suggests that ${\rm NH}_3$ enrichment preferentially suppresses the fourth-harmonic ITA mode $f_1$ , while leaving the first-harmonic ITA mode relatively unaffected.

The dynamics of quasiperiodic oscillations are also clearly reflected in the phase-resolved chemiluminescence images shown in figure 13(a). These images depict one full cycle of the dominant low-frequency mode ( $\approx$ 12 Hz) for a fuel blend of $P_{\textrm{CH}_4}$ = 0.4, $P_{\textrm{H}_2}$ = 0.3 and $P_{\textrm{NH}_3}$ = 0.3. This frequency corresponds to $|f_2-f_1|$ , as demonstrated by the PSD of time-resolved flame images in figure 13(b). Over the course of this cycle, a compact flame initially anchored near the dump plane undergoes contraction and reaches its minimum length before gradually elongating to its maximum extent. This cyclic variation in the flame length represents a process occurring on a slow time scale during which the flame undergoes near-quenching followed by re-ignition events. The coexistence of dynamics occurring on both slow and fast time scales during quasiperiodicity stands in stark contrast to the more stationary flame structure observed during limit cycle oscillations at $P_{\textrm{CH}_4}$ = 1, where the mean flame shape remains largely unchanged. Similar flame dynamics, marked by transitions between near-extinction and full combustion, flame elongation, and pinch-off, have been reported in other thermoacoustic systems, including both laminar and turbulent premixed combustors (Kabiraj & Sujith Reference Kabiraj and Suijith2012; Liu et al. Reference Liu, Yang, Ruan, Yu, Nan, Li and Lu2024). When the ${\rm CH}_4{-}{\rm H}_2$ fuel blend is further enriched with ${\rm NH}_3$ at $P_{\textrm{NH}_3}$ = 0.6, the system transitions from a quasiperiodic state to a periodic state, as evidenced by the time trace (figure 12 b). Figure 12(e) shows sharp spectral peaks at the $f_2$ and its harmonics without any trace of the $f_1$ mode in the PSD. This transition marks a shift from quasiperiodic to periodic oscillations, now governed by the first harmonic of the ITA mode. These results demonstrate that ${\rm NH}_3$ enrichment selectively suppresses the fourth-harmonic ITA mode while leaving the first-harmonic ITA mode largely unaffected. This selective suppression mechanism reveals fundamental differences in how fuel composition affects different harmonic modes within the ITA family. The oscillation amplitudes, however, remain considerably higher than those observed during the periodic state driven by the fourth-harmonic ITA mode, as seen in pure ${\rm CH}_4$ or ${\rm CH}_4{-}{\rm NH}_3$ combustion.

Figure 13.

Sequence of phase-averaged flame images during one time period of the first dominant mode in quasiperiodic oscillations. The flame length shrinks to the minimum at $t/T = 2/9$ and grows through re-ignition events at $t/T = 3/9$ and $t/T = 4/9$ . Flame length exhibits the highest variation at the slow time scale. This time period corresponds to the beating frequency of the amplitude modulations at quasiperiodicity.

Finally, we examine how the system responds when the ${\rm CH}_4{-}{\rm H}_2$ fuel blend with $P_{\textrm{H}_2}$ = 0.5 is enriched with ${\rm NH}_3$ through incremental increases in $P_{\textrm{NH}_3}$ . As shown earlier (§ 3.3), the system operating with ${\rm CH}_4{-}{\rm H}_2$ fuel blend at $P_{\textrm{H}_2}$ = 0.5 exhibits period-3 oscillations, characterised by sharp spectral peaks at $f_2$ , $2f_2$ and $3f_2$ of comparable amplitude in the PSD. Upon the introduction of ${\rm NH}_3$ at $P_{\textrm{NH}_3}$ = 0.1, the system exhibits notable changes in both the frequency and oscillation amplitude. The dominant spectral peak shifts from 145 to 135 Hz, while the r.m.s. of $p'$ rises from 2100 to 3000 Pa (not shown here for brevity). Further increases in $P_{\textrm{NH}_3}$ (up to 0.5) result in minimal changes in the amplitude, indicating that the system enters a saturation regime. However, the dominant frequency gradually shifts back towards 145 Hz, approaching the original value observed in the absence of ${\rm NH}_3$ enrichment.

Figure 14(a) shows phase-averaged flame images at nine time instants spanning one cycle of the dominant mode (135 Hz) under period-3 oscillations. Unlike in the quasiperiodic case, in which the dominant flame frequency differs from that of $p'$ , the dominant frequency observed in the PSD of time-resolved flame images (figure 14 b) closely matches the dominant peak in the PSD of $p'$ (figure 12 f), further confirming the period-3 state. The phase-averaged images reveal a flatter and shorter flame with an M-shaped structure, which contrasts markedly with the V-shaped flame observed during period-1 oscillations. This transition of the flame macrostructure from a V-shaped to an M-shaped configuration has been previously observed upon the addition of hydrogen to turbulent swirl flames. This transition coincides with the onset of high-amplitude pressure oscillations (Schefer et al. Reference Schefer, Wicksall and Agrawal2002). While the flame length varies as it evolves through the cycle, the highest heat release rate remains concentrated in a narrow region close to the flame root. This behaviour is characteristic of higher ${\rm H}_2$ enrichment levels due to the greater reactivity of hydrogen. The intense heat release rate generates large-amplitude pressure oscillations, driven by a strong positive feedback coupling between unsteady heat release rate ( $q'$ ) and acoustic pressure $p'$ – a hallmark of robust thermoacoustic instability. This coupling mechanism is further corroborated by the spatially resolved Rayleigh index distribution shown in figure 15(b). This map quantifies the local spatial distribution of thermoacoustic driving and damping by measuring the time-integrated correlation between $q^\prime$ and $p^\prime$ at each point in the flame. A positive RI indicates a thermoacoustically active driving region (where $q^\prime$ and $p^\prime$ are in phase), while a negative RI indicates a damping region. Figure 15 reveals that the RI map of the ${\rm CH}_4$ flame exhibiting limit cycle oscillations is strikingly different from the ${\rm H}_2$ enriched flame that exhibits period-3 oscillations. Specifically, the spatial location of the driving region differs profoundly. For instance, during limit cycle oscillations, alternating driving and damping regions in the RI map of the pure ${\rm CH}_4$ flame (figure 15 $a$ ) are distributed along the shear layer of the elongated V-shaped flame, with the zone of driving located near the flame root. The simultaneous occurrence of alternating in-phase and out-of-phase regions of $q^\prime$ and $p^\prime$ along the shear layer results in local cancellation and a moderate net integrated driving, leading to lower-amplitude oscillations (Ahn et al. Reference Ahn, Lee, Jung and Kim2018). In contrast, for the period-3 state (figure 15 $b$ ), the positive RI is concentrated in a narrow, intense band close to the dump plane, enclosed by comparatively weak damping regions upstream and downstream. The flat and compact RI map for the ${\rm H}_2$ enriched flame is attributed to the higher reactivity and diffusivity of ${\rm H}_2$ . Furthermore, figure 15 also shows that the magnitude of the Rayleigh index is substantially larger for the period-3 state than for the limit cycle state. This elevated RI reflects the stronger positive feedback between $p^\prime$ and $q^\prime$ , leading to large-amplitude oscillations that are approximately one order of magnitude larger than those observed for the pure ${\rm CH}_4$ flame. The fundamental difference between these two states is also visible in the balance between the driving and damping regions of the corresponding RI map. Notably, both positive and negative RI regions are distributed symmetrically for the period-1 limit cycle state, reflecting a near-equilibrium balance between driving and dissipation characteristics. This balance is evidenced by the modest $p'$ oscillation amplitudes during the limit cycle state. However, during the period-3 state, the positive RI substantially dominates the overall RI map, while the negative RI regions are either spatially confined to a narrow zone close to the flame anchoring location or weakly present at the flame tip. This pronounced asymmetry between driving and damping results in less local cancellation and a high integrated thermoacoustic driving, resulting in significantly large-amplitude thermoacoustic oscillations characteristic of the period-3 state. Furthermore, the RI map corresponding to the period-3 state also exhibits a degree of asymmetry across the constant $z=0$ line that separates the top and bottom flame regions, suggesting lateral heat release oscillations that can arise due to the excitation of the azimuthal modes of the ${\rm H}_2$ enriched flame. Collectively, these observations highlight that the compact M-shaped flame driven at $f_2$ of the ${\rm H}_2$ enriched flame is a more efficient thermoacoustic amplifier than the elongated V-shaped flame driven at $f_1$ of the pure ${\rm CH}_4$ flame. The transition from the $f_1$ limit cycle to the period-3 state represents not merely a quantitative increase in oscillation amplitude, but a qualitative change in the spatial organisation of thermoacoustic energy exchange.

Figure 14.

Phase-averaged images during one time period of the dominant mode ( $f_2 = 133$ Hz) at period-3 cycle. The mean flame has a flat and ’M’ shape at this 50 $\,\%$ ${\rm H}_2$ -rich condition.

Figure 15.

Rayleigh index for ( $a$ ) a limit cycle ( $P_{\textrm{H}_2}$ = 0) and ( $b$ ) a period-3 cycle ( $P_{\textrm{H}_2}$ = 0.5), computed from $p^\prime$ and time-resolved chemiluminescence images oscillating at the dominant frequencies $f_1 = 179$ and $f_2 = 133$ Hz, respectively. The Rayleigh index for a period-3 cycle is higher than that of a limit cycle, much like $p^\prime$ is higher for a period-3 than for a limit cycle.

3.4.1. Limit cycle

Figure 16(a) shows the SPOD eigenvalue spectra as a function of frequency for the combustor operating with pure ${\rm CH}_4$ fuel ( $P_{\textrm{CH}_4}$ = 1) under limit cycle oscillations. The periodic nature of the thermoacoustic system is evident from the sharp dominant peak at $f_1$ = 179 Hz in the SPOD spectrum, indicating that most of the energy is concentrated in mode 1, which exerts a greater influence on the flow field compared with other modes. The SPOD spectra also show a second dominant mode at the first harmonic of $f_1$ , denoted as $2f_1$ . This finding is consistent with the trend seen in the PSD of $p'$ (figure 1), confirming the periodic characteristics. Figures 16(b) and 16(c) show the spatial distribution of the leading SPOD modes at $f_1$ and $2f_1$ , respectively. These dominant spatial modes manifest as a series of alternating positive and negative convective wavepackets, exhibiting axial oscillatory characteristics. Notably, these wavepackets are localised along the shear layers. The leading mode at $2f_1$ is similar to the mode at $f_1$ , particularly in its quasi-symmetric structure along the flame centreline. The wavepackets of both modes display a conical spatial distribution that mirrors the V-shaped flame geometry observed in high-speed chemiluminescence images (figure 2). Despite these similarities, there are also notable differences between the two modes. Specifically, the wavepackets at $f_1$ are longer and wider than those at $2f_1$ , reflecting differences in the characteristic length scales of the respective oscillatory patterns. Furthermore, the wavepackets associated with the $f_1$ mode are energetically more active near the flame root and extend downstream up to $x \lt 50$ mm. By contrast, the wavepackets corresponding to the $2f_1$ mode are more energetically dominant in the intermediate downstream region $15\lt x\lt 30$ mm, while their intensity fluctuations appear significantly reduced near the flame root. This spatial differentiation suggests that the fundamental mode primarily governs flame dynamics in the near-field region, where flame anchoring and initial heat release occur, while the harmonic mode influences flame behaviour in the intermediate downstream region, where flame development and convective processes dominate. These distinct spatial signatures provide insights into the multiscale nature of thermoacoustic coupling in combustion systems operating under periodic limit cycle conditions

Figure 16.

SPOD eigenvalue spectra as a function of frequency for ( $a$ ) a limit cycle ( $P_{\textrm{H}_2}$ = 0), ( $d$ ) a quasiperiodic state ( $P_{\textrm{H}_2}$ = 0.2) and ( $g$ ) a period-3 cycle ( $P_{\textrm{H}_2}$ = 0.5). The SPOD mode shapes at first two dominant modes for ( $b{,}c$ ) a limit cycle ( $P_{\textrm{H}_2}$ = 0), ( $e{,}f$ ) a quasiperiodic state ( $P_{\textrm{H}_2}$ = 0.2) and ( $h{,}i$ ) a period-3 cycle ( $P_{\textrm{H}_2}$ = 0.5).

3.4.2. Quasiperiodicity

We now analyse the SPOD spectrum and coherent flame structures during the quasiperiodic oscillations that emerge with increasing hydrogen enrichment in the range $0.3\lt P_{\textrm{H}_2}\lt 0.5$ . Figure 16(d–f) show results analogous to figure 16(a–c), but for the quasiperiodic state. The SPOD spectrum in figure 16(d) for the quasiperiodic state (for $P_{\textrm{H}_2} =$ 0.3, $P_{\textrm{NH}_3}$ = 0.2 and $P_{\textrm{CH}_4}$ = 0.5) shows a dominant peak at 11 Hz, identified as beating frequency $f_B = |f_2-f_1|$ . Subsequent energetic modes appear at $f_1$ , $f_2$ and several other frequencies that are linear combinations of $f_1$ and $f_2$ , a trend consistent with those observed in the PSD of $p^\prime$ . Notably, the spectral amplitude of the SPOD mode at $f_2$ exceeds that at $f_1$ , reflecting the increasing dominance of the first-harmonic ITA mode with ${\rm H}_2$ enrichment. The significant difference in eigenvalues between the SPOD mode at $f_B$ and those at $f_1$ or $f_2$ signifies that the $f_B$ mode is the most energetically dominant mode during quasiperiodicity, governing the low-frequency quasiperiodic dynamics. Figure 16(e–f) shows the spatial distribution of coherent flame structures for SPOD modes at $f_B$ and $f_2$ , respectively. The convective wavepackets corresponding to the $f_B$ mode (figure 16 e) extend downstream up to $x=80$ mm, nearly spanning the entire combustor length $L=80$ mm and exhibiting characteristics of lateral oscillations. These coherent structures indicate that the zone of pronounced heat-release-rate oscillations occurs near the downstream end of the combustor. This spatial distribution indicates that the elongated flame structure observed in figure 13 is linked to the beating frequency $f_B$ , with oscillation dynamics governed by a slow-time-scale process. In contrast, figure 16(f) shows that the wavepacket corresponding to the $f_2$ mode does not extend to the entire combustor length; instead, it remains confined to the downstream region $x\lt 60$ mm. The alternating positive and negative wavepackets observed for the $f_2$ mode are characteristic of dominant axial oscillations. However, traces of low-amplitude lateral oscillations are also observed around $x\approx 20$ mm. While wavepackets near the flame root appear conical and symmetric about the flame centreline, those located farther downstream become increasingly asymmetric. This asymmetric distribution indicates non-uniform heat-release-rate oscillations around the centreline during quasiperiodic oscillations, reflecting the complex three-dimensional flame dynamics that emerge from the interaction between multiple incommensurate frequencies.

3.4.3. Period-3 cycle

Figure 16(g–i) shows the SPOD eigenvalue spectra and coherent flame structures of the two leading SPOD modes for a combustor exhibiting period-3 oscillations. This corresponds to a condition in which the combustor operates with a fuel blend of methane ( ${\rm CH}_4$ ) enriched with a sufficiently higher hydrogen concentration but a lower ammonia concentration: $P_{\textrm{H}_2}$ = 0.5, $P_{\textrm{NH}_3}$ = 0.2 and $P_{\textrm{CH}_4}$ = 0.3. The SPOD spectra reveal that the two dominant peaks occur at $f_2$ and its harmonics $2f_2$ , and additional peaks correspond to its higher-order harmonics of $f_2$ , indicating periodic dynamics. Figure 16(e–f) shows that the leading SPOD modes at $f_2$ and $2f_2$ share several key characteristics with those observed in the periodically oscillating combustor operating with pure ${\rm CH}_4$ fuel. These similarities include symmetrically distributed wavepackets along the flame centreline, and the presence of alternating positive and negative wavepackets for both leading SPOD modes at $f_2$ and $2f_2$ , suggesting the dominance of axial oscillations. Furthermore, the convective wavepackets corresponding to $2f_2$ are shorter than those at $f_2$ , consistent with the trend observed for their counterparts from the periodic state of figure 16(b,c). However, there are also notable disparities in the spatial distribution of coherent flame structures between these two different periodic states. Most prominently, the convective wavepackets corresponding to both $f_2$ and $2f_2$ appear substantially flatter, markedly different from the V-shaped spatial distribution of wavepackets observed for the periodic case operating with pure ${\rm CH}_4$ , as seen in figure 16(a–c). The flatter wavepacket geometry is consistent with the compact M-shaped flame observed in phase-averaged chemiluminescence images and indicates fundamental changes in the spatial organisation of heat-release-rate fluctuations. Furthermore, the presence of both positive and negative wavepackets along the lateral direction near the flame root for both SPOD modes, as seen in figure 16(g–h), suggests the occurrence of lateral oscillations in the heat-release-rate near the flame base. The inverted U-shaped wavepackets in the leading SPOD mode at $f_2$ indicate a complex interaction between axially and laterally oscillating coherent flame structures arising during the period-3 dynamical state, which further distinguishes this state from the purely periodic case. Notably, the SPOD eigenvalue spectra do not show a dominant second harmonic of $f_2$ (at 3 $f_2$ ) with a comparable amplitude to that of $f_2$ or $2f_2$ . This is because the period-3 dynamics are damped by the ${\rm NH}_3$ enrichment, as evident from the time trace in figures 6 and 7.

In summary, we have shown that hydrogen enrichment not only significantly modifies the thermoacoustic behaviour, but also fundamentally alters the coherent flame structures. These changes fundamentally alter the spatial distribution of the heat release rate within the combustor, with direct consequences for the thermoacoustic feedback mechanism. We also find that the convective wavepackets associated with the flame morphology can exhibit distinct characteristics even between ostensibly similar periodic states, depending on the underlying instability mechanism.