# Self-coupling: An Effective Method to Mitigate Thermoacoustic Instability

Sneha Srikanth  Department of Mechanical Engineering, Indian Institute of
Technology Madras, Chennai 600036, India  Ankit Sahay  Samadhan A. Pawar
<EMAIL_ADDRESS>Department of Aerospace Engineering, Indian Institute
of Technology Madras, Chennai 600036, India  Krishna Manoj  Department of
Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA
02139, USA  R. I. Sujith  Department of Aerospace Engineering, Indian
Institute of Technology Madras, Chennai 600036, India

###### Abstract

The presence of undesirable large-amplitude self-sustained oscillations in
combustors resulting from thermoacoustic instability can lead to performance
loss and structural damage to components of gas turbine and rocket engines.
Traditional feedback controls to mitigate thermoacoustic instability possess
electromechanical components, which are expensive to maintain regularly and
unreliable in the harsh environments of combustors. In this study, we
demonstrate the quenching of thermoacoustic instability through self-coupling
– a method wherein a hollow tube is used to provide acoustic self-feedback to
a thermoacoustic system. Through experiments and modeling, we identify the
optimal coupling conditions for attaining amplitude death, i.e., complete
suppression of thermoacoustic instabilities, in a horizontal Rijke tube. We
examine the effect of both system and coupling parameters on the occurrence of
amplitude death. We thereby show that the parametric regions of amplitude
death occur when the coupling tube length is close to an odd multiple of the
length of the Rijke tube. The optimal location of the coupling tube for
achieving amplitude death is near the anti-node of the acoustic standing wave
in the Rijke tube. Furthermore, we find that self-coupling mitigates
thermoacoustic instability in a Rijke tube more effectively than mutual
coupling of two identical Rijke tubes. Thus, we believe that self-coupling can
prove to be a simple, cost-effective solution for mitigating thermoacoustic
instability in gas turbine combustors.

††preprint: APS/123-QED

##  I Introduction

The presence of undesirable self-sustained oscillations in many practical
systems has been a matter of concern for researchers over the years. Notorious
examples of such oscillations include wobbling bridges [1], fluttering
aircraft wings [2], rumbling combustors [3, 4], oscillatory prey-predator
populations [5, 6], and stock market fluctuations [7]. The presence of these
oscillations can have catastrophic consequences such as destruction of
bridges, structural damage to aircraft and combustors, extinction of species,
and financial crisis, respectively. It is thus vital to develop methods to
quench these oscillations.

In the present study, we investigate the mitigation of thermoacoustic
instability, which refers to the occurrence of ruinously high amplitude self-
sustained oscillations in a combustor due to the reinforcing interaction
between the heat release rate fluctuations from the heat source and the
acoustic field of the combustor [3, 8]. The manifestation of thermoacoustic
instability can have several catastrophic consequences. It can cause thrust
oscillations in rocket engines, which could jeopardize space missions.
Thermoacoustic instability can cause rocket and gas turbine engine components
to vibrate at high amplitudes, leading to catastrophic structural damage. The
increased heat transfer due to thermoacoustic instability can overwhelm the
thermal protection system and the large amplitude acoustic oscillations can
damage electronic components in the engines [9, 8].

Traditionally, control strategies to suppress thermoacoustic instability are
classified into passive and active controls [3, 10, 11, 9, 12]. Passive
control strategies reduce the sensitivity of the combustor to acoustic
disturbances by altering the geometry of the combustor or by installing
components such as baffles and Helmholtz resonators [3, 13]. However, these
strategies are effective only over a limited range of frequencies and the
design changes involved in their implementation are costly and time-consuming.
On the other hand, in most active control strategies, an actuator perturbs the
system dynamics and breaks the coupling between the pressure and heat release
rate fluctuations in the combustor, thereby leading to the suppression of
thermoacoustic instability [11]. In this method, if the parameters of the
forcing are controlled by the operator and are not directly dependent on the
performance of the combustor, the control strategy is referred to as an open-
loop control [14]; otherwise it is referred to as closed-loop control.

In closed-loop control, also known as active feedback control, the pressure
signal acquired from a thermoacoustic system is processed and used as input
for a controller, which then accordingly instructs an actuator to modify
system parameters (such as the inlet boundary conditions or the fuel flow
rate). This, in turn, disrupts the flame-acoustic coupling and quenches
thermoacoustic instability in the system [15, 16]. The processing of the input
pressure signal commonly involves phase-shifting and amplification. Though
several studies have demonstrated the effectiveness of active feedback control
in quenching thermoacoustic instability in various thermoacoustic systems [15,
16, 17, 11], the design, implementation, and maintenance of active feedback
control systems is highly cumbersome. Moreover, the applicability of the
feedback control method is restricted due to the lack of reliability of
sensors and actuators for operating in the harsh environment of practical
combustors, limited space for installing actuators, and high power
requirements for operating feedback control units [12].

The aforementioned disadvantages of traditional passive and active control
methodologies make it clear that there is a need to develop a control strategy
that is cheap, simple to implement on practical combustion systems, and is
effective in mitigating thermoacoustic instability. Towards this purpose, we
propose the method of self-coupling in which we provide acoustic self-feedback
using a tube. We refer to the quenching of limit cycle oscillations to a
common fixed point due to the coupling of one or more oscillators as
‘amplitude death’ (AD) [18, 19]. In self-coupling, the acoustic wave takes a
finite time to propagate through the coupling tube and affect the acoustic
field of the Rijke tube [20]. This leads to a delay in the self-feedback of
acoustic oscillations of the system. Such delayed self-feedback has generally
been implemented in the past to stabilize steady states in several systems,
including the Van der Pol oscillator [21], the Rössler oscillator [22], and in
electrochemical and optomechanical [23, 24] systems. However, the practical
implementation of these self-feedback techniques involve electronic components
to capture the delayed signal and to amplify and feed it back to the system.
This is resolved in the method of self-coupling used in our study where the
acoustic pressure signal from a thermoacoustic system is directly fed to
itself, without any explicit signal processing or modifying the inlet flow
conditions. Thus, this method completely removes the requirement for any
electro-mechanical components such as signal processors, sensors, and
actuators used in traditional active and self-feedback controls. Additionally,
the connecting tube is easier to design and implement as compared to
traditional passive controls. Though methods similar to self-coupling, such as
the Herschel-Quincke tube and the Infinity tube have been used in the past to
suppress electrically driven acoustic pressure oscillations [20, 25], their
application in thermoacoustic systems is yet to be examined.

In the present study, we address the following questions: (i) Can self-
coupling mitigate thermoacoustic instability? (ii) What are the optimal values
of coupling parameters (length, diameter, and location of the coupling tube)
for attaining amplitude death in a thermoacoustic system? (iii) How does the
amplitude of the limit cycle oscillations prior to introducing self-coupling
affect its quenching? (iv) What is the nature of the transition between steady
state and oscillatory state in the self-coupled thermoacoustic system? Towards
this purpose, we systematically perform experimental and theoretical
investigation on a horizontal Rijke tube subjected to self-coupling.

The Rijke tube is a convenient prototype of a thermoacoustic oscillator
traditionally used to study the occurrence and mitigation of thermoacoustic
instability [26, 27, 28]. It consists of a duct with a heat source (e.g., an
electrically heated wire mesh) present inside. The interaction between the
heat released by the heat source and the acoustic field of the Rijke tube can
lead to thermoacoustic instability. Through experiments, numerical
simulations, and approximate analytical solutions, we demonstrate that self-
coupling can mitigate thermoacoustic instability in a horizontal Rijke tube at
optimal conditions. We observe that self-coupling causes amplitude death for
wider parametric regions when the length of the coupling tube is close to an
odd multiple of the length of the Rijke tube. The occurrence of amplitude
death is easier for larger diameters of coupling tube and smaller amplitude of
limit cycle oscillations prior to coupling. We show that the transition
between the states of limit cycle oscillations and amplitude death due to
self-coupling is explosive and hysteretic for a Rijke tube that exhibits
subcritical Hopf bifurcation in the absence of coupling. We formulate a model
for the self-coupled Rijke tube which qualitatively captures the experimental
findings. We also find that the optimal coupling location for attaining
amplitude death is around the centre of the Rijke tube, which corresponds to
the antinode of the standing acoustic pressure wave developed in the Rijke
tube prior to coupling.

Recent studies have demonstrated the occurrence of amplitude death by mutual
coupling of two Rijke tubes using one or two connecting tubes [29, 30, 31,
32]. Here, we also compare the effectiveness of suppressing limit cycle
oscillations in a single Rijke tube via self-coupling against that obtained
via mutual coupling of two identical Rijke tubes. We demonstrate that limit
cycle oscillations of significantly greater amplitudes can be easily
suppressed through self-coupling of a single Rijke tube as compared to mutual
coupling of two such identical Rijke tubes.

The rest of the paper is organized as follows. In Sec. II, we provide details
of the experimental setup. Following this, Sec. III presents our results and
discussions on the self-coupled Rijke tube. Within Sec. III we experimentally
investigate the effect of self-coupling on the mitigation of thermoacoustic
instability in a horizontal Rijke tube in Sec. III.1. Subsequently, in Sec.
III.2, we present a model of the self-coupled Rijke tube and investigate it
numerically and analytically. In Sec. III.3, we compare the quenching of limit
cycle oscillations due to the self-coupling of a Rijke tube with that obtained
through mutual coupling of two such identical Rijke tubes. Finally, we present
our conclusions from the study in Sec. IV.

##  II  Experimental setup of the self-coupled Rijke tube

In Fig. 1, we show a schematic representation of the horizontal Rijke tube
used in the present study. The Rijke tube is a long duct with a rectangular
cross section of $9.3$ cm $\times$ 9.4 cm and a length ($L_{\text{duct}}$) of
$104$ cm, similar to the ones used in [33, 29, 30]. An electrically heated
wire mesh, powered by an external DC power supply, acts as a compact heat
source. The heated wire mesh is located at $27$ cm downstream from the inlet
in the duct. Air flow is supplied to the Rijke tube through a mass flow
controller (MFC, Alicat Scientific) of uncertainty $\pm$(0.8% of the measured
reading + 0.2% of the full scale reading). A decoupler of dimensions 102 cm
$\times$ 45 cm $\times$ 45 cm is attached to the inlet of the Rijke tube. The
decoupler is a large chamber used to dampen out the fluctuations in the
incoming air flow so that a steady air flow enters the Rijke tube. During
experiments, the heater power supplied to the wire mesh is increased so that
the system behavior transitions from steady state to limit cycle oscillations
via subcritical Hopf bifurcation for the given air flow rate [34].

Figure 1:  Schematic of a horizontal Rijke tube self-coupled using a
connecting tube. The axial location (normalized by the length of the Rijke
tube, $L_{\text{duct}}$) of the heater mesh and the connecting tube are
denoted by $x_{f}$ and $x_{c}$, respectively. Ball-type coupling valves are
manually opened to establish acoustic feedback in the system.

In order to quench the limit cycle oscillations, we implement self-coupling to
acoustically couple the Rijke tube to itself using a single vinyl tube of
length $L$ and internal diameter $d$. The self-coupling tube is attached to
two sides of the Rijke tube at an axial distance of $57$ cm from the inlet of
the Rijke tube (the axial location of the connecting tube on normalizing by
$L_{\text{duct}}$ is $x_{c}=0.55$), unless otherwise specified. Ball-type
valves are manually operated to control the self-coupling in the system. The
length of the connecting tube ($L$) is varied from 92 cm to 362 cm in steps of
5 cm. Coupling tubes of diameters ($d$) 4 mm to 12 mm in steps of 2 mm are
considered in this study.

To measure the acoustic pressure fluctuations in the system, a piezoelectric
pressure transducer (PCB 103B02, sensitivity $217.5$ mV/kPa, uncertainty $\pm
0.15$ Pa) is mounted along the length of the duct at 57 cm from the inlet. The
pressure data is acquired from the Rijke tube at a sampling rate of $10$ kHz
for a duration of $3$ s for each parametric condition using a data acquisition
system (NI USB 6343). The resolution of frequency in the power spectrum of the
signal is equal to 0.2 Hz. To measure the acoustic damping in the Rijke tube,
we send an acoustic pulse into the Rijke tube using a loudspeaker in the
absence of air flow and heating and calculate its decay rate. We observe the
decay rate for the Rijke tube to be $15.8\pm 2$ $\text{s}^{-1}$. To ensure
consistency of the experimental conditions and repeatability of the results,
the experiments are conducted only when the measured acoustic decay rates lie
in the aforementioned range.

Before starting any experiment, the Rijke tube is preheated for $10$ minutes
in the steady state regime of operation by supplying DC power at $1$ V to the
wire mesh. The preheating ensures a steady temperature profile inside the
Rijke tube. The experimental setup for the mutually coupled Rijke tubes
(discussed in Sec. III.3) is similar to the above description, except that we
couple two identical Rijke tubes with the connecting tube. For more
experimental details on mutually coupled Rijke tubes, readers may refer to
[29, 30].

##  III Results and Discussion

###  III.1  Experimental investigation of the self-coupled Rijke tube

Figure 2:  Variation of the RMS value of acoustic pressure fluctuations with
heater power in the forward and reverse paths for (a) the uncoupled Rijke tube
($p^{\prime}_{\text{0,rms}}$) and (b) the self-coupled Rijke tube
($p^{\prime}_{\text{rms}}$). The normalized length of the connecting tube is
$L/L_{\text{duct}}=1.17$ and its internal diameter is $d=8$ mm in (b). The
introduction of self-coupling shifts the Hopf point [marked as ‘H’ in (a)] and
the fold point [marked as ‘F’ in (a)] to higher values of heater power. Air
flow rate of 120 SLPM (or 0.002 m3/s) is maintained in the Rijke tube for both
the plots. Figure 3:  The amplitude response of the Rijke tube when self-
coupling is induced for (a) $p^{\prime}_{\text{0,rms}}=120$ Pa, and (b)
$p^{\prime}_{\text{0,rms}}=320$ Pa. The normalized change in the amplitude
suppression $\Delta p^{\prime}_{\text{rms}}/p^{\prime}_{\text{0,rms}}$ is
measured for different values of $L/L_{\text{duct}}$ and $d$. A large
parametric region of amplitude death (AD) is observed for lower value of
$p^{\prime}_{\text{0,rms}}$ in (a); while for higher values of
$p^{\prime}_{\text{0,rms}}$ in (b), the region of amplitude death shrink and
is observed only for around $L/L_{\text{duct}}\approx 1$ and
$L/L_{\text{duct}}\approx 3$. Air flow rate is maintained at 80 SLPM (or
0.00133 m3/s) in the Rijke tube.

In this section, we first discuss the effect of self-coupling on the
bifurcation characteristics of the Rijke tube as observed in our experiments.
In Fig. 2(a), we show the variation in the root-mean-square (RMS) value of
acoustic pressure fluctuations (in Pa) as a function of the heater power (in
W) in the absence of self-coupling in the system. We denote the RMS value
using $p^{\prime}_{\text{0,rms}}$ for the uncoupled case and
$p^{\prime}_{\text{rms}}$ for the coupled case. In the forward path
(increasing heater power), we observe that the transition from fixed point to
limit cycle oscillations of RMS value around 290 Pa happens via subcritical
Hopf bifurcation; while in the reverse path (decreasing heater power), the
system attains fixed point via fold bifurcation when
$p^{\prime}_{\text{0,rms}}$ is around 200 Pa. Thus, the transition between the
steady state and the limit cycle oscillatory state is explosive (subcritical)
and hysteretic. In Fig. 2(a), ‘H’ denotes the Hopf bifurcation point and ‘F’
denotes the fold bifurcation point. We notice a bistable region in between the
two bifurcation points. When the Rijke tube is self-coupled with a single
coupling tube of dimensions $L/L_{\text{duct}}=1.17$ and $d=8$ mm in Fig.
2(b), we observe that the system preserves the bifurcation characteristics
that we observed in the uncoupled case [see Fig. 2(a)]. However, we notice
that the introduction of self-coupling shifts the Hopf and the fold
bifurcation points to higher values of the heater power than those values
observed for the uncoupled oscillator. This, in turn, indicates that self-
coupling enhances the parameter space of steady state in the system; in other
words, it delays the transition to limit cycle oscillations in the system.

Figure 4:  The suppression of acoustic pressure fluctuations in the self-
coupled Rijke tube is shown for different values of the normalized length of
connecting tube ($L/L_{\text{duct}}$) and the RMS value of oscillations in the
uncoupled state ($p^{\prime}_{\text{0,rms}}$) when the internal diameter of
coupling tube ($d$) is fixed at (a) $4$ mm, and (b) $8$ mm. Air flow rate is
maintained at 80 SLPM (or 0.00133 m3/s) in the Rijke tube.

Next, we study the effect of variation in the parameters of the connecting
tube (i.e., length $L$ and diameter $d$) and the RMS value of limit cycle
oscillations in the uncoupled state ($p^{\prime}_{\text{0,rms}}$) on the
suppression characteristics in the self-coupled Rijke tube oscillator. We note
that prior to implementing self-coupling, we establish limit cycle
oscillations in the system. As discussed in Sec. II, the value of
$p^{\prime}_{\text{0,rms}}$ can be increased by increasing the value of the
heater power. To study the behavior of the self-coupled Rijke tube over a
larger range of $p^{\prime}_{\text{0,rms}}$ than that shown in Fig. 2(a), we
maintain the air flow rate at a value of 80 SLPM (or 0.00133 m3/s) in our
subsequent experiments. Lowering the air flow rate reduces the value of
$p^{\prime}_{\text{0,rms}}$ at the Hopf point of the Rijke tube [34]. At this
air flow rate, we vary the value of $p^{\prime}_{\text{0,rms}}$ for a larger
range of 120 Pa to 320 Pa by varying the heater power. We refer to the
dimensions of the coupling tube (i.e., $L$ and $d$) as coupling parameters and
the RMS value ($p^{\prime}_{\text{0,rms}}$) and frequency ($f_{0}$) of limit
cycle oscillations prior to coupling as system parameters in the subsequent
discussion of the paper.

Figure 3 illustrates the amplitude response of the self-coupled Rijke tube for
different coupling parameters. Here, we show the amplitude response in terms
of the normalized change in the RMS value of limit cycle oscillations in the
Rijke tube due to self-coupling, $\Delta
p^{\prime}_{\text{rms}}/p^{\prime}_{\text{0,rms}}=(p^{\prime}_{\text{0,rms}}-p^{\prime}_{\text{rms}})/p^{\prime}_{\text{0,rms}}$,
as a function of $L/L_{\text{duct}}$ and $d$. The color bar illustrates values
of $\Delta p^{\prime}_{\text{rms}}/p^{\prime}_{\text{0,rms}}$ ranging from $0$
to $1$, where $\Delta p^{\prime}_{\text{rms}}/p^{\prime}_{\text{0,rms}}\approx
1$ corresponds to complete suppression of limit cycle oscillations (i.e.,
amplitude death) and $\Delta
p^{\prime}_{\text{rms}}/p^{\prime}_{\text{0,rms}}\approx 0$ indicates the
absence of any suppression of limit cycle oscillations in the Rijke tube due
to self-coupling. We study the response of the self-coupled Rijke tube
oscillator for two different RMS values of limit cycle oscillations in the
uncoupled state: $p^{\prime}_{\text{0,rms}}=120$ Pa [see Fig. 3(a)], and
$p^{\prime}_{\text{0,rms}}=320$ Pa [see Fig. 3(b)].

In Fig. 3(a), we note that when the RMS value of the limit cycle oscillations
is low ($p^{\prime}_{\text{0,rms}}=120$ Pa), self-coupling of the Rijke tube
causes a significant reduction in the amplitude of acoustic pressure
oscillations for a vast parametric region of coupling parameters. However, as
the value of $p^{\prime}_{\text{0,rms}}$ is increased to a higher value of
$320$ Pa in Fig. 3(b), we observe the region of amplitude death only around
odd values $L/L_{\text{duct}}$ (i.e., $L/L_{\text{duct}}\approx 1$ and $3$).
We also notice that the range of $L/L_{\text{duct}}$ for which the Rijke tube
exhibits amplitude death increases as the internal diameter of the coupling
tube ($d$) is increased.

Figure 5:  Schematics of the Rijke tube and the time series of the acoustic
pressure fluctuations ($p^{\prime}$) depict the effect of coupling location on
the suppression of thermoacoustic instability for: (a) the uncoupled Rijke
tube, (b,c) and the self-coupled Rijke tube when the coupling tube is fixed at
a distance ($x_{c}$) of 0.82 and 0.55 from the inlet, respectively. All
distances are non-dimensionalized by the length of the Rijke tube,
$L_{\text{duct}}=1.04$ m. Other parameters are maintained as
$p^{\prime}_{\text{0,rms}}=200$ Pa, $L_{\text{duct}}=95$ cm, and $d=4$ mm in
all of the plots.

Next, we examine in detail the amplitude response of the self-coupled Rijke
tube on variation of the RMS value of the limit cycle oscillations in the
uncoupled state ($p^{\prime}_{\text{0,rms}}$) and the normalized length of the
connecting tube ($L/L_{\text{duct}}$). The internal diameter of the coupling
tube is kept constant at $4$ mm and $8$ mm in Fig. 4(a) and 4(b),
respectively. Note that the ordinate representing the
$p^{\prime}_{\text{0,rms}}$ values is unevenly distributed. In Fig. 4, we
observe that limit cycle oscillations of low RMS values (i.e., for
$p^{\prime}_{\text{0,rms}}\leq 140$ Pa), and thus low amplitudes, can be
easily suppressed for a large range of $L/L_{\text{duct}}$. As
$p^{\prime}_{\text{0,rms}}$ is increased above $140$ Pa, we notice that the
amplitude death region shrinks and occurs only in a narrow range of
$L/L_{\text{duct}}$. For a low value of $d$ in Fig. 4(a), we observe amplitude
death to occur near $L/L_{\text{duct}}=1$ when $p^{\prime}_{\text{0,rms}}$ is
greater than 140 Pa. In contrast, we notice two distinct regions of amplitude
death in Fig. 4(b) when the value of $d$ is increased to $8$ mm. Here, the two
regions of amplitude death occur at around $L/L_{\text{duct}}=1$ and $3$,
respectively, which is similar to what we observed in Fig. 3(b). Thus, from
Figs. 3 and 4, we infer that self-coupling can quench small amplitude limit
cycle oscillations for a large range of coupling parameters. On the other
hand, the quenching of large amplitude limit cycle oscillations is possible
only for a critical range of the length of the coupling tube and it is
comparatively easy when the diameter of the coupling tube is bigger.

So far, we have investigated the behavior of the self-coupled Rijke tube for a
fixed location of the coupling tube. We next examine how varying the coupling
location affects the quenching of limit cycle oscillations by tracking the
temporal variation of acoustic pressure oscillations in the system. We observe
that as compared to the uncoupled state [depicted in Fig. 5(a)], coupling the
Rijke tube at a location towards the end of the duct does not cause
significant suppression of limit cycle oscillations [Fig. 5(b)]. On the other
hand, when the location of the coupling tube along the duct of the Rijke tube
is closer to the center [Fig. 5(c)], the oscillations are quenched and
amplitude death is achieved. This suggests that the optimal axial location of
the coupling tube for mitigating thermoacoustic instability is around the
center of the Rijke tube. We discuss this behavior of the self-coupled Rijke
tube through modeling in the next section.

###  III.2  Theoretical analysis of the self-coupled Rijke tube oscillator

In this section, we theoretically analyze a model of the self-coupled Rijke
tube oscillator. We build on the reduced-order model of the horizontal Rijke
tube proposed by Balasubramanian and Sujith [35]. We derive the governing
equations of the self-coupled Rijke tube in the following manner. First, we
consider the non-dimensionalized linearized momentum and energy equations for
the acoustic field of the duct by neglecting the effect of mean flow (zero
Mach number approximation [36]) and mean temperature gradient:

$\displaystyle\gamma M\dfrac{\partial u^{\prime}}{\partial t}+\dfrac{\partial
p^{\prime}}{\partial x}$ $\displaystyle=0,$ (1) $\displaystyle\dfrac{\partial
p^{\prime}}{\partial t}+\gamma M\dfrac{\partial u^{\prime}}{\partial x}+\zeta
p^{\prime}$
$\displaystyle=(\gamma-1)\dot{Q}^{\prime}\delta(x-x_{f})+C\delta(x-x_{c}).$
(2)

Here, $x$ is the axial distance non-dimensionalized by the length of the duct
$L_{\text{duct}}$. $t$ is the time non-dimensionalized by the ratio
$L_{\text{duct}}/c_{0}$, where $c_{0}$ is the speed of sound. $u^{\prime}$ and
$p^{\prime}$ are the acoustic velocity and acoustic pressure, non-
dimensionalized by the steady state velocity ($u_{0}$) and pressure ($p_{0}$),
respectively, of the air flowing through the Rijke tube. $\gamma$ is the ratio
of specific heats, while $M$ is the Mach number ($M=u_{0}/c_{0}$). $\zeta$
captures the damping in the model. $\dot{Q}^{\prime}$ is the non-dimensional
heat release rate fluctuations per unit area from the heat source. $C$ is the
self-coupling term added to account for the effects induced by the connecting
tube. $\dot{Q}^{\prime}$ is multiplied by a Dirac delta function to indicate
that the heat source is concentrated at the non-dimensional heater location
$x_{f}$ in the Rijke tube. Similarly, the two ends of the connecting tube are
located at the same coupling location $x_{c}$ along the axial length of the
Rijke tube. We formulate $\dot{Q}^{\prime}$ using the correlation given by
Heckl [37]:

$\displaystyle\dot{Q}^{\prime}=\frac{2L_{w}(T_{w}-T_{0})}{c_{0}p_{0}S\sqrt{3}}$
$\displaystyle\sqrt{\pi\lambda_{T}C_{v}u_{0}\rho_{0}r_{w}}$
$\displaystyle\times\left[\sqrt{\left\lvert\frac{1}{3}+u^{\prime}(x,t-\tau_{h})\right\rvert}-\sqrt{\frac{1}{3}}\right],$
(3)

where $r_{w}$, $T_{w}$, and $L_{w}$ are the radius, the temperature, and the
length of the heated wire, respectively. $T_{0}$ and $\rho_{0}$ are the steady
state temperature and density of the medium. $S$ is the cross-sectional area
of the duct. $\lambda_{T}$ and $C_{v}$ are the thermal conductivity and the
specific heat at constant volume of the medium, respectively.
$u^{\prime}(x,t-\tau_{h})$ is the acoustic velocity at time $t-\tau_{h}$.
Here, the time lag, $\tau_{h}$, accounts for the thermal inertia of heat
transfer in the medium [38].

We use time-delay coupling to capture the effect of delayed interaction of
acoustic waves propagating through the connecting tube with the acoustic field
of the duct. The coupling term is described by the following expression:

$\displaystyle C=K_{\tau}\left[p^{\prime}(x,t-\tau)-p^{\prime}(x,t)\right],$
(4)

where $K_{\tau}$ is the coupling strength and $\tau$ is the coupling delay
characterizing the time it takes for the acoustic waves to propagate through
the connecting tube.

The normalized length of the connecting tube ($L/L_{\text{duct}}$) in the
experiment can be related to the coupling delay ($\tau$) in the model in the
following manner:

$\displaystyle\tau$
$\displaystyle=\dfrac{\tilde{\tau}}{L_{\text{duct}}/c_{0}},$ (5) where
$\tilde{\tau}$ is the dimensional coupling delay. Assuming that the acoustic
waves propagate through the connecting tube at speed $c_{0}$, the time taken
by the sound waves to travel through connecting tube is
$\displaystyle\tilde{\tau}$ $\displaystyle=L/c_{0}.$ (6) Hence,
$\displaystyle\tau$
$\displaystyle=\dfrac{L/c_{0}}{L_{\text{duct}}/c_{0}}=L/L_{\text{duct}}.$ (7)

Substituting Eq. (3) and (4) into the energy equation Eq. 2, we get the
following equation:

$\displaystyle\dfrac{\partial p^{\prime}}{\partial t}$ $\displaystyle+\gamma
M\dfrac{\partial u^{\prime}}{\partial x}+\zeta p^{\prime}$
$\displaystyle=\frac{K}{2}\left[\sqrt{\left\lvert\frac{1}{3}+u^{\prime}(t-\tau_{h})\right\rvert}-\sqrt{\frac{1}{3}}\right]\delta(x-x_{f})$
$\displaystyle\quad+K_{\tau}\left[p^{\prime}(x,t-\tau)-p^{\prime}(x,t)\right]\delta(x-x_{c}),$
(8)

where $K$ is the non-dimensional heater power given by:

$\displaystyle K=\frac{4(\gamma-1)L_{w}}{\gamma
Mc_{0}p_{0}S\sqrt{3}}(T_{w}-T_{0})\sqrt{\pi\lambda_{T}C_{v}u_{0}\rho_{0}r_{w}}.$
(9)

We now employ the Galerkin technique [39] to decompose the partial
differential equations given by Eq. (1) and (8) into a set of ordinary
differential equations. To that end, we choose the basis functions of
$u^{\prime}$ and $p^{\prime}$ to be the natural acoustic modes of the duct in
the absence of the heater [35]. The duct of the Rijke tube is open at both
ends, where the total pressure ($p$) must equal the ambient pressure
($p_{0}$). As a result, the acoustic pressure fluctuations
($p^{\prime}=p-p_{0}$) are absent at the ends of the tubes. So, the basis
functions must satisfy the boundary conditions
$p^{\prime}(0,t)=p^{\prime}(1,t)=0$. Thus, $u^{\prime}$ and $p^{\prime}$ are
decomposed into their modes as:

$\displaystyle u^{\prime}(x,t)=\sum_{j=1}^{N}\eta_{j}\cos(j\pi x),$ (10)
$\displaystyle p^{\prime}(x,t)=-\sum_{j=1}^{N}\frac{\gamma
M}{j\pi}\dot{\eta}_{j}\sin(j\pi x),$ (11)

where $\eta_{j}$ and $\dot{\eta}_{j}$ are the time-varying coefficients of the
$j$th mode of $u^{\prime}$ and $p^{\prime}$, respectively, and $N$ is the
total number of Galerkin modes considered in the expansion. On employing the
Galerkin technique, we obtain the set of ordinary differential equations from
the partial differential equations Eq. (1) and (8) as:

$\displaystyle\ddot{\eta}_{j}$ $\displaystyle+2\zeta
j\pi\dot{\eta}_{j}+(j\pi)^{2}\eta_{j}$ $\displaystyle=$ $\displaystyle-j\pi
K\left[\sqrt{\left\lvert\frac{1}{3}+u^{\prime}_{f}(t-\tau_{h})\right\rvert}-\sqrt{\frac{1}{3}}\right]\sin(j\pi
x_{f})$ $\displaystyle-\frac{j\pi}{\gamma
M}K_{\tau}\left[p^{\prime}_{c}(t-\tau)-p^{\prime}_{c}(t)\right]\sin(j\pi
x_{c}),$ (12)

where $u^{\prime}_{f}(t-\tau_{h})=u^{\prime}(x_{f},t-\tau_{h})$ and
$p^{\prime}_{c}(t-\tau)=p^{\prime}(x_{c},t-\tau)$. $\zeta_{j}$ is the
frequency dependent damping given by [28]:

$\zeta_{j}=\dfrac{1}{2\pi}\left(c_{1}\dfrac{\omega_{j}}{\omega_{1}}+c_{2}\sqrt{\dfrac{\omega_{1}}{\omega_{j}}}\right),$
(13)

where $c_{1}$ and $c_{2}$ are the damping coefficients. Equation (12)
represents the set of governing equations for the self-coupled Rijke tube
oscillator.

Before performing numerical simulations on the model, we attempt to derive the
conditions for attaining amplitude death in the system analytically. Towards
this purpose, we determine the stability of the steady state in the system and
locate the parameter values where the solution $\eta_{j}=0$ loses its
stability. We first simplify Eq. (12) by considering only the first mode and
linearizing the heat release rate term under the assumption of small
amplitudes [40]. We subsequently drop the subscript $j$ and perform algebraic
manipulations to yield the following differential equation:

$\displaystyle\ddot{\eta}+2\zeta\pi\dot{\eta}+\pi^{2}\eta=-\frac{\sqrt{3}}{4}K\sin(2\pi
x_{f})\eta(t-\tau_{h})$
$\displaystyle+K_{\tau}\left[\dot{\eta}(t-\tau)-\dot{\eta}(t)\right]\sin^{2}(\pi
x_{c}).$ (14)

We now substitute a solution of the form $\eta=e^{qt}$ into Eq. (14), where
$q=p+i\omega$ is an eigenvalue of the differential equation. Here, $p$ is the
real part of the eigenvalue and it describes the rate at which perturbations
to the steady state decay or grow. $\omega$ is the imaginary part of the
eigenvalue and it describes the angular frequency with which the perturbations
oscillate while decaying or growing in magnitude [41]. Thus, when $p<0$, the
steady state is stable in the self-coupled system and so amplitude death can
be achieved. In this manner, we detect the critical parameter values for which
$p$ changes its sign (i.e., crossing $p=0$) and obtain the expressions for the
boundary demarcating the amplitude death region as:

$\displaystyle\tan\left(\frac{\omega\tau}{2}\right)$
$\displaystyle=\dfrac{\sqrt{3}K\pi\sin(2\pi
x_{f})\sin(\omega\tau_{h})-8\pi\zeta\omega}{\sqrt{3}K\pi\sin(2\pi
x_{f})\cos(\omega\tau_{h})+4(\pi^{2}-\omega^{2})},$ (15) $\displaystyle
K_{\tau}$ $\displaystyle=\frac{8\pi\zeta\omega-\sqrt{3}K\pi\sin(2\pi
x_{f})\sin(\omega\tau_{h})}{4\sin^{2}(\pi
x_{c})\left(\omega\cos(\omega\tau)-1\right)},$ (16)

where $\omega$ is the imaginary part of $q$. Since the natural frequency of
the first mode of the duct in the absence of the heater is $\pi$, we vary the
value of $\omega$ around this value to yield the parametric curve describing
the boundary of the amplitude death region, as illustrated later on in Fig.
6(a).

Figure 6:  (a) Two-parameter bifurcation plot between coupling strength
($K_{\tau}$) and coupling delay ($\tau$) showing multiple regions of amplitude
death (AD) in the model of the self-coupled Rijke tube oscillator. The black
curve denotes the boundary between the regions of AD and limit cycle
oscillatory state (LCO) determined using linear stability analysis. Inset
containing a part of the bifurcation plot shows the presence of LCO for low
coupling strength. (b) One-parameter bifurcation plot between
$p^{\prime}_{\text{rms}}$ and $K_{\tau}$ for $\tau=1$ illustrates the
occurrence of explosive transition and hysteresis between LCO and AD in the
system. The non-dimensional heater power $K$ is maintained at 1.2 in (a) and
(b). (c) Two-parameter bifurcation between $K$ and $\tau$ shows the
oscillating boundary of the amplitude death region with $\tau$ in the system.
The black curve denotes the boundary between the AD and LCO regions obtained
analytically using the method of averaging. (d) One-parameter bifurcation plot
between $p^{\prime}_{\text{rms}}$ and $K$ for $\tau=1$. $K_{\tau}$ is
maintained at 0.05 in (c) and (d).

To further simplify the conditions for obtaining AD, we employ the method of
averaging on the linearized equation Eq. (14) by assuming additionally small
values of $K$, $K_{\tau}$, $\tau_{h}$, and $\tau$. This gives us the condition
for amplitude death as:

$\displaystyle\cos(\omega\tau)<1-\dfrac{\sqrt{3}K\sin(2\pi
x_{f})\sin(\omega\tau_{h})-8\pi\zeta}{4K_{\tau}\sin^{2}(\pi x_{c})}.$ (17)

We use the above condition subsequently in Fig. 6(c) to demarcate the
parametric regions of amplitude death and compare it with our numerical and
experimental results.

Next, we discuss the effect of self-coupling on the occurrence of amplitude
death in the model of the horizontal Rijke tube. We primarily vary four
parameters in the model: (i) coupling strength ($K_{\tau}$), (ii) coupling
delay ($\tau$), (iii) non-dimensional heater power ($K$), and (iv) coupling
location ($x_{c}$). We utilize the approximate analytical solutions shown in
Eqs. (15)-(17) to visualize the optimal conditions for achieving amplitude
death in the self-coupled Rijke tube. We also perform numerical simulations by
numerically integrating the governing equation of the model Eq. (12) using the
inbuilt function dde23 of MATLAB${}^{\tiny{\text{\textregistered}}}$ [42]. In
all of our simulations, we measure the dynamics of the acoustic pressure
fluctuations in the Rijke tube at the heater location $x_{f}$. We use the
first 5 modes for our numerical simulations since we observe negligible
changes in the dynamics of the system on consideration of higher modes [43].
In Eq. (12), we fix the values of $\tau_{h}$, $\gamma$, $c_{0}$, $c_{1}$ and
$c_{2}$ at 0.2, 1.4, 340 m/s, 0.1 and 0.06, respectively, based on previous
theoretical studies [43, 44]. According to the experiments in the present
study, we maintain the value of $M$ at $4.49\times 10^{-4}$ and $x_{f}$ at
$0.256$. We restrict the range of our system parameter $K$ to 1.2 so that we
primarily excite the first mode of the Rijke tube and obtain period-1 limit
cycle oscillations in the uncoupled state of the system.

First, we examine the effect of varying the coupling strength ($K_{\tau}$) and
coupling delay ($\tau$) on the dynamical behavior of the self-coupled Rijke
tube. In Fig. 6(a), we show the two-parameter bifurcation diagram between the
coupling parameters $K_{\tau}$ and $\tau$ for a high value of non-dimensional
heater power ($K=1.2$). The coupling location $x_{c}$ is fixed at $0.55$,
which is the same as in our experiments. We observe that self-coupling can
subdue high amplitude thermoacoustic oscillations in the Rijke tube for
particular ranges of coupling delay and these ranges are roughly centered
around odd numbers of $\tau$. Additionally, we notice that the islands of
amplitude death decrease in size with an increase $\tau$. We compare Figs.
3(b) with 6(a) using the relation given in Eq. (7). This shows that our model
matches with the experimental observation that the optimal values of
connecting tube length for achieving amplitude death are odd multiples of the
length of the Rijke tube.

In Fig. 6(b), we track the variation of the RMS value of the limit cycle
oscillations ($p^{\prime}_{\text{rms}}$) in the self-coupled Rijke tube over a
small range of coupling strength for a fixed value of coupling delay
($\tau=1$). We observe that the transition from limit cycle oscillatory state
to amplitude death occurs through fold bifurcation on increasing $K_{\tau}$ in
the forward path, while the transition back to the state of limit cycle
oscillations occurs through subcritical Hopf bifurcation in the reverse path.
As a result, we observe hysteresis or bistability between the states of limit
cycle oscillations and amplitude death in the self-coupled Rijke tube. The
bistable region surrounds each island of amplitude death, as seen in Fig.
6(a). The bistable zone is significantly wider for higher values of coupling
strength. The boundary demarcating the regions of amplitude death derived
using linear stability analysis [obtained from Eq. (15) and (16)] matches well
with the outer boundary of the bistable zone. This is because linear stability
analysis predicts the local stability of the fixed point $\eta=0$, and thus
finds the Hopf point of the system [41]. Slight differences exist between the
analytical and numerical results due to our assumption of single mode.

We next examine how variation in the non-dimensional heater power influences
the behavior of the self-coupled Rijke tube. Figure 6(c) demonstrates the
effect of the non-dimensional heater power $K$ on the acoustic pressure
fluctuations in the system as $\tau$ is varied for a particular coupling
strength ($K_{\tau}$ = 0.05). As we know, an increase in $K$ corresponds to an
increase in the amplitude of limit cycle oscillations in the uncoupled state.
We observe amplitude death for lower values of $K$, i.e., for low amplitudes
of limit cycle oscillations in the uncoupled state. For sufficiently high
values of $K$, i.e., high amplitudes of limit cycle oscillations in the
uncoupled state, the oscillations continue to exist even on introducing self-
coupling, regardless of the value of $\tau$. Furthermore, we observe that the
boundary of the region of amplitude death oscillates with $\tau$, with the
peaks at around $\tau=1,\ 3,\ 5,\ ...$. On increasing $K$ at these values of
$\tau$, the amplitude death region tapers out and subsequently disappears at
high values. This behavior of the model qualitatively resembles the
experimental results presented in Fig. 4(b). We also note that the transition
between the states of limit cycle oscillations and amplitude death on varying
$K$ is explosive and hysteretic in Fig. 6(d), similar to what we observe in
experiments [refer to Fig. 3(b)]. In Fig. 6(c), we see that the boundary of
the amplitude death region obtained using the method of averaging [from Eq.
(17)] matches well with the numerical results, especially for small values of
$\tau$ where our simplifying assumptions are valid.

Figure 7:  (a) One-parameter bifurcation plot depicting the variation in
$p^{\prime}_{\text{rms}}$ with the location of the self-coupling tube on the
duct ($x_{c}$) for the self-coupled Rijke tube. The black (red) curve denotes
the case when the Rijke tube is coupled after (before) it reaches limit cycle
oscillatory state. The region highlighted in yellow denotes the values of
$x_{c}$ for which amplitude death is achieved if the Rijke tube is coupled at
the start but not on coupling after oscillations are established. (b) Standing
wave pattern obtained by plotting the maximum and minimum values of the
acoustic pressure oscillations ($p^{\prime}_{0}$) in the uncoupled Rijke tube
at different positions ($x$) along the duct. The antinode of the acoustic
standing wave is observed at the middle of the duct, which is the optimal
coupling location for achieving amplitude death in the self-coupled Rijke
tube, as seen in (a). $K_{\tau}=0.05$, $\tau=1$, and $K=0.8$ are maintained in
both the plots. Figure 8:  Trends in the relative suppression of the acoustic
pressure oscillations ($\Delta p^{\prime}/p^{\prime}_{\text{0,rms}}$) in two
mutually coupled identical Rijke tubes and in a self-coupled Rijke tube with
change in (a) the normalized length of the connecting tube
($L/L_{\text{duct}}$) in experiments ($d=8$ mm and
$p^{\prime}_{\text{0,rms}}=320$ Pa) and (b) the coupling delay ($\tau$) in the
model ($K_{\tau}=0.05$ and $K=0.81$). The highlighted green region indicates
the range of $L/L_{\text{duct}}$ or $\tau$ where the self-coupled Rijke tube
exhibits greater amplitude suppression than the mutually coupled system.

We now investigate the effect of the position of the coupling tube along the
length of a Rijke tube on the suppression of limit cycle oscillations in the
model. The position of the coupling tube is denoted as $x_{c}$ in the right-
hand-side of Eq. (12). Figure 7 illustrates the change in the RMS value of the
limit cycle oscillations ($p^{\prime}_{\text{rms}}$) on variation of $x_{c}$
while all other parameters are kept fixed in the system. We observe a region
of amplitude death when the connecting tube is around midway along the duct in
the model. On the other hand, when $x_{c}$ is towards the ends of the duct
(i.e., $x_{c}$ is close to 0 or 1), we notice that the self-coupled Rijke tube
retains the state of limit cycle oscillations. Thus, the model substantiates
our observations from experiments (Fig. 5 of Sec. III.1) where we noted the
optimal coupling location for achieving amplitude death to be around the
middle of the Rijke tube.

In order to understand why the optimal value of $x_{c}$ leading to amplitude
death is around the middle of the Rijke tube, we examine how the acoustic
standing wave pattern looks in the Rijke tube prior to coupling [refer Fig.
7(b)]. Since the first acoustic mode is primarily excited in the Rijke tube,
the acoustic standing wave pattern has nodes at the end of the duct and an
antinode at around the middle of the duct (at $x\approx 0.5$). Thus, on
coupling the Rijke tube around the antinode of the acoustic standing wave, the
Rijke tube receives the strongest acoustic feedback, easing the occurrence of
AD. In Fig. 7(a), we consider two cases of self-coupling: (i) when we let the
Rijke tube attain limit cycle oscillations and then induce self-coupling, and
(ii) when we couple the Rijke tube at the start when the transients are small,
i.e., before limit cycle oscillations are established in the duct. We observe
wider region of amplitude death in the second case when self-coupling is
induced to begin with.

Thus, through experiments and modeling, we showed that self-coupling can bring
about amplitude death of limit cycle oscillations in a thermoacoustic system
for optimal values of coupling and system parameters. Recent studies have
exploited the method of mutual coupling to quench limit cycle oscillations in
two thermoacoustic oscillators [44, 29, 45, 31, 30, 32]. Hence, we next
compare how effective self-coupling is in quenching high amplitude limit cycle
oscillations in a thermoacoustic oscillator as compared to mutual coupling of
two such identical oscillators, through both experiments and modeling.

###  III.3 Comparison between self-coupled and mutually coupled Rijke tube
oscillators

In order to compare the amplitude suppression characteristics of limit cycle
oscillations in a mutually coupled system of identical Rijke tubes and a
single self-coupled Rijke tube, we set the same system and coupling parameters
and measure the fractional change in the RMS value of limit cycle oscillations
after coupling is induced in the system ($\Delta
p^{\prime}_{\text{rms}}/p^{\prime}_{\text{0,rms}}$). We repeat this for
different values of the normalized coupling tube length ($L/L_{\text{duct}}$)
in experiments or the coupling delay ($\tau$) in the model. In both the
experiments and the model [Fig. 8(a) and 8(b), respectively], we observe that
in the green regions present around odd values of $L/L_{\text{duct}}$ or
$\tau$, the self-coupled system suppresses limit cycle oscillations to a
greater extent than the mutually coupled system and even induces amplitude
death. Away from these optimal values of $L/L_{\text{duct}}$ or $\tau$, the
Rijke tube exhibits almost the same magnitude of amplitude suppression for
both self and mutual couplings. Here, we note that previous studies have
introduced frequency mismatch to quench high amplitude limit cycle
oscillations in mutually coupled Rijke tubes [29, 30]. We do not consider such
mismatch in the current study. Thus, we emphasize that a self-coupled system
performs significantly better in suppressing the oscillations in a nonlinear
system, compared to the amplitude suppression in a mutually coupled system of
identical oscillators.

##  IV Conclusions

In this study, we demonstrated that limit cycle oscillations can be quenched
in a thermoacoustic system through the method of self-coupling, wherein we
couple the acoustic field of a thermoacoustic oscillator to itself using a
connecting tube. This method removes the need for electromechanical components
that are present in traditional feedback control strategies. Through
experiments and modeling, we found that the optimal lengths of the connecting
tube to achieve amplitude death are around odd multiples of the length of the
Rijke tube. Increasing the diameter of the connecting tube increases the
ranges of the connecting tube length over which amplitude death can be
achieved. Additionally, we observed that oscillations of low amplitude are
more easily quenched as compared to those of high amplitude. Furthermore, our
experiments and model indicate the optimal axial location of the connecting
tube to be around the antinode of the acoustic standing wave in the uncoupled
Rijke tube. We observed that the transition between steady state and
oscillatory state is explosive and hysteretic in the self-coupled Rijke tube,
similar to the behavior of the uncoupled Rijke tube. Furthermore, we
demonstrated that self-coupling is more effective in achieving amplitude death
in a Rijke tube as compared to mutual coupling of two such identical Rijke
tubes. Thus, we anticipate self-coupling to be a simple, cost-effective
alternative to traditional feedback controls for mitigating thermoacoustic
instability in single and multiple thermoacoustic systems, such as those
practical gas turbine and rocket combustors. It would be interesting to
explore the mitigation of thermoacoustic instability through the combined
application of self and mutual couplings in multiple combustion systems.
Quenching transverse thermoacoustic instabilities in a combustor through self-
coupling is also a scope for future studies.

###### Acknowledgements.

S. S. is grateful to Prof. Preeti Aghalayam and other members of the Young
Research Fellow Program of Indian Institute of Technology Madras (Project ID:
202025), India. A. S. gratefully acknowledges the Half-Time Research
Assistantship (HTRA) from the Ministry of Education. R. I. S. gratefully
acknowledges the J. C. Bose Fellowship (No. JCB/2018/000034/SSC) from the
Department of Science and Technology (DST), and the IoE initiative
(SB/2021/0845/AE/MHRD/002696) for the financial support. We are thankful to
Mr. Rohit R. for helping out with the experiments.

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