3.1. Single fuel jet observations and dynamics

Earlier studies (Sim et al. Reference Sim, Vargas, Ahn and Karagozian2020b) explore the dynamics of single jet diffusion flames exposed to external acoustic excitation in the present acoustic waveguide, but at a single jet Reynolds number, $Re = 65$. The present studies extended the experiments to a wider range of jet Reynolds numbers, amplitudes of excitation and at a higher camera framing rate, enabling quantification of transitions in the flame dynamics (e.g. from SOC to PLOR to BO) for a larger parameter space with improved temporal resolution. At very low Reynolds numbers for the single jet, e.g. for $Re = 20$, even relatively high amplitude acoustic excitation (e.g. local $p' = 30.8$ Pa, corresponding to $p'_{max} = 170$ Pa) produced relatively small oscillations in the flame cap formed over the injector, demonstrating SOC behaviour without periodic lift-off. For these conditions, figure 3 shows the following: in figure 3(a), a sequence of 6 instantaneous flame images over a single acoustic cycle, with repetitive back-and-forth flame oscillations in response to velocity perturbations; in figure 3(b), the first four POD mode shapes and associated modal energy content, with consistency to the back-and-forth flame oscillations and relatively strong energetic content in the first and second modes; in figure 3(c), plots of POD mode coefficients with respect to one another, creating phase portraits from the sequence of images, with fairly clear and symmetric periodic behaviour; and in figure 3(d), power spectral density (PSD) plots extracted from each of the four POD modes, demonstrating peaks at the applied frequency (332 Hz) and its higher harmonics. While the phase portraits here appeared a bit more smeared than for the higher jet Reynolds number conditions (as in Sim et al. (Reference Sim, Vargas, Ahn and Karagozian2020b) and in cases described below), this in part was a result of the higher camera frame rate with a lower relative image intensity than in Sim et al. (Reference Sim, Vargas, Ahn and Karagozian2020b). Yet the clear periodicity in the coupling of the dominant dynamical modes, and the relative symmetry in the phase portraits, even in the higher and less energetic modes, suggested a lock-in to the applied excitation, with clear higher-mode coupling. For this very low $Re = 20$ case, excitation conditions beyond the amplitude shown in figure 3 produced flame BO, with no substantial detection of PLOR behaviour prior to BO. This was an unusual feature of very low Reynolds number single jet flames.

Figure 3. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the single jet burner with $Re=20$ and local perturbation pressure $p'=30.8$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

At the more moderate $Re = 40$ condition, where the unforced flame enveloped the injector more completely (figure 2a), the flame transitions became more clearly visible with increasing amplitude. With increased excitation amplitude, the single jet flame transitioned from low-level SOC to moderate SOC with a clearly deflected flame to PLOR, as shown, for example, in figures 4, 5 and 6, respectively. For the low-level excitation producing SOC in figure 4, as at lower Reynolds numbers, there was consistency in the POD mode shapes with the nearly symmetric back-and-forth oscillations in the bulk-deflected flames, and relatively strong energetic content in the first and second POD modes, as also seen in prior single jet experiments (Sim et al. Reference Sim, Vargas, Ahn and Karagozian2020b). The phase portraits in figure 4(c) showed fairly clear and periodic behaviour in terms of mode coupling, with either symmetric or antisymmetric shapes, again consistent with lock-in to the applied acoustic excitation and simple oscillations in the flame. There was rather interesting periodic coupling at higher modes (e.g. in modes 3 and 4), and in this and other cases, it was likely that the phase portraits reflected the interactions of the applied frequency (332 Hz) and its higher harmonics, as observed in the spectra shown in figure 4(d). As the excitation amplitude increased, it produced more significant transverse flame oscillations with a slight bulk deflection away from the PN, consistent with the orientation of the acoustic radiation force (Tanabe et al. Reference Tanabe, Morita, Aoki, Satoh, Fujimori, Sato, Tanabe, Aoki and Fujimori2000; Sevilla-Esparza et al. Reference Sevilla-Esparza, Wegener, Teshome, Rodriguez, Smith and Karagozian2014). The flame dynamics documented in figure 5 displays slight reductions in energetic content of the first two modes (figure 5b) and more asymmetric and complex phase portraits (figure 5c), with the latter representing continuing alterations in the Lissajous figures consistent with lock-in prior to dynamical transitions (Li & Juniper Reference Li and Juniper2013).

Figure 4. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the single jet burner with $Re =40$ and local perturbation pressure $p'=18.1$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

Figure 5. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the single jet burner with $Re=40$ and local perturbation pressure $p'=29.9$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

Figure 6. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the single jet burner with $Re =40$ and local perturbation pressure $p'=37.1$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

For single jet flames at $Re = 40$ but at a higher excitation amplitude, periodic lift-off and reattachment was observed, as shown, for example, in the instantaneous images taken over one acoustic cycle in figure 6(a). At amplitudes somewhat above a local perturbation pressure of $p' = 37.1$ Pa, the flame could no longer stabilize on the single jet burner and experienced BO. For PLOR, as shown in figure 6(a), the deflected flame ceased to simply oscillate back and forth, but rather periodically detached from the burner edge closest to the PN, later reattaching to the burner. While SOC caused the flame to oscillate in direct response to the applied acoustic excitation, when the forcing amplitude was increased to the extent shown in figure 6, the flame began to lift-off from the side of the injector opposite to the direction of bulk flame deflection, but in a periodic fashion. It is likely that the local lift-off occurred due to an increased degree of local flame strain, hence when the local strain field relaxed periodically, but at a different time scale from that of the acoustics, the flame became anchored again at the burner edge. The time scale at which the PLOR occurred thus resulted in the PLOR's frequency being far smaller than that of the acoustic cycle. Hence the sequence of images in figure 6(a) did not constitute a single PLOR cycle. In prior phase-locked chemiluminescence imaging of such single flames, synchronized to the forcing frequency, this phenomenon produces smeared OH* images (Sim et al. Reference Sim, Vargas, Ahn and Karagozian2020b). The POD analysis applied to high-speed visible imaging of this PLOR condition revealed other clear influences of the periodicity in the lift-off process. The POD mode shapes in figure 6(b) show that the energetic content in POD mode 1 was significantly diminished, having been distributed to the higher POD modes. Mode shapes were also different from those with lower amplitude excitation, reflecting the alterations in flame shape during the coupling to both the applied frequency and the lift-off frequency, which, according to the spectra in figure 6(d), was $f_l \approx 16$ Hz. The POD mode coefficient plots/phase portraits in figure 6(c) showed much larger periodic structures, corresponding to the lower frequency oscillations, but with smaller-scale oscillations (appearing to be smeared in the coefficient plots) associated with the applied acoustic frequency. The fact that both the applied frequency and periodic lift-off frequency, as well as linear combinations thereof, were visible in figure 6(d), suggested a quasi-periodic response and coupling in the single flame during the altered dynamics. Animations corresponding to the single jet flame oscillations and the phase portraits in figures 4, 5 and 6 are shown in the supplementary materials for this paper available at https://doi.org/10.1017/jfm.2023.714, in the files named movie 1, movie 2 and movie 3, respectively. The moving red dot on the animation in the phase portrait corresponds to the individual visible image frame shown, and so oscillations associated with PLOR (figure 6) can be more clearly distinguished from those for SOC (figures 4 and 5). This transition in the phase portrait to larger-scale motions, but with smearing associated with the higher frequency oscillations on top of the lower frequency motion, was consistent with PLOR in prior single jet studies (Sim et al. Reference Sim, Vargas, Ahn and Karagozian2020b). It was similarly seen in different coaxial injector geometries in separate experiments (Vargas Reference Vargas2022), representing a characteristic signature quite distinct from that for SOC behaviour. Hence, the time scales for liftoff and reattachment were dependent on many different factors, including injector geometry, applied frequency and, clearly, fuel characteristics.

Extensive additional datasets, similar to those in figures 3–6, were taken for the single jet case, involving Reynolds numbers up to 100 and for a range of excitation conditions (Vargas Reference Vargas2022). For the most part, these flames exhibited similar dynamical behaviour to that documented in figures 4 and 5 for SOC, and in figure 6 for PLOR. For the range of Reynolds numbers examined, $Re=20\unicode{x2013}100$, the amplitudes that led to a transition from SOC to PLOR and eventually to BO may be extracted from datasets such as the above. Findings for these conditions are summarized in figure 7, representing the local perturbation pressure $p'$ required to achieve either PLOR or BO for single jets at different Reynolds numbers. The data points fit linear trends for the approximate boundaries for these different regimes of flame response. It was observed that higher $Re$ flames required a greater amplitude of forcing to trigger the transition from SOC to PLOR, and that the amplitude that produced BO also linearly increased with $Re$ but at a greater rate, leading to an increasingly wider regime of amplitudes that could accommodate PLOR for higher amplitude excitation. In contrast, for the lowest Reynolds value ($Re=20$), transition to PLOR was not observed during the present experiments.

Figure 7. Points of transition based on forcing amplitude for the single jet for $Re = 20\unicode{x2013}100$, corresponding to acoustic forcing $f_a=332$ Hz and a relative flame location $x/\lambda = +0.029$. Red circle markers represent forcing amplitude $p'$ creating the transition from SOC to PLOR, and blue crosses represent the amplitude causing flame BO/extinction. The dashed lines represent a linear fit of these boundaries.

Not only were POD analyses such as those documented in those in figures 3–6 useful in quantifying characteristic signatures for flame transitions, they also provided insights into the essential features of the dynamics required to be captured in ROMs. Expanding on the observation that energetic content in the first few modes was reduced as excitation amplitude and thus flame–acoustic coupling increased, one can explore the variation in cumulative energy for the different types of flame dynamical behaviours. As an example, figure 8 shows the cumulative mode energy distribution for the single fuel jet under two different excitation amplitudes, producing SOC and PLOR, in addition to the unforced case, for $Re = 40$. The plot represents the number of POD modes required to achieve a given cumulative mode energy percentage. The flame experiencing SOC required as few as two POMs to represent 90 % of the cumulative mode energy for that condition, while a flame undergoing PLOR required more than 10 modes to create 90 % of the cumulative energetic content. This latter observation, consistent with observations in figures 3–6 for spectra and mode shapes, suggested that a ROM representing the PLOR dynamics would require significantly greater complexity and number of modal components than a model representing simple SOC behaviour. In the unforced flame, for which POD did not recover a clear dynamics, in theory even more POD modes would be required to represent flame features, as compared with PLOR conditions, per figure 8. Yet the lack of detectable natural oscillations using the present diagnostics, in this case, would not make such an analysis particularly fruitful.

Figure 8. Cumulative POD modal energetic content as a function of mode number for the single fuel jet under unforced and forced conditions with $f_a=332$ Hz and flame relative location $x/ \lambda = +0.029$. Forcing conditions creating SOC correspond to local excitation amplitude $p' = 23.5$ Pa, while PLOR corresponds to excitation at $p' = 37.1$ Pa.

3.2. Triple fuel jet observations and dynamics

Similar experiments were performed for the response of triple fuel jets to external acoustic perturbations for a fixed excitation frequency and jet position (relative to the PN), and for a range of jet Reynolds numbers and increasing excitation amplitudes. For example, time-series snapshots in figure 9 for $Re = 40$, taken over approximately 30 acoustic cycles, demonstrated for all cases a bulk deflection away from the PN, consistent with the orientation of the acoustic radiative force (Tanabe et al. Reference Tanabe, Morita, Aoki, Satoh, Fujimori, Sato, Tanabe, Aoki and Fujimori2000). Yet different amplitudes of excitation affected the flames associated with the three microjets to different degrees. At the local pressure perturbation $p' =31.7$ Pa, for example, shown in figure 9(a), the leftmost jet flame (of the three) was observed to undergo PLOR, with severe curvature changes that caused clipping of the group flame, although the majority of the oscillating group flame remained attached to the right two jets. Increasing the local forcing amplitude further, to $p' = 37.7$ Pa, led to even greater bulk deflection of the flame structures (figure 9b), where the group flame enveloping the two rightmost injectors was periodically swept away by the bulk flow, although a small flame persisted, near the rightmost jet, enabling periodic full reignition of the group flame. As with the single jet flame, the reignition process was dependent on a time scale that was much slower than the applied acoustic time scale. While the flame response behaviour in figure 9(a) could not be considered to be full-scale PLOR, the periodic lift-off by the leftmost flame indicated a transitional behaviour somewhere between SOC and PLOR for these multi-jet flames. Behaviour in figure 9(b), in contrast, exhibited full-scale PLOR, with a greater degree of complexity in the flame structure arising from multiple fuel injectors. The modal decomposition analysis discussed below enabled the identification of additional frequencies as well as dynamical signatures of the highly perturbed flame.

Figure 9. Representative time-series images of triple jet diffusion flames with $Re=40$ under acoustic forcing with $f_a=332$ Hz, located at $x/ \lambda = +0.029$ and local pressure perturbation amplitudes (a) 31.7 Pa and (b) 37.7 Pa. Panels show (a) $p'_{max}=175$ Pa (or $p' = 31.7$ Pa); (b) $p'_{max}=205$ Pa (or $p' = 37.7$ Pa).

The POD analysis applied to the time-series images for the acoustically coupled triple fuel jet flames enabled additional insights to be derived, including alterations in the flame dynamics as excitation amplitude was systematically increased. For example, figures 10(a)–10(c) show the results from POD analysis applied to imaging of the triple jet flame responses to relatively low forcing amplitudes, $p'=9$, 18.1 and 26.2 Pa, all of which exhibited SOC behaviour, but the latter of which narrowly preceded transitional behaviour. In figures 10(a)–10(c), POD mode shapes for the four most energetic modes are shown, in addition to three-dimensional phase portraits representing the POD mode coefficient plots for the first three modes, $a_1$, $a_2$ and $a_3$. Even at very low amplitude acoustic excitation, as shown in figure 10(a), there was a clear dominance of the first mode based on the relative percentage of energetic content. The mode shape of POD mode 1 represented clear transverse oscillations, while the higher modes (2–4) with much lower percentages relate to minor changes along the flame boundary. Unlike the single jet, the triple jet had the potential to exhibit a locally emerging dynamics associated with the individual jets. Gradual growth in the asymmetric nature of the phase portraits in various dimensions may be seen in the three-dimensional plots in figure 10, in addition to the size of the trajectories, reflective of an increase in the magnitude of the flame oscillations and thus the POD mode coefficients with increasing excitation amplitude.

Figure 10. The POD analysis results for 332 Hz excitation at different forcing amplitudes for the triple jet ($Re=40$) at the location $x/ \lambda = +0.029$, for local pressure amplitudes (a) $p'=9.0$ Pa, (b) $p'=18.1$ Pa and (c) $p'=26.2$ Pa. Results include the first four POD mode structures with the associated percentage of energetic content and sample plots of the POD mode coefficients against one another for the first three modes.

A comparison of the results of POD analysis for various triple jet excitation cases, with one another and with the single jet dynamics, provides useful insights. For relatively low amplitude acoustic excitation, the group flame oscillated back and forth with no substantial bulk deflection or partial flame lift-off at any of the injectors, as shown in figure 10(a). At successively increasing amplitudes of excitation, initiation of a small asymmetry was observed in the POD mode shapes and three-dimensional phase portraits in figures 10(b) and 10(c). For these relatively low amplitude excitation conditions, the phase portraits were relatively clean, showing clear periodicity among the various modes. A more detailed example of the periodicity is shown in figure 11, representing the triple jet flame at the excitation amplitude corresponding to the case in figure 10(c) ($p'= 26.2$ Pa), including: (a) visible images over one acoustic cycle, (b) the first four POD mode shapes, (c) POD-based two-dimensional phase portraits involving coefficients for modes 1–4 and (d) spectral characteristics for each of the four modes. The spectra showed clear peaks at only the excitation frequency, 332 Hz, and at higher harmonics of this frequency. This spectral character, in addition to the shapes of the phase portraits, clearly showed behaviour that can be categorized as SOC. Such characteristics were similar to those for the single jet at a moderate amplitude of excitation, producing SOC lock-in to the excitation, as in figure 4. Yet for the triple jet, the POD coefficient plots/phase portraits had additional loops and exhibited initiation of asymmetry in the coefficient trajectories, suggestive of the beginning of a transition to increased complexity in the dynamics of flame–acoustic coupling, due in part to the more complex burner geometry.

Figure 11. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the triple jet burner with $Re=40$ and local perturbation pressure $p'=26.2$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

Indeed, when the amplitude of excitation was increased to just above that producing SOC in figure 11, a slight periodic lift-off by the leftmost flame ensued, similar to the behaviour discussed previously in connection with figure 9(a). At a slightly higher amplitude of excitation, e.g. when $p'= 27.1$ Pa as shown in figure 12, a transitional behaviour lying somewhat between SOC and PLOR was observed for the multiple jet flames, where there was a small degree of lift-off at the left side of the burner and simple oscillations of the group flame at the right side. This flame dynamics, with partial periodic lift-off of an individual jet, similar to the oscillating flame sequence in figure 9(a), created a quasi-periodicity involving both applied and lift-off frequencies, with the former more dominant overall. These phenomena were manifested in more distorted POD mode shapes and a significant reduction in the energetic content of the first two POD modes, seen in figure 12(b), with evidence of the slower periodic lift-off process in the lower frequency peaks, along with linear combinations of the lift-off and applied frequencies, appearing in PSD plots (figure 12d). The small increase in applied perturbation amplitude created a rather abrupt transition in the flame dynamics, and this behaviour continued at higher amplitudes of excitation, e.g. for $p'= 31.7$ Pa in figure 13. The flame response in figures 13(a)–13(d), with partial periodic lift-off of an individual jet at the left side of the burner, corresponded to the sequence in figure 9(a), demonstrating a quasi-periodicity involving both applied and lift-off frequencies. The phase portraits resulting from that additional lower frequency periodicity in the flow appeared to be more smeared, with the larger trajectories reflecting the lower frequency lift-off and the smaller oscillations associated with the applied frequency (figure 13c). This result was somewhat unexpected in that the actual degree of periodic lift-off for this condition was not as great as for single jets (for example, compare figures 6(a) and 13(a) for single and triple jets, respectively). Nevertheless, the alteration in the phase portraits provided a clear dynamical signature for the SOC–PLOR transition condition in flame–acoustic coupling, which for the triple jet at $Re = 40$ occurred at roughly $p' = 27.1$ Pa.

Figure 12. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the triple jet burner with $Re=40$ and local perturbation pressure $p'=27.1$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

Figure 13. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the triple jet burner with $Re=40$ and local perturbation pressure $p'=31.7$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

Full-scale PLOR in the triple jet flame manifested itself in much more vigorous flame distortion, as seen previously for $p' = 37.7$ Pa in figure 9(b); this is shown as well for $p' = 37.1$ Pa in the sequences of images over one cycle in figure 14(a). There continued to be multiple frequencies apparent in the POD-based modal spectra (figure 14d), associated with periodic lift-off and applied excitation and their harmonics and linear combinations, evidence of quasi-periodic behaviour but with greater dominance of the lift-off phenomenon. Interestingly, the energetic content of the first POD mode was larger here (figure 14b) than for the transitional case in figure 13(b), revealing the increasing dominance of the lift-off process and its lower frequency in the overall dynamics. Moreover, the phase portraits revealed much larger-scale trajectories associated with the larger-scale periodic lift-off, but still with evidence of the smaller-scale oscillations associated with applied excitation. Beyond this amplitude of excitation, the triple jet flame experienced a full-scale flame BO. Video animations of the oscillating triple flames and associated POD coefficient plots/phase portraits may be seen in the supplementary files named movie 4, movie 5, movie 6 and movie 7, for the results shown in figures 11, 12, 13 and 14, respectively.

Figure 14. Visible images over one period (a) and POD results for modes 1–4 (b–d) for the triple jet burner with $Re=40$ and local perturbation pressure $p'=37.1$ Pa. (a) Visible imaging over one cycle. (b) First four POD modes. (c) The POD-based phase portraits. (d) The POD mode spectra.

The extensive range of datasets involving the triple fuel jet burner enabled characterization of the flame transition boundaries (from SOC to transitional SOC/PLOR to full-scale PLOR to BO) as well as quantification of the variation in cumulative energetic content for the different types of flame dynamical behaviours. Plots of the local perturbation amplitudes required us to create transitional SOC/PLOR, PLOR and BO for the triple jet flame as a function of jet Reynolds number and are shown in figure 15. An example of the cumulative energetic content for the different dynamical behaviours reflected in figures 11–14, in addition to other conditions, may be found in figure 16.

Figure 15. Points of transition based on forcing amplitude for the triple fuel jet for $Re = 20\unicode{x2013}80$, corresponding to acoustic forcing $f_a=332$ Hz and a relative flame location $x/\lambda = +0.029$. Green circle markers represent forcing amplitude $p'$ creating the transition from SOC to SOC/PLOR behaviour, red circles represent the transition for relatively full-scale PLOR and blue crosses represent the amplitude causing flame BO/extinction. The dashed lines represent a fit of the points representing these boundaries.

Figure 16. Cumulative POD modal energetic content as a function of mode number for the triple fuel jet under various amplitudes of excitation associated with $f_a= 332$ Hz and flame relative location $x/ \lambda = +0.029$. Forcing conditions corresponding to a transition between SOC and the SOC/PLOR behaviour occurred between $p' = 26.2$ and 27.1 Pa.

In the transition boundary plot of figure 15 one sees a monotonic increase in the forcing amplitude required to achieve transition with increasing jet Reynolds number, similar to the single jet case in figure 7. But of course the ‘transitional’ SOC/PLOR behaviour, in which one fuel jet demonstrated lift-off and the others did not, was unique to the triple jet. Overall, both single and triple fuel jet configurations exhibited a similar flame dynamics at similar amplitudes of excitation, although close to BO the single jet flame appeared to withstand higher excitation amplitudes during PLOR behaviour than the triple jet flames, which were influenced by each other's complexity and localized lift-off.

In figure 16, there was a clear difference in the cumulative modal energy associated with SOC behaviour as compared with transitional SOC/PLOR and full-scale PLOR behaviour. For SOC cases, whose characteristic signature was clearly delineated in phase portraits, more than 90 % of the energetic content could be represented by as few as 4 POD modes. In contrast, for PLOR or even transitional SOC/PLOR cases for the triple jet, 10 POD modes were barely capable of capturing 70 % of the flame dynamics’ energy. Hence, not only were characteristic signatures in phase portraits quite different for different types of acoustic–flame coupling, but the requirements for characterizing the flame dynamics in terms of numbers of modal components were quite different as well.