4.1 Synchronization map

The response dynamics of the acoustic pressure oscillation is presented in the form of 1 : 1 (the ratio between the forcing frequency and the natural frequency at the synchronized state) forced synchronization map (figure 2 a) by varying the forcing amplitude ( $A_{f}$ ) and the normalized forcing frequency ( $f_{f}/f_{n0}$ ) one at a time. Further, the response dynamics obtained from the model is shown in figure 2(b). The different regimes of forced synchronization such as regimes of phase locking, phase trapping and phase drifting (Li & Juniper Reference Li and Juniper2013c ) are identified and illustrated in figures 2(a) and 2(b).

The characterization of the different states of synchronization is based on the dynamics of the instantaneous phase difference ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ ) between the forcing signal ( $\unicode[STIX]{x1D719}_{F}$ ) with the signal of the forced system ( $\unicode[STIX]{x1D719}_{p^{\prime }}$ ), i.e. $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=|\unicode[STIX]{x1D719}_{F}-\unicode[STIX]{x1D719}_{p^{\prime }}|$ . The instantaneous phases are calculated through Hilbert transform of the respective signals. During the phase-locking state, the instantaneous phases ( $\unicode[STIX]{x1D719}$ ) as well as the frequencies ( $\unicode[STIX]{x1D714}$ ) of the forcing and the response signals show a perfect locking, i.e. $|\unicode[STIX]{x0394}\unicode[STIX]{x1D719}|=\text{const.}$ and $\unicode[STIX]{x0394}\unicode[STIX]{x1D714}=|\unicode[STIX]{x1D714}_{F}-\unicode[STIX]{x1D714}_{p^{\prime }}|=0$ (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). In contrast, during the phase trapping state, the relative phase ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D719}$ ) between the response signal and the forcing signal shows a bounded oscillatory behaviour, which results in the locking of mean frequencies ( $\unicode[STIX]{x1D714}=\langle \text{d}\unicode[STIX]{x1D719}/\text{d}t\rangle$ ) of the signals without exhibiting their instantaneous phase locking (Thévenin et al. Reference Thévenin, Romanelli, Vallet, Brunel and Erneux2011; Li & Juniper Reference Li and Juniper2013c ). Phase drifting, on the other hand, corresponds to the unbounded growth or decay of the unwrapped relative phase between the forcing and the response signals with time.

With an increase in the forcing amplitude, the region of 1 : 1 synchronization increases, resulting in a V-shaped synchronization region (I, figure 2 a,b), called the Arnold tongue (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003; Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). The model qualitatively captures the phenomenon of 1 : 1 synchronization (figure 2 b) observed in the experiments (figure 2 a). We notice a region of phase trapping (II, figure 2 a,b) between the regions of phase drifting (III, figure 2 a,b) and phase locking (I, figure 2 a,b) for both $f_{f}/f_{n0}<1$ and $f_{f}/f_{n0}>1$ .

In the Arnold tongue obtained from experiments (figure 2 a), we observe an asymmetry about $f_{n0}$ . The curves are less steep for $f_{f}<f_{n0}$ as compared to that observed for $f_{f}>f_{n0}$ . Such an asymmetry in the Arnold tongue is due to the nonlinearity involved in the periodic oscillator (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). A similar asymmetry in the Arnold tongue has been observed in forced low density jets (Hallberg & Strykowski Reference Hallberg and Strykowski2008) and hydrodynamically self-excited jets (Li & Juniper Reference Li and Juniper2013a ,Reference Li and Juniper c ). However, in another study on a jet diffusion flame, the asymmetry showed the opposite trend, having steeper curves for $f_{f}<f_{n0}$ (Li & Juniper Reference Li and Juniper2013b ). We further observe a small region in the parameter space where phase drifting directly transitions to phase locking without phase trapping when the forcing frequency is very close to the $f_{n0}$ (shown as a dashed circle in figure 2 a,b), corresponding to the mechanism of phase locking (here we refer to the route and not the state) towards forced synchronization (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). Whereas, the state of phase trapping is exhibited between phase drifting and phase locking in the regime corresponding to the mechanism of suppression.

Having described the overall synchronization behaviour of a forced Rijke tube oscillator, let us look at the transition in the forced dynamics and synchronization state as we vary the forcing parameters individually.

4.2 Synchronization states achieved as the frequency and amplitude of forcing are varied independently in experiments

4.2.1 Effect of the variation of forcing frequency

First, we vary the forcing frequency ( $f_{f}=90$  Hz–220 Hz) across the natural frequency of the system ( $f_{n0}=168.8$  Hz) keeping the forcing amplitude ( $A_{f}$ ) fixed at 5 mV. The value of $A_{f}$ is chosen such that we are able to look into different states of forced synchronization occurring through the mechanism of suppression when $f_{f}$ is varied. We observe different features of the forced synchronization and the response dynamics of the acoustic pressure. Representative states of synchronization and corresponding forced dynamics are shown for four different values of $f_{f}$ (figure 3).

Figure 3.

As $f_{f}$ is varied towards $f_{n0}$ (168.8 Hz), we observe different dynamical states in the response signal (I) and different states of synchronization (II, inset) such as phase drifting (a, $f_{f}=90$  Hz), intermittent phase locking (b, $f_{f}=161.5$  Hz), phase trapping (c, $f_{f}=164.3$  Hz) and phase locking (d, $f_{f}=164.7$  Hz). Dynamics is presented by plotting the return map (I) and the corresponding time series (inset of I). Amplitude spectrum (forcing frequency is indicated by red dots) of the response signals (II) along with the relative phase dynamics (inset of II) are shown to identify the state of synchronization. The response frequency ( $f_{r}$ ) corresponding to natural oscillations is marked with a dashed line in the amplitude spectra.

The response dynamics of the acoustic pressure signals subjected to harmonic forcing can be inferred from the first return map (figure 3I) and the time series shown in figure 3I (inset). Further, the spectral properties such as frequency of the response signal (figure 3II) and instantaneous relative phase (calculated through Hilbert transform) between forcing and response signals are shown in figure 3II (inset). The forcing frequencies are indicated by red dots in the amplitude spectra (figure 3II).

When $f_{f}$ is 90 Hz (far from $f_{n0}$ , figure 3 aII), the dynamics of the response signal is insensitive to forcing (inset of figure 3 aI). The relative phase shows a continuous drift with time (inset of figure 3 aII), indicating that the forcing and the response oscillations are desynchronized.

When the value of $f_{f}$ ( $=$  161.5 Hz) is close to $f_{n0}$ (figure 3 bII), the amplitude of the response signal shows modulation, known as beats (inset of figure 3 bI) with a beating frequency of $|\,f_{r}-f_{f}|$ , where $f_{r}$ is the response frequency. First return map of such modulated oscillations shows an oval shaped loop around the diagonal line, indicating two frequency oscillation. On the other hand, a staircase-like structure in the dynamics of the relative phase (inset of figure 3 bII) indicates a state of intermittent phase locking, where the spans of the phase locked states are separated by jumps (also referred to as phase slips) of integer multiples of $2\unicode[STIX]{x03C0}$ radians.

When $f_{f}$ ( $=$  164.3 Hz) is very near to $f_{n0}$ , however, outside the phase-locking boundary (figure 3 cII), the relative phase indicates the presence of a phase trapping state, as inferred from the bounded and oscillatory behaviour of the relative phase. Such an oscillatory relative phase has a frequency equal to the beating frequency of the response signal (inset of figure 3 cII). Although the mean frequencies ( $\unicode[STIX]{x1D714}=\langle \text{d}\unicode[STIX]{x1D719}/\text{d}t\rangle$ ) of the forcing and the response signals are perfectly locked, the amplitude spectrum of the response signal still shows two distinct frequency peaks corresponding to $f_{f}$ and  $f_{r}$ . This reflects in the beating structure of the response signal (inset of figure 3 cI), and a closed-loop structure along the diagonal line in the first return map (figure 3 cI).

During the onset of phase-locking state ( $f_{f}=164.7$  Hz), the frequency peak corresponding to $f_{r}$ gets completely suppressed, thus showing a single peak at the forcing frequency (figure 3 dII). During this state, the relative phase fluctuates around a constant phase shift (inset of figure 3 dII) indicating a perfect locking of the instantaneous phases. The first return map corresponding to this state, hence, shows a cluster of points congregated along the diagonal line (figure 3 dI).

When $f_{f}$ is varied in such a way that it crosses the right boundary of the synchronization region (see Arnold tongue in figure 2), the observations of forced synchronization states such as phase trapping, intermittent phase locking and phase drifting are repeated in the response dynamics (not shown here for brevity).

4.2.2 Effect of the variation of forcing amplitude

We further investigate the effect of forcing amplitude on the response dynamics. Towards that purpose, we vary the forcing amplitude, maintaining the forcing frequency constant at $f_{f}=155$  Hz (significantly away from  $f_{n0}$ , chosen to look into different states of forced synchronization occurring through the mechanism of suppression, as shown in figure 4) and at $f_{f}=167.5$  Hz (very close to  $f_{n0}$ , chosen to look into different states of forced synchronization occurring through the mechanism of phase locking, as shown in figure 5).

Figure 4.

The plots show frequency spectra of the response signal, and temporal variation of the relative phase between the response dynamics and the forcing signal (insets) at different values of (a)  $A_{f}=3$  mV, (b)  $A_{f}=12$  mV, (c)  $A_{f}=18$  mV and (d)  $A_{f}=24$  mV, when $f_{f}$ is maintained constant at 155 Hz, which is sufficiently away from $f_{n0}$ (168.8 Hz). The red dot in the frequency spectra corresponds to the location of the forcing frequency ( $f_{f}$ ).

For $f_{f}=155$  Hz, when the amplitude of forcing is very small ( $A_{f}=3$  mV), the external forcing does not affect the natural oscillations, resulting in a behaviour of phase drifting between the forcing and the response signals (inset of figure 4 a). With the increase of forcing amplitude to 12 mV, we observe a beating phenomenon (not shown here) in the amplitude of the response signal. This corresponds to a state of intermittent phase locking, confirmed from the observation of a wavy staircase-like structure in the relative phase dynamics (inset of figure 4 b). As we further increase $A_{f}$ ( $A_{f}=18$  mV), we notice a shift in dominance of the peak frequencies from  $f_{r}$ to  $f_{f}$ in the amplitude spectrum (see figure 4 c). This results in the bounded nature of the relative phase, where the phase difference between the forcing and the response signals is trapped between finite values and oscillates in a periodic manner (inset of figure 4 c). When the forcing amplitude is sufficiently high ( $A_{f}=24$  mV), we observe a perfect locking of instantaneous phases of the signals (inset of figure 4 d), wherein the natural oscillations get completely suppressed and the system oscillates with the forcing frequency (figure 4 d). Therefore, at $f_{f}=155$  Hz, forced synchronization occurs through the mechanism of suppression of natural oscillation (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009).

For $f_{f}=167.5$  Hz, when $A_{f}$ is very small (0.4 mV, figure 5 a), the plot of relative phase shows a phase drifting behaviour, which subsequently transitions to a staircase-like structure at a higher value of the forcing amplitude ( $A_{f}=0.8$  mV in figure 5 b). When $A_{f}=1.2$  mV, we notice a shift in the response frequency ( $f_{r}$ ) towards the forcing frequency ( $f_{f}$ ) as shown in figure 5(c). We, further, observe an intermittent phase-locking behaviour in the dynamics of the relative phase (inset of figure 5 c). When the amplitude of forcing is high ( $A_{f}=1.7$  mV), the natural frequency eventually gets locked with the forcing frequency (inset of figure 5 d). Here, we notice that the transition from phase drifting to phase locking does not happen through phase trapping. The region of the Arnold tongue for which the state of phase trapping does not appear at the synchronization boundary (shown with a dashed circle in figure 2) is referred to as a phase locking region as forced synchronization occurs through the mechanism of phase locking in this region. The mechanisms of phase locking and suppression correspond to saddle-node and torus-death bifurcation, respectively (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). In the present study, we report the experimental observation of both the routes in a Rijke tube. Having characterized the different forced dynamics and synchronization behaviour, we now shift our attention to understand the effect of harmonic forcing on the amplitude of the response signal.

Figure 5.

Plots showing amplitude spectra of the response signal, and instantaneous relative phase variation between the response and the forcing signal (insets), obtained for different values of $A_{f}=0.4$  mV (a), 0.8 mV (b), 1.2 mV (c) and 1.7 mV (d), when $f_{f}$ (167.5 Hz) is very close to $f_{n0}$ (168.8 Hz).

4.3 Effect of harmonic forcing on response amplitude

The overall amplitude of the forced system is referred to as the response amplitude from here on for brevity. Depending upon the choice of forcing parameters (i.e. $f_{f}$ and  $A_{f}$ ), the amplitude response of the system could be different. Typical time series of the acoustic pressure oscillations obtained in experiments for different values of forcing frequency are shown in figure 6. The time instances when the forcing is switched on and off are marked with violet and red dashed lines, respectively. The unforced pressure oscillation in the Rijke tube is periodic (see inset of figure 6 a) with an amplitude of nearly 215 Pa and a natural frequency of 168.8 Hz.

Figure 6.

The time series of the acoustic pressure oscillations obtained from experiments showing the effect of forcing on the amplitude of the response signal for (a)  $f_{f}=90$  Hz, (b)  $f_{f}=120$  Hz and (c)  $f_{f}=155$  Hz, keeping the forcing amplitude fixed at 40 mV. Depending upon the forcing frequency, the response oscillations can be quenched or amplified.

We vary $f_{f}$ starting from 90 Hz towards the natural frequency (168.6 Hz) keeping the forcing amplitude constant (40 mV). We find both reduction and increase in the response amplitude of forced oscillation depending upon the frequency of forcing. The acoustic pressure oscillations at forcing frequencies of 90, 120 and 155 Hz are shown in figure 6. When we force the system with a forcing frequency around 90 Hz, we find a maximum reduction in the response amplitude where the dynamics of the forced system does not exhibit sinusoidal oscillations (see inset of figure 6 a), indicating the existence of nonlinear interaction between the forced and the natural oscillations. We further note that the epoch of the initial transient in the signal after the forcing is switched on is long, i.e. around 5 s at $f_{f}=90$  Hz. Once the forcing is switched off, the system takes an even longer time to get back to the original unforced state of oscillation. At this forcing frequency, the thermoacoustic oscillations get destabilized due to the disturbance induced by the external forcing (the analysis is given in § 4.7). Therefore, once the forcing is switched off, the re-establishment of thermoacoustic oscillations takes a long time.

We find a reduction in the response amplitude even when $f_{f}=120$  Hz (figure 6 b). However, the reduction in amplitude is less compared to the previous case. Therefore, we believe that a low frequency forcing seems to be beneficial for mitigating thermoacoustic instability in practical systems to the extent that the system behaves like a periodic oscillator with a single dominant frequency. However, an experimental validation on more complicated thermoacoustic systems with many degrees of freedom needs to be performed. We further observe that the transient is short compared to the previous case and is aperiodic in nature (inset of figure 6 b). On the other hand, the forcing with a frequency ( $f_{f}=155$  Hz) close to the natural frequency ( $f_{n0}=168.8$  Hz) of the system leads to an amplification of the response signal (figure 6 c). This suggests that choosing a forcing frequency close to the natural frequency for controlling the thermoacoustic instabilities can be even more dangerous for the life of the systems.

Therefore, from a practical point of view, it is important to explore the effect of harmonic forcing on the response signal. The phenomena of reduction and amplification of the response signal can be understood with a careful analysis of the forcing and the self-excited oscillation separately.

4.4 Effect of harmonic excitation on the amplitudes of self-excited and forcing oscillations

We further analyse the effect of harmonic excitation on the amplitudes of the self-excited and the forcing oscillations separately. To that end, we obtain amplitudes of the self-excited oscillation ( $A_{n}$ corresponding to the response frequency, $f_{r}$ ) and that of the forcing ( $A_{f}$ corresponding to the forcing frequency, $f_{f}$ ) from the amplitude spectrum (by fast Fourier transform) of the response pressure signal. The normalized amplitudes ( $A_{n}/A_{0}$ and $A_{f}/A_{0}$ ) are plotted with the normalized forcing frequency ( $f_{f}/f_{n0}$ ), where $A_{0}$ is the unforced spectral amplitude of the self-excited oscillations from both experimental data and model (figure 7). For this plot, the forcing frequency is varied, keeping the forcing amplitude constant (5 mV in experiment and 0.35 in model). We further plot the r.m.s. value of pressure oscillations obtained from experiments (figure 7 a) and from the model (figure 7 a) as we vary  $f_{f}$ . We observe an asymmetry across the natural frequency in the amplitude response plots of the forced limit cycle oscillations. We further notice two distinct phenomena occurring simultaneously while $f_{f}/f_{n0}$ is close to 1.

Figure 7.

Variation of spectral amplitude of self-excited oscillation ( $A_{n}/A_{0}$ , in red squares) and of forcing ( $A_{f}/A_{0}$ , in blue circles) when the forcing frequency is varied across the natural frequency of the Rijke tube at a constant value of forcing amplitude ( $A_{f}=5$  mV in experiment and, $A_{f}=0.35$ in model), obtained from experiment (a) and model (b). Synchronous quenching of self-excited oscillations occurs when the forcing frequency ( $f_{f}$ ) is close to the natural frequency. This is associated with resonance amplification of the forcing signal. Due to the combined effects of synchronous quenching of the natural oscillation and resonant amplification of the forcing, there is an overall amplification in the $P_{rms}/P_{rms}^{0}$ value of the response signal (a,b) where $P_{rms}^{0}$ is the r.m.s. value of unforced pressure oscillation.

The amplitude corresponding to the self-excited oscillations ( $A_{n}/A_{0}$ ) shows a gradual decrease as $f_{f}/f_{n0}$ approaches 1. Such reduction in the amplitude of self-excited oscillations occurs due to the phenomenon of synchronous quenching until the onset of phase locking is observed (figures 7 a and 7 b, represented in red squares). In the regime of frequency or phase locking of the Arnold tongue (shown in figure 2), the spectral amplitude of self-excited oscillations is completely suppressed. In the same regime of frequency ratio, the amplitude of forced oscillation ( $A_{f}/A_{0}$ ) initially shows a gradual increase as $f_{f}/f_{n0}$ approaches 1 (figures 7 a and 7 b, represented in blue circles) and then exhibits a sudden jump to a very large amplitude at the onset of phase locking due to the effect of resonance amplification (Odajima et al. Reference Odajima, Nishida and Hatta1974; Ohe & Takeda Reference Ohe and Takeda1974; Abel et al. Reference Abel, Ahnert and Bergweiler2009). Thus, the simultaneous occurrence of synchronous quenching of natural oscillations and resonance amplification of the forcing signal makes the response signal dominated by the forcing signal. This, in turn, results in the phase locking between the forcing and the response signals. Further, due to the combined effects of synchronous quenching of natural oscillations and resonant amplification of forcing, the r.m.s. value of the response pressure oscillation shoots up to a very high value, compared to the unforced amplitude of the signal, in the synchronization region and then exhibits a gradual decrease with the further increase in $f_{f}/f_{n0}$ (violet curve in figure 7 a). We call this shooting up of the response amplitude (around $f_{f}=f_{n0}$ ) synchronance (synchronization–resonance). Such simultaneous occurrence of synchronization and resonance in the frequency locking region has also been reported by Abel et al. (Reference Abel, Ahnert and Bergweiler2009) who conducted experiments with a self-excited organ tube forced externally using a loudspeaker. Further, we notice that when the value of forcing frequency is far away from  $f_{n0}$ , the amplitude of natural oscillations is not affected by the forcing, and thereby, the r.m.s. value of the response signal is also unaltered.

4.5 Effect of forcing on the response amplitude

We further examine the r.m.s. value of response pressure oscillation when $A_{f}$ is varied keeping $f_{f}$ fixed at different values. Four values of $f_{f}$ are chosen in such a manner that two of them are close to $f_{n0}$ and the other two are relatively further away from  $f_{n0}$ . We observe that the response amplitude of the acoustic pressure first decreases to a minimum value prior to exhibiting a linear increase, which is primarily due to the gradual increase in the forcing amplitude. This behaviour is observed for $f_{f}<f_{n0}$ (figure 8 a,b) and also for $f_{f}>f_{n0}$ only when the frequency detuning is small (figure 8 c). This reduction in the amplitude of the acoustic pressure signal is not observed when $f_{f}$ is significantly larger than $f_{n0}$ (figure 8 d), wherein the response amplitude shows a continuously increasing trend without any minima. From the analysis of relative phase between the forcing and the forced oscillations, as shown in figures 4 and 5, we notice that the minimum in the plot of response amplitude (see figure 8) is close to the boundary of the phase-locking regime (shown as the shaded portion in figure 8 a–c). In the region of perfect phase locking, the response amplitude of the acoustic pressure oscillations demonstrates almost a linear growth with an increase in  $A_{f}$ . Such a growth of the response amplitude of pressure signal in the synchronization region is significantly steep when $f_{f}$ is close to $f_{n0}$ (figure 8 b,c). This rapid amplification of the r.m.s. values of response pressure oscillations when $f_{f}/f_{n0}$ approaches 1 happens due to the phenomenon of synchronance as explained earlier in figure 7. However, the growth in the response amplitude of the pressure signal in the synchronization region is moderate when $f_{f}$ is much lower than $f_{n0}$ (figure 8 a). We further notice that the minimum value of the response amplitude, observed prior to the onset of perfect phase-locking state, increases as $f_{f}$ is increased.

Figure 8.

The variation of response amplitude ( $P_{rms}$ ) of the acoustic pressure signals obtained from experiments as $A_{f}$ is varied for different $f_{f}=100$  Hz (a), 155 Hz (b), 175 Hz (c) and 200 Hz (d). Reduction in response amplitude is observed for all forcing frequencies except when $f_{f}$ is sufficiently higher than $f_{n0}$  (d). The shaded region indicates the zone of phase locking.

4.6 Characterization of maximum reduction in the response amplitude due to harmonic forcing

The reduction in the r.m.s. value of the response pressure signal due to harmonic forcing is shown in figure 9 when we vary $f_{f}$ across  $f_{n0}$ , and for each value of $f_{f}$ we further vary $A_{f}$ to obtain the minimum amplitude for that particular  $f_{f}$ . Suppression ( $P_{0}-P_{min}$ ) in the amplitude of the response signal is normalized with the r.m.s. value of the unforced natural oscillation ( $P_{0}$ ), where $P_{min}$ represents the minimum r.m.s. value of the response signal achieved while $A_{f}$ is varied for each value of  $f_{f}$ . We observe the maximum reduction in the amplitude of the response signal when $f_{f}$ is significantly lower than the natural frequency ( $f_{n0}$ ). As we move towards  $f_{n0}$ , the amount of amplitude reduction gradually decreases (figure 9 a). This also suggests that the effectiveness of forcing in suppressing the amplitude of the unforced oscillation decreases as $f_{f}$ approaches  $f_{n0}$ .

Figure 9.

Amplitude reduction as a function of normalized forcing frequency ( $f_{f}/f_{n0}$ ) obtained from experiments (a) and model (b) representing the maximum decrease in overall response amplitude of the acoustic pressure signal when $f_{f}$ is varied across  $f_{n0}$ . The amplitude reduction ( $P_{0}-P_{min}$ ) is normalized with the r.m.s. value of the unforced amplitude ( $P_{0}$ ), where $P_{min}$ represents the minimum r.m.s. value achieved while $A_{f}$ is varied for a particular value of  $f_{f}$ .

We do not observe any reduction in the response amplitude when $f_{f}>f_{n0}$ in experiments (figure 9 a). In other words, when $f_{f}>f_{n0}$ , the r.m.s. value of the response signal does not reduce at all as $A_{f}$ is varied (see figure 8 d). Similar observation has been mentioned in Lubarsky et al. (Reference Lubarsky, Shcherbik, Zinn, McManus, Fric and Srinivasan2003), Bellows et al. (Reference Bellows, Hreiz and Lieuwen2008) and Abel et al. (Reference Abel, Ahnert and Bergweiler2009). Amplification of the forcing signal has been found in a system of self-excited organ pipe for $f_{f}>f_{n0}$ (Abel et al. Reference Abel, Ahnert and Bergweiler2009) and in a laminar flame combustor for $f_{f}<f_{n0}$ as well as $f_{f}>f_{n0}$ (Guan et al. Reference Guan, Gupta, Kashinath and Li2018). Thus, forcing the limit cycle oscillation at a frequency greater than the natural frequency of the oscillations might not lead to a significant suppression of the overall amplitude of the limit cycle, depending on the thermoacoustic system involved.

When $f_{f}/f_{n0}$ is around 0.6, i.e. the forcing frequency is near the subharmonic frequency of natural oscillations ( $f_{n0}/2$ ), the reduction in response amplitude of the limit cycle oscillations in the system reaches almost 80 % of their unforced amplitude (figure 9 a). Such a suppression in natural oscillation that occurs when $f_{f}$ is significantly away from $f_{n0}$ is due to the phenomenon of asynchronous quenching (Minorsky Reference Minorsky1967; Staubli Reference Staubli1987). During asynchronous quenching, we notice that the minimum amplitude achieved through forcing is close to that of the imposed forcing amplitude. Therefore, we suggest that the occurrence of asynchronous quenching through the proper choice of forcing parameters can be effective in mitigating the amplitude of thermoacoustic instability while forced at a frequency much lower than the natural frequency.

On performing a similar analysis in the model, we observe a reduction in the response amplitude of as high as 95 % of the unforced amplitude when $f_{f}/f_{n0}$ is ${\sim}0.03$ (figure 9 a). Therefore, the model chosen in this study is able to capture the phenomenon of amplitude reduction due to asynchronous quenching qualitatively, but not quantitatively. Further, the amplitude reduction of the response signal is found even when the system is forced with $f_{f}>f_{n0}$ (figure 9 b). The phenomenon of asynchronous quenching occurring at both $f_{f}/f_{n0}<1$ and $f_{f}/f_{n0}>1$ is not observed in the present experimental set-up. However, using a laminar flame combustor, Guan et al. (Reference Guan, Gupta, Kashinath and Li2018) showed the existence of asynchronous quenching for $f_{f}<f_{n0}$ as well as $f_{f}>f_{n0}$ , which qualitatively corroborates the forced Duffing–van der Pol oscillator with weak nonlinearity.

4.7 Physical explanation for the amplitude suppression due to asynchronous quenching

As the phenomenon of thermoacoustic instability is a consequence of positive coupling between the acoustic pressure ( $p^{\prime }$ ) and the heat release rate ( $\dot{q}^{\prime }$ ) oscillations, it is critical to explore the effect for harmonic forcing on such thermoacoustic coupling. As we do not have a means to measure $\dot{q}^{\prime }$ (3.10) in the present experimental facility of the Rijke tube, we try to find a plausible explanation from the model that governs the essential features of this system.

Figure 10.

Cycle-averaged value of the acoustic power production ( $\langle p^{\prime }\dot{q}^{\prime }\rangle _{t}$ ; $t=512$  × time period) is plotted for a low forcing frequency ( $f_{f}/f_{n0}=0.06$ ) and by varying the amplitude of forcing ( $A_{f}$ ) in the model. Both $p^{\prime }$ and $\dot{q}^{\prime }$ are normalized to zero mean and unit variance, and their product is averaged over 512 cycles. Amplitude spectra of $p^{\prime }$ and $\dot{q}^{\prime }$ and the relative phase between them are shown for two different $A_{f}$ (indicated as red dots) corresponding to the instants of ‘no reduction’ and ‘maximum reduction’.

The phenomena of synchronous quenching and asynchronous quenching are well captured in the present model of the Rijke tube (refer to figures 7 b and 9 b, respectively). Further, the association of resonance amplification of the forcing signal with the synchronous quenching of self-excited oscillations is also well predicted in the model (figure 7 b). Therefore, in the model, we look into the normalized and cycle-averaged acoustic power production, $\langle p^{\prime }\dot{q}^{\prime }\rangle _{t}$ (where $t=512$  × time period), which is responsible for the sustenance of thermoacoustic instability (Lieuwen & Yang Reference Lieuwen and Yang2005).

The variation of $\langle p^{\prime }\dot{q}^{\prime }\rangle _{t}$ with $A_{f}$ is shown in figure 10. Here, we choose a forcing frequency ( $f_{f}/f_{n0}=0.06$ ) at which the model shows significant reduction in the r.m.s. value of the pressure oscillations (figure 9 b). With an increase in  $A_{f}$ , $\langle p^{\prime }\dot{q}^{\prime }\rangle _{t}$ decreases (figure 10) and becomes minimum when we achieve maximum reduction in the amplitude of pressure oscillation. The low value of acoustic driving ( $\langle p^{\prime }\dot{q}^{\prime }\rangle _{t}$ ) explains the reduction in the amplitude of self-excited oscillation observed during thermoacoustic instability. Further, we observe that, at the instant of no amplitude reduction, the acoustic pressure and the heat release rate oscillate at the natural frequency (figure 10i) and their relative phase oscillates within 0 and $\unicode[STIX]{x03C0}/2$ rad (figure 10ii), satisfying the Rayleigh criterion (Rayleigh 1878). On the other hand, at the instant of maximum amplitude reduction, both the acoustic pressure and the heat release rate oscillations have a dominant peak at the forcing frequency (figure 10iii), and therefore, they are phase locked, which can be inferred from the bounded relative phase shown in figure 10(iv). Interestingly, at this condition of forcing, the relative phase between $p^{\prime }$ and $\dot{q}^{\prime }$ oscillates around $\unicode[STIX]{x03C0}/2$ rad, which according to the Rayleigh criterion will lead to an alternative acoustic driving and damping (figure 10iv). This, in turn, is manifested as a very low value ( $=$  0.045) of $\langle p^{\prime }\dot{q}^{\prime }\rangle _{t}$ (corresponding to an average phase difference, $\langle \unicode[STIX]{x0394}\unicode[STIX]{x1D719}_{p^{\prime }\dot{q}^{\prime }}\rangle _{t}$ , of  $0.49\unicode[STIX]{x03C0}$ ) compared to the value (0.63) of $\langle p^{\prime }\dot{q}^{\prime }\rangle _{t}$ during thermoacoustic instability (figure 10). This observation suggests a plausible physical mechanism behind the quenching of thermoacoustic instabilities due to forcing, which mainly happens due the destruction of the positive interaction between $p^{\prime }$ and $\dot{q}^{\prime }$ when a high amplitude forcing is applied at the low value of frequency. In summary, asynchronous quenching, which is a characteristic phenomenon of a harmonically forced limit cycle oscillator, appears promising in practical and more complex thermoacoustic systems to suppress the unwanted high amplitude pressure oscillations.