3.1.1 Combustion noise

Instantaneous fields of local acoustic power production $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ , are illustrated in figure 3(a–d) at different instants of time for $\unicode[STIX]{x1D719}=0.98$ . The instantaneous field of local acoustic power production is obtained as the product of the instantaneous pressure fluctuations, $p^{\prime }(t)$ and the instantaneous local heat release rate fluctuations, $\dot{Q}^{\prime }(x,y,t)$ obtained from the high-speed camera. Lieuwen & Zinn (Reference Lieuwen and Zinn2000) showed that if the length scale of the reaction zone is much less than the acoustic length scale, the variation of pressure across the reaction zone is small. Nair (Reference Nair2014) had also confirmed this experimentally in the same combustor used for this study. Further, as described in § 2, the chemiluminescence images are acquired for a small section of the combustion chamber where we do not expect $p^{\prime }$ to vary. Thus, $p^{\prime }(t)$ used for the calculation of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ is acquired using a single piezoelectric transducer near the dump plane.

Figure 3.

(a–d) Instantaneous fields of local acoustic power production ( $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ ) during the occurrence of combustion noise at $\unicode[STIX]{x1D719}=0.98$ for different instants of time $t$ . (e) The corresponding $p^{\prime }$ and $\dot{q}^{\prime }$ (obtained from PMT). Note that the colour bars used to indicate the local acoustic power production are different for the different time instants. The instantaneous fields of local acoustic power production appear seemingly grainy. The inset displayed in (a) represents a zoomed in view. $\dot{Q}^{\prime }(x,y,t)$ is obtained by subtracting the temporal mean at each location from the instantaneous value at that pixel location.

During the occurrence of combustion noise, the spatial distribution of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ appears grainy without any evident or distinct pattern for different instants of time. In other words, the value of $p^{\prime }\dot{Q}^{\prime }$ at one location and at one instant of time has non-similar values in its neighbourhood, which makes the field of local acoustic power production appear grainy. We call these grainy structures disordered patterns or incoherent patterns in the spatial dynamics of the turbulent reacting field. Even though the acoustic pressure fluctuations are chaotic and we observe irregular spatial dynamics during the occurrence of combustion noise, we cannot conclusively state the dynamical nature of the spatial structures as spatial chaos. However, spatio-temporal chaos has been observed in turbulent fluid systems which at certain conditions exhibit coherent behaviour. For example, Manneville (Reference Manneville2010) suggested that spatial chaos can be characterized by coherence at the local scale and incoherence at the global scale. There appears to be coherence or similar values for immediate neighbours when zooming in (refer to inset of figure 3 a) to very small scales while the spatial dynamics appears irregular at the larger scales. This behaviour during the occurrence of combustion noise suggests that the spatio-temporal dynamics could be spatio-temporal chaos.

Figure 4.

(a–d) Instantaneous velocity vector fields during the occurrence of combustion noise, overlaid on the vorticity field. The region marked as grey is representative of the region occupied by the bluff body and areas with high reflection. The contours of the vortices detected by the vortex detection algorithm are coloured purple. We do not observe any large scale vortices upstream of the buff body in the shear layer region during the occurrence of combustion noise. The shear layer upstream of the bluff body can be identified by higher vorticity strength.

The instantaneous fields of velocity vectors corresponding to the same conditions as in figure 3(a–d) are shown in figure 4(a–d) overlaid on the vorticity field. These fields represent an area approximately 23 mm downstream of the dump plane. We do not observe any large scale vortices in figure 4. It should be noted that the smallest vortex that can be captured is limited by the resolution of the PIV. Some small scale vortices are observed just above the bluff body. These structures could form as a result of boundary layer separation from the bluff body.

Figure 5.

(a–d) Instantaneous field of local acoustic power production $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ at different instants of time $t$ during a periodic epoch of intermittency. (e) Simultaneous $p^{\prime }(t)$ and $\dot{q}^{\prime }(t)$ (obtained from PMT) corresponding to (a–d) are marked. Here, $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ comprises of regions of coherent dynamics. During the periodic epochs, the values of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ are one order of magnitude higher than those during combustion noise. At $\unicode[STIX]{x1D719}=0.83$ , the emergence of correlated regions illustrates the emergence of coherence in the spatial dynamics of the turbulent thermoacoustic system. (f–i) Instantaneous field of local acoustic power production during the aperiodic regime of the intermittency. (j) Simultaneous $p^{\prime }(t)$ and $\dot{q}^{\prime }(t)$ (obtained from PMT). In general, we observe grainy structures upstream and downstream of the bluff body with clustered regions of low magnitude negative $p^{\prime }\dot{Q}^{\prime }$ downstream of the bluff body.

3.1.2 Intermittency

Figure 5(a–d) shows the instantaneous fields of local acoustic power production $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ during a periodic epoch of intermittency at $\unicode[STIX]{x1D719}=0.83$ . Regions of coherent behaviour with large magnitude begin to emerge in the field of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ (refer to figure 5 d). One can observe a region upstream of the bluff body with a cluster of similar values of $p^{\prime }\dot{Q}^{\prime }$ . This clustering phenomenon observed in the spatial dynamics hints at self-organization and the emergence of localized coherent spatial structures amidst incoherent regions (grainy regions) (Mondal et al. Reference Mondal, Unni and Sujith2017). In other words, the local interactions at the small scale in the field of $p^{\prime }\dot{Q}^{\prime }$ , enabled by the favourable local flame speeds, result in coherence at the large scale. Similar behaviour of the emergence of localized patterns was observed in an experimental study on pattern formation in the transverse acoustic pressure field (Sánchez-Morcillo et al. Reference Sánchez-Morcillo, Martínez-Mora, Pérez-Arjona, Espinosa and Alonso2010). They observed that the localized structure emerges due to the effect of spatial coupling. Figure 5(f–i) shows the fields of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ obtained during an aperiodic epoch of intermittency. For all the instants of time depicted here, there appears to be very low acoustic power production. Coherence is observed at a large scale in figure 5(f) downstream of the bluff body although its magnitude is very low. Incoherent regions or grainy structures are observed upstream and downstream of the bluff body. Here, there are clustered regions of negative $p^{\prime }\dot{Q}^{\prime }$ around the bluff body. Further, one should note that in general, these regions of negative $p^{\prime }\dot{Q}^{\prime }$ during the aperiodic epochs of intermittency have higher magnitude than that of the state of combustion noise.

Figure 6.

(a–d) Instantaneous velocity vector fields which are overlaid on the vorticity field during a large amplitude periodic epoch of intermittency. A large scale coherent structure with high vorticity strength is detected upstream of the bluff body in (c). (e–h) Instantaneous velocity fields corresponding to the aperiodic epoch of intermittency. Here, we do not observe large scale coherent structures upstream of the bluff body. The vorticity strength during these aperiodic oscillations is slightly higher than the strength observed during combustion noise in the shear layer region. These images correspond to the same instants of time as those shown in figure 5.

The instantaneous velocity fields during the periodic epoch of intermittency corresponding to the figure 5(a–d) are shown in figure 6(a–d). In these images, one can observe the emergence of relatively large vortices upstream of the bluff body (refer to figure 6 c). Large scale coherent structures promote entrainment and stirring at the macro-scale (Eckart Reference Eckart1948). Upon impingement or breakdown of these large structures, the fine scale turbulence enhances the diffusivity and, subsequently, molecular scale mixing (Eckart Reference Eckart1948). In this manner, in the turbulent combustor, the large coherent structures promote entrainment, stirring and, subsequently, mixing between the fresh reactants and hot products upon impingement. This initiates the combustion of the entrained fresh reactants causing heat release rate to occur in unison and to be localized. We conjecture that, at intermittency, the observed coherence or ordered dynamics of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ is due to this coherent heat release rate at an appropriate phase of pressure fluctuations resulting in the periodic pattern of the acoustic pressure oscillations.

During the aperiodic epochs of intermittency, we do not observe any large scale vortex in figure 6(e–h) similar to that observed during the periodic epochs of intermittency. However, we observe small scale vortices downstream and upstream of the bluff body. These vortices either do not grow in size or they vanish in consecutive time instants. Acoustic perturbations or heat release rate fluctuations could disrupt the vortex growth and cause the suppression of the growth of these structures (Renard et al. Reference Renard, Thevenin, Rolon and Candel2000). Similarly, in our turbulent combustor, it is probable that the shear layer is perturbed by the acoustic fluctuations and the baroclinic vorticity generated by the flame, leading to the suppression of the growth of unstable modes in the shear layer (Renard et al. Reference Renard, Thevenin, Rolon and Candel2000).

Figure 7.

(a–f) Instantaneous fields of local acoustic power production $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ at different instants of time ( $t$ ) during the occurrence of thermoacoustic instability for one periodic cycle. (g) Simultaneous $p^{\prime }(t)$ and $\dot{q}^{\prime }(t)$ (obtained from PMT). Here, we observe that the fields of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ comprise of regions of coherent dynamics.

3.1.3 Thermoacoustic instability

Figure 7(a–f) shows the fields of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ for one periodic cycle during the occurrence of thermoacoustic instability at $\unicode[STIX]{x1D719}=0.63$ . We observe large regions of coherent dynamics similar to that observed during the periodic epochs of intermittency. These coherent regions are observed mainly on top of the bluff body and downstream of the bluff body. The magnitude of the positive $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ in these coherent regions is higher compared to that observed during the occurrence of intermittency. Further, the magnitude of the positive acoustic power in these coherent regions is significantly higher than their negative counterparts found in other regions of the reactive flow field.

The instantaneous velocity fields with detected vortices overlaid on the vorticity field in figure 8 show the emergence of large scale coherent structures upstream of the bluff body. In addition, the vorticity fields obtained during thermoacoustic instability show higher strength. Large vortices are also observed downstream of the bluff body, which are likely due to vortex shedding from the bluff body. At these low equivalence ratios, where thermoacoustic instability is observed, the temperature differences between the unreacted mixture and the hot products become small. It is possible that the resulting lower strength of baroclinic vorticity does not hinder the growth of the stronger large scale vortices. The large structures in figure 8(d,e) are detected with data smoothing of the velocity field using a Gaussian weighted kernel, prior to the calculation of the $d_{2}$ field in addition to the post-processing steps described in § 2 before the vortex detection procedure. This is to ensure a better eduction of the large scale structure observed in the velocity vector field (Varun et al. Reference Varun, Balasubramanian and Sujith2008). One should note that when comparing figure 7(e,f) with the corresponding vorticity field in figure 8(e,f), regions with highest values of the local acoustic power production exist at regions of high vorticity downstream of the bluff body. One should also note that this occurs after the formation of the large vortex observed in figure 8(d) upstream of the bluff body and its subsequent impingement. Here, the strong interaction between the flow patterns and the spatial structure of the flame results in self-sustained periodic acoustic pressure oscillations.

A previous study by Ho & Nosseir (Reference Ho and Nosseir1981) observed the phenomenon of ‘collective interaction’ of small scale coherent structures impinging on a flat plate. These studies established that this process, termed collective interaction, closes the feedback loop between the convected coherent structures and the propagating pressure waves generated by the impingement of the coherent structures resulting in self-sustained oscillations. Additionally, they observed that due to this collective interaction, there is an emergence of a single dominant frequency. The observed large scale vortices in figure 6(c) during the periodic epoch of intermittency and that in figure 8(d) during thermoacoustic instability are likely to be the result of this collective interaction of multiple small scale vortices. This collective interaction process in the turbulent combustor is illustrated in detail in the next section.

Figure 8.

(a–f) Instantaneous velocity vector fields during thermoacoustic instability ( $\unicode[STIX]{x1D719}=0.63$ ) overlaid on the field of vorticity. We can observe the emergence of a large vortex upstream of the bluff body larger than the vortices observed during the periodic epochs of intermittency. These images correspond to the same conditions as figure 7.

3.1.4 Self-organization leading to emergence of coherent structures in the flow field

Figure 9.

(a) Perturbation of the flame edge (green curve) during aperiodic oscillations. Small scale vortices (marked as purple circles) during the occurrence of aperiodic oscillations wherein collective interaction does not occur. (b) Schematic representing the collective interaction in fluid flow wherein small scale vortex interactions lead to self-organization at large scales (dark blue circle) during periodic oscillations of intermittency and thermoacoustic instability. Here, the small scale structures are intact within the large scale structure.

The illustration shown in figure 9(b) describes the phenomenon of collective interaction in the bluff body combustor wherein small scale vortices interact with each other, resulting in the self-organization and order at the larger scales. In figure 9(a), even though small scale vortices are observed in the flow field, collective interaction does not occur, in the sense, the multiple small scale vortices do not form a large structure by themselves, which could be the case during the occurrence of the aperiodic oscillations.

Figure 10.

(a–t) Mie scattering images corresponding to the states of combustion noise, intermittency and thermoacoustic instability (combustion noise: (a–e), periodic epoch of intermittency: (f–j), aperiodic epoch of intermittency: (k–o), thermoacoustic instability: (p–t)). The circles marked in green qualitatively represent the vortices in the flow field. The regions marked in purple indicate the bluff body, the central shaft and the dump plane. (u–x) The corresponding pressure fluctuations during the states of (u) combustion noise, (v) periodic epoch of intermittency, (w) aperiodic epoch of intermittency and (x) thermoacoustic instability.

We perform separate experiments to acquire planar Mie scattering images using oil droplets and simultaneous acoustic pressure fluctuations at $\unicode[STIX]{x1D719}=0.98$ , 0.83 and 0.63 and the results are shown in figure 10. We mark the regions possibly occupied by vortices with circles based on the observed perturbation of the flame front. Since the spatial resolution is not lost, which usually occurs for PIV evaluation (e.g. we obtain a vector corresponding to an interrogation window of size, say $32\times 32$ pixels), the Mie scattering images could provide better insight in detecting collective interaction. During the occurrence of combustion noise (refer to figure 10 a–e), we find that the flame front contains small scale vortices which remain relatively small in size as they proceed toward the bluff body. The small scale vortices appear to pass through the reactive flow field as illustrated in figure 9(a). Here, collective interaction does not occur. However, during the periodic epoch of intermittency, one can observe small vortices interacting to form a larger vortex. It is likely that collective interaction between small scale vortices is occurring in figure 10(h,j). The phenomenon of collective interaction is clearer in figure 10(r,s) during thermoacoustic instability. Here, within the large vortex, the structure of the small scale vortices is intact (refer to figure 10 q–s). During the aperiodic epoch of intermittency (refer to figure 10 k–o), we observe small scale vortices similar to those observed during combustion noise. During intermittency (periodic epochs) and thermoacoustic instability, the emergence of large scale vortices from the interaction of small scale vortices points toward the phenomenon of pattern formation through self-organization in the thermoacoustic system.

One could adopt the view that spatio-temporal patterns are the effect of the coupling between the flame, the flow field and the acoustic field. It is seen that as the equivalence ratio is reduced, small scale interactions within the flow field take place, resulting in the formation of large scale structures. The process by which small scale interactions take place within a subsystem itself, resulting in the emergence of order at the macro-scales is the pattern forming process. This should be distinguished from the interactions across the subsystems, which are referred to as the coupling. Once the patterns form through small scale interactions, the coupling between the subsystems is more ordered. In the absence of patterns, the coupling between the subsystems is disordered. In reality, it is probable that a two-way process occurs through which the coupling among the subsystems results in the patterns and the patterns in turn influence the coupling between the subsystems. The notion that these events occur simultaneously likely points to self-organization in the thermoacoustic system.

It would be interesting to investigate how the self-organized patterns in the flow affect the flame speeds to understand the interaction between the flow patterns and the flame structure in a better way. Since a partially premixed flame is used in this study, direct correlations with other studies on variations in flame speed due to the emergence of patterns in the flow field would be misleading. However, being a partially premixed flame, there are similarities in the dynamics with that of fully premixed flames. Using network theory, Singh et al. (Reference Singh, Belur, Chaudhuri and Sujith2017) observed the emergence of hubs in folded regions of the flame. Essentially, these hubs are locations in the flame where the connectivity with other parts of the flame is high. These folded regions are characterized by a large local displacement flame speed (Chaudhuri Reference Chaudhuri2015; Uranakara et al. Reference Uranakara, Chaudhuri, Dave, Arias and Im2016) due to its curvature. In figure 10, during the occurrence of combustion noise, the folded regions appear to be spread out over the flame front. This distribution could result in local flame speed fluctuations and spatial disorder of the heat release rate. With the emergence of the large vortex due to small scale interactions, it is probable that even a greater connectivity exists between the parts of the folded region. Future investigations are needed to show conclusively show that the self-organized structure results in lesser fluctuations of the local flame speed and leads to spatial order of the heat release rate.

3.4.1 Spatial variance

One of the measures which indicate the formation of patterns in the spatial domain is the spatial variance which measures the variability of a signal in space (Guttal & Jayaprakash Reference Guttal and Jayaprakash2009). We investigate the transition from combustion noise to thermoacoustic instability by calculating the spatial variance of the field of $p^{\prime }\dot{Q}^{\prime }(x,y,t)$ . Spatial variance is calculated as

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D70E}(t)=\mathop{\sum }_{i=1}^{m}\mathop{\sum }_{j=1}^{n}(p^{\prime }{\dot{Q}^{\prime }}_{ij}(t)-\overline{p^{\prime }\dot{Q}^{\prime }}(t))^{2},\end{eqnarray}$$

where $p^{\prime }{\dot{Q}^{\prime }}_{ij}(t)$ is the variable of interest at the location ( $i,j$ ) of the field of instantaneous local acoustic power production; $\overline{p^{\prime }\dot{Q}^{\prime }}(t)$ is the spatial average of $p^{\prime }{\dot{Q}^{\prime }}_{ij}(t)$ at that time instant and $m$ , $n$ are the number of pixels in the horizontal and vertical direction respectively. Spatial variance has been used as a statistical measure to study patterns in ecology, biology, weather etc. Strong fluctuations about the mean due to increased recovery time can cause an increase in the spatial variation of a system prior to a transition (Kéfi et al. Reference Kéfi, Guttal, Brock, Carpenter, Ellison, Livina, Seekell, Scheffer, van Nes and Dakos2014). It has been used in a recent study on the transition to thermoacoustic instability in a combustor with a swirl-stabilized flame (George et al. Reference George, Unni, Raghunathan and Sujith2018).

The spatial variance increases due to the formation of spatial patterns amidst incoherent/disordered regions. Figure 14 shows the spatial variance calculated for the instantaneous fields of local acoustic power production at equivalence ratios $\unicode[STIX]{x1D719}=0.98$ (g), 0.83 (h) and 0.63 (i). The corresponding time traces of $p^{\prime }$ and $\dot{q}^{\prime }$ are also shown. The spatial variance increases as the combustion dynamics transitions from combustion noise to thermoacoustic instability. Even during the occurrence of intermittency, there is an increase in the spatial variance in the field of acoustic power, indicating the intermittent emergence of spatial patterns. The maximum value of spatial variance is observed during thermoacoustic instability. Spatial variance is large for both the peaks and troughs of the pressure fluctuations. However, we also observe large values of spatial variance alternating with values of low spatial variance. The lower value of spatial variance is observed during the transition from positive to negative and negative to positive cycles of acoustic pressure.

3.4.2 Spatial correlation

Another spatial indicator which illustrates the emergence of patterns in the spatial dynamics is the spatial correlation. Dakos et al. (Reference Dakos, van Nes, Donangelo, Fort and Scheffer2010) used spatial correlation as an indicator to study catastrophic transitions in ecological systems. Their study states that as a control parameter is varied, the spatial correlation between neighbouring cells tends to increase well before the transition. In our combustor, during periodic oscillations, it is likely that the collective interaction of the vortices and the subsequent ordered reaction causes neighbouring regions to behave in the same manner. In our analysis, we use Moran’s $I$ correlation (Moran Reference Moran1950) to measure the spatial correlation. It is calculated as

(3.3) $$\begin{eqnarray}I=\frac{N}{\displaystyle \mathop{\sum }_{i=1}^{m}\mathop{\sum }_{j=1}^{m}w_{ij}}\frac{\displaystyle \mathop{\sum }_{i=1}^{m}\mathop{\sum }_{j=1}^{m}w_{ij}(X_{i}-\overline{X})(X_{j}-\overline{X})}{\displaystyle \mathop{\sum }_{i=1}^{m}(X_{i}-\overline{X})^{2}},\end{eqnarray}$$

where $N$ is the number of spatial units indexed by $i$ and $j$ ( $N=m\times n$ ). $X$ is the value of $p^{\prime }\dot{Q}_{ij}^{\prime }$ at the corresponding value of the matrix at ( $i,j$ ) and $\overline{X}$ is the spatial average of $X$ ; and $w_{ij}$ is a component of a matrix of spatial weights. Values of Moran’s $I$ correlation range from $-1$ to $+1$ where negative values indicate negative spatial autocorrelation and positive values indicate positive spatial autocorrelation. A zero value indicates a random spatial pattern. In our study, we calculate the spatial correlation statistics only for regions where the maximum heat release rate is observed to reduce the computational effort. Dakos et al. (Reference Dakos, van Nes, Donangelo, Fort and Scheffer2010) used a spatial weight of 1 for the neighbouring units of a cell and 0 otherwise. However, there are many schemes available in the literature to choose the spatial weights matrix (Cliff & Ord Reference Cliff and Ord1973; Getis Reference Getis2009; Getis & Aldstadt Reference Getis and Aldstadt2010). In our study, the spatial weight for the calculation is input as 1 if the distance between the two points is under a threshold and 0 if it is above. In our analysis, we have used a threshold of half the diameter of the bluff body. This threshold is sufficient to capture the formation of the emerging spatially coherent structures.

Figure 15.

(a–c) Pressure fluctuations corresponding to $\unicode[STIX]{x1D719}=0.98$ (a), 0.83 (b) and 0.63 (c). (d–f) Global heat release rate oscillations corresponding to $\unicode[STIX]{x1D719}=0.98$ (d), 0.83 (e) and 0.63 (f). (g–i) Spatial correlation for different equivalence ratios $\unicode[STIX]{x1D719}=0.98$ (g), 0.83 (h) and 0.63 (i). As the equivalence ratio is reduced, the spatial correlation (Moran’s $I$ correlation) increases due to the increase in correlation among nearby locations in space. This is characteristic of formation of spatial patterns. Here, the spatial correlation calculated is two-dimensional.

In figure 15, we observe an increase in the spatial correlation with a reduction in the equivalence ratio which indicates the emergence of patterns in the spatial dynamics as the system transitions from combustion noise to thermoacoustic instability. Spatial correlation of the acoustic power field is able to detect the emergence of patterns because during periodic oscillations, there is coherence in the field of instantaneous local acoustic power production. The increased coherence in the spatial dynamics during thermoacoustic instability and periodic epochs of intermittency is due to the heat release rate in unison upon impingement, which we have explained earlier in § 3.2.