# Mitigation of limit cycle oscillations in a turbulent thermoacoustic system
via delayed acoustic self-feedback

Ankit Sahay<EMAIL_ADDRESS>Abhishek Kushwaha  Samadhan A. Pawar
Midhun P. R  Jayesh M. Dhadphale  R. I. Sujith  Department of Aerospace
Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu
600036, India.

###### Abstract

In this study, we report the suppression of limit cycle oscillations in a
bluff body stabilized turbulent combustor through delayed acoustic self-
feedback. Such feedback control is achieved by coupling the acoustic field of
the combustor to itself through a single coupling tube attached near the anti-
node position of the acoustic standing wave. We observe that the amplitude and
dominant frequency of the limit cycle oscillations gradually decrease as the
length of the coupling tube is increased. Complete suppression of these
oscillations is observed when the length of the coupling tube is nearly $3/8$
times the wavelength of the fundamental acoustic mode of the combustor.
Meanwhile, as we approach the suppression of thermoacoustic instability, the
dynamical behavior of acoustic pressure changes from the state of limit cycle
oscillations to low amplitude aperiodic oscillations via intermittency. We
also study the change in the nature of the coupling between the unsteady flame
dynamics and the acoustic field during the suppression of thermoacoustic
instability. We find that the temporal synchrony between these oscillations
changes from the state of synchronized periodicity to desynchronized
aperiodicity through intermittent synchronization. Furthermore, we reveal that
the application of delayed acoustic self-feedback with optimum feedback
parameters completely disrupts the positive feedback loop between
hydrodynamic, acoustic, and heat release rate fluctuations present in the
combustor during thermoacoustic instability, thus mitigating the instability.
We anticipate this method to be a viable and cost-effective option to mitigate
thermoacoustic oscillations in turbulent combustion systems used in practical
propulsion and power systems.

††preprint: AIP/123-QED

> Thermoacoustic instabilities are large amplitude, self-sustained periodic
> oscillations observed in practical gas turbines and rocket engines used for
> propulsion and power generation applications. The thermoacoustic instability
> results from positive feedback between the heat release rate and acoustic
> pressure fluctuations. The presence of such large amplitude acoustic
> pressure oscillations can severely affect the structural integrity of a
> combustor. Over the years, various active and passive control strategies
> have been developed to mitigate thermoacoustic instability. Recently, time
> delayed feedback involving the use of a connecting tube has been shown to
> suppress limit cycle oscillations in different laminar systems such as
> acoustic pipelines and Rijke tubes. However, unlike laminar thermoacoustic
> systems, a turbulent thermoacoustic system is a complex system where the
> local closed-loop interaction between hydrodynamic, acoustic, and heat
> release rate fluctuations leads to an emergence of periodic (ordered)
> oscillations during the onset of thermoacoustic instability. In the present
> study, we show the use of time delayed feedback to suppress limit cycle
> oscillations in a single bluff body stabilized turbulent combustor. We
> couple (feedback) the acoustic field of the combustor to itself through a
> connecting tube. The length of the connecting tube is proportional to the
> time delay in the feedback; thus, the method used to control instability
> using such a connecting tube is named the delayed acoustic self-feedback in
> this study. We show that at the appropriate length and diameter of the
> coupling tube, the proposed method completely disrupts the complex
> interactions inside the reaction field of the turbulent combustor during
> thermoacoustic instability, leading to the suppression of amplitude of limit
> cycle oscillations by more than 90%. During the state of suppression, the
> amplitude of acoustic pressure fluctuations is comparable to that observed
> for the state of stable operation (i.e., combustion noise) of the combustor.

##  I Introduction

Thermoacoustic instabilities have proven to be a major impediment to the
development of low-emission gas turbine engines used for propulsion and power
generation applications Lieuwen and Yang (2005). Such instabilities lead to
ruinously large amplitude pressure oscillations that are established when
positive feedback develops between the acoustic field and the heat release
rate fluctuations in the reaction field of the combustor Sujith and Pawar
(2021). The presence of these instabilities results in serious performance
losses, structural damages, and reduced operational range Culick (2006).
Therefore, it is necessary to find ways to mitigate thermoacoustic instability
in the course of developing new dynamically stable combustion systems.

Traditionally, different mechanisms of closed-loop and open-loop active
controls have been developed for suppressing thermoacoustic instability
Dowling and Morgans (2005). In these methods, the dynamics of a thermoacoustic
system is forced using external perturbations of specific amplitudes and
frequencies. However, these methods suffer from several limitations, such as
the use of complex electromechanical components, lack of reliability of
sensors while working in the hostile conditions of practical combustors, and
high maintenance and replacement costs. Another way to mitigate thermoacoustic
instability is to use passive damping devices such as perforated liners,
quarter and half-wave resonators, Helmholtz resonators Zhao and Li (2015),
Herschel-Quincke tubes Rajaram and Ritland (2012), modification of the fuel
injector geometry/location Steele  _et al._ (2000); Smith (2005), and
secondary air/fuel injections Schadow and Gutmark (1992); Huang and Yang
(2009). Engine manufacturers have used such passive damping devices to
suppress thermoacoustic instability in practical combustors Lieuwen and Yang
(2005); Bellucci  _et al._ (2004).

Recently, a concept from synchronization theory, called mutual coupling of
oscillators, has been adopted to suppress limit cycle oscillations in two or
more thermoacoustic systems Sujith and Pawar (2021); Moon  _et al._ (2020a).
At appropriate values of coupling parameters, the coupled systems approach the
same steady state of oscillation quenching, known as amplitude death Zou  _et
al._ (2021). Through rigorous experimental and numerical analysis, amplitude
death Thomas  _et al._ (2018a); Hyodo, Iwasaki, and Biwa (2020) and partial
amplitude death Dange  _et al._ (2019); Srikanth  _et al._ (2022a) have been
studied in coupled laminar thermoacoustic systems Doranehgard, Gupta, and Li
(2022). Here, partial amplitude death is a phenomenon in which some of the
oscillators are in the suppressed state, but the others continue their
oscillations Atay (2003). In addition, a few studies have investigated the
dynamics of coupled turbulent combustors Thomas  _et al._ (2018b); Jegal  _et
al._ (2019); Moon  _et al._ (2020b); Guan  _et al._ (2021); Pedergnana and
Noiray (2022); Fournier  _et al._ (2021), and have reported the presence of
amplitude death Jegal  _et al._ (2019). Here, the mutual coupling between the
systems is achieved by using one (or more than one) connecting tube of a fixed
length and diameter. In all the studies, it is often assumed that an increase
in the length of the coupling tube correspondingly increases the delay time in
the coupling of the acoustic fields in the systems Dange  _et al._ (2019);
Sahay  _et al._ (2021).

Till now, concepts from synchronization theory have been proposed to mitigate
thermoacoustic instability in multiple coupled thermoacoustic systems. A
majority of these studies have extensively focused on suppressing
thermoacoustic instability in laminar systems. In a laminar system such as a
Rijke tube, the system loses stability and becomes unstable via a Hopf
bifurcation Etikyala and Sujith (2017). The nature of a Hopf bifurcation is
determined by the stability of the emerging branch of the limit cycle
oscillations. A subcritical Hopf bifurcation occurs when a system quickly
transitions from a stable, steady solution to an unstable, steady solution
accompanied by a sudden rise in the amplitude of oscillations Pikovsky,
Rosenblum, and Kurths (2003). In contrast, a turbulent combustion system
transitions from disorder (combustion noise) to ordered dynamics (limit cycle
oscillations during thermoacoustic instability) via intermittency. The
intermediate state of intermittency exhibits the characteristics of both
combustion noise and thermoacoustic instability. Intermittency is
characterized by epochs of low-amplitude aperiodic oscillations interspersed
with seemingly random bursts of high-amplitude periodic oscillations. In a
turbulent combustor, interactions between hydrodynamic, acoustic, and heat
release rate fluctuations lead to a variety of dynamics, such as chaos,
intermittency, and ordered dynamics. The interplay of many different system
components can can lead to the emergence of collective behavior. The process
of larger entities, patterns, and regularities emerging through interactions
between constituent entities that are unable to exhibit these features on
their own is known as "emergence," and it is a key component of complex
systems. Large coherent vortices and an ordered acoustic field develop from
the background of turbulent combustion during the onset of thermoacoustic
instability in a turbulent combustion system. As a result, a turbulent
thermoacoustic system is a complex system in which aperiodic (disordered)
fluctuations disappear, and periodic (ordered) oscillations emerge at the
onset of thermoacoustic instability.

In this paper, we demonstrate the effectiveness of coupling the acoustic field
of a single turbulent combustor with itself to break the closed-loop
interaction between heat release rate fluctuations and acoustic pressure
fluctuations and thus, mitigating thermoacoustic instability when it has
occurred in the combustor. This method of coupling the acoustic field with
itself is referred to as delayed acoustic self-feedback Biwa  _et al._ (2016);
Srikanth  _et al._ (2022b). As compared to the traditional closed-loop active
controls where the acoustic field in the combustor is subjected to its own
feedback after processing it via digital delay lines and power amplifiers to
alter the inlet conditions of the combustor Annaswamy and Ghoniem (2002), the
method of self-feedback is realized by continuously making the acoustic field
in the system interact with itself through a connecting tube. The connecting
tube induces a time delay in the feedback, where the time delay is
proportional to the length of the tube. Time delayed feedback, introduced
electronically, has been successfully used in diverse experimental systems,
such as lasers Basso  _et al._ (1998), gas-discharge Pierre, Bonhomme, and
Atipo (1996), hydrodynamic Lüthje, Wolff, and Pfister (2001), electrochemical
Parmananda  _et al._ (1999), and ferromagnetic Benner and Just (2002) systems.
Recently, time delayed feedback through the use of a connecting tube has also
been used to suppress limit cycle oscillations in the acoustic field of
different systems, such as electroacoustic system Biwa  _et al._ (2016),
acoustic pipeline Lato, Mohany, and Hassan (2019), and horizontal Rijke tube
Srikanth  _et al._ (2022b), that does not involve turbulent flow. These
studies have shown that subjecting a system to delayed acoustic self-feedback
affects its bifurcation characteristics Srikanth  _et al._ (2022b) and the
suppression of limit cycle oscillations is realized only when a tube of a
length close to the odd multiple of the half-wavelength of the anticipated
acoustic standing wave is used Biwa  _et al._ (2016); Lato, Mohany, and Hassan
(2019). We here discuss the application of this method to suppress limit cycle
oscillations in a single turbulent combustion system.

As discussed above, the phenomenon of thermoacoustic instability in a
turbulent combustion system is a complex process and happens due to nonlinear
interaction between the flame, the flow, and the acoustic field of a combustor
Sujith and Pawar (2021). As per the Rayleigh criterion Rayleigh (1878), such
instability occurs only when an in-phase synchrony is developed between the
acoustic pressure and the heat release rate fluctuations, and the acoustic
driving is greater than acoustic damping in the combustor Lieuwen and Yang
(2005). Since the coupling between the acoustic pressure and the heat release
rate oscillations plays a crucial role in the genesis of thermoacoustic
instability, it is important to understand how the temporal and spatiotemporal
interactions between the heat release rate and the acoustic pressure
oscillations change when the system is subjected to delayed acoustic self-
feedback. Toward this purpose, we systematically address the following
questions in this paper - (i) how effective is delayed acoustic self-feedback
in disrupting the complex interactions between the acoustic pressure and the
heat release rate fluctuations and consequently mitigating thermoacoustic
instability in a turbulent combustor? (ii) What is the nature of the
transition of acoustic pressure fluctuations during the suppression of
thermoacoustic instability? Moreover, (iii) how do the temporal and
spatiotemporal couplings affect the heat release rate and acoustic pressure
oscillations due to delayed acoustic self-feedback?

We conduct experiments on a bluff body stabilized turbulent combustor to
answer the aforementioned questions. We report the mitigation of
thermoacoustic instability in the combustor through delayed acoustic self-
feedback when the connecting tube is attached at the acoustic pressure anti-
node of the combustor duct. Previous studies have demonstrated that the
optimal location for pressure actuation is near the pressure maxima of the
acoustic standing wave Magri and Juniper (2013); Srikanth  _et al._ (2022b);
Skene and Taira (2022). The amplitude of acoustic pressure fluctuations in the
suppressed state is comparable to that observed during the stable operation of
the combustor. We find that the suppression of thermoacoustic instability
depends on various parameters, i.e., the length and diameter of the coupling
tube and the amplitude of thermoacoustic instability observed prior to the
initiation of coupling. As we approach the state of suppression, acoustic
pressure fluctuations change their behavior from limit cycle oscillations to
low amplitude aperiodic oscillations via intermittency. Also, the coupled
behavior of acoustic pressure and heat release rate fluctuations transitions
from the state of phase synchronization to desynchronization via intermittent
synchronization. Furthermore, the coherent regions of acoustic power
production observed in the spatial field of the combustor during the state of
thermoacoustic instabilities disintegrate completely during the state of
significant suppression.

The rest of the paper is outlined as follows. In Sec. II, we describe the
experimental setup of a bluff body stabilized turbulent combustor used in this
study. In Sec. III.1 \- III.2, we perform time series analyses of global heat
release rate and acoustic pressure fluctuations for different conditions of
delayed acoustic self-feedback. In Sec. III.3, we compare the spatiotemporal
changes in the combustor as limit cycle oscillations are suppressed. Finally,
we summarize the key findings from the study in Sec. IV.

##  II Experimental Setup

Figure 1:  The schematic of a turbulent bluff body stabilized combustor
subjected to delayed acoustic self-feedback using a single coupling tube.

We experimentally demonstrate the application of delayed acoustic self-
feedback of the acoustic field to mitigate thermoacoustic instability in a
turbulent bluff body stabilized dump combustor (Fig. 1). The combustor has a
cross-section of $90\times 90$ $\text{mm}^{2}$ and a length of
$L_{\text{duct}}=1060$ mm. Air at ambient conditions first enters the settling
chamber, which ensures that the flow entering the combustor is isolated from
the upstream disturbances. The fuel (liquefied petroleum gas - 60% butane +
40% propane) is injected just upstream of the dump plane (approx. 59 mm
upstream of the bluff body) through four holes of 1 mm diameter drilled
symmetrically around the circular shaft that holds the bluff body in the
combustor. A disk-shaped bluff body of thickness 10 mm and diameter 47 mm is
used as the flame stabilizer and is positioned 30 mm downstream of the dump
plane. The combustion products are exhausted through a long duct into the
atmosphere via a decoupler. The decoupler is a large chamber used to keep the
combustor exit at an acoustically open boundary state and to decrease acoustic
losses to the environment brought on by acoustic radiation.

Digital mass flow controllers (Alicat Scientific, MCR Series) are used to
control the flow rates of both air and fuel separately. During the
experiments, we fix the fuel flow rate at a particular value and increase the
air flow rate until thermoacoustic instability is established in the system.
The increase in the air flow rate causes a decrease in the global equivalence
ratio. The global equivalence ratio is calculated as
$\phi=(\dot{m_{f}}/\dot{m_{a}})_{\text{actual}}/(\dot{m_{f}}/\dot{m_{a}})_{\text{stoichiometry}}$.
The global equivalence ratio is varied from 1.00 to approximately 0.6, with an
uncertainty of $\pm 0.02$. The air flow rate varies from 527 to 900 SLPM
(standard liter per minute), with a maximum uncertainty in the flow rate of
air being $\pm 10.24$ SLPM. The fuel flow rate is kept constant at 34 SLPM,
and the maximum uncertainty in the fuel flow rate is $\pm 0.44$ SLPM. The
Reynolds number of the air flow increases in the range of 14,500 to 25,000,
with a maximum uncertainty of $\pm 400$. The Reynolds number was computed
using the expression $Re=4\dot{m}D_{1}/\pi\mu D^{2}_{0}$, where $\dot{m}$ is
the mass flow rate of the fuel-air mixture, $D_{0}$ is the diameter of the
burner, $D_{1}$ is the diameter of the circular bluff body, and $\mu$ is the
dynamic viscosity of the fuel-air mixture at standard operating conditions. To
calculate the Reynolds Number, we use the following expression to calculate
the viscosity of the binary gas mixture Wilke (1950); Krieger (1951):

$\mu=\sum_{i=1}^{2}\frac{\mu_{i}}{1+\dfrac{1}{x_{i}}\hskip
3.0pt{{\sum}}_{\begin{subarray}{c}j=1\\\ j\neq i\end{subarray}}^{j=2}\hskip
4.0pt\cfrac{x_{j}\left[1+\left(\mu_{i}/\mu_{j}\right)^{1/2}\left(M_{j}/M_{i}\right)^{1/4}\right]^{2}}{2\sqrt{2}\left(1+M_{i}/M_{j}\right)^{1/2}}}\hskip
2.0pt,$ (1)

where $x_{i}$ denotes the mole-fraction, $\mu_{i}$ denotes the viscosity, and
$M_{i}$ denotes the molecular weight of the gas constituent $i$ in the binary
gas mixture. The air-fuel mixture is ignited at the dump plane using a spark
plug connected to an 11 kV ignition transformer.

Delayed acoustic self-feedback is established in the combustor using a single
coupling tube made of flexible stainless steel braided hose of length
$L_{\text{c}}$ and internal diameter $d_{\text{c}}$ (refer to Fig. 1). The
value of $L_{\text{c}}$ is varied from 1000 to 2000 mm in steps of 100 mm,
while the value of $d_{\text{c}}$ is varied from 6.35 to 25.4 mm
($d_{\text{c}}=6.35$, $9.525$, $12.7$, $19.05$, and $25.4$ mm). The coupling
tube is attached at an axial distance of 70 mm from the dump plane on two
opposite faces of the combustor walls. This position is near the anti-node of
the acoustic pressure standing wave that, in turn, induces stronger acoustic
feedback in the coupled system. As mentioned in Sec. I, previous studies have
shown that the optimal location for pressure actuation is near the pressure
maxima of the acoustic standing wave Magri and Juniper (2013); Srikanth  _et
al._ (2022b); Skene and Taira (2022). We have not used any other position on
the combustor duct to attach the coupling tube. Ball-type valves are manually
operated to switch on and off the delayed self-feedback of the acoustic field
in the system. We first establish thermoacoustic instability in the system and
then switch on the delayed feedback by opening the valves of a coupling tube
of specific length $L_{\text{c}}$ and internal diameter $d_{\text{c}}$.

Figure 2:  Time series of acoustic pressure $p^{\prime}$ illustrating the
effect of delayed acoustic self-feedback on the amplitude of limit cycle
oscillations. The enlarged portions show the dynamics of $p^{\prime}$ during
different stages of delayed acoustic self-feedback. The dimensions of the
coupling tube are $L_{\text{c}}=1400$ mm and $d_{\text{c}}=25.4$ mm.

The acoustic pressure fluctuations $p^{\prime}$ are measured using a PCB103B02
piezoelectric transducer (sensitivity: 217.5 mV/kPa and uncertainty: $\pm$0.15
Pa) mounted on the combustor wall at 20 mm from the dump plane. A
photomultiplier tube (PMT, Hamamatsu H10722-01) equipped with a CH* filter
(wavelength of 430 nm and 12 nm FWHM) is used to capture the global heat
release rate fluctuations in the flame $\dot{q}^{\prime}$. We simultaneously
acquired both $p^{\prime}$ and $\dot{q}^{\prime}$ signals for 3 seconds at a
sampling rate of 10 kHz using an A/D card (NI-6143, 16 bit). The frequency bin
size in the power spectrum is 0.3 Hz. High-speed CH* chemiluminescence images
of the flame are simultaneously captured with $p^{\prime}$ and
$\dot{q}^{\prime}$ signals at 2000 fps for 3 seconds using a CMOS camera
(Phantom - V12.1 with a ZEISS 50 mm camera lens). The image of the flow-field
from the dump plane spans 90 mm $\times$ 120 mm with a resolution of 574
$\times$ 764 pixels. To ensure consistency in experimental conditions for
different trials, the experiments were performed only when the decay rate of
periodic perturbations in the acoustic pressure field, measured in the absence
of flow, remained constant at $11\pm 1.5$ s-1. This decay rate is ensured in
the combustor irrespective of the presence of the coupling tube.

##  III Results and Discussion

In this section, we characterize the effect of delayed acoustic self-feedback
on the suppression of thermoacoustic instability in the turbulent combustor.
We first discuss the dynamical behavior of the acoustic pressure signal
($p^{\prime}$) alone and then present the change in the coupled behavior of
both heat release rate ($\dot{q^{\prime}}$) and acoustic pressure
($p^{\prime}$) fluctuations during the suppression of thermoacoustic
instability.

Figure 3:  Effect of delayed acoustic self-feedback, induced by increasing the
length of the coupling tube $L_{\text{c}}$, on the suppression of acoustic
pressure fluctuations $\Delta\bar{p}$. The regions marked in I, II, and III
denote the partial suppression, complete suppression, and no suppression,
respectively, of limit cycle oscillations ($p^{\prime}_{0,\text{rms}}=3200$
Pa). The values of $d_{\text{c}}$ is fixed at 25.4 mm.

###  III.1 Route from thermoacoustic instability to the state of suppression

In Fig. 2, we show a representative time series of the acoustic pressure
fluctuations under the influence of an optimal delayed acoustic self-feedback
induced using a coupling tube of length $L_{\text{c}}=1400$ mm and an internal
diameter of $d_{\text{c}}=25.4$ mm. Prior to the initiation of feedback, we
observe limit cycle oscillations with a root-mean-square (RMS) value of
$p^{\prime}_{0,\text{rms}}=4390$ Pa and a frequency of $164$ Hz in the
combustor. The subscript $0$ in $p^{\prime}_{0,\text{rms}}$ indicates the
state of thermoacoustic instability. The application of delayed acoustic self-
feedback causes a reduction in the RMS value of the limit cycle oscillations
to about $p^{\prime}_{\text{rms}}=230$ Pa, where the acoustic pressure
fluctuations exhibit aperiodic oscillations during the state of amplitude
death. The RMS value of acoustic pressure fluctuations is $120\pm 50$ Pa
during the stable operation. The removal of delayed acoustic self-feedback at
around $t=7.6$ second revives the limit cycle oscillations, and the combustor
returns to the original state of thermoacoustic instability.

In Fig. 3, we show the percentage change in the RMS value of $p^{\prime}$
signals, i.e., $\Delta\bar{p}$, as a function of the length of the coupling
tube ($L_{\text{c}}$) when its internal diameter is kept constant at 25.4 mm.
Here, $\Delta\bar{p}$ indicates the normalized difference between the RMS
values of the $p^{\prime}$ signal during the state of thermoacoustic
instability when the delayed acoustic self-feedback is off
($p^{\prime}_{0,\text{rms}}$) and that of $p^{\prime}$ signals when the
delayed acoustic self-feedback is switched on, i.e.,
$\Delta\bar{p}=(p^{\prime}_{0,\text{rms}}-p^{\prime}_{\text{rms}})/p^{\prime}_{0,\text{rms}}$.
In the absence of suppression of thermoacoustic instability,
$\Delta\bar{p}\rightarrow 0\%$, while for complete suppression,
$\Delta\bar{p}\rightarrow 100\%$ (region II in Fig. 3). We emphasize that the
combustor is not perfectly silent during its stable state of operation, i.e.,
$p^{\prime}_{0,\text{rms}}=120\pm 50$ Pa during the stable operation. Thus, we
do not observe $100\%$ reduction in the RMS value of acoustic pressure
fluctuations during the state of significant suppression of thermoacoustic
instability. Post the region of complete suppression (i.e., in region III of
Fig. 3), $\Delta\bar{p}$ dips suddenly to a lower value as thermoacoustic
instabilities are nearly unaffected by the presence of the coupling tube.
During the state of suppression (region II in Fig. 3), we note that the
amplitude of $p^{\prime}$ signal ($p^{\prime}_{\text{rms}}\sim 200$ Pa) is
comparable to that observed for the state of stable operation (i.e.,
combustion noise) of the combustor ($p^{\prime}_{\text{rms}}\sim 150$ Pa).
Furthermore, we find that the optimum value of $L_{\text{c}}$ corresponding to
the suppression of thermoacoustic instability is around 1.5 times the length
of the combustor (i.e., $L_{\text{c}}/L_{\text{duct}}\approx 1.5$). Since the
combustor’s duct can be modeled as an open-closed acoustic duct and the
combustor exhibits the first acoustic mode during the state of thermoacoustic
instability, the optimum value of $L_{\text{c}}$ corresponding to the complete
suppression is approximately equal to $\dfrac{3}{2}\times
L_{\text{duct}}=\dfrac{3}{2}\times\dfrac{\lambda}{4}=\dfrac{3\lambda}{8}$,
where $\lambda$ is the wavelength of the acoustic field self-excited in the
combustor duct during the state of thermoacoustic instability.

Figure 4:  (i) Time series, (ii) power spectrum, and (iii) scalograms of
$p^{\prime}$ as the system behavior transitions from a state of (a)
thermoacoustic instability to (e) complete suppression of oscillations via
(b-d) intermittency. A vertical dotted red line in (ii) indicates the natural
frequency of thermoacoustic instability in the absence of delayed acoustic
self-feedback. Zoomed regions of plots are shown in insets, where The green
and red shaded regions in i-(d) highlight the aperiodic and periodic epochs of
$p^{\prime}$ during the state of intermittency.

Figure 4 shows the change in the characteristics of $p^{\prime}$ signal in the
absence of coupling (Fig. 4(a)) and when the combustor is subjected to delayed
acoustic self-feedback for different lengths of the coupling tube (Figs.
4(b-e)). In the absence of such feedback, during thermoacoustic instability
(Fig. 4(a)), we observe large amplitude limit cycle oscillations in the
$p^{\prime}$ signal (Fig. 4(a)-(i)) with a sharp spectral peak at 163.9 Hz
corresponding to the fundamental mode of the combustor (Fig. 4(a)-(ii)). The
scalogram of the $p^{\prime}$ signal shows continuous distribution of the
spectral power throughout the time in a narrow frequency band around 163.9 Hz
(Fig. 4(a)-(iii)). The introduction of delayed acoustic self-feedback leads to
a significant change in the dynamical behavior of the $p^{\prime}$ signal. As
the suppression of thermoacoustic instability is approached on increasing
$L_{\text{c}}$, the dynamics of $p^{\prime}$ signal changes from the state of
large amplitude limit cycle oscillations (Fig. 4(b)) to low amplitude
aperiodic oscillations (Fig. 4(e)) via a regime of intermittent oscillations
(Figs. 4(c,d)). During the occurrence of intermittent oscillations, epochs of
low amplitude aperiodic fluctuations appear amidst epochs of large amplitude
periodic oscillations in an apparently random manner (see inset in Fig.
4(d)-(iii)). With an increase in $L_{c}$, we also notice that the dominant
frequency ($f_{d}$) of $p^{\prime}$ signal shifts gradually towards lower
values (compare Fig. 4(a)-(ii) to Fig. 4(e)-(ii)). An increase in the length
of the coupling tube $L_{c}$ introduces acoustics of corresponding lower
frequency in the combustor, thus reducing the dominant frequency of the
$p^{\prime}$ signal. Also, the spectral magnitude of the dominant frequency
decreases continuously such that the power spectrum appears broadband during
the state of suppression of thermoacoustic instability. The scalogram plots
during this transition show an increase in discontinuities in the variation of
dominant spectral amplitude of the signal (Fig. 4(b)-(iii) to Fig.
4(d)-(iii)), and these discontinuities increase during complete suppression of
thermoacoustic instability (Fig. 4(e)-(iii)).

Figure 5:  The percentage suppression of $p^{\prime}$ signals
($\Delta\bar{p}$) when delayed acoustic self-feedback is applied to (a)
thermoacoustic instability of different amplitudes
($p^{\prime}_{0,\text{rms}}$) for constant values of $L_{\text{c}}=1400$ mm
and $d_{\text{c}}=25.4$ mm, and (b) thermoacoustic instability of two
amplitudes ($p^{\prime}_{0,\text{rms}}\approx 3400$ Pa and $4100$ Pa) for
different coupling tube diameters ($d_{\text{c}}=6.35$, $9.525$, $12.7$,
$19.05$, and $25.4$ mm) at a fixed value of $L_{\text{c}}=1400$ mm. Figure 6:
CWT plots between $p^{\prime}$ and $\dot{q}^{\prime}$ signals and the temporal
variation of their phase difference $\Delta\phi$, calculated at the dominant
frequency $f_{d}$ (indicated by a horizontal red line and an arrow in each CWT
plot), for (a) the state of thermoacoustic instability in the absence of
delayed acoustic self-feedback and (b-d) in the presence of delayed acoustic
self-feedback in the combustor with increasing values of $L_{\text{c}}$. The
inset in (d) shows a magnified view of a phase jump. The value of
$d_{\text{c}}$ is kept constant at 25.4 mm.

Since limit cycle oscillations observed in the practical turbulent systems
could be of different amplitudes, we next examine the effectiveness of delayed
acoustic self-feedback on the suppression of such oscillations of different
amplitudes ($p^{\prime}_{0,\text{rms}}$) in the system (Fig. 5(a)). As
mentioned before, the subscript $0$ in $p^{\prime}_{0,\text{rms}}$ indicates
the state of thermoacoustic instability. The values of $L_{\text{c}}=1400$ mm
and $d_{\text{c}}=25.4$ mm are fixed during these experiments, as they
correspond to the regime of complete suppression of LCO
($p^{\prime}_{0,\text{rms}}=3200$ Pa) in Fig. 3. Here, the amplitude of LCO
before initiating the feedback is increased by increasing the fuel flow rate
in the reactant mixture while keeping the equivalence ratio constant at the
same time. We observe that limit cycle oscillations of amplitudes
$p^{\prime}_{0,\text{rms}}\leq 5000$ Pa (green-colored region on left side of
Fig. 5(a)) can be completely suppressed with an optimum size of the coupling
tube, while quenching of larger amplitude limit cycle oscillations (i.e.,
$p^{\prime}_{0,\text{rms}}>5000$ Pa, red-colored region on right side of Fig.
5(a)) is not observed with this coupling tube. By "optimum size", we imply the
dimensions (length and minimum internal diameter) of the connecting tube that
are sufficient to induce maximum suppression of limit cycle oscillations in
the system. In the red-colored region on the right side of Fig. 5(a), we
observe a small decrease in the amplitude of limit cycle oscillations due to
delayed acoustic self-feedback. Thus, we infer that a coupling tube with
optimal values of $L_{\text{c}}$ and $d_{\text{c}}$ can mitigate
thermoacoustic instability lesser than a certain critical value of
$p^{\prime}_{0,\text{rms}}$.

Furthermore, we investigate the effects of the internal diameter of the
coupling tube on the suppression of limit cycle oscillations. For two
amplitudes of thermoacoustic instability (i.e.,
$p^{\prime}_{0,\text{rms}}=3400\pm 150$ and $4100\pm 150$ Pa), we perform
experiments with coupling tubes having five different internal diameters
ranging from $d_{\text{c}}=6.35$ to $25.4$ mm, keeping the length of the
coupling tube constant at $L_{\text{c}}=1400$ mm. We observe that the
suppression of limit cycle oscillations depends on $d_{\text{c}}$, where a
larger diameter coupling tube can quench such oscillations easily for the
optimum value of $L_{\text{c}}$. Furthermore, limit cycle oscillations with
higher amplitude ($p^{\prime}_{0,\text{rms}}=4100$ Pa) undergoes lower
suppression than those with lower amplitudes ($p^{\prime}_{0,\text{rms}}=3400$
Pa) for the same coupling parameters.

###  III.2  Temporal investigation of changes in the coupled acoustic pressure
and global heat release rate oscillations

As discussed in Sec. I, thermoacoustic instability occurs due to a closed-loop
interaction between the heat release rate ($\dot{q}^{\prime}$) and acoustic
pressure ($p^{\prime}$) fluctuations. Several studies have recently shown that
the onset of such instability in turbulent combustors is a consequence of
mutual synchronization between these $p^{\prime}$ and $\dot{q}^{\prime}$
fluctuations Pawar  _et al._ (2017); Godavarthi  _et al._ (2018); Kasthuri
_et al._ (2022); Murayama and Gotoda (2019); Guan  _et al._ (2019); Dutta,
Ramachandran, and Chaudhuri (2019); Guan  _et al._ (2022). Therefore,
quenching of thermoacoustic instability can be achieved by breaking the
synchrony between these two oscillations. Thus, in this section, we
investigate the effect of delayed acoustic self-feedback on the coupled
behavior of the global heat release rate and the acoustic pressure
fluctuations during the suppression of thermoacoustic instability. We use
different tools from the synchronization theory to study the coupled behavior
of these oscillations. In Fig. 6, we examine the locking of the instantaneous
phases and frequencies of $p^{\prime}$ and $\dot{q}^{\prime}$ signals using a
cross-wavelet transform (CWT) Grinsted, Moore, and Jevrejeva (2004); Pawar
_et al._ (2019). In this plot, the regions of common spectral power for both
the signals are highlighted by the bright color, and the corresponding wrapped
instantaneous relative phases between them are indicated by arrows. When the
signals are synchronized, their CWT shows a larger magnitude of the spectral
power throughout the time and a constant alignment of arrows at a particular
phase difference. In contrast, desynchronization of signals is indicated in
the CWT by a near random distribution of the common spectral power and an
arbitrary alignment of arrows in time.

Figure 7:  The variation of PLV and Pearson’s correlation coefficient $\rho$
between the global heat release rate and acoustic pressure fluctuations under
the influence of delayed acoustic self-feedback through connecting tubes of
different lengths. The internal diameter of the coupling tube is kept constant
at $d_{\text{c}}=25.4$ mm, and $p^{\prime}_{0}$ is kept constant at 3400 $\pm$
150 Pa.

In Fig. 6(a), we notice that, in the absence of delayed acoustic self-feedback
in the system, a common frequency band occurs around the dominant frequency of
163.9 Hz in the CWT for the entire duration of the $p^{\prime}$ and
$\dot{q}^{\prime}$ signals, and the relative phase ($\Delta\phi$) between them
is observed to stay constant around a mean value of $35.3^{\circ}$. Here, the
instantaneous phase of the signals is calculated using the analytic signal
method with the Hilbert transform (Pikovsky, Rosenblum, and Kurths, 2003).
These behaviors of the frequency and phase indicate the existence of phase
synchronization between the global heat release rate and acoustic pressure
oscillations Pawar  _et al._ (2019); Kasthuri  _et al._ (2022). When the
system is subjected to delayed acoustic self-feedback for lower values of
$L_{\text{c}}$ (see for $L_{\text{c}}=1200$ mm in Fig. 6(b)), i.e., in the
regime I of partial suppression of Fig. 3, we observe similar properties of
phase synchronization between $p^{\prime}$ and $\dot{q}^{\prime}$ signals as
that witnessed in the absence of delayed acoustic self-feedback (Fig. 6(a)).
Just prior to the complete suppression of thermoacoustic instability (i.e.,
for $L_{\text{c}}=1400$ mm in Fig. 6(c)), we notice the presence of
discontinuities in the common frequency bands of CWT of $p^{\prime}$ and
$\dot{q}^{\prime}$ signals. These discontinuities can be easily seen in the
plot of $\Delta\phi$, where we notice the number of phase jumps about a mean
phase difference of these signals. During each phase jump, the value of
$\Delta\phi$ shows a sudden change of $2\pi$ rad, and the phase jump happens
due to the desynchronization of the signals Pikovsky, Rosenblum, and Kurths
(2003). Furthermore, we find that these instances of desynchronized
oscillations coincide with low amplitude aperiodic oscillations observed in
$p^{\prime}$ and $\dot{q}^{\prime}$ signals (Fig. 4(d)). Thus, in this
condition of delayed acoustic self-feedback, these signals exhibit a property
of intermittent phase synchronization.

During the state of complete suppression of thermoacoustic instability (i.e.,
for $L_{\text{c}}=1600$ mm in Fig. 6(d)), we notice an increase in the
discontinuities in the common frequency bands of CWT of $p^{\prime}$ and
$\dot{q}^{\prime}$ signals as compared to that seen in Fig. 6(c), which can
also be confirmed from the $\Delta\phi$ plot where we notice an increase in
the number of phase jumps about a mean phase difference. This indicates the
desynchronization of $p^{\prime}$ and $\dot{q}^{\prime}$ signals in the
system, which further suggests the loss of positive feedback between these
oscillations during the suppression of thermoacoustic instability.

In addition to CWT, we also use two other measures, called phase locking value
(PLV) and Pearson’s correlation coefficient ($\rho$), to detect the
synchronization behavior of $p^{\prime}$ and $\dot{q}^{\prime}$ signals for
different values of $L_{\text{c}}$ shown in Fig. 3. PLV helps us to detect
phase synchronization between two signals and is calculated as:

$\text{PLV}=\dfrac{1}{N}\Bigg{\lvert}\sum\limits_{n=1}^{N}\text{exp}(i\Delta\phi)\Bigg{\lvert},$
(2)

where $N$ is the number of data points in the two signals, and $\Delta\phi$ is
the instantaneous phase difference between the $p^{\prime}$ and
$\dot{q}^{\prime}$ signals. On the other hand, Pearson’s correlation
coefficient $\rho$ aids in finding the amplitude correlation between the
signals Gonzalez-Miranda (2002), and is calculated as:

$\rho=\dfrac{\text{Cov}(p^{\prime},\dot{q}^{\prime})}{\sigma_{p^{\prime}}\sigma_{\dot{q}^{\prime}}},$
(3)

where Cov represents the covariance between two signals, and $\sigma$ denotes
the standard deviation of a signal. We present the values of these measures in
Fig. 7. We find that in the regime of partial suppression of thermoacoustic
instability, both $p^{\prime}$ and $\dot{q}^{\prime}$ signals are phase
synchronized, confirmed from the PLV near 1. Inside the regime of suppression
marked as II in Fig. 7, as the signals are aperiodic where the phase of the
oscillations is not physically defined, we do not calculate PLV Pikovsky,
Rosenblum, and Kurths (2003); Sujith and Pawar (2021). The presence of a high
correlation between $p^{\prime}$ and $\dot{q}^{\prime}$ signals in the regimes
of oscillatory states is confirmed by the high positive value (between 0.55
and 0.7) of $\rho$. The value of $\rho$ drops to 0.25 in the regime of
suppression of thermoacoustic instability (region II in Fig. 7), which happens
due to the lack of correlation between desynchronized $p^{\prime}$ and
$\dot{q}^{\prime}$ signals Pawar  _et al._ (2017).

Figure 8:  (i) Time instants of acoustic pressure time series, and (ii - vii)
instantaneous spatial distribution of local Rayleigh Index for the state of
(a) thermoacoustic instability in the absence of delayed acoustic self-
feedback, and for (b-e) different states of delayed acoustic self-feedback.
The internal diameter of the coupling tube is kept constant at
$d_{\text{c}}=25.4$ mm, and $p^{\prime}_{0}$ is kept constant around 3400 Pa.

Thus, we observe that the extent of synchronization between the coupled
$p^{\prime}$ and $\dot{q}^{\prime}$ signals decreases during the mitigation of
thermoacoustic instability in a turbulent combustor, where the signals change
their behavior from the state of phase synchronized periodicity to
desynchronized aperiodicity via intermittent synchronized oscillations. Next,
we look at the effectiveness of delayed acoustic self-feedback to disrupt the
complex closed-loop interactions between hydrodynamic, acoustic, and heat
release rate fluctuations that exists inside the turbulent combustor during
the state of thermoacoustic instability.

###  III.3  Spatiotemporal analysis of acoustic power production in the
turbulent flow field during different states of self-delayed feedback

In this section, we compare the spatiotemporal changes in the reaction field
of the combustor as thermoacoustic instability is suppressed through delayed
acoustic self-feedback. In Fig. 8, we examine the local acoustic power
$p^{\prime}(t)\dot{q^{\prime}}(x,y,t)$ field (see panels ((ii-vii)) at some
representative time instances marked on the time series of acoustic pressure
fluctuations (see panel (i)). Regions with
$p^{\prime}(t)\dot{q^{\prime}}(x,y,t)$ $>$ $0$ represent acoustic power
sources, and regions with $p^{\prime}(t)\dot{q^{\prime}}(x,y,t)$ $<$ $0$
represent acoustic power sinks. Due to the periodic formation of large-scale
vortical structures at the dump plane, we observe the coherent production of
acoustic power sources over large clusters during thermoacoustic instability
(Fig. 8(a)). Additionally, vortices are shed from the tip of the bluff body.
In these coherent regions of acoustic power, the magnitude of the acoustic
power sources is significantly greater than their negative counterparts in
regions of acoustic power sinks in the reactive flow field. The regions of
acoustic power sources grow in size when the acoustic pressure fluctuations
approach local extrema (Figs. 8(a)-(iii, vi)), and they decrease in their
spatial size when the acoustic pressure fluctuations $p^{\prime}$ approach the
mean value of zero (Figs. 8(a)-(ii, iv)).

Similar trend of change in the size of acoustic power sources is observed when
the turbulent combustor is subjected to delayed acoustic self-feedback by
coupling it with a connecting tube of length $L_{\text{c}}=1200$ mm (Figs.
8(b)-(ii - vii)), and during a periodic epoch of intermittency observed in
Fig. 8(c) ($L_{\text{c}}=1400$ mm). Coherence is observed at a large scale in
Fig. 8(b)-(vi) downstream of the bluff body. However, the magnitude and
spatial coherence of acoustic power sources observed in Fig. 8(b) are low
compared to that observed during the state of thermoacoustic instability. The
spatial coherence and magnitude of the acoustic power sources further decrease
during the periodic epoch of intermittency ($L_{\text{c}}=1400$ mm in Fig.
8(c)). This is accompanied with an increase in the suppression of
thermoacoustic instability from Fig. 8(b) to Fig. 8(c). During the aperiodic
epoch of intermittency, we do not observe any large coherent regions of
acoustic power in Fig. 8(d), similar to that observed during the periodic
epochs of intermittency. However, we observe small scale patches of acoustic
power sources downstream (Figs. 8(d)-(i, vii)) and upstream (Fig. 8(d)-(iv))
of the bluff body. We believe these small scale acoustic power sources result
from small scale vortices shed from the dump plane and the tip of the bluff
body that do not grow in size and combust shortly after they are formed.

Finally, when the length of the connecting tube is increased to
$L_{\text{c}}=1600$ mm, we notice further suppression of the limit cycle
behavior of thermoacoustic instability, where the acoustic pressure exhibits
low amplitude aperiodic oscillations as seen in Fig. 8(e)-(i). During this
state of significant suppression of thermoacoustic instability, the spatial
distribution of $p^{\prime}(t)\dot{q^{\prime}}(x,y,t)$ seems granular and
devoid of a discernible pattern for different instants of time. These grainy
structures can be considered as disordered patterns or incoherent patterns in
the spatial dynamics of the turbulent reacting field. We hypothesize that the
presence of several small-scale vortices in the flow field generate
disorganized and chaotic flow dynamics, resulting in the incoherent acoustic
power production. As discussed in Sec. I, the occurrence of thermoacoustic
instability in a turbulent combustion system is a complex process and happens
due to nonlinear interaction between the flame, the flow, and the acoustic
field of a combustor. The application of optimal delayed acoustic self-
feedback disrupts this complex interaction in the turbulent combustor, thus
mitigating thermoacoustic instability.

##  IV  Conclusions

In summary, we demonstrate the mitigation of thermoacoustic instability (i.e.,
limit cycle oscillations) in a bluff body stabilized turbulent combustor by
inducing a delayed acoustic self-feedback in the system. The complex closed-
loop coupling between acoustics, hydrodynamics, and flame dynamics present
inside the turbulent combustor during thermoacoustic instability is disrupted
when optimal delayed acoustic self-feedback is introduced in the system. This
self-feedback is achieved by coupling the acoustic field of the system to
itself, using a connecting tube attached near the pressure anti-node position
of the acoustic standing wave. We observe a significant suppression of
thermoacoustic instability in the combustor when the length of the coupling
tube is approximately 3/8 times the wavelength of the fundamental acoustic
mode established in the combustor during the state of thermoacoustic
instability. The amplitude of the acoustic pressure fluctuations during the
state of suppression is comparable to that observed for the state of stable
operation (combustion noise) in the system. We find that the mitigation of
thermoacoustic instability is associated with a gradual decrease in the
amplitude of acoustic pressure oscillations and a shift in their dominant
frequency toward lower values. Furthermore, we notice that the dynamical
behavior of acoustic pressure fluctuations changes from the state of limit
cycle oscillations to aperiodic oscillations via intermittent oscillations
during the suppression of thermoacoustic instability. The synchronization of
the global heat release rate and acoustic pressure fluctuations, observed
during the uncoupled state of thermoacoustic instability, breaks down
gradually and the oscillations become desynchronized as the system approaches
the state of a significant suppression. During this state, we do not observe
any coherent structures of acoustic power production in the reaction field as
witnessed during thermoacoustic instability. We also notice the disintegration
of acoustic power sources in the flow field during the state of suppression of
oscillations, which happens due to the destruction of the local coupling
between acoustic pressure and heat release rate fluctuations in the spatial
field of the combustor.

Thus, we show that delayed acoustic self-feedback achieved through a single
connecting tube provides a promising control mechanism to suppress
thermoacoustic instability in turbulent combustors. Unlike in traditional
active closed-loop controls used in thermoacoustic studies, we do not need to
preprocess the acoustic pressure signal acquired from the combustor prior to
feedback using any electromechanical devices in the method of delayed acoustic
self-feedback. Therefore, we believe this methodology based on delayed
acoustic self-feedback opens up novel, cost-effective ways to mitigate
thermoacoustic instability in turbulent combustion systems used in practical
propulsion and gas turbine engines.

###### Acknowledgements.

We thank Mr. S. Thilagaraj and Mr. Anand for their help in conducting the
experiments. This work is supported by the J. C. Bose Fellowship (No.
JCB/2018/000034/ SSC) and the IoE initiative (SB/2021/0845/AE/MHRD/002696)
from the Department of Science and Technology, Government of India.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Instability of Near-Extreme Solutions to the Whitham Equation

John D. Carter

###### Abstract

The Whitham equation is a model for the evolution of small-amplitude,
unidirectional waves of all wavelengths on shallow water. It has been shown to
accurately model the evolution of waves in laboratory experiments. We compute
$2\pi$-periodic traveling-wave solutions of the Whitham equation and
numerically study their stability with a focus on solutions with large
steepness. We show that the Hamiltonian oscillates as a function of wave
steepness when the solutions are sufficiently steep. We show that a
superharmonic instability is created at each extremum of the Hamiltonian and
that between each extremum the stability spectra undergo similar bifurcations.
Finally, we compare these results with those from the Euler equations.

##  1 Introduction

###  1.1 The Euler equations

The irrotational motion of an inviscid, incompressible, homogeneous, two-
dimensional fluid with a free surface over a horizontal impermeable bed can be
modeled by a system of partial differential equations known as the Euler
equations, see for example [18, 21]. Stokes [28] established the existence of
periodic traveling-wave solutions to the Euler equations and determined the
first few terms in series expansions of these solutions using asymptotic
Fourier expansions. These solutions are now referred to as Stokes waves.
Stokes [29] conjectured that the solution with largest wave height on an
infinite-depth fluid has a crest angle of $2\pi/3$. Many years later, Amick
_et al._ [16] and Plotnikov [25] independently proved this conjecture. Amick
_et al._ [16] also proved that the solution with largest height on a finite-
depth fluid also has a crest angle of $2\pi/3$. Mitchel [23] estimated that
the most extreme Stokes wave on a fluid of infinite depth has a steepness of
$s=0.142$ where steepness is defined by $s=H/L$ and $H$ and $L$ are the wave
height and wavelength of the wave, respectively. Dyachencko _et al._ [12]
determined that the solution on an infinite-depth fluid with maximal steepness
has steepness $s=0.14106348\dots$ using very precise computations.

Benjamin & Feir [2] predicted that periodic traveling-wave solutions to the
Euler equations on an infinite-depth fluid are linearly unstable with respect
to perturbations with long wavelengths. Whitham [32] and Benjamin [1]
predicted that periodic traveling-wave solutions to the Euler equations on a
finite-depth fluid are unstable provided the nondimensional wavenumber, $k$,
and nondimensional still fluid depth, $h$, satisfy $kh>1.363$. Bridges &
Mielke [3] rigorously proved this finite-depth result. These long-wavelength
instabilities, in infinite- or finite-depth, are known as Benjamin-Feir or
modulational or subharmonic instabilities.

Longuet-Higgins [22] established that sufficiently steep periodic solutions of
the Euler equations on an infinite depth fluid are linearly unstable with
respect to perturbations of the same wavelength as the solution. Instabilities
that have the same period as the unperturbed solution are known as
superharmonic or co-periodic instabilities. Tanaka [30] established that the
transition from superharmonic stability to superharmonic instability on an
infinite-depth fluid occurs when the Hamiltonian is maximized. Saffman [26]
proved that this stability transition occurs when the Hamiltonian is maximized
as a function of wave height (or steepness). Zufiria & Saffman [35] extended
Saffman’s result to the finite-depth case.

Recently, Korotkevich _et al._ [20] and Deconinck _et al._ [8] examined the
stability spectra of near-extreme periodic traveling-wave solutions to the
Euler equations on an infinite-depth fluid. The main purpose herein is to show
that the Whitham equation, which was proposed as a small-amplitude
approximation of the Euler equations on a finite-depth fluid, shares a
striking number of properties with the Euler equations in the large-amplitude
limit.

###  1.2 The Whitham equation

Whitham [33, 34] proposed the following nondimensional model of waves on a
shallow (finite-depth) fluid with horizontal bathymetry

$u_{t}+\mathcal{K}*u_{x}+\frac{3}{2}uu_{x}=0,$ (1)

where $u=u(x,t)$ represents the free-surface displacement of the fluid, $x$ is
the horizontal coordinate, $t$ is the temporal coordinate, $*$ represents
convolution, and $\mathcal{K}$ is the Fourier multiplier defined by

$\hat{\mathcal{K}}(k)=\sqrt{\frac{\tanh(k)}{k}}.$ (2)

All variables are nondimensional. We assume $2\pi$-periodic boundary
conditions in the $x$-dimension and that the fluid has a nondimensional
undisturbed depth of $h=1$. Equation (1) reproduces the _uni_ directional
phase speed of the Euler equations for all wavenumbers. It is known as the
Whitham equation for water waves. Moldabayev _et al._ [24] identify a scaling
regime in which the Whitham equation can be derived from the Hamiltonian
theory of surface water waves. The Whitham equation has been shown to
accurately model the evolution of long waves of depression in laboratory
experiments, see for example [31, 4]. Particular generalizations of the
Whitham equation that allow for nonhorizontal bathymetry have been shown to
accurately model the evolution of waves over nonhorizontal bathymetry [5].

The Whitham equation has three known conserved quantities: the solution mean,

$\mathcal{M}=\frac{1}{2\pi}\int_{-\pi}^{\pi}u~{}dx,$ (3)

the $\mathcal{L}_{2}$-norm,

$\mathcal{L}_{2}=\int_{-\pi}^{\pi}u^{2}~{}dx,$ (4)

and its Hamiltonian,

$\mathcal{H}=\frac{1}{2}\int_{-\pi}^{\pi}\left(u\mathcal{K}*u+\frac{1}{2}u^{3}\right)dx.$
(5)

Note that the Euler equations also conserve the solution mean and the
$\mathcal{L}_{2}$-norm of the surface displacement, along with its Hamiltonian
(though the Hamiltonian for the Euler equations has a different form than that
of the Whitham equation).

Ehrnström and Kalisch [13] proved that the Whitham equation admits periodic
traveling-wave solutions and computed some of these solutions. Ehrnström &
Wahlén [15] proved that the Whitham equation has a solution with maximal wave
height and that this solution is cusped.

Hur & Johnson [17] proved that small-amplitude periodic traveling-wave
solutions of the Whitham equation are stable with respect to the modulational
instability if $k<1.146$ and are unstable with respect to the modulational
instability if $k>1.146$. Sanford _et al._ [27] numerically corroborated this
result and established that all moderate- to large-amplitude solutions are
unstable with respect to the modulational instability. Carter _et al._ [6]
numerically studied the superharmonic instability in the Whitham equation and
made comparisons with results from the finite-depth Euler equations. They
found that Whitham solutions steeper than $s=0.062$ are unstable with respect
to the modulational instability and that solutions steeper than $s=0.104$ are
unstable with respect to the superharmonic instability.

The remainder of the paper is outlined as follows. Section 2 presents some
$2\pi$-periodic traveling-wave solutions of the Whitham equation and a few of
their properties. Section 3 contains a stability analysis of solutions to the
Whitham equation, with a focus on near-extreme solutions. Section 4 contains
comparisons of the Whitham stability results with the Euler stability results.
Finally, Section 5 contains a summary of the main results presented herein.

##  2 Traveling-wave solutions

We compute $2\pi$-periodic traveling-wave solutions of the Whitham equation of
the form

$u(x,t)=f(x-ct)=f(\xi),$ (6)

using a Fourier basis and the branch-following method described in [14, 7].
Without loss of generality, we consider solutions with zero mean (i.e.
$\mathcal{M}=0$). Figure 1 contains plots of four zero-mean $2\pi$-periodic
traveling-wave solutions of the Whitham equation. Figure 2 contains a plot of
wave height, $H$, versus wave speed, $c$, for the branch of zero-mean
$2\pi$-periodic solutions of the Whitham equation. The values of the solution
parameters at the colored dots are included in Table 1. Note that the curve
has turning points at the orange and red dots. The existence of the first of
these turning points was first explored by Kalisch _et al._ [19]. We
hypothesize that the speed of the solutions continues to oscillate as the wave
height increases. However, we were unable to further extend the branch due to
the large number of Fourier modes required to accurately resolve the nearly
cusped solutions.

Figure 3 contains a plot comparing the solution corresponding to the red dot
in Figure 2 with the asymptotic formula for the shape of the steepest solution
derived by Ehrnström & Wahlén [15] that is valid near the crest

$f\sim\frac{4}{3}\left(\frac{c}{2}-\sqrt{\frac{\pi}{8}}~{}|\xi|^{1/2}\right),\hskip
11.38092pt\text{as }|\xi|\rightarrow 0.$ (7)

The two curves show good agreement in the vicinity of $\xi=0$. The computed
solution was obtained using $2^{16}$ Fourier modes and its parameter values
are listed in Table 1. The inset plot demonstrates that the computed solution
is smooth, while the solution with maximal steepness is not.

Figure 1: Plots of four zero-mean $2\pi$-periodic traveling-wave solutions to the Whitham equation. The heights of the solutions are included in the legend. Figure 2: Plot of wave height versus wave speed for the branch of zero-mean $2\pi$-periodic solutions to the Whitham equation. Figure 3: Plot comparing the crest of the solution corresponding to the red dot in Figure 2 (solid curve) with the asymptotic form of the steepest solution near the peak given in equation (7) (dashed curve). Color | $c$ | $H$ | $s$ | $\mathcal{H}$ | $\mathcal{L}_{2}$
---|---|---|---|---|---
Green | 0.9252 | 0.3916 | 0.0623 | 0.0422 | 0.0940
Yellow | 0.9747 | 0.6564 | 0.1045 | 0.0691 | 0.1508
Orange | 0.9760 | 0.6995 | 0.1113 | 0.0685 | 0.1497
Red | 0.9753 | 0.7639 | 0.1216 | 0.0678 | 0.1482
Table 1: Parameter values for the solutions that correspond to the colored
dots that appear in many of the figures. The parameters $c$, $H$, $s$,
$\mathcal{H}$, and $\mathcal{L}_{2}$ represent the wave speed, wave height,
wave steepness, Hamiltonian, and $\mathcal{L}_{2}$-norm, respectively.

Figure 4 contains a plot of the Hamiltonian, $\mathcal{H}$, versus the wave
steepness for the branch of zero-mean $2\pi$-periodic traveling-wave solutions
to the Whitham equation. The Hamiltonian has local minima at the origin and at
the red dot. It has a local maximum at the yellow dot. The yellow dot is the
global maximum on the interval examined. Under-resolved numerical simulations
suggest that the Hamiltonian has a second local maximum after the red dot.
However, due to the resolution required near this wave height, we are not able
to ensure the accuracy of this statement. We hypothesize that the Hamiltonian
continues to oscillate as wave steepness further increases.

Figure 5 shows that the plot of the $\mathcal{L}_{2}$-norm versus wave
steepness is qualitatively similar to the plot of Hamiltonian versus wave
steepness. Just like the Hamiltonian, the $\mathcal{L}_{2}$-norm has local
minima at the origin and at the red dot, and a local maximum at the yellow
dot. Under-resolved numerical simulations suggest that the
$\mathcal{L}_{2}$-norm has a second local maximum after the red dot.

Figure 4: Plot of the Hamiltonian versus wave steepness for the branch of
zero-mean $2\pi$-periodic traveling-wave solutions to the Whitham equation.
Figure 5: Plot of the $\mathcal{L}_{2}$-norm versus wave steepness for the
branch of zero-mean $2\pi$-periodic traveling-wave solutions to the Whitham
equation.

Figure 6 contains a plot of the Hamiltonian versus wave speed for the branch
of zero-mean $2\pi$-periodic traveling-wave solutions of the Whitham equation.
This plot shows that the Hamiltonian has a global maximum at the yellow dot.
The curve has turning points at the orange and red dots. The plot of the
$\mathcal{L}_{2}$-norm versus $c$ is qualitatively similar to the plot of
$\mathcal{H}$ versus $c$ and is therefore omitted. These plots show that the
solution with maximal speed (the orange dot) has steepness higher than the
solution with maximal Hamiltonian and $\mathcal{L}_{2}$-norm (yellow dot).
Finally, note that the maximum of the Hamiltonian (yellow dot) occurs at a
speed less than the maximal speed (orange dot).

Figure 6: Plot of the Hamiltonian versus wave speed for the branch of zero-
mean $2\pi$-periodic traveling-wave solutions to the Whitham equation.

##  3 Stability

In order to study stability of traveling-wave solutions to the Whitham
equation, enter a coordinate frame moving with the speed of the solution by
letting $\xi=x-ct$. This converts the Whitham equation to

$u_{t}-cu_{\xi}+\mathcal{K}*u_{\xi}+\frac{3}{2}uu_{\xi}=0.$ (8)

We consider perturbed solutions of the form

$u_{pert}(\xi,t)=u(\xi)+\epsilon v(\xi,t)+\mathcal{O}(\epsilon^{2}),$ (9)

where $u$ is a traveling-wave solution, $\epsilon$ is a small constant, and
$v$ is the leading-order term of the perturbation. Substituting (9) into (8)
and linearizing gives

$v_{t}-cv_{\xi}+\mathcal{K}*v_{\xi}+\frac{3}{2}\left(uv\right)_{\xi}=0.$ (10)

We use the Fourier-Floquet-Hill method [9] to solve this equation. Without
loss of generality, assume

$v(\xi,t)=\mbox{e}^{i\mu\xi}V(\xi)\mbox{e}^{\lambda t}+c.c.,$ (11)

where $\mu\in[-\frac{1}{2},\frac{1}{2})$ is known as the Floquet parameter,
$\lambda$ is a complex constant, $c.c.$ stands for complex conjugate, and
$V(\xi)$ is a $2\pi$-periodic complex-valued function with Fourier series

$V(\xi)=\sum_{j=-N}^{N}\hat{V}(j)\mbox{e}^{ij\xi},$ (12)

where $N$ is a large positive integer. If $\mu=0$, then the perturbation has
the same $\xi$-period as the unperturbed solution. If $\mu\neq 0$, then the
perturbation has a period larger than that of the unperturbed solution.

Substituting (11) and (12) into (10) gives the following matrix eigenvalue
problem that depends on the parameter $\mu$

$\hat{\mathcal{L}}\hat{\mathcal{V}}=\lambda\hat{\mathcal{V}},$ (13)

where $\hat{\mathcal{V}}=(\hat{V}(-N),\hat{V}(-N+1),\dots,\hat{V}(N))^{T}$ and
the elements of the matrix $\hat{\mathcal{L}}$ are given by

$\hat{\mathcal{L}}_{mn}=\begin{cases}i\left(\mu+m\right)\left(c-\hat{\mathcal{K}}\left(\mu+m\right)\right),&m=n,\\\
-\frac{3}{2}i\left(\mu+m\right)\hat{u}(n-m),&m\neq n,\end{cases}$ (14)

where the $\hat{u}$’s are the Fourier coefficients of $u$. Given a solution of
the Whitham equation, we choose an equally-spaced sampling of $\mu$ values and
find the corresponding eigenvalues and eigenvectors of $\mathcal{L}$ using a
standard eigensolver (“eigen” in the Julia programming language). If there
exists a perturbation with $\mu=0$ and $\Re(\lambda)>0$, then the solution is
said to be linearly unstable with respect to the superharmonic instability. If
there exists a perturbation with $0<\mu\ll 1$ and $\Re(\lambda)>0$, then the
solution is said to be linearly unstable with respect to the modulational
instability. The solution is said to be spectrally stable if there exist no
$\mu$ such that $\Re(\lambda)>0$.

Solutions to the Whitham equation with steepness $s<0.062$ (and period $2\pi$)
are stable. This is consistent with the Hur & Johnson [17] result that small-
amplitude solutions of the Whitham equation are stable with respect to the
modulational instability when $k<1.146$. When the steepness of the solutions
surpasses $s=0.062$, they become unstable with respect to the modulational
instability and their stability spectra contain a figure eight centered at the
origin, see plot (i) in Figure 7. This transition corresponds to the green dot
in Figures 2 and 4-6. As wave steepness increases, the figure eight increases
in size and morphs into “mushrooms” with lobes that bend and eventually
intersect the $\Re(\lambda)$-axis away from the origin, see plot (ii) in
Figure 7. After the lobes intersect the $\Re(\lambda)$-axis, the spectra
include a figure infinity centered at the origin surrounded by a vertical
peanut-like shape, see plot (iii). As the steepness increases further, the
figure infinity decreases in size and the vertical peanut transforms into a
horizontal peanut, see plot (iv). These four solutions are all unstable, but
are stable with respect to the superharmonic instability.

Solutions to the Whitham equation become unstable with respect to the
superharmonic instability once their steepness surpasses $s=0.1045$. In other
words, solutions to the Whitham equation become unstable with respect to the
superharmonic instability once their steepness is large enough that the first
local maximum of the Hamiltonian (the yellow dot in Figure 4), or
$\mathcal{L}_{2}$-norm (the yellow dot in Figure 5), is reached. As this
maximum is attained, the figure infinity collapses to the origin and the
horizontal peanut pinches off into two oval-like shapes centered on the
$\Re(\lambda)$-axis, see plot (v) in Figure 8. There is no figure eight or
figure infinity in this spectrum. The spectrum is confined to the imaginary
axis other than the two ovals centered on the $\Re(\lambda)$-axis. Regardless,
all values of $\mu$ give instabilities as part of the ovals. As the wave
steepness continues to increase, the ovals decrease in size and move away from
the origin while remaining centered on the $\Re(\lambda)$-axis, see plot (vi).
Eventually a new figure eight centered at the origin develops (in between the
two ovals), see plot (vii). Note that the ovals centered at $\lambda\approx\pm
1.0976$ are omitted from plot (vii) so that the details near the origin are
more apparent. As the solution steepness further increases, the new figure
eight undergoes transitions similar to those experienced by the first figure
eight (i.e. the figure eight that was created when the steepness reached
$s=0.062$) while remaining between the two ovals centered on the
$\Re(\lambda)$-axis that continue to decrease in size and move away from the
origin. When the local minima of the Hamiltonian and $\mathcal{L}_{2}$-norm at
the red dots in Figures 4 and 5 are reached, a second superharmonic
instability is created and the spectra are composed of four ovals centered on
the $\Re(\lambda)$-axis. Additional under-resolved simulations suggest that
this pattern (the creation of a figure eight centered at the origin which
bifurcates into two ovals once the solution becomes steep enough) is repeated
as the steepness is increased beyond the local minimum of the Hamiltonian at
the red dot.

Figure 7: Stability spectra for four solutions to the Whitham equation with
differing wave steepnesses before the onset of the superharmonic instability.
Figure 8: Stability spectra for three solutions to the Whitham equation with
differing wave steepnesses above the onset of the superharmonic instability.
Plot (vii) also includes two small ovals centered near $\lambda=\pm 1.0976$
that are omitted so the finer details near the origin are visible.

##  4 Comparison with results from the Euler equations

Although the Whitham equation was proposed as a simplification of the finite-
depth Euler equations in the small-amplitude limit, it shares a surprising
number of similarities to the Euler equations in the large-amplitude limit.
These similarities (and some differences) are discussed in this section. The
following comments assume that the solutions to both the Whitham and Euler
equations have nondimensional period $2\pi$ and correspond to a nondimensional
depth of $h=1$, except when noted.

  1. 1.

For large values of wave steepness, the plot of the Whitham Hamiltonian
oscillates as a function of wave steepness similarly to how the plot of the
Euler Hamiltonian oscillates as a function of steepness.

  2. 2.

The Whitham solution with maximal speed has a steepness that is larger than
the steepness of the solution corresponding to the maximum of the Hamiltonian.
This is similar to the Euler result.

  3. 3.

Small-amplitude solutions of the Whitham equation are stable while small-
amplitude solutions to the finite-depth Euler equations are unstable.
Deconinck & Oliveras [10] showed that small-amplitude solutions to the Euler
equations are unstable with respect to “high-frequency” instabilities, but are
stable with respect to the modulational instability. The Whitham equation does
not admit these high frequency instabilities. Deconinck & Trichtchenko [11]
showed that this apparent discrepancy is due to the unidirectional nature of
the Whitham equation.

  4. 4.

Moderately steep solutions to both the Euler and Whitham equations are
unstable with respect to the modulational instability. The modulational
instability onset for the Whitham equation occurs at $s=0.062$ while the onset
in the Euler equations with dimensionless depth $h=1$ occurs at $s=0.085$, see
[6].

  5. 5.

Solutions to both the Euler and Whitham equations with sufficiently large
steepness are unstable with respect to superharmonic instabilities. The
superharmonic instability onset for the Whitham equation occurs at $s=0.1045$
while the onset in the Euler equations with dimensionless depth $h=1$ occurs
at $s=0.099$, see [6].

  6. 6.

The onset of the superharmonic instability in the Whitham equation occurs at
the first maximum of the Hamiltonian, just as in the finite-depth Euler
equations, see [35].

  7. 7.

The Whitham equation admits additional superharmonic instabilities each time
an extremum of the Hamiltonian is achieved. Similar behavior is seen in the
infinite-depth Euler equations, see [8]. To the best of our knowledge, there
are not yet any published related results for the finite-depth Euler
equations.

  8. 8.

When the superharmonic instability is created, the most unstable mode
corresponds to $\mu=\pm 0.5$. This means that the most unstable mode has a
period that is twice that of the unperturbed solution. It appears that the
$\mu=\pm 0.5$ mode remains the most unstable mode as the solution steepness
increases. This is different than the infinite-depth Euler result where the
$\mu=0$ and $\mu=\pm 0.5$ perturbations alternate between being the most
unstable.

##  5 Summary

We have shown that the Whitham equation, an equation proposed as a model for
small-amplitude waves on shallow water, has a number of similarities with the
Euler equations in the large-amplitude limit. In particular, (i) the
Hamiltonian oscillates as a function of solution steepness once the steepness
is large enough and (ii) superharmonic instabilities are created at each
nonzero critical point of the Hamiltonian.

## Acknowledgements

The author thanks Anastassiya Semenova and Bernard Deconinck for especially
helpful conversations. The author also thanks Malek Abid, Eleanor Byrnes,
Sergey Dyachenko, Marc Francius, Henrik Kalisch, and Christian Kharif for
additional helpful discussions.

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