A 2D low-order thermoacoustic network model of annular combustor with ba ffl es in the plenum

Liang Ji School of Astronautics, Beihang University BeiJing 102206 China

Yuanqi Fang School of Aeronautics and Astronautics, Zhejiang University Zhejiang 310027 China

Jingxuan Li 1

School of Astronautics, Beihang University BeiJing 102206 China Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University Ningbo 315100 China

Gaofeng Wang School of Aeronautics and Astronautics, Zhejiang University Zhejiang 310027 China

Lijun Yang School of Astronautics, Beihang University BeiJing 102206 China Aircraft and Propulsion Laboratory, Ningbo Institute of Technology, Beihang University Ningbo 315100 China

ABSTRACT Experimental results showed that configuring di ff erent ba ffl es (a regular ba ffl e, a single- or double- layer perforated plate and a Helmholtz resonator) in the plenum has an impact on the self-excited oscillation in annular combustor. In this paper, a 2D low-order network model is proposed to describe this phenomenon and predict the modal frequencies and growth rates of longitudinal and circumferential thermoacoustic modes for annular combustor with di ff erent ba ffl es. In the plenum, only the circumferential acoustic propagation and the influence of ba ffl es on it are accounted for as the longitudinal geometry size is much smaller than the acoustic wavelength of the longitudinal mode. Pure longitudinal acoustic propagation is considered in burners and both longitudinal and circumferential ones are accounted for in the combustion chamber. These three modules are connected by jump conditions. Compared with the experimental results, Ba ffl es using Helmholtz

1 jingxuanli@buaa.edu.cn

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

resonators feature the best damping performance for specific modes. The circumferential mode is converted to two unfolded non-degenerate modes when configuring perforated plates ba ffl es. And these two modes are orthogonal to each other.

1. INTRODUCTION

In order to achieve low NOx emission, the aero-engine with lean premixed combustion is prone to lead combustion instability, which impact is often catastrophic and making the design work challenging. How to control and predict the instability modes in the engine has been the focus of research. For finding stable operating condition, the full-scale numerical calculation or experiment of the engine is expensive, which is unacceptable for design. The low-order model based on solving linearized conservation equations can well describe the acoustic characteristics, combining linear or nonlinear flame model [1] [2] [3] [4]and di ff erent boundary condition which makes the stability prediction of the engine based on a small calculation possible. The low-order model for annular combustor has been widely studied [2] [3] [4] [5] [6] [7] [4] [8]. Some studies [2] [4]conducted 2D annular combustor models which considered the axial and circumferential propagation and neglect the radius ones, besides some conducted 3D models [9]. However these models can only be applied to modes with circumferential symmetry. Bauerheim et al. [5] [6] [7] simplifies the annular combustor by focusing on the circumferential wave in plenum and chamber and neglecting the axial ones which could capture the e ff ect of the circumferential mean flow and the coupling modes for plenum and chamber. Similarly, in our low-order model only circumferential modes in the plenum are discussed. In addition, in order to suppress combustion instability, several passive control schemes are often used including perforated plates, Helmholtz resonators and ba ffl es. And the perforated plates and HRs could absorb acoustic energy to suppress the combustion instability Perforated plates usually have a broadband acoustic absorption, and ones with bias flow is better than ones without bias flow [10]. Helmholtz resonator is di ff erent from perforated plates which structure leads to good sound absorption e ff ect only for specific frequency [11]. Ba ffl es could truncate the acoustic propagation, which is more common in rocket engines, for instance, F-1 rocket engine is equipped with ba ffl es to damp the combustion instability [12]. But in practical engineering, the di ff erent parameters and locations of these devise cause big changes in control e ff ect. Serval research established the perforated plates [13] [14] [15], HRs [16] and ba ffl es [17] models providing important help for the design work. The perforated model [18] [19] [15] based on the orifices’ impedance considering the influence between holes could easily couple with low-order network model. Besides Dupere et al. [16] established a HRs model which gives the nonlinear impendence for HRs without bias flow and well describes the acoustic absorption e ff ect. Acoustically, Ba ffl es can be considered as a solid boundary condition. Wicker and Wicker et al. [17] partitioned the combustion chamber and established ba ffl e model. The pressure and heat release rate disturbances caused by combustion instability are in the combustion chamber where devices often installed, and numerous works have described their e ff ect of damping to the combustion system. However, in the experiment, we found that there is a circumferential mode in the annular plenum which also a ff ect the combustion. In this study, we try to conduct a model which absorption devices are installed in the plenum analysis the e ff ect for this mode. The structure of the article is as follows. A model for annular combustor based on low-order network is established in Sec II. In this model, only plenum and burners are considered, and di ff erent ba ffl es installed in radial direction coupling with models. The modes changing after ba ffl es installed are discussed in Sec III. Di ff erent locations of ba ffl es are the focus of our attention. In Sec IV, the conclusions are made on the key findings of the present study.

2. MODEL

In this model, ignoring the influence of entropy and vortex waves, the annular combustor model with ba ffl es in plenum is established as shown in Figure 1. In the model, the structure of the valve chamber is simplified as a ring, while the structure of buners is simplified as a cylinder. Figure 1 shows the annular combustor model expanded circumferentially at the average radius R m . Because the main research is the circumferential mode in plenum, the structure of combustion chamber is ignored in the model.In addition, due to its structural shape, the plenum is considered to be a narrow ring, only the circumferential propagation A + and A − in two directions of sound waves exists, and the radial and axial ones are ignored. Besides, in the model, M burners are arranged at equal intervals along the axial direction. Only acoustic wave C + and C − propagating along the axial direction are considered. The outlet boundary of the burners is open. There is a flame near the outlet at L B 1 , and away from the inlet of the burners at L B 2 .Ba ffl ed with di ff erent structures (HRs, perforated plates and regular flat) are installed between two burners and arranged along the radial direction. The circumferential cross area of plenum and axial cross area of burner is S p and S b respectively.

Figure 1: Annular combustor with di ff erent ba ffl es in the plenum.

2.1. Acoustic wave in plenum and burners Based on linearization, the waves in the plenum and burners should be satisfied with the wave equation,

1 c 2 D 2

D t 2 p ′ −∇ 2 p ′ = 0 , (1)

Where D / D t = ∂/∂ t + ¯u · ∇ , and[¯] and[ ] ′ represents mean and perturbation value.The M burners and N ba ffl es divides the plenum into the M + N sections. The mean flow Mach number in the plenum is very small which is neglected. In each section of plenum, pressure and velocity perturbation is,

p ′ A = ˆ p A ( l ) e i ω t , u ′ A = ˆ u A ( l ) e i ω t (2)

ˆ p A ( l ) = A + e − ikl + A − e ikl (3)

ˆ u A ( l ) =  A + e − ikl − A − e − ikl  /ρ c (4)

Where, wave number k = ω/ c , l = R m θ . Base on equation above,the transition matrix T T can be obtained,

− 1  e − ikl i 0

 (5)

 1 1





 1 1

 , T t =

 ˆ p i ˆ u i

 = T t

 ˆ p i + 1 ˆ u i + 1

1 /ρ c − 1 /ρ c

1 /ρ c − 1 /ρ c

0 e ikl i

The same method could obtain the acoustic wave in burners.

ˆ p C ( x ) = C + e − ikx + C − e ikx (6)

ˆ u C ( x ) =  C + e − ikx − C − e − ikx  /ρ c (7)

In the model, the flame is assumed to be infinitely thin, and there is a step of acoustic wave passing through the flame as is shown in Figure 1 (e). According to the conservation equation of energy and mass, the step condition can be written as follows,

ˆ p m , c 2 = ˆ p m , c 3 (8)

γ ¯ pS b γ − 1 ˆ u m , c 2 + ˆ Q m = γ ¯ pS b

γ − 1 ˆ u m , c 3 (9)

The relationship between heat release rate perturbation ˆ Q m and velocity perturbation ˆ u m , c 2 can be linked by flame transfer function. In order to simplify the model, the classical linear n − τ model is used in this paper,

ˆ Q m ¯ Q m = n e i ωτ ˆ u m , c 2

¯ u m , c 2 + = γ ¯ pS b

γ − 1 ˆ u m , c 3 (10)

In addition, the boundary condition of burners is opening,

ˆ p C ( L B 1 + L B 2 ) = 0 (11)

When acoustic wave cross m th burners as Figure 1(d) shown, the momentum and mass conservation equations should be satisfied [7], ˆ p m = ˆ p m + 1 / 2 (12)

ˆ u m = ˆ u m + 1 / 2 + θ ˆ u m , b (13)

where, θ = S b / S p . Combining Eqs.(2-14), we could get the transformation matrix T m between just before and after the burners,  ˆ p m + 1 ˆ u m + 1

 = T m

 ˆ p m ˆ u m

 (14)

2.2. Acoustic wave impact Ba ffl es Helmholtz resonators without bias flow mainly rely on the viscosity of the orifice and the resonance e ff ect of the back cavity to dissipate acoustic energy. When the ba ffl e is a HR, as shown in Figure 1 (a), according to the HR model established by Dowling [16], the impedance boundary condition at the HR orifice side can be written as the following expression,

ˆ u h , 1 = β S p

− α ¯ ρ  ˆ u h , 1  − i ωρ l e f f − c 2 ρ S n

! (15)

Z h = ˆ p h , 1

i ω V h

d 2 h

Where beta is the area correction coe ffi cient. In this paper, β = 0 . 8. d is the diameter of the HR hole, α is the flow coe ffi cient of the orifice , L ef f = L h + 0 . 68 d is the corrected length of the hole and V h is the volum of the resonant cavity. At wall side the boundary condition is,

ˆ u h , 2 = 0 (16)

When the ba ffl es is a regular flat, as shown in Fig. 1 (b), the boundary condition of circumferential acoustic propagation can be easily obtained,

ˆ u b , 1 = 0 , ˆ u b , 2 = 0 (17)

The acoustic characteristics of the perforated plate without bias flow can be described by the impedance of the perforated plate, that is, the ratio of the di ff erence between the acoustic pressure perturbation on both sides of the perforated plate and the velocity perturbation of the perforated plate. In the experiment, a double-layer perforated plate is adopted. Therefore in the model, we also connect two single-sided perforated plates in series. The acoustic impedance for perforated plates [15] is written as

ˆ u p , 1 i ω c σ C D t p + 2 δ f int

Z p = ˆ p p , 2 − ˆ p p , 1

Ψ v ,

3 π (18)

Ψ v = J 2 ( k s d p / 2 ) J 0 ( k s d p / 2 ) , k s = q

− i ωρ

µ , δ = 4 d p

Where, t p is the thickness of single-layer perforated plate, d is the hole size, σ = π d 2 / 4 b 2 is porosity of the plate and b is the distance between the holes.Therefore, the transfer matrix T p of acoustic wave passing through perforated plate is  ˆ p p , 2 ˆ u p , 2

 = T p

 ˆ p p , 1 ˆ u p , 1

 , T p =

 1 − 2 Z P 0 1

 (19)

2.3. Eigenvalue system Firstly, the plenum with n regular ba ffl es is divided into n spaces, and each region has corresponding boundary conditions. Combined with the transfer matrix obtained in the above section, the eigenvalue system can be obtained,

M n ( ω ) X = 0 (20)

the elements of the matrix are shown in appendix A. The eigenvalue of system can be obtained by solving, Y

n det( M n ) = 0 (21)

Secondly, for nonlinear system with HRs, the plenum is also divided into many regions. Di ff erent from the regular ba ffl es, the Eq (21) can be solved only after the pressure amplitude in plenum is given. According to the amplitude measured by the experiment is about 30Pa, we set the pressure perturbation amplitude on the wall side of HRs ˆ p h , 2 = 30Pa. When no ba ffl e is installed or perforated plates are installed in the plenum, according to the transfer matrix, the transfer matrix of the sound wave propagating along the circumferential direction for one cycle can be written as, M z = Y T t T M . . . T p . . . T M (22)

The equation of eigenvalue system can be transformed into,

det ( M z − I ) = 0 (23)

The obtained mode, ω = 2 pif + i ω i ,where ω i is mode growth rate. ω i > 0 represents stable.

3. RESULTS

According to the experiment, the temperature before and after the flame is T b and T a = 1500K respectively. The label 1B represents there is one ba ffl e in the plenum. When two ba ffl es are installed, we define the labels according to the numbers of burners between two ba ffl es. For example, label 2B-2 represents the angel between the two ba ffl es is π/ 4, and there are two burners between them. The labels for three or four ba ffl es are shown in Figure 2. In this paper, the selection of parameters is shown in Table 1.

Table 1: Paramerters of plenum, burners and ba ffl es Parameters Values Parameters Values Parameters Values

S p 3000mm 2 t p 2.5mm d h 3mm

S b 18.55mm 2 d p 2mm L h 2.5mm

R m 125mm σ 12.6% V h 16.53mm 3

Figure 2: The modes after installing ba ffl es.(a) is regular ba ffl es and (b) is HRs

Set flame upstream temperature T b = 410K (obtained by measure the plenum wall), the mode without ba ffl e can be calculated through the model established above, ω 0 = 2 π · 545 . 2Hz + 1 . 11s − 1 .

And solving the Eq(21), we persent the modes for the regular ba ffl es and HRs at di ff erent positions, as shown in Figure 2. For the regular ba ffl es, the splitting of modes can be observed in Figure 2(a). After the ba ffl es installed, the plenum is divided into di ff erent sections, and each section has its mode. However, the experiment result are fixed at 560Hz, which may be the circumferential mode in the combustion chamber propagating into the plenum through the burners. The growth rates of regular ba ffl es cases are beside 0, which represents the regular ba ffl es may cause the system instability. The HRs are designed with the resonance frequency f h = 500 − 600Hz. The HRs cases with the same phenomenon of modes splitting has the same reason. Compared with the regular ba ffl es the growth rates are much greater than 0 for some specific modes, which improves the stability of the system. The modes in plenum with perforated plates are shown in Figure 3. After the perforated plate installed , the circumferential mode is converted to two unfolded modes, which is di ff erent from the splitting in the regular ba ffl es and HRs cases. The perforated plates maintained the connectivity of circumferential space in the plenum and the circumferential propagation of acoustic waves. This phenomenon of modes division is also observed in the experiment. Among the cases with two perforated plates, the two modes gradually become similar and then separate with the enlargement of the angle between the two ba ffl es. The cases with di ff erent T b are also calculated. Compare with Figure(a) and (b), as the temperature rise, modes frequencies increase because of the sound speed changes and the growth rates increase slightly.

Figure 3: The modes after installing the perforated plates.(a) T b = 410K and (b) T b = 500K

oo ee

In order to further study the phenomenon of mode division, we plot the pressure perturbation amplitude along the circumferential direction, as shown in Figure 4. There are two maximum points of the two pressure perturbation amplitudes, and the two modes are of the same order. When the acoustic wave passes through the perforated plate, its pressure disturbance usually takes a step. At the same time, the introduction of perforated plates causes the asymmetry of acoustic pressure perturbation. In addition, we found that the distribution of two modes in the plenum is almost orthogonal to each other, which is consistent with the phenomenon observed in the experiment.

Figure 4: The amplitude of pressure perturbation in the plenum with di ff erent perforated plates locations

4. CONCLUSIONS

In this paper, a low-order network model for annular combustor with di ff erent ba ffl es in plenum are conducted which only circumferential wave in plenum and axial ones in burners are considered. When install regular ba ffl es or HRs, the plenum are divided into sections and each section has its mode. And this model have some defect which the e ff ects of chamber are considered. Perforated plates in the plenum convert the circumferential mode into two unfolded non-degenerate modes which are orthogonal to each other consistent with the experiment. The perforated plate brings asymmetry to the plenum.

APPENDIX A

The eigenvalue system for installing regular ba ffl es or HRs in plenum,









ˆ p b , h , 2 , n ˆ u b , h , 2 , n ˆ p b , h , 1 , n + 1 ˆ u b , h , 1 , n + 1

M b 1 0

0 1

M n ( ω ) =

0 1

0 0

0 0

0 1

where, M b = Y T t T M . . . T M T t

ACKNOWLEDGEMENTS

The authors would like to gratefully acknowledge financial support from the Chinese National Natural Science Funds for National Natural Science Foundation of China (Grant Nos. 11927802 and U1837211)

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