Prediction of combustion instabilities based on the Green’s function in a three-dimensional annular combustor with multiple flame interac- tion Lei Qin 1 Research Institute of Aero-Engine, Beihang University 100191, Beijing, China Xiaoyu Wang 2 Research Institute of Aero-Engine, Beihang University 100191, Beijing, China Guangyu Zhang 3 Research Institute of Aero-Engine, Beihang University 100191, Beijing, China Xiaofeng Sun 4 School of Energy and Power Engineering, Beihang University 100191, Beijing, China

ABSTRACT The prediction of combustion instabilities is essential for modern annular combustors fed by multiple flames. A model based on the Green’s function is established in a three-dimensional (3-D) geometry with interactions of different flame responses. The flames are described by the flame trans- fer function (FTF), which are considered as the multiple sources to produce the pressure in the ple- num and chamber. With equality of the pressure at the neighboring sectors in the combustor, the eigenfunction is obtained in frequency domain. The relevant eigenvalue is used to give the unstable frequency and judge the stability behavior of combustion system. The effects of non-identical flame responses on complicated slanted mode under the condition of asymmetric are studied with the pre- sent model. In addition, the combustion instabilities of radial mode can also be predicted by this model when the radial position distributions of the heat sources move in the case of small hub ratio.

1. INTRODUCTION

Combustion instability is a thorny problem that receives much concern in modern gas turbine combustors. When the phase between unsteady heat release and pressure is suitable, energy will be transferred from the flame to the pressure resulting in the onset of combustion instability [1]. In fact,

1 LeiQin@buaa.edu.cn

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great thermal and mechanical stresses will even cause the engine damage once the combustion insta- bility occurs. Therefore, it is still a challenge to predict the combustion instability more accurately and to prevent its onset by taking more effective measures.

As a physical object we often encounter, plenum is connected to combustion chamber through multiple burners in modern gas turbine annular combustors. Due to the effect of the flow, combustion condition and other factors, the flame responses could be different in the burners [2]. Therefore, ef- fects of the interaction of multiple different flame responses on combustion instability were the focus of many studies [2-5]. Moreover, there are multiple nozzles in the can combustor in industry [6, 7], which also lead to the existence of the interaction of multiple flame responses. Acharya and Liuwen [6] found that combustion instability of the high-frequency transverse mode could be prevented by reasonably rearranging the position of the nozzles. This suggests that the distribution of different flames could be used as s passive tool in addition to the perforated liner [8, 9]. Consequently, it can be expected that the effect of the interference between the multiple different flames will affect the prediction and the control strategy of combustion instability in can combustors.

Many theoretical or modelling investigations have been conducted to study the interaction of the multiple different flame responses. Parmentier [3] and Bauerheim [4] established the annular network reduction method using a network discretization along the azimuthal direction. It is found that the distribution of different flame responses along the azimuthal direction leads to azimuthal asymmetry. As a result, the prediction of the combustion instability is shifted and the onset of the combustion instability can be suppressed by appropriately modulating flame responses. Evesque and Polifke [5] expressed the mode explicitly as a function of the azimuthal angle under in a two-dimensional model. This low- order acoustic modelling could be used to analyze the modal coupling with the considera- tion of the azimuthal asymmetry. The study found that without the heat release, the amplitude of the coupled eigenmodes of an annular combustor is changed due to the variation of the cross-section area of the burners. Moreover, Beaulieu and Noiray [10] developed the slow-flow dynamics on the Bloch sphere, and studied both rotational and reflectional explicit symmetry-breaking bifurcations of the azimuthal modes.

It is shown that the above works put all emphasis on the combustion instabilities of the azimuthal modes [11, 12]. The reason is that the length along the azimuthal direction is much longer than that along the axial direction, which makes the combustion instability of the azimuthal modes is easier to occur in the modern annular gas turbine annular combustors. However, it is still necessary to develop the different model including more factors in order to better to predict and control the combustion instabilities with the effect of multiple different flame responses. For instance, the annular network reduction method [3, 4] is one-dimensional, which makes that the boundary condition cannot be con- tained. Meanwhile, in addition to the azimuthal modes, the combustion instability of other modes, such as slanted modes of azimuthal mode involving an axial dependence cannot be predicted by the use of such model. The low order acoustic modelling is to use the conservation conditions at the position of the burners, as a result, the number of the modes considered will be limited by the number of the burners. Moreover, it is also difficult to apply the one-dimensional and two-dimensional mod- els to predict the combustion instability involving three-dimensional geometry as a key factor, such as can combustors. Therefore, an analytical model is developed in this paper to study the combustion instability with the interaction of multiple different flame responses in a 3-D geometry based on the Green’s function.

The advantages of using Green’s function to study the interaction of multiple different flames is that the position and the strength of the flame can be contained arbitrarily, as well as the coupling of

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multiple modes is involved naturally. In this way, it provides us with a clearer physical understanding for the problem.

This paper is structured as follows. In Section 2, a brief derivation of this model is given for two configurations. One configuration is that the combustor consists of an annular plenum connected to an annular chamber through multiple burners, and the other is that an annular chamber fed by burners. The combustion instabilities of azimuthal mode and slanted mode affected by multiple different flame responses are studied, and analysis is carried out in Section 3. The effects of the radial position of heat sources and different flame responses in the can combustor are discussed in Section 4. Finally, the conclusion is given in Section 5.

2. THE MODEL

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There are two configurations studied as shown in Fig.1. They represent the PBC configuration where the plenum and the chamber are related via the burners, and the BC configuration where the chamber is fed by multiple burners. These two models are established based on the different geome- try, which are described separately in the following sections.

(a) (b)

Fig. 1. (Colour online) Schematic of combustors. (a) PBC Configuration; (b) BC Configuration.

2.1. The unsteady heat release In order to contain the effect of the unsteady heat release, we assume that the combustion takes place completely in the burners. Due to the thin geometry of the burners, only the planar waves the propagate inside that. Therefore, the pressure 𝑝ư 1 and velocity 𝑢ư 1 at the upstream of the flame can be written as

𝑝ư 1 = 𝑒 𝑖𝜔𝑡 ൫𝑝Ƹ 1𝑑 𝑒 −𝑖𝜔𝑥𝑐ҧ 1 Τ + 𝑝Ƹ 1𝑢 𝑒 𝑖𝜔𝑥𝑐ҧ 1 Τ ൯ , (1)

𝜌ഥ 1 𝑐ҧ 1 𝑒 𝑖𝜔𝑡 ൫𝑝Ƹ 1𝑑 𝑒 −𝑖𝜔𝑥𝑐ҧ 1 Τ −𝑝Ƹ 1𝑢 𝑒 𝑖𝜔𝑥𝑐ҧ 1 Τ ൯ , (2)

1

𝑢ư 1 =

Similarly, the pressure 𝑝ư 2 and the velocity 𝑢ư 2 at the downstream of flame can be described in the same form. The jump conditions through the flame are imposed on the assumption of the low Mach number and the compact flame, which leads to the equality of pressure and flow rate discontinuity related to unsteady heat release

𝑝ư 1 ȁ 𝑥=𝑏 = 𝑝ư 2 ȁ 𝑥=𝑏 , (3)

𝛾−1

𝑢ư 1 ȁ 𝑥=𝑏 −𝑢ư 2 ȁ 𝑥=𝑏 = −

𝜌ഥ 1 𝑐ҧ 1 2 𝑞ư , (4)

where 𝑏 is the axial position of the flame and 𝛾 the ratio of specific heats. The unsteady heat release is given by the FTF [13]

𝛾−1

𝜌ഥ 1 𝑐ҧ 1 2 𝑞ư = 𝑘 𝑓 𝑢ư 1 ሺ𝑡−𝜏ሻ , (5)

where the interaction index 𝑘 𝑓 and the time delay 𝜏 are the data to describe the unsteady heat release.

2.2. The model for PBC configuration

The typical combustor in gas turbine is composed of an annular plenum related to an annular chamber via burners. The effect of the unsteady heat release has been given in the last section. With all the considerations mentioned above, the velocity at the inlet and the outlet of the burners can be illustrated as a function of the pressure at the inlet and outlet of the burners

𝑢ư 1,𝑗 ห 𝑥=0 = 𝛼 𝑗 𝑝ư 1,𝑗 ห 𝑥=0 + 𝛽 𝑗 𝑝ư 2,𝑗 ห 𝑥=𝑙 𝑗 , (6)

𝑢ư 2,𝑗 ห 𝑥=𝑙 𝑗 = 𝜀 𝑗 𝑝ư 1,𝑗 ห 𝑥=0 + 𝜂 𝑗 𝑝ư 2,𝑗 ห 𝑥=𝑙 𝑗 , (7)

where 𝑙 𝑗 means the length of the 𝑗 𝑡ℎ burner. Moreover, the flames are introduced as the monopoles at the cross-section area of the burners connecting plenum to combustion chamber which leads to the pressure disturbance written as

𝜕𝐺൫𝑟Ԧ,𝑟Ԧ ሶ 𝑗 ,𝑡,𝑡 ሶ 𝑗 ൯

𝜕𝑡 ሶ 𝑗 𝑑𝑟Ԧ ሶ 𝑗 𝑑𝑡 ሶ 𝑗 , (8)

𝑝ư 𝑝 ሺ𝑟Ԧ, 𝑡ሻ= σ ׭ 𝜌ҧ 𝑝 𝑁 𝑗=1 𝑢ư 1,𝑗

𝜕𝐺൫𝑟Ԧ,𝑟Ԧ ሶ 𝑗 ,𝑡,𝑡 ሶ 𝑗 ൯

𝜕𝑡 ሶ 𝑗 𝑑𝑟Ԧ ሶ 𝑗 𝑑𝑡 ሶ 𝑗 , (9)

𝑝ư 𝑐 ሺ𝑟Ԧ, 𝑡ሻ= −σ ׭ 𝜌ҧ 𝑐 𝑁 𝑗=1 𝑢ư 2,𝑗

where 𝐺൫𝑟Ԧ, 𝑟Ԧ ሶ , 𝑡, 𝑡 ሶ ൯ is the Green’s function for a 3-D hard wall annular geometry , with the description in time domain

𝜕𝐺൫𝑟Ԧ,𝑟Ԧ ሶ ,𝑡,𝑡 ሶ ൯

𝜕𝑡 = −𝑐ҧ 2 σ σ σ 𝛩 𝑖,𝑗 ൫𝑒 𝑖𝜔 𝑚𝑛ℎ ሺ𝑡−𝑡ሶሻ + 𝑒 −𝑖𝜔 𝑚𝑛ℎ ሺ𝑡−𝑡ሶሻ ൯ ∞ ℎ=0 ∞ 𝑛=0 ∞ 𝑚=−∞ , (10)

where

𝜓 𝑚𝑛 ሺ𝜇 𝑚𝑛 𝑟ሻ𝑒 𝑖𝑚𝛳 cos ቀ ℎ𝜋

𝑙 𝑥ቁ𝜓 𝑚𝑛 ሺ𝜇 𝑚𝑛 𝑟ሶሻ𝑒 −𝑖𝑚𝛳ሶ cos ቀ ℎ𝜋

𝑙 𝑥ሶቁ

𝛩 𝑖,𝑗 =

2𝛤 𝑚𝑛ℎ , (5)

where 𝑁 is the number of the burners. The azimuthal, radial and axial mode number is denoted as 𝑚, 𝑛 and ℎ , respectively. 𝜓 𝑚𝑛 and 𝜇 𝑚𝑛 are the eigenfunction and eigenvalue of the annular cavity under the consideration of the hard wall, respectively. 𝜔 𝑚𝑛ℎ is the allowed frequency and 𝛤 𝑚𝑛ℎ the volume integral of products of the eigenfunctions. Therefore, the continuity of the pressure is satisfied at the interface of the neighbouring sectors

𝑝ư 1,𝑗 ห 𝑥=0 = 𝑝ư 𝑝,𝑗 ห 𝑥=0 , (12)

𝑝ư 2,𝑗 ห 𝑥=𝑙 𝑗 = 𝑝ư 𝑐,𝑗 ห 𝑥=𝑙 𝑗 , (13)

In order to obtain the complex thermoacoustic frequencies, we perform Laplace transform 𝑃ሺ𝑠ሻ= 𝐿ሾ𝑝ưሺ𝑡ሻሿ on the both sides of Eq. (12) and (13), with 𝑠= 𝑖𝜔 . Accordingly, the pressure in the frequency domain can be written in the form of

2 σ σ σ 𝛩 𝑖,𝑗 ൬

2𝑖𝜔

2 −𝜔 2 ൰ ∞ ℎ=0 ∞ 𝑛=0 ∞ 𝑚=−∞ 𝑁 𝑗=0 ቀ𝛼 𝑗 𝑃 𝑝,𝑗 ሺ𝑠ሻ+ 𝛽 𝑗 𝑃 𝑝,𝑖 ሺ𝑠ሻቁ𝑠 𝑏,𝑗 , (14)

𝑃 𝑝,𝑖 ሺ𝑠ሻ= σ 𝜌ҧ 𝑝 𝑐ҧ 𝑝

𝜔 𝑚𝑛ℎ

2 σ σ σ 𝛩 𝑖,𝑗 ൬

2𝑖𝜔

2 −𝜔 2 ൰ ∞ ℎ=0 ∞ 𝑛=0 ∞ 𝑚=−∞ 𝑁 𝑗=0 ቀ𝜀 𝑗 𝑃 𝑝,𝑗 ሺ𝑠ሻ+ 𝜂 𝑗 𝑃 𝑝,𝑖 ሺ𝑠ሻቁ𝑠 𝑏,𝑗 , (15)

𝑃 𝑐,𝑖 ሺ𝑠ሻ= −σ 𝜌ҧ 𝑐 𝑐ҧ 𝑐

𝜔 𝑚𝑛ℎ

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Therefore, the pressure at the observation of the 𝑖 𝑡ℎ flame has been given as depicted at Eq. (14) and (15). Similarly, the pressure at any position of the plenum and chamber can be obtained in the same form. Then the eigenfunction combined plenum and chamber via the burners can be written as

൬ቂ 𝐀 𝐁 𝐗 𝐇 ቃ−ቂ𝐈 𝐝 𝟎 𝟎 𝟎 ቃ+ ൤𝟎 𝟎 𝟎 𝐈 𝐝 ൨൰൤𝐏 𝟏

𝐏 𝟐 ൨= 0 , (16)

𝐈 𝐝 is the 𝑁 -by- 𝑁 identity matrix. The vectors 𝐏 𝟏 and 𝐏 𝟐 denote the complex value of acoustic pressure in the plenum and the chamber, respectively. In addition, the submatrix 𝐀 , 𝐁 , 𝐗 and 𝐇 con- tain the coupling parameters relating the plenum to the chamber.

2.3. The model for BC configuration

It is worth noting that the configuration composed of burners and chamber is analysed by Parmentier et al. [3] to identify the effect of different flame responses on combustion instability of azimuthal mode. Under this assumption, naturally, the eigenfunction Eq. (16) described in the last section can be simplified to describe the configuration without containing the plenum. While the inlet boundary condition of the burner is needed, which can be described as the impedence condition

𝑧 𝑗 ሺ𝜔ሻ= 𝑝ư 1,𝑗 ห 𝑥=0 𝑢ư 1,𝑗 ห 𝑥=0 ൗ , (17)

Then we have

𝑧 𝑗 ሺ𝜔ሻ൫𝜌ഥ 1,𝑗 𝑐ҧ 1,𝑗 ൯ ൗ +1

𝑧 𝑗 ሺ𝜔ሻ൫𝜌ഥ 1,𝑗 𝑐ҧ 1,𝑗 ൯ ൗ +1 𝑝Ƹ 1𝑢,𝑗 = 𝑅 0,𝑗 𝑝Ƹ 1𝑢,𝑗 , (18)

𝑝Ƹ 1𝑑,𝑗 =

For instance, if a closed end is introduced at the inlet of the burners, the value of 𝑧 𝑗 ሺ𝜔ሻ is infinite which leads to 𝑅 0,𝑗 = 1 . Then the velocity can also be described by the pressure like Eq. (7) at the outlet of the burner

𝑢ư 2,𝑗 ห 𝑥=𝑙 𝑗 = 𝜆 𝑗 𝑝ư 2,𝑗 ห 𝑥=𝑙 𝑗 , (19)

As depicted in Eq. (19), the velocity is only related to the pressure in the combustion chamber for BC configuration, then the pressure generated by the burners can be written in the same form as that in Eq. (9). With the equality of the pressure at the outlet cross-section of the burners as Eq. (13), the pressure at the outlet of each burner for BC configuration can be given in time domain. Finally, the eigenfunction in frequency domain is obtained with the Laplace transform and described as

ሺ𝐓+ 𝐈 𝐝 ሻ𝐏= 0 , (20)

where the submatrix 𝐓 represents the coupling parameters, and 𝐈 𝐝 is the identity matrix. The dimen- sions of these two submatrices are 𝑁 -by- 𝑁 .

The eigenfunctions for the PBC configuration and BC configuration are established in the form of a matrix in Eq. (16) and Eq. (20). The complex thermoacoustic frequency can be obtained if and only if its determinant is null. The real part is the frequency predicted, and the imaginary part deter- mines the stability.

3. AZIMUTHAL MODE IN ANNULAR CHAMBER

The study in Ref [3] presented a passive control method by using different types of burners to break the symmetry of the system in order to damp azimuthal modes. In a 3-D combustor, there are other complex modes in addition to the azimuthal modes. For instance, the modes that are azimuthal but also involve an axial dependence, i.e., slanted modes. This section will investigate whether the slanted mode is stable with the different flame responses in burners when azimuthal modes have been

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controlled. The parameters are fixed as shown in Table 1, except for other statements. With the con- sideration of the effects of narrow tubes, the standard-length correction is described as 𝑙= 𝑙 0 + 2 × 0.4ඥ4𝑠 𝑏 𝜋 Τ , where 𝑙 0 means the length of burner without correction.

3 4 5

3 4 5

2

6

6

2

1

7

7

1

12

8

8

12

Type “1”

11

9

9

11

10

10

Type “2”

(a) (b)

Fig. 2. (Colour online) Distributions of flame responses. Configuration studied (a) in section 3.1 and (b) in section 3.2. Table 1: Parameters for discussion.

Plenum Chamber

Half perimeter 𝐿 1 6.59 m Half perimeter 𝐿 2 6.59 m

Section in circumference 𝑠 𝑝 0.6 m 2 Section in circumference 𝑠 𝑐 0.6 m 2

Unburnt gases Burnt gases

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Mean pressure 𝑝ҧ 1 2000000 pa Mean pressure 𝑝ҧ 2 2000000 pa

Mean temperature 𝑇 ത 1 700 K Mean temperature 𝑇 ത 2 1800 K

Mean density 𝜌ҧ 1 9.79 kg m 3 Τ Mean density 𝜌ҧ 2 3.81 kg m 3 Τ

Mean sound speed 𝑐ҧ 1 743 m s Τ Mean sound speed 𝑐ҧ 2 1191 m s Τ

Burner Flame parameters

Length 𝑙 0 0.6 m Interaction index 𝑘 𝑓 variable

Section 𝑠 𝑏 0.03 m 2 Time delay τ variable

3.1. Identical type of flame response

Fig. 3. (Colour online) Stability maps of the azimuthal mode with first order, azimuthal mode with second order, and the slanted mode in annular chamber. The BC configuration, including 12 identical burners with closed inlet ends and an annular chamber is presented. The interaction index is 𝑘 𝑓 = 0.5 .

Figure. 3 shows that the azimuthal mode with first order (1A0L) is stable when 𝜏< 0.5𝜏 𝑐 . 𝜏 𝑐 = 2𝑐 ҧ 2

𝐿 𝑐 , which means the period of the unperturbed first-order azimuthal mode in the chamber. Azimuthal

mode with second order (2A0L) and slanted mode (1A1L) in annular chamber show a feature of multiple periodicity at a revolution time 𝜏 𝑐 , when the flame responses are distributed as established in Fig. 2 (a). It is stable for 1A0L, 2A0L, and 1A1L in an annular chamber when the time delay is 𝜏= 1.2 ms~2.4 ms . We assume that the time delay is 𝜏= 5.9 ms , corresponding to 𝜏= 0.53𝜏 𝑐 , which is marked with a red star in Fig. 3. Then, this case is linked to the condition that azimuthal mode with first order (1A0L) and slanted mode (1A1L) in annular chamber is unstable, azimuthal mode with second order (2A0L) is marginally stable. We rearrange the flame responses of burners that “1” and “4” burners are replaced by type “1” burners with ൫𝑘 𝑓1 , 𝜏 1 ൯ to break the symmetry, as illustrated in Fig. 2 (b), and the corresponding stability map is illustrated in Fig. 4.

3.2. Different types of flame responses

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Fig. 4. (Colour online) Stability maps of the azimuthal mode with first order, azimuthal mode with second order, and the slanted mode in annular chamber. The BC configuration, including 12 burners with closed inlet ends and an annular chamber is presented. There are two types of different burners, ten identical burners with interaction index 𝑘 𝑓2 = 0.5 , time delay 𝜏 2 = 5.9 ms ≈0.53𝜏 𝑐 , “1” and “4” burners with interaction index 𝑘 𝑓1 = 3 and variable time delay.

The stability of the combustor can be realized with the change of 𝜏 1 as shown in Fig. 4. The first order (1A0L) and the second order (2A0L) is stable when time delay 𝜏 1 = 0.2 ms~3.0 ms , while the slanted mode (1A1L) could be unstable as the design of the value 𝜏 1 mentioned above. When the time delay 𝜏 1 is between 1.4 ms and 2.3 ms , all modes become stable. These results suggest that pas- sive control of thermoacoustic modes by associating different burners can work while the time delay will be assessed over-optimistic when neglecting the stability of the slanted mode (1A1L).

Furthermore, there are two frequencies of azimuthal mode with second order (2A0L) in the cir- cumstances of different burners distribution discussed above. In addition to the stability maps shown in Fig. 4, the second thermoacoustic frequency with the second-order azimuthal mode remains un- changed and the growth rate keeps negative in the revolution time 𝜏 𝑐 .

3.3. Rayleigh index analysis

The Rayleigh index [14] of each burner is given in order to analyze the stability behaviors of burners, and four typical cases as shown in Figs. 3 and 4 are discussed to draw further conclusions of thermoacoustic modes in the combustor.

Case 1: the identical flame responses are presented, with interaction index 𝑘 𝑓 = 0.5 and time delay 𝜏= 5.9 ms , which corresponds to the condition marked by red stars in Fig.3.

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Case 2: the flame responses in “1” and “4” burners are different from others, with interaction index 𝑘 𝑓1 = 3.0 and time delay 𝜏 1 = 1.7 ms . The azimuthal mode with first order (1A0L), azimuthal mode with second order (2A0L), and the slanted mode (1A1L) are all stable under this situation, as shown in Fig.4.

Case 3: the flame responses in “1” and “4” burners are different from others, with interaction index 𝑘 𝑓1 = 3.0 and time delay 𝜏 1 = 2.6 ms . In this case, Fig.4 shows that the azimuthal mode with first order (1A0L) and azimuthal mode with second order (2A0L) are stable, while the slanted mode (1A1L) is unstable.

Case 4: the flame responses in “1” and “4” burners are different from others, with interaction index 𝑘 𝑓1 = 3.0 and time delay 𝜏 1 = 5.5 ms . It corresponds to the situation illustrated in Fig.4 that these three modes are all unstable.

(a) (b)

(c) (d)

Fig. 5. (Colour online) Relative pressure amplitude and Rayleigh index of the azimuthal mode with first order, azimuthal mode with second order, and slanted mode in annular chamber. The BC con- figuration, including 12 burners with closed inlet ends and an annular chamber is presented. There are two types of different burners, ten identical burners with interaction index 𝑘 𝑓2 = 0.5 , time delay 𝜏 2 = 5.9 ms ≈0.53𝜏 𝑐 . “1” and “4” burners with (a) 𝑘 𝑓1 = 0.5 , 𝜏 1 = 5.9 ms ; (b) 𝑘 𝑓1 = 3.0 , 𝜏 1 = 1.7 ms ; (c) 𝑘 𝑓1 = 3.0 , 𝜏 1 = 2.6 ms ; (d) 𝑘 𝑓1 = 3.0 , 𝜏 1 = 5.5 ms .

The relative pressure amplitude and the normalized Rayleigh index of four cases are demon- strated in Fig.5. For case 1, the local Rayleigh index of each burner is over zero for the azimuthal mode with first order (1A0L) and the slanted mode (1A1L), which leads to instabilities of all burners as well as combustor. However, the local Rayleigh index of certain burners are less than zero for azimuthal mode with second order (2A0L). Namely, azimuthal mode with second order (2A0L) is stable. With the similar analysis mentioned above, the local Rayleigh index of “1” and “4” burners

are negative, and leading to the sum of the local Rayleigh index of these three modes negative in case 2, although other burners are unstable. Additionally, the local Rayleigh indexes of azimuthal mode with first order (1A0L) and azimuthal mode with second order (2A0L) in “1” and “4” burners remain negative and motivate stability in case 3. However, the local Rayleigh index of the slanted mode (1A1L) is evidently positive and is unstable under this situation. In case 4, the local Rayleigh index of the azimuthal mode with first order (1A0L), azimuthal mode with second order (2A0L), and the slanted mode (1A1L) are almost positive, as a result, theses three modes are all unstable. Furthermore, the relative pressure amplitude is in similar behaviors at each burner for azimuthal mode with first order (1A0L) and slanted mode (1A1L) when the time delay changing. While there is a phase differ- ence between slanted mode (1A1L) and azimuthal mode with first order (1A0L) as shown in Fig.5(d). 4. RADIAL MODE IN CAN COMBUSTOR

The existing investigations shown that the combustion instabilities of high-frequency transvers modes could occur in can combustor [6, 7]. This model could be used to study the combustion insta- bility in can combustor when the chamber is considered as a cylinder. Under this assumption, the 3- D effects, for instance, the radial position of the flames can be illustrated. The relative parameters are presented in Table 2, and the other parameters not shown remain the same as that in Table 1.

3

3

2

4

4

2

1

5

1

5

8

6 Type “1”

6

8

7

7

Type “2”

(a) (b)

Fig. 6. (Colour online) Distributions of flame responses. Configuration studied (a) in section 4.1 and (b) in section 4.2.

4.1. Radial position of the flames Table 2: Parameters for can combustor.

Chamber Burner

Radius 𝑅 𝑐 0.24 m Correction length 𝑙 0.16 m

Axial length 𝐿 𝑐 0.5 m Section in circumference 𝑠 𝑐 0.000254 m 2

(a) (b) Fig. 7. (Colour online) Frequencies and Growth rates versus time delay and non-dimensional radius. 8 identical nozzles with closed inlet ends and a cylinder chamber is presented, with interaction index 𝑘 𝑓1 = 6.0 .

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𝜏 𝑟 is the period of the unperturbed radial mode with first order in the chamber. With the identical nozzles distributed along the azimuthal direction, the frequencies and the growth rates of radial modes with first order have a periodical trend when the time delay is changing, as illustrated in Fig.7. Mean- while, frequencies and growth rates have an evident change versus time delay, when the nozzles are located at non-dimensional radius 𝑟 𝑎 = 𝑟𝑅 𝑐 = Τ 0.1 . As the nozzles move outward from 𝑟 𝑎 = 0.1 to 𝑟 𝑎 = 0.9 , the absolute value of the frequencies and growth rates decrease to a minimum and then increase gradually.

4.2. Different types of flame responses

The effects of the radial positions of the flames on the thermoacoustic stabilities of radial mode with first order were studied in the case of identical flame responses, as shown in Fig. 6(a) in the last section. It is shown in Fig. 7(b) that the values of the growth rates are affected by the radial positions of flames, while the sign remains unchanged with the fixed flame responses and azimuthal distribu- tions of flames. Meanwhile, as the discussion presented in section 3, the stability behaviors are related to the flame responses, especially, the time delay of flame transfer function. Similarly, the complex thermoacoustic frequencies of the radial mode with first order are shown in Fig. 8 when the interac- tion of different types of flame responses at 𝑟 𝑎 = 0.4 are studied with the same parameters fixed in section 4.1.

(a) (b)

Fig. 8. Frequencies and Growth rates versus time delay. There are 8 nozzles with closed inlet ends and a cylinder chamber, with the interaction index 𝑘 𝑓1 = 6.0 , variable time in first burner, and inter- action index 𝑘 𝑓2 = 6.0 ,time delay 𝜏 2 = 0.17 ms in the other burners.

It is worth noting that the complex thermoacoustic frequencies are the same when the flame response situated at any position is changed under the consideration of the first-order radial mode. Therefore, a consistent conclusion can be obtained with the variation of flame responses at any noz- zle. As depicted in Fig. 6(b), there are two types of flame responses, type “1” with interaction index 𝑘 𝑓1 = 6.0 , variable time, and type “2” interaction index 𝑘 𝑓2 = 6.0 ,time delay 𝜏 2 = 0.17 ms . Figure 7 (b) demonstrates that the radial mode with first order is unstable, when the time delay is 𝜏= 0.17 ms which corresponds to 𝜏= 0.515𝜏 𝑟 . As illustrated in Fig. 8(b), the radial mode with first order of the combustor is stable when a flame time delay is 𝜏= 0.15𝜏 𝑟 ~0.35𝜏 𝑟 . The minimum and maximum of growth rate correspond to the time delay 𝜏= 0.25𝜏 𝑟 and 𝜏= 0.75𝜏 𝑟 , which are linked to the most stable and unstable states of combustor, respectively.

5. CONCLUSIONS

This paper presents a model to predict the combustion instabilities of multiple different flames in 3-D combustors based on the Green’s function. Two models are developed for the study of the configuration with or without the plenum. The flames are described by the flame transfer function (FTF), and the eigenfunction is established with the consideration that the pressure in the plenum and the chamber are produced by the burners. The conclusions can be drawn as follows:

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Firstly, the prediction of the combustion instability in an annular chamber is presented. It is shown that although the different flame response will bring great challenges to the combustor, the different types of flame responses, for example the various time delay, can destroy the circumferential symmetry distribution which leads to the damp of azimuthal mode instability. However, it could be overly optimistic for the designed time delay when neglecting the thermoacoustic modes of azimuthal mode involve an axial dependence.

Secondly, the effect of the radial position and different flame responses on radial mode instabil- ity is studied. When the nozzles located at 0.6 𝑅 𝑐 corresponding to a pressure node, the first-order radial mode is the most stable, while it is also affected by the different flame response distributions.

6. ACKNOWLEDGEMENTS

The study was supported by the National Natural Science Foundation of China (Grant Nos. 51790514 and 52106038).

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