4.1. Turbulent networks during combustion noise

Low amplitude aperiodic acoustic pressure oscillations characterize the state of combustion noise. In figure 2, we examine the velocity field (see panel (b)) and the local acoustic power ($p^\prime \dot q^\prime$) field (see panel ((c)) at some arbitrary time instances ($A$–$E$) marked on (see panel (a)) the time series of acoustic pressure fluctuations. The black arrows in panel (b) represent the velocity vectors and the contour plot represents the vorticity ($\omega$) field. The white colour rectangles represent the outline of the bluff body shaft and the bluff body. The grey colour rectangles represent the mask used in the analysis of PIV. The colour bar at the bottom left side represents the range of $\omega$. Positive values of $\omega$ correspond to anticlockwise vorticity and negative values correspond to clockwise vorticity, respectively. High values of positive and negative vorticity correspond to the small discrete vortices present along the outer and inner shear layers, respectively. The colour bar at the bottom right-hand side represents the range of $p^\prime \dot {q}^\prime$. We have represented the magnitude of the local acoustic power in the logarithmic scale ($\pm \log _{10}(|p^\prime \dot {q}^\prime |+1)$). Red colour regions represent acoustic power sources ($p^\prime \dot {q}^\prime >0$) and blue colour regions represent the acoustic power sinks ($p^\prime \dot {q}^\prime \leqslant 0$). The grainy pattern in the $p^\prime \dot {q}^\prime$ field reveals that acoustic power production happens in an incoherent manner during the occurrence of combustion noise. We can infer that the presence of many small-scale vortices in the flow field would induce a disorganized and chaotic flow dynamics leading to an incoherent acoustic power production.

Figure 2.

(b) The local velocity and (c) the local acoustic power fields (in the logarithmic scale) at time instants $A, B, C, D$ and $E$ marked on the time series of acoustic pressure during combustion noise ($\phi =0.97$) in (a). The contour plots in panel (b) are the corresponding vorticity ($\omega$) fields. We observe small-scale vortices in the outer shear layer as well as in the inner shear layers. The local acoustic power production happens in an incoherent manner.

In figure 3, we examine the spatial distribution of the node strength ($s$) at the corresponding time instances marked in figure 2(a). The colour bar at the bottom left-hand side indicates the range of $s$. Since we are considering only the magnitude of the induced velocity for the network construction, $s$ is positive throughout the flow field. Comparing the spatial distribution of $\omega$ (figure 2b) with that of $s$ (figure 3a), we observe that regions with high value of $\omega$ coincide with regions with high value of $s$. The maximum value of $s$ corresponds to the vortices being shed from the tip of the bluff body (figure 2b at time instances D and E). We plot the probability distribution of $s$ in figure 3(b). Here, $P(s)$ is the probability that a given node has a strength $s$. We observe power laws with an exponent $2\leqslant \gamma \leqslant 3$, thus unravelling the weighted scale-free nature of the vorticity dynamics during the occurrence of combustion noise (figure 3b). We analysed 100 turbulent networks during the occurrence of combustion noise and observed the weighted scale-free topology in 81 networks (refer to appendix A in the supplementary material available at https://doi.org/10.1017/jfm.2021.166). Thus, the weighted scale-free topology appears, disappears and reappears during the state of combustion noise.

Figure 3.

(a) The spatial variation of the node strength, $s$, during the state of combustion noise, at the corresponding time instants shown in figure 2. Regions with high values of $s$ coincide with regions of high $\omega$ at the tip of the bluff body. (b) The probability distribution of $s$ in the log–log scale. The power law exponents show that the vorticity dynamics is weighted scale-free during combustion noise with the vortices shed from the tip of the bluff body acting as the primary hubs of the networks. One primary (red) and one secondary (green) hub is encircled.

The weighted scale-free nature of the vorticity dynamics has two implications. First, there are many vortices with weak interaction strength, i.e. they induce low velocities. Second, there are a few vortices with very high interaction strength (seen as red in figure 3a); these vortices induce very high velocities. Such vortices are called the ‘hubs’ of the network. The vortices that are shed from the tip of the bluff body, which has the highest node strength, are the primary hubs of the network (one of them is encircled in figure 3a, panel B). The vortices present in the outer shear layer (seen as green in figure 3a) have node strength values lower than the that for the vortices shed from the tip of the bluff body. These vortices are the secondary hubs of the network (one of them is encircled in figure 3a, panel A). They are not as dominant as the primary hubs. The primary hubs (fluid elements with very high vorticity) have stronger influence on all other fluid elements compared with the secondary hubs (fluid elements with moderately high vorticity). There are many smaller, weaker eddies (represented by blue in figure 3a), which have an influence only in their local vicinity. Thus, we can say that the vortices shed from the tip of the bluff body play a vital role in determining the spatiotemporal dynamics of the flow field during the occurrence of combustion noise. To obtain the power-law fit, we ensure that the coefficient of determinism $R^2 \geqslant 0.90$ (Taira et al. Reference Taira, Nair and Brunton2016; Murayama et al. Reference Murayama, Kinugawa, Tokuda and Gotoda2018). Also, we ensure that at least $50\,\%$ of the total data points are used for curve fitting. The same procedure is followed for all the plots reported in the present study.

4.2. Turbulent networks during intermittency

Bursts of periodic oscillations embedded randomly amidst low amplitude aperiodic oscillations characterize the acoustic pressure during the state of intermittency. Here, we examine the vorticity interaction during an aperiodic epoch and a periodic epoch of acoustic pressure oscillations, respectively. In figure 4, we examine the flow field (see panel (b)) and the local $p^\prime \dot {q}^\prime$ field (see panel (c)) during the aperiodic epoch of intermittency (see panel (a)). We observe small-scale vortices present along the outer shear layer downstream of the dump plane as well as in the inner shear layer at the bluff body tip (figure 4b), as found during the state of combustion noise. The grainy structure in the $p^\prime \dot {q}^\prime$ field (figure 4c) shows that acoustic power production happens in an incoherent manner as observed during the state of combustion noise.

Figure 4.

(b) The local velocity and (c) the local acoustic power fields, in the logarithmic scale, at different time instants $A, B, C, D$ and $E$ marked on the time series of acoustic pressure during an aperiodic epoch of intermittency ($\phi =0.81$) in (a). We observe small-scale vortices in the outer shear layer as well as in the inner shear layer. The local acoustic power production happens in an incoherent manner as observed during the state of combustion noise.

In figure 5(b), we find that the vorticity dynamics during the aperiodic epoch of intermittency is weighted scale-free, as observed during the state of combustion noise. Further, the primary hubs of the network are the vortices shed from the tip of the bluff body (figure 5a, at time instants $A$, $D$ and $E$). However, in contrast to the case of combustion noise, at certain time instances (figure 5a, at time instants $A$ and $D$), we observe primary hubs in the outer shear layer as well. This implies that these vortices also have strong influence over the entire network. Hence, the vortices in the outer and inner shear layers influence the spatiotemporal dynamics during the aperiodic epoch of intermittency. Thus, we discern that the vortices in the outer shear layer play a stronger role during the aperiodic epoch of intermittency when compared with that during combustion noise. This is not apparent when we compare only the flow fields in figures 2(b) and 4(b), respectively.

Figure 5.

(a) The spatial variation of the node strength, $s$, during the aperiodic epoch of intermittency, at the corresponding time instants shown in figure 4. Regions with high values of $s$ coincide with regions of high $\omega$ at the tip of the bluff body as well as in the outer shear layer. (b) The power law exponents show that the vorticity dynamics is weighted scale-free during the aperiodic epoch of intermittency. The primary hubs in the outer and inner shear layers are encircled.

Next, we examine the periodic epoch of intermittency (figure 6a). During this regime, we observe the periodic formation of large-scale vortical structures downstream of the dump plane. These vortical structures begin to form at the local minima of $p^\prime$ (figure 6b, at time instant $A$). The large-scale vortical structure then convects towards the bluff body (figure 6b at time instants $B$ and $C$). Recently, Krishnan et al. (Reference Krishnan, Sujith, Marwan and Kurths2019b) have shown that the formation of large-scale vortical structure leads to the conglomeration of acoustic power sources in the flow field. Hence, we observe a coherent acoustic power production during the formation of the vortical structure (figure 6c at time instants $A$ and $B$). During the periodic bursts, the flame moves back and forth, in sync with the periodic formation of the large-scale vortical structures. Hence, the presence of acoustic sinks upstream of the bluff body could be due to the absence of flame upstream of the bluff body (at time instants $C$ and $D$). Further, we observe the shedding of a bigger vortex at the tip of the bluff body compared with the states of combustion noise and aperiodic epoch of intermittency (at time instant $B$).

Figure 6.

(b) The local velocity and (c) the local acoustic power fields, in the logarithmic scale, at different time instants $A, B, C, D$ and $E$ marked on the time series of acoustic pressure during a periodic epoch of intermittency ($\phi =0.81$) in (a). We observe large-scale vortical structures downstream of the dump plane. The local acoustic power production happens in a coherent manner during the formation of the large-scale vortical structure.

In figure 7(b), we observe that the node strength distributions display a weighted scale-free behaviour ($2 \leqslant \gamma \leqslant 3$). This implies that the fluid elements with high vorticity behave as the hubs of the network (figure 7a at time instants $B$ and $C$) and hence have strong influence over the entire network. This weighted scale-free behaviour is consistently observed during the high amplitude bursts. We note that the vortices shed from the tip of the bluff body also act as hubs of the network (at time instants $B$, $C$ and $E$). Thus, the vorticity interactions during both the aperiodic and periodic epochs of intermittency have a weighted scale-free behaviour. We note that the magnitude of $\omega$ during both the aperiodic and periodic regimes of intermittency is of the same order as that observed during the state of combustion noise. However, the magnitude of $p^\prime \dot {q}^\prime$ is one order higher than that observed during the state of combustion noise. We observe that the weighted scale-free network topology during the aperiodic and periodic regimes of intermittency is intermittent in nature, i.e. the weighted scale-free topology appears, disappears and reappears (refer to appendix A in the supplementary material). Out of the 100 networks analysed each during the aperiodic and periodic regimes of intermittency, we observed weighted scale-free behaviour in only 52 and 43 networks, respectively. Thus, the weighted scale-free network topology is not as dominant as that observed during the state of combustion noise.

Figure 7.

(a) The spatial variation of the node strength, $s$, during the periodic epoch of intermittency, at the corresponding time instants shown in figure 6. Regions with high values of $s$ coincide with regions of high $\omega$ at the tip of the bluff body as well as in the outer shear layer. (b) The power law exponents suggest that the vorticity dynamics during the high-amplitude periodic burst is weighted scale-free. The vortical structures present downstream of the dump plane and at the tip of the bluff body are the primary hubs (encircled).

4.3. Turbulent networks during thermoacoustic instability

Large amplitude periodic acoustic pressure oscillations mark the onset of thermoacoustic instability. Figure 8 shows the flow field (see panel (b)) and the $p^\prime \dot {q}^\prime$ field (see panel (c)) at five different time instants (see panel (a)) during one acoustic cycle of thermoacoustic oscillations. During thermoacoustic instability, we observe the periodic formation of large-scale vortical structures downstream of the dump plane (figure 8b, at time instants $A$ and $B$). The strength and the size of these vortical structures are higher than those observed during the periodic epoch of intermittency (George et al. Reference George, Unni, Raghunathan and Sujith2018). The formation of these large-scale vortical structures leads to the coherent production of acoustic power sources over large clusters (Krishnan et al. Reference Krishnan, Sujith, Marwan and Kurths2019b), as observed in figure 8(c). A large vortex is also shed from the tip of the bluff body (figure 8b, at time instant $C$).

Figure 8.

(b) The local velocity and (c) the local acoustic power fields, in logarithmic scale, at time instants $A, B, C, D$ and $E$ marked on the time series of acoustic pressure during one acoustic cycle of thermoacoustic instability ($\phi =0.62$) in (a). We observe formation of large-scale vortical structures downstream of the dump plane and shedding of large vortices from the tip of the bluff body. The acoustic power production happens over large clusters.

The frequency of the formation of the large-scale vortical structure as well as the shedding of the large vortex from the bluff body tip equals the acoustic frequency during thermoacoustic instability (142 Hz). The impingement of the large-scale vortical structures on the top wall of the combustor occurs when the pressure is at local maxima (figure 8b, at time instants $C$). This leads to a sudden increase in the heat release rate. Consequently, we observe the maximum acoustic power production at the local maxima of $p^\prime$ (figure 8c, at time instant $C$). The maximum value of $s$ coincides with the maximum value of $\omega$ in the vortical structure ( figure 9a, at time instant $B$). The power-law distribution shows that the vorticity dynamics is weighted scale-free during the presence of large-scale vortical structures downstream of the dump plane and the large vortex downstream of the bluff body (figure 9b). These two large-scale vortical structures thus act as the hubs of the turbulent network and control the spatiotemporal dynamics during thermoacoustic instability. Here also, the weighted scale-free topology appears, disappears and reappears (refer to appendix A in the supplementary material). Out of the 100 turbulent networks analysed, we observed weighted scale-free topology in only 46 networks. Thus, we note that the occurrence of weighted scale-free turbulent network topology observed during the state of thermoacoustic instability is not as dominant as observed during the state of combustion noise. We note that the magnitudes of $\omega$ during the occurrence of thermoacoustic instability is one order of magnitude higher than that during the state of combustion noise. However, the magnitude of $p^\prime \dot {q}^\prime$ is two orders of magnitude higher than that during the occurrence of combustion noise.

Figure 9.

(a) The spatial variation of the node strength, $s$, during the occurrence of thermoacoustic instability, at the corresponding time instants shown in figure 8. Regions with high values of $s$ coincide with regions with high values of $\omega$ in the vortical structures. (b) The power law exponents suggest that the vorticity dynamics during thermoacoustic oscillations is weighted scale-free. The primary (red) and secondary (green) hubs are encircled.

To focus on the formation of the large-scale vortical structure downstream of the dump plane better, we perform high-speed PIV just downstream of the dump plane (refer to § 1 in supplementary material). Again, we observe that these large-scale vortical structures act as the hubs of the weighted scale-free networks (figure 1c, at time instant $A$, in § 1 of the supplementary material).

Thus, during thermoacoustic instability, there are two primary hubs. One just downstream of the dump plane and the other at the tip of the bluff body. The large-scale vortical structure that is formed downstream of the dump plane at the local minima of $p^\prime$, convects upwards to the top of the bluff body as $p^\prime$ grows in amplitude. As the large-scale vortex convects towards the top combustor wall, its size as well as circulation increases. Hence, the vortex remains as the primary hub during its convection towards the top combustor wall (at time instant $B$ in figure 9a). Near the local maxima of $p^\prime$, the large-scale vortical structure impinges upon the combustor wall near the top of the bluff body. At the same time instant, a vortex is shed from the tip of the bluff body due to the separation of the inner shear layer. As $p^\prime$ decreases in amplitude, this vortex convects downstream of the bluff body. During this period, it increases in size but the circulation decreases. Hence, the large-scale vortex acts as a primary hub at the tip of the bluff body (at time instant $B$ in figure 9a) but changes to a secondary hub as it convects downstream of the bluff body (at time instants $C$, $D$ and $E$ in figure 9a). Now, we ask the following question: Can we suppress thermoacoustic oscillations, if we target these hubs?

4.4. Control of thermoacoustic oscillations by disturbing the hubs

There are mainly two approaches to control thermoacoustic instability namely, (i) active control and (ii) passive control. Active control measures include the use of actuators such as loudspeakers or solenoid valves and oscillatory air/fuel injection to suppress the oscillatory instabilities (McManus et al. Reference McManus, Poinsot and Candel1993). Even though active control approaches are successful over a wide range of frequencies, the practical implementation of these methods adds complexity to the system as they involve the use of electro-mechanical components. Passive control, on the other hand, includes simple geometrical modifications such as the use of acoustic liners, baffles and Helmholtz or quarter-wave resonators, or modification of the injector geometry, injector location or the use of microjet injection or fuel staging to suppress thermoacoustic oscillations (Schadow & Gutmark Reference Schadow and Gutmark1992; Huang & Yang Reference Huang and Yang2009). Since passive control approaches are simple and robust, they find applications in real practical systems.

In the present study, we mitigate thermoacoustic oscillations by disturbing the hubs. We use a steady air-jet injection to disturb the hubs. The injection locations are shown in figure 10. Each of the ports is of 5 mm diameter. The injection location marked as 1 is at $60^{\circ }$ with respect to the dump plane. The injection locations marked as 2 is at a distance of 15 mm from the dump plane. The distance between each of the ports at the top and bottom walls is 10 mm. We investigate the effect of eight different cases of steady air-jet injection, namely through a pair of ports at the dump plane marked as 1 and through two pairs of ports at the top and bottom walls.

Figure 10.

The schematic diagram of the cross-section of the combustion chamber with the spatial location of the secondary steady air-jet injection ports marked as 1, 2, 3, 4, 5, 6, 7 and 8. All the dimensions are in mm.

In figure 11, we show the variation of $p^\prime _{rms}$ with respect to the total momentum flux ratio, $({v_j}/{v_a})^2$, of the secondary air-jet injection for the eight different cases. Here, ${v_j}$ represents the velocity of the secondary air jet through a single port and ${v_a}$ represents the velocity of the main air at the combustor inlet. The normalization factor $p^\prime _{rms ({\rm baseline})}$ corresponds to $p^\prime _{rms}$ during thermoacoustic instability in the absence of a secondary air-jet injection into the combustion chamber ($({v_j}/{v_a})^2=0$). We increase the air flow rate through each of the ports in steps of 5 SLPM to a maximum of $25\pm 1.2$ SLPM.

Figure 11.

The variation of $p\prime _{rms}$ (normalized) of thermoacoustic oscillations due to a secondary air-jet injection at different locations with respect to the total momentum flux ratio $({v_j}/{v_a})^2$. The vertical error bars represent the standard deviation. The regions below (i) ports 2 and 3 and (ii) ports 4 and 5 correspond to the location of formation of primary hubs during thermoacoustic instability.

We obtain a suppression of 86.4 %, 82.2 % and 79.7 % in $p^\prime _{rms}$ for an air jet injection at (i) the dump plane through port 1, (ii) ports 2 and 3 and (iii) ports 4 and 5, respectively. The optimum total momentum flux ratio at which we observe the suppression of thermoacoustic oscillations for dump plane injection is exactly half of that observed for the other two cases. Even though we observe a maximum suppression of thermoacoustic oscillations at the same total momentum flux ratio, the decrease in $p^\prime _{rms}$ is rather sudden for the secondary air-jet injection at ports 4 and 5 when compared with the injection at ports 2 and 3. The $p^\prime _{rms}$ (${\approx }200$ Pa) after suppression, for all three cases of secondary air-jet injection, is close to the value observed during the state of combustion noise ($p^\prime _{rms}\approx 100$ Pa).

The location of ports 2 and 3 coincide with the region downstream of the dump plane where the large-scale vortical structures are formed. The region below ports 4 and 5 is the location where the large-scale vortical structure impinges, resulting in the sudden spike in the heat release rate. Also, another large-scale vortical structure forms below the ports 4 and 5, at the tip of the bluff body. It should be noted that we did not obtain any suppression of the thermoacoustic oscillations when we injected steady air jets individually through ports 2, 3, 4 and 5 (refer to figure 2 in appendix B in the supplementary material for the effect of steady air jets through each of the seven ports along the length of the combustor). This could be due to the fact that the size of the large-scale vortical structures that form during thermoacoustic instability span a considerable area. So, to successfully prevent the formation of the large-scale vortices downstream of the dump plane and at the tip of the bluff body, we may require the injection of the air jets through a pair of ports instead of a single port.

Even though the region below ports 3 and 4 lies in the convection path of the large-scale vortical structure, the suppression is not that effective (41.68 %). Also, we do not obtain any suppression of thermoacoustic oscillations when we inject air jets downstream of the bluff body (through (i) ports 5 and 6, (ii) ports 6 and 7 and (iii) ports 7 and 8). It should be noted that the secondary hubs during thermoacoustic instability are formed below the regions of ports 6, 7 and 8 (figure 9a at time instants $C$ and $D$). This shows that it is imperative to distinguish between the primary and secondary hubs for the control strategy to be successful. Hence, we clearly see that the regions where the two large-scale vortical structures form periodically during thermoacoustic instability – the one downstream of the dump plane and the one at the tip of the bluff body – are indeed the two optimal locations to implement passive control strategies to mitigate thermoacoustic instability.

Next, we examine the vorticity interaction during the secondary air-jet injection at the dump plane and compare it with that observed during the state of combustion noise. In figure 12(a), we observe that the magnitude of $p^\prime$ is comparable to that observed during combustion noise. Figure 12(b) shows the spatial distribution of the node strength distribution when thermoacoustic oscillations are suppressed (at $(v_j/v_a)^2=6.41$). We still observe small-scale vortices in the outer and inner shear layers. We find that the spatiotemporal dynamics of the turbulent network during the stable state obtained with the injection of steady air jets is also intermittently weighted scale-free. Out of 100 turbulent networks, only 36 networks were weighted scale-free during the stable state obtained with the injection of steady air jets as compared with 81 networks during the occurrence of combustion noise (refer to appendix A in the supplementary material). Thus, the weighted scale-free property observed during the stable state obtained with the injection of steady air jets is not as dominant as observed during the state of combustion noise.

Figure 12.

(b) The spatial distribution of the node strength, $s$, and (c) its probability distribution during the time instants $A-E$ marked on the time series of acoustic pressure with steady secondary air injection at the dump plane in (a). The total momentum flux ratio between the secondary air jets and the main air flow rate is 6.41.

Recently, we have used network theory to identify the optimal location to implement a passive control strategy (Krishnan et al. Reference Krishnan, Manikandan, Midhun, Reeja, Unni, Sujith, Marwan and Kurths2019a). We constructed weighted spatial correlation networks from the velocity field and used network measures to identify the optimal location for secondary air-jet injection. We referred to this optimal location as the ‘critical region’, i.e. the region that controls the spatiotemporal dynamics during thermoacoustic instability on an average sense. The critical region corresponds to the region on top of the bluff body shaft, just downstream of the dump plane, where the fluid parcels exhibit coherent periodic motion. It is this coherent pumping action of the flow at the combustor inlet that leads to the periodic formation of large-scale vortical structures downstream of the dump plane at the onset of thermoacoustic instability. The secondary air injection through port 1 at the dump plane as well as through ports 2 and 3 at the top and bottom walls of the combustor target this critical region. When the critical region is disturbed by the steady air jets, the periodic coherent motion is disrupted.

During the state of combustion noise, the acoustic power production happens over small fragmented islands in an ocean of acoustic sinks. At the onset of thermoacoustic instability, the acoustic power production happens over large islands or clusters (Krishnan et al. Reference Krishnan, Sujith, Marwan and Kurths2019b). The secondary air-jet injections during thermoacoustic instability, targeting the critical region over the bluff body shaft, break apart these large clusters of acoustic power sources at an optimal total momentum flux ratio (Krishnan et al. Reference Krishnan, Manikandan, Midhun, Reeja, Unni, Sujith, Marwan and Kurths2019a). Hence, the secondary air-jet injections through (i) port 1 and (ii) ports 2 and 3 prevent the formation of the large-scale vortical structures downstream of the dump plane, which eventually stop the coherent production of acoustic power sources over large clusters or islands in the turbulent reactive flow field. As the coherent acoustic power production ceases, the acoustic damping outweighs the acoustic driving leading to the suppression of large amplitude thermoacoustic oscillations.

To examine whether the hub of the turbulent network, identified in the current study, coincides with the critical region identified by Krishnan et al. (Reference Krishnan, Manikandan, Midhun, Reeja, Unni, Sujith, Marwan and Kurths2019a), we compare the two networks in figure 13. In figure 13(a), we plot the spatial distribution of the node strength of the weighted correlation network during thermoacoustic instability obtained from weighted correlation network analysis. The region above the bluff body shaft, just downstream of the dump plane, where the node strength is the maximum is the critical region. In figure 13(b), we plot the spatial distribution of the node strength of the turbulent network constructed from the phase averaged vorticity field. The phase averaging is performed over 10 cycles of thermoacoustic instability at the local pressure minima (the phase at which the formation of the large-scale vortical structure happens downstream of the dump plane). We plot the probability distribution of the node strength of the phase-averaged turbulent network in figure 13(c). The power-law exponent suggests that the phase-averaged turbulent network is weighted scale-free with the primary hubs of the network coinciding with the phase-averaged vorticity at the lip of the dump plane and the tip of the bluff body. Thus, the spatial location of the primary hubs of the phase averaged turbulent networks during the occurrence of thermoacoustic instability are indeed the two optimal locations for implementing passive control strategies. Clearly, we note that the hub of the turbulent network is above the critical region obtained from the weighted spatial correlation network analysis. However, the steady secondary air jets through (i) port 1 and (ii) ports 2 and 3 target both the hub (at the lip of the dump plane) as well as the critical region obtained from the weighted spatial correlation network analysis.

Figure 13.

(a) The spatial distribution of the node strength during thermoacoustic instability from the weighted correlation network analysis performed in Krishnan et al. (Reference Krishnan, Manikandan, Midhun, Reeja, Unni, Sujith, Marwan and Kurths2019a). (b) The spatial distribution of the node strength of the phase averaged turbulent network during thermoacoustic instability and (c) the probability distribution of the node strength of the phase averaged turbulent network. The phase averaged turbulent network is weighted scale-free and the primary hubs of the phase averaged turbulent networks (encircled) are the two optimal locations for implementing passive control strategies.

Next, we examine a plausible mechanism of suppression of thermoacoustic instability with secondary air-jet injection through ports 4 and 5. During thermoacoustic instability, the maximum acoustic power production occurs above the bluff body at the local maximum of the acoustic pressure oscillations (Krishnan et al. Reference Krishnan, Sujith, Marwan and Kurths2019b). At the local maxima of $p^\prime$, the sudden spike in the acoustic power happens due to the impingement of the large-scale vortical structures against the walls of the combustor. The secondary air-jet injections through ports 4 and 5 suppress this sudden increase in heat release (as observed in the high-speed chemiluminescence images, not shown here). The suppression of the sudden spike in heat release, in turn, could be reducing the acoustic driving leading to the decrease in the amplitude of $p\prime$. Also, we can argue that the secondary air injection at ports 4 and 5 would prevent the shear layer separation from the tip of the bluff body, thus preventing the formation of large vortices downstream of the bluff body.

Now, we highlight the novelty of the current study. Since the large clusters of acoustic power sources, which are observed during thermoacoustic instability, span a considerable area, we may have to inject microjets at multiple locations to suppress thermoacoustic oscillations. Such an approach cannot be considered as smart and efficient. More importantly, occasionally there are instances during the state of stable operation when we observe the acoustic power production occurring in large clusters (Krishnan et al. Reference Krishnan, Sujith, Marwan and Kurths2019b). However, the magnitude of the acoustic power production is two orders of magnitude lower than that during thermoacoustic instability. Hence, a passive control strategy based only on the spatial distribution of acoustic power sources is very inefficient. Further, the heat release rate is not easy to measure for many conditions even with a perfectly accessible combustor. Also, it must be noted that chemiluminescence is a surrogate and works only in certain cases.

In contrast to chemiluminescence, PIV measurements are less ambiguous. Using spatial network analysis based only on the physics of the flow field inside the combustion chamber, we are able to identify two optimal locations for implementing passive control strategies. This shows that the control strategy using network analysis based on the velocity field is more robust compared with the one based on the spatial distribution of local acoustic power sources.

Further, we note that we are able to identify two optimal locations based on the vorticity network analysis as compared with the single optimal location, which we had identified using the velocity correlation network analysis (Krishnan et al. Reference Krishnan, Manikandan, Midhun, Reeja, Unni, Sujith, Marwan and Kurths2019a; Unni et al. Reference Unni, Nair, Krishnan, Marwan and Kurths2021). Hence, we can say that the vorticity network analysis used in the current study is better compared with the correlation network analysis performed by us in our earlier study (Krishnan et al. Reference Krishnan, Manikandan, Midhun, Reeja, Unni, Sujith, Marwan and Kurths2019a).

Earlier, Murayama et al. (Reference Murayama, Kinugawa, Tokuda and Gotoda2018) had constructed turbulent networks during thermoacoustic instability in a swirl stabilized combustor. They reported the presence of scale-free behaviour in the vorticity dynamics with the ring vortices near the injector exit acting as the hubs of the turbulent network. However, they did not characterize the vorticity dynamics during the states of combustion noise and intermittency. Later, Murayama & Gotoda (Reference Murayama and Gotoda2019) studied the suppression of thermoacoustic oscillations using steady air-jet injection in the same swirl stabilized combustor that was used in Murayama et al. (Reference Murayama, Kinugawa, Tokuda and Gotoda2018), from the viewpoint of complex networks and synchronization. They proposed a measure called the synchronization index, based on the phase synchronization parameter and the joint probability of recurrence plots, to capture the mutual coupling between the pressure and the heat release rate oscillations. With a steady secondary air-jet injection through the centre body of the axial swirler, they showed that the periodicity of noisy periodic oscillations in the flow velocity field is lost. This decrease in the periodicity significantly affects the mutual coupling between the pressure and heat release rate oscillations, resulting in the suppression of thermoacoustic oscillations. In light of our analysis, we can interpret their result as follows: the suppression of thermoacoustic instability occurs due to the prevention of the formation of the ring vortices (hubs of the turbulent network) by the air injection through the centre body of the axial swirler.

In network theory, researchers have shown that the scale-free networks are highly resilient to random node removal, but vulnerable to targeted attack (Barabási & Bonabeau Reference Barabási and Bonabeau2003). Thus, a scale-free network disintegrates when we target the hubs. Due to this feature, the hub of a scale-free network is also known as the Achilles heel. Similarly, here, we can argue that the large-scale vortical structures are the Achilles heel of the turbulent network during thermoacoustic instability as we are able to ‘kill’ thermoacoustic oscillations by targeting the large-scale vortical structures with secondary air injections.