Experiments are performed in a partially premixed bluff body-stabilized turbulent combustor by varying the mean flow velocity. Simultaneous measurements obtained for unsteady pressure, velocity, and heat release rate are used to investigate the dynamic regimes of intermittency (10.1 m/s) and thermoacoustic instability (12.3 m/s). Using wavelet analysis, we show that during intermittency, modulation of heat release rate occurring at the acoustic frequency fa by the heat release rate occurring at the hydrodynamic frequency fh results in epochs of heat release rate fluctuations where the heat release rate is phase locked with the acoustic pressure. We also show that the flame position during intermittency and thermoacoustic instability are essentially dictated by saddle point dynamics in the dump plane and the leading edge of the bluff body.
Gas turbine manufacturers invest in different technologies to make the combustor more stable while operating in lean conditions. Lean combustion is essential to reduce emissions due to stringent regulations in recent years. However, land-based gas turbine power plants incur significant losses due to power outages and maintenance costs arising due to thermoacoustic instabilities . Thus, active research is carried out on methods to forewarn and mitigate their occurrence in practical combustion systems.
A bluff body is commonly used as a stabilization mechanism in combustor flow fields. However, the complex flow patterns that arise as a result make the analysis of the instability mechanisms extremely difficult. Mixing of the reactants and heat transfer between unburnt reactants and burnt products is enhanced by the vortices present in the flow field [2,3]. In such combustors, a large periodic heat release rate is usually associated with impingement of hot combustion products in the vortices on the combustor walls. Such periodic impingement can force an acoustic mode inside the combustion chamber, which in turn modifies the flow and consequently, the mixing and heat release rates. This results in a positive feedback loop under certain conditions leading to thermoacoustic instability . Besides impingement, vortex interaction, merging, and collision are all mechanisms that can initiate this feedback loop depending on the geometry of the combustor. Understanding the vortex dynamics is therefore critical to describe the flame dynamics and mechanism of sound production in large classes of combustors.
The bluff body-stabilized combustor used in this study has been shown to display the intermittency route to thermoacoustic instability . As the flow rates are increased from stable operating conditions (reduction of equivalence ratio), periodic bursts of oscillation are observed in pressure traces intermittently amidst a state of aperiodic fluctuations. When the equivalence ratio is further decreased, a complete transition to periodic high amplitude oscillations emerges . To forewarn thermoacoustic instability, several precursors to instability are developed that quantify the burst states in intermittent regime: 0–1 test [7,8], recurrence tests , generalized Hurst exponent test [9,10], to name a few. Although useful, these precursors do not provide a way to identify the underlying dynamics responsible for this intermittent transition to instability. We employ the framework of Lagrangian coherent structures (LCS) to describe the flow patterns primarily responsible for sound production both during thermoacoustic instability and during the precursor stage of intermittency.
Lagrangian coherent structures or LCS, introduced by Haller  are locally the most repelling, attracting, and shearing material lines in the flow field. LCS has been useful in many fields such as aerodynamics , aeroacoustics , biological feeding , and oceanography . Recently, LCS has also been employed to study reacting flow fields in combustors [16,17]. These material lines can be extracted by tracking the ridges in the field of finite time Lyapunov exponents (FTLEs). In this study, we specifically track Lagrangian saddle points, identified as common points to the attracting and repelling LCS, to describe the role of large-scale vortices and acoustics in flame dynamics. To identify and isolate the acoustic component of the flow field, we utilize a well-known flow-decomposition technique called dynamic mode decomposition (DMD) . These techniques in conjunction with pressure and heat release rate measurements are used to identify the acoustic signatures present in the flow field during the bursting stages of intermittency.
The paper is organized as follows: in the next section (Experimental setup), the experimental configuration and the data acquisition techniques employed in the study are described. Then, post-processing techniques applied to the pressure, velocity and heat release rate data are presented in the s section Methodology. Salient results are illustrated in the next subsequent section Results and discussion followed by a section on Conclusions.
Experiments are conducted on a backward-facing-step combustor with a circular bluff body as the flame holder (Fig. 1). The bluff body located 45 mm from the dump plane is a circular disk of 47 mm diameter and has a thickness of 10 mm. The burner was provided with a hollow central shaft of 16 mm diameter to mount the bluff body. The central shaft is also used to deliver the fuel into the chamber through four radial injection holes of 1.7 mm diameter located 110 mm upstream of the dump plane. Air is supplied using a high pressure tank, which then passes through moisture separators before entering the plenum chamber and mixes with the fuel. Fuel used is Liquefied Petroleum Gas (LPG), with a composition of 60% propane and 40% butane by volume. The combustible mixture is ignited by a spark plug mounted on the dump plane that is connected to an 11 kV ignition transformer. The combustion chamber has a square cross section of 90 × 90 mm2 with a total length of 1100 mm. To minimize the acoustic radiation losses, a decoupler of large chamber size 1000 mm × 500 mm × 500 mm is used at the end of combustion chamber. The acoustic boundary condition of an open duct () is also approximately achieved in the decoupler.
Mass flow controllers (Alicat Scientific, MCR Series, Tucson, AZ), which have an measurement uncertainty of ±(0.8% of the reading + 0.20% of full scale), are used to measure and control the supply of air and fuel into the combustion chamber. This results in an uncertainty of ±0.02 in the equivalence ratio. The unsteady pressure fluctuations are measured using a piezoelectric transducer (PCB103B02, Depew, New York-uncertainty ±0.15 Pa) mounted on the combustor wall at a distance of 40 mm from the dump plane. The transducers are mounted on specially made pressure ports with Teflon adapters. The adapters result in an acoustic phase delay of 5 deg, not large enough to affect the analysis and results presented in this study. To capture the global unsteady heat release rate (), a photomultiplier tube module is mounted with an OH* filter (narrow bandwidth filter centered at 308 nm and 12 nm full width at half maximum (FWHM) having a field of view of 70 deg, which covers the viewing region. The signals from the piezoelectric transducer and the photomultiplier tube module are acquired at a sampling rate of 10 kHz for 3 s using a 16 bit analog-to-digital (A–D) conversion card (NI-6143, Austin, TX).
Spatial variations of unsteady heat release rate fluctuations are captured using high-speed CH* chemiluminescence images. The images are captured at a frame rate of 2 kHz using a high-speed CMOS camera (Phantom-V 12.1, Wayne, NJ). The repetition rate of the camera was fixed as 2000 fps with a spatial resolution of 800 × 600 pixels to ensure that the spatiotemporal flame dynamics is captured well. The exposure time is fixed at 499 μs. The camera is mounted with a ZIESS 50 mm camera lens at f/2 aperture. The flame images are captured using a narrow bandwidth filter centered at 435 nm (10 nm FWHM). A total of 5034 images are captured at each equivalence ratio studied.
Simultaneous velocity measurements are obtained using a single cavity-double pulsed Nd-YLF laser (Photonics, Ronkonkoma, USA) of operating wavelength 527 nm. The laser is operated at a repetition rate of 2 kHz. Using right angle prisms and convex lenses, the laser is steered toward the combustion chamber and expanded into a laser sheet of 2 mm thickness, using a 600 mm spherical lens and a −16 mm cylindrical lens. The sheet is then transmitted into the combustion chamber through a horizontal slit of 5 mm width made of quartz. TiO2 particles of approximately 1 μm are used as seeding particles. The Mie scattered light is captured using a high-speed CMOS camera. The camera is fitted with a ZEISS 100 mm lens with an aperture of f/5.6 and a short bandpass optical filter, centered at 527 nm. The camera is focused on a test section area of 32 mm × 64 mm and is operated at a frame rate of 2 kHz with a resolution of 510 × 1024 pixels. The time delay between two laser pulses is set between 15 and 25 μs.
The maximum displacement test is used with appropriate thresholds for different flow rates to detect outliers . Postprocessing algorithms used to detect and replace the outliers ensure less than 1% of the total velocity vectors are spurious, which are replaced using the bilinear interpolation method. The size of the final interrogation window is 32 × 32 for low flow rates and 48 × 48 for high flow rates. As a result, the uncertainty of velocity measurements can rise up to 5% for high flow rates whereas for low flow rates, the uncertainty is between 1.25% and 2.5%.
Finite Time Lyapunov Exponent.
For a flow field, which is time independent, the material surfaces along which the fluid parcels converge are termed unstable manifolds and the material surfaces along which fluid parcels diverge are termed stable manifolds. Fluid parcels stretch across the stable manifolds and assume the shape of the unstable manifolds with the passage of time. Ridges in the field of FTLEs, defined for an explicitly time dependent flow field, are essentially the finite time, unsteady versions of the stable and unstable manifolds [11,21].
In Eq. (2), we essentially quantify the rate of separation of neighboring trajectories of fluid parcels, which were initially close using with . The ridges or the maximum values in the FTLE field when the displacement of fluid parcels is tracked forward in time (hereafter referred to as fFTLE field) represent repelling LCS over the finite time interval T. When the displacement of fluid parcels is tracked backward in time, the ridges of the FTLE field (hereafter referred to as bFTLE field) represent attracting LCS. If one were to overlay the attracting LCS over the repelling LCS, one can identify certain common points of dynamic significance termed Lagrangian saddle points. Due to the approximate nature of FTLE analysis and the limitations in spatial and temporal resolution of data acquisition, we shall use the common points only as a heuristic indicator of the location of saddle points. Moreover, in turbulent flows, the attracting and repelling LCS can criss-cross multiple times in certain regions when overlaid. In such regions, we choose the saddle points heuristically by tracking common points near the location of the vortex cores in the braids of which one expects to find a saddle point of interest.
Dynamic Mode Decomposition.
where represent the time coefficients of the DMD modes or the strength of the DMD mode at time t. The time coefficients for each DMD mode can be computed using the least squares method by solving equation (3) at various time instants ti.
While calculating the coefficients, eigenvectors are considered orthogonal to each other, which is an approximation [23,24]. However, this orthogonality often holds true as can be verified by performing a Fourier transform on the time coefficients. In this study, we observe that the characteristic frequency of the DMD mode is retained in the time coefficients even after assuming orthogonality. Interested readers may refer Ref.  for details on implementation of the modified DMD algorithm on the flow field of the combustor under study.
In the section Results and Discussion, the implementation of these methods to the velocity fields obtained from the bluff body-stabilized combustor is discussed in detail.
Results and Discussion
Temporal Dynamics of Heat Release Rate During Intermittency.
Experiments are performed at different air flow rates corresponding to different mean burner velocities V0 by keeping the fuel flow rate at a constant value of 0.5 g/s. Unsteady pressure, velocity, and CH* chemiluminescence images were obtained simultaneously at sampling rates of 10 kHz, 1 kHz, and 2 kHz, respectively, for a duration of 1.364 s. The OH* chemiluminescence data are also obtained simultaneously at a sampling rate of 10 kHz.
The pressure, heat release rate (OH* chemiluminescence), and acoustic velocity fluctuations for the intermittent regime (, V0 = 10.1 m/s) are shown in the top three panels of Fig. 2. The acoustic component of velocity fluctuations (DMD time coefficient of the acoustic mode aac of the velocity data) is obtained using the procedure outlined earlier. The dominant frequencies in pressure and unfiltered velocity data are 176.4 Hz and 30.7 Hz (not shown), which will hereafter be referred to as the acoustic frequency (fa) and the hydrodynamic frequency (fh), respectively. The heat release rate fluctuations have three dominant frequencies: fh, fa and an intermediate frequency Hz.
We examined the phase relationship between acoustic velocity and the unsteady pressure and heat release rate fluctuations by performing a Hilbert transform (see the bottom two panels of Fig. 2). On examination, time windows can be identified when the phase difference between the acoustic pressure and the acoustic velocity aac has a constant value. We define these windows as burst events. The observed phase relationship between pressure and acoustic velocity during a burst event is intuitively obvious, as the pressure fluctuations in a burst must predominantly be acoustic. However, the interesting feature is that there is a continuous drift in the phase difference between the heat release rate and aac punctuated occasionally by regions of constant phase difference. This is suggestive of a highly nonlinear mechanism.
To investigate this feature further, we obtained the frequency–time plots of pressure and heat release rate signals through a wavelet transform (Fig. 3). It is seen that the acoustic frequency fa is present only during a burst event in the frequency–time plot of the pressure signal (shaded boxes above the panels). Further, we also notice the weak presence of the hydrodynamic mode fa in the pressure signal during such burst events. The wavelet transform of the heat release rate shows that both the frequencies fa and fh are present and dominant during burst events. However, as shown in the inset, there is a punctuated presence of the acoustic mode during burst events, whereas the hydrodynamic mode is present continuously. The time scale between successive heat release rate peaks at frequency fa is approximately equal to . This spacing and pattern indicates that there is a modulation of the heat release rate fluctuations at the acoustic mode of frequency fa by the heat release rate fluctuations at the slower hydrodynamic mode of frequency fh during burst events. This modulation results in repeated waxing and waning of the heat release rate at fa thereby acting as a punctuated heat source to the acoustic pressure.
Although the frequency of the heat release rate fluctuations and the acoustic pressure fluctuations match in the short time interval of intense heat release rate, Rayleigh criterion may not always be satisfied as a favorable phase relationship between pressure and heat release rate need not exist. We have already seen in Fig. 2 that there is an observable drift in the phase difference between the heat release rate and acoustic velocity aac even during burst events. The frequency–time plot of heat release rate fluctuations now confirms that this phase drift is due to the presence of the hydrodynamic mode in heat release rate. Further, the instants of peak heat release rate at fa coincide with the local peak in the global heat release rate signal. These instances also correspond to the short time windows of constant phase difference we observed between the pressure and heat release rate fluctuations (see Fig. 2). This shows that there is phase synchronization or a phase lock-in between the pressure and heat release rate over short time instances during a burst event. If the operating conditions are such that this lock-in happens with a favorable phase relationship over long time durations, instability can be established as Rayleigh criterion is satisfied. Such a model of thermoacoustic instability has previously been proposed by Nair and Sujith  and Matveev and Culick  and the experimental data we obtained support that model.
An additional observation in the frequency–time plot of the heat release rate is that there is a continuous frequency band from fh to fa during the burst states (see the inset of Fig. 3). The presence of a band suggests that an energy transfer from the lower hydrodynamic mode to the acoustic mode is potentially responsible for the burst events. The continuous band also indicates that there can be a drift of the instantaneous phase in between peaks of heat release rate at fa, which was already established earlier. When the heat release rate at the acoustic mode reaches a peak, the phase difference between the heat release rate fluctuations and the acoustic pressure remains constant (same frequency fa) for a brief time instant before drifting out of phase again due to the modulation by the hydrodynamic mode at fh. If during these short time windows, the phase relationship is favorable, Rayleigh criterion is satisfied, and we can have amplification of pressure fluctuations producing bursts.
Based on these observations, we can convincingly surmise that the heat release rate affects the pressure indirectly through velocity fluctuations. As has been reported in several works prior to this, the heat release rate in such combustors depends primarily on the dynamics of velocity fluctuations . The velocity fluctuations, in the regime that we explored, consist of a dominant hydrodynamic mode at fh and a weak acoustic mode at fa. This results in strong heat release rate fluctuations at fh, which can produce heat release rate fluctuations at the acoustic mode of frequency fa either directly through velocity fluctuations at fa or through system nonlinearities. The simultaneous presence of two dominant frequencies fh and fa in heat release rate fluctuations results in epochs of intense heat release rate fluctuations at the acoustic mode, which if phased properly with the acoustic pressure, can result in burst events during intermittency via Rayleigh's criterion.
Flame Dynamics During Intermittency.
We determined the Lagrangian saddle points in the velocity data acquired during the intermittent regime by using the FTLE framework outlined earlier. Interpolation is carried out on the velocity data to improve the spatial resolution ten times that of the acquired data prior to tracking fluid parcel displacements. A reference snapshot of the LCS data to be presented in this section is shown in Fig. 4. The top half shows the ridges of bFTLE field (lines in the foreground) along with CH* chemiluminescence (contour in the background). The bottom half of the Fig. 4 shows the ridges of both bFTLE (line contour representing attracting LCS) and fFTLE (filled contour representing repelling LCS) fields. The circles in both halves of the figure indicate the approximate location of the Lagrangian saddle points. These saddle points of interest are selected based on their proximity to the vortex cores.
In Fig. 5, CH* chemiluminescence along with the attracting and repelling LCS is shown for one periodic cycle of the bursting state in the intermittency regime in a time window when the heat release rate at fa is close to a peak (constant phase difference with the acoustic mode). The corresponding pressure fluctuations and heat release rate fluctuations are also shown at the bottom of the figure. From the figure, we find that there are two sets of critical saddle points labeled S and T that dictate flame dynamics. The saddle points labeled S emerge and propagate along the shear layer from the upstream dump plane. The saddle points labeled T emerge and propagate along the separating shear layer from the edge of the bluff-body closest to the upstream dump plane. These two sets of saddle points are seen to be continuously generated at the acoustic frequency fa. We also find that the flame is preferentially aligned with the bFTLE fields and are concentrated in the vortex cores identified by the bFTLE fields.
Initially, the saddle point S1 moves toward the leading edge of the bluff body carrying the flame in the region between S1 and S2 (see Figs. 5(a)–5(c)). This interface, which represents the vortex core located along the shear layer, is part of the attracting LCS (bFTLE ridges). After reaching the contraction near the bluff body, the flame now transitions to the wake of the bluff body behind the saddle point T1 (Fig. 5(d)). Subsequently, the saddle point T1 convects downstream and carries the flame along with it (Figs. 5(e) and 5(f)). This passage of the flame from the upstream dump plane to regions downstream of the bluff body and the generation of saddle points S and T happen at the frequency fa as long as the magnitude of heat release rate is near its peak.
We also computed the FTLE ridges and heat release rates in a time window when bursts are absent (see Fig. 6). The chosen window has no dominant frequency components as identified from the frequency–time plot of the pressure signal (see Fig. 3). We see that the heat release rate is extremely feeble. Further, the saddle point sequences S and T are now absent. This clearly indicates that it is the periodic passage of saddle points dictated by the frequencies fh and fa that govern the location and nature of heat release rate. Since it is ultimately the dynamics of the velocity field that governs the dynamics of saddle points, it makes perfect sense to model the heat release rate purely as function of the velocity fluctuations in such systems.
Flame Dynamics During Thermoacoustic Instability.
To identify common features observed in the dynamics of intermittent bursts and thermoacoustic instability, we also performed the Lagrangian saddle point analysis over a cycle of thermoacoustic instability. During instability, unlike during intermittency, the phase difference between the pressure and heat release rate fluctuations must necessarily be favorable via Rayleigh's criterion and we expect heat release rate fluctuations to continuously feed energy into acoustic oscillations. For the combustor studied, instability happens at a mean flow velocity of 12.3 m/s (). The principal dominant frequency in pressure, velocity, and heat release rate fluctuations during instability is f =141.9 Hz. The pressure spectra have a dominant first harmonic at 3f (not shown) and the velocity shows a dominant secondary peak at 2f. This indicates that the combustor resembles an acoustically closed-open boundary with the dump plane representing the close boundary.
CH* chemiluminescence images along with the FTLE ridges and saddle points over a cycle of instability are shown in Fig. 7. The corresponding pressure and heat release rate are also shown in the two bottom panels of the figure. We see that although there is a small phase difference between the pressure and heat release rate fluctuations, there is sufficient overlap for instability to be excited. In other words, Rayleigh criterion is satisfied. During the compression phase (a–d), the regions of intense heat release rate mostly lie ahead of the bluff body, whereas during the expansion phase (e–g), these regions mostly lie in the wake of the bluff body. This fairly good correlation between the flame location and the acoustic pressure is due to the small and constant phase difference between acoustic pressure and heat release rate fluctuations. We could not make such conclusions on the flame location based on pressure fluctuations during intermittency due to the arbitrary nature of the phase relationship between pressure and heat release rate fluctuations during a burst event. We once again observe two saddle sequences S and T just as we did for intermittent burst oscillations. The saddle points labeled S carry the flame from the dump plane to the leading edge of the bluff body and the saddle points labeled T carries the flame from the leading edge of the bluff body to further downstream.
Thus, we find that the mechanism behind the generation of sound is intimately linked to the dynamics of saddle points in the flow field, which determine the location of intense heat release rate fluctuations. The flow field, through the presence of the hydrodynamic mode, also dictates the phase alignment of heat release rate with the acoustic field. Hence, we find that studying the flow field in conditions that presage an impending instability is quite useful to shed light on the processes that generate and sustain large-amplitude, coherent pressure oscillations.
Measurements of unsteady pressure, heat release rate fluctuations, and two-dimensional velocity field were acquired from a bluff body-stabilized backward-facing-step combustor in the regimes of intermittency and thermoacoustic instability. Analyzing the frequency–time dynamics of heat release rate reveals that the modulation of heat release rate occurring at the acoustic frequency fa by the heat release rate occurring at the hydrodynamic frequency fh results in epochs of intense heat release rate fluctuations at fa. During these epochs, we find that the heat release rate is phase locked with the acoustic pressure.
Investigating the spatial dynamics of the flow field using the FTLE framework reveals that two sets of saddle points, one emerging from the dump plane shear layer (labeled S) and another emerging from the leading edge of the bluff body (labeled T), dictate the location of the flame and its dynamics. In an acoustic cycle during a burst event in intermittency, the flame is carried by S toward the bluff body from where it is carried by T toward the wake of the bluff body. New saddle points are generated every acoustic cycle and the process is repeated. The two sets of saddle points are also seen during thermoacoustic instability. Due to a favorable phase relationship between pressure and heat release rate, the flame lies predominantly upstream of the bluff body during the compression phase when the flame is carried by saddle point S and predominantly downstream of the bluff body in the expansion phase when the flame is carried by saddle point T.
We conclude by stating that since the phase of the heat release rate fluctuations drift continuously during intermittency, Rayleigh's criterion, which is a time-averaged measure, may not be an appropriate descriptor of stability in such regimes. A description of the growth of pressure oscillations during burst events therefore requires a more elaborate statistical study of the cycle to cycle variability of the phase difference between pressure and heat release rate fluctuations, which will be taken up in future studies.
C. P. Premchand and Vineeth Nair would like to thank IRCC, IIT Bombay for funding the study (Grant no. 16IRCCSG006). N. B. George, M. Raghunathan, V. R. Unni, and R. I. Sujith are thankful to Office of Naval Research Global (ONRG) for their financial support.
Industrial Research and Consultancy Centre, IIT Bombay (Grant No. 16IRCCSG006; Funder ID: 10.13039/501100011627).
Office of Naval Research Global (Grant No. N62909-18-1-2061; Funder ID: 10.13039/100007297).