Amplitude Death and Growth in a Pair of Non-identical Thermoacoustic Oscillators Interacting via Time-delay or Dissipative Coupling

Mohammad Hossein Doranehgard 1

Hong Kong University of Science and Technology Clear Water Bay, Hong Kong

Vikrant Gupta 2

Southern University of Science and Technology Shenzhen, China

Larry K. B. Li 3

Hong Kong University of Science and Technology Clear Water Bay, Hong Kong

ABSTRACT We numerically explore the e ff ect of non-identical heater powers on the quenching and amplification of self-excited thermoacoustic oscillations in two Rijke tubes interacting via time-delay or dissipative coupling. On applying either type of coupling separately, we find that the presence of non-identical heater powers can shrink the regions of amplitude death in both oscillators, while producing new regions of amplitude amplification in the weaker oscillator. We find that the magnitude of amplitude amplification increases with the heater power mismatch and with the total power input. This study highlights the critical role that non-identical thermal loads can play in determining the amplitude response of coupled thermoacoustic systems, facilitating the design of control strategies for coupled oscillator-like devices such as gas turbines.

1. INTRODUCTION

When two or more oscillators interact via coupling, they can develop a variety of collective multi- scale behavior, as manifested through adjustments in their phase and amplitude dynamics [1]. If the coupling is weak, mutual synchronization can occur, leading to frequency or phase locking between identical or non-identical oscillators [2,3]. By contrast, if the coupling is strong, oscillation quenching can occur, leading to amplitude death – a state in which all the oscillators of the system stop, with their phase trajectories converging to the same stable fixed point [4–6].

1 mhd@connect.ust.hk

2 vikrant@sustech.edu.cn

3 larryli@ust.hk

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

The collective behavior of a coupled oscillator system depends not only on the coupling strength but also on the coupling type. Previous studies have shown that amplitude death can be induced by various types of coupling, such as dissipative coupling, time-delay coupling, conjugate coupling, and nonlinear coupling [7]. Unsurprisingly, this has led to amplitude death being detected in various systems, such as electronic circuits [8], thermo-kinetic systems [9], and chemical reactions [10]. Besides the coupling parameters, the individual oscillator parameters may also influence the emergent behavior of coupled oscillator systems [3], but their e ff ect remains a topic of active research, particularly in the field of thermoacoustics [11]. In thermoacoustic systems, heat-release-rate and pressure fluctuations can interact in positive feedback to generate self-excited flow oscillations via the Rayleigh mechanism [4]. At high amplitudes, such thermoacoustic oscillations can exacerbate thermal and mechanical stresses [12] and induce flame blowout and flashback [13], reducing the e ffi ciency, operability and reliability of devices such as gas turbines, domestic boilers, and rocket engines [14, 15]. It is therefore important to identify the system and coupling parameters for which thermoacoustic oscillations can be suppressed or amplified [16–19]. Previous work on passive and active control of thermoacoustic oscillations has focused mostly on systems with isolated acoustic chambers (e.g. single combustors) [14, 16, 20–27]. This is because such systems exhibit relatively simple geometries and well-defined boundary conditions, facilitating simulation, experimentation and analysis [28–30]. By contrast, the thermoacoustics of coupled oscillator systems has been studied less extensively, despite their relevance to practical devices such as can-annular gas turbines [31–35]. Crucially, it is widely acknowledged that the collective behavior of a coupled thermoacoustic system cannot simply be inferred from the individual behavior of its constitute oscillators in isolation [18, 19]. This is because bidirectional acoustic interactions can occur between adjacent oscillators, producing complex collective dynamics that can be di ffi cult to predict within a single oscillator framework based on a reductionist approach [11,18,19]. By adopting a coupled oscillator framework, several researchers have been able to identify states of quenching suitable for the suppression of thermoacoustic oscillations [18,19]. Prime among these is amplitude death, which, as noted earlier, occurs when all the oscillators of a coupled system become quenched to a common steady state [5]. Amplitude death has proved to be an e ff ective mechanism for suppressing thermoacoustic oscillations, by disrupting the feedback loop between sound waves and heat-release-rate fluctuations. Experiments on thermoacoustics within the past decade have shown that amplitude death can be found in both laminar systems (e.g. those powered by Bunsen flames [36], electrically heated meshes [37] and porous stacks [38,39]) and turbulent systems (e.g. those powered by lean-premixed flames [40, 41]). Besides amplitude death, other complex nonlinear phenomena have also been reported, such as partial amplitude death, phase-flip bifurcations, and in-phase / anti- phase synchronization [37]. Despite the contributions from experiments, surprisingly detailed insight can be gained by analyzing low-order models of coupled oscillator systems. For example, modeling studies have shown that achieving amplitude death requires a finite detuning if dissipative coupling is applied alone [42,43]. However, amplitude death can also arise without any detuning (i.e. in identical oscillators) if time-delay coupling is applied alone, provided that the coupling strength and delay time are in appropriate ranges [43,44]. Importantly, if dissipative and time-delay coupling are applied simultaneously, then the parameter region corresponding to amplitude death grows in size [38,43]. The above review would suggest that a potential way to suppress thermoacoustic oscillations might be to induce amplitude death by carefully adjusting the system parameters (e.g. detuning) and the coupling parameters (e.g. type and strength). However, most previous studies on coupled thermoacoustic systems have focused on the idealized case of identical oscillators, i.e. oscillators with identical limit-cycle frequencies and amplitudes before coupling is applied. Although some studies – most notably that by [43] – have considered the e ff ect of non-identical limit-cycle characteristics, the full extent of the parameter space has yet to be systematically explored. This is an important task because in practical devices, even oscillators built with identical dimensions and identical materials will rarely exhibit identical limit-cycle amplitudes and frequencies. Besides amplitude death, another

state worth examining is when the limit-cycle amplitude of an uncoupled oscillator becomes amplified due to coupling – a state we refer to here as amplitude amplification. This state is just as important as amplitude death itself because in some devices (e.g. pulse combustors [45] and solid-state lasers [46]), it is not su ffi cient to only avoid amplitude death but also necessary to maintain or amplify the existing self-excited oscillations. Consequently, it is important to understand how the introduction of mismatches in the initial (uncoupled) limit-cycle characteristics of a coupled oscillator system can influence its quenching and amplification behavior. In this numerical study, we explore the e ff ect of non-identical heater powers on the quenching and amplification of two self-excited thermoacoustic oscillators, each modeled as a prototypical Rijke tube. We couple the two oscillators together, first via dissipative coupling only and then via time- delay coupling only. We show that irrespective of the coupling type, increasing the heater power mismatch can shrink the regions of amplitude death in both oscillators, while creating new regions of amplitude amplification in the weaker oscillator. This study highlights the important role that non- identical thermal loads can play in determining the parameter space over which amplitude death and amplitude amplification occur in coupled thermoacoustic systems.

2. LOW-ORDER THERMOACOUSTIC MODEL

We consider a prototypical thermoacoustic system consisting of two Rijke tube oscillators interacting via time-delay or dissipative coupling. Following [47] and [43], we start with the linearized forms of the momentum and energy equations for an acoustic field with a negligible mean temperature gradient:

ρ ∗ ∂ u ′∗

∂ t ∗ + ∂ p ′∗

∂ x ∗ = 0 , (1)

∂ p ′∗

∂ t ∗ + γ p ∗ ∂ u ′∗

∂ x ∗ = ( γ − 1) ˙ Q ′∗ δ x ∗ − x ∗ h
, (2)

where the superscript ∗ denotes dimensional quantities. Here t ∗ , x ∗ and x ∗ h denote time, the position along the duct, and the position of the heat source, respectively. Moreover, p ′∗ , u ′∗ , γ and ρ ∗ denote the acoustic pressure, the acoustic velocity, the specific heat ratio, and the mean fluid density, respectively. The heat source is modeled as a compact cylinder producing HRR fluctuations per unit area ˙ Q ′∗ that are localized in space by the Dirac delta function δ . The heat source is positioned at x ∗ h = l ∗ / 4 ( l ∗ ≡ the duct length), which is ideal for producing self-excited thermoacoustic oscillations in a Rijke tube. The acoustic duct is open at both ends, implying that p ′∗ = 0 at the system boundaries ( x ∗ = 0 and l ∗ ), with the instantaneous pressure p ∗ there being equal to the time-averaged pressure p ∗ . To parameterize the governing equations (Equation 1 and Equation 2), we define the following dimensionless variables (without the superscript ∗ ):

x ≡ x ∗

l ∗ , t ≡ t ∗

l ∗ / c ∗ 0 , u ′ ≡ u ′∗

u ∗ 0

p ∗ c ∗ 0 , M ≡ u ∗ 0 c ∗ 0 , (3)

p ∗ , ˙ Q ′ ≡ ˙ Q ′∗

p ′ ≡ p ′∗

where M , u 0 and c 0 denote the mean-flow Mach number, the steady state velocity and the speed of sound, respectively. Substituting the variables from Equation 3 into Equation 1 and Equation 2, we obtain the following dimensionless momentum and energy equations:

γ M ∂ u ′

∂ t + ∂ p ′

∂ t = 0 , (4)

∂ p ′

∂ t + γ M ∂ u ′

∂ x + ζ p ′ = ( γ − 1) ˙ Q ′ δ ( x − x h ) , (5)

where ζ is a damping coe ffi cient whose value will be set later. We use a modified version of King’s law to model the quasi-steady heat transfer from the heat source (a round cylinder) to its surrounding fluid [48,49]:

˙ Q ′ ( t ) = 2 L h  T h − T 

p

πλ C v u 0 ρ R h

S √

3 c 0 p

r

r

 , (6)



| 1

1 3

3 + u ′ h ( t − τ h ) | −

×

where T h , R h and L h de not e the temperature, radius and length of the heat source, respectively. Furthermore, C v , λ and T denote, respectively, the constant-volume specific heat, the thermal conductivity and the mean fluid temperature in the acoustic duct of cross-sectional area S . Following [50] and [49], we model the thermal-inertia of the heater with the time lag parameter τ h acting on the acoustic velocity, u ′ h ( t − τ h ). By inserting the HRR model (Equation 6) into the energy equation (Equation 5), we get [43,50]:

∂ x + ζ p ′ = ( γ − 1) 2 L h  T h − T 

∂ p ′

∂ t + γ M ∂ u ′

p

πλ C v u 0 ρ R h

S √

3 c 0 p

r

r

 δ ( x − x h ) . (7)



| 1

1 3

3 + u ′ h ( t − τ h ) | −

×

Next we use the Galerkin technique to simplify the system of partial di ff erential equations (PDEs, Equation 4 and Equation 7) into a system of ordinary di ff erential equations (ODEs). To do this, we expand the acoustic velocity and pressure in terms of basis functions (ten Galerkin modes, N = 10) representing the natural acoustic duct modes with no heat input [43,50]:

N X

u ′ =

j = 1 η j cos ( j π x ) , (8)

N X

j = 1 ˙ η j γ M

p ′ = −

j π sin ( j π x ) , (9)

where η j and ˙ η j denote, respectively, the expansion coe ffi cients for the acoustic velocity ( u ′ ) and the acoustic pressure ( p ′ ). By substituting the Galerkin expansion (Equation 8 and Equation 9) into the PDE system (Equation 4 and Equation 7), we obtain the following ODE system [43,47]:

d η j

dt = ˙ η j , (10)

d ˙ η j

dt + 2 ζ j ω j ˙ η j + ω 2 j η j =

r

r



 sin ( j π x h ) , (11)

| 1

1 3

− j π K

3 + u ′ h ( t − τ h ) | −

where the j th duct mode (Galerkin mode) has angular frequency ω j = j π . We account for frequency dependent dissipation via the damping coe ffi cient [51,52]:

" c 1 ω j ω 1 + c 2 r ω 1

# , (12)

ζ j = 1

2 π

ω j

Figure 1: Pressure amplitude ( p ′ rms ) and limit-cycle frequency ( ω ) as functions of the heater power ( K ).

where the parameter values ( c 1 = 0 . 1, c 2 = 0 . 06) are chosen based on the analysis by [43]. The dimensionless heater power is defined as:

K ≡ 4 ( γ − 1) L h ( T h − T )

p

πλ C v u 0 ρ R h , (13)

M γ S √

3 c 0 p

where all the parameter values are chosen based on the analysis by [47]. To establish a reference condition, we first consider a single (uncoupled) Rijke tube oscillator. We examine its temporal evolution by numerically solving the ODE system described above (Equation 10 to Equation 13). Figure 1 shows the bifurcation diagram, with K as the bifurcation parameter. Along the forward path (increasing K ), the root-mean-square pressure fluctuation p ′ rms increases abruptly from around zero to a high value at a critical heater power ( K = 0 . 62). The high-amplitude state is self-excited and periodic in time (see the insets of Fig. 1), indicating that the system has transitioned from a fixed point to a limit cycle via a Hopf bifurcation, with K = 0 . 62 being the Hopf point. Along the backward path (decreasing K ), the system does not immediately revert to its initial fixed- point state, but instead remains on the limit-cycle branch until the heater power drops below the saddle-node point ( K = 0 . 52). The di ff erence between the forward and backward paths leads to a hysteretic bistable regime (gray shading in Fig. 1), which indicates that the Hopf bifurcation is subcritical. In the limit-cycle regime, the system oscillates self-excitedly at a natural frequency of ω = 2 π f = 3 . 25. As K increases, this limit-cycle frequency changes only slightly, but the limit-cycle amplitude increases significantly, with the pressure waveform becoming less sinusoidal owing to the emergence of harmonics (see the insets of Fig. 1). Next we couple two Rijke tube oscillators together, using superscripts A and B to refer to tubes A and B, respectively. For tube A, the modified system of ODEs from Equation 10 and Equation 11 becomes [43]: d η A j dt = ˙ η A j , (14)

d ˙ η A j dt + 2 ζ j ω j ˙ η A j + ω 2 j η A j =

r

r

− j π K A 

 sin ( j π x h )

| 1

1 3

3 + u ′ A h ( t − τ h ) | −

+ κ τ  ˙ η B j ( t − τ ) − ˙ η A j ( t ) 

+ κ d  ˙ η B j − ˙ η A j 

, (15)

| {z } Time-delay coupling

| {z } Dissipative coupling

_° oe 0.0016 * sora ee * coorgl WN 510 510 > > Forward \ \ aa 3.50 < Backward pe 0.001 E ~= 0.0005

where switching the superscripts A and B gives the governing equations for tube B. The strengths of the time-delay and dissipative coupling terms are denoted by κ τ and κ d , respectively. In this study, both κ τ and κ d remain constant between the two tubes. Moreover, the coupling delay time τ also remains constant between the two tubes. This implies that both the time-delay and dissipative coupling terms are symmetric. We adjust the heater powers ( K A and K B ) independently in a parameter space defined by α ≡ K A + K B = { 2 , 3 , 4 } and β ≡ K B / K A = { 1 , 1 . 5 , 2 . 2 , 3 } . Thus, α quantifies the total power input in the two tubes, while β quantifies the di ff erence in their heater powers. To introduce detuning, we fix ω A at 3.25, but vary ω B such that 0 . 75 ⩽ ω B /ω A ⩽ 1 . 34. To quantify the changes in the steady state amplitude due to coupling, we use the normalized oscillator amplitude, defined as the ratio of the root-mean-square pressure fluctuation with coupling to the same quantity without coupling, ϵ A ≡⟨ p ′ A rms ⟩ / p ′ A rms and ϵ B ≡⟨ p ′ B rms ⟩ / p ′ B rms , where the angle brackets ⟨⟩ denote the presence of coupling.

3. RESULTS AND DISCUSSION

We consider two coupling schemes: (Sec. 3.1) dissipative coupling only, and (Sec. 3.2) time-delay coupling only.

3.1. Dissipative coupling only First we examine how α and β a ff ect ϵ A and ϵ B when the two tubes interact via dissipative coupling only ( κ d > 0, κ τ = 0). Figure 2 shows ϵ A and ϵ B in a parameter space defined by κ d and ω B /ω A . Before discussing the results, we note that when uncoupled, tube A at α = 2 and β = 3 is at a fixed point (Fig. 1: K A = 0 . 5) rather than a limit cycle, implying that ϵ A is undefined; this specific case is therefore omitted from all the figures. In Fig. 2, four distinct types of amplitude response can be identified: (cyan) amplitude death, ϵ A or ϵ B ⩽ 0 . 01; (blue) amplitude reduction, 0 . 01 < ϵ A or ϵ B < 1; (white) neutral response, ϵ A or ϵ B = 1; and (red) amplitude amplification, ϵ A or ϵ B > 1. When the heater powers are identical ( β = 1), both tubes exhibit amplitude-death regions on either side of ω B /ω A = 1 (Fig. 2). Increasing β from 1 causes these regions to shrink, although this e ff ect is more pronounced for ω B /ω A > 1 than for ω B /ω A < 1. Figure 3 shows time traces of the pressure fluctuation for a sample case where amplitude death occurs in both tubes as a result of dissipative coupling. Before the coupling is applied ( t < 200), both tubes exhibit period-1 self- excited pressure oscillations of di ff erent limit-cycle amplitudes ( β = 1 . 5). However, after the coupling is applied at t = 200, the oscillations in both tubes rapidly quench to a negligible amplitude, which is a characteristic feature of amplitude death. After the coupling is removed at t = 400, tube B takes a relatively long time to return to its initial period-1 state, while tube A remains quenched, indicating hysteresis. A key finding is that introducing non-identical K can hinder the emergence of amplitude death. Indeed when both the heater power ratio and the total heater power are high ( β = 3, α = 4), neither tube exhibits amplitude death for the present test conditions. Intriguingly, Fig. 2 also shows that when β > 1, amplitude amplification occurs, but only in the weaker oscillator (tube A). When β increases for a fixed α , the magnitude of amplitude amplification in tube A increases, without much a ff ecting the magnitude of amplitude reduction in tube B, apart from a shrinkage in the regions of amplitude death. The shape of the amplitude-amplification region resembles that of the classic 1:1 Arnold tongue found in unidirectionally coupled systems undergoing forced synchronization [3]. Here the strength of the dissipative coupling ( κ d ) acts as an e ff ective forcing amplitude, causing the region of amplitude amplification to widen on both sides of ω B /ω A = 1. When α increases for a fixed β > 1, the magnitude of amplitude amplification grows in tube A, while the regions of amplitude death shrink in both tubes, vanishing entirely at α = 4 and β = 3 for the present range of frequency ratios.

Figure 2: Dissipative coupling only ( κ τ = 0): the normalized oscillator amplitude ( ϵ A ≡⟨ p ′ A rms ⟩ / p ′ A rms , ϵ B ≡⟨ p ′ B rms ⟩ / p ′ B rms ) shown in a parameter space defined by the dissipative coupling strength ( κ d ) and the frequency ratio ( ω B /ω A ). Focus is placed on the e ff ect of the heater power ratio ( β ≡ K B / K A ) and the total heater power ( α ≡ K A + K B ).

B=2.2 B=15 Kad Kd Kd 0.5 0.5 0.5 0 a=4 Tube B Tube A Tube B 0.8 1 1.22 08 1 1.2 08 1 12 08 1 12 08 1 1.2 08 1 1.2 w/w w/w wE/wA wE/wA wB/wA — wP/wA €4 CGC — 0.0 05 410 1.5 2.0 2.5

Figure 3: Time traces of the pressure fluctuation for a sample case where amplitude death occurs in both tubes as a result of dissipative coupling only. The system and coupling parameters are α = 2, β = 1 . 5, κ d = 0 . 5, κ τ = 0 and ω B /ω A = 1 . 2.

3.2. Time-delay coupling only Next we examine how α and β a ff ect ϵ A and ϵ B when the two tubes interact via time-delay coupling only ( κ d = 0, κ τ > 0) with a symmetric coupling delay time ( τ ) and ω A = ω B . Figure 4 shows ϵ A and ϵ B in a parameter space defined by κ τ and τ . For β = 1 (Fig. 4: identical oscillators), both tubes exhibit a region of amplitude death, with no evidence of amplitude amplification. As α increases, the amplitude-death region shrinks but remains centered at τ ≈ T / 2 ≈ 1 ( T is the oscillation period), which is consistent with the numerical simulations of [43] and the lab experiments of [36]. For β > 1 (Fig. 4: non-identical oscillators), the region of amplitude death seen in both tubes at β = 1 splits into separate islands centered at τ ≈ T / 4 ≈ 0 . 5 and τ ≈ 3 T / 4 ≈ 1 . 5. As α increases, these islands remain centered at the same values of τ , but shrink until disappearing altogether at α = 4. As is the case with dissipative coupling (Sec. 3.1), we find that amplitude amplification occurs

— Tube A — Tube B Coupling off In ul 200 400 600 800

Figure 4: Time-delay coupling only ( κ d = 0): the normalized oscillator amplitude ( ϵ A ≡⟨ p ′ A rms ⟩ / p ′ A rms , ϵ B ≡⟨ p ′ B rms ⟩ / p ′ B rms ) shown in a parameter space defined by the time-delay coupling strength ( κ τ ) and the coupling delay time ( τ ) with ω A = ω B . Focus is placed on the e ff ect of the heater power ratio ( β ≡ K B / K A ) and the total heater power ( α ≡ K A + K B ).

B=2.2 B=3 B=1.5 £02 T T 0 0.5115205115205115205 11.5 205 1 T A -B T ee Ex UE 0.0 0.5 1.0 1.5 2.1 T 1.5 20.5 1 1.5 2 T

Figure 5: Time traces of the pressure fluctuation for a sample case where amplitude amplification occurs in tube A, while amplitude reduction occurs in tube B, both as a result of time-delay coupling. The system and coupling parameters are α = 4, β = 2 . 2, κ d = 0, κ τ = 0 . 3, ω A = ω B and τ = 1.

only in the presence of a heater power mismatch ( β > 1) and only in the weaker oscillator (tube A). Increasing β or α is found to increase the magnitude of amplitude amplification without much a ff ecting the coupling delay times at which it occurs, τ ≈ T / 2 ≈ 1 and τ ≈ T ≈ 2. Figure 5 shows time traces of the pressure fluctuation for a sample case where amplitude amplification occurs in tube A, while amplitude reduction occurs in tube B. Before the application of time-delay coupling ( t < 200), both tubes exhibit self-excited pressure oscillations of di ff erent limit-cycle amplitudes ( β = 2 . 2). Compared with the case of Fig. 3, here the values of K A and K B are higher, which causes the pressure waveforms in both tubes to undergo period doubling. This results in period-2 oscillations, similar to those observed numerically by [43] and experimentally by [53] at high heater powers. Following the application of coupling at t = 200, the oscillations in tube A become amplified, while those in tube B become attenuated. After the removal of coupling at t = 400, both tubes return to their initial

—Tube A — Tube B 0.04 Coupling off Coupling on Imm -0.04 100 200 300 400 500 600

limit-cycle states, with no sign of hysteresis. Crucially, we note that a coupling delay time ( τ ≈ 1) that induces amplitude death in identical oscillators ( β = 1) could induce amplitude amplification in non-identical oscillators ( β > 1). Thus, if time-delay coupling is used alone to suppress thermoacoustic oscillations via amplitude death, then it is essential to account for any possible mismatches in the oscillator characteristics. Otherwise, a state of amplitude death could turn into a state of amplitude amplification if a mismatch in K were to emerge, either via unintentional processes or by design.

4. CONCLUSIONS

In this numerical study, we have investigated the e ff ect of non-identical heater powers on the quenching and amplification of two self-excited thermoacoustic oscillators, each modeled as a prototypical Rijke tube. By parameterizing the system in terms of the heater power ratio ( β ) and the total heater power ( α ), we were able to identify several key findings. When only dissipative coupling is applied, introducing non-identical heater powers ( β > 1) is found to produce a central region of amplitude amplification in the weaker oscillator (tube A), resembling the shape of the classic 1:1 Arnold tongue found in unidirectionally coupled systems undergoing forced synchronization. As β or α increases, the magnitude of amplitude amplification grows in the weaker oscillator (tube A), while the regions of amplitude death shrink in both oscillators. When only time-delay coupling is applied, increasing β or α is found to shrink the amplitude-death regions in both oscillators. Switching from identical oscillators ( β = 1) to non-identical oscillators ( β > 1) causes a amplitude-death region to split into multiple islands centered at τ = T / 4 and 3 T / 4. Meanwhile, amplitude-amplification regions emerge at τ = T / 2 and T and grow in magnitude as β increases. However, the range of τ over which amplitude amplification occurs remains largely constant as β and α are varied for non-identical oscillators. Collectively, these findings show that introducing non-identical heater powers in a coupled thermoacoustic system can have profound and unexpected e ff ects on the overall amplitude response. In particular, we have shown that although increasing β can generally shrink the regions of amplitude death, it can also induce amplitude amplification, particularly at high α . In some coupled thermoacoustic systems optimized for stable operation (e.g. can-annular gas turbines), the occurrence of amplitude amplification may lead to flow oscillations strong enough to accelerate cyclic fatigue and degrade system e ffi ciency and reliability. Knowing the system and coupling parameters at which amplitude death and amplitude amplification occur can aid the design of these and other coupled thermoacoustic systems.

ACKNOWLEDGEMENTS

This work was supported by the Research Grants Council of Hong Kong (Project Nos. 16210418, 16210419, 16200220 and 16215521) and the Guangdong–Hong-Kong–Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications (Project No. 2020B1212030001). V.G. was supported by the National Natural Science Foundation of China (Grant Nos. 91752201, 12002147 and 12050410247), the Department of Science and Technology of Guangdong Province (Grant Nos. 2019B21203001 and 2020B1212030001), and the Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory, Guangzhou (Grant No. GML2019ZD0103).

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