Open-loop control of periodic thermoacoustic oscillations: Experiments and low-order modelling in a synchronization framework

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Abstract

Open-loop forcing is known to be an effective strategy for controlling self-excited thermoacoustic oscillations, but the details of this synchronization process have yet to be comprehensively explored. In this study, we experimentally examine the synchronization dynamics of a laminar conical premixed flame in a tube combustor subjected to periodic acoustic forcing. We compare the response of this forced self-excited system with that of a forced Duffing–van der Pol oscillator, and find many similarities but also some differences. The similarities include (i) a torus-birth bifurcation from periodicity to quasiperiodicity at low forcing amplitudes, producing a stable ergodic T2 torus attractor in phase space; (ii) a transition from T2 quasiperiodicity to lock-in above a critical forcing amplitude, which increases linearly as the forcing frequency ff deviates from the natural frequency f1; (iii) two distinct routes to lock-in, one via a torus-death bifurcation if ff is far from f1 and one via a saddle-node bifurcation if ff is close to f1; and (iv) asynchronous quenching (AQ), which coincides with a torus-death bifurcation to lock-in and reduces the oscillation amplitude – by up to 90% in the combustor. There are, however, quantitative differences between the two systems, which pertain mainly to (i) the magnitude of the amplitude reduction achieved by AQ and (ii) the size of the AQ region in the ff/f1–forcing-amplitude plane.

This study has three main contributions. First, it shows that studying open-loop control from a synchronization perspective can provide valuable insight into the optimal forcing conditions. Second, it shows that the optimal forcing condition for weakening thermoacoustic oscillations is that which causes the onset of lock-in via a torus-death bifurcation, as this is where AQ occurs. Third, it shows that the synchronization dynamics of a real combustor can be qualitatively modelled with a low-order universal oscillator. This suggests that it may be possible to develop and test new control strategies by analyzing the solutions to such an oscillator.

Introduction

Despite extensive research, thermoacoustic instability continues to hinder the development of lean-burn gas turbines, leading to crippling reductions in their safe operating envelopes and preventing full exploitation of their NOx-reduction potential [1]. There are two main strategies for mitigating thermoacoustic instability in gas turbines: passive control and active control.

In passive control, the aim is to weaken the driving mechanisms or strengthen the damping mechanisms, either by good initial design or by adding a passive device to an existing system [2]. However, to use passive control effectively, it is necessary to understand why the system oscillates. This is not trivial because of the complex interplay among the various possible driving mechanisms, such as mixture-strength coupling [3], hydrodynamic coupling [4] and entropy-wave coupling [5].

An alternative strategy is active control, which can be implemented in two different ways [6]: closed-loop control and open-loop control. In closed-loop control, the system is acted on by actuators (e.g. loudspeakers or fuel-modulation valves) receiving real-time instructions from a controller monitoring the system via one or more sensors [7]. This type of control works well in simple thermoacoustic systems but is challenging in industrial systems because of the limited reliability, accuracy and bandwidth of sensors and actuators that can withstand the harsh environment of a combustor [8]. It is also difficult to design controllers that can operate reliably with input signals contaminated by noise and the possible coexistence of multiple stable and unstable thermoacoustic modes, all coupled via different time delays [9]. This makes closed-loop control unacceptably risky for some applications, such as aircraft propulsion.

In open-loop control, the system is still acted on by actuators, but without instructions from controllers or sensors. The actuator signal is therefore independent of the state of the system [1]. Previous examples have used loudspeakers to modify the acoustic boundary conditions and to impart acoustic velocity perturbations to the flame [10], [11]. Others have used fast-response valves to modulate the flow rates of air [12] and fuel [13], [14] so as to disrupt the feedback loop between the acoustic pressure and the flame’s heat release rate (HRR).

Compared to closed-loop control, open-loop control may be less accommodating to changes in operating load, atmospheric conditions, and fuel composition [7]. Nevertheless, open-loop control still has several advantages. First, it is easier to implement because it does not require controllers or sensors. Second, because the phase of the actuator signal is allowed to drift relative to that of the thermoacoustic oscillations, the mechanical specifications of the actuator can be relaxed, enabling less costly and more robust equipment to be used [13]. Third, closed-loop control may be ineffective against the highly nonlinear oscillations found in some systems because most feedback algorithms are designed to stabilize a nominally unstable system in the limit of infinitesimal (linear) fluctuations, i.e. when the system is near a fixed point in phase space [7]. Open-loop control, by contrast, is inherently nonlinear, relying on high-amplitude forcing to stabilize the system [14]. In fact, open-loop control has been shown to be more effective than closed-loop control when the forcing frequency is far from the natural instability frequency [15]. For these reasons, and because the design of most feedback controllers requires some prior knowledge of the open-loop response [16], we will focus on open-loop control.

Lubarsky et al. [15] have shown that modulation of the fuel injection rate in a pressurized combustor can reduce the thermoacoustic amplitude by up to 90%, even when the fuel modulation frequency (250–330 Hz) is far from the natural instability frequency (386 Hz). Richards et al. [13] used 12 injectors to periodically modulate the equivalence ratio of a gas-turbine combustor and found that this could reduce the thermoacoustic amplitude by more than 50%. However, they remarked that this strategy tends to be effective only when the system is near its stability boundary, which suggests that it may not be as effective on highly nonlinear oscillations.

Besides fuel modulation, acoustic forcing has also been shown to be an effective strategy for weakening thermoacoustic oscillations. Bellows et al. [10] applied acoustic forcing to a self-excited turbulent premixed combustor and found that (i) the system can lock into high-amplitude forcing at off-resonance frequencies, leaving no sign of the original natural mode, (ii) the thermoacoustic power can be reduced by up to 90%, but (iii) weak forcing ( < 10% of the burner velocity) has no significant effect on the amplitude or frequency of the oscillations. Recently, Balusamy et al. [11] carried out experiments similar to those of Bellows et al. [10] but interpreted the results within a synchronization framework using tools from nonlinear dynamics [17], [18]. They showed that a forced self-excited thermoacoustic system can exhibit a rich range of nonlinear features, including multiple bifurcations, quasiperiodicity and frequency pulling/pushing during forced synchronization. They proposed that such features could be modelled with a low-order universal oscillator based on a van der Pol kernel, but did not attempt to do so.

Section snippets

Contributions of the present study

In this study, we perform experiments and low-order modelling to investigate the open-loop control of a self-excited thermoacoustic system – a laminar conical premixed flame in a tube combustor. Our aim is to explore open-loop control within a synchronization framework using concepts from nonlinear dynamics [17], [18] and to model this control process phenomenologically with a universal oscillator containing just a few degrees of freedom. The key advantage of this approach over classical

Experimental setup

Experiments are performed on a laminar conical premixed flame in a tube combustor. Shown in Fig. 1, the setup consists of a stainless steel burner (inner diameter, ID: 16.8 mm; length: 800 mm), a quartz tube combustor with double open ends (ID: 44 mm; length: L= 860 mm), an acoustic decoupler (ID: 180 mm; length: 200 mm), and a loudspeaker for acoustic forcing. For flame stability, a copper extension tube (ID: 12 mm; length: 30 mm) with a fine-mesh screen is mounted at the burner exit. The fuel

Low-order model

Previous studies have shown that the forced response of various self-excited systems can be modelled with low-order universal oscillators [20], [21], [22]. Inspired by those studies, we use a forced DVDP oscillator [19] to phenomenologically model our combustor, gaining insight into the control dynamics from a synchronization perspective. The forced DVDP oscillator is a generic model of a forced self-excited oscillator with linear driving and nonlinear damping [19]. We choose this particular

Results: Experiments and low-order modelling

To quantify the effectiveness of the control strategy, we use the non-dimensional parameter ηpRMS[σ(pf)σ(puf)]/σ(puf), where σ(pf) and σ(puf) denote the root mean square of the pressure fluctuation when the system is forced and unforced, respectively. Thus, the pressure oscillations are weakened by the forcing when ηpRMS<0 but are amplified by the forcing when ηpRMS>0. An analogous parameter is defined for the DVDP oscillator: ηxRMS[σ(xf)σ(xuf)]/σ(xuf).

Conclusions

To identify effective strategies for open-loop control of self-excited thermoacoustic systems, we have experimentally investigated the synchronization dynamics of a laminar conical premixed flame in a tube combustor subjected to periodic acoustic forcing. We modelled this forced self-excited system as a forced DVDP oscillator, one of the simplest low-order universal oscillators with self-excited temporal solutions. We compared the synchronization dynamics of the two systems and found many

Acknowledgments

This work was funded by the Research Grants Council of Hong Kong (Projects 16235716 & 26202815).

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