Lagrangian coherent structures during combustion instability in a premixed-flame backward-step combustor

This paper quantitatively examines the occurrence of large-scale coherent structures in the flow field during combustion instability in comparison with the flow-combustion-acoustic system when it is stable. For this purpose, the features in the recirculation zone of the confined flow past a backward-facing step are studied in terms of Lagrangian coherent structures. The experiments are conducted at a Reynolds number of 18600 and an equivalence ratio of 0.9 of the premixed fuel-air mixture for two combustor lengths, the long duct corresponding to instability and the short one to the stable case. Simultaneous measurements of the velocity field using time-resolved particle image velocimetry and the $\mathrm{C}{\mathrm{H}}^{*}$ chemiluminescence of the flame along with pressure time traces are obtained. The extracted ridges of the finite-time Lyapunov exponent (FTLE) fields delineate dynamically distinct regions of the flow field. The presence of large-scale vortical structures and their modulation over different time instants are well captured by the FTLE ridges for the long combustor where high-amplitude acoustic oscillations are self-excited. In contrast, small-scale vortices signifying Kelvin-Helmholtz instability are observed in the short duct case. Saddle-type flow features are found to separate the distinct flow structures for both combustor lengths. The FTLE ridges are found to align with the flame boundaries in the upstream regions, whereas farther downstream, the alignment is weaker due to dilatation of the flow by the flame's heat release. Specifically, the FTLE ridges encompass the flame curl-up for both the combustor lengths, and thus act as the surrogate flame boundaries. The flame is found to propagate upstream from an earlier vortex roll-up to a newer one along the backward-time FTLE ridge connecting the two structures.

Figure 1

Schematic of the backward-facing step combustor showing the inlet section (a) of length 365 mm and the downstream section (b) of length $L_d=1350$ and 480 mm for the long and short combustors, respectively.

Figure 2

Time series of pressure fluctuations $p\prime$ (shown in black) for (a) $L_d=1350\mathrm{mm}$ (long duct) and (b) $L_d=480\mathrm{mm}$ (short duct). Shown in red are the time series of the fluctuating component of the spatially integrated chemiluminescence signal $I{}_{\mathrm{CH}}^{\prime }$ for the corresponding experiments. The red circles denote the instants at which we plot the FTLE.

Figure 3

(a) f-FTLE, (b) b-FTLE, and (c) $\mathrm{C}{\mathrm{H}}^{*}$ chemiluminescence for $L_d=1350\mathrm{mm}$ (long duct) at instant 3 [Fig. 2], and (d) f-FTLE, (e) b-FTLE, and (f) $\mathrm{C}{\mathrm{H}}^{*}$ chemiluminescence for $L_d=480\mathrm{mm}$ (short duct) at instant 1 [Fig. 2]. The black arrow in each plot indicates the direction of the mean flow.

Figure 4

Overlapped f-FTLE (green), b-FTLE (red), and the velocity fields for (a) the long duct experiment at instant 3 [Fig. 2] and (b) the short duct experiment at instant 1 [Fig. 2]. The white regions correspond to the magnitude of b-FTLE and f-FTLE being less than $80{\mathrm{s}}^{-1}$.

Figure 5

b-FTLE ridges superimposed on the corresponding chemiluminescence field for the long duct case for the instants indicated in Fig. 2. Regions marked as S1, S2, and M (yellow and green circles) are the same as those in Fig. 4, at all instants.

Figure 6

f-FTLE ridges superimposed on the corresponding chemiluminescence field for the long duct case for the instants indicated in Fig. 2. Regions marked as S1, S2, and M (yellow and green circles) are the same as those in Fig. 4, at all instants.

Figure 7

b-FTLE ridges superimposed on the corresponding chemiluminescence field are shown for the short duct case for the instants indicated in Fig. 2. Regions marked as S1, S2, and S3 (yellow circles) are the same as those in Fig. 4, at all instants.

Figure 8

f-FTLE ridges superimposed on the corresponding chemiluminescence field are shown for the short duct case for the instants indicated in Fig. 2. Regions marked as S1, S2, and S3 (yellow circles) are the same as those in Fig. 4, at all instants.