2. Linear analysis with a low-order model

The geometry considered in this study is that of a generic annular combustion chamber in which $N$ flames are located. Each flame is stabilised by a tubular burner. The geometry is assumed to have a rotational symmetry of degree $N$ and a mirror symmetry. A schematic of this geometry, adapted to resemble the experimental set-up used in § 3, is shown in figure 2. The propagation of azimuthal acoustic waves in the annular set-up can be modelled in the Laplace domain, with Laplace variable $s\equiv \sigma +\text {i}\omega$ – where $\sigma$ and $\omega$ designate, respectively, the growth rate and the angular frequency – by

(2.1) \begin{equation} s^2\hat {p}(\theta )-\frac {c^2}{R^2}\frac {\partial ^2 \hat {p}}{\partial \theta ^2}- s \mathfrak {F}(\theta ,s)\hat {p}(\theta )=0, \end{equation}

where $c$ is the speed of sound, assumed to be uniform, $R$ the mean radius of the annular chamber, and $\mathfrak {F}(\theta ,s)$ combines the response of the heat released by the flame to acoustic perturbations, $F(\theta ,s)$ , and the response of the tubes via

(2.2) \begin{equation} \mathfrak{F}(\theta ,s)=\sum _{j=1}^{N}\delta (\theta -\theta _j)\left (\gamma _FF(\theta ,s)-\gamma _T\frac {1}{Z_T(s)}\right ), \end{equation}

where $\theta _j = 2\unicode{x03C0} (j-1)/N$ is the azimuthal position of the $j$ th tube, $Z_T$ designates the tube impedance, $\gamma _F$ is a complex-valued frequency-dependent factor, and $\gamma _T$ is a real-valued constant that depends on the geometry. Here, $\delta (\theta )$ denotes the Dirac delta function. More details about $Z_T$ , $\gamma _T$ and $\gamma _F$ are given in Appendix A. If all the flame responses are identical, then also the equations exhibit rotational and mirror symmetry in the azimuthal direction. On the other hand, by choosing different flame responses at different positions, the rotational symmetry is reduced and the mirror symmetry may be broken.

To construct a low-order model that encompasses the effect of symmetry breaking on the spectral properties, we follow the two-mode projection approach of Noiray et al. (Reference Noiray, Bothien and Schuermans2011). For our analysis, it is convenient to express the pressure oscillations of a nearly degenerate mode in the spinning wave basis as

(2.3) \begin{equation} \hat {p}(\theta ,s)=\hat {a}_+(s)\text {e}^{-\textit{im}\theta }+\hat {a}_-(s)\text {e}^{\textit{im}\theta }, \end{equation}

where $\hat {a}_-$ and $\hat {a}_+$ are, respectively, the complex amplitudes of the clockwise (CW) and counter-clockwise (CCW) spinning azimuthal modes of order $|m|\neq 0$ . After substituting (2.3) into (2.1), we project (2.1) onto the spinning mode shapes $\text {e}^{\mp \textit{im}\theta }$ , which yields the eigenvalue problem

(2.4) \begin{equation} \left[\begin{array}{c@{\quad}c} s^2+\omega _m^2-s\mathfrak {F}_0 & -s\mathfrak {F}_{2\mathrm{m}}\\ -s\mathfrak {F}_{-2\mathrm{m}} & s^2+\omega _m^2-s\mathfrak {F}_0 \end{array}\right] \begin{bmatrix} \hat {a}_+ \\ \hat {a}_- \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix}. \end{equation}

where $\omega _m\equiv |m|c/R$ , and

(2.5) \begin{equation} \mathfrak {F}_k\equiv \left \{ \begin{array}{l@{\quad}l} \gamma _FF_k-\frac {\gamma _T}{Z_T} & \mbox {if } k = nN, \,\,\, n\in \mathbb {Z} \\ \gamma _FF_k & \mbox {otherwise } \end{array} \right . \end{equation}

with

(2.6) \begin{equation} F_{_{k}}(s)\equiv \frac {1}{N}\sum _{j=1}^{N}F(\theta _j,s)\text {e}^{\textit{ik}\theta _j}. \end{equation}

Choosing a flame response distribution for which $\mathfrak {F}_{2\mathrm{m}}=\mathfrak {F}_{-2\mathrm{m}}=0$ maintains the degeneracy, as can be seen from the fact that the matrix in (2.4) becomes diagonal with identical elements, and the dispersion relation reads

(2.7) \begin{equation} \left|\begin{array}{c@{\quad}c} s^2+\omega _m^2-s\mathfrak {F}_0 & 0\\ 0 & s^2+\omega _m^2-s\mathfrak {F}_0 \end{array}\right| =(s^2+\omega _m^2-s\mathfrak {F}_0)^2=0. \end{equation}

On the other hand, choosing a flame response distribution for which $\mathfrak {F}_{2\mathrm{m}}$ and $\mathfrak {F}_{-2\mathrm{m}}$ are non-zero, which implies reducing or breaking the rotational symmetry of the system, lifts the degeneracy. This has clear connections to the $C_{2\mathrm{m}}$ criterion first derived in Noiray et al. (Reference Noiray, Bothien and Schuermans2011). If the mirror symmetry is preserved, then one has that $\mathfrak {F}_{2\mathrm{m}}=\mathfrak {F}_{-2\mathrm{m}}$ . On the contrary, $\mathfrak {F}_{2\mathrm{m}}\neq \mathfrak {F}_{-2\mathrm{m}}$ implies that the staging pattern breaks the mirror symmetry. Notice that the eigenvalue problem expressed in equation (15) of Humbert et al. (Reference Humbert, Orchini, Paschereit and Noiray2022) in the standing wave basis is as generic as that expressed here in the spinning wave basis, (2.4). The latter has, however, the advantage of highlighting the link between the breaking of the mirror symmetry in the physical system and the asymmetry between the off-diagonal components in the eigenvalue problem itself via the inequality $\mathfrak {F}_{2\mathrm{m}}\neq \mathfrak {F}_{-2\mathrm{m}}$ .

2.1. Symmetry-breaking induced EPs

We now use the low-order model just presented to prove that symmetry-breaking perturbations in annular combustors can be used to construct EPs. We investigate the scenario in which the flame response distribution lifts the degeneracy of the mode $|m|$ , which generally causes $(\mathfrak {F}_{2\mathrm{m}},\mathfrak {F}_{-2\mathrm{m}})\neq (0,0)$ . More specifically, we wish to identify a perturbation pattern for which $\mathfrak {F}_{2\mathrm{m}}=0$ and $\mathfrak {F}_{-2\mathrm{m}}\neq 0$ (or vice versa), resulting in the following dispersion relation:

(2.8) \begin{equation} \left|\begin{array}{c@{\quad}c} s^2+\omega _m^2-s\mathfrak {F}_0 & 0\\ -s\mathfrak {F}_{-2\mathrm{m}} & s^2+\omega _m^2-s\mathfrak {F}_0 \end{array}\right| =(s^2+\omega _m^2-s\mathfrak {F}_0)^2=0. \end{equation}

The quadratic nature of the dispersion relation (2.8) implies that the eigenvalues’ algebraic multiplicity is 2. Importantly, we note that the dispersion relation takes the same form as for the rotationally symmetric case (2.7), for which both off-diagonal elements vanish ( $\mathfrak {F}_{\pm 2m}=0$ ). However, since $\mathfrak {F}_0$ depends on the asymmetry, the eigenvalue is modified by the latter. Because of the shape of the chosen symmetry-breaking pattern, with $\mathfrak {F}_{-2\mathrm{m}}\neq 0$ , the resulting matrix is defective and has a single eigenvector $\boldsymbol {a}$ . The eigenvector can be found by substituting (2.8) and $\mathfrak {F}_{2\mathrm{m}}=0$ into (2.4), which yields

(2.9) \begin{equation} \left[\begin{array}{c@{\quad}c} 0 & 0\\ -s\mathfrak {F}_{-2\mathrm{m}} & 0 \end{array}\right] \begin{bmatrix} \hat {a}_+ \\ \hat {a}_- \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix}, \end{equation}

resulting in $\boldsymbol {a} = [0,1]^{\textrm{T}}$ , which corresponds to the CW spinning mode $\text {e}^{\textit{im}\theta }$ , associated with this degenerate eigenvalue. This implies that the geometric multiplicity of the eigenvalue is 1. Because the algebraic multiplicity is larger than the geometric multiplicity, the identified degenerate eigenvalue forms an EP. This analysis proves that, if $s$ is an eigenvalue which satisfies $\mathfrak {F}_{2\mathrm{m}}(s)=0$ and $\mathfrak {F}_{-2\mathrm{m}}(s)\neq 0$ , then $s$ forms an EP associated with a CW spinning mode. For this condition to be verified, two parameters, e.g. the flame gain and time delay, need to lift the mode’s degeneracy and ensure that the mirror symmetry is broken. Under these necessary conditions, if one is able to vary continuously and independently the two parameters defining the asymmetry, then one is able to obtain a symmetry-breaking induced EP. For symmetry reasons, if $\mathfrak {F}_{-2\mathrm{m}}(s)=0$ and $\mathfrak {F}_{2\mathrm{m}}(s)\neq 0$ at the eigenvalue a symmetry-breaking induced EP is formed and corresponds to a CCW spinning mode.

3. Identification of symmetry-breaking induced EPs in an experimentally determined state-space model of an annular combustor

Figure 2.

Acoustic set-up characterised experimentally (inside the dashed rectangle).

The theoretical predictions are now validated on a state-space model fitted on experimentally measured acoustic data. The data were collected in an annular lab-scale set-up, which has been used in various studies to investigate azimuthal thermoacoustic modes (Humbert et al. Reference Humbert, Orchini, Paschereit and Noiray2022). The experimental set-up whose acoustic response was employed to fit a state-space model consists of an annular cavity connected to $N=12$ tubes, as shown in figure 2, mimicking an annular combustion chamber connected to $N$ tubular burners. Each tube is equipped with a microphone recording the pressure signal. A loudspeaker is used to force acoustic waves in the set-up, mimicking the acoustic monopole produced by unsteady heat release rate in combustion systems (Dowling & Stow Reference Dowling and Stow2003). The input microphone signals and output loudspeaker signals are regulated by a dSPACE 1103 board interfaced to a Simulink model. With this procedure, the (open-loop) acoustic response of the annular set-up is obtained and can be exploited to fit an acoustic state-space model. In real annular combustors, the heat released by each flame fluctuates in reaction to acoustic perturbations present in the corresponding burner. More specifically, the flame heat release rate fluctuations are strongly coupled to the acoustically induced fuel flow rate oscillations, and/or to the acoustically induced vortices that impinge the flame (Lieuwen Reference Lieuwen2012). This unsteady heat release rate response to acoustic and flow fluctuations closes a feedback loop between the acoustics, the flow dynamics and the flame. The linear response of the $j^{\textit{th}}$ flame model is defined in the Laplace domain by

(3.1) \begin{equation} F(\theta _j,s)\equiv \frac {\hat {q}_j}{\hat {y}_j} = \beta (\theta _j)H(s)\text {e}^{-s\tau (\theta _j)}, \hspace {5mm} H(s)=\frac {2\omega _{c}\zeta s}{s^2+2\omega _{c}\zeta s+{\omega _{c}}^2}, \end{equation}

where we set $\omega _c/(2\unicode{x03C0} )=330$ Hz, $\zeta =0.707$ , and $\beta (\theta _j)$ and $\tau (\theta _j)$ are the flame gain and time delay, $\hat {y}_j$ and $\hat {q}_j$ respectively denote the microphone output voltage and the voltage sent to the loudspeaker (see figure 2). This model mimics the experimentally determined linear response of flames, which typically shows a delayed response to acoustic perturbations due to the finite convection time required by the acoustic/flow induced fluctuations to reach the flame – modelled by the exponential term $\exp (-s\tau )$ – and a high gain only in a small frequency band (Durox et al. Reference Durox, Schuller, Noiray and Candel2009) – modelled by the band-pass filter $H(s)$ . The gain $\beta (\theta _j)$ models the sensitivity of the flame unsteady heat release rate to the acoustic disturbances. Note that in real combustors, the larger the temperature jump across the flame the more intense the flame reaction to acoustic fluctuations. The flame response parameters are functions of their azimuthal positions $\theta _j$ for $j=1,\ldots,N$ . By varying their values non-uniformly, one can induce specific symmetry-breaking patterns in the system’s response.

Similarly to Humbert et al. (Reference Humbert, Orchini, Paschereit and Noiray2022), we choose non-uniform distributions of the flame gain and of the flame time delay as

(3.2) \begin{equation} \beta (\theta _j)=\beta _0(1+b_d\cos [d(\theta _j-\theta _{b_d})]), \quad \tau (\theta _j)=\tau _0(1+\varepsilon _d\cos [d(\theta _j-\theta _{\varepsilon _d})]), \end{equation}

where $\beta _0$ and $\tau _0$ are the mean gain and mean time delay. In practice, a non-uniform distribution of the flame gain can be achieved by feeding the burners with different fuel flow rates (Kokanović et al. Reference Kokanović, Guidati, Torchalla and Schuermans2006; Bothien, Noiray & Schuermans Reference Bothien, Noiray and Schuermans2015). A non-uniform distribution of the time delay, instead, can be achieved by arranging burners with different convection times, e.g. by varying the axial location of the injectors or by using different flame holder lengths (Berenbrink & Hoffmann Reference Berenbrink and Hoffmann2000; Krüger et al. Reference Krüger, Hüren, Hoffmann, Krebs, Flohr and Bohn2000). The coefficients $b_d$ and $\varepsilon _d$ are the $d$ Fourier components of the flame gain and time delay, respectively, which quantify the asymmetry in terms of flame gain and time delay. The angles $\theta _{b_d}$ and $\theta _{\varepsilon _d}$ define the orientations of the gain and time-delay staging patterns, respectively. The integer $d$ needs to be a divisor of $N=12$ . In the analysis considered here, we choose to reduce the rotational symmetry so that $d=4$ . With this choice, for $(b_d,\varepsilon _d)\neq (0,0)$ , this lifts the degeneracy of any mode with azimuthal wavenumber $|m|$ that is multiple of $d/2$ by generating non-zero $\mathfrak {F}_{2\mathrm{m}}$ and $\mathfrak {F}_{-2\mathrm{m}}$ . Although the theory presented in § 2.1 is applicable to any such mode, the example application illustrated here focuses on $|m|=d/2=2$ only. For this purpose, we found that choosing $\beta _0=12$ and $\tau _0=4$ ms ensured the mode with $m=2$ to be linearly unstable in the symmetric configuration.

For $(\varepsilon _d,b_d)=(0,0)$ the system has a nominal rotational symmetry of degree $N$ and a mirror symmetry. As discussed in detail in Humbert et al. (Reference Humbert, Orchini, Paschereit and Noiray2022), when $\varepsilon _d$ and/or $b_d$ are non-zero, the degree of rotational symmetry is decreased to $d$ and the eigenvalue degeneracy for the mode $|m|=d/2=2$ is lifted. If, in addition, the gain and time-delay staging patterns are not aligned (i.e. $\theta _{\varepsilon _d}-\theta _{b_d}$ is not a multiple of $\unicode{x03C0} /d$ ), the mirror symmetry is broken. We note that the mirror symmetry is also broken if $\theta _{\varepsilon _d}$ or $\theta _{b_d}$ are not a multiple of $\unicode{x03C0} /N$ . Since the degree of rotational symmetry of the thermoacoustic system is reduced from $N$ to $d$ due to the chosen asymmetric distribution of flame responses, the thermoacoustic eigenvalue problem is block-diagonalised (Gray Reference Gray2006; Mensah, Campa & Moeck Reference Mensah, Campa and Moeck2016), i.e. reduced to $d$ independent eigenvalue problems forming $d$ matrices of size $(N/d)\times (N/d)$ . For each matrix, the thermoacoustic eigenvalues can be found by solving a dispersion relation of the form $\det [\mathbf {G}_b^{-1}(s)-\mathbf {F}(s) ]=0$ , where $\mathbf {G}_b$ is the acoustic state-space model associated with the Bloch wavenumber $b$ , and the matrix $\mathbf {F}$ contains the flame responses (3.1) on the main diagonal and zeros elsewhere.

As a reference, the degenerate eigenvalue corresponding to the symmetric case, $(\varepsilon _4,b_4)=(0,0)$ , is indicated by the red triangle in figure 3. To construct a symmetry-breaking induced EP, we proceed as follows. First, we lift the degeneracy of the mode of interest, $|m|=2$ , by setting $\varepsilon _4 = 0.05$ while $\theta _{\varepsilon _4}=\unicode{x03C0} /8$ . The eigenvalues of the mode $|m|=2$ split. These split eigenvalues are indicated by the green square markers in figure 3. Second, the value of $\varepsilon _4$ is left unchanged, and we wish to restore the degeneracy of the two eigenvalues that have split without restoring the fundamental symmetries of the system by varying (at least) two other parameters. In our model, we chose to set $\theta _{b_4}=0$ and the EP is constructed by varying $b_4$ and $\theta _{\varepsilon _4}$ . More specifically, the four pairs of eigenvalue trajectories depicted in figure 3 correspond to four closely spaced values of $\theta _{\varepsilon _4}$ and to a continuous variation of $b_4$ . For the four values of $\theta _{\varepsilon _4}$ , when the value of $b_4$ is continuously increased, the two eigenvalues get closer and closer to each other. When further increasing $b_4$ , the eigenvalue trajectories avoid crossing each other, and exhibit the strong veering behaviour that is characteristic in the vicinity of an EP (Kato Reference Kato1980). For even larger values of $b_4$ , the two trajectories turn by $90^\circ$ and move further away from each other. In this example, the EP – indicated by the black star in figure 3 – is close to the eigenvalue of the symmetric system because $\mathfrak {F}_0$ is close to its value in the symmetric case, due to the fact that the asymmetry is moderate. At this EP, as for the symmetric case, the mode is linearly unstable since it has a positive linear growth rate. In the vicinity of the EP, since the eigenvalues are very close to one another, one can qualify them as nearly degenerate. However, the eigenspace is fundamentally different from that of the symmetric case since at the EP the two eigenvectors coalesce. The single remaining eigenvector corresponds to a spinning mode. Moreover, the finite values of the parameters quantifying the asymmetry at the considered EP, $\varepsilon _4=0.05$ and $b_4\approx 0.229$ , allow us to highlight how a quantification of the asymmetry based on measuring the eigenvalue splitting could lead to wrong conclusions.

Figure 3.

Trajectories of a pair of non-degenerate eigenvalues – corresponding to the mode of azimuthal order $|m|=2$ – in the vicinity of a symmetry-breaking induced EP (whose location is indicated by the black star) for $\varepsilon _4=0.05$ and $\theta _{b_4}=0$ . The red triangle represents the thermoacoustic eigenvalue of the axisymmetric system. The green squares correspond to the two eigenvalues for $b_4=0$ and $\varepsilon _4=0.05$ . We use four values of $\theta _{\varepsilon _4}= ({\unicode{x03C0} }/{8})(1+\delta )$ , with $\delta =[-0.008,-0.004,0,0.004]$ , and for each value $b_4$ is increased continuously. For each trajectory, the lighter the marker colour, the larger $b_4$ . The EP is found for $b_4\approx 0.229$ and $0.996 ({\unicode{x03C0} }/{8})\lt \theta _{\varepsilon _4}\lt ({\unicode{x03C0} }/{8})$ .