PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Amplitude death in Coupled Thermoacoustic Oscillators withFrequency Detuning
Hiroaki HYODO1; Tetsushi BIWA1
1Tohoku University, JapanABSTRACT
Thermoacoustic oscillation is a harmful vibration in combustors of a gas turbine engine because it candamage the combustors. Development of a suppression method of thermoacoustic oscillation is urgent
necessity. Recently, a coupling-induced amplitude death phenomenon was reported experimentally andnumerically in thermoacoustic oscillators of Sondhauss type. Since the coupling was established only byusing hollow tubes that connects two oscillators, this method can be a simple and reliable method to stop theoscillations, compared to active control method involving electronics and acoustic drivers. In this study, thetube coupling is further tested for the coupled thermoacoustic oscillators with frequency detuning, in order toexplore the possibility of stronger suppression effect. The linear stability of the coupled oscillators isnumerically obtained based on hydrodynamic equations. We describe that the amplitude death is realizedwith much smaller tube diameter than that of coupled thermoacoustic oscillators with no detuning. Also, wedescribe that the tube length resulting in the amplitude death is different from that of coupled identicaloscillators. These numerical results are compared with experiments to confirm the validity.
Keywords: Amplitude death, Thermoacoustics, Coupling

1. INTRODUCTION
Thermoacoustic oscillation is a self-sustained gas oscillation when a heat source exists in a duct.
Combustion oscillation (1) is a thermoacoustic oscillation in combustors of a gas turbine engine. Lean
premixed combustion method is employed to reduce the emission of NOx in gas turbine engine. In thismethod, the emergence of combustion oscillation is a troublesome issue because the combustion
oscillation leads to serious damage to the combustors. Suppression methods of thermoacoustic
oscillation such as active control method have been studied. These suppression methods, however,sometimes can be complex because of requirement of the equipment to control an external forcing.
Development of simple and reliable suppression method of thermoacoustic oscillation is an urgent
necessity.
The stabilization of oscillation can be realized by the coupling of two or more oscillators. The
annihilation of the oscillation is called amplitude death. The presence of the amplitude death has beenreported theoretically and experimentally in various oscillation systems (2-4). Recently, thestabilization of coupled Sondhauss type thermoacoustic oscillators has been reported (5). Two
identical Sondhauss oscillators were coupled using a hollow tube. The Sondhauss oscillator weremodeled based on the hydrodynamic equations. The stability analysis has shown the amplitude deathoccurs when the coupling tube lengths are one-quarter or three-quarters of the wavelength offundamental acoustic oscillations in uncoupled oscillators.
In this study, the tube coupling is further tested for the coupled Sondhauss type thermoacoustic
oscillators with frequency detuning, in order to explore the possibility of stronger suppression effect.
Frequency detuning is introduced by using two oscillators with different resonance tube length. The
conditions realizing amplitude death is numerically obtained and presented on the parameter plane onthe detuning and coupling tube length. We have found that the tube length resulting in the amplitude
death is different from that of coupled identical oscillators. We also have found that the coupling tube
diameter required to cause the amplitude death is much smaller tube diameter than that of coupled
thermoacoustic with no detuning.

1

hyodo@amsd.mech.tohoku.ac.jp5627

2. COUPLED THRERMOACOUSTIC OSCILLATORS
2.1 Thermoacoustic Oscillator
A Sondhauss type thermoacoustic oscillator used in this study is made of a resonance tube, a stack,and cold and hot heat exchangers. The working gas is air at atmospheric pressure. A circular tube ofdiameter D = 40 mm is used as a resonance tube. The stack is made of many circular pores of 1.12 mm
diameter and 40 mm length. The porosity of the stack is 0.78. The stack is located in the resonance tube
with its central position separated from the end by 180 mm. The stack is sandwiched between hot andcold heat exchangers, each of which is made of parallel plates with 0.5 mm thickness and 20 mm length,
placed with 1 mm spacing. The temporal mean gas temperature, TC , in the cold heat exchanger isconstant at room temperature (297 K). The temperature in the hot heat exchanger is a control parameterto induce thermoacoustic oscillations. In the other parts of the oscillator, the temporal mean
temperature is equal to T C.

Figure 1 – Coupled thermoacoustic oscillators
2.2 Coupled Oscillators
Two thermoacoustic oscillators are coupled using a connecting tube, as shown in Figure 1. Twooscillators, Oscillator A and Oscillator B, are different only in the resonance tube length. Theresonance tube length, L A, of Oscillator A is fixed at L A = 860 mm, whereas the tube length, L B, ofOscillator B is a control parameter to make a frequency detuning. The temperature T HA in the hot heatexchanger of Oscillator A is fixed at THA = 537 K. The temperature T HB in the hot heat exchanger ofOscillator B is controlled as explained in section 3. These two oscillators are coupled at their ends byusing the connecting tube, which is modeled by a circular channel with diameter d and length L.

3. CALCULATION METHOD3.1 Basic Equation
The hydrodynamic equations are used as basic equations for the analysis. The equations ofcontinuity, momentum, and energy are linearized for a purely oscillating flow with angular frequency
ω (6). Taking the cross-sectional average along a radial direction of a flow channel, these basicequations are reduced to a pair of quasi-one-dimensional differential equations of complex pressure
amplitude P and complex velocity amplitude U as
dPdx
dUdx

 ZU ,(1)YP  GU .

(2)

In these equations, Z, Y, and G are given asYGUm1,A 1  FQZiZiZ

A ª¬1  J  1 FD º¼J Pm(3),FD  FQ
1 dTm,
1  FQ 1  V Tm dx

(4)
(5)

where ρ m, Pm , and Tm respectively denote the temporal mean density, pressure and temperature of the5628

gas. Also, γ and σ are the specific heat ratio and Prandtl number of the gas.
For the gas with thermal diffusivity α and kinematic viscosity ν, the thermoacoustic function χ j(j=α or ν) is given using the first-order and zeroth-order Bessel functions J 1 and J 0 of the first kind asFj

2 J1 ª j  1 ZW j º¬¼.
i  1 ZW j J 0 ª j  1 ZW j º¬¼(6)In equation (6), τj is given

Wj

r2 / 2 j ,(7)where r is the radius of the flow channel.
The differential equations (1) and (2) are solved analytically. The solution gives the relation of Pand U between two different points x and x+l asªP x  l ºªP x º(8)«» M«»,¬«U x  l ¼»¬«U x ¼»
where M is the transfer matrix given asª GMsinh O  cosh OGl « b2e ««¬«

2Ysinh Obº»».
Gsinh O  cosh O »b¼»
2Zsinh Ob(9)G 2  4YZ and O bl / 2 .
In equation (9), b
In the following, the transfer matrices M A and M B denote the matrix of Oscillator A and Oscillator
B, respectively. These matrices are created by taking a product of the transfer matrices of the circulartube, cold heat exchanger, stack, hot heat exchanger, and the circular tube. The transfer matrix M Tdenotes that of the coupling tube.

3.2 Frequency Equation
Transfer matrices M A and M B link acoustic variables (P A,B(0), U A,B(0)) at x A,B = 0 with (P A,B(L A,B),U A,B(L A,B)) at x A,B = L A,B on both ends of the oscillator:ª PA, B LA, B º«»
«U A, B LA, B »¬¼ªP0 ºM A, B « A, B».
¬«U A, B 0 ¼»(10)

Transfer matrix M T links the acoustic variables (P T (0), UT (0)) with (P T (L), UT (L)) at the ends of thecoupling tube asª PT L ºª PT 0 º(11)«» MT «».
¬«UT L ¼»¬«UT 0 ¼»
When the two oscillators are uncoupled, the boundary conditions require U A,B = 0 at the closedends (x A,B = 0 and x A,B = L A,B ). To have a nonzero solution,
M A, B

210(12)

must be satisfied. This equation serves as the frequency equation of the uncoupled oscillator, where
the two subscript numbers denote the transfer matrix component.
For the coupled case, the connecting conditions between the oscillators and the connecting tube
are the continuity of the pressure amplitude
PA LAPT 0 , PB LBPT L(13)and the continuity of the velocity amplitude:
U A LA U T 0 , U B L U T L .
(14)
As a condition to have a nonzero solution of P and U, we obtain the frequency equation of coupled5629

oscillators as
M T 21 M A 11 M B 11  M T 11 M A 11 M B 21
 M T 22 M B 11 M A 21  M T 12 M A 21 M B 210(15)When THA, THB, d, L, and L B are given, equations (12) and (15) can be solved for the complexfrequency f = (ω/2π). The real part of solution represents the frequency of oscillation. The imaginary
part represents the stability of the equilibrium; a positive imaginary part means that the equilibrium
state is linearly stable, whereas a negative one means that the equilibrium is unstable. In this way, thestability of the uncoupled oscillator is investigated. By solving equation (12) with changing the hot
end temperature, the critical temperature above which the oscillations are excited are searched. The
critical temperature ΔT A = T HA − TC of the uncoupled Oscillator A is obtained as ΔT A = 190 K. When
ΔT A is set to 240 K, the complex frequency fA (= fRA + i fIA) of the uncoupled Oscillator A is obtained asfRA = 206 Hz and fIA = −1.20 (1/s) which indicates the excitation of the fundamental acoustic
oscillations whose wavelength is essentially twice the length of L A, and the characteristic number ofcycles QA(= fRA / fIA) is 172 before the oscillation amplitude grows substantially. When the temperature
ΔT B (= T HB − TC ) and the resonance tube length L B is given, the complex frequency fB (= fRB + i fIB) ofthe uncoupled Oscillator B can be obtained as well. When the resonance tube length L B differs from L A,
also the frequency fRB differs from fRA although the oscillation mode is persistently the fundamental
one. The frequency detuning ΔfR = fRB − fRA is chosen as a parameter. It is to be noted that the linear
stability fIB is also affected by changing L B. Therefore, the temperature ΔT B is controlled so that theQ B(=fRB / fIB) of Oscillator B becomes the same value as Q A.
Figure 2 presents the resonance tube length L B dependence of the frequency fRB of uncoupled
Oscillator B. The red curve represents fRB and the horizontal black line represents the frequency withL B = L A. The frequency fRB monotonically decreases as the tube length L B increases. The difference
between the red curve and black line represents the frequency detuning ΔfR. In the following
calculation, the length L B is changed in the range 790 mm < L B < 950 mm so that the frequency
detuning range becomes −20 Hz < ΔfR < 20 Hz.

Figure 2 – Resonance tube length L B dependence of oscillation frequency fRB

4. RESULTS AND DISCUSSION
4.1 Calculation Result
By solving equation (15), the stability of the coupled oscillators is investigated by changing L B andthe coupling tube length L while keeping the coupling tube diameter d = 8 mm. Figure 2 presents acontour map of fI on the plane of ΔfR and L. Broken curves represent the contour line with fI = 0. Insideof the broken curves, the fI is positive. In these areas, the amplitude death occurs, i.e. both twooscillators are stabilized. The death regions are located near L = 0.85 m and L = 1.7 m. Two lengths L= 0.85 m and L = 1.7 m are close to a half or one wavelength of 206 Hz fundamental acoustic waves .
Near L = 1.7 m, the death regions are created when the detuning ΔfR is larger than 10 Hz which is 5%of the fundamental oscillation frequency. Near L = 0.85 m, the detuning ΔfR must be larger than 15 Hz
to realize the amplitude death. These death regions are slightly asymmetric. In the center of the death5630

Figure 3 – Contour map of fI of the coupled thermoacoustic oscillators
region with L = 1.7, ΔfR = −16 Hz, fR and fI are fR = 198 Hz, fI = 1.16 (1/s). This means that theoscillation ceases after 171 cycles Q(=fR / fI =170.7). Outside of the death regions, the value of fI is not
extremely smaller than that of uncoupled oscillator, which means that the instability is not enhanced bythe coupling.
It should be noted here that the frequency detuning reduces the tube diameter and changes thetube length necessary for the amplitude death in the coupled thermoacoustic oscillators. By changing
and comparing the contour maps of fI when d value is reduced less than 8 mm, we have found thatthe smallest d that stabilize the oscillators is d = 3mm. On the other hand, our previous study (5),conducted using the same oscillator as Oscillator A in figure 1, demonstrated that two identical
thermoacoustic oscillators without detuning were stabilized only when the coupling tube diameter islarger than 25 mm. The tube diameter difference of 3 mm and 25 mm is significantly large,
considering that the resonance tube diameter is 40 mm. In this study, the death regions appeared with
L = 0.85 m and L = 1.7 m. These lengths correspond to a half wavelength and one wavelength of thefundamental oscillation. In the case with zero detuning, the necessary length s were L =0.42 and L =1.27 m, corresponding to one-quarter and three-quarters of the acoustic wavelength. The coupled
thermoacoustic oscillators with frequency detuning is tested by experiments in the next section, toconfirm the amplitude death conditions.
4.2 Comparison with Experimental Result
Two thermoacoustic oscillators shown in figure 1 were constructed in the same way as Reference(5). In Oscillator B, one of the ends of the resonance tube was replaced with the movable plug tochange L B. When the resonance tube length L B was changed, not only the frequency fB of uncoupled
Oscillator B but also the pressure amplitude of uncoupled Oscillator B was changed. Therefore, thetemperature difference ΔT B was controlled so that the pressure amplitude of oscillator A and B became
the same before the coupling was introduced. The two oscillators were connected by thick-walled
flexible tube with the diameter 8 mm.
Figure 3 presents the oscillation states after the coupling on the plane of the frequency detuning Δf = f B - fA and coupling tube length L. The amplitude death regions (crosses) appeared with L = 0.8 m
and L = 1.6 m when the frequency detuning was employed. For L = 0.8 m, the frequency detuning

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Figure 4 – Experimental bifurcation diagram
required to cause the amplitude death is larger than that of L = 1.6 m. We observed the oscillation statesoutside of the death regions. When the frequency detuning was small, the two oscillators weresynchronous (circles). When the frequency detuning was increased, the synchronization was broken(horizontal lines). The pressure amplitude of coupled oscillators did not greatly exceed that ofuncoupled oscillators. These experimental results are consistent with the calculation results. The
stability analysis is useful for examining the amplitude death conditions even if the frequency detuning
is introduced.

5. CONCLUSIONS
The amplitude death conditions in the coupled thermoacoustic oscillators of the Sondhauss typewith frequency detuning were explored with respect to the coupling tube length and the detuning. We
found that the tube coupling with frequency detuning realize the amplitude death with much smaller
tube diameter than that of coupled thermoacoustic oscillators with no detuning. The frequency
detuning was found to greatly reduce the tube diameter necessary to cause the amplitude death,indicating that the stronger suppression effect is expected than in the coupling with zero detuning.

REFERENCES1. Lieuwen TC, Yang V. Combustion instabilities in gas turbine engines: operational experience,fundamental mechanisms, and modeling. Progress in Aeronautics and Astronautics series – Vol 210.
American Institute of Aeronautics and Astronautics, 1801 Alexander Bell Drive, Reston, VA
20191-4344, USA; 2006.
2. Reddy DVR. Time delay induced death in coupled limit cycle oscillators. Phys. Rev. Lett. 1998; 80:5109.
3. Kumar P, Prasad A, Ghosh R. Stable phase-locking of an external-cavity diode laser subjected toexternal optical injection. J. Phys. B. 2008; 41: 135402.
4. Nagao R, Zou W, Kurth J. Restoring oscillatory behaviour from amplitude death with anti-phasesynchronization patterns in networks of electrochemical oscillations. Chaos. 2016; 85: 3381.
5. Hyodo H, Biwa T. Stabilization of thermoacoustic oscillators by delay coupling. Phys. Rev. E. 2018; 98:052223.
6. Rott N. Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z. Angew. MathPhys. 1969; 20: 230.

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