2.1. Model description

This study focuses on a BC configuration where an annular chamber is fed by $N$ burners (figure 3). An impedance $Z$ is imposed at the upstream end of each burner. The mean density and sound speed are denoted ${\it\rho}^{0}$ and $c^{0}$ in the annular chamber and ${\it\rho}_{u}^{0}$ and $c_{u}^{0}$ for the unburnt mixture in the $N$ burners. The perimeter and the cross-sectional area (perpendicular to the azimuthal direction) of the annular chamber are denoted $2L_{c}=2{\rm\pi}R_{c}$ and $S_{c}$ respectively. The length and section of the $i$ th burner are $L_{i}$ and $S_{i}$ . The position along the annular cavity is given by the angle ${\it\theta}$ , defining an abscissa $x_{c}=R_{c}{\it\theta}$ . The location of the flame is similar in all burners and is given by the normalized abscissa ${\it\alpha}=z_{f,i}/L_{i}$ (figure 3).

This model corresponds to situations where pressure fluctuations in the combustion chamber depend on the angle ${\it\theta}$ (or the azimuthal position $x$ ) but not on the axial direction $z$ in the chamber (they depend on the coordinate $z$ only in the burners). This case can be observed in combustors terminated by choked nozzles which acoustically behave almost like a rigid wall (i.e. $u^{\prime }=0$ under the low-upstream-Mach-number assumption (Marble & Candel Reference Marble and Candel1977)). Since the chamber inlet is also close to a velocity node, modes which have no variation along $z$ can develop in the chamber, as shown by recent LES in real engines (Wolf et al. Reference Wolf, Staffelbach, Roux, Gicquel, Poinsot and Moureau2009).

The model provides the analytical expression of eigenfrequencies for a general asymmetric case for any mode order $p$ and any number of burners $N$ as well as general rules on stability for annular combustors. Results on the structure and nature of azimuthal modes (spinning, standing or mixed) will be derived using this analytical study to show how asymmetry can promote specific modes and control instabilities (Moeck et al. Reference Moeck, Paul and Paschereit2010; Noiray et al. Reference Noiray, Bothien and Schuermans2011; Gelbert et al. Reference Gelbert, Moeck, Paschereit and King2012; Worth & Dawson Reference Worth and Dawson2013a ,Reference Worth and Dawson b ).

2.2. Annular network reduction

Network models that account for one annular cavity connected to $N$ burners usually require a large number of unknown variables (acoustic pressure and velocity in each network tube) and a large matrix describing the system (typically a $2N\times 2N$ matrix). To reduce the size of the system (i.e. a matrix of size $2\times 2$ , independent of the number $N$ of burners), the ANR (annular network reduction) methodology proposed in Bauerheim et al. (Reference Bauerheim, Parmentier, Salas, Nicoud and Poinsot2014c ) is applied: the full annular combustor is split into $N$ sectors which differ only in the burner/chamber junction (figures 4 and 5). Between each sector, the propagation of azimuthal waves (along ${\it\theta}$ or $x$ ) can be modelled by a transfer matrix $\unicode[STIX]{x1D64D}_{i}$ , as proposed by Parmentier et al. (Reference Parmentier, Salas, Wolf, Staffelbach, Nicoud and Poinsot2012) (figure 5, propagation),

(2.1) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{i+1/2}=\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64D}_{i}\end{array}\right]\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{i},\quad \text{where}\;\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64D}_{i}\end{array}\right]=\left[\begin{array}{@{}cc@{}}W & 0\\ 0 & \displaystyle \frac{1}{W}\end{array}\right].\end{eqnarray}$$

Here, $q^{\pm }=p^{\prime }\pm {\it\rho}^{0}c^{0}u^{\prime }$ , $W=\text{e}^{2jkL_{c}/N}$ and the wavenumber $k={\it\omega}/c^{0}$ .

Figure 4.

3D view of a BC configuration with $N=4$ burners (a), zoom on the $i$ th sector (b) and model of the whole BC configuration (c), where ${\rm\Gamma}_{i}$ represents the burner/chamber interaction (Parmentier et al. Reference Parmentier, Salas, Wolf, Staffelbach, Nicoud and Poinsot2012).

Figure 5.

The ANR methodology: each sector is decomposed into a free propagation of azimuthal waves (characteristic length $2L_{c}/N$ ) and a compact burner/chamber interaction (characteristic length $a\ll {\it\lambda}$ ) modelled by the coupling parameter ${\rm\Gamma}_{i}$ .

The area where the $i$ th burner is connected to the annular chamber (- - - in figure 4 b) was investigated by O’Connor et al. (O’Connor & Lieuwen Reference O’Connor and Lieuwen2012a ,Reference O’Connor and Lieuwen b ,Reference O’Connor and Lieuwen c ) and can be assumed to be compact: $a\ll {\it\lambda}$ , where $a=2\sqrt{S_{i}/{\rm\pi}}$ is the burner diameter and ${\it\lambda}=2L_{c}/p$ is the acoustic wavelength, leading to the compactness criterion $p\ll L_{c}\sqrt{{\rm\pi}/S_{c}}\simeq 116$ . As shown in figure 6, using the equations of acoustic propagation in the cold ( $0<z<{\it\alpha}L_{i}$ ) and hot ( ${\it\alpha}L_{i}<z<L_{i}$ ) parts of the burner as well as the jump conditions through the $i$ th flame ( $z={\it\alpha}L_{i}$ ), the effect of the whole $i$ th burner on the annular chamber can be obtained by a translated impedance from $z=0$ (impedance $Z$ ) to the burner/chamber junction at $z=L_{i}$ (impedance $Z_{tr}=(p_{b,i}^{\prime }(z=L_{i})/{\it\rho}^{0}c^{0}w_{b,i}^{\prime }(z=L_{i}))$ (Blimbaum et al. Reference Blimbaum, Zanchetta, Akin, Acharya, O’Connor, Noble and Lieuwen2012; Bauerheim et al. Reference Bauerheim, Parmentier, Salas, Nicoud and Poinsot2014c ):

(2.2) $$\begin{eqnarray}Z_{tr}=\frac{\mathbb{F}S_{1-{\it\alpha}}^{k}[\,jC_{{\it\alpha}}^{k_{u}}-S_{{\it\alpha}}^{k_{u}}Z]+C_{1-{\it\alpha}}^{k}[C_{{\it\alpha}}^{k_{u}}Z+jS_{{\it\alpha}}^{k_{u}}]}{\mathbb{F}C_{1-{\it\alpha}}^{k}[\,jS_{{\it\alpha}}^{k_{u}}Z+C_{{\it\alpha}}^{k_{u}}]+S_{1-{\it\alpha}}^{k}[\,jC_{{\it\alpha}}^{k_{u}}Z-S_{{\it\alpha}}^{k_{u}}]},\end{eqnarray}$$

where $\mathbb{F}=({\it\rho}^{0}c^{0}/{\it\rho}_{u}^{0}c_{u}^{0})(1+n_{i}\text{e}^{\,j{\it\omega}{\it\tau}_{i}})$ , the notations for the sine and cosine functions are $C_{{\it\alpha}}^{k_{u}}=\cos ({\it\alpha}k_{u}L_{i}),S_{{\it\alpha}}^{k_{u}}=\sin ({\it\alpha}k_{u}L_{i}),C_{1-{\it\alpha}}^{k}=\cos ((1-{\it\alpha})kL_{i}),S_{1-{\it\alpha}}^{k}=\sin ((1-{\it\alpha})kL_{i})$ and the wavenumbers are $k={\it\omega}/c^{0}$ and $k_{u}={\it\omega}/c_{u}^{0}$ . It should be noted that the $n$ – ${\it\tau}$ model can be replaced by more complex flame descriptions such as flame describing functions (FTFs) (Noiray et al. Reference Noiray, Durox, Schuller and Candel2008) or transfer matrices (Polifke et al. Reference Polifke, Poncet, Paschereit and Doebbeling2001).

Figure 6.

Equivalent impedance of the whole $i$ th burner (which includes the $i$ th active flame) near the burner/chamber interaction zone (figure 5). The translated impedance $Z_{tr}$ at $z=L_{i}$ takes into account the upstream impedance $Z$ at $z=0$ , the propagation in the cold ( $0<z<{\it\alpha}L_{i}$ ) and hot ( ${\it\alpha}L_{i}<z<L_{i}$ ) parts of the burner as well as the active flame effect via the FTF ( $n_{i},{\it\tau}_{i}$ ).

The jump conditions at the burner/chamber junction at null Mach number read (Davies Reference Davies1988; Dowling Reference Dowling1995; Poinsot & Veynante Reference Poinsot and Veynante2011; Bauerheim, Nicoud & Poinsot Reference Bauerheim, Nicoud and Poinsot2014b )

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle p_{i+1/2}^{\prime }=p_{i+1}^{\prime }=p_{b,i}^{\prime }(z=L_{i}) & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle u_{i+1/2}^{\prime }S_{c}+\underbrace{w_{b,i}^{\prime }(z=L_{i})}_{=\frac{p_{i+1/2}^{\prime }}{{\it\rho}^{0}c^{0}Z_{tr}}}S_{i}=u_{i+1}^{\prime }S_{c}. & \displaystyle\end{eqnarray}$$

Consequently, a transfer matrix $\unicode[STIX]{x1D64F}_{i}^{\ast }$ for the interaction part of figure 5 can be deduced:

(2.5) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}p^{\prime }\\ {\it\rho}^{0}c^{0}u^{\prime }\end{array}\right]_{i+1}=\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64F}_{i}^{\ast }\end{array}\right]\left[\begin{array}{@{}c@{}}p^{\prime }\\ {\it\rho}^{0}c^{0}u^{\prime }\end{array}\right]_{i+1/2}=\left[\begin{array}{@{}cc@{}}1 & 0\\ 2j{\rm\Gamma}_{i} & 1\end{array}\right]\left[\begin{array}{@{}c@{}}p^{\prime }\\ {\it\rho}^{0}c^{0}u^{\prime }\end{array}\right]_{i+1/2},\end{eqnarray}$$

where the coupling parameter ${\rm\Gamma}_{i}$ (Palies Reference Palies2010; Parmentier et al. Reference Parmentier, Salas, Wolf, Staffelbach, Nicoud and Poinsot2012; Schuller et al. Reference Schuller, Durox, Palies and Candel2012; Bauerheim et al. Reference Bauerheim, Parmentier, Salas, Nicoud and Poinsot2014c ) (figure 5, burner/chamber interaction) is directly linked to the equivalent admittance $1/Z_{tr}$ of the whole $i$ th burner:

(2.6) $$\begin{eqnarray}{\rm\Gamma}_{i}=-\frac{j}{2}\frac{S_{i}}{S_{c}Z_{tr}}.\end{eqnarray}$$

When a velocity node ( $Z=\infty$ ) or a pressure node ( $Z=0$ ) is imposed at the upstream end of each burner and flames are located at the burner/chamber junction ( ${\it\alpha}=1$ ), the coupling parameters ${\rm\Gamma}_{i}$ reduce to (using (2.2) and (2.6))

(2.7) $$\begin{eqnarray}{\rm\Gamma}_{i}=\frac{1}{2}\frac{S_{i}{\it\rho}^{0}c^{0}}{S_{c}{\it\rho}_{u}^{0}c_{u}^{0}}\tan (k_{u}L_{i})\left(1+n_{i}\text{e}^{\,j{\it\omega}{\it\tau}_{i}}\right)\quad \text{when }Z=\infty\end{eqnarray}$$

or

(2.8) $$\begin{eqnarray}{\rm\Gamma}_{i}=-\frac{1}{2}\frac{S_{i}{\it\rho}^{0}c^{0}}{S_{c}{\it\rho}_{u}^{0}c_{u}^{0}}\;\text{cotan}(k_{u}L_{i})\left(1+n_{i}\text{e}^{\,j{\it\omega}{\it\tau}_{i}}\right)\quad \text{when }Z=0,\end{eqnarray}$$

where $k_{u}={\it\omega}/c_{u}$ and $(n_{i},{\it\tau}_{i})$ are the interaction index and the time delay of the FTF for the $i$ th flame (Crocco Reference Crocco1951). It should be noted that the coupling term in (2.5) is $2j{\rm\Gamma}_{i}p_{i+1/2}^{\prime }$ , thus burners and flames located at a pressure node have no effect on the acoustic mode, except in specific situations where ${\rm\Gamma}_{i}$ takes infinite values. For instance, ${\rm\Gamma}_{i}\rightarrow \infty$ in (2.8) when the burner length $L_{i}$ goes to zero. This corresponds to the Blimbaum exception case (Blimbaum et al. Reference Blimbaum, Zanchetta, Akin, Acharya, O’Connor, Noble and Lieuwen2012) where the burner impedance attempts to force a pressure node at the burner/chamber junction. These specific situations require 3D acoustic considerations and are out of the scope of this study: since ${\rm\Gamma}_{i}\rightarrow \infty$ , the low-coupling-factor assumption $\Vert {\rm\Gamma}_{i}\Vert \ll 1$ (further described in (3.3)) is not satisfied meaning that no analytical solution can be provided.

Finally, (2.5) can be recast to relate characteristic waves $q^{\pm }=p^{\prime }\pm {\it\rho}^{0}c^{0}u^{\prime }$ instead of primitive variables $p^{\prime }$ and $u^{\prime }$ leading to the scattering matrix $\unicode[STIX]{x1D64F}_{i}$ :

(2.9) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{i+1}=\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64F}_{i}\end{array}\right]\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{i+1/2},\quad \text{where }\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64F}_{i}\end{array}\right]=\left[\begin{array}{@{}cc@{}}1+j{\rm\Gamma}_{i} & j{\rm\Gamma}_{i}\\ -j{\rm\Gamma}_{i} & 1-j{\rm\Gamma}_{i}\end{array}\right].\end{eqnarray}$$

The waves at both ends of the $i$ th sector are connected by the $\unicode[STIX]{x1D648}_{i}=\unicode[STIX]{x1D64F}_{i}\unicode[STIX]{x1D64D}_{i}$ scattering matrix using (2.1) and (2.9) (figure 5):

(2.10) $$\begin{eqnarray}\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{i+1}=\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64F}_{i}\end{array}\right]\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{i+1/2}=\underbrace{\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64F}_{i}\end{array}\right]\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D64D}_{i}\end{array}\right]}_{\unicode[STIX]{x1D648}_{i}}\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{i}.\end{eqnarray}$$

Use of the periodicity condition $\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{N+1}=\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{1}$ and (2.10) leads to

(2.11) $$\begin{eqnarray}\left(\mathop{\prod }\limits_{i=N}^{1}\unicode[STIX]{x1D648}_{i}\right)\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{1}=\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{1}.\end{eqnarray}$$

The system defined by (2.11) has non-trivial solutions only if its determinant is null. Therefore, the ANR methodology provides an implicit analytical dispersion relation for the pulsation ${\it\omega}$ for a general non-symmetric BC configuration:

(2.12) $$\begin{eqnarray}\det \left(\mathop{\prod }\limits_{i=N}^{1}\unicode[STIX]{x1D648}_{i}-\unicode[STIX]{x1D644}_{d}\right)=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D644}_{d}$ is the $2\times 2$ identity matrix.

3.1. Unperturbed annular cavity (without burners and flames)

First, an annular chamber with no burner (i.e. ${\rm\Gamma}_{i}=0$ , for all $i\in [1,N]$ ) is studied as a reference case (figure 7 a). The sound speed field corresponds to a reactive case: $c=c^{0}$ in the annular chamber. The transfer matrix of each sector (2.10) reduces to $\unicode[STIX]{x1D648}_{i}=\unicode[STIX]{x1D64D}_{i}$ since $\unicode[STIX]{x1D64F}_{i}=\unicode[STIX]{x1D644}_{d}$ : only azimuthal propagation occurs. Consequently (2.11) reduces to

(3.1) $$\begin{eqnarray}\left[\begin{array}{@{}cc@{}}W^{N} & 0\\ 0 & \displaystyle \frac{1}{W^{N}}\end{array}\right]\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{1}=\left[\begin{array}{@{}c@{}}q^{+}\\ q^{-}\end{array}\right]_{1}.\end{eqnarray}$$

The dispersion relation is therefore $W^{N}=1$ , where $W=\text{e}^{2jkL_{c}/N}$ . The $N$ solutions of (3.1) are $W_{0}=\text{e}^{2jp{\rm\pi}/N}$ and correspond to real eigenfrequencies of the unperturbed problem:

(3.2) $$\begin{eqnarray}kL_{c}=p{\rm\pi},\quad \text{so that }f=\frac{pc^{0}}{2L_{c}},\text{ for all }p\in \mathbb{N}.\end{eqnarray}$$

As expected, (3.2) corresponds to a family of azimuthal modes where the first one (  $p=1$ ) is the first azimuthal mode at frequency $c^{0}/2L_{c}$ which is the mode observed in many practical cases. Equation (3.1) also provides the eigenvectors $V$ associated with the eigenfrequencies $f$ given by (3.2). In this situation, the generated eigenspace $\{V\}$ is two-dimensional: all azimuthal modes are degenerate and can be either standing, spinning or mixed. All modes are neutral since no acoustic dissipation is included (zero growth rate: $\text{Im}(\,f)=0$ ).

3.2. Non-symmetric BC configuration with active flames in the low-coupling limit

The use of a combustor with non-identical burners is a promising approach for the control of azimuthal modes (figure 7 d). An asymptotic expansion of the dispersion relation (2.12) can be used to study this case. Since all burners can be different, all coupling parameters ${\rm\Gamma}_{i}$ (2.6) can be different. A fully analytical solution can be formulated when the solution is ‘close’ to the unperturbed annular cavity case of § 3.1. This is obtained by assuming small coupling parameters ${\rm\Gamma}_{i}$ :

(3.3) $$\begin{eqnarray}{\rm\Gamma}_{i}\ll 1,\quad \text{for all }i\in [1,N].\end{eqnarray}$$

Under this assumption, a Taylor expansion of the transfer matrix of the whole system ( $\unicode[STIX]{x1D648}=\prod _{i=N}^{1}\unicode[STIX]{x1D64F}_{i}\unicode[STIX]{x1D64D}_{i}$ ) at second order gives

(3.4) $$\begin{eqnarray}\unicode[STIX]{x1D648}=\left[\begin{array}{@{}cc@{}}\displaystyle W^{N}[1+j{\rm\Sigma}-Q(1)]+Q(W)+o({\rm\Gamma}_{i}^{2}) & \displaystyle j\mathop{\sum }\limits_{i=1}^{N}{\rm\Gamma}_{i}\left(\frac{1}{W}\right)^{N-2i+2}+o({\rm\Gamma}_{i})\\ \displaystyle -j\mathop{\sum }\limits_{i=1}^{N}{\rm\Gamma}_{i}W^{N-2i+2}+o({\rm\Gamma}_{i}) & \displaystyle \frac{1}{W^{N}}[1-j{\rm\Sigma}-Q(1)]+Q\left(\frac{1}{W}\right)+o({\rm\Gamma}_{i}^{2})\end{array}\right],\end{eqnarray}$$

where

(3.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\rm\Sigma}=\mathop{\sum }\limits_{i=1}^{N}{\rm\Gamma}_{i},\\ \displaystyle Q(x)=\mathop{\sum }\limits_{i=1}^{N-1}\mathop{\sum }\limits_{j=i+1}^{N}{\rm\Gamma}_{i}{\rm\Gamma}_{j}x^{N-2(j-i)}\end{array}\right\}\end{eqnarray}$$

and the Landau notation $o(x)$ , called ‘little-o’, is used to designate any quantity that is negligible compared with $x$ .

From (3.4), the dispersion relation at second order is

(3.6) $$\begin{eqnarray}\displaystyle \det \left(M-Id\right) & {\approx} & \displaystyle -\frac{W^{2N}-2W^{N}+1}{W^{N}}-\frac{j{\rm\Sigma}(W^{2N}-1)}{W^{N}}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }\limits_{i=1}^{N-1}\mathop{\sum }\limits_{j=i+1}^{N}{\rm\Gamma}_{i}{\rm\Gamma}_{j}[W^{2N}-W^{N-2(j-i)}-W^{2(j-i)}+1]=0.\end{eqnarray}$$

Equation (3.6) is a dispersion relation which involves terms ( $W=\text{e}^{2jkL_{c}/N}$ and ${\rm\Gamma}_{i}(k)$ ) depending on the wavenumber $k={\it\omega}/c^{0}$ . Under the low-coupling assumption (3.3), the wavenumber $k$ is close to the wavenumber of the unperturbed problem $k^{0}=p{\rm\pi}/L_{c}$ (§ 3.1): $k\approx k^{0}+{\it\epsilon}/L_{c}$ . A proper asymptotic expansion of $W^{\pm }=\text{e}^{2jkL_{c}/N}=\text{e}^{2j(p{\rm\pi}+{\it\epsilon}^{\pm })/N}\approx \text{e}^{2jp{\rm\pi}}(1+2j{\it\epsilon}^{\pm }/N)$ in terms of the wavenumber perturbations ${\it\epsilon}^{+}$ and ${\it\epsilon}^{-}$ gives

(3.7) $$\begin{eqnarray}W^{\pm }=(1+\mathscr{E}^{\pm })W_{0}+o(\mathscr{E}^{\pm }),\quad \text{ i.e. }kL_{c}=p{\rm\pi}+{\it\epsilon}^{\pm }\text{or }f^{\pm }=\frac{pc^{0}}{2L_{c}}+\frac{c^{0}}{2{\rm\pi}L_{c}}{\it\epsilon}^{\pm }.\end{eqnarray}$$

Here, $W_{0}=\text{e}^{2jp{\rm\pi}/N}$ is the solution of the unperturbed problem and corresponds to $kL_{c}=p{\rm\pi}$ (i.e. $f=pc^{0}/2L_{c}$ ), where $p$ is the mode order, $\mathscr{E}^{\pm }=2j({\it\epsilon}^{\pm }/N)$ and $j^{2}=-1$ . It should be noted that the two components $V^{+}$ and $V^{-}$ of the azimuthal mode do not necessarily have the same wavenumber perturbation ${\it\epsilon}^{\pm }$ . Therefore, the notation $W^{\pm }$ is used since the azimuthal propagation of waves $W$ depends on the wavenumber perturbation ${\it\epsilon}^{\pm }$ .

The coupling parameters ${\rm\Gamma}_{i}$ also depend on the frequency and therefore on $W^{\pm }$ (or  ${\it\epsilon}^{\pm }$ ) and can be approximated by

(3.8) $$\begin{eqnarray}{\rm\Gamma}_{i}(W)\approx \underbrace{{\rm\Gamma}_{i}(W=W_{0})}_{{\rm\Gamma}_{i}^{0}}+\;\mathscr{E}^{\pm }W_{0}\underbrace{\left(\frac{\partial {\rm\Gamma}_{i}}{\partial W}\right)_{W=W_{0}}}_{{\rm\Gamma}_{i}^{1}}\approx {\rm\Gamma}_{i}^{0}+\frac{2j{\it\epsilon}^{\pm }W_{0}}{N}{\rm\Gamma}_{i}^{1}.\end{eqnarray}$$

Using (3.7) and (3.8), a Taylor expansion of the terms $W({\it\epsilon}^{\pm })$ and ${\rm\Gamma}_{i}({\it\epsilon}^{\pm })$ in the dispersion relation (3.6) at second order ( $o({\it\epsilon}^{2})$ , knowing that ${\rm\Gamma}_{i}^{0}$ is of order ${\it\epsilon}^{\pm }$ ) gives

(3.9) $$\begin{eqnarray}A-4B{\it\epsilon}^{\pm }+4C{{\it\epsilon}^{\pm }}^{2}=0,\end{eqnarray}$$

where

(3.10) $$\begin{eqnarray}\left.\begin{array}{@{}rcl@{}}A\, & =\, & \displaystyle -[W_{0}^{2N}-2W_{0}^{N}+1]-j{\rm\Sigma}_{0}\left[W_{0}^{2N}-1\right]\\ \, & \, & +\,\displaystyle \mathop{\sum }\limits_{i=1}^{N-1}\mathop{\sum }\limits_{j=i+1}^{N}{\rm\Gamma}_{i}^{0}{\rm\Gamma}_{j}^{0}[W_{0}^{2N}-W_{0}^{N-2(j-i)}-W_{0}^{2(j-i)}+1],\\ B\, & =\, & \displaystyle \frac{j}{2}\left[W_{0}^{2N}-1\right]\left[1+\frac{{\rm\Sigma}_{1}}{N}\right]-\frac{{\rm\Sigma}_{0}}{2}\left[W_{0}^{2N}+1\right],\\ C\, & =\, & \displaystyle \frac{1}{N^{2}}\left[\binom{N}{N-2}W_{0}^{2N}+\binom{N+1}{N-1}\right].\end{array}\right\}\end{eqnarray}$$

Here, ${\rm\Sigma}_{0}=\sum _{i=1}^{N}{\rm\Gamma}_{i}^{0}$ and ${\rm\Sigma}_{1}=\sum _{i=1}^{N}{\rm\Gamma}_{i}^{1}$ , knowing that ${\rm\Gamma}_{i}^{\pm }({\it\omega})\simeq {\rm\Gamma}_{i}^{0}+(2j{\it\epsilon}^{\pm }/N)W_{0}{\rm\Gamma}_{i}^{1}$ (see (3.8)). It should be noted that the analytical resolution of the dispersion relation will lead to the solution ${\it\epsilon}^{\pm }\propto {\rm\Gamma}_{i}^{0}$ , which proves that ${\rm\Gamma}_{i}^{0}$ is a first-order term and ${\rm\Gamma}_{i}^{0}{\rm\Gamma}_{j}^{0}$ and ${\rm\Gamma}_{i}^{0}{\it\epsilon}^{\pm }$ are second-order terms.

From § 3.1, $W_{0}^{N}=1$ , which leads to simplifications of the coefficients $A,B$ and $C$ :

(3.11) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}A=\displaystyle 4\mathop{\sum }\limits_{i=1}^{N-1}\mathop{\sum }\limits_{j=i+1}^{N}{\rm\Gamma}_{i}^{0}{\rm\Gamma}_{j}^{0}\left[\sin \left(\frac{2p{\rm\pi}}{N}(j-i)\right)\right]^{2},\\ B=-{\rm\Sigma}_{0},\\ C=1.\end{array}\right\}\end{eqnarray}$$

Eigenfrequencies are deduced from the quadratic equation (3.9):

(3.12) $$\begin{eqnarray}{\it\epsilon}^{\pm }=\frac{B\pm \sqrt{B^{2}-AC}}{2C},\end{eqnarray}$$

which leads to a simple expression for the wavenumber perturbations in the case of a general non-symmetric BC configuration:

(3.13) $$\begin{eqnarray}{\it\epsilon}^{\pm }=-\frac{1}{2}\left({\rm\Sigma}_{0}\pm \sqrt{{\rm\Sigma}_{0}^{2}-A}\right),\end{eqnarray}$$

where ${\rm\Sigma}_{0}=\sum _{i=1}^{N}{\rm\Gamma}_{i}^{0}$ will be called the ‘coupling strength’ while $A$ is the non-symmetric part defined in (3.11) depending on the number of burners $N$ and the mode order $p$ .

The term $\mathscr{S}_{0}=\sqrt{{\rm\Sigma}_{0}^{2}-A}$ in (3.13) is called the ‘splitting strength’ because it separates the two eigenvalues ${\it\epsilon}^{+}$ and ${\it\epsilon}^{-}$ . It can be recast for simplification (see appendix A) and highlights the key role of the $2p$ th complex Fourier coefficients ${\it\gamma}(\pm 2p)$ of the azimuthal coupling factor distribution ${\rm\Gamma}^{0}=[{\rm\Gamma}_{1}^{0},\dots ,{\rm\Gamma}_{N}^{0}]$ :

(3.14) $$\begin{eqnarray}\mathscr{S}_{0}^{2}={\rm\Sigma}_{0}^{2}-A=\mathop{\sum }\limits_{i,j=1}^{N}{\rm\Gamma}_{i}^{0}{\rm\Gamma}_{j}^{0}\cos \left(\frac{4p{\rm\pi}}{N}(\,j-i)\right)={\it\gamma}(2p)\times {\it\gamma}(-2p),\end{eqnarray}$$

where ${\it\gamma}(k)=\sum _{i=1}^{N}{\rm\Gamma}_{i}^{0}\text{e}^{-j2k{\rm\pi}i/N}$ is the $k$ th Fourier coefficient of the coupling factor azimuthal distribution ${\rm\Gamma}^{0}$ .

Equation (3.13) is a generalization of the results of Noiray et al. (Reference Noiray, Bothien and Schuermans2011) and Parmentier et al. (Reference Parmentier, Salas, Wolf, Staffelbach, Nicoud and Poinsot2012) to an annular chamber connected by $N$ burners with active flames. It shows the following.

1. (i) The stability of the $N$ burner combustor is controlled to first order by the imaginary part of the coupling strength ${\rm\Sigma}_{0}=\sum _{i=1}^{N}{\rm\Gamma}_{i}^{0}$ . This coupling strength depends only on the sum of the individual coupling parameters ${\rm\Gamma}_{i}$ , not on the pattern used to distribute these burners when they differ.

2. (ii) The splitting strength $\mathscr{S}_{0}$ defined by (3.14) controls the nature of the modes: if $\mathscr{S}_{0}=0$ the modes are degenerate (i.e. ${\it\epsilon}^{+}={\it\epsilon}^{-}$ ) and if $\mathscr{S}_{0}\neq 0$ they are not.

Noiray et al. (Reference Noiray, Bothien and Schuermans2011) obtained a similar result where the mode was controlled by ${\it\gamma}_{HR}(2p)$ , the $2p$ th Fourier coefficient of the heat-release distribution. Equation (3.14) is a generalization of such a result: the present network model developed in this paper shows that the mode is controlled by the azimuthal distribution of the coupling parameter ${\rm\Gamma}_{i}^{0}$ , which includes the active flame ( $n$ – ${\it\tau}$ model), but also by the geometry characteristics, the upstream impedance $Z$ of the burners as well as the density and temperature differences between cold and burnt gases. All these features can affect the asymmetry of the system and therefore the stability; they cannot be neglected when studying combustion instabilities in annular chambers.

A summary of this analytical method providing the frequencies and the stability map of the $p$ th azimuthal mode in a chamber with $N$ burners is given in appendix B.

4.1. Description of the configurations

The 3D geometries correspond to BC set-ups with $N=3$ or $N=24$ burners (figure 8), similar to figure 3 (the physical and geometrical parameters are defined in table 1). The burner/chamber interfaces are placed at $z=0$ and the flames are on the burner side. The flame width is equal to 2 mm, which guarantees its compactness with respect to the acoustic wavelength. The boundary conditions correspond to impermeable walls everywhere except at the upstream end of the burners where an impedance $Z=0$ (i.e. $p^{\prime }=0$ ) is imposed to mimic a connection to a large plenum. For the $N=3$ burner configuration, two cases are investigated (table 2): first with identical burners and then with two types of burners with different time delays ${\it\tau}_{1}$ and ${\it\tau}_{2}$ . The interaction index of the flames is set to the same value $n=1.0$ (knowing that typical low-frequency values for $n$ are around $T^{0}/T_{u}^{0}-1\simeq 1.57$ here (Poinsot & Veynante Reference Poinsot and Veynante2011)) in each burner.

Figure 8.

Toy models to validate the ATACAMAC methodology. (a) Perfect annular chamber with $N=3$ , (b) $N=24$ cylindrical burners, (c) burner/chamber configuration.

Table 1.

The parameters used for numerical applications. They correspond to a typical large-scale industrial gas turbine.

Table 2.

The BC configurations investigated with both the 3D Helmholtz solver AVSP and the analytical approach ATACAMAC. ○: burner without CBO; ●: burner with CBO.

For the $N=24$ configuration (figure 8 b), two types of burners with different time delays are mixed to mimic the combustion chamber where burners can be equipped (or not) with CBOs to modify their flame response (Krueger et al. Reference Krueger, Hueren, Hoffmann, Krebs, Flohr and Bohn2000; Berenbrink & Hoffmann Reference Berenbrink and Hoffmann2001). Table 2 displays the circumferential patterns (● for CBO burners and ○ for burners without CBOs) that are considered.

4.2. Description of the 3D acoustic code

The assumptions and results of ATACAMAC can be validated using a full 3D acoustic solver called AVSP (Selle et al. Reference Selle, Benoit, Poinsot, Nicoud and Krebs2006; Nicoud et al. Reference Nicoud, Benoit, Sensiau and Poinsot2007; Sensiau et al. Reference Sensiau, Nicoud and Poinsot2009; Silva et al. Reference Silva, Nicoud, Schuller, Durox and Candel2013), which solves the Helmholtz equation in a reactive flow without the assumptions used in ATACAMAC (Nicoud et al. Reference Nicoud, Benoit, Sensiau and Poinsot2007), but of course at a higher cost. The AVSP solver takes into account the interaction between combustion and acoustics. It solves the eigenvalue problem issued from the discretization on unstructured meshes of the Helmholtz equation at zero Mach number. The meshes contain approximately two million cells (corresponding to the ratio of the wavelength to the longest cell length ${\it\lambda}/{\rm\Delta}h_{max}\simeq 250$ ), which is sufficient considering the simplicity of the geometry and the wavelength of the first azimuthal mode. Source terms due to flames are modelled using FTFs (Crocco Reference Crocco1951). The local heat-release fluctuations in the burner $i$ are expressed as

(4.1) $$\begin{eqnarray}q_{i}^{\prime }=n_{u,i}\,\text{e}^{j{\it\omega}{\it\tau}_{i}}\,\boldsymbol{u}^{\prime }(\boldsymbol{x}_{ref,i})\boldsymbol{\cdot }\boldsymbol{n}_{ref,i},\end{eqnarray}$$

where $\boldsymbol{x}_{ref,i}$ is a reference point upstream of the flame in burner $i$ .

The local interaction index $n_{u,i}$ describes the local flame–acoustic interactions. The values of $n_{u,i}$ are assumed to be constant in the flame zone $i$ (figure 8) and are chosen to recover the global value of the interaction index $n_{i}$ corresponding to the infinitely thin flame when integrated over the flame zone $i$ (Nicoud et al. Reference Nicoud, Benoit, Sensiau and Poinsot2007). For the sake of simplicity, they are also assumed to be independent of frequency. These assumptions allow one to use AVSP to check the precision of the analytical techniques developed in ATACAMAC.

In annular configurations with multiple burners, the heat-release fluctuations in burner $i$ are assumed to be driven by velocity fluctuations at the reference point $\boldsymbol{x}_{\mathit{ref},\mathit{i}}$ . This assumption, called ISAAC (independence sector assumption in annular combustor) in Sensiau et al. (Reference Sensiau, Nicoud and Poinsot2009), was validated by an LES of a full annular combustor (Staffelbach et al. Reference Staffelbach, Gicquel, Boudier and Poinsot2009) and is used in the present study. In the infinitely thin flame model used in ATACAMAC the reference points are chosen at the flame locations $z_{f,i}$ . The normalized abscissa of the flame is set to ${\it\alpha}\simeq 0.91$ (Bauerheim et al. Reference Bauerheim, Parmentier, Salas, Nicoud and Poinsot2014c ). Three-dimensional effects near the burner/chamber junctions can be accounted for (Pierce Reference Pierce1981) using a standard length correction in the low-frequency range for a flanged tube (Silva et al. Reference Silva, Guillemain, Kergomard, Mallaroni and Norris2009), which is applied at the downstream burner end ( ${\rm\Delta}L_{i}=0.4\sqrt{4S_{i}/{\rm\pi}}$ ). In AVSP, the reference points are placed a few millimetres upstream of the flames (figure 8) due to numerical accuracy issues (Silva et al. Reference Silva, Nicoud, Schuller, Durox and Candel2013).

5.1. Symmetric case with $N=3$ identical burners

In an axisymmetric configuration where the burners are the same for all sectors (i.e. ${\rm\Gamma}_{i}={\rm\Gamma}$ , for all $i\in [1,3]$ ), only two different mode types exist. Indeed, the splitting strength $\mathscr{S}_{0}$ in (3.14) simplifies depending on the mode order $p$ and the number of burners $N$ (appendix A):

(5.1) $$\begin{eqnarray}\left.\begin{array}{@{}l@{}}\text{if }p=3m,\quad \text{for}\ m\in \mathbb{N}\text{ then }\mathscr{S}_{0}=3{\rm\Gamma}^{0},\\ \text{for any other mode of order }p\!:\mathscr{S}_{0}=0.\end{array}\right\}\end{eqnarray}$$

Consequently, only two different classes of modes can develop in annular BC configurations with $N=3$ identical burners.

1. (i) Non-degenerate singlets. If $p=3m,m\in \mathbb{N}$ the splitting strength is not null ( $\mathscr{S}_{0}={\rm\Sigma}_{0}=3{\rm\Gamma}^{0}$ , (3.14)) and the azimuthal mode is split into two components  $V^{-}$ and  $V^{+}$ with different wavenumber perturbations ( ${\it\epsilon}^{\pm }$ ) and frequencies ( $f^{\pm }$ ) (3.7):

   (5.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\it\epsilon}^{-}=0,\\ {\it\epsilon}^{+}=-3{\rm\Gamma}^{0},\end{array}\right\}\quad \text{corresponding to }\left.\begin{array}{@{}c@{}}f^{-}=\displaystyle \frac{pc^{0}}{2L_{c}},\\ f^{+}=\displaystyle \frac{pc^{0}}{2L_{c}}-\frac{3}{2}\frac{c^{0}{\rm\Gamma}^{0}}{{\rm\pi}L_{c}}.\end{array}\right\}\end{eqnarray}$$
   Figure 10 displays the mode structure associated with $f^{+}$ and $f^{-}$ : due to symmetry considerations, these modes (e.g. $p=3$ ) can lock on burners ( $N=3$ ) (see Bauerheim et al. (Reference Bauerheim, Cazalens and Poinsot2014a ) for an analytical proof). Their wavelength corresponds to the chamber perimeter (or half of it in the case of a configuration with an even number of burners). The first mode $V^{-}$ with frequency $f^{-}$ is standing and imposes a pressure node at every burner; therefore, it is unperturbed by them ( ${\it\epsilon}^{-}=0$ ); the mode is neutral. The second mode $V^{+}$ at the frequency $f^{+}$ is also standing but imposing an azimuthal velocity node (i.e. a pressure anti-node) at every burner ( ${\it\epsilon}^{+}=-3{\rm\Gamma}^{0}$ ).

2. (ii) Degenerate doublets. All other azimuthal modes ( $p\neq 3m,m\in \mathbb{N}$ ) are composed of two eigenmodes $V^{\pm }$ which have the same frequencies (degenerate modes) because the splitting strength $\mathscr{S}_{0}=0$ :

   (5.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\it\epsilon}^{-}=\displaystyle -\frac{3}{2}{\rm\Gamma}^{0},\\ {\it\epsilon}^{+}=\displaystyle -\frac{3}{2}{\rm\Gamma}^{0},\end{array}\right\}\quad \text{corresponding to }\left.\begin{array}{@{}c@{}}f^{-}=\displaystyle \frac{pc^{0}}{2L_{c}}-\frac{3}{4}\frac{c^{0}{\rm\Gamma}^{0}}{{\rm\pi}L_{c}},\\ f^{+}=\displaystyle \frac{pc^{0}}{2L_{c}}-\frac{3}{4}\frac{c^{0}{\rm\Gamma}^{0}}{{\rm\pi}L_{c}}.\end{array}\right\}\end{eqnarray}$$

   In this configuration, the transfer matrix of the whole system ( $\unicode[STIX]{x1D648}$ defined in (3.4)) is equivalent to the null matrix. The mode nature is undetermined, as pointed out by Noiray et al. (Reference Noiray, Bothien and Schuermans2011): a standing, spinning or mixed mode can develop. Noiray et al. (Reference Noiray, Bothien and Schuermans2011) have shown that nonlinearities on the FTF can, however, promote one of these natures, a phenomenon that cannot be described by ATACAMAC since it is based on linear FTFs.

Figure 10.

The pressure structure of the two components of the azimuthal mode in an $N=3$ burner configuration: a standing mode imposing a pressure node at every burner location (—–, $f^{-}$ ) and a standing mode imposing a pressure anti-node at every burner location (- - -, $f^{+}$ ).

Figures 11 and 12 provide two validation points (marked by ●) of ATACAMAC for symmetric configurations using the full 3D acoustic solver AVSP: modes denoted $A^{\pm }$ when ${\it\tau}_{1}/{\it\tau}_{c}^{0}={\it\tau}_{2}/{\it\tau}_{c}^{0}=0.23$ and $E^{\pm }$ when ${\it\tau}_{1}/{\it\tau}_{c}^{0}={\it\tau}_{2}/{\it\tau}_{c}^{0}=0.68$ . Good agreement is obtained between the acoustic code (AVSP) and ATACAMAC.

Figure 11.

(a) Real and (b) imaginary parts of the frequencies $f^{+}$ and $f^{-}$ of the two components of the first mode (  $p=1$ ) in the B3_C1 configuration with the pattern 121 (○ ● ○) and a fixed ${\it\tau}_{2}/{\it\tau}_{c}^{0}=0.23$ . —–: ATACAMAC (numerical resolution of (2.12));  $\Box$ : ATACAMAC (analytical formula (5.6)); $\times$ : AVSP; - - -: trajectory of the mode average $(\,f^{+}+f^{-})/2$ ; ●: symmetric case where ${\it\tau}_{1}/{\it\tau}_{c}^{0}={\it\tau}_{2}/{\it\tau}_{c}^{0}=0.23$ , ${\it\tau}_{c}^{0}$ corresponds to $1/f_{0}\simeq 11\ \text{ms}$ .

Figure 12.

(a) Real and (b) imaginary parts of the frequencies $f^{+}$ and $f^{-}$ of the two components of the first mode (  $p=1$ ) in the B3_C1 configuration with the pattern 121 (○ ● ○) and a fixed ${\it\tau}_{2}/{\it\tau}_{c}^{0}=68$ . —–: ATACAMAC (numerical resolution of (2.12));  $\Box$ : ATACAMAC (analytical formula (5.6)); $\times$ : AVSP; - - -: trajectory of the mode average $(\,f^{+}+f^{-})/2$ ; ●: symmetric case where ${\it\tau}_{1}/{\it\tau}_{c}^{0}={\it\tau}_{2}/{\it\tau}_{c}^{0}=0.68$ , ${\it\tau}_{c}^{0}$ corresponds to $1/f_{0}\simeq 11\ \text{ms}$ .

5.2. Symmetry breaking with $N=3$ different burners

This section discusses the behaviour of azimuthal modes when one of the three burners has a different FTF corresponding to a different value of ${\rm\Gamma}_{i}^{0}$ (see figure 9 b). In particular, the observation in Perrin & Charnley (Reference Perrin and Charnley1973) and Sensiau et al. (Reference Sensiau, Nicoud and Poinsot2009) will be investigated: circumferential variations with specific patterns obtained by distributing different burner types along the azimuthal direction could split nominally degenerate doublets into non-degenerate singlets, as observed by Sensiau et al. (Reference Sensiau, Nicoud and Poinsot2009). If two burners have a coupling factor ${\rm\Gamma}_{1}^{0}$ and the third one ${\rm\Gamma}_{2}^{0}$ , (3.13) can be solved with $N=3$ and gives the following solution.

1. (i) Non-degenerate singlets. Azimuthal modes with $p=3m,m\in \mathbb{N}$ are non-degenerate singlets characterized by $\mathscr{S}_{0}={\rm\Sigma}_{0}=2{\rm\Gamma}_{1}^{0}+{\rm\Gamma}_{2}^{0}$ (3.14) with wavenumber perturbations:

   (5.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\it\epsilon}^{-}=0,\\ {\it\epsilon}^{+}=-{\rm\Sigma}_{0}=-2{\rm\Gamma}_{1}^{0}-{\rm\Gamma}_{2}^{0},\end{array}\right\}\quad \text{corresponding to }\left.\begin{array}{@{}c@{}}f^{-}=\displaystyle \frac{pc^{0}}{2L_{c}},\\ f^{+}=\displaystyle \frac{pc^{0}}{2L_{c}}-\frac{c^{0}(2{\rm\Gamma}_{1}^{0}+{\rm\Gamma}_{2}^{0})}{2{\rm\pi}L_{c}},\end{array}\right\}\end{eqnarray}$$
   where ${\rm\Sigma}_{0}=\sum _{i=1}^{N}{\rm\Gamma}_{i}^{0}$ . These modes, as in the symmetric cases, impose a pressure node or pressure anti-node at each burner location leading to two modes with different frequencies: $f^{+}\neq f^{-}$ .

2. (ii) Nearly degenerate singlets. For other azimuthal modes ( $p\neq 3m,m\in \mathbb{N}$ ), (3.13) leads to nearly degenerate singlets (Perrin & Charnley Reference Perrin and Charnley1973): the degenerate doublet encountered in symmetric configurations (denoted DD with ${\it\epsilon}_{DD}=-{\rm\Sigma}_{0}/2=-{\rm\Gamma}_{1}^{0}-{\rm\Gamma}_{2}^{0}/2$ , (5.3)) is split depending on the splitting strength $\mathscr{S}_{0}={\rm\Gamma}_{1}^{0}-{\rm\Gamma}_{2}^{0}$ (3.14) for the $N=3$ case with the pattern 121 (○ ● ○, table 2):

   (5.5) $$\begin{eqnarray}{\it\epsilon}^{\pm }=\underbrace{-{\textstyle \frac{1}{2}}{\rm\Sigma}_{0}}_{{\it\epsilon}_{DD}}\ \pm \underbrace{{\textstyle \frac{1}{2}}\mathscr{S}_{0}}_{\text{Splitting}},\end{eqnarray}$$
   so that
   (5.6a,b ) $$\begin{eqnarray}{\it\epsilon}^{-}=-{\textstyle \frac{1}{2}}\left({\rm\Gamma}_{1}^{0}+2{\rm\Gamma}_{2}^{0}\right)\quad \text{and}\quad {\it\epsilon}^{+}=-{\textstyle \frac{3}{2}}{\rm\Gamma}_{1}^{0}.\end{eqnarray}$$

These results were validated in figures 11 and 12 for the first azimuthal mode (  $p=1$ ) of the configuration B3_C1 with the pattern 121 (○ ● ○, table 2), where the coupling parameters are defined by (2.8).

When $\mathscr{S}_{0}\neq 0$ , the nominally doublet mode is split into two dissimilar azimuthal modes (e.g. modes denoted $B^{+}$ and $B^{-}$ in figure 11, corresponding to ${\it\tau}_{1}/{\it\tau}_{c}^{0}=0.55$ and ${\it\tau}_{2}/{\it\tau}_{c}^{0}=0.23$ ) with close frequencies and different growth rates, as mentioned in Perrin & Charnley (Reference Perrin and Charnley1973) and Sensiau et al. (Reference Sensiau, Nicoud and Poinsot2009). The term $\mathscr{S}_{0}={\rm\Gamma}_{1}^{0}-{\rm\Gamma}_{2}^{0}$ for the pattern 121 (○ ● ○) measures the differences in flame response between the two burner types and controls the degree of degeneracy of the azimuthal mode. It should be noted that some asymmetry could still give degenerate doublets (i.e. $\mathscr{S}_{0}=0$ ); for instance, the first-order mode (  $p=1$ ) of an $N=6$ burner BC configuration with the pattern (● ○ ● ○ ● ○) or (● ● ● ○ ○ ○) is a doublet with ${\it\epsilon}=-{\textstyle \frac{3}{2}}({\rm\Gamma}_{1}^{0}+{\rm\Gamma}_{2}^{0})$ .

Figure 13 displays the associated mode structure of the symmetric (mode $A$ ) and asymmetric (mode $B$ ) cases. For symmetric configurations, the mode structure is undetermined so that both spinning and standing modes can occur (figure 13 a, top). However, breaking the symmetry using two different burner types (e.g. mode $B$ with ${\it\tau}_{1}/{\it\tau}_{c}^{0}=0.55$ and ${\it\tau}_{2}/{\it\tau}_{c}^{0}=0.23$ ) leads to standing modes only (figure 13 a, bottom).

Figure 13.

Three-dimensional representation and isolines of the pressure modulus (a) and modulus and phase of the acoustic pressure (b) of the first azimuthal modes (  $p=1$ ) of the asymmetric case B3_C1 with the pattern 121 (○ ● ○, table 2) in two situations: mode $A$ ( ${\it\tau}_{1}/{\it\tau}_{c}^{0}={\it\tau}_{2}/{\it\tau}_{c}^{0}=0.23$ ) and mode $B$ ( ${\it\tau}_{1}/{\it\tau}_{c}^{0}=0.55$ and ${\it\tau}_{2}/{\it\tau}_{c}^{0}=0.23$ ). ○: $A^{\pm }$ (spinning); ●: $A^{\pm }$ (standing); —–: $B^{\pm }$ (standing).

5.3. Conclusion on symmetry breaking in the $N=3$ case

Sections 5.1 and 5.2 show that the splitting strength $\mathscr{S}_{0}$ defined by (3.14) controls both the stability and the mode structure of an annular chamber ( $N=3$ ) where two types of burners are installed. Nevertheless, a ‘necessary condition’ for stability can be derived independently of the splitting strength value. Indeed, the imaginary part of the mode average $\text{Im}({\it\epsilon}^{+}+{\it\epsilon}^{-})/2$ does not depend on the splitting strength $\mathscr{S}_{0}$ but only on the total coupling strength ${\rm\Sigma}_{0}=\sum _{i=1}^{N}{\rm\Gamma}_{i}^{0}$ , yielding a necessary condition for stability:

(5.7) $$\begin{eqnarray}{\textstyle \frac{1}{2}}\text{Im}({\it\epsilon}^{+}+{\it\epsilon}^{-})=-{\textstyle \frac{1}{2}}\text{Im}({\rm\Sigma}_{0})<0.\end{eqnarray}$$

If this condition is not fulfilled (figure 14 a), there is no hope of stabilizing the mode since at least one of the two components of the azimuthal mode ( $V^{+}$ or $V^{-}$ ) will remain unstable (e.g. $A^{\pm }$ and $B^{+}$ in figure 11).

Figure 14.

Mode stability for an annular chamber with $N$ burners depending on the condition (5.7) and the splitting strength $\text{Im}(\mathscr{S}_{0})/2$ .  $\times$ : ${\it\epsilon}^{-}$ ; $\Box$ : ${\it\epsilon}^{+}$ . The splitting strength is $\mathscr{S}_{0}^{2}=\sum _{i,j=1}^{N}{\rm\Gamma}_{i}^{0}{\rm\Gamma}_{j}^{0}\cos ((4p{\rm\pi}/N)(j-i))$ (3.14) and measures the difference between the two burner types 1 and 2. Shaded areas correspond to unstable zones. (a) All modes are unstable, (b) all modes are stable and (c) symmetry breaking makes one mode unstable: the splitting strength $\mathscr{S}_{0}$ must be reduced to stabilize both modes as in the situation (b).

For a symmetric case where the splitting strength $\mathscr{S}_{0}$ is zero, (5.7) is a necessary and sufficient condition to have a stable mode. However, when the symmetry is broken, satisfying condition (5.7) cannot guarantee stability (figure 14 b,c). In this case, the necessary and sufficient condition becomes

(5.8) $$\begin{eqnarray}\max (\text{Im}({\it\epsilon}^{+}),\text{Im}({\it\epsilon}^{-}))<0,\end{eqnarray}$$

because the splitting introduced by symmetry breaking (measured by the splitting strength $\mathscr{S}_{0}$ ) has to be taken into account. For weak splitting (figure 14 b) the two modes $V^{+}$ and $V^{-}$ remain stable (e.g. eigenmodes $D^{\pm }$ in figure 12), but for higher splitting (figure 14 c) one mode can become unstable (e.g. mode $C^{+}$ in figure 12). It should be noted that $\max (\text{Im}({\it\epsilon}^{+}),\text{Im}({\it\epsilon}^{-}))>\text{Im}({\it\epsilon}^{+}+{\it\epsilon}^{-})/2$ and for a symmetric configuration ${\it\epsilon}^{+}={\it\epsilon}^{-}$ .