# Assessing Transfer Matrix Models and Measurements Using Acoustic Energy Conservation Principles

Véranika Latour, Preethi Rajendram Soundararajan, Daniel Durox, Antoine Renaud, Sébastien Candel

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Véranika Latour, Preethi Rajendram Soundararajan, Daniel Durox, Antoine Renaud, Sébastien Candel. Assessing Transfer Matrix Models and Measurements Using Acoustic Energy Conservation Principles. Journal of Engineering for Gas Turbines and Power, 2024, 146 (1), pp.011021-1. 10.1115/1.4063636. hal-05235143

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ASSESSING TRANSFER MATRIX MODELS
AND MEASUREMENTS USING ACOUSTIC
ENERGY CONSERVATION PRINCIPLES

Véranika Latour*

Laboratoire EM2C,
CNRS,

CentraleSupélec,
Université Paris-Saclay,
3, rue Joliot Curie,

Gif-sur-Yvette cedex 91192, France

e-mail: veranika.latour@centralesupelec.fr

Preethi Rajendram Soundararajan†

Laboratoire EM2C,
CNRS,

CentraleSupélec,
Université Paris-Saclay,
3, rue Joliot Curie,

Gif-sur-Yvette cedex 91192, France

Daniel Durox

Laboratoire EM2C,
CNRS,

CentraleSupélec,
Université Paris-Saclay,
3, rue Joliot Curie,

Gif-sur-Yvette cedex 91192, France

Antoine Renaud

Laboratoire EM2C,
CNRS,

CentraleSupélec,
Université Paris-Saclay,
3, rue Joliot Curie,

Gif-sur-Yvette cedex 91192, France

Sébastien Candel

Laboratoire EM2C,
CNRS,

CentraleSupélec,
Université Paris-Saclay,
3, rue Joliot Curie,

Gif-sur-Yvette cedex 91192, France

ABSTRACT

Acoustic transfer matrices are widely used in the analysis of combustion dynamics of

<footnote>*Corresponding author</footnote>

<footnote>†Present address: Department of Engineering, University of Cambridge, Cambridge, England CB2 1PZ, United Kingdom.</footnote>
gas turbines. The reliability of the analysis thus depends on the quality of the determina-
tion of the transfer matrices of the individual acoustic elements composing the system's
acoustic network. These matrices are, in some simple cases, deduced analytically using
one-dimensional acoustic modeling. For more complex elements, such as swirlers, perfo-
rated plates, or injection units, the transfer matrix has to be obtained experimentally using
an impedance tube setup. There are, however, uncertainties in the experimental determi-
nation of the transfer matrix coefficients and in the modeling of key elements like injection
units. It is thus worth examining experimental data and assessing models using acoustic
energy conservation principles. The general idea is to consider the acoustic power flow in
the element represented by the transfer matrix *T* and compare the power input to the power
output. This is best accomplished by obtaining a representation in terms of a scattering
matrix *S*, which may be deduced from the transfer matrix *T*. It is first shown that standard
models like those corresponding to a constant area duct or an area change comply with
acoustic energy conservation. This analysis is then employed to assess the *L* – *ζ* model,
widely used to describe injection unit dynamics. Acoustic conservation principles are then
used to assess transfer matrices of a family of injectors determined experimentally and
check that the data complies with these principles.

NOMENCLATURE

T, T<sub>i,j</sub> Transfer matrix and corresponding coefficients

$S, S_{i,j}$ Scattering matrix and corresponding coefficients

f, g Forward & backward propagating Riemann invariants

$p', v'$ Acoustic pressure and velocity

M Mach number

$\rho, c$ Density and speed of sound

$\omega, k$ Angular frequency and wavenumber $k = \omega/c$

A Surface area

$\eta$ Area ratio between inlet and outlet
$\xi$ Specific impedance ratio

$\mathcal{P}$ Acoustic power flow

$F$ Integrated acoustic flux

$\psi_{1 \to 2}, \psi_{1 \leftarrow 2}$ Acoustic energy dissipation

$E_1/E_2, E_2/E_1$ Energy density ratios

$l_{eff}, l_{red}$ Reduced lengths used in the $L-\zeta$ model

$S_N$ Swirl number

$\Delta p$ Pressure drop

$\zeta_b$ Pressure loss coefficient

## INTRODUCTION

Thermo-acoustic instabilities, resulting from the coupling between combustion and the combustor acoustic modes, can occur in high power devices such as aircraft engines or gas turbines. Their prediction at an early stage of combustor design remains difficult and requires a thorough understanding of the acoustics of the system as well as of the flame dynamics induced by acoustic disturbances.

Reduced order models used to investigate combustion dynamics in gas turbines are often based on an acoustic network description. The system is represented by a set of transfer matrices, each of which represents the acoustic response of an element in the network. The transfer matrix $T$ links the downstream acoustic state, expressed in terms of primitive variables $p'/(\rho c)$ and $v'$, to the upstream variables, with $p'$ and $v'$, the acoustic pressure and velocity, $\rho$, the density, and $c$, the speed of sound. In various circumstances it is more convenient to work with Riemann invariants representing waves propagating in the system. This is achieved by defining

$$f = (1/2)[p'/(\rho c) + v'] \qquad (1)$$

$$g = (1/2)[p'/(\rho c) - v'] \qquad (2)$$
One can then establish another transfer matrix, $U$, that connects upstream and downstream waves. It is also physically interesting to employ scattering matrices, $S$, which relate incoming and outgoing waves. The matrices $T, U$, and $S$ may be derived analytically in some simple cases, but are most often determined experimentally for more complex elements. This is the case in particular for injection units which include area contractions and expansions in combination with swirlers.

When the system is geometrically complex, one generally considers that its transfer matrix is best obtained by measuring its coefficients experimentally. This is usually accomplished by placing the element one wishes to characterize in an impedance tube equipped with multiple microphones. This system is then modulated by driver units placed at one of its ends, and a “two-source” or a “two-load” [1] scheme is used to retrieve the four transfer matrix coefficients.

Examples of such experimental determinations of transfer or scattering matrices for complex acoustic elements as well as of acoustic networks for combustors can be found in the literature. The method was extensively developed by Paschereit and his co-workers in a series of articles dealing with the modeling of lean premixed combustion systems found in modern gas turbines [2, 3, 4, 5]. In these studies, transfer matrices are determined experimentally and integrated in the acoustic network of the system. Detailed comparisons between measured and modeled transfer matrix coefficients for a family of swirl burners are reported by Fischer *et al.* [6]. Gaudron *et al.* [7] investigated the impact of the forcing level on the transfer matrix coefficients of an acoustic network model of a turbulent swirling combustor. In a more recent addition to the literature, Van Horn and Scarborough [8] report scattering matrices for an axial swirler and a perforated plate and consider the effect of a bias flow on the transmission and reflection coefficients of these elements. The numerical determination of scattering and transfer matrices of burners or elements like swirlers is also possible [9, 10, 11, 12].

There are however uncertainties in the experimental procedures described previously and it is worth making sure that the data gathered complies with acoustic energy conservation principles, like ensuring that a passive element will not produce more energy than it receives. Similarly, this analysis can be extended to models representing complex acoustic elements. The objective of the
present article is hence to assess transfer matrix models and experimental data using acoustic energy conservation principles.

This work is organized into three parts. Practical expressions of acoustic conservation are first derived by examining fluxes of acoustic energy on the two sides of a general element (Section 1). It is shown that this is best accomplished by making use of the scattering matrix formulation. These theoretical expressions are next applied to elements that are modeled analytically (Section 2). The standard $L - \zeta$ model, which is widely used to represent injector dynamics [2, 6, 13], is then considered in this framework, and conditions are derived to ensure that this model complies with acoustic energy conservation principles. This analysis is then used in Section 3 to assess transfer matrix coefficients determined experimentally in the case of a family of acoustically weakly transparent injectors. To finish, this data is compared with the analytical coefficients of the $L - \zeta$ model to see if this model may suitably represent these injectors.

# 1 THEORETICAL ANALYSIS OF POWER FLOW IN ONE DIMENSIONAL ACOUSTIC SYSTEMS

It is first convenient to derive an expression of the acoustic energy flux in terms of Riemann invariants, $f$ and $g$, introduced in Eqns. (1) and (2). The instantaneous flux is given by $\mathcal{F} = p'v'$. This flux integrated over a period of oscillation may be conveniently obtained from a complex representation of pressure and velocity fluctuations through

$$F = \frac{1}{T} \int_{0}^{T} \mathcal{F} dt = \frac{1}{2} \mathrm{Re}\{p'v'^*\} \quad (3)$$

Inserting $p' = \rho c(f + g)$ and $v' = f - g$, this expression becomes:

$$F = \frac{1}{2} \mathrm{Re}\{p'v'^*\} = \frac{1}{2} \rho c \mathrm{Re}\{(f + g)(f - g)^*\} = \frac{1}{2} \rho c [|f|^2 - |g|^2] \quad (4)$$
The last expression is obtained by noting that $f^*g - fg^*$ is purely imaginary and does not contribute to the period average acoustic energy flux. Now, since the area of the various components forming the system is not constant, it is appropriate to consider the acoustic power flow obtained by multiplying the flux by the surface area $A$

$$ \mathcal{P} = \frac{1}{2} A \rho c [ |f|^2 - |g|^2 ] \qquad (5) $$

Let us now consider the power flow in an element defined by a scattering matrix $S$. The outgoing waves are given in terms of the incoming waves by the following expression

$$ \begin{bmatrix} f_2 \\ g_1 \end{bmatrix} = \begin{bmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{bmatrix} \begin{bmatrix} f_1 \\ g_2 \end{bmatrix} \qquad (6) $$

Conservation of power flowing through the system requires that

$$ A_1 \rho_1 c_1 [ |f|^2 - |g|^2 ]_1 = A_2 \rho_2 c_2 [ |f|^2 - |g|^2 ]_2 \qquad (7) $$

In a passive element, power may be conserved or dissipated. This may be expressed by comparing the incident power to that leaving the element

$$ A_1 \rho_1 c_1 |f_1|^2 + A_2 \rho_2 c_2 |g_2|^2 \geq A_2 \rho_2 c_2 |f_2|^2 + A_1 \rho_1 c_1 |g_1|^2 \qquad (8) $$
This expression may be written in a simpler form by introducing the area ratio $\eta = A_1/A_2$ and the specific impedance ratio $\xi = \rho_1 c_1 / \rho_2 c_2$. One obtains

$$ \eta\xi|f_1|^2 + |g_2|^2 \geq |f_2|^2 + \eta\xi|g_1|^2 \quad (9) $$

To exploit this inequality, it is natural to consider two canonic problems. In the first case, one assumes that a wave is incident from the upstream side while $g_2 = 0$, so that $f_2 = S_{11}f_1$ and $g_1 = S_{21}f_1$. In this case, the acoustic power flowing in section 1 should be greater or equal to the acoustic power flowing in section 2 which translates into

$$ \eta\xi[1 - |S_{21}|^2] - |S_{11}|^2 \geq 0 \quad (10) $$

In the second case, one assumes that a non-zero wave is incident from the downstream side while $f_1 = 0$ on the upstream side. One then has: $f_2 = S_{12}g_2$ and $g_1 = S_{22}g_2$. The second condition reads

$$ [1 - |S_{12}|^2] - \eta\xi|S_{22}|^2 \geq 0 \quad (11) $$

These two conditions ensure that the acoustic power expressed in Eqn. (9) is conserved, when the left-hand sides in the previous two expressions vanish, or that power is dissipated, when they are positive. It is convenient to measure the power dissipated in the system with respect to the power associated with the incoming wave. These ratios are easily deduced from Eqns. (10) and (11)

$$ \psi_{1 \to 2} = [1 - |S_{21}|^2] - [1/(\eta\xi)]|S_{11}|^2 \quad (12) $$

$$ \psi_{1 \leftrightarrow 2} = [1 - |S_{12}|^2] - (\eta\xi)|S_{22}|^2 \quad (13) $$
## 2 ACOUSTIC POWER FLOW OF MODELED TRANSFER MATRICES

It is important to note that in accordance with the standard convention, waves in the system have the common harmonic factor $exp(-i\omega t)$ and keep in mind that this choice differs from that used in many articles cited as references.

It is first natural to examine the scattering matrix coefficients corresponding to a constant section duct of length $l$ and see if they ensure energy conservation using expressions derived previously. With the present convention, the transfer matrix takes the form

$$T = \begin{bmatrix} \cos kl & i \sin kl \\ i \sin kl & \cos kl \end{bmatrix} \qquad (14)$$

where $k = \omega/c$ is the wavenumber, and $\omega$, the angular frequency. The corresponding scattering matrix is

$$S = \begin{bmatrix} \exp(ikl) & 0 \\ 0 & \exp(ikl) \end{bmatrix} \qquad (15)$$

and one finds that the two left-hand sides in Eqns. (10) and (11) vanish identically indicating that the acoustic power flowing in the duct is conserved.

A less trivial example is provided by an area change element. The transfer matrix is in this case

$$T = \begin{bmatrix} 1 & 0 \\ 0 & \eta \end{bmatrix} \qquad (16)$$
where $\eta = A_1/A_2$. The scattering matrix now writes

$$S = \begin{bmatrix} \frac{2\eta}{1+\eta} & \frac{1-\eta}{1+\eta} \\ -(1-\eta)/(1+\eta) & \frac{2}{1+\eta} \end{bmatrix} \qquad (17)$$

A few calculations show that $\eta[1 - |S_{21}|^2] = |S_{11}|^2$ and $[1 - |S_{12}|^2] = \eta|S_{22}|^2$, and one finds again that acoustic energy conservation is satisfied.

It is now interesting to examine the $L - \zeta$ model, formulated in [2, 6, 13]. The model, derived from unsteady Bernoulli equations for a compact acoustic element, is generally expressed in the form of a transfer matrix

$$T = \begin{bmatrix} 1 & B_{12} \\ 0 & A_1/A_2 \end{bmatrix} \qquad (18)$$

where $B_{12}$ relates the difference in $p'/(\rho c)$ between the output and input of the injection unit to the velocity fluctuation $v'_1$, and is expressed in the form

$$B_{12} = M[1 - \zeta_b - (A_1/A_2)^2] + i\omega l_{eff}/c \qquad (19)$$

In this expression, $M$, $\zeta_b$, $\omega$, and $l_{eff}$ are the mean Mach number in the injector, the pressure loss coefficient, the angular frequency, and a reduced length. One may first consider the case where $A_1$ differs from $A_2$ and use $\eta = A_1/A_2$ to represent the area ratio. It is also convenient to define $\delta = (1 + \eta)/2$ and $b = B_{12}/2$. It is then a simple matter to obtain the scattering matrix

$$S = \begin{bmatrix} \frac{\eta}{\delta - b} & \frac{(\delta - b - \eta)}{(\delta - b)} \\ -(\delta + b - \eta)/(\delta - b) & \frac{1}{\delta - b} \end{bmatrix} \qquad (20)$$
To analyze the power flow in the injection unit, it is easier to assume that the entrance and exhaust areas $A_1$ and $A_2$ are equal so that $\eta = 1$ and $\delta = 1$. The scattering matrix becomes in this particular case

$$S = \begin{bmatrix} 1/(1-b) & -b/(1-b) \\ -b/(1-b) & 1/(1-b) \end{bmatrix} \qquad (21)$$

Let us consider the low frequency range and assume that $\omega l_{eff}/c \ll 1$ so that $B_{12} \approx -M\zeta_b$ is real and negative.

One may then deduce $\psi_{1 \to 2}$ and $\psi_{1 \leftarrow 2}$. Some calculations lead to

$$\psi_{1 \to 2} = \psi_{1 \leftarrow 2} = \frac{|b|}{(1 + |b|)^2} \qquad (22)$$

One concludes that acoustic energy is dissipated in the system and that the proportion of power dissipated in the injector does not depend on the direction of the incident wave. Under standard conditions, the Mach number is relatively small so that $2|b| = |B_{12}| \approx M\zeta_b$ and the proportion of power dissipated in the system is approximately given by $\psi_{1 \to 2} = \psi_{1 \leftarrow 2} \approx M\zeta_b$.

In complement to the power flow analysis carried out above, it is also instructive to consider the energy density on the input and output sections of an element. The instantaneous energy density is given by

$$\mathcal{E} = \frac{1}{2} \frac{p'^2}{\rho c^2} + \frac{1}{2} \rho v'^2 \qquad (23)$$

The energy density integrated over a period is

$$E = \frac{1}{T} \int_{0}^{T} \mathcal{E} dt = \frac{1}{2} \left[ \frac{1}{2} \frac{|p'|^2}{\rho c^2} + \frac{1}{2} \rho |v'|^2 \right] \qquad (24)$$
Inserting $p'/(\rho c) = f + g$ and $v' = f - g$, one obtains after a few calculations

$$E = \frac{1}{2} \rho (|f|^2 + |g|^2) \qquad (25)$$

Considering case 1, where the system is submitted to a wave $f_1$ while $g_2 = 0$, one finds that the ratio of the energy density in section 2 to the energy density in section 1 is given by

$$\frac{E_2}{E_1} = \frac{|S_{11}|^2}{1 + |S_{21}|^2} \qquad (26)$$

In case 2, where the incident wave is $g_2$ while $f_1 = 0$, the ratio of energy densities is now given by

$$\frac{E_1}{E_2} = \frac{|S_{22}|^2}{1 + |S_{12}|^2} \qquad (27)$$

In the case of an area change, one obtains

$$\frac{E_2}{E_1} = \frac{2\eta^2}{1 + \eta^2}, \quad \frac{E_1}{E_2} = \frac{2}{1 + \eta^2} \qquad (28)$$

The apparent asymmetry in these expressions may be eliminated by replacing $\eta$ by its expression $A_1/A_2$. One obtains

$$\frac{E_2}{E_1} = \frac{2A_1^2}{A_1^2 + A_2^2}, \quad \frac{E_1}{E_2} = \frac{2A_2^2}{A_1^2 + A_2^2} \qquad (29)$$

which are perfectly symmetric. Analytical expressions of the energy density ratios may also be obtained for the $L-\zeta$ model by considering the low frequency limit and assuming that $b$ is real and
negative. One finds

$$ \frac{E_2}{E_1} = \frac{\eta^2}{\delta^2 + (\eta - \delta)^2 + 2|b|\eta + 2|b|^2} \quad (30) $$

$$ \frac{E_1}{E_2} = \frac{1}{\delta^2 + (\eta - \delta)^2 + 2|b|\eta + 2|b|^2} \quad (31) $$

One may verify that the last two expressions tend to those found previously for an area change element (Eqn. (28)) in the $b = 0$ limit.

## 3 MEASURED TRANSFER MATRICES AND ACOUSTIC POWER FLOW

The analytical expressions derived in the previous section are now used to examine scattering matrices determined experimentally. This section begins with a brief description of the experimental test facility and data processing. The measurement method is demonstrated in the case of a constant area channel and transfer and scattering matrices are measured for a set of injectors. This data is then assessed with the energy analysis exposed previously: the proportion of power dissipated in the injector units is estimated using Eqns. (12) and (13) and the energy density ratios $E_2/E_1$ and $E_1/E_2$ are examined using Eqns. (26) and (27). In the last part of this section, experimental results are compared to the $L - \zeta$ model.

### 3.1 Experimental determination of transfer matrices

The acoustic transfer matrices are determined in an impedance tube, that is usually designated as a Kundt's tube. In the experimental setup shown in Fig. 1, the element one wishes to characterize is placed in a channel of constant diameter, here 5 cm. A driver unit, connected to a sinewave generator, placed on the left end of the tube, generates harmonic modulations in a range of frequencies extending from 250 to 850 Hz. Brüel & Kjær microphones are flush-mounted upstream (microphones M1, M2, and M3) and downstream of the sample (microphones M4, M5, M6, and M7) to measure the acoustic pressure and velocity, using the multi-microphone method [14, 15]. Microphones are separated by 5 or 5.5 cm from each other, a spacing well below the
wavelengths associated with the frequency range of interest. Microphone signals are acquired with a sampling frequency $f_s=16 384$ Hz. Air is supplied to the system through an aperture in the upstream section of the tube and the flow rate is controlled through a Bronkhorst EL-FLOW mass flow controller with a relative accuracy of 1%. The bias flow velocity is kept low to preserve a good signal-to-noise ratio by reducing turbulent flow noise. The flow velocity in the channel, upstream of the injector, is 0.25 m s⁻¹, inducing a bulk velocity at the injector outlet $U_b = 12$ m s⁻¹.

The “two-load” method is used in order to obtain two acoustically independent states and measurements are made in two configurations, (a) and (b). This way, one gets two independent acoustic fields, and therefore four equations for the four unknown transfer matrix coefficients. For this study, measurements in configuration (a) correspond to an open end and for those in configuration (b), a perforated plate is placed at the right end of the channel. Other studies using the two-load method report the use of alternative ends, such as an open end and an additional channel extending the test section [7].

The transfer matrix coefficients are then determined by solving the system below

$$ \begin{bmatrix} (p'_2)^a/\rho c \\ (v'_2)^a \\ (p'_2)^b/\rho c \\ (v'_2)^b \end{bmatrix} = \begin{bmatrix} (p'_1)^a/\rho c & (v'_1)^a & 0 & 0 \\ 0 & 0 & (p'_1)^a/\rho c & (v'_1)^a \\ (p'_1)^b/\rho c & (v'_1)^b & 0 & 0 \\ 0 & 0 & (p'_1)^b/\rho c & (v'_1)^b \end{bmatrix} \begin{bmatrix} T_{11} \\ T_{12} \\ T_{21} \\ T_{22} \end{bmatrix} \quad (32) $$

where the superscripts a and b designate measurements in configuration (a) and (b) respectively.

Measurements are carried out for a set of equally-spaced discrete frequencies between 250 Hz and 850 Hz, and a loudspeaker input voltage of 3.5 V (peak to peak). Hilbert transforms are used to deduce analytical signals from the real signals recorded by the microphones. Coherence between upstream and downstream signals was checked to be higher than 0.9 to ensure a good signal-to-noise ratio. Acoustic pressure and velocity are then determined at a position upstream and downstream of the element using the multi-microphone method, and the transfer matrix coef-
ficients are found by solving Eqn. (32). The transfer matrix of the element is extracted by removing
the portions of straight channel between the acoustic variables measurement location and the in-
jector inlet and outlet. The area changes between the impedance tube and the injector inlet and
outlet also have to be taken into account.

The data processing methodology is first validated for the straight duct test case. A straight duct of length l=31 cm, corresponding to the distance between the last microphone on the upstream side of the test section and the first microphone on the downstream side of the test section, is considered. The measurements are carried out in the presence of a bias flow velocity of 0.25 m s⁻¹ in the tube. The experimental and theoretical results are displayed in Fig. 2. A good match is observed between experimental data and theoretical predictions.

3.2 Transfer and scattering matrices of injector units and acoustic power flow analysis

The injector unit investigated is presented in Fig. 3. It comprises an air distributor, also called “main body”, formed by 6 cylindrical channels guiding the air into a radial swirler, which imparts a tangential velocity to the air in the clockwise direction. Air exits the injector through a conical hole in the terminal plate. Changing the swirler geometry (channels diameter and orientation) changes the injection unit pressure drop and swirl number. Three different swirlers are investigated, and their pressure drop and swirl number, measured under cold flow conditions, with an air flow rate of 2.6 g s⁻¹ [16], are gathered in Tab. 1.

The investigation of the transfer matrices of these different injector configurations is motivated
by recent experiments revealing the influence of the injector unit on thermo-acoustic instabilities in
the single-injector configuration SICCA-Spray [17, 18] and in the multiple-injector annular system
MICCA-Spray [19]. It is therefore important to get a reliable representation of this unit that can
be used in thermo-acoustic reduced-order models. Previous studies [20] showed that this family
of injectors induces a non-negligible dissipation of acoustic energy, due to the area changes and
high pressure drop values induced by the swirler unit. They are for this reason deemed as “weakly
transparent to acoustic waves”. Such conditions are less well-documented in the literature, since
many studies consider axial swirlers having lower pressure drop levels and a flow pathway al-
lowing acoustic propagation with a lesser degree of dissipation. The present study thus provides
indications on the case of radial swirlers that are less transparent to acoustic waves.

The transfer matrix coefficients of the injector equipped with swirlers 712, 716, and 726, obtained experimentally, are plotted in Fig. 4. In order to assess the role played by the swirler in the acoustic response of the injector, the results are compared to the case of the injector without swirler (denoted "MB-TP" for "Main Body-Terminal Plate" in the following figures). The effect of the swirler is mainly visible in the $T_{12}$ plot, for frequencies above 400 Hz, where the modulus takes lower values for the injector without swirler than in the case where the injector is equipped with one of the swirlers. The differences observed in the moduli and phases corresponding to the different swirling injectors are relatively small. This could hint that the variations in flame dynamics observed in MICCA-Spray and SICCA-Spray equipped with the different swirling injectors result from modifications in the flame response to acoustic disturbances rather than from differences in the acoustic characteristics of the injectors.

The corresponding scattering matrices are deduced from transfer matrix coefficients using the equations given, for example, in [21]. The scattering matrix representation is physically meaningful since it links Riemann invariants upstream and downstream of the element and enables a causal interpretation of the coefficients [21]. In this formalism, $S_{11}$ is the transmission coefficient from upstream to downstream and $S_{22}$ from downstream to upstream. $S_{12}$ and $S_{21}$ are reflection coefficients of waves originating from downstream and upstream respectively.

The scattering matrix coefficients for the injector equipped with swirlers 712, 716, and 726, and the injector without swirler are shown in Fig. 5. The modulus of the transmission coefficient $S_{11}$ is slightly higher in the case of the injector without swirler than in the complete injector case, hinting that the swirler plays a non-negligible role in impeding the passage of acoustic waves propagating from upstream and through the injector. The value of the transmission coefficient $S_{22}$ is below 0.5 for the range of frequencies tested, showing that the transmission of acoustic waves from downstream is weak throughout the range of frequencies tested. The modulus of reflection coefficient $S_{12}$ is around 0.5 for low frequencies and increases with frequency to reach values close to 0.9 above 800 Hz, indicating that the reflection of waves coming from the downstream side of the injector is augmented as frequency increases. The modulus of coefficient $S_{21}$ is nearly
constant and equal to 0.5 in the range of frequencies studied, with a slight decrease and then
increase after 600 Hz.

The scattering matrix coefficients obtained above are now used to calculate the acoustic en-
ergy density ratios E₂/E₁ and E₁/E₂, defined in Eqns. (26) and (27), which are displayed in Fig.
6. The ratio E₂/E₁ can be interpreted as the proportion of acoustic energy transmitted to the
downstream side, in a case where the incoming wave propagates from upstream. This ratio in-
creases up to 350 Hz and then decreases with frequency, reflecting the evolution of the modulus
of coefficient S₁₁. The values taken by the ratio E₂/E₁ are higher in the case of the injector without
swirler, which once more highlights the role played by the swirler in the transmission of acoustic
waves originating from upstream. One can also observe that E₂/E₁ takes values higher than one
for frequencies below 450 Hz. This is due to the fact that the area ratio between the inlet and
the outlet of the injector, η = A₁/A₂, is equal to 3.37, making the acoustic energy density higher
on the downstream side of the injector than on the upstream side. If one now examines the ratio
E₁/E₂, which corresponds to the proportion of acoustic energy originating from downstream and
that is transmitted upstream, one can see that the acoustic energy is only partially transmitted from
downstream to the upstream side of the injector, and this is so for the whole range of frequencies
tested. Thus, during a self-sustained instability, in which acoustic perturbations are generated by
unsteady heat release rate fluctuations and approach the injector from the downstream side, the
combustion chamber is decoupled from the upstream manifold by the injection unit.

To complete the analysis, one needs to verify that the coefficients determined experimentally
comply with the acoustic energy conservation principles. To this end, the relative acoustic power
flows, expressed in terms of $\psi_{1\to2}$ and $\psi_{1\leftarrow2}$, introduced in Eqns. (12) and (13), are plotted in
Fig. 7, for the injector without swirler and for the injector equipped with the three different swirlers.
These two quantities represent the power dissipated in the system with respect to the incoming
acoustic power flow from upstream for $\psi_{1\to2}$ and from downstream for $\psi_{1\leftarrow2}$. Both quantities are
positive for the range of frequencies tested, which complies with the acoustic energy conservation
principles expressed in Eqns. (10) and (11). Energy dissipation depends on the frequency, and
one observes an increase followed by a decrease from 700 Hz for both $\psi_{1\to2}$ and $\psi_{1\leftarrow2}$. The effect
of the swirler is mostly visible on $\psi_{1\leftarrow 2}$, the latter taking lower values when the swirler is absent. Figure 7 shows that $\psi_{1\rightarrow 2}$ is higher than $\psi_{1\leftarrow 2}$. This can be explained by the presence of a bias flow. A mode conversion takes place at the exit of the injector through the generation of vortices convected in the downstream region. In this region, the convective vorticity mode captures a part of the incident acoustic energy. This is investigated in [22] in a case where the incident acoustic wave impinges on a diaphragm placed in a channel. A large eddy simulation [12] carried out in the case of a radial swirler also indicates that vortex formation and shedding are the main acoustic energy loss mechanisms. The asymmetry observed between $\psi_{1\rightarrow 2}$ and $\psi_{1\leftarrow 2}$ may then be linked to a difference in the mode conversion efficiency when the incident wave is traveling in the flow direction and when it travels in the opposite direction. This is only an inference that needs to be confirmed.

### 3.3 Comparison with the $L-\zeta$ model

It is now worth comparing the experimental data to the $L-\zeta$ model, for the transfer matrix and scattering matrix coefficients as well as from an acoustic energy analysis point of view.

For compact acoustic elements, in addition to the standard expression presented in Eqn. (18), another form of the $L-\zeta$ model can be found in [6, 13], where coefficient $T_{21}$ is non zero and takes into account the capacitance effects. The corresponding transfer matrix reads:

$$T = \begin{bmatrix} 1 & [1 - \zeta_b - (\frac{A_u}{A_d})^2]M_u + i\frac{\omega}{c}l_{eff} \\ i\frac{\omega}{c}l_{red} - M_d & A_u/A_d \end{bmatrix} \quad (33)$$

where $A_u$, $A_d$, $M_u$, and $M_d$, are the upstream and downstream sections of the injector, the Mach numbers in the upstream and downstream sections respectively, and $\zeta_b$, the element pressure loss coefficient. Two characteristic lengths, $l_{red}$ and $l_{eff}$, appear to represent capacitance and inductance effects respectively, and are defined as follows [13]
$$l_{red} = \int_{u}^{d} \frac{A(x)}{A_{d}} dx, \quad l_{eff} = \int_{u}^{d} \frac{A_{u}}{A(x)} dx \qquad (34)$$

These quantities are determined for the injector equipped with swirler 716. Two cases are considered in the study: the standard $L - \zeta$ model, where $T_{21} = 0$, as in Eqn. (18), and the $L - \zeta$ model with capacitance effects taken into account, where $T_{21} \neq 0$ and expressed as shown in Eqn. (33). The latter will be referred to as “$L - \zeta$ model with capacitance effects”.

The transfer matrix coefficients measured experimentally are first compared to those obtained with the $L - \zeta$ model in Fig. 8. The $L - \zeta$ model compares well to the experimental data in the low frequency range (below 400 Hz), but differences are observed when the frequency increases. These differences can be explained by the fact that the model does not capture the acoustic dissipation effects that are more pronounced at higher frequencies, as pointed out by Paschereit et al. [4] and Van Horn and Scarborough [8], who also observed a good agreement of the model with experimental values for frequencies below 400 Hz, but differences at higher frequencies. In the present case, the modulus of $T_{11}$, after taking values close to 1 up to 400 Hz, decreases to reach values below 0.5 around 800 Hz. This behavior might be related to the compactness hypothesis losing validity when frequency increases, and acoustic elements becoming less compact due to wavelengths decrease with frequency, as discussed in [8]. The increase of $T_{22}$ with frequency is not captured by the $L - \zeta$ model, although the low-frequency limit is suitably predicted. The $L - \zeta$ model with capacitance effects retrieves the increase of $|T_{21}|$ with frequency, but the model values deviate from the experimental data for frequencies above 400 Hz. In comparison, the standard $L - \zeta$ model sets $T_{21}$ to 0 and does not reproduce the experimental trend for this coefficient. To finish, a good match is observed between the $L - \zeta$ model and experiments for the phase angle, but the phase of $T_{21}$ is well retrieved only when capacitance effects are included.

When one examines the corresponding scattering matrix coefficients displayed in Fig. 9, in the case where capacitance effects are taken into account ($T_{21} \neq 0$), which corresponds to the black curve in Fig. 9), the $L - \zeta$ model overpredicts the transmission coefficients $S_{11}$ and predicts an
increase with frequency whereas a decrease is observed experimentally. The transmission coefficient $S_{22}$ predicted by the model suitably captures the trend observed in the experimental data. The reflection coefficient $|S_{12}|$ is underpredicted for frequencies above 400 Hz and its increase observed in the experimental data is not captured, while the reflection coefficient $|S_{21}|$ is slightly overpredicted in the range from 400 and 750 Hz. If one now considers the case where capacitance effects are not accounted for ($T_{21} = 0$), which corresponds to the red curve in Fig. 9), the main differences appear in coefficients $|S_{11}|$ and $|S_{22}|$, which exhibit significant discrepancies with the experimental results. The phase angles obtained with both models are close to the experimental observations. Only the phase of coefficient $S_{21}$ calculated with the standard $L-\zeta$ model is slightly off, especially at high frequencies.

It is instructive to compare the acoustic power flow predictions of the $L-\zeta$ model with those deduced from the experimental data, shown in Fig. 10. One can see that the model underestimates the level of dissipation and does not retrieve the evolution with frequency. In the case where capacitance effects are not included ($T_{21} = 0$), one finds that $\psi_{1\to2} = \psi_{1\leftarrow2}$ and their common values, corresponding to the red line in Fig. 10, are nearly constant with respect to frequency, and well below the experimental levels. $\psi_{1\to2}$ and $\psi_{1\leftarrow2}$ are however above 0 in the range of frequency investigated, which confirms that this model complies with acoustic energy conservation principles. When capacitance effects are included, the acoustic energy conservation is not verified since $\psi_{1\to2}$ takes negative values throughout the range of frequencies studied. This can be attributed to the overprediction of the transmission coefficient $S_{11}$, its modulus taking values above 1.5. This model also predicts opposite trends for $\psi_{1\to2}$ and $\psi_{1\leftarrow2}$, whereas similar trends but different values are observed in the experimental data. The asymmetry in the dissipation of acoustic energy has to be captured if one wishes to reproduce the injector contribution to the damping in a thermo-acoustic system.

## 4 CONCLUSION

Acoustic energy conservation principles are first used to assess transfer matrix models that arise in combustion instability analysis. It is found that acoustic energy is conserved by some
standard models and dissipated by the $L-ζ$ model, which can be used to represent injector units. Transfer and scattering matrices are then determined experimentally for a family of acoustically weakly transparent injectors and analyzed from an energetic standpoint. Measurements show that for these injectors, acoustic energy dissipation is more important for acoustic waves propagating from the upstream side of the injector. Finally, the experimentally measured transfer matrix coefficients are compared to predictions from the $L-ζ$ model. Experimental results match with the model in the low-frequency limit, but differences are found when frequency increases. It also appears that, in order to correctly represent the injector dynamics, the $L-ζ$ model needs to include capacitance effects, to capture the trend observed in the experimental measurements of coefficient $T_{21}$. An energetic analysis shows that the $L-ζ$ model underestimates the level of acoustic energy dissipation that the injector unit induces. Differences in energy dissipation observed experimentally when the acoustic wave originates from the upstream or downstream sides are not retrieved either. More generally, this study shows that it is useful to estimate acoustic power flows in combustion components like injectors and to check that they comply with acoustic energy dissipation when these components are passive. The impact of these findings on the low-order modeling of instabilities and their predictions needs further consideration.

## ACKNOWLEDGEMENTS

This work was partially supported by the FlySAFe project of the Agence Nationale de la Recherche ANR-22-CE05-0022-02.

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TABLES

<table><caption>Table 1: Swirl number, pressure drop and pressure drop coefficient of the swirlers used in the study, measured under cold flow conditions with an air flow rate of 2.6 g s⁻¹ (adapted from [19]).</caption><thead><tr><th>Swirler</th><th>S<sub>N</sub></th><th>Δp (kPa)</th><th>ζ<sub>b</sub></th></tr></thead><tbody><tr><td>712</td><td>0.59</td><td>4.50</td><td>4.10</td></tr><tr><td>716</td><td>0.70</td><td>5.74</td><td>5.23</td></tr><tr><td>726</td><td>0.74</td><td>6.00</td><td>5.47</td></tr></tbody></table>
FIGURES

<figure><img src="image_2.png" /><figcaption>Fig. 1: Kundt's tube setup for transfer matrix measurements. The two-load method is used and measurements are carried out with an open end in configuration (a) and a perforated plate in configuration (b), shown here. Microphones are labelled MX.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Transfer matrix of a straight channel of length $l=31$ cm measured in the impedance tube with the two-load method and with a bias flow of $0.25 \text{ m s}^{-1}$. Experimental data (●) and theoretical model (---).</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Fig. 3: An exploded view of the injector unit investigated in this study, with its various components together with a schematic drawing of the swirler.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4: Transfer matrices of the injector equipped with swirlers 712 (■), 716 (♦), and 726 (★), and of the injector without swirler, MB-TP (○), measured in the impedance tube with the two-load method and a bias flow of 12 m s⁻¹ at the injector outlet.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 5: Scattering matrices of the injector equipped with swirlers 712 (■), 716 (♦), and 726 (★), and of the injector without swirler, MB-TP (○), measured in the impedance tube with the two-load method and with a bias flow of 12 m s⁻¹ at the injector outlet.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6: Acoustic energy density ratios for the injector equipped with swirlers 712 (♦,●), 716 (♦,●), and 726 (♦,●), and the injector without swirler, MB-TP, (◇,○), obtained from scattering matrix measurements with a bias flow of 12 m s⁻¹ at the injector outlet. Diamond symbols (♦) correspond to E₂/E₁ and circular symbols (●) to E₁/E₂.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 7: Acoustic energy dissipation for the injector equipped with swirlers 712 (♦,●), 716 (♦,●), and 726 (♦,●), and the injector without swirler, MB-TP,(◇,○), obtained from scattering matrix measurements with a bias flow of 12 m s⁻¹ at the injector outlet. Diamond symbols (♦) correspond to ψ₁→₂ and circular symbols (●) to ψ₁↔₂.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8: Transfer matrix of the injector equipped with swirler 716 (♦) measured in the impedance tube with the two-load method and a bias flow of 12 m s⁻¹ at the injector outlet, and comparison with the standard L – ζ model (—) and the L – ζ model with capacitance effects (—).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 9: Scattering matrix of the injector equipped with swirler 716 (♦) measured in the impedance tube with the two-load method and with a bias flow of 12 m s⁻¹ at the injector outlet, and comparison with the L – ζ model (—) and the L – ζ model with capacitance effects (---).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10: Acoustic energy dissipation for the injector equipped with swirler 716 (♦,●) obtained from scattering matrix measurements with a bias flow of 12 m s⁻¹ at the injector outlet. Diamond symbols (♦) correspond to $\psi_{1\to2}$ and circular symbols (●) to $\psi_{1\leftarrow2}$. Comparison with the L – ζ model ($\psi_{1\to2} = \psi_{1\leftarrow2}$, —) and the L – ζ model with capacitance effects ((--) for $\psi_{1\to2}$ and (—) for $\psi_{1\leftarrow2}$).</figcaption></figure>