2.1. The MICCA-spray annular system

The MICCA-spray annular combustion chamber, shown in figure 1, is formed by two quartz walls of the same length, $l=400$ mm, of outer diameter 300 mm for the inner cylinder and inner diameter 400 mm for the outer quartz cylinder. The backplane comprises 16 regularly spaced injection units, which deliver liquid fuel, heptane in the present experiments, as a hollow cone spray of droplets mixed with air (Vignat et al. Reference Vignat, Rajendram Soundararajan, Durox, Vié, Renaud and Candel2021). The air mass flow rate is controlled through two Bronkhorst EL-FLOW mass flow meters and the fuel mass flow rate through a Bronkhorst CORI-FLOW mass flow rate controller. Eight Brüel & Kjær microphones, flush-mounted on waveguides to protect them from the hot gas temperature environment, enable to record pressure fluctuations at the combustion chamber backplane, with a sampling frequency $f_s=32\,768$ Hz.

Figure 1. (a) MICCA-Spray test rig and array of the eight photomultipliers used to measure OH* emission from the flames (these sensors are employed in § 4.2 and Appendix C). (b) Top view of MICCA-Spray and microphones positions. (c) MICCA-Spray equipped with eight $U$-injectors placed on one half of the annulus and eight $S$-injectors on the other half. The white dashed line separates $U$ and $S$ injection units.

The stability maps of MICCA-Spray equipped with these two types of injectors are presented by Rajendram Soundararajan et al. (Reference Rajendram Soundararajan, Durox, Renaud and Candel2022a). In that reference, as already mentioned, ‘injector $S$’ is denoted ‘707’ and ‘injector $U$’, ‘727’. The operating point chosen for the present study corresponds to a thermal power value $\mathcal {P}=118$ kW and a global equivalence ratio $\phi =0.9$. Under these conditions, the chamber is unstable when it is equipped with $U$-injectors and stable when it is equipped with $S$-injectors.

2.2. Injectors’ characteristics and flame describing functions

An exploded view of the injection units used in this study is presented in figure 2(c). These injectors are composed of four main elements: an air distributor, a liquid fuel atomizer, a swirler and a terminal plate with a convergent nozzle. Changing the swirler changes the swirl number and pressure drop through the injection unit.

Figure 2. (a) SICCA-Spray set-up, (b) SICCA-Spray configuration for FDF measurements and (c) exploded view of the injector along with a top view of the swirler.

The characteristics of injectors $U$ and $S$ (swirl number, pressure drop and pressure drop coefficient) are gathered in table 1. Injector $S$ presents a lower swirl number and a lower pressure drop value than injector $U$.

Table 1. Injectors’ characteristics: swirl number ($S_N$), pressure drop ($\Delta p$) and pressure drop coefficient ($\sigma$), obtained from measurements under cold conditions for an air mass flow rate of $2.3\ {\rm g}\ {\rm s}^{-1}$ (Vignat et al. Reference Vignat, Rajendram Soundararajan, Durox, Vié, Renaud and Candel2021).

It is also interesting to examine the effects of the injector characteristics on flame dynamics through the measurement of flame describing functions (FDFs). FDFs characterize the response of flames to acoustic disturbances and the FDF framework (Dowling Reference Dowling1997; Noiray et al. Reference Noiray, Durox, Schuller and Candel2008), which is the nonlinear extension of flame transfer functions, is now widely used in combustion system stability analysis. They enable to capture saturation effects on the flame response and explain nonlinear phenomena such as triggering, mode hopping or hysteresis. FDFs are the subject of theoretical investigations (Ghirardo et al. Reference Ghirardo, Ćosić, Juniper and Moeck2015; Orchini, Illingworth & Juniper Reference Orchini, Illingworth and Juniper2015) and are commonly used as an input in reduced-order models to interpret experimental observations (Schuermans et al. Reference Schuermans, Guethe, Pennell, Guyot and Paschereit2010; Paliès et al. Reference Paliès, Durox, Schuller and Candel2011; Han, Li & Morgans Reference Han, Li and Morgans2015; Schuller, Poinsot & Candel Reference Schuller, Poinsot and Candel2020).

FDF measurements are carried out in a single-injector configuration, SICCA-Spray, displayed in figure 2, representing one sector of the annular system MICCA-Spray. The system can be modulated from the upstream side of the injector by two driver units placed at the bottom of the plenum and connected to a sinewave generator. The operating point in SICCA-Spray is chosen so that it corresponds to the thermal power value of one injector in MICCA-Spray, that is to say, 1/16th of the total thermal power in MICCA-Spray.

With the type of injectors used, equivalence ratio fluctuations are small compared with velocity fluctuations, so that the flame behaves in a quasi-premixed fashion (Rajendram Soundararajan et al. Reference Rajendram Soundararajan, Durox, Renaud, Vignat and Candel2022b). FDFs are thus defined as the ratio of relative heat release rate fluctuations to relative volume flow rate fluctuations

(2.1)\begin{equation} {\mathcal{F}}(f, q')=\frac{{\dot{Q}}'/\bar{\dot{Q}}}{q'/\bar{q}}= G(f,q')\exp({{\rm i}\varphi(f,q')}). \end{equation}

Relative heat release rate fluctuations are measured with a photomultiplier equipped with an OH* filter centred at 310 nm. The injector being weakly transparent to acoustic waves, relative velocity fluctuations measured upstream of the injector are not the same as those measured on its downstream side. Hence, velocity fluctuations are measured with a laser Doppler anemometry system (Dantec Dynamics), in the combustion chamber, at a position ($z=2.5$ mm above the backplane and at $r=3.5$ mm from the centre for injector $S$ and at $r=4.0$ mm for injector $U$) where relative velocity fluctuations match relative volume flow rate fluctuations, to comply with the FDF definition (Rajendram Soundararajan et al. Reference Rajendram Soundararajan, Durox, Renaud, Vignat and Candel2022b). The fuel droplets, featuring a mean diameter below $5\ \mathrm {\mu }{\rm m}$ at the measurement positions, can serve as flow tracers in the frequency range of interest (250–850 Hz). Measurements are carried out for seven levels of modulation amplitudes, corresponding to different voltage outputs of the sinewave generator, and leading to different values of relative velocity fluctuations depending on the frequency and the level of the modulation amplitude.

FDFs measured in SICCA-Spray equipped with injectors $U$ and $S$ are displayed in figure 3. FDF gain values are substantially lower for injector $S$ than for injector $U$, and this is particularly true in a range of frequencies around 800 Hz, close to the eigenfrequency of the 1A1L (1st azimuthal–1st longitudinal) mode of the MICCA-Spray annular combustion chamber, involved in the combustion/acoustics coupling, as will be detailed in the next section. Differences in phases are also observed, with higher phase values for injector $S$ than injector $U$ in the range of frequencies investigated. Figure 3 also displays flame images obtained from the Abel inversion of the images captured with a charge-coupled device (CCD) camera, to show the effect of the injector pressure drop and swirl number on the flame shape. The ‘$V$’ shaped flame obtained with injector $S$ is longer and narrower than that obtained with injector $U$, a hollow ‘$M$’ shape flame which is more compact, due to a higher swirl number of the injection unit.

Figure 3. (a,d) Flame describing function gain, (b,e) flame describing function phase, and (c, f) OH* chemiluminescence distributions obtained after Abel inversion of the average images recorded by an intensified CCD camera, under stable operation in SICCA-Spray: (a–c) $S$-injector, (d–f) $U$-injector.

Another effect that deserves to be considered is that injectors $U$ and $S$ have different pressure drop coefficients. When these injectors are mixed in various proportions, the air flow velocities through these units will not be the same, and since the mass flow rate of fuel is unaffected, the equivalence ratios will also differ for the two types of injectors and their operating point will be shifted with respect to the case where all injection units are the same. Calculations reported in Appendix A indicate that the operating point for $U$-injectors are shifted towards higher equivalence ratio values, while those of $S$-injectors change towards lower values and that the changes depend on the number of $U$ and $S$ injectors. It is shown however in Appendix A, through FDF measurements for different equivalence ratio values, that the variations encountered in this study only have a slight effect on the FDF gain and phase values, and hence on the flame dynamics of the individual flames.

2.3. Modal structures and frequencies

At this point, it is worth briefly describing the acoustic mode structures that can be observed in annular systems, specify those that are effectively observed experimentally and obtain an estimate of the corresponding eigenfrequencies. In the present configuration, the annulus mean radius $R$ is large compared with the width of the annulus, and one may neglect any radial variation. Then, the large ratio between the surface area of the backplane and that of the injector outlet induces a decoupling of the plenum and the combustion chamber: the combustion chamber modes are weakly influenced by the plenum acoustics, as shown by Schuller et al. (Reference Schuller, Durox, Palies and Candel2012), where conditions on temperatures and area ratios to obtain decoupled cavities are discussed. Furthermore, the injectors used are weakly transparent to acoustic waves, accentuating the decoupling (Rajendram Soundararajan et al. Reference Rajendram Soundararajan, Durox, Renaud, Vignat and Candel2022b; Latour et al. Reference Latour, Rajendram Soundararajan, Durox, Renaud and Candel2024). Hence, the chamber backplane acts like a rigid wall and the exhaust is open to the atmosphere. In this configuration, the acoustic coupling modes are of the mixed axial-azimuthal type and may be designated by $mA,nL$. The pressure field can be cast in the following form:

(2.2)\begin{equation} \tilde{p}_{mn}(x, \theta,t) = [A^+_{mn}\exp({\rm i} m\theta-{\rm i}\omega_{mn} t) +A^-_{mn} \exp(-{\rm i} m \theta-{\rm i} \omega_{mn} t)] \psi_n(x). \end{equation}

In this expression, $m$ and $n$ are integers, with $m \geqslant 0$ and $n \geqslant 1$, $\theta$ is the azimuthal angle considered positive in the CCW direction, and $\omega _{mn}$ is the angular frequency. The amplitudes $A^+_{mn}$ and $A^-_{mn}$ correspond to CCW and CW spinning waves, respectively, and $\psi _n(x)$ is the axial wavefunction satisfying the boundary conditions on the chamber backplane and at its exhaust. In the present case, the backplane is rigid and the outlet is open to the atmosphere, i.e. corresponds to a pressure node (in fact, the pressure does not quite vanish at the outlet but at a short distance from that section, and this may be accommodated by adding an end correction $\delta _a$ to the length of the chamber and replacing this length by $l'=l+\delta _a$). The longitudinal eigenfunctions are, in this case: $\psi _n(x)=\cos [(2n-1){\rm \pi} x/(2l')]$. The eigenfrequencies corresponding to the previous modes are then given by a standard expression:

(2.3)\begin{equation} f_{mn}= \left[m^2\left(\frac{c}{{\mathcal{P}}_a}\right)^2+ (2n-1)^2\left(\frac{c}{4l'}\right)^2 \right]^{1/2}, \end{equation}

where ${\mathcal {P}}_a= 2 {\rm \pi}R$ is the mean perimeter of the system and $c$, the speed of sound. In all the present experiments, one observes the 1A1L mode corresponding to $m=1$ and $n=1$, and one may simplify the notation by neglecting the indices $m$ and $n$ in the pressure field expression:

(2.4)\begin{equation} \tilde{p}(x, \theta,t) = [A^+\exp({\rm i} \theta-{\rm i} \omega t) +A^- \exp(-{\rm i} \theta-{\rm i} \omega t)] \psi(x). \end{equation}

It is worth mentioning that this expression of the pressure field does not capture nearfield modifications associated with the flames. One may also think that the purely azimuthal structure will be perturbed by the different staging patterns. However, these effects are expected to be small and this is confirmed by experimental measurements of the annular distributions of pressure signals as will be seen later on. Thus, (2.4) constitutes a reasonable approximation of the pressure field.

Finally, the eigenfrequency of this mode writes

(2.5)\begin{equation} f_{11}= \left[\left(\frac{c}{{\mathcal{P}}_a}\right)^2+ \left(\frac{c}{4l'}\right)^2 \right]^{1/2}. \end{equation}

It is now worth estimating the resonant frequency using the last expression. In the MICCA-Spray experiments, the perimeter ${\mathcal {P}}_a \simeq 1.1$ m. The end correction for an annular system is not well known theoretically, but some pressure measurements near the chamber exhaust indicate that $\delta _a \simeq 0.09$ m so that $l'\simeq 0.49$ m. Assuming an average temperature in the chamber $T\simeq 1500$ K and defining a speed of sound $c=761\ {\rm m}\ {\rm s}^{-1}$, the eigenfrequency of the 1A1L mode is $f_{11}\simeq 808$ Hz. We will see that this value is close to the peak frequencies observed experimentally, indicating that the coupling mode is essentially defined in this set-up by the combustion chamber modes and that most oscillations are of the 1A1L type.

For more complex geometries (for example, in the case of a coupling between the plenum and the combustion chamber), the eigenfrequencies of the modes involved in the combustion/acoustics coupling can be found using a Helmholtz solver, where the geometry and temperatures in the various cavities can be specified (Bourgouin et al. Reference Bourgouin, Durox, Moeck, Schuller and Candel2013). It is important to stress that the presence of the flame acting as an unsteady heat release source modifies the resonant loop, possibly shifting the oscillation frequency of the system with respect to the modal eigenfrequency (Durox, Schuller & Candel Reference Durox, Schuller and Candel2002; Schuller et al. Reference Schuller, Poinsot and Candel2020). However, the acoustic method exposed previously provides a reasonable first approximation of the thermoacoustic oscillation frequency, which can then be refined with a perturbation/correction procedure.

2.4. Modal identification

The signals recorded by the eight waveguided microphones plugged on the chamber backplane can be used to determine the frequency of oscillation of the modes coupling the instability. The peak frequency is obtained from the power spectral density (PSD) of these microphone signals. The PSD is calculated using Welch's method applied to 63 blocks of 8192 samples with a 50 % overlap and a Hamming window weighting. With these parameters, the frequency resolution is $\Delta f=4$ Hz. For all the unstable configurations investigated, the power spectral densities, which will be discussed further in § 4.3, only feature a main peak at the fundamental frequency. Cases leading to non-degenerate modes characterized by close eigenfrequencies, whose difference is below the resolution frequency, will also be examined.

The complex wave amplitudes $A_+$ and $A_-$ may be retrieved from the eight microphone signals by solving an overdetermined system of linear equations using a least squares algorithm. One may then deduce the instability amplitude

(2.6)\begin{equation} A = [{(A^+)^2+(A^-)^2 }]^{1/2}, \end{equation}

which corresponds to a value averaged in time and space, and is independent of the instability mode.

3.1. Side-by-side arrangements of $S$-injectors

A first category of arrangements is investigated and will be referred to as ‘${\mathcal {C}}$ arrangements’. Here, $N_s$ $S$-injectors are placed side-to-side, with $N_s$ ranging from 0, corresponding to the case where there are no $S$-injectors in the system, to $N_s=8$, where half of the injectors is of the $U$ type while the other half is of the $S$ type. Pressure traces are also presented for a configuration comprising 16 $S$-injectors ($N_s=16$) to complete the comparison. The eight investigated arrangements of this kind are displayed in figure 4.

Figure 4. ‘Side-by-side’ (type $\mathcal {C}$) configurations investigated.

Pressure time series for $N_s=0, 4, 8$ and 16 are displayed in figure 5. When $N_s=0$, the combustor exhibits high pressure fluctuation levels and when 16 $S$-injectors are mounted ($N_s=16$), the system operates in a stable fashion. Replacing an increasing number of $U$-injectors by $S$-injectors diminishes the level of pressure fluctuations in the chamber. When $N_s=8$, pressure oscillations reach levels similar to those recorded in configuration ${\mathcal {C}}_{16}$ (around 100 Pa).

Figure 5. Pressure time series in the MICCA-Spray annular combustor for four configurations: (a) ${\mathcal {C}}_0$, (b) ${\mathcal {C}}_4$, (c) ${\mathcal {C}}_8$ and (d) ${\mathcal {C}}_{16}$. Experiments are carried out at a thermal power ${\mathcal {P}}=118$ kW and a global equivalence ratio $\phi =0.9$.

Figure 6 shows the level of pressure fluctuations in the chamber and the frequency of the instability as a function of the number of $S$-injectors. A gradual decrease in the pressure level is observed as $N_s$ is increased. If one now looks at the frequencies, the values found are around 800 Hz, indicating that the mode involved in the combustion/acoustics coupling is the 1A1L mode of the annular chamber. The instability frequency decreases from 820 Hz to 790 Hz as the number of $S$-injectors in the chamber is augmented. The presence of the flames shifts the frequencies with respect to the eigenfrequency of the system without flames (which, in this case, was estimated to be 808 Hz for the 1A1L mode by taking a burnt gas temperature $T=1500$ K). The change in frequency with the number of $S$-injectors in the system reflects modifications in the combustion/acoustics coupling linked to a progressively weaker combustion response as the number of $S$-injectors, characterized by FDFs that differ from those of $U$-injectors, is increased. This might also possibly be explained by a non-uniform azimuthal temperature distribution associated with the variation in equivalence ratio values due to the introduction of $S$-injectors in the system.

Figure 6. (a) Instability amplitude and (b) frequency as a function of the number $N_s$ of $S$-injectors in MICCA-Spray, operated at a thermal power ${{\mathcal {P}}}=118$ kW and a global equivalence ratio $\phi =0.9$, for type ${\mathcal {C}}$ configurations.

3.2. Effects of the $S$-injector azimuthal locations

In what follows, the number of $S$-injectors is fixed, and the effects of the relative position of these units are investigated by considering arrangements with 2, 4, 6 and 8 $S$-injectors. A first group of arrangements shown in figures 7(a)–7(d), denoted ${\mathcal {L}}_2$, ${\mathcal {L}}_4$, ${\mathcal {L}}_6$ and ${\mathcal {L}}_8$, respectively, corresponds to cases where the $S$-injectors are divided into two groups of equal size and placed on opposite sides of the annulus. For $N_s=4$ and $N_s=8$, a second category of arrangements, shown in figures 7(e) and 7( f), is also investigated, and will be denoted ${\mathcal {A}}_4$ and ${\mathcal {A}}_8$ to designate an alternation of $U$- and $S$-injectors.

Figure 7. (a–d) Type ${\mathcal {L}}$ and (e, f) type $\mathcal {A}$ configurations investigated.

Figures 8 and 9 show the arrangements investigated together with the pressure time series. These two figures indicate that, for a given number of $S$-injectors, ‘face-to-face’ (type ${\mathcal {L}}$) arrangements lead to higher levels of pressure fluctuations than ‘side-by-side’ (type ${\mathcal {C}}$) arrangements. In this respect, the case with eight $S$-injectors is particularly interesting (figure 8c–e). Arrangements where the eight $S$-injectors are gathered on one side of the chamber (${\mathcal {C}}_8$) or for which there is an alternation between injectors $S$ and $U$ (${\mathcal {A}}_8$) lead to stable operation while arrangement ${\mathcal {L}}_8$ induces large amplitude oscillations, with pressure fluctuation levels similar to those recorded in the cases without $S$-injectors. For configurations with four $S$-injectors in the system (figure 9c–e), arrangement ${\mathcal {A}}_4$ enables a more efficient reduction of the oscillations amplitude than ${\mathcal {C}}_4$.

Figure 8. Pressure time series in the annular combustor MICCA-Spray, operated at a thermal power ${{\mathcal {P}}}=118$ kW and a global equivalence ratio $\phi =0.9$, for different configurations: (a) ${\mathcal {C}}_6$ and (b) ${\mathcal {L}}_6$, and (c) ${\mathcal {C}}_8$, (d) ${\mathcal {L}}_8$ and (e) ${\mathcal {A}}_8$.

Figure 9. Pressure time series in the annular combustor MICCA-Spray, operated at a thermal power ${{\mathcal {P}}}=118$ kW and a global equivalence ratio $\phi =0.9$, for different configurations: (a) ${\mathcal {C}}_2$ and (b) ${\mathcal {L}}_2$, and (c) ${\mathcal {C}}_4$, (d) ${\mathcal {L}}_4$ and (e) ${\mathcal {A}}_4$.

Figure 10 shows the amplitude of pressure fluctuations and the instability frequency as a function of the number of $S$-injectors for all the configurations tested. Compared with ${\mathcal {C}}$ configurations, ${\mathcal {L}}$ arrangements lead to a level of pressure fluctuations in the chamber higher than or very close to the reference level corresponding to the case where there are no $S$-injectors in the chamber ($N_s=0$). For a fixed value of $N_s$, $\mathcal {A}$ configurations lead to a lower level of pressure fluctuations than ${\mathcal {C}}$ configurations. Instability frequencies are very close for ${\mathcal {C}}$ and ${\mathcal {L}}$ configurations, although the amplitudes of pressure fluctuations are relatively different.

Figure 10. (a) Instability amplitude and (b) frequency as a function of the number of $S$-injectors in MICCA-Spray, $N_s$, for different configurations: ${\mathcal {C}}$ (blue), ${\mathcal {L}}$ (orange) and $\mathcal {A}$ (yellow). Experiments are carried out at an operating point of thermal power ${{\mathcal {P}}}=118$ kW and an equivalence ratio $\phi =0.9$.

3.3. Effects of symmetry breaking on modal dynamics

This section is aimed at examining the consequences of symmetry breaking on the modal dynamics in the annular system. It is worth, at this stage, to complement the review of the literature by briefly examining articles that deal with azimuthal mode dynamics and more specifically with the switching taking place between spinning, standing and mixed modes (Krebs et al. Reference Krebs, Flohr, Prade and Hoffmann2002; Bourgouin et al. Reference Bourgouin, Durox, Moeck, Schuller and Candel2013; Worth & Dawson Reference Worth and Dawson2013,  Reference Worth and Dawson2017; Prieur et al. Reference Prieur, Durox, Schuller and Candel2017; Mazur et al. Reference Mazur, Nygård, Dawson and Worth2019). It is also found that this switching prevails in numerical simulations (Wolf et al. Reference Wolf, Staffelbach, Gicquel, Müller and Poinsot2012) and that the frequency of this phenomenon is relatively low compared with the acoustic frequency scale (Worth & Dawson Reference Worth and Dawson2017).

These questions are actively investigated on the theoretical level, and models are developed to explain why some modes are preferred. Ghirardo & Juniper (Reference Ghirardo and Juniper2013) extended the model proposed by Noiray et al. (Reference Noiray, Bothien and Schuermans2011) by taking into account the effects of acoustic azimuthal velocity on the flames. Noiray & Schuermans (Reference Noiray and Schuermans2013) showed that mode switching can be explained by introducing stochastic noise forcing in analytical models of annular systems. Another question of interest concerns the preferred location of the nodal line in rotationally symmetric annular combustors when the standing mode prevails. An attempt to explain these observations is made by considering spontaneous breaking of rotational and reflectional symmetries in the system (Indlekofer et al. Reference Indlekofer, Faure-Beaulieu, Dawson and Noiray2022). Another theoretical description of the dynamics of azimuthal waves is proposed by Faure-Beaulieu & Noiray (Reference Faure-Beaulieu and Noiray2020) who show that the nature of the mode can be portrayed by making use of three slow-state variables, namely the amplitude, nature angle and nodal line angle. Symmetry breaking in azimuthal combustion dynamics is also analysed by Ghirardo et al. (Reference Ghirardo, Nygård, Cuquel and Worth2021) by making use of the method of averaging and considering small departures in the flame response gain with respect to baseline symmetric values.

Effects of symmetry breaking on modal dynamics in an annular system and on the nodal line position have been investigated in laboratory scale annular configurations. Experiments in the annular Rijke tube showed that the position of the nodal line coincides with the position of the heating grids with the lowest power for some staging patterns (Moeck et al. Reference Moeck, Paul and Paschereit2010). Dawson & Worth (Reference Dawson and Worth2015) observed that placing baffles in the NTNU annular combustor chamber led to preferred standing modes whereas a configuration without baffles featured a permanent switching between standing and spinning modes. Symmetry breaking experiments in MICCA equipped with matrix burners showed that the orientation of the nodal line is controlled by the asymmetry pattern introduced in the system (Aguilar et al. Reference Aguilar, Dawson, Schuller, Durox, Prieur and Candel2021).

It is also worth recalling that modal dynamics can be described using the spin ratio, $s_r$, introduced by Bourgouin et al. (Reference Bourgouin, Durox, Moeck, Schuller and Candel2013), which can be deduced from the wave amplitudes $A_+$ and $A_-$:

(3.1)\begin{equation} s_r=\frac{ |A_+ | - |A_- |}{ |A_+ |+ |A_- |}. \end{equation}

A pure standing mode corresponds to the case where $A_+=A_-$, such that $s_r=0$. For a spinning mode in the clockwise (respectively counterclockwise) direction, $A_+=0$ (respectively $A_-=0$) and the corresponding spin ratio is $s_r=-1$ (respectively $s_r=+1$). This spin ratio definition is close to that formulated by Evesque, Polifke & Pankiewitz (Reference Evesque, Polifke and Pankiewitz2003) based on acoustic energy considerations, but this earlier proposal does not distinguish between clockwise and counterclockwise spinning motions. Another representation based on the quaternion variables (Ghirardo & Bothien Reference Ghirardo and Bothien2018) is also possible. It can be shown that the spin ratio is related to the ‘nature angle’, $\chi$, by $s_r=\tan \chi$.

An interesting visualization of the modal characteristics consists in plotting the joint probability density function ( j.p.d.f.), $p(A_+,A_-)$, calculated over a time series of the pressure field in the annular system. This is exemplified in figure 11 where this j.p.d.f. is shown for different configurations of $U$ and $S$ injection units.

Figure 11. Joint probability density functions of $A_+$ and $A_-$, $p(A_+,A_-)$, determined from pressure time series: (a) ${\mathcal {C}}_0$, (b) ${\mathcal {C}}_1$, (c) ${\mathcal {C}}_2$, (d) ${\mathcal {C}}_4$, (e) ${\mathcal {C}}_6$, ( f) ${\mathcal {L}}_2$, (g) ${\mathcal {L}}_4$, (h) ${\mathcal {L}}_6$ and (i) ${\mathcal {L}}_8$. Dashed lines, corresponding to $A_+/A_- =1/2$ and $A_+/A_- =2$, separate the standing mode from the spinning modes in the clockwise and counterclockwise directions.

In these plots, the distinction between standing and spinning modes is that proposed by Worth & Dawson (Reference Worth and Dawson2013).

1. (i) If $A_+/A_-<1/2$ (i.e. $-1< s_r<-1/3$), the mode is deemed as clockwise spinning.

2. (ii) If $1/2< A_+/A_-<2$ (i.e. $-1/3< s_r<1/3$), the mode is deemed as standing.

3. (iii) If $A_+/A_->2$ (i.e. $1/3< s_r<1$), the mode is deemed as counterclockwise spinning.

When there are no $S$-injectors in the system (configuration ${\mathcal {C}}_0$), $A_+$ and $A_-$ take a wide range of values, and a broad region of the $\{A_+,A_-\}$ plane is explored by the system: the mode switches from spinning to standing at a frequency of the order of a few Hz. When one $S$-injector is added (configuration ${\mathcal {C}}_1$), a preferred spinning mode in the counterclockwise direction is observed. When two or more $S$-injectors are present (configurations ${\mathcal {C}}_2$, ${\mathcal {C}}_4$, ${\mathcal {C}}_6$, ${\mathcal {L}}_2$, ${\mathcal {L}}_4$, ${\mathcal {L}}_6$ and ${\mathcal {L}}_8$), a standing mode is preferred, as points of highest probability density function value lie within the two dashed lines separating the standing mode from the spinning modes.

The location of the $S$-injectors imposes the orientation of the nodal line, as can be seen from the pressure recordings for configurations ${\mathcal {C}}_0$, ${\mathcal {C}}_8$, ${\mathcal {C}}_4$ and ${\mathcal {L}}_4$ presented in figure 12. For configuration ${\mathcal {C}}_0$, which constitutes a reference unstable case, all the microphones record similar levels of pressure fluctuations. For configurations ${\mathcal {C}}_4$ and ${\mathcal {L}}_4$, the positions where the microphones recorded the lowest pressure fluctuations coincide with those of the $S$-injectors. Using these microphone recordings, the pressure field can be reconstructed using (2.4), and the position of the nodal line (where the pressure fluctuations are the lowest) can be determined.

Figure 12. Pressure time series at different microphone positions for configurations: (a) ${\mathcal {C}}_0$; (b) ${\mathcal {C}}_8$; (c) ${\mathcal {C}}_4$; and (d) ${\mathcal {L}}_4$.

Nodal line distributions are displayed in figure 13 for configurations ${\mathcal {C}}_2$, ${\mathcal {C}}_4$, ${\mathcal {L}}_2$ and ${\mathcal {L}}_4$. They show narrower distributions for configurations ${\mathcal {C}}_4$ and ${\mathcal {L}}_4$ than for ${\mathcal {C}}_2$ and ${\mathcal {L}}_2$. Another way to analyse the nodal line position consists in representing the probability of the nodal line location in the annular geometry, determined from the nodal line distribution, as illustrated in figure 13( f–i). For configuration ${\mathcal {C}}_2$, the nodal line passes through the $S$-injectors, and for configurations ${\mathcal {C}}_4$ and ${\mathcal {L}}_4$, the nodal line shows a preferred location between two injectors, closer to an injector than to another neighbour of the same type. When there is a single $S$-injector or when an odd number of $S$-injectors is in a ${\mathcal {C}}$ configuration, the nodal line passes respectively through the unique injector or through the central unit in the ${\mathcal {C}}$ arrangement. These findings generally correspond to what one might expect on a physical basis, but it is worth trying to explain what is observed by calculating the growth rates associated with the different configurations and showing that the nodal line location is that corresponding to a maximum in the growth rate (see § 4.3).

Figure 13. Nodal line positions for four staging patterns. (a) Origin and sign of the angles. Nodal line distribution (with a bin size of 4.5 $^{\circ }$) for configurations: (b) ${\mathcal {C}}_2$; (c) ${\mathcal {C}}_4$; (d) ${\mathcal {L}}_2$; and (e) ${\mathcal {L}}_4$. Probability of the nodal line position for configurations: ( f) ${\mathcal {C}}_2$; (g) ${\mathcal {C}}_4$; (h) ${\mathcal {L}}_2$; and (i) ${\mathcal {L}}_4$.

4.1. Acoustic energy balance framework

One seeks an analytical expression for the growth rate, as a function of the FDFs of the two types of injectors and their relative position in the annulus. The starting point of this analysis is the acoustic energy balance equation, which reads

(4.1)\begin{equation} \frac{\partial {\mathcal{E}}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{{\mathcal{F}}}_a = {\mathcal{S}} - {\mathcal{D}}, \end{equation}

where ${\mathcal {E}}$, $\boldsymbol {{\mathcal {F}}}_{a}$, $\mathcal {S}$ and $\mathcal {D}$ are the instantaneous acoustic energy density, the acoustic energy flux, the Rayleigh source term and the volumetric damping term, respectively,

(4.2)$$\begin{gather} {\mathcal{E}}=\frac{1}{2} \frac{p'^2}{\rho_0 c^2} + \frac{1}{2} \rho_0 v'^2 \end{gather}$$

(4.3)$$\begin{gather}{}\boldsymbol{{\mathcal{F}}}_a= p'\boldsymbol{v}' \end{gather}$$

(4.4)$$\begin{gather}{\mathcal{S}}=\frac{\gamma-1}{\rho_0 c^2}p' {\dot{q}}' \end{gather}$$

with $p', \boldsymbol{v}'$ the acoustic pressure and velocity, ${\dot {q}}'$ the heat release rate fluctuations, $\rho _0$ the ambient density, $c$ the speed of sound and $\gamma$ the specific heat ratio.

In the case of harmonic oscillations, spatial and temporal variations may be decoupled, and one can write

(4.5a–c)\begin{equation} p'=Re\{\tilde{p} (t) {\rm e}^{-{\rm i}\omega t}\},\quad \boldsymbol{v}'=Re\{\tilde{\boldsymbol{v}} (t) {\rm e}^{-{\rm i}\omega t}\},\quad {\dot{q}}'=Re\{\tilde{\dot{q}} (t) {\rm e}^{-{\rm i}\omega t}\}, \end{equation}

where $\tilde {p}(t)$, $\tilde {\boldsymbol {v}} (t)$ and $\tilde {{\dot {q}}} (t)$ are slowly varying functions, and $\omega =2{\rm \pi} f$, with $f$ the frequency of oscillation.

Integrating the balance equation over a period of oscillation yields an equation linking period averages:

(4.6)\begin{equation} \frac{\partial {E}}{\partial t} + \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{F} = {S} - {D}, \end{equation}

where $E$, $\boldsymbol {F}$ and $S$ are the period averages of ${\mathcal {E}}$, $\boldsymbol {{\mathcal {F}}}_a$ and ${\mathcal {S}}$. These quantities are conveniently determined with (4.7), valid if the slowly varying variables can be assumed constant over a cycle of oscillation

(4.7)\begin{equation} \frac{1}{T}\int_{[T]} a'b' \,{\rm d}t= \frac{1}{2} Re\{\tilde{a}\tilde{b}^* \} \end{equation}

with $a'=Re\{\tilde {a}(t) {\rm e}^{-{\rm i}\omega t}\}$, $b'=Re\{\tilde {b}(t) {\rm e}^{-{\rm i}\omega t}\}$ and $^*$ designating the complex conjugate. One hence obtains

(4.8)$$\begin{gather} {E}= \frac{1}{4} \frac{\tilde{p} \tilde{p}^*}{\rho_0 c^2} + \frac{1}{4} \rho_0 \tilde{\boldsymbol{v}} \boldsymbol{\cdot} \widetilde{\boldsymbol{v}}^*, \end{gather}$$

(4.9)$$\begin{gather}{\boldsymbol{F}}= \tfrac{1}{2} Re\{\tilde{p} \tilde{\boldsymbol{v}}^*\}, \end{gather}$$

(4.10)$$\begin{gather}{S}=\frac{\gamma-1}{\rho_0 c^2}\frac{1}{2} Re\{ \tilde{p} \tilde{{\dot q}}{^*}\}. \end{gather}$$

Taking the average of the balance equation over the volume $V$ of the combustor, one gets

(4.11)\begin{equation} \frac{{\rm d}{\langle{E\rangle}}}{{\rm d} t}= \langle{S\rangle} -\langle{D\rangle}-\langle{F\rangle} \end{equation}

with $\langle {E\rangle }$, $\langle {S\rangle }$ and $\langle {D\rangle }$ the integrals over the volume $V$ of $E$, $S$ and $D$, and $\langle {F\rangle }=\int _{A} \boldsymbol {F}\boldsymbol {\cdot }\boldsymbol {n} \,{\rm d}A$ the surface integral of the normal energy flux.

Assuming that $p'$, $\boldsymbol{v}'$ and ${\dot {q}}'$ have a common term $\exp (-{\rm i}\omega t)$, with $\omega = \omega _r+ {\rm i}\omega '_i$, where $\omega _r$ is the angular frequency and $\omega '_i$ represents the effective growth or decay rate of the oscillations, one can divide (4.11) by $2\langle {E\rangle }$ to identify a growth rate $\omega _i$ and a damping rate $\alpha$:

(4.12)$$\begin{gather} \omega_i= \frac{1}{2\langle{E\rangle}}\langle{S\rangle} \end{gather}$$

(4.13)$$\begin{gather}\alpha= \frac{1}{2\langle{E\rangle}}[\langle{D\rangle}+\langle{F\rangle}] \end{gather}$$

such that $\omega '_i= \omega _i -\alpha$ represents the effective growth rate, determining whether the oscillation will grow or decay. Here, $\omega _i$ corresponds to the growth rate in the absence of damping and this term can take positive or negative values; $\alpha$ is the damping rate, resulting from sources of acoustic energy dissipation in the control volume and acoustic energy fluxes escaping the volume and inducing a rate of change of the acoustic energy in the volume. Assuming that the boundaries are only passive (that the acoustic flux leaving the volume is always positive or null), the damping rate $\alpha$ is always positive and the oscillation will grow if $\omega _i > \alpha$.

To determine $\omega _i$, one now has to calculate the period average of the Rayleigh source term

(4.14)\begin{equation} \langle{S\rangle}=\frac{\gamma-1}{\rho_0 c^2} \frac{1}{T} \int_{[T],V} p' {\dot{q}}' \,{\rm d}V \,{\rm d}t. \end{equation}

Introducing the volume integrated heat release rate fluctuations ${\dot{Q}}=\int _{V} {\dot {q}}'\,{\rm d}V$, and assuming that the pressure is constant in the region where there are heat release rate fluctuations, i.e. that the flames are compact with respect to the acoustic wavelength, one gets

(4.15)\begin{equation} \langle{S\rangle}={\dot{Q}}_0\frac{\gamma-1}{\gamma} \frac{1}{T} \int_{[T]} \frac{p'}{p_0} \frac{{\dot{Q}}'}{{\dot{Q}}_0} \,{\rm d}t, \end{equation}

which can be rewritten as

(4.16) \begin{equation} \langle{S\rangle}={\dot{Q}}_0\frac{\gamma-1}{\gamma} \frac{1}{2} Re \left\{\frac{\tilde{p}}{p_0} \left[\frac{{\widetilde{{\dot{Q}}'}}}{{\dot{Q}}_0}\right]^* \right\}. \end{equation}

One now seeks an analytical expression for the relative heat release rate fluctuations, ${\widetilde{{\dot{Q}}'}}/{\dot {Q}}_0$. Introducing the flame describing function, ${\mathcal {F}}$, linking relative heat release rate fluctuations to relative velocity fluctuations,

(4.17)\begin{equation} {\mathcal{F}}= \frac{{\widetilde{{\dot{Q}}'}}/{{\dot{Q}}_0}}{\tilde{{v}}/{\bar{v}}} \end{equation}

and using the effective impedance, $\zeta$, to link relative velocity fluctuations to relative pressure fluctuations at the flame position, one gets

(4.18)\begin{equation} \frac{{\widetilde{{\dot{Q}}'}}}{{\dot{Q}}_0}={\mathcal{F}}\frac{1}{\bar{v}}\frac{\tilde{p}}{\rho_0 c \zeta}= \frac{\mathcal{F}}{\zeta} \frac{1}{\gamma M} \frac{\tilde{p}}{p_0}, \end{equation}

where $M$ is the Mach number of the flow at the flame.

Introducing ${\mathcal {F}}=G_F {\rm e}^{{\rm i}\varphi _F}$ and $\zeta =G_\zeta {\rm e}^{{\rm i}\varphi _\zeta }$, one can write the relative heat release rate fluctuations in the following form:

(4.19)\begin{equation} \frac{{\widetilde{{\dot{Q}}'}}}{{\dot{Q}}_0}= \frac{G_F}{G_\zeta} \frac{1}{\gamma M} \frac{\tilde{p}}{p_0} \exp({{\rm i}\varphi_F-{\rm i}\varphi_\zeta}). \end{equation}

In the case of an annular combustor, assuming that the 1A1L mode is involved in the coupling and considering a mixed azimuthal structure, pressure fluctuations can be cast in the form

(4.20)\begin{equation} \tilde{p}(x,\theta)= [a \exp({{\rm i}\theta-{\rm i}\omega t})+b \exp({-{\rm i}\theta-{\rm i}\omega t})]\psi(x) . \end{equation}

To simplify notation, the complex amplitudes $A_+$ and $A_{-}$ of (2.4) have been replaced by $a$ and $b$. As already indicated in § 2, (4.20) constitutes an approximate representation of the pressure field that neglects distortions induced by circumferential staging in the 1A azimuthal modal structure.

The axial wavefunction corresponding to the 1L mode is $\psi (x)= \cos [{\rm \pi} x/(2l)]$. It is assumed that the flames are compact with respect to the acoustic wavelength, and one may neglect any variation in the axial direction and assume that the axial wavefunction takes a constant value at the flame position, $\psi (x) \simeq \psi (x_f)$, and simply designate this value by $\psi _f$.

The volume integrated period average contribution of the $j$th flame, located at an azimuthal angle $\theta _j$, to the source term using (4.16) writes:

(4.21)\begin{equation} \langle{S_j\rangle}=\frac{1}{2} \frac{(\gamma-1)}{\gamma} {\dot{Q}}_0 \frac{1}{\gamma p_0^2} \frac{ 1}{ M} \frac{G_F}{G_\zeta} \psi_f^2 Re(J), \end{equation}

where

(4.22)\begin{equation} J=(a {\rm e}^{{\rm i}\theta}+b {\rm e}^{-{\rm i}\theta})(a^* {\rm e}^{-{\rm i}\theta}+b^* {\rm e}^{{\rm i}\theta}){\rm e}^{-{\rm i}(\varphi_j-\varphi_\zeta)}. \end{equation}

Rewriting

(4.23a,b)\begin{equation} a=|a| {\rm e}^{{\rm i}\varPhi_+}, \quad b=|b| {\rm e}^{{\rm i}\varPhi_-} \end{equation}

and introducing

(4.24)\begin{equation} \theta_0=(\varPhi_{-} -\varPhi_+)/2, \end{equation}

which corresponds to the ‘spatial phase’ term determining the pressure anti-node location in the quaternion representation (Ghirardo & Bothien Reference Ghirardo and Bothien2018) (this is easily verified, for example, in the case of a standing mode by taking $|a|=|b|$), one obtains, after a few calculations,

(4.25)\begin{align} \langle{S_j\rangle}=\frac{1}{2} \frac{(\gamma-1)}{\gamma} {\dot{Q}}_0 \frac{1}{\gamma p_0^2} \frac{ 1}{ M} \frac{G_F}{G_\zeta} \psi_f^2 \cos(\varphi_j-\varphi_\zeta)\left[|a|^2+|b|^2+2|a||b| \cos(2(\theta - \theta_0))\right]. \end{align}

Now the acoustic energy integrated over a period and the combustor volume reads

(4.26)\begin{equation} \langle{E\rangle}=\int_V \left[\frac{1}{4} \frac{\tilde{p} \tilde{p}^*}{\rho_0 c^2} + \frac{1}{4} \rho_0 \tilde{\boldsymbol{v}} \boldsymbol{\cdot} \tilde{\boldsymbol{v}}^*\right]\,{\rm d}V, \end{equation}

which can be simplified into

(4.27)\begin{equation} \langle{E\rangle}=\frac{1}{4} \frac{1}{\rho_0 c^2} [|a|^2 + |b|^2] V. \end{equation}

Using the growth rate expression (4.12) and making use of the assumption that the different injectors operate independently from each other, one may write

(4.28)\begin{equation} \omega_i= \frac{1}{2\langle{E\rangle}}{\sum_{j=1}^{N} \langle{S_j\rangle}}, \end{equation}

which yields, after insertion of (4.25) and (4.27):

(4.29)\begin{align} \omega_i = \frac{({\gamma-1})}{\gamma} \frac{{\dot{Q}}_0}{p_0 V} \psi_f^2 \sum_{j=1}^{N} \frac{G_{Fj}}{M G_{\zeta j}}\cos (\varphi_{Fj}-\varphi_{\zeta j}) \left[1+\frac{2|a||b|}{|a|^2+|b|^2} \cos (2(\theta_j-\theta_0)) \right]. \end{align}

The growth rate appears like a weighted sum of the contributions of the individual flames depending on the position of the injectors with respect to the pressure anti-nodal line position ($\theta _0$), appearing in the $\cos (2(\theta _j-\theta _0))$ term. It also depends on the phase difference between the flame describing functions and the phase of the specific impedance that links the pressure and velocity disturbances acting on the flame through the $\cos (\varphi _{Fj}-\varphi _{\zeta j})$ term, and on the gain of the FDFs of the various types of injectors.

It is first interesting to consider a case where all injectors have the same FDF and impedance gain and phase values, i.e. when $G_{Fj}=G_F$, $\varphi _{Fj}=\varphi _F$, $G_{\zeta j}=G_\zeta$ and $\varphi _{\zeta j}=\varphi _\zeta$. In that case, one finds that the second term in the brackets in expression (4.29) does not contribute to the sum over the $N$ injectors and one obtains

(4.30)\begin{equation} \omega_i = \frac{({\gamma-1})}{\gamma} \frac{N{\dot{Q}}_0}{p_0 V} \psi_f^2 \frac{G_{F}}{MG_\zeta} \cos (\varphi_F-\varphi_\zeta). \end{equation}

In this particular case of identical flame responses for all the injectors, one finds that the growth rate does not depend on the nature of the mode that assures the coupling, and that $\omega _i$ is the same for mixed, CCW or CW spinning, and standing modes. This definitely explains experimental observations where these various modes appear and continuously switch between the various possible amplitudes of the CCW and CW composing the azimuthal structure. This ceases to be true when injectors feature different responses characterized by differences in the gains $G_F$ and phases $\varphi _F$.

It is also interesting to consider two limiting cases corresponding to a standing mode and a pure spinning mode. In the first case, $|a|=|b|$, and (4.29) becomes

(4.31)\begin{equation} \omega_i = {2}({\gamma-1}) \frac{{\dot{Q}}_0}{\gamma p_0 V} \psi_f^2 \sum_{j=1}^{N} \frac{G_{Fj}}{M G_{\zeta j}}\cos (\varphi_{Fj}-\varphi_{\zeta j}) \cos^2 (\theta_j-\theta_0). \end{equation}

In the latter expression, the $G_{Fj}\cos (\varphi _{Fj}-\varphi _{\zeta j})$ term associated with each flame is weighted by the $\cos ^2 (\theta _j-\theta _0)$ term, depending on the position of the flame with respect to the anti-nodal line.

In the second case, one of the wave amplitudes $a$ or $b$ vanishes and the growth rate takes the form:

(4.32)\begin{equation} \omega_i = ({\gamma-1}) \frac{{\dot{Q}}_0}{\gamma p_0 V} \psi_f^2\sum_{j=1}^{N} \frac{G_{Fj}}{M G_{\zeta j}}\cos (\varphi_{Fj}-\varphi_{\zeta j}). \end{equation}

If the mode is spinning, the contribution of each flame to the total growth rate is independent of its position in the annular system.

4.2. Validation of the analytical framework

The objective of the present section is to validate the analytical framework with two test cases before using it to determine the growth rates for all the configurations investigated. To do so, it is convenient to work with the dimensionless quantity $\langle {S_j \rangle }/{\dot {Q}}_0$, the source term at flame $j$ divided by the mean heat release rate in one flame, since it can be determined experimentally from simultaneous pressure and photomultiplier recordings using (4.15) recalled below as

(4.33)\begin{equation} \langle{S_j \rangle}/{\dot{Q}}_0=\frac{\gamma-1}{\gamma} \frac{1}{T} \int_{[T]} \frac{p'_j}{p_0} \frac{{\dot{Q}}'}{{\dot{Q}}_0} \,{\rm d}t. \end{equation}

The flames behaving in a quasi-premixed fashion (see § 2.2), the relative heat release rate fluctuations can be approximated by the relative intensity fluctuations recorded by a photomultiplier equipped with an OH* filter, $I'/I \approx \dot {Q}'/\dot {Q}_0$.

The source term at the $j$th flame can also be determined analytically and expressed as a function of the level of peak pressure fluctuations in the chamber at injector $j$, $|{p'_j}/{p_0}|^2$, by combining (4.16) and (4.19) as

(4.34)\begin{equation} \langle{S_j\rangle}/{\dot{Q}}_0 =\frac{1}{2} \frac{\gamma -1}{\gamma^2} \frac{G_{Fj}}{G_{\zeta j}} \frac{1}{M} {\left|\frac{p'_j}{p_0}\right|}^2 \cos (\varphi_{Fj}-\varphi_{\zeta j}). \end{equation}

The level of peak pressure fluctuations at injector $j$, ${|{p'_j}/{p_0}|}^2$, constitutes a data input from experiments for this validation case.

The experimental set-up used for this study is presented in figure 14(a,d). Eight photomultipliers record the relative light intensity fluctuations emitted by eight adjacent flames on one half of the annulus. Pressure fluctuations are reconstructed at each injector position using pressure recordings and (2.4). The values of $\langle {S_j \rangle }/{\dot {Q}}_0$ thus obtained experimentally and analytically for eight flames will be compared for two configurations leading to a well-defined standing mode with the nodal line location controlled by the position of $S$-injectors: ${\mathcal {L}}_4$ and a variation of configuration ${\mathcal {L}}_4$, denoted ${\mathcal {L}}_{4+1}$, in which an additional $S$-injector was placed close to the pressure anti-node.

Figure 14. Experimental validation of the analytical framework: (a–c) configurations ${\mathcal {L}}_4$ and (d–f) ${\mathcal {L}}_{4+1}$. (a,d) Experimental set-up, with the positions of the $U$- and $S$-injectors and the photomultipliers (denoted ‘PMX’). (b,e) Peak pressure fluctuations ${|{p'_j}/{p_0}|}^2$ at the eight photomultiplier locations. (c,f) Comparison between $\langle {S_j\rangle }/{\dot {Q}}_0$ determined experimentally (blue) and analytically (red) at the eight photomultiplier positions.

The different quantities appearing in the source term expression and the methodology used to determine them are gathered in table 2. The gain and phase values of the FDF are obtained from the FDF measurements presented in § 2.2. FDF gain and phase are evaluated at the frequency $f_{11}=808$ Hz, corresponding to the eigenvalue of the 1A1L mode of the MICCA-Spray combustion chamber calculated by (2.3). The impedance phase $\varphi _\zeta$ and the product $MG_\zeta$ are determined with the procedure described in Appendix B. It is also shown that the impedances $\zeta _U$ and $\zeta _S$ of the different injection units are quite close. Although the respective pressure drop values of injectors $U$ and $S$ are different, their transfer matrices (discussed in Appendix B) are mainly determined by the air distributor and terminal plate which are common to the two types of injectors, the swirler only having a minor influence on the coefficients of these matrices. As a consequence, $\zeta _S \simeq \zeta _U$, and one may use the same effective impedance $\zeta$.

Table 2. Parameters used for the growth rate calculations and procedure employed to determine these quantities.

The comparison between experimental data and the analytical determination of the source term is presented in figure 14(c, f). Experimental measurements for the source term show that the flames located at the pressure node do not contribute significantly to the total source term. The flames bringing the highest contribution are those located at the pressure anti-node. Placing a $S$-injector close to the pressure anti-node (configuration ${\mathcal {L}}_{4+1}$, figure 14d–f) induces a drop in the source term value at the position of the $S$-injector facing photomultiplier ‘PM4’. This can be linked to the smaller FDF gain value measured for injector $S$, compared with that for injector $U$. A good qualitative and quantitative agreement between the experimental measurements and the evaluation of the analytical expression is found. The drop in the value of the source term observed for configuration ${\mathcal {L}}_{4+1}$ at the position of the $S$-injector facing photomultiplier ‘PM4’ is also well retrieved by the analytical expression of the source term.

4.3. Nodal line locking

The analytical expression of the growth rate is now used to interpret mode selection and nodal line locking phenomena.

At this point, it is first useful to recall theoretical results on how breaking the symmetry in an annular system changes the associated eigenvalues and modal structures. A perfectly symmetric annular combustor features degenerate eigenvalues. In this case, the mode structure is spinning, standing or of mixed type. Symmetry breaking induced by the staging of different injectors (or the introduction of Helmholtz resonators) splits the initially degenerate eigenvalue into a couple of distinct eigenvalues, characterized by different frequencies and growth rates (Stow & Dowling Reference Stow and Dowling2003; Bauerheim et al. Reference Bauerheim, Salas, Nicoud and Poinsot2014,  Reference Bauerheim, Cazalens and Poinsot2015). It is found by Bauerheim et al. (Reference Bauerheim, Salas, Nicoud and Poinsot2014) that staging two types of injectors in an annular system gives rise to standing modes with perpendicular nodal lines and introduces a ‘splitting strength’, controlling the difference between the eigenfrequencies and growth rates associated with the two modes. The latter quantity depends on the flame dynamical characteristics and the staging pattern.

One may now interpret the experimental results obtained in MICCA-Spray in light of these theoretical findings in combination with the analysis of the growth rate expressions for the different configurations. To that end, figure 15 shows the growth rate as a function of the nodal line location for all the configurations investigated. Power spectral densities (PSDs) of the pressure signals, together with nodal line angular positions and spin ratio time series, are also reported for unstable configurations in Appendix C to support the analysis carried out in this section.

Figure 15. Growth rate $\omega _i$ as a function of the nodal line position $\theta _n$ for all the configurations investigated. Red vertical lines correspond to the position of $S$-injectors in the annular system. Although only two possible standing modes with perpendicular nodal line positions are theoretically predicted for cases ${\mathcal {C}}_2, {\mathcal {C}}_4, {\mathcal {C}}_6, {\mathcal {L}}_2, {\mathcal {L}}_4, {\mathcal {L}}_6$ and ${\mathcal {L}}_8$, the growth rate $\omega _i$ is evaluated for a range of nodal line positions $\theta _n$ to examine the change in shape of the function for these different staging patterns.

Let us start with configurations ${\mathcal {C}}_0$ and ${\mathcal {C}}_{16}$, where all the injectors are of one type. Theoretically, the mode is degenerate. The power spectral densities of the pressure signals feature a single peak and mixed modes are observed experimentally with the nodal line position sweeping a broad range of locations (see Appendix C). The associated growth rate given by (4.31) shows no dependence on the nodal line position, which is coherent with the experimental observations. For case ${\mathcal {C}}_8$, the growth rate also does not show any variations with the nodal line location $\theta _n$, and its value is half-way between those found for ${\mathcal {C}}_0$ and ${\mathcal {C}}_{16}$, and probably lower than the damping rate since this case is found to be stable in the tests.

For configuration ${\mathcal {C}}_1$ where an $S$-injector replaces a $U$-injector, the mode is of mixed type and a single frequency peak is observed in the power spectral density (see the corresponding plots reported in Appendix C). The mathematical evaluation of (4.31) indicates a small difference between the minimum and maximum values of the growth rate. One may infer that, for this staging pattern, the splitting strength is low, giving rise to a small loss of degeneracy, with a frequency difference between the two eigenmodes below the frequency resolution of the PSD ($\Delta f=4$ Hz). In addition, the beating observed in the pressure signal at a frequency around 2 Hz, displayed in Appendix C, is another indication of this slight loss of degeneracy. It is also interesting to note that, in this particular configuration, the mixed modal structures are preferentially of the CCW spinning wave type. One might think that this preferred spinning direction could be linked to the tangential flow induced by the injection unit swirlers which are all oriented in the same direction. However, this global rotating flow is slow (Durox et al. Reference Durox, Prieur, Schuller and Candel2015) and rotates in the CW direction. The statistical preference observed for the CCW mode in configuration ${\mathcal {C}}_1$ is probably not linked to the bulk swirl, as already discussed by Rajendram Soundararajan et al. (Reference Rajendram Soundararajan, Durox, Renaud and Candel2022a).

For alternate configurations ${\mathcal {A}}_4$ and ${\mathcal {A}}_8$, the growth rate $\omega _i$ is independent of the nodal line location and the mode is degenerate, as a result of the remaining discrete rotational symmetry. Arrangement ${\mathcal {A}}_8$ leads to stable operation and the oscillations in configuration ${\mathcal {A}}_4$ correspond to mixed modes at a reduced amplitude level (see Appendix C).

Equation (4.31) highlights interesting features for configurations ${\mathcal {C}}_2, {\mathcal {C}}_4, {\mathcal {C}}_6, {\mathcal {L}}_2, {\mathcal {L}}_4, {\mathcal {L}}_6$ and ${\mathcal {L}}_8$. Theoretically, these configurations, for which the splitting strength is expected to take high values, present two distinct eigenvalues associated with two different growth rates and eigenfrequencies. The two resulting modes are standing and the associated nodal lines are perpendicular. Depending on the growth rate values, one mode can prevail and be effectively observed in practice. Power spectral densities of the pressure signals for these cases, reported in Appendix C, feature a single well-defined peak. A standing mode with a fixed nodal line is observed and the other analytically possible mode, with a nodal line perpendicular to the first mode, is not observed experimentally. Examining the mathematical properties of (4.31), two extrema appear for two perpendicular nodal line positions (it is worth stressing at this point that, in figure 15, although only two possible standing modes are predicted theoretically for these cases, the growth rate $\omega _i$ is evaluated for a range of nodal line positions $\theta _n$ to examine the change in shape of the function for these different staging patterns). For configurations ${\mathcal {C}}_2$, ${\mathcal {C}}_4$ and ${\mathcal {C}}_6$, (4.31) indicates that the nodal line leading to a maximum of the growth rate passes at equal distances from the $S$-injectors which is consistent with the experimental observations. For type ${\mathcal {L}}$ configurations, the corresponding nodal lines are aligned with the diameter joining the centres of the two groups of $S$-injectors placed on opposite sides of the annular chamber. Hence, of the two possible nodal line orientations for these non-degenerate cases, the nodal line leading to a maximum growth rate is selected and indeed observed experimentally. To finish, the difference between the maximum and minimum value of the growth rate increases with the number of $S$-injectors for these cases, and this can be interpreted as reflecting an increase in the value of the splitting strength.

4.4. Growth rate calculations and estimation of the damping rate

It is now instructive to consider the growth rates associated with different configurations and use their values to derive an estimate of the damping rate corresponding to instabilities coupled by the 1A1L mode.

The damping rate $\alpha$ plays a key role in thermoacoustic analysis but its value is generally not measured or quoted. A theoretical estimate is difficult to obtain because the processes that give rise to damping are not always easy to model. One possibility consists in using resonance techniques and measuring the quality factor of the system under stable operation. This only gives an effective damping rate $\alpha '$ that needs to be corrected for the flame influence (Paliès et al. Reference Paliès, Durox, Schuller and Candel2011). A second approach relies on system identification methods applied to simultaneous recordings of pressure and heat release rate fluctuations (Boujo et al. Reference Boujo, Denisov, Schuermans and Noiray2016). A third possibility consists in estimating the Rayleigh source term under self-sustained oscillations at a limit cycle (Durox et al. Reference Durox, Schuller, Noiray, Birbaud and Candel2009; Vignat Reference Vignat2020). Starting from (4.11),

(4.35)\begin{equation} \frac{{\rm d}{\langle{E\rangle}}}{{\rm d} t}= \langle{S\rangle} -\langle{D\rangle}-\int_{A} \boldsymbol{F}\boldsymbol{\cdot}\boldsymbol{n} \,{\rm d}A \end{equation}

and assuming steady self-sustained oscillations, the ${{\rm d}{\langle {E\rangle }}}/{{\rm d} t}$ term vanishes, and one gets

(4.36)\begin{equation} \langle{D\rangle}+\int_{A} \boldsymbol{F}\boldsymbol{\cdot}\boldsymbol{n} \,{\rm d}A= \langle{S\rangle}. \end{equation}

Hence, under steady self-sustained oscillations, the source term is in equilibrium with the acoustic losses, which result from damping and outgoing acoustic energy fluxes. The source term $\langle {S\rangle }$ was evaluated by Durox et al. (Reference Durox, Schuller, Noiray, Birbaud and Candel2009) and Vignat (Reference Vignat2020) through simultaneous photomultipliers and microphone recordings. It is shown by Vignat (Reference Vignat2020) that the latter method and the approach based on system identification proposed by Boujo et al. (Reference Boujo, Denisov, Schuermans and Noiray2016) yield comparable results in the case of instabilities coupled by a purely axial mode.

In this section, growth rates $\omega _i$ are calculated using the analytical framework presented in § 4.1 for configurations ${\mathcal {C}}_0$, ${\mathcal {C}}_2$, ${\mathcal {C}}_4$, ${\mathcal {C}}_6$, ${\mathcal {C}}_7$ and ${\mathcal {C}}_8$, and then for ${\mathcal {L}}_2$, ${\mathcal {L}}_4$, ${\mathcal {L}}_6$, ${\mathcal {L}}_8$, ${\mathcal {L}}_{10}$ and ${\mathcal {L}}_{12}$, and finally for ${\mathcal {A}}_{4}$ and ${\mathcal {A}}_{8}$. These results are then used to estimate the damping rate. For a given configuration, the growth rate can be determined for any combination of $|a|$ and $|b|$, and one can then see which combination provides the highest growth rate. In many situations, equal values of these amplitudes ($|a|=|b|$) yield the maximum value. This is consistent with the experimental observations, as these configurations give rise to standing modes. Equation (4.29) with $|a|=|b|$ will hence be used to determine the growth rates. The nodal line location is the one leading to the maximum value of the growth rate.

It is worth recalling that the growth rates obtained with the analytical expression are determined for low levels of modulation amplitude, since FDF data are only available for levels of relative velocity fluctuations around 10 % in the frequency range of interest (around 800 Hz). They hence correspond to linear growth rates. The values of the growth rate will change with the level of modulation amplitude, $\omega _i=\omega _i(v'=0)g(v'/\bar {v})$, where $g$ can be increasing or decreasing for intermediate values of $v'/\bar {v}$ and then becomes a decreasing function of $v'/\bar {v}$ at high amplitude levels.

Figure 16 shows the evolution of the growth rate as a function of the number of $S$-injectors for ${\mathcal {C}}$, ${\mathcal {L}}$ and $\mathcal {A}$ configurations. For the three types of configurations investigated, the calculated growth rate decreases with the number of $S$-injectors in the system, and for a fixed number of $S$-injectors, growth rates for type ${\mathcal {C}}$ and $\mathcal {A}$ staging patterns are lower than those found for ${\mathcal {L}}$ arrangements.

Figure 16. (a) Growth rates determined with the analytical model and (b) experimental instability amplitudes $A$ as a function of the effective growth rate $\omega _i'=\omega _i-\alpha$ for configurations ${\mathcal {C}}_0$, ${\mathcal {C}}_2$, ${\mathcal {C}}_4$, ${\mathcal {C}}_6$, ${\mathcal {C}}_7$ and ${\mathcal {C}}_8$ (blue); ${\mathcal {L}}_2$, ${\mathcal {L}}_4$, ${\mathcal {L}}_6$, ${\mathcal {L}}_8$, ${\mathcal {L}}_{10}$ and ${\mathcal {L}}_{12}$ (orange); and ${\mathcal {A}}_{4}$ and ${\mathcal {A}}_{8}$ (yellow). The grey band appearing in panel (a) separates stable points from unstable points, providing an indication for the lower and upper values of the damping rate.

The calculated values of the growth rates enable to get an estimate of the damping rate of the system: stable and unstable points enable to identify upper and lower bounds of the damping rate. If a point is unstable, its growth rate is higher than the damping rate of the system. If a point is stable, its growth rate is lower than the damping rate of the system. Hence, the unstable point with the lowest growth rate constitutes the upper limit of the damping rate, and the stable point with the highest growth rate, the lower limit. To distinguish between stable and unstable points, the two criteria defined by Latour et al. (Reference Latour, Durox, Rajendram Soundararajan, Renaud and Candel2023) will be used.

1. (i) The root mean square (r.m.s.) pressure is at least 3 times higher than a baseline level corresponding to the r.m.s. pressure level of noise under combustion and flow, measured for a stable operating point, which is typically 80 Pa.

2. (ii) The power spectral density exhibits a peak, $SPL_{\Delta f}^m$, that exceeds the maximum level measured in the baseline configuration, $SPL_{\Delta f}^{bl,m}$, by at least 25 dB: $SPL_{\Delta f}^m \geqslant SPL_{\Delta f}^{bl,m} + \Delta S$ (with $\Delta S =25$ dB). This condition ensures that the signal is dominated by a well-defined frequency tone.

The upper and lower limits thus determined correspond to the grey band plotted in figure 16(a). Configuration ${\mathcal {C}}_7$, which is marginally unstable (the criterion for the r.m.s. pressure level is fulfilled, but not that for the power spectral density peak), is inside the grey band. One finds that the damping rate is between 310 and $390\ {\rm s}^{-1}$. This value is consistent with that found by Vignat (Reference Vignat2020), through source term evaluation from simultaneous photomultiplier and pressure measurements during self-sustained oscillations coupled by a longitudinal mode in the annular combustor MICCA-Spray.

One may ask whether the limit cycle amplitude can be linked in some way to the effective growth rate calculated in the linear range. Intuitively, one expects that high values of this rate will lead to high levels of oscillation. This possible link is examined in figure 16(b), where the instability amplitude $A$ is plotted as a function of the effective growth rate $\omega _i'=\omega _i-\alpha$, determined analytically from lower and upper estimates of $\alpha$ (310 and $390\ {\rm s}^{-1}$, corresponding to the grey band in figure 16a). One can see that configurations associated with high positive values of the effective growth rate correspond to high levels of limit cycle instability amplitude. Configurations associated with negative values of the effective growth rate correspond to stable operation in MICCA-Spray. It is also interesting to point out that for similar values of the growth rate, the level of pressure oscillations at limit cycle also depends on the type of configuration (${\mathcal {C}}$, ${\mathcal {L}}$ or $\mathcal {A}$). This may be due to the fact that gain saturation (reflected by the $g$ function in the expression $\omega _i=\omega _i(v'=0)g(v'/\bar {v})$) depends on the configuration or that the damping rate changes with the configuration.