Study of the control mechanism on combustion instabilities using perforated injector mounting surface

Abstract

This study analyzes the key factors affecting the control of combustion instabilities on the basis of getting insight into the intrinsic physical mechanisms, focusing on the application of the perforated injector mounting surface and the outside perforated liner. The interactions between multiple injectors and the perforated screens are modeled using equivalent monopole sources, facilitating an analysis of perforated screens on thermoacoustic instabilities. Results show that lowering the mean temperature in the cooling cavities increases the phase difference of acoustic pressure across perforated screens, enhancing acoustic dissipation and stabilizing azimuthal modes. Additionally, the perforated screens and the flame modify thermoacoustic frequencies, necessitating adjustments of the perforated parameters to suppress unstable modes at various time delays. Changes in thermoacoustic frequency affect both the acoustic energy gained from the flame and the acoustic energy dissipated on the perforated screens, with both factors playing distinct roles in controlling combustion instability at different perforation ratios. Moreover, while maintaining the mass flow entering the combustion chamber primarily through the injectors, the perforated injector mounting surface effectively suppresses combustion instability within the limits of constrained flow adjustment. This work provides a theoretical framework for understanding the control mechanisms associated with perforated screens.

Keywords

Combustion (thermoacoustic) instability perforated screens control mechanism analysis temperature difference theoretical model

Introduction

Self-excited combustion (thermoacoustic) instability poses significant risks to structural safety, prompting considerable research on mitigation strategies in practical applications.^1–3 Azimuthal combustion instability, especially prevalent in annular chambers, has garnered attention due to its unique waveform characteristics,^4,5 leading to numerous studies on nonlinear effects^6–8 and symmetry-breaking phenomena associated with azimuthal modes.^9,10 These investigations have yielded valuable insights into the underlying physical mechanisms.

According to the Rayleigh criterion, the onset of combustion instability is primarily influenced by two factors:¹¹ the phase difference between acoustic pressure and unsteady heat release rate, and the impedance walls. Devices like Helmholtz resonators¹² and side perforated liners¹³ are commonly used in combustors to enhance acoustic dissipation. Helmholtz resonators provide effective control within a constrained broadband range near the resonant frequencies. By using a damper body housing multiple interconnected volumes, adequate adjustment of governing parameters has been demonstrated to achieve a broadband damping characteristic at low frequencies.¹⁴ Meanwhile, the side perforated liner can realize broadband control of combustion instability through acoustic-to-vortical energy conversion at its apertures, with subsequent dissipation, while simultaneously cooling the chamber walls to maintain temperatures below material limits.¹⁵ These combined advantages have led to the widespread implementation of side perforated liners in annular combustors for aero-engines.

In the absence of the heat sources, experimental studies confirm the intrinsic acoustic dissipation capability of the perforated liner, suggesting its potential for combustion instability suppression.^13,16,17 When accounting for combustion heat release, a simplified one-dimensional model demonstrates that unstable thermoacoustic modes can be stabilized by optimizing liner porosity and bias flow.¹⁸ Furthermore, theoretical models incorporating the three-dimensional (3D) acoustic scattering characteristics of side perforated liners have been developed. Systematic parametric analyses of these control methods, specifically targeting highly susceptible azimuthal modes, reveal the stabilization effects achievable with both uniform and non-uniform liner configurations.^19,20 These results elucidate the coupled interactions between heat sources and perforated liners, providing insights for developing effective suppression technologies. As axial lengths of combustion chambers decrease, challenges in studying azimuthal instabilities intensify, particularly regarding the effectiveness of side perforated liners. It is evident that reduced axial length weakens acoustic dissipation, undermining stability control measures.

To enhance acoustic dissipation at the impedance wall within a fixed cavity, it is essential not only to optimize the side perforated liner parameters to increase its acoustic energy dissipation but also to explore strategies for increasing the acoustic impedance boundary area in other regions of the combustion chamber. This approach offers a promising avenue for improving overall acoustic dissipation.

Similar to side perforated liners, the injector mounting surface features small perforations that cool the combustion chamber wall and facilitate acoustic dissipation. Numerical analysis reveals that the perforated plate adjacent to the injectors significantly enhances acoustic dissipation through enhanced damping areas.²¹ Experimentally, optimized acoustic damping of the front panel on the second reheat stage injector effectively suppressed combustion instability within Alstom’s test power plant.²² Meanwhile, our previous study investigated the effects of the parameters of the perforated injector mounting surface, specifically perforation ratio, aperture radius, and bias flow, on suppressing combustion instability.²³ Collectively, these results demonstrate the efficacy of the perforated injector mounting surface for instability suppression across multiple methodologies. However, the physical mechanisms underlying its effective control of combustion instability require further in-depth analysis.

This article extends our prior investigation of the perforated injector mounting surface,²³ aiming to identify key factors and elucidate the physical mechanisms involved in controlling combustion instability through this application. The principal contents of this study are structured as follows: methods section presents a 3D theoretical model of combustion instability, which integrates perforated screens (including both injector mounting surfaces and side perforated liners) and multiple injector flames. Detailed derivations build upon our prior work.²³ Results section investigates critical parameters that affect the capability of perforated screens to suppress combustion instability and elucidates underlying physical mechanisms. Finally, conclusion section synthesizes the concluding insights of the study.

3D thermoacoustic model with perforated screens

The annular combustor under investigation consists of a diffuser, multiple injectors, and a combustion chamber. These components may be considered as a series of annular and cylindrical cavities, as illustrated in Figure 1. The injector mounting surface serves as the interface between the diffuser and the combustion chamber, featuring eight uniformly distributed injectors and damping apertures. Additionally, a perforated liner is located on the outside wall of the combustion chamber.

Figure 1.

Sketch and equivalent monopole sources of the simplified annular combustor.

It should be noted that our previous work²³ developed a 3D analytical model to investigate combustion instability in the model combustor. This model specifically assessed coupling effects between perforated screens and injector flames. The validation of the model was conducted through two complementary approaches: comparison with existing solutions under hard-walled boundary conditions that account for the effects of injector flames, and comparison with 3D finite element method that incorporates perforated screens in the absence of injector flames. Furthermore, we investigated the effects of both uniform and radially nonuniform perforated injector mounting surfaces on the first-order azimuthal mode (1A), focusing on variations in perforation ratio, aperture radius, and Mach number of bias flows on stability maps. For brevity, a concise overview of the model development is provided below, while the detailed derivation and validation can be found by Qin et al.²³

Acoustic pressure disturbances related to source terms

As depicted in Figure 1, the acoustic field within the combustor is affected by flame responses in the injectors, the outside perforated liner, and the perforated injector mounting surface. Physically speaking, the volumetric expansion generated by combustion in the injector is equivalent to a monopole source. Similarly, perforated screens exert an influence equivalent to that of distributed monopoles. Namely, the acoustic disturbances generated by these components can be characterized as monopole sources. This mathematical representation has been validated and applied in various studies within the fields of aeroacoustics and combustion instabilities.^20,23,24 Accordingly, the pressure disturbances induced by these source terms²⁵ within the combustion chamber can be described as follows:

\begin{aligned} {\tilde{p}}_{(f, x, s)} (r_{i}, t_{i}) = & - \sum_{j = 1}^{N_{(f, x, s)}} {\int \int}_{S} {\bar{ρ}}_{c} {\tilde{u}}_{(f, x, s)} (r_{j}^{'}, t_{j}^{'}) \\ \frac{\partial G_{c} (r_{i}, r_{j}^{'}, t_{i}, t_{j}^{'})}{\partial t_{j}^{'}} d S (r_{j}^{'}) d t_{j}^{'} \end{aligned}

(1)

where symbols

\bar{[]}

and

\tilde{[]}

represent the mean and fluctuating quantities in the time domain, respectively. Subscripts

c

f

s

, and

x

denote the quantities associated with the combustion chamber, injector flames, injector mounting surface, and the outside perforated liner, respectively.

{\tilde{p}}_{f}

{\tilde{p}}_{s}

, and

{\tilde{p}}_{x}

represent the pressure disturbances generated by the injector flames, the perforated injector mounting surface, and the outside perforated liner, respectively.

N_{f}

N_{s}

, and

N_{x}

denote the number of equivalent monopoles for these components. The acoustic velocity disturbances at the injector outlet, injector mounting surface, and outside perforated liner are denoted by

{\tilde{u}}_{f}

{\tilde{u}}_{s}

, and

{\tilde{u}}_{x}

, respectively.

G (r_{i}, r_{j}^{'}, t_{i}, t_{j}^{'})

is a 3D Green’s function tailored for the annular cavities, with

r^{'}

and

r

indicating the cylindrical coordinate for source and observation points. The details of the Green’s function are shown in the Appendix. Additionally,

\bar{ρ}

S

, and

t

represent the mean density, area, and time coordinate, respectively, while subscripts

i

(and

j

) denote the serial number of the monopoles used to characterize injectors, the injector mounting surface, and the outside perforated liner.

The Green’s function approach has been employed as a mathematical tool to examine combustion instability involving multiple injectors.²⁶ In this study, the effects of the injectors and the perforated screens are modeled as equivalent distributed sources, thereby facilitating an investigation of the coupling effects between the injectors and the non-locally reacting perforated liner of finite length by using Green’s function. This ensures that the orthogonal conditions of the 3D modes are maintained under the discontinuous distributed impedance boundary conditions.

The pressure disturbance within the combustion chamber is the cumulative result of the pressure disturbances generated by injector flames, as well as those induced by the impedances associated with the injector mounting surface and outside perforated liner. Therefore, the pressure disturbance in the combustion chamber can be expressed as follows:

{\tilde{p}}_{(c_{f}, c_{s}, c_{x})} (r_{i}, t_{i}) = {\tilde{p}}_{f} (r_{i}, t_{i}) + {\tilde{p}}_{x} (r_{i}, t_{i}) + {\tilde{p}}_{s} (r_{i}, t_{i})

(2)

where

{\tilde{p}}_{c_{f}}

{\tilde{p}}_{c_{s}}

, and

{\tilde{p}}_{c_{x}}

represent the acoustic pressure in the combustion chamber, with the location at the injector, injector mounting surface, and outside perforated liner, as shown in Figure 1. As an approximation of the effects of the compressor outlet, the diffuser inlet can be regarded as a hard-walled boundary condition in the acoustic field. If it is further assumed that the volume of the injectors is negligible compared with that of the diffuser, then the effects of the diffuser can be described through an annular cavity. The pressure disturbances generated by the perforated screens in the diffuser

{\tilde{p}}_{e_{s}}

and liner backing cavity

{\tilde{p}}_{a_{x}}

can be described as follows:

\begin{aligned} {\tilde{p}}_{(e_{s}, a_{x})} (r_{i}, t_{i}) = & - \sum_{j = 1}^{N_{(s, x)}} {\int \int}_{S} {\bar{ρ}}_{(e, a)} {\tilde{u}}_{(s, x)} (r_{j}^{'}, t_{j}^{'}) \\ \frac{\partial G_{(e, a)} (r_{i}, r_{j}^{'}, t_{i}, t_{j}^{'})}{\partial t_{j}^{'}} d S (r_{j}^{'}) d t_{j}^{'} \end{aligned}

(3)

with subscripts

e

and

a

denoting the quantities associated with the diffuser and liner backing cavity, respectively. For instance,

G_{e}

and

G_{a}

represent the 3D Green’s functions for the diffuser and liner backing cavity, respectively. Both share the mathematical form as the Green’s function in equation (1), with detailed expressions provided in the Appendix.

Unsteady heat release responses in the injectors

Thermoacoustic instabilities arise due to the coupling between heat input and flow perturbations. Previous studies show that unsteady heating generates acoustic disturbances that, in turn, affect the heat input rate.^27,28 To characterize this feedback, the classical flame transfer function model is employed.²⁹ In practical applications, flame surfaces are generally distributed within the combustion chamber. Considering that the flame length is acoustically compact relative to the wavelength of mode 1A, we model the flame as a concentrated source positioned at the injector outlet. This simplification captures key flame dynamics and has been validated against numerical simulations in the prior work.²⁹ Given the small radius of the injectors, it is reasonable to assume that only plane waves propagate within them. The jump condition at the injector outlet $x = l_{f}$ of the compact flame indicates a continuity of pressure accompanied by a discontinuity in the unsteady volume flow rate, leading to

{\tilde{p}}_{c_{f}} (r_{j}^{'}, t) |_{x = l_{f, j}, θ = θ_{f, j}, r = r_{f, j}} = {\tilde{p}}_{g, j} (t) |_{x = l_{f, j}}

(4)

{\tilde{u}}_{f} (r_{j}^{'}, t) |_{x = l_{f, j}, θ = θ_{f, j}, r = r_{f, j}} = {\tilde{u}}_{g, j} (t) |_{x = l_{f, j}} + k_{f, j} {\tilde{u}}_{g, j} (t - τ_{j}) |_{x = l_{f, j}}

(5)

where subscript

g

represents the quantities upstream the compact flame.

k_{f}

represents the interaction index and

τ

is the time delay. We specify

k_{f} = 0.5

to represent the magnitude of unsteady heat release rate, a value held constant throughout this study. A closed inlet boundary condition for the injector is assumed, which yields

{\tilde{u}}_{g, j} |_{x = 0} = 0

(6)

Therefore, the velocity disturbance in the $j$ th injector is linked to the pressure disturbance at the injector outlet ( $x = l_{f}$ ), resulting in

{\tilde{u}}_{f} (r_{j}^{'}) = ξ_{j} {\tilde{p}}_{c_{f}} (r_{j}^{'})

(7)

where

ξ_{j}

represents a coefficient reflecting the relationship between the acoustic pressure and acoustic velocity at the

j

th injector outlets. Under the condition of a compact flame located at the injector outlet and a closed boundary at the injector inlet, the coefficient

ξ_{j}

in equation (7) is

ξ_{j} = - \frac{i (1 + k_{f, j} e^{- i ω τ_{j}})}{{\bar{ρ}}_{g, j} {\bar{c}}_{g, j}} \tan (\frac{ω l_{f, j}}{{\bar{c}}_{g, j}})

(8)

with

i

denoting the imaginary unit.

Impedance boundary conditions of the perforated screens

For the perforated injector mounting surface and the outside perforated liner, the impedance boundary conditions are satisfied to describe the energy conversion between acoustics and vortices across the perforated screens, which can be written as follows:

\begin{aligned} {\tilde{u}}_{x} (r_{j}^{'}) = & \frac{{\tilde{p}}_{a_{x}} (r_{j}^{'}) - {\tilde{p}}_{c_{x}} (r_{j}^{'})}{Z_{x} (r_{j}^{'})} \\ {\tilde{u}}_{s} (r_{j}^{'}) = & \frac{{\tilde{p}}_{e_{s}} (r_{j}^{'}) - {\tilde{p}}_{c_{s}} (r_{j}^{'})}{Z_{s} (r_{j}^{'})} \end{aligned}

(9)

where

Z_{s}

and

Z_{x}

denote the impedances at the perforated injector mounting surface and the outside perforated liner, respectively. As shown in equation (9), the impedances

Z_{s}

and

Z_{x}

are spatially dependent functions, enabling the incorporation of non-uniform perforated screens. Building on this framework, our previous study separately investigated the suppression effects of radially non-uniform perforated injector mounting surfaces and azimuthally non-uniform side perforated liners on mode 1A, including stability maps and mode nature.^19,23 While the present work assumes uniform perforated screens, we retain the spatial impedance formulation to preserve the analytical capability for future studies of non-uniform boundary conditions.

Assuming negligible mean flow velocity within the combustion chamber and treating the cooling flow as the bias flow, we use the impedance model,³⁰ which incorporates both the thickness of the perforated plate $t^{v}$ and the Mach number of the bias flow $M^{v}$ , expressed as follows:

Z = \frac{i \bar{ρ} ω}{ς} = \bar{ρ} \bar{c} \bar{\bar{z}}

(10)

where

ω

denotes the complex frequencies of the modes in the investigated systems;

\bar{\bar{z}}

represents the nondimensional acoustic impedance; and

ς

signifies the compliance of the perforated plates, accounting for the effects of both the bias flow and the perforated plate thickness. This relationship can be characterized by the vortex-acoustic energy transfer mechanism,³⁰ which is written as follows:

\frac{1}{ς} = \frac{π (r^{v})^{2}}{σ} \frac{1}{K} + \frac{t^{v}}{σ}

(11)

with

r^{v}

and

σ

representing the aperture radius and perforation ratio. The Rayleigh conductivity

K

of a single aperture with bias flow can be found by Howe.³¹

Eigenvalue equations for the thermoacoustic analysis

To obtain the indicator for assessing the stability of the investigated modes, the equations (7) and (9) are substituted into equations (1) to (3). Consequently, the eigenvalue equations in the frequency domain can be derived using the Laplace transform $\hat{p} (s) = L [\tilde{p} (t^{'})]$ , with $s = i ω$ , which yields

[M - I] P = 0

(12)

with

M = [\begin{matrix} A_{c_{f}} & A_{c_{x}} & A_{c_{s}} & A_{a_{x}} & A_{e_{s}} \\ 0 & B_{c_{x}} & 0 & B_{a_{x}} & 0 \\ 0 & 0 & D_{c_{s}} & 0 & D_{e_{s}} \end{matrix}]

(13)

P = {[{\hat{p}}_{c_{f}} {\hat{p}}_{c_{x}} {\hat{p}}_{c_{s}} {\hat{p}}_{a_{x}} {\hat{p}}_{e_{s}}]}^{T}

(14)

where

I

and

0

represent the identity and zero matrix, respectively. The vectors

{\hat{p}}_{c_{f}}

{\hat{p}}_{c_{s}}

, and

{\hat{p}}_{c_{x}}

correspond to the complex pressure at the injectors, injector mounting surface, and the outside perforated liner within the combustion chamber, respectively. Additionally,

{\hat{p}}_{a_{x}}

represents the set of complex pressures located upstream of the injector mounting surface in the diffuser, while

{\hat{p}}_{e_{s}}

denotes the sub-vector of complex values in the liner backing cavity. The submatrices

A

B

, and

D

denote the coefficients related to observation points in the combustion chamber, liner backing cavity, and diffuser, respectively. For instance,

A_{c_{f}}

represents the coefficient matrix associated with the acoustic pressure

{\hat{p}}_{c_{f}}

when the observation points are located in the combustion chamber. Similar representations are also found in other coefficient matrices.

The complex frequency is derived when the determinant is equal to zero. The real component of the complex frequency corresponds to the frequency, while the imaginary component (which represents the negative of the growth rate) indicates the stability behaviors. Furthermore, once the complex frequency is established, the complex pressure at the source terms can be determined. This enables the calculation of pressure moduli in the combustion chamber, liner backing cavity, and diffuser, respectively. Further details regarding this model, including its validation, can be found in our previous work.²³

Results and discussions

In this study, a simplified annular combustor is examined as a case study to facilitate an analysis of the key factors and physical mechanisms that influence the effectiveness of perforated screens. The relevant parameters associated with the annular combustor and the perforated screens can be found by Qin et al.²³

The effect of time delay on optimal design parameters of perforated screens

Flexibility in both load and fuel composition is of increasing importance, which suggests that the representative time delay may vary. Therefore, it is essential to investigate the impact of time delay on the optimal design parameters of the perforated screens. Figure 2 shows the frequencies and growth rates versus time delay over a period $τ_{c}$ , with perforation ratios of the perforated liner $σ_{x}$ and injector mounting surface $σ_{s}$ at $σ_{s} = σ_{x} = 0$ and $σ_{s} = σ_{x} = 0.005$ , where $τ_{c}$ represents the time period of the unperturbed mode 1A. The stability of this mode is ensured throughout the entire period $τ_{c}$ at $σ_{s} = σ_{x} = 0.005$ . As depicted in Figure 2(b), the time delay corresponding to the maximum growth rate varies; Under hard wall conditions ( $σ_{s} = σ_{x} = 0$ ), the growth rate relatively peaks at $0.75 τ_{c}$ . In contrast, for $σ_{s} = σ_{x} = 0.005$ , this peak shifts to $\sim 0.95 τ_{c}$ . This shift is attributed to a frequency reduction in the mode 1A caused by the perforated screens and multiple injectors, as shown in Figure 2(a), necessitating a longer time delay to minimize the phase difference between acoustic pressure and the unsteady heat release rate ( $Δ ϕ$ ). Further details are provided in Figure 3.

Figure 2.

Thermoacoustic frequencies of mode 1A against the time delay: (a) frequencies; and (b) growth rates.

Figure 3.

Sketch of phase difference between acoustic pressure and unsteady heat release rate ( $Δ ϕ$ ): (a) constant natural frequency; and (b) frequency with the effects of injector flames and perforated screens.

Figure 3(a) shows that, assuming a natural frequency ( $\sim$ 360 Hz for mode 1A), the phase difference $Δ ϕ$ exceeds $π / 2$ during the time delay from $0 τ_{c}$ to $0.5 τ_{c}$ , maintaining stability in the mode. However, as the time delay progresses from $0.5 τ_{c}$ to $0.75 τ_{c}$ , $Δ ϕ$ decreases and reaches $Δ ϕ = 0$ at $0.75 τ_{c}$ , corresponding to the point of maximum instability. Beyond this threshold, $Δ ϕ$ increases, which leads to a decline in the growth rates of mode 1A.

Nevertheless, Figure 2(a) indicates that the frequency of mode 1A decreases due to the injector flames and perforated screens. Consequently, the condition $Δ ϕ = 0$ is not satisfied at $0.75 τ_{c}$ , as shown in Figure 3(b). Furthermore, the value of $Δ ϕ$ under perforated screen conditions is lower than that observed under hard wall conditions for a fixed time delay. This leads to an increased growth rate when the delay time exceeds $0.75 τ_{c}$ under perforated conditions in Figure 2(b). If the time delay extends beyond $τ_{c}$ in Figure 2, $Δ ϕ$ will continue to increase, leading to a decrease in growth rates.

Since the growth rate reflects the degree of combustion instability, the optimal damping parameters are expected to vary across time periods. Taking $0.75 τ_{c}$ as example (Figures 4(a) and (b)), the range of optimal damping parameters to mitigate instability of mode 1A increases when the time delay is below $0.75 τ_{c}$ . Conversely, this range decreases for time delay exceeding $0.75 τ_{c}$ . These trends are supported by the stability maps in Figures 4(d) and (f), which correspond to $0.67 τ_{c}$ and $0.95 τ_{c}$ , respectively.

Figure 4.

Thermoacoustic frequencies of mode 1A when the perforation ratios of the injector mounting surface and the side perforated liner change: (a) and (b) $τ = 0.75 τ_{c}$ ; (c) and (d) $τ = 0.67 τ_{c}$ ; and (e) and (f) $τ = 0.95 τ_{c}$ .

From an engineering perspective, ensuring robust and reliable prevention of combustion instability necessitates stabilizing mode 1A across variable time delay. Consequently, adopting a conservative design for damping parameters that accommodates variable time delays may represent the optimal strategy for enhancing engine safety under variable operating conditions.

The sensitivities of the gas temperature inside the backing cavity on the stability

To our knowledge, the mean gas temperature significantly influences combustion instability through several mechanisms. On one hand, the sound speed $\bar{c}$ and wavelength are temperature-dependent within the combustion chamber, liner baking cavity, and diffuser. This affects frequency and phase difference $Δ ϕ$ , thereby affecting stability. Additionally, temperature variations result in wavelength changes across these components, leading to differences in pressure phase and acoustic dissipation across the perforated screens. On the other hand, the temperature in the combustion chamber and liner baking cavity (diffuser) influences the impedance of the perforated screens, as described by the impedance model in equation (10), which includes both perforated plate thickness and bias flow Mach number. Due to few literature on acoustic impedance under high-temperature conditions and temperature differences across aperture, we assume that the temperature through the apertures of perforated screens is the average temperature between the diffuser (liner baking cavity) and the combustion chamber. This assumption is made for the following considerations: The model assumes adiabatic cavity walls and uniform internal temperatures in its baseline formulation. In practical applications, however, heat conduction through the wall preheats the neighboring cooling air, while convective heating occurs as the gas flows through apertures. To characterize these thermal effects, we approximate gas temperature in the aperture as the average of neighboring cavity temperatures (cooling air and hot gas).

Figure 5 shows the sensitivity of stability on the mean temperature of the diffuser ${\bar{T}}_{e}$ and liner backing cavity ${\bar{T}}_{a}$ (with ${\bar{T}}_{e} = {\bar{T}}_{a}$ ), when the mean temperature in the combustion chamber is ${\bar{T}}_{c} = 2000$ K. It is depicted that the increase of temperature in these regions weakens the ability to prevent combustion instability for perforation ratios $σ_{x} = σ_{s} = 0.005$ and $σ_{x} = σ_{s} = 0.007$ , respectively. This suggests that the temperatures of the backing cavity and diffuser are critical for controlling combustion instability within a certain range of lower temperatures. These findings can be further explained from an acoustic energy perspective,²³ as plotted in Figure 6. Wherein, $E_{f}$ is the acoustic energy gained from the flame, $E_{s}$ and $E_{x}$ are the acoustic energies dissipated across the injector mounting surface and the outside perforated liner, $E_{t}$ is the total energy in the system, with $E_{t} = E_{f} - E_{s} - E_{x}$ . The data in Figure 6(b) represent the normalized values, defined as ${\bar{\bar{E}}}_{(s, x, t)} = \frac{E_{(s, x, t)}}{E_{f}}$ . Under the condition of $E_{f}$ is positive, the sign of ${\bar{\bar{E}}}_{t}$ is the indicator of the stability behavior: positive values denote the instability, while negative values indicate the stability.

Figure 5.

Thermoacoustic frequencies of mode 1A against the mean temperature in the diffuser ${\bar{T}}_{e}$ and liner backing cavity ${\bar{T}}_{a}$ : (a) frequencies; and (b) growth rates. The time delay is fixed as $τ = 0.75 τ_{c}$ and the mean temperature in the combustion chamber is ${\bar{T}}_{c} = 2000$ K.

Figure 6.

Relevant analysis at perforation ratios $σ_{x} = σ_{s} = 0.005$ : (a) non-dimensional acoustic impedance of the perforated screens; and (b) normalized acoustic energy.

We present the perforation ratios at $σ_{x} = σ_{s} = 0.005$ as an example to illustrate the temperature dependence of stability in Figure 6. Results show that the non-dimensional acoustic resistance of the perforated injector mounting surface ( $Re ({\bar{\bar{z}}}_{s})$ ) and outside perforated liner ( $Re ({\bar{\bar{z}}}_{x})$ ) increase with temperature, while the non-dimensional acoustic reactance ( $Im ({\bar{\bar{z}}}_{s})$ and $Im ({\bar{\bar{z}}}_{x})$ ) decrease within a relatively narrow range of magnitudes. Moreover, higher temperatures lead to a notable reduction in acoustic loss across the injector mounting surface ( ${\bar{\bar{E}}}_{s}$ ) and a minor decrease in acoustic energy on the outside perforated liner ( ${\bar{\bar{E}}}_{x}$ ), which facilitates the onset of combustion instability over ${\bar{T}}_{e} = {\bar{T}}_{a} = 600$ K.

As presented in Figure 6, ${\bar{\bar{E}}}_{s}$ has a more pronounced impact on the stability. Therefore, the focus is placed on the changes of ${\bar{\bar{E}}}_{s}$ with the temperature.

Figure 7 displays the non-dimensional pressure modulus $| \bar{\bar{p}} |$ and phase $ϕ (\bar{\bar{p}})$ in the diffuser and combustion chamber at ${\bar{T}}_{e} = {\bar{T}}_{a} = 400$ K and ${\bar{T}}_{e} = {\bar{T}}_{a} = 900$ K, respectively. It could be found that increased temperature in the diffuser reduces the temperature gradient between the diffuser and the combustion chamber, resulting in a smaller phase difference in acoustic pressure at ${\bar{T}}_{e} = {\bar{T}}_{a} = 900$ K. This, in turn, decreases acoustic dissipation across the perforated injector mounting surface compared to ${\bar{T}}_{e} = {\bar{T}}_{a} = 400$ K, thereby affecting the stability of mode 1A.

Figure 7.

Non-dimensional pressure in the combustion chamber and diffuser: (a) acoustic pressure modulus; and (b) acoustic pressure phase. The perforation ratios of the injector mounting surface and the outer perforated liner are $σ_{s} = σ_{x} = 0.005$ , respectively.

Indeed, the impedance model applicable to high temperatures or temperature differences across the apertures in gas turbines is still an open question and worthy of further study. It could be expected that the development of an accurate impedance model for practical applications will facilitate a more robust optimal design.

Effects of damping and phase difference on growth rates

It is known that the frequency affects the phase difference $Δ ϕ$ , which in turn affects the growth rates. To distinguish the contributions to changes in growth rates attributable to damping versus those resulting from the phase difference $Δ ϕ$ , the analysis from the perspective of acoustic energy is presented. Specifically, the damping effects related to the outside perforated liner and the perforated injector mounting surface are characterized by acoustic dissipation $E_{d}$ , with $E_{d} = E_{s} + E_{x}$ , while the effect of phase difference $Δ ϕ$ is represented by the acoustic energy gained from the flame $E_{f}$ . The relative difference in acoustic energy is defined as follows:

\begin{aligned} Δ E_{f} & = E_{f} |_{perforated screens} - E_{f} |_{hard walls} \\ Δ E_{d} & = E_{d} |_{perforated screens} - E_{d} |_{hard walls} \\ Δ \bar{\bar{E}} & = \frac{Δ E_{d}}{Δ E_{f}} \end{aligned}

(15)

where

Δ E_{f}

(

Δ E_{d}

) represents the relative difference in acoustic energy gained from the flame (acoustic dissipation) between the conditions of the perforated screens and hard walls. A positive value of

Δ E_{f}

indicates that the acoustic energy gained from the flame increases in the presence of perforated screens compared to hard walls, and vice versa. Given

E_{d} = 0

for the hard-walled boundary conditions,

Δ E_{d}

is the positive under the conditions of perforated screens. The magnitude

| Δ \bar{\bar{E}} |

allows for the analysis of changes in growth rate attributed to acoustic damping and phase difference

Δ ϕ

. Specifically,

| Δ \bar{\bar{E}} | > 1

indicates the acoustic damping at the perforated walls dominates growth rate modifications, whereas

| Δ \bar{\bar{E}} | < 1

signifies the phase difference as the primary driver of these changes.

Taking Figure 4(a) and (b) as an example, the frequencies and $| Δ \bar{\bar{E}} |$ against $σ_{s}$ are plotted in Figure 8. The perforation ratio of the perforated liner is set as $σ_{x} = 0$ . The choice of $σ_{x} = 0$ is that the hard-walled boundary condition is obtained when the perforation ratio of the perforated injector mounting surface is $σ_{s} = 0$ , thereby facilitating a direct comparison with the hard-walled scenario. Meanwhile, Figure 4 shows that variations in the $σ_{s}$ significantly influence combustion instabilities and frequencies, suggesting that changes in the $σ_{s}$ will substantially affect both $Δ E_{f}$ and $Δ E_{d}$ .

Figure 8.

Frequency and $| Δ \bar{\bar{E}} |$ against the perforation ratio of the injector mounting surface $σ_{s}$ . The time delay is fixed as $τ = 0.75 τ_{c}$ and the perforation ratio of the outside perforated liner is set as $σ_{x} = 0$ .

Figure 8 illustrates that frequency decreases as the perforation ratio of injector mounting surface increases, except at $σ_{s} = 0.001$ . Meanwhile, the value of $| Δ \bar{\bar{E}} |$ is negative, also with the exception of $σ_{s} = 0.001$ . This indicates that $E_{f}$ under the perforated screens is less than that observed under the hard-walled boundary conditions, excluding the specific case of $σ_{s} = 0.001$ . The magnitude $| Δ \bar{\bar{E}} |$ peaks at $σ_{s} = 0.003$ , nearly corresponding to the minimum growth rates observed in Figure 4. For $σ_{s} \leq 0.015$ , $| Δ \bar{\bar{E}} | > 1$ confirms that acoustic damping dominates growth rate modifications. However, with further increases in the perforation ratio, $| Δ \bar{\bar{E}} | < 1$ , the phase difference $Δ ϕ$ becomes increasingly significant in determining the growth rates.

The ratio of air mass flow from damping apertures and injectors

The parameters of a perforated injector mounting surface, such as perforation ratio and Mach number of bias flow, which aim to suppress combustion instability, must simultaneously satisfy the demands of thermal management and structural integrity. Consequently, these design parameters can only be modified within a limited practical range. To assess the ratio of the bias flow through the perforated injector mounting surface to that through the injectors, and thereby evaluate whether combustion instability can be stabilized with only marginal adjustments to the bias flow through the injector mounting surface, the following analysis is conducted.

In terms of the low Mach number of mean flow in a real combustion chamber, the mean flow is neglected in the present investigation. This assumption is commonly adopted in both analytical and numerical analyses.²⁹ For a conservative estimate, we assume that the Mach number of the air flow in the injectors featuring $M_{g} = 0.017$ in the following analyses, the other parameters needed can be found by Qin et al.²³ Consequently, the Mach number within the combustion chamber is

\begin{aligned} M_{c} & = \frac{{\bar{ρ}}_{g} M_{g} {\bar{c}}_{g} S_{f} N_{f}}{{\bar{ρ}}_{c} {\bar{c}}_{c} S_{c}} \\ = \frac{11.61 \times 0.017 \times 491 \times 0.016 \times 8}{3.48 \times 896.4 \times π ({0.5}^{2} - {0.3}^{2})} = 0.0079 \end{aligned}

(16)

where

{\bar{ρ}}_{g}

M_{g}

{\bar{c}}_{g}

, and

S_{f}

represent the mean density, Mach number of the mean flow, sound speed, and cross-sectional area of the injector, respectively. Similarly,

{\bar{ρ}}_{c}

M_{c}

{\bar{c}}_{c}

, and

S_{c}

represent the mean density, Mach number of the mean flow, sound speed, and cross-sectional area of the combustion chamber, respectively. It could be seen that the assumption of neglecting the Mach number of mean flow within the combustion chamber is suitable. Under this condition, the mass flow rate of air through the injectors is

{\dot{m}}_{g} = {\bar{ρ}}_{g} M_{g} {\bar{c}}_{g} S_{f} N_{f} = 12.404 kg/s

(17)

The mass flow rate of air from the damping apertures is

\begin{aligned} {\dot{m}}_{s} & = {\bar{ρ}}_{e} M_{s}^{v} {\bar{c}}_{s} S_{c} σ_{s} \\ = 5.807 \times 694.38 \times π ({0.5}^{2} - {0.3}^{2}) \times 0.01 M_{s}^{v} \\ = 20.27 M_{s}^{v} kg / s \end{aligned}

(18)

where

{\bar{ρ}}_{e}

M_{s}^{w}

{\bar{c}}_{e}

, and

σ_{s}

represent the mean density, Mach number of the bias flow, sound speed, and perforation ratio related to the perforated injector mounting surface, respectively. Therefore, the ratio of the mass flow rate of air exiting through the damping apertures to that flowing through the injectors is expressed as follows:

{\bar{\bar{m}}}_{β} = \frac{{\dot{m}}_{s}}{{\dot{m}}_{g}}

(19)

As depicted in Figure 9(a), the mode 1A can be stabilized within the investigated range of the Mach number of the bias flow. Meanwhile, the data of ${\bar{\bar{m}}}_{β}$ in Figure 9(b) indicate that air mass flow through the injectors is the predominant contributor to the total mass flow into the combustion chamber, in contrast to the flow through the damping apertures on the injector mounting surface. This indicates that suppressing combustion instability by altering the Mach number of bias flow through the injector mounting surface does not affect the fact that the airflow in the combustion chamber mainly comes from the injectors. Nevertheless, there exists a possibility that the flame dynamics may be influenced by these apertures. Consequently, it is essential to consider combustion characteristics of the flames when undertaking impedance design for the injector mounting surface from an engineering perspective.

Figure 9.

Growth rate and mass flow against Mach number of bias flow: (a) growth rate; and (b) mass flow of air through damping apertures over the injector mounting surface ${\dot{m}}_{s}$ and the ratio of the mass flow of air from the damping apertures and through the injectors ${\bar{\bar{m}}}_{β}$ .

Conclusion

This work investigates the control mechanisms of key parameters of perforated screens on combustion instability by modeling multiple injectors and perforated screens in an annular combustor as equivalent monopole sources. Interactions among these sources are analyzed using a 3D Green’s function approach. Conclusions are mainly derived from following respects:

The temperature difference across the perforated plate influences acoustic energy dissipation at the screens by affecting the phase difference of acoustic pressure. For instance, a decrease in temperature within the cooling cavities increases the phase difference of mode 1A between neighboring cavities, thereby enhancing acoustic energy dissipation at the perforated screens.

Perforated screens and flames affect the thermoacoustic frequency, leading to changes in the optimized parameters of the perforated screens over various time delays. Therefore, design of the perforated screens, accounting for the periodic time delays of the flame, is essential to optimize parameters that suppress unstable modes.

The control mechanism of thermoacoustic instability is affected by two aspects: the variation in acoustic energy gained from the flame due to changes in the thermoacoustic frequency, which modifies the phase difference between acoustic pressure and unsteady heat release rate, and the acoustic energy dissipated by the perforated screens. This study demonstrates that the dominance of these mechanisms varies with the perforation ratio. Specifically, for perforation ratio $σ_{s} \leq 0.015$ , acoustic dissipation on the perforated screens is predominant, while for perforation ratio $σ_{s} \geq 0.015$ , changes in phase difference play a more significant role in suppressing combustion instability.

This study found that mass flow primarily enters the combustion chamber through the injectors, with a minor contribution from the damping apertures on the injector mounting surface. This demonstrates that combustion instability can be suppressed by adjusting the bias flow through the perforated injector mounting surface within a limited range, while maintaining the injectors as the primary airflow pathway into the combustion chamber.

Footnotes

ORCID iDs

Lei Qin

Guangyu Zhang

Xiaoyu Wang

Lei Li

Xiaofeng Sun

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (grant no. 52406037), China Postdoctoral Science Foundation (grant nos. GZB20240930 and 2025M774244) and the Fundamental Research Funds for the Central Universities.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

Under the low-Mach-number assumption, both the inlet and outlet of the combustion chamber are treated as acoustically closed boundary conditions. Consequently, for annular cavities with finite axial length, the 3D Green’s function in the time domain can be expressed as follows: (\rm A.1)

\begin{aligned} G (r, r^{'}, t, t^{'}) = - i {\bar{c}}^{2} \sum_{m = - \infty}^{\infty} \sum_{n = 0}^{\infty} \sum_{h = 0}^{\infty} ψ_{m n} (μ_{m n} r) e^{i m θ} \\ \cos (\frac{h π}{l} x) ψ_{m n} (μ_{m n} r^{'}) e^{- i m θ^{'}} \cos (\frac{h π}{l} x^{'}) \\ \frac{(e^{i ω_{m n h} (t - t^{'})} - e^{- i ω_{m n h} (t - t^{'})})}{2 ω_{m n h} Γ_{m n h}} \end{aligned}

with (\rm A.2)

\begin{aligned} \frac{\partial G (r, r^{'}, t, t^{'})}{\partial t^{'}} & = - {\bar{c}}^{2} \sum_{m = - \infty}^{\infty} \sum_{n = 0}^{\infty} \sum_{h = 0}^{\infty} ψ_{m n} (μ_{m n} r) e^{i m θ} \\ \cos (\frac{h π}{l} x) ψ_{m n} (μ_{m n} r^{'}) e^{- i m θ^{'}} \cos (\frac{h π}{l} x^{'}) \\ \frac{(e^{i ω_{m n h} (t - t^{'})} + e^{- i ω_{m n h} (t - t^{'})})}{2 Γ_{m n h}} \end{aligned}

where

i

denotes the imaginary unit,

\bar{c}

is the sound speed, and

r = (θ, r, x)

represents the spatial position vector in azimuthal, radial, and axial coordinates, respectively. The corresponding mode numbers are denoted by

m

n

, and

h

. Additionally,

μ_{m n}

ψ_{m n}

ω_{m n h}

, and

Γ_{m n h}

represent the radial eigenvalue, radial eigenfunction, allowed frequency, and volume integration, respectively, associated with corresponding mode numbers for annular cavities under hard-walled boundary conditions.

Applying the Laplace transform ( $s = i ω$ ) to the time-domain Green’s function (equation (A.2)) yields the frequency-domain Green’s function, expressed as follows: (\rm A.3)

\begin{aligned} L [\frac{\partial G (r, r^{'}, t, t^{'})}{\partial t^{'}}] \\ = - {\bar{c}}^{2} \sum_{m = - \infty}^{\infty} \sum_{n = 0}^{\infty} \sum_{h = 0}^{\infty} \frac{2 i ω}{ω_{m n h}^{2} - ω^{2}} ψ_{m n} (μ_{m n} r) e^{i m θ} \\ \frac{\cos (\frac{h π}{l} x) ψ_{m n} (μ_{m n} r^{'}) e^{- i m θ^{'}} \cos (\frac{h π}{l} x^{'})}{2 Γ_{m n h}} \end{aligned}

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