Low-order network modelling of the e ff ect of Helmholtz resonators on nonlinear thermoacoustic modes in annular combustors

Liming Yin 1

Department of mechanics and aerospace engineering, Southern University of Science and Technology Shenzhen, China

Dong Yang 2

Department of mechanics and aerospace engineering, Southern University of Science and Technology Shenzhen, China

ABSTRACT Modern aero-engines and land-based gas turbines often use annular combustors where many burners are installed in the circumferential direction. These combustors are often operated under lean- premixed pre-vaporized (LPP) conditions in which the flames are more susceptible to acoustic perturbations than traditional combustion systems. The resulted thermoacoustic instabilities may involve mode patterns varying in both longitudinal and circumferential directions. When nonlinear flame models are considered, our previous work proved that a 2-D low-order network model can capture limit cycle oscillations involving uncoupled and nonlinearly coupled modes, including longitudinal, circumferentially spinning / standing, and slanted modes. This kind of low-order network modelling tool has been recognised as a computationally e ffi cient way of analysing thermoacoustic instabilities in annular combustors. Helmholtz resonators (HRs) are widely used to damp acoustic oscillations. For thermoacoustic instabilities in annular combustors, the presence of the HRs causes modal coupling and mode shape change. For nonlinearly coupled thermoacoustic modes, it is di ffi cult to predict the mode change accurately. In this paper, HRs are incorporated into the aforementioned low-order network model. The e ff ect of HRs on di ff erent linear and nonlinear thermoacoustic mode patterns are studied in detail. This provides a powerful low-order network modelling tool for studying the damping performance of HRs on nonlinear thermoacoustic modes in annular combustors.

1. INTRODUCTION

Due to stringent NOx emission requirements, modern aero-engine and ground-based gas turbines are regularly worked under lean burning circumstances; makes the flame combustion more susceptible

1 12032392@mail.sustech.edu.cn

2 yangd3@sustech.edu.cn

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

to acoustic influences [1]. These oscillations are caused by a feedback between the flame and the acoustics, which results in high pressure and heat release oscillations in the chamber, and might cause the entire propulsion system to fail [2]. Predicting and suppressing thermoacoustic instabilit in annular combustion chambers remains an important research topic. Two dimensional low-order network models can naturally handle the coexistence of circumferential and longitudinal modes. In our previous work, we proposed a model based on this idea which can be used for systematically predicting thermoacoustic limit cycle patterns on the coupling between di ff erent circumferential modes, and between longitudinal and circumferential modes. It’s ability to predict the frequency, amplitude and spatial pattern evolution between linearly unstable modes and limit cycles have been systematically studied and validated. Helmholtz resonators (HRs) can be used as passive dampers for acoustic oscillations, and of particular interest in current work is their use in damping thermoacoustic oscillations [3–5]. The issue of incorporating HRs into annular combustors was studied by Stow and Dowling, using a low-order network model which expressed perturbations in terms of modal expansions [6]. A similar low-order network modeling tool has been successfully applied to the optimization of resonators and is used to systematically improve the design and location of several HRs attached to an annular duct [7]. However, the e ff ect of HRs on axial and circumferential modal coupling, and on circumferential modal coupling with nonlinear flame models needs further studies. In this paper, using the standard approach of low-order network modeling [8–11], HRs are incorporated into the aforementioned low-order network model. A new model called LONMH was obtained. The developed network tool is then applied to a laboratory annular combustor geometry with a given mean flow and a prescribed simple but generic nonlinear flame model. The e ff ect of HRs on the first longitutal and standing thermoacoustic modes are preliminarlily studied.

2. LOW-ORDER NETWORK MODEL WITH HRS(LONMH)

2.1. Low-order Network Model Low-order network model is an e ffi cient and widely used tool to obtain coupled prediction of thermoacoustic oscillations. It simplifies the combustor as a network of connected “modules”, each with fixed cross sectional area and uniform mean flows. The heat release zone is assumed short compared to the acoustic wavelength and is treated as a discontinuity of the acoustic field. By combing linear perturbations in each module, jump condition across modules, flame model and acoustic boundary conditions, thermoacoustic modes of the system can be calculated. Similar to Ref. [12, 13], as shown in Figure 1, in the present study, we consider a typical annular combustor system which can be reduced to an annular-shaped plenum with multiple premix ducts connecting it to an annular-shaped combustor. Premix ducts are equidistant from each other and can be considered to be cylindrical duct [14–16]. For the linear perturbations in an annular duct, the Navier-Stokes equations controlling fluid motion reduce to the following equations for linear perturbations when viscosity, heat input, and heat conduction are ignored, and a stable uniform single-component mean flow and constant heat capacity are assumed [12]: 1

¯ c 2 ¯ D 2

Dt 2 −∇ 2 ! p ′ = 0 (1a)

¯ ρ ¯ T ¯ Ds ′

Dt = 0 (1b)

¯ D ξ ′

Dt = 0 (1c)

where ¯ D / Dt = ∂/∂ t + ¯ u · ∇ . c , ρ, T are the sound speed, density and temperature respectively, and p , s , ¯ u , ξ are pressure,entropy, velocity and vorticity respectively. An overbar, ¯ [] , denotes time-

average and a prime, [] ′ , denotes small perturbations. Linear flow perturbations can be thought of as the sum of three types of disturbances away from the combustion zone and for constant flow area, as shown in equation (1) : 1) an acoustic disturbance that is isentropic and irrotational, 2) an entropic disturbance that is incompressible and irrotational, and 3) a vortical disturbance that is incompressible and isentropic [17]. We will consider perturbations with complex frequency ω and hence write p ′ = Re h ˆ p ( x , θ ) e i ω t i , with p ′ ( x , θ ) = P ∞ n = −∞ ˆ p n ( x ) e in θ , and similarly for the orther perturbation variables. The ˆ p n

represents the component with circumferential wavenumber n , when | n | large ( | n | > N ) the component will be highly cuto ff , hence can be ignored. We can therefore approximate by taking p ′ ( x , θ ) = P N n = − N ˆ p n ( x ) e in θ , etc [18]. For a detailed derivation process of the perturbation equations in premix duct and the transverse perturbation propagation matrix, please refer to Ref. [13].

Figure 1: Low-order network schematic diagram

2.2. The Model of Helmholtz Resonators When HRs are incorporated into a thermoacoustic system, as shown by the blue dot in Figure 1, their geometrical size is normally much smaller than the acoustic wave length, so they can be considered to be acoustically compact. As shown in [14, 5], each HR can be modelled as a spatial delta input to the linearised mass and energy equations for the main duct. We made two reasonable assumptions about the introduction of the HR model: 1) the HRs’ mean flow mass flux is assumed to be substantially lower than that of the annular duct, with the modest contribution taken into account when calculating the uniform mean flow characteristics in succeeding annular sections; 2) the HRs are assumed to be perpendicular to the duct, resulting in radial fluctuating neck flows that only contribute unstable mass and energy fluxes — they do not contribute longitudinal or circumferential momentum fluxes [7].

3. RESAULT AND DISCUSSION

We discussed two di ff erent geometric cases. The first case considers a thin annular plenum linked to a thin annular combustor with the same mean radius via one or more identical burners. This is referred to as PBC. We then replace the burners with an annular duct with the same length and total cross-sectional area to connect the plenum and chamber, and fix the other parameters, this is referred to as PAC. The mean plenum / combustor radius is ¯ R = 0 . 175 m , the plenum has length 0 . 07 m and cross-sectional area 0 . 077 m 2 and the combustor has physical length 0 . 4 m and cross-sectional area

0 . 055 m 2 . Each of the burners has length 0 . 019 m and their total cross-sectional area 0 . 013685 m 2 . At the input, the mean pressure, temperature, and velocity are 10 5 Pa , 300 K , and 0 . 094 ms , while the mean temperature following the flame is 1500 K . The plenum inlet is assumed to be acoustically closed, whereas the combustor output is assumed to be acoustically open.

3.1. The E ff ect of Burner Numbers on Thermoacoustic Unstable Modes

We firstly consider a linear flame model, which is a simple n − τ mode, T L ( ω ) = n L f e i ωτ L f with n L f = 1 and τ L f = 1 ms . For the PBC combustor geometry with 16 burners, as shown in Figure 2(a), the predicted thermoacoustic modes below 700 Hz include two unstable modes: (506 Hz , 980 s − 1 ), (603 Hz , 874 s − 1 ).

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Figure 2: Contour plot for the two di ff erent geometric models

The e ff ect of the burners on the unstable mode of the system under a linear flame was studied. For the PAC combustor geometry, as shown in Figure 2(b), the predicted thermoacoustic modes below 700 Hz include two unstable modes: (506 Hz , 980 s − 1 ), (604 Hz , 869 s − 1 ). It is clear that compared to the PAC case, considering 16 burners does not change the first mode that propagates only along the axial direction, but increases the growth rate of the second mode. Under the PAC case, the axial position of the linear flame coincides with the position of the flame in the PBC structure, where the flame is added as a continuous flame throughout the system. But under the PBC structure, the flame exists the outlet of each burner; the flame at this case is added to the entire system as a discrete flame model and this is one of the reasons for the di ff erence. Ensuring that the total cross-sectional area remains unchanged, and constantly increasing the number of the burner, the e ff ect of discrete flame tends to be the same as that of continuous flame. With the increase of the number of premixed ducts, their distance in the circumferential direction decreases, and the flame continuity in the circumferential direction increases. As shown in Figure 3, the frequency and growth rate of the second unstable thermoacoustic mode under PBC structure gradually approach the solution under PAC structure as the number increases, but even after the converged result is reached, the growth rate under the two structures still has a di ff erence of 3 s − 1 , and the frequency also sees an about 0 . 5 Hz di ff erence. The reason for this di ff erence is that when acoustic perturbations propagate in a thin annular duct, it is assumed to have both longitudinal and circumferential variations. However, only planar-wave acoustic perturbations in a premix duct can be sustained at low frequencies. As shown in Figure 4, if the length of the burner decreases continuesly to zero, the di ff erence between PAC and the PBC with

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(b) Growth rate change trend

Figure 3: The second mode changes with the increase in the number of burners

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Figure 4: The second mode changes with the dcrease in the length of burners

160 burners gradually disappear.

3.2. The Impact of Adding a Single HR Considering a Linear Flame Model We now add an HR model to the PBC model with 16 burners. The HR have the neck length l h = 0 . 03 m , neck cross-sectional area S h = 3 . 5 e − 3 m 2 , mean bias flow Mach number M h = 8 . 2 e − 3 , mean cavity temperature T h = 1000 K , cavity volume V h = 1 . 3 e − 3 m 3 , circumferential position θ = 0, and longitudinal locati ons x h = 0 . 289 m – all of these parameters are fixed. From the well-known expression f re f = c √ S n / ( Vl ) / 2 π , it is possible to find the frequency of the resonator is about 550 Hz . As discussed in Ref. [18], with only a single HR attached to the annular duct, the HR will not dampen one of the non-degenerate modes that result – the clockwise and anti-clockwise modal components become coupled to set up a standing wave with a pressure node at the HR location. For this case, as shown in figure 5, the original two unstable modes become four modes: (500 Hz , − 6 s − 1 ),

(510 Hz , 955 s − 1 ), (603 Hz , 874 s − 1 ), and (605 Hz , 856 s − 1 ).

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Figure 5: Count plot of the model with HR

In thi s case, we firstly consider the mode wh i ch was a t (5 06 Hz , 980 s − 1 ). For this mode, placi n g a single H elmholtz resonator on th e duct (say a t θ = 0) ha s a slight damping e ff ect, moving it t o (510 Hz , 9 55 s − 1 ). The growth rate of this unsta b le mode has only dropped by 25 s − 1 , but the mo d e nature h as changed. As shown by Figure 6, after a d ding a H e l m holtz resonator, the purely longitudin a l mode be comes a slanted mode which propagates both lon gitudinally and circumferentially.

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(a) n = 0 component of LONMH

(b) n = ± 1 components of LONMH

Figure 6: Axial variation of the pressure mode shape for the mode at 510 Hz

For the mode at 510 Hz , the pressure amplitude of the n = ± 1 components are much less than the pressure amplitude of the n = 0 component. The pressure amplitude of each component and the resonator parameters and their damping e ff ects need to be further explored.

3.3. The Impact of Adding a Single HR Considering a Nonlinear Flame Model In this work, we use an infinitely small Helmholtz resonator to explore the e ff ect of adding a single resonator on nonlinear unstable mode. Adjust the neck area of the resonator to 3 . 5 e − 6 m 2 , provided that the frequency of the resonator is 550 Hz , then calculate the e ff ect on the thermoacoustic mode with this resonator. We found that with the linear flame model, the e ff ect of the resonator is intuitively invisible. The original two unstable modes (506 Hz , 980 s − 1 ) and (603 Hz , 874 s − 1 ) are not damped. However, besause of symmetry breaking induced by the resonator, azimuthal modes have become non-degenerate. Because the damping e ff ect is too small, the two non-degenerate modes cannot be distinguished from each other.

With a nonlinear flame model, by tracking the evolusion of the mode from unstable to limit cycle, we can simultaneously solve for frequency and modal amplitudes, thus distinguishing the two non- degenerate modes. The nonlinear model has the form [8],

T N ω, | u ′ / ¯ u |
= L n L f e i ω  τ L f + τ N (1 −L ) f  (2)

where L is a function of | u ′ / ¯ u | with the relation

= Z | u ′ / ¯ u |

L | u ′ / ¯ u
  u ′

1 1 + ( x + α ) β dx (3)

¯ u

0

τ N f = 0 . 1 ms , α = 0 . 9 and β = 40 are used. For low-order network models that do not include resonators, the (603 Hz , 874 s − 1 ) mode is degenerate, so the n = ± 1 components can exist independently and the mode shape in the circumferential direction is undetermined – it can be spinning, standing, or mixed [19–21]. With the nonlinear flame model, the essential e ff ect of the resonator on the eliminating the mode degeneracy can be found from the di ff erence in the limit cycles. With the nonlinear flame mode, the (603 Hz , 874 s − 1 ) mode can be either spinning or standing with the latter’s pressure node at any burner position. As shown in Figure 7, after adding a resonator, there are two standing modes which

0 175 350 524 699 874 0.00 157.16 314.32 471.48 628.64 785.80

0 175 350 524 699 874 0.00 157.16 314.32 471.48 628.64 785.80

0 175 350 524 699 874 565.92 573.27 580.63 587.98 595.33 602.69

0 175 350 524 699 874 565.92 573.27 580.63 587.98 595.33 602.69

1 2 3 4 5 6 7 8 9 10111213141516 0.00 191.85 383.70 575.55 767.39 959.24

1 2 3 4 5 6 7 8 9 10111213141516 0.00 191.91 383.83 575.74 767.66 959.57

1 2 3 4 5 6 7 8 9 10111213141516 0.00 7.91 15.82 23.73 31.64 39.55

1 2 3 4 5 6 7 8 9 10111213141516 0.00 7.91 15.82 23.74 31.65 39.56

(a) Standing mode1

(b) Standing mode2

Figure 7: Pressure and velocity perturbationsat each of the 16 burner outlets and evolution of two di ff erent standing modes.

are gradually stabilized towards limit cycles. For mode1, the pressure and velocity nodes are at the 5th and 13th burners respectively, while their anti-nodes are at the 1st and 9th burners. For mode2, the pressure and velocity nodes are at the 1th and 9th burners, and their anti-nodes are at the 5st and

13th burners. It should be noted that the 1st burner is at θ = 0, corresponding to the angular position of the HR. The addition of the resonator gives two fixed standing limit cycle solutions with di ff erent node / anti-node distributions.

4. CONCLUSIONS

The present work uses a low-order network modeling tool to study the damping performance of HRs on nonlinear and linear thermoacoustic modes in annular combustors. This model could be used when designing LPP combustors with HRs to damp combustion instabilities and to investigate the e ff ect of varying parameters such as combustor temperature or duct lengths. The model has been used to investigate the e ff ect of burner numbers on linearly unstable thermoacoustic modes. It shows that compared to the case using an annulus to replace the burners, the number of burners and planar wave propagation within the buners could both slightly change the mode frequency and growth rate. Later, the e ff ect of adding a resonator considering a linear flame model on the unstable thermoacoustic model was analyzed. Besides being damped by the resonator, the original pure longitudinal mode becomes a slanted mode. Finally, the e ff ect of an infinitesimal HR on the nonlinear thermoacoustic modes with nonlinear flame model is studied. Two fixed standing limit cycle solutions with fixed node / anti-node distributions were identified, with the circumferential position of the resonator at the node of one standing mode and the anti-node of the other standing mode. For other nonlinear thermoacoustic modes, damping e ff ect of HRs is under further study.

ACKNOWLEDGEMENTS

The authors would like to thank the National Natural Science Foundation of China (project No. 52106159) for supporting this research.

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