3.1. Azimuthal aeroacoustic instability

In the range $60 < \dot {m} < 100\ {\rm g}\ {\rm s}^{-1}$, which corresponds to $42< U_x < 70\ {\rm m}\ {\rm s}^{-1}$, the system presents an aeroacoustic instability involving the first azimuthal acoustic mode and a shear layer mode of azimuthal order 1. It is characterised by the shedding of large vortices that span across the cavity's width $W$ (see Reference Faure-Beaulieu, Xiong, Pedergnana and NoirayPart 1). The whistling frequency does not vary much over the whole range of bulk velocities: it starts at 773 Hz when $U_x = 42\ {\rm m}\ {\rm s}^{-1}$ and reaches 809 Hz when $U_x = 70\ {\rm m}\ {\rm s}^{-1}$. Figure 2(b) shows the acoustic pressure field associated with the first pure azimuthal mode in the analogue geometry of a cylindrical chamber. In this simplified case, the Helmholtz eigenvalue problem can be solved analytically and the shapes of the first pair of degenerate azimuthal eigenmodes are $p(r,\varTheta )={\rm J}_1(rZ_1/R)\cos (\varTheta )$ (represented in the figure) and $p(r,\varTheta )={\rm J}_1(rZ_1/R)\sin (\varTheta )$, where ${\rm J}_1$ is the Bessel function of the first kind and order 1, and $Z_1$ is the first zero of ${\rm J}_1'$. The corresponding frequency is 786 Hz, which is in agreement with the experimental results.

Figure 2. (a) Measured acoustic pressure signals for $\dot {m}=84\ {\rm g}\ {\rm s}^{-1}$ ($U_x=59\ {\rm m}\ {\rm s}^{-1}$) and $\dot {m}_s=0$. (b) Instantaneous acoustic pressure field associated with the first pure azimuthal acoustic mode of a cylindrical cavity, obtained analytically. The eigenfrequency is 786 Hz.

Figure 2(a) shows acoustic signals recorded simultaneously by three cavity microphones at different azimuthal positions and equal distance from the axis: the colours of the lines correspond to the colours of the microphones in the sketch 1(c). These signals were recorded for $\dot {m} = 84\ {\rm g}\ {\rm s}^{-1}$ ($U_x=59\ {\rm m}\ {\rm s}^{-1}$) and $\dot {m}_s=0$, a condition at which a high-amplitude aeroacoustic limit cycle at 790 Hz is present, reaching an acoustic level of 165 dB. The long time interval displayed in this figure does not allow us to see the details of the quasi-sinusoidal oscillations at 790 Hz, but it shows the fluctuations of the acoustic pressure envelope which are largely caused by the forcing from turbulence. On the yellow time trace ($\varTheta =332^\circ$), one can see at $t\approx 15$ s some quasi-periodic modulation of the acoustic envelope, corresponding to a beating azimuthal mode, i.e. cyclic alternation of clockwise (CW) and CCW mixed modes with a period lasting a few hundreds of acoustic periods. This deterministic phenomenon of self-sustained beating modes was recently discovered and explained in the case of thermoacoustic instabilities in combustion chambers by Faure-Beaulieu et al. (Reference Faure-Beaulieu, Indlekofer, Dawson and Noiray2021). The yellow time trace is also characterised by sporadic high-amplitude bursts which can reach 8000 Pa for a few seconds.

3.2. Modal projection of the acoustic field

Interpreting the raw acoustic signals is not straightforward. It is therefore convenient to use the quaternion projection proposed by Ghirardo & Bothien (Reference Ghirardo and Bothien2018) on the basis of the work of Flamant, Le Bihan & Chainais (Reference Flamant, Le Bihan and Chainais2017), in order to unravel the modal dynamics. This projection can be used to describe the acoustic pressure field associated with an azimuthal wave of any order and it is particularly suited to providing a straightforward and unambiguous interpretation of the corresponding thermoacoustic time series (Ghirardo & Bothien Reference Ghirardo and Bothien2018). It writes as

(3.1)\begin{align} p_a=A\cos(m(\varTheta-\theta))\cos(\chi)\cos(\omega t + \varphi) + A\sin(m(\varTheta-\theta))\sin(\chi)\sin(\omega t + \varphi),\end{align}

where $m$ is the azimuthal order of the mode – in the present study, it is always equal to 1. In (3.1), $\varTheta$ is the azimuthal coordinate, $t$ is the time and $\omega$ is the angular frequency of the oscillations. This ansatz is referred to as the ‘quaternion projection’ because it is the real part of a quaternion analytical signal (Flamant et al. Reference Flamant, Le Bihan and Chainais2017)

(3.2)\begin{equation} p_a={\rm Re}\big(A \mathrm{e}^{\mathrm{i} m(\varTheta-\theta)}\mathrm{e}^{-\mathrm{k}\chi}\mathrm{e}^{\mathrm{j}(\omega t + \varphi)}\big), \end{equation}

where $\mathrm {i}$, $\mathrm {j}$ and $\mathrm {k}$ are the basic quaternions and $A(t)$, $\chi (t)$, $\theta (t)$ and $\varphi (t)$ are the four slow variables which define the instantaneous state of the mode. The positive variable $A$ is the amplitude of the acoustic pressure. The nature angle $\chi$ indicates the type of mode: if $\chi =0$, the mode is purely standing; if $\chi ={\rm \pi} /4$, it is purely CCW spinning; if $\chi =-{\rm \pi} /4$, it is purely CW spinning; if $0<\chi <{\rm \pi} /4$, it is a CCW mixed mode that can be decomposed as the superposition of a pure standing mode and a pure CCW spinning mode; if $0>\chi >-{\rm \pi} /4$, it is CW mixed mode. The angle $\theta$ indicates the orientation of the maximal acoustic pressure amplitude: for a standing mode, $\theta$ is simply the antinodal direction. For a mixed mode, it is the antinodal direction of the standing component of the mode. For a pure spinning mode, $\theta$ transforms into a temporal phase information. The last variable, $\varphi$, is a temporal phase that can be associated with small fluctuations of the acoustic period $2{\rm \pi} /\omega$. The four variables $A$, $\chi$, $\theta$ and $\varphi$ are useful to represent variations of the instantaneous mode state, which are slow compared with the acoustic period, i.e. $\dot {A}/A$, $\dot {\chi }$, $\dot {\theta }$ and $\dot {\varphi }$ are much smaller than $\omega$.

To perform the quaternion projection, the prescribed frequency $\omega$ must be close to the frequency of the instability. It is usually not appropriate to define $\omega$ as the frequency of the pure acoustic mode, because it can noticeably differ from the aeroacoustic mode frequency due to the time delay of the shear layer response and the presence of small flow asymmetries (Noiray, Bothien & Schuermans Reference Noiray, Bothien and Schuermans2011; Kim et al. Reference Kim, John, Adhikari, Wu, Emerson, Acharya, Isono, Saitoh and Lieuwen2022). The most adequate option is usually to choose $\omega$ as the frequency of the dominant peak in the power spectral density.

Furthermore, the expression (3.1) describes the dependence of the acoustic field along the spatial coordinate $\varTheta$ only. This one-dimensional (1-D) description is readily applicable to the case of thin annular combustors (Faure-Beaulieu & Noiray Reference Faure-Beaulieu and Noiray2020), and it can also be used for azimuthal modes exhibiting non-uniform distribution of the acoustic pressure in the axial and radial direction, i.e. $A=A(t,x,r)$. Indeed, in such situation, the three-dimensional (3-D) acoustic field can be spatially averaged over constant $\varTheta$ planes as explained in Appendix A. All the six microphones used for the quaternion projection are located at the same radial and axial positions and the full 3-D acoustic field can be reconstructed from the measured amplitudes and the knowledge of the mode shape.

3.3. Effect of an imposed swirl on the aeroacoustic mode

Acoustic measurements were performed for different levels of swirl by increasing $\dot {m}_s$ from 0 to $21\ {\rm g}\ {\rm s}^{-1}$ by steps of $3\ {\rm g}\ {\rm s}^{-1}$ while keeping a constant total mass flow $\dot {m}_t=84\ {\rm g}\ {\rm s}^{-1}$ (Reynolds number ${Re}=1.5\times 10^{5}$). The corresponding mean azimuthal velocity remains moderate compared with the axial velocity: PIV measurements in the cavity show a maximal local value of $15\ {\rm m}\ {\rm s}^{-1}$ when $\dot {m}_s=18\ {\rm g}\ {\rm s}^{-1}$, while the mean axial bulk velocity is $U_x=59\ {\rm m}\ {\rm s}^{-1}$. This gives a swirl number of 0.18, which is relatively small compared with the values usually found in the literature on swirling flows (e.g. Gallaire & Chomaz Reference Gallaire and Chomaz2003b; Qadri, Mistry & Juniper Reference Qadri, Mistry and Juniper2013; Oberleithner et al. Reference Oberleithner, Paschereit and Wygnanski2014; Tammisola & Juniper Reference Tammisola and Juniper2016).

The frequency of the aeroacoustic instability is not substantially affected by the imposed swirl and remains between 790 and 797 Hz. To characterise the influence of the imposed swirl on the aeroacoustic limit cycle, the state variables $A$, $\chi$, $\theta$ and $\varphi$ are extracted from the acoustic time series following the procedure described by Ghirardo & Bothien (Reference Ghirardo and Bothien2018). Figure 3 presents the evolution of the state variables for $\dot {m}_s=0$, 3, 6 and $9\ {\rm g}\ {\rm s}^{-1}$ (from left to right). The cases $\dot {m}_s>9\ {\rm g}\ {\rm s}^{-1}$, not shown here, behave similarly as for $\dot {m}_s=9\ {\rm g}\ {\rm s}^{-1}$. Let us now describe the dynamics of the slow state variable at these four conditions.

Figure 3. Evolution of the slow state variables $A$, $\chi$, $\theta$ and $\varphi$ extracted from the experimental acoustic measurements, for different swirl numbers imposed on the flow upstream of the axisymmetric cavity. The four tangential mass flows $\dot {m}_s$ considered in columns (a–d) are equal to 0, 3, 6 and $9\ {\rm g}\ {\rm s}^{-1}$ respectively for a total mass flow $\dot {m}_t$ kept constant and equal to $84\ {\rm g}\ {\rm s}^{-1}$.

For $\dot {m}_s=0$, which is shown in figure 3(a), the evolution of the nature angle $\chi$ indicates intermittent transitions between a CCW mixed mode with $\chi \approx {\rm \pi}/8$ and a beating mode characterised by quasi-periodic changes of the spinning direction that manifest themselves by triangular oscillations of $\chi$. These two regimes can last several seconds, which is long compared with the acoustic period ($\approx$1.25 ms) and the characteristic time of growth of an aeroacoustic instability. The amplitude $A$ displays large bursts in mixed mode regime, and stays around 5 kPa in the beating regime. Faure-Beaulieu et al. (Reference Faure-Beaulieu, Indlekofer, Dawson and Noiray2021) showed that beating can be caused by small non-uniformities in the geometry, the impedance of the cavity walls or the flow. The small asymmetry of the mean axial velocity profile reported earlier is thus a possible explanation for the occurrence of beating in the present configuration. In the mixed mode regime, the preferred orientation $\theta$ undergoes small stochastic fluctuations around a fixed angle defined modulo ${\rm \pi}$, while during the beating regime, it shows periodic square oscillations.

When $\dot {m}_s=3\ {\rm g}\ {\rm s}^{-1}$, the mixed mode regime and its high-amplitude bursts disappear, and only the beating mode regime is observed (figure 3b). The fast drift of $\varphi$ is explained by the mismatch between the frequency $\omega$ chosen for the quaternion projection and the actual dominant aeroacoustic frequency during the considered time interval. For this case, the aeroacoustic dynamics shows no preference for the CW or CCW spinning direction, which is confirmed by the probability density function (p.d.f.) of $\chi$ in figure 4. The symmetry of the aeroacoustic dynamics is recovered because the small imposed swirl compensates the effect of the small inherent reflectional asymmetry of the system. The reflectional symmetry of the aeroacoustic dynamics, which manifests itself here as a robust beating mode without preference for the CW or the CCW spinning direction, is therefore recovered by imposing a small mean swirl, i.e. imposing a small reflectional asymmetry to the system. In summary, inherent imperfections are present in the system, as in the case of thermoacoustic instabilities investigated by Faure-Beaulieu et al. (Reference Faure-Beaulieu, Indlekofer, Dawson and Noiray2021). They lead to a dominant CCW mixed mode in absence of mean swirl ($\dot{m}_s= 0\ {\rm g}\ {\rm s}^{-1}$), and this dominance is eliminated by imposing a small swirl on the incoming flow ($\dot{m}_s= 3\ {\rm g}\ {\rm s}^{-1}$, and $\dot{m}_s/\dot{m}_t = 3.5\,\%$).

Figure 4. Experimental p.d.f. of the nature angle $\chi$ for the same conditions as in figure 3, for increasing imposed swirler mass flow $\dot {m}_s=0$, 3, 6 and $9\ {\rm g}\ {\rm s}^{-1}$ (darker shades correspond to stronger swirl). Each p.d.f. is obtained from acoustic time traces of 100 s. Values of $-{\rm \pi} /4$, $0$ and ${\rm \pi} /4$ correspond respectively to the states of pure CW spinning wave, pure standing wave and pure CCW spinning wave.

When the swirler mass flow is further increased to $\dot {m}_s=6\ {\rm g}\ {\rm s}^{-1}$ (figure 3c), the dynamics of the state variables become similar to the case $\dot {m}_s=0$, except that the preferred spinning direction is CW instead of CCW. Indeed, a sporadic alternance of mixed CW regime and beating regime is observed, indicating that the imposed swirl tends to promote the CW, i.e. counterswirl direction. Finally, when $\dot {m}_s=9\ {\rm g}\ {\rm s}^{-1}$, the beating regime disappears, leaving only a CW spinning mode of amplitude of 8 kPa (figure 3d). For higher values of $\dot {m}_s$, the mode keeps spinning in the same counterswirl direction.

Figure 4 summarises the evolution of the p.d.f. of $\chi$ for the four experimental conditions presented in figure 3. Darker lines indicate higher swirler mass flows $\dot {m}_s$. Without swirl, the p.d.f. is bimodal and asymmetric, with a higher peak for a positive value of $\chi$ and a smaller peak for a negative $\chi$. At $\dot {m}_s=3\ {\rm g}\ {\rm s}^{-1}$, the p.d.f. has two symmetric peaks indicating that there is no longer a preference for one spinning direction. The p.d.f. for $\dot {m}_s=6\ {\rm g}\ {\rm s}^{-1}$ is almost the mirror image of the case without swirl, with a preference for CW mixed modes. For $\dot {m}_s=9\ {\rm g}\ {\rm s}^{-1}$, the p.d.f. has a single, sharp peak, very close to $\chi =-{\rm \pi} /4$ (quasi-pure CW spinning mode). In order to explain these observations, it is therefore key to separately investigate the effects of the mean flow on the aeroacoustic wave and the ones of the wave on the mean flow. The latter mechanism is the topic of Reference Faure-Beaulieu, Xiong, Pedergnana and NoirayPart 1, and the former is treated in the present Part 2.

5.1. Decomposition of the aeroacoustic field

The coherent velocity fluctuations $\tilde {\boldsymbol {u}}$ are decomposed into an irrotational acoustic part $\boldsymbol {u}_a$, and a hydrodynamic part $\boldsymbol {u}_h$ which corresponds to the incompressible vortical motion of the shear layer: $\tilde {\boldsymbol {u}}=\boldsymbol {u}_a + \boldsymbol {u}_h$. Similarly, the coherent pressure fluctuations are written as $\tilde {p} = p_a + p_h$ where $p_a$ is the acoustic pressure and $p_h$ is the pseudo-sound associated with the coherent, incompressible and rotational velocity fluctuations. In the next paragraphs, we apply this decomposition to the present system and analyse the fields $\boldsymbol {u}_a$ and $p_h$.

Phase averaging applied to PIV measurements gives access to $\tilde {\boldsymbol {u}}$ in the cavity's centre and the shear layer (see Reference Faure-Beaulieu, Xiong, Pedergnana and NoirayPart 1), while the microphones on the cavity walls give access to $\tilde {p}$ at discrete locations. To estimate the respective contributions of acoustics and hydrodynamics to $\tilde {p}$ and $\tilde {\boldsymbol {u}}$ at different locations, we consider the experimental results, together with the LNSE results and Helmholtz solver computations.

For $U_x = 59\ {\rm m}\ {\rm s}^{-1}$, $\tilde {\boldsymbol {u}}$ can reach $50\ {\rm m}\ {\rm s}^{-1}$ in the shear layer. The pseudo-noise pressure $p_h$, which is of the order of $\rho U_x u_h$, is approximated by $\rho U_x \tilde {u}$. This leads to $p_h\approx 3$ kPa in the shear layer, which is not negligible compared with the typical pressure fluctuation amplitude of 5 kPa measured by the microphones. Figure 7(a) shows the distribution of $p_h$ associated with the first hydrodynamic shear layer mode of azimuthal order $m=-1$, scaled to correspond to velocity fluctuations of the order of $50\ {\rm m}\ {\rm s}^{-1}$. The pseudo-sound $p_h$ indeed reaches values of 3 kPa in the shear layer, but vanishes away from the shear layer and is negligible at the microphone's location, indicated with a black rectangle. Figure 7(b) shows the $p_a$ field of the first azimuthal (pure) acoustic mode, obtained from a Helmholtz solver simulation. Unlike $p_h$, $p_a$ reaches its maximal values at the outer wall of the cavity. Therefore, it can be assumed that the microphones measure only the acoustic contribution $p_a$.

Figure 7. (a) Pseudo-sound field associated with the first shear layer mode, from the incompressible LNSE analysis around the simulated RANS mean flow without swirl. The grey rectangle indicates the position of the microphones at $r=90$ mm. (b) Acoustic pressure field of the first CCW spinning azimuthal acoustic mode at a frequency 793 Hz, obtained from a Helmholtz solver. The acoustic amplitude was imposed to match the measured data at the microphone location. The phase $\phi =0$ is chosen as the phase reference for which the acoustic pressure reaches its positive maximum in the upper half of the cavity. At $\phi ={\rm \pi} /2$, the slice shown in the figure coincides with the acoustic mode nodal plane and the acoustic pressure is 0 everywhere. The vertical dashed line indicates the position of the axial slices shown in figure 8. (c) Out-of-plane component $u_{az}$ of $\boldsymbol {u}_a$ for the same mode, at the same instant of the oscillation cycle. The acoustic pressure is in phase with the azimuthal acoustic velocity $u_{a\varTheta }$, which is characteristic of a travelling wave. (d,e) Vertical and axial components of $\boldsymbol {u}_a$ for the first CCW spinning azimuthal acoustic mode, with a phase difference ${\rm \pi} /2$ with respect to (b,c). Indeed, $u_{ay}$ and $u_{ax}$ are phase shifted by ${\rm \pi} /2$ with respect to $p_a$ and $u_{a\varTheta }$. Therefore, at $\phi =0$, they vanish in this slice.

Figure 8. Slices of the acoustic pressure and acoustic velocity fields (respectively colour and arrows) in the transverse plane of the axisymmetric configuration (see dashed line in figure 7) for a CCW spinning mode and a standing mode with orientation $\theta =0$ at two different phase instants: $\phi =0 \mod 2{\rm \pi}$ and $\phi ={\rm \pi} /2 \mod 2{\rm \pi}$, with $\phi =\omega t + \varphi$ the instantaneous acoustic phase. These pure acoustic modes are obtained with a Helmholtz solver computation.

The figures 7(c), 7(d) and 7(e) show the three velocity components of the first azimuthal acoustic mode. They are represented in Cartesian coordinates, rather than in cylindrical coordinates, to facilitate their interpretation. The out-of-plane and vertical velocity components $u_{az}$ and $u_{ay}$ are respectively related to the azimuthal and radial velocity components $u_{a\varTheta }$ and $u_{ar}$. A propagating 1-D acoustic wave of amplitude $p_a\sim 5\ {\rm kPa}$ has an acoustic velocity $u_a=p_a/\rho c=13\ {\rm m}\ {\rm s}^{-1}$, which matches well with the magnitude of the out-of-plane acoustic velocity fluctuations shown in figure 7(c), although the present configuration cannot be approximated by a simple 1-D closed waveguide as it can be done for thin annular combustion chambers (Faure-Beaulieu & Noiray Reference Faure-Beaulieu and Noiray2020). The 3-D nature of the acoustic field can also be seen in figure 8, which shows, on the left, the acoustic pressure and velocity fields of a spinning pure acoustic mode at phases $\phi =0$ and $\phi ={\rm \pi} /2$ of the cycle, and, on the right, these fields when the acoustic mode is standing. The vertical dashed black line in figure 8 corresponds to the slices shown in figures 7(b) and 7(c). At phase $\phi =0$, the velocity field in the vertical plane is orthogonal to this plane, and it is thus purely azimuthal. At phase ${\rm \pi} /2$, the vertical slice corresponds to the nodal line of the acoustic pressure, i.e. $p_a=0$ in this plane, and the velocity vector field corresponds to a downwards motion towards the bottom half of the cavity (see also figure 7d,e). This CCW spinning mode can be interpreted as the superposition of two standing modes with orientations $\theta =0$ and $\theta ={\rm \pi} /2$, which exhibit a phase lag of ${\rm \pi} /2$. It is interesting to stress that for the spinning mode, the acoustic azimuthal velocity and pressure are in phase, while for the standing mode, they exhibit a phase shift of ${\rm \pi} /2$.

One can also see in figure 7(d) that $u_{ax}$ and $u_{ay}$ exhibit maxima of about $40\ {\rm m}\ {\rm s}^{{-1}}$ around the corners of the cavity openings, which are far above the magnitude of $u_{az}$. This effect, although real, is overestimated due to the inviscid fluid assumption of the present Helmholtz solver computation which leads to singularities in the potential field at sharp corners. The resolvent analysis performed by Boujo et al. (Reference Boujo, Bauerheim and Noiray2018) on the 2-D counterpart of the present axisymmetric geometry shows that the optimal volumetric forcing of the shear layer is located in the vicinity of the upstream corner. Therefore, the fast acoustic motion around the corner perturbs the shear layer very efficiently, if not optimally. The resulting hydrodynamic oscillations are then advected and amplified along the shear layer.

5.2. Aeroacoustic wave equation

The starting point of the derivation of a low-order model of the present axisymmetric aeroacoustic system is the 3-D wave equation with unsteady velocity source terms, presented in Appendix A of the paper by Faure-Beaulieu & Noiray (Reference Faure-Beaulieu and Noiray2020)

(5.1)\begin{equation} \dfrac{\partial^2 \tilde{p}}{\partial t^2} + 2\bar{\boldsymbol{u}}\boldsymbol{\cdot}\boldsymbol{\nabla}\dfrac{\partial \tilde{p}}{\partial t} - c^2\nabla^2 \tilde{p} = 2\rho c^2\boldsymbol{\nabla} \bar{\boldsymbol{u}}:\boldsymbol{\nabla}\tilde{\boldsymbol{u}}^\top, \end{equation}

where the air density $\rho$ and speed of sound $c$ are assumed uniform in the cavity, and $:$ is the double dot product between two tensors, with the same convention as in Faure-Beaulieu & Noiray (Reference Faure-Beaulieu and Noiray2020). The focus of this previous study was on thermoacoustics, and therefore, the acoustic source term of interest was the one of the coherent heat release fluctuations $(\gamma -1)\partial \dot {\mathcal {Q}}/\partial t$, while the aeroacoustic source term $2\rho c^2\boldsymbol {\nabla } \bar {\boldsymbol {u}}:\boldsymbol {\nabla } \tilde {\boldsymbol {u}}^\top$ was neglected. Conversely, in the present study, only this term is considered. The acoustic pressure is described with the quaternion formalism

(5.2)\begin{align} p_a&={\rm Re}\big(A \mathrm{e}^{\mathrm{i}(\theta-\varTheta)} \mathrm{e}^{-\mathrm{k}\chi}\mathrm{e}^{\mathrm{j}(\omega t + \varphi)}\big)\equiv{\rm Re}(Ah)\nonumber\\ &=\tfrac{1}{2}( A h + A h^*)\nonumber\\ &=\tfrac{1}{2}\big( A \mathrm{e}^{\mathrm{i}(\theta-\varTheta)}\mathrm{e}^{-\mathrm{k}\chi}\mathrm{e}^{\mathrm{j}(\omega t + \varphi)} + A\mathrm{e}^{-\mathrm{j}(\omega t + \varphi)}\mathrm{e}^{\mathrm{k}\chi}\mathrm{e}^{-\mathrm{i}(\theta-\varTheta)} \big), \end{align}

where $h^*$ is the conjugate of the unitary quaternion $h=\mathrm {e}^{\mathrm {i}(\theta -\varTheta )} \mathrm {e}^{-\mathrm {k}\chi }\mathrm {e}^{\mathrm {j}(\omega t + \varphi )}$. This non-commutative quaternion formalism can be directly used to describe an azimuthal wave in a 1-D wave guide, in which case $A,\chi,\theta,\varphi$ are solely functions of time and slowly vary with respect to the acoustic period $2{\rm \pi} /\omega$. For the present axisymmetric geometry, a spatial averaging operation detailed in Appendix A allows us to reduce the 3-D problem with $A=A(t,r,z)$ to a 1-D equation. The spatial averaging is performed over an elementary volume $(r,\varTheta,x)\in [0,R]\times [\varTheta -\delta \varTheta /2,\varTheta +\delta \varTheta /2]\times [-X_\infty,X_\infty ]$, with $X_\infty$ chosen sufficiently large to be out of the zone of influence of the acoustic mode, which is always possible due to the evanescent character of the mode in the duct. Taking the limit $\delta \varTheta \rightarrow 0$, a 1-D azimuthal wave equation is obtained for the acoustic pressure averaged over a slice $[0,R]\times [-X_\infty,X_\infty ]$, which is denoted by $\langle p_a\rangle$ and depends solely on $t$ and $\varTheta$

(5.3)\begin{equation} \dfrac{\partial^2 \langle p_a\rangle}{\partial t^2} + \alpha\dfrac{\partial \langle p_a\rangle}{\partial t} + 2\dfrac{U_{\varTheta\,{eff}}}{R}\dfrac{\partial \langle p_a \rangle}{\partial \varTheta\partial t} - \omega_a^2\dfrac{\partial^2 \langle p_a\rangle}{\partial \varTheta^2} = \mathcal{S} ,\end{equation}

which has a damping term $\alpha \partial {\cdot }/\partial t$, a convective term $2(U_{\varTheta \,{eff}}/R)\partial ^2 {\cdot }/\partial \varTheta \partial t$ and a source term $\mathcal {S}$ resulting from the spatial averaging of the aeroacoustic source term $2\rho c^2\boldsymbol {\nabla } \bar {\boldsymbol {u}}:\boldsymbol {\nabla } \tilde {\boldsymbol {u}}^\top$.

5.3. Model of the aeroacoustic source term

The hydrodynamic component of the velocity $\boldsymbol {u}_h$ provides the main contribution to the source term $\mathcal {S}$: indeed, the ratio between the spatial derivatives of $\boldsymbol {u}_a$ and $\boldsymbol {u}_h$ has the order of magnitude of the ratio between the shear layer thickness ($\sim$1 cm) and the acoustic wavelength ($\sim$0.4 m). Additionally, the zones of strong gradients of $\overline{\boldsymbol{u}}$ and $\boldsymbol {u}_h$ coincide, so that the product $\boldsymbol {\nabla }\bar {\boldsymbol {u}}:\boldsymbol {\nabla }\bar {\boldsymbol {u}}_h^\top$ is even greater compared with $\boldsymbol {\nabla }\bar {\boldsymbol {u}}:\boldsymbol {\nabla }\bar {\boldsymbol {u}}_a^\top$. Therefore, only the contribution of $\boldsymbol {u}_h$ is considered from now on. The developed expression of $\boldsymbol {\nabla } \bar {\boldsymbol {u}}:\boldsymbol {\nabla } \boldsymbol {u}_h^\top$ in cylindrical coordinates is

(5.4)\begin{align} \boldsymbol{\nabla}\bar{\boldsymbol{u}}:\boldsymbol{\nabla} \boldsymbol{u}_h^\top &= \dfrac{\partial\bar{u}_x}{\partial x}\dfrac{\partial u_{hx}}{\partial x} + \dfrac{\partial\bar{u}_r}{\partial x}\dfrac{\partial u_{hx}}{\partial r} + \dfrac{\partial\bar{u}_\varTheta}{\partial x}\dfrac{\partial u_{hx}}{r\partial \varTheta}\nonumber\\ &\quad + \dfrac{\partial\bar{u}_x}{\partial r}\dfrac{\partial u_{hr}}{\partial x} + \dfrac{\partial\bar{u}_r}{\partial r}\dfrac{\partial u_{hr}}{\partial r} + \dfrac{\partial\bar{u}_r}{r \partial \varTheta}\dfrac{\partial u_{hr}}{r\partial \varTheta}\nonumber\\ &\quad + \dfrac{\partial\bar{u}_x}{r\partial \varTheta}\dfrac{\partial u_{h\varTheta}}{\partial x} - \dfrac{\partial\bar{u}_\varTheta}{r\partial \varTheta}\dfrac{\partial u_{h\varTheta}}{\partial r} - \dfrac{\partial\bar{u}_\varTheta}{\partial r}\dfrac{\partial u_{h\varTheta}}{r\partial \varTheta}. \end{align}

It is obtained by writing the expression of $\boldsymbol {\nabla }\bar {\boldsymbol {u}}:\boldsymbol {\nabla } \tilde {\boldsymbol {u}}^\top$ in Cartesian coordinates in which $\partial /\partial y$ and $\partial /\partial z$ were respectively substituted with $\cos (\varTheta )\partial /\partial r -\sin (\varTheta )\partial /r\partial \varTheta$ and $\sin (\varTheta )\partial /\partial r +\cos (\varTheta )\partial /r\partial \varTheta$, and $u_y$ and $u_z$ with $\cos (\varTheta ) u_r -\sin (\varTheta )u_\varTheta$ and $\sin (\varTheta ) u_r +\cos (\varTheta )u_\varTheta$. When the aeroacoustic source term is averaged over the small volume defined in § 5.2 and $\delta \varTheta \rightarrow 0$, one obtains

(5.5)\begin{equation} \mathcal{S}=\psi_1\langle u_{hx} \rangle + \psi_2\langle u_{hr} \rangle + \psi_3\langle u_{h\varTheta} \rangle. \end{equation}

The constants $\psi _{1,2,3}$ depend only on $\langle \boldsymbol {u}\rangle$. They can have a small dependency $\psi _{1,2,3}(\varTheta )$ in the azimuthal coordinate if the mean flow is not perfectly axisymmetric.

The vortices in the shear layer are periodically detached from the upstream corner of the cavity through the action of the vertical oscillations of the acoustic velocity $u_{ar}$. Therefore, $\langle u_{hx,r,\varTheta } \rangle$ oscillate with some constant phase with respect to $u_{ar}$, which has itself a constant phase difference ${\rm \pi} /2$ with $p_a$. Therefore, the source term $\mathcal {S}$ can be expressed as a linear combination of $p_a$ and ${\partial \langle p_a\rangle }/{\partial t}$ accounting respectively for the in-phase and delayed components of the shear layer's response

(5.6)\begin{equation} \mathcal{S}=\beta\dfrac{\partial \langle p_a\rangle}{\partial t} - \mu \langle p_a\rangle + \varXi(\varTheta,t) . \end{equation}

The effects $\psi _{1,2,3}$ are encapsulated in $\beta$ and $\mu$. In the presence of rotational asymmetries, $\beta$ and $\mu$ depend on $\varTheta$. When these asymmetries are small, the relative spatial variations of $\beta$ and the variations of $\mu$ with respect to $\omega _a^2$ are also small. The term $\varXi (\varTheta,t)$ is a random additive forcing at the cavity opening, which is due to the broadband noise generated by the highly turbulent flow in the wind channel and its air supply line, as in the 2-D counterpart of the present axisymmetric configuration (Bourquard et al. Reference Bourquard, Faure-Beaulieu and Noiray2021). Unlike the stochastic forcing of thermoacoustic modes in combustion chambers mainly produced by turbulence-driven heat release rate fluctuations, the present stochastic forcing originates from turbulent vorticity fluctuations. The minus sign in front of $\mu$ ensures consistency with the conventions of Faure-Beaulieu et al. (Reference Faure-Beaulieu, Indlekofer, Dawson and Noiray2021), but $\mu$ can be of any sign. In presence of large hydrodynamic oscillations, the mean shear layer thickens (Boujo et al. Reference Boujo, Bauerheim and Noiray2018), so that the gradients of $\bar {\boldsymbol {u}}$ decrease. As a consequence, $\psi _{1,2,3}$ decrease. This is accounted for in the model (5.6) by making the coefficient of $\partial \langle p_a\rangle /\partial t$ amplitude dependent

(5.7)\begin{equation} \mathcal{S}=(\beta(\varTheta)-g(\lvert u_{ar}\rvert))\dfrac{\partial \langle p_a\rangle}{\partial t} - \mu(\varTheta) \langle p_a\rangle + \varXi(\varTheta,t), \end{equation}

where $g$ is an increasing function and $g(0)=0$. One assumes that $g(\lvert u_{ar}\rvert )$ is smooth when $u_{ar}$ oscillates around 0, which allows us to write the Taylor series: $g(\lvert u_{ar}\rvert )=(1/2)g''(0)u_{ar}^2+O(u_{ar}^4)$, where only even order terms are present because the function is even. This approximation to a limited order is sufficient to account for the nonlinear saturation of the shear layer's response to high-amplitude forcing. Equation (5.7) becomes

(5.8)\begin{equation} \mathcal{S}=\biggl(\beta(\varTheta)-\dfrac{\kappa}{\omega^2}\biggl(\dfrac{\partial \langle p_a\rangle}{\partial t}\biggr)^2\biggr)\dfrac{\partial \langle p_a\rangle}{\partial t} - \mu(\varTheta) \langle p_a\rangle + \varXi(\varTheta,t), \end{equation}

where $\omega ^2$ was included so that the definition of the nonlinear term's coefficient $\kappa$ is consistent with the constant $\kappa$ used in the literature on low-order modelling of azimuthal thermoacoustic instabilities, where a slightly different source term is derived (e.g. Noiray et al. Reference Noiray, Bothien and Schuermans2011; Noiray & Schuermans Reference Noiray and Schuermans2013; Faure-Beaulieu & Noiray Reference Faure-Beaulieu and Noiray2020). An amplitude dependency could also be considered for $\mu$, which would change slightly the aeroacoustic frequency in limit cycle. For simplicity, this effect is omitted. The aeroacoustic equation then becomes

(5.9)\begin{align} \dfrac{\partial^2 \langle p_a\rangle}{\partial t^2} + (\alpha-\beta)\dfrac{\partial \langle p_a\rangle}{\partial t} + 2\dfrac{U_{\varTheta\,{eff}}}{R}\dfrac{\partial \langle p_a \rangle}{\partial \varTheta\partial t} - (\omega_a^2+\mu)\dfrac{\partial^2 \langle p_a\rangle}{\partial \varTheta^2} =-\dfrac{\kappa}{\omega^2}\left(\dfrac{\partial \langle p_a\rangle}{\partial t}\right)^3 + \varXi, \end{align}

where the identity $\langle p_a \rangle = - \partial ^2 \langle p_a \rangle /\partial \varTheta ^2$ was used to group the reactive terms. The variables defining the linear response of the shear layer, $\beta$ and $\mu$, are functions of $\varTheta$ and therefore periodic. They are written as Fourier series

(5.10a,b)\begin{align} \beta(\varTheta)=\beta_0+\sum_{n\geq 0} b_n \cos(\varTheta-\varTheta_{\beta n}),\quad \mu(\varTheta)=\mu_0 + \omega^2\sum_{n\geq 0} m_n \cos(\varTheta-\varTheta_{\mu n}). \end{align}

The average aeroacoustic growth rate $(\beta _0-\alpha )/2$ determines the stability of the system: when it is positive, the acoustic response of the shear layer exceeds the acoustic losses modelled in the term $\alpha$ and any perturbation will grow exponentially until the nonlinear loss term in the right-hand side of (5.9) becomes sufficiently large to lead to amplitude saturation. The coefficients $b_n$ account for the presence of non-uniformities in the shear layer's resistive response around the cavity. The average reactive response $\mu _0$ causes a frequency drift from the pure acoustic frequency $\omega _a$ to a reference aeroacoustic frequency $\omega =\sqrt {\omega _a^2+\mu _0}$. The terms $m_n$ account for the presence of non-uniformities of this reactive response and can be interpreted as a spatial modulation of the effective speed of sound around the cavity, which can result from a non-homogeneous temperature field, geometrical asymmetries, or non-uniformities of the mean flow velocity leading to a modulation of the convective time delay around the cavity.

5.4. Equations for the state variables

The goal is now to extract the equations for the slowly evolving state variables $\langle A\rangle,\chi,\theta,\varphi$, following the procedure described in Faure-Beaulieu & Noiray (Reference Faure-Beaulieu and Noiray2020): the wave equation (5.9) and the slow time scale constraint (A9) are projected onto $\mathrm {e}^{\mathrm {i}\varTheta }$, leading to a complex oscillator equation and a complex constraint equation. This gives a system of four equations linear in $\dot {\langle A\rangle },\dot {\chi },\dot {\theta },\dot {\varphi }$, which is inverted, leading to the following equations:

Amplitude equation

(5.11)\begin{align} \dot{A}&=\dfrac{1}{2}\left(\beta_0-\alpha\right)A+\dfrac{b_2}{4}\cos(2(\theta-\varTheta_{\beta 2}))\cos(2\chi)A\nonumber\\ &\quad -\dfrac{3\kappa}{64}(5+\cos(4\chi))A^3 +\dfrac{ 3\varGamma }{ 4\omega^2 A } + \zeta_A. \end{align}

Nature angle equation

(5.12)\begin{align} \dot{\chi} &= \dfrac{3\kappa}{64}A^2\sin(4\chi)-\dfrac{b_2}{4}\cos(2(\theta-\varTheta_{\beta 2}))\sin(2\chi)\nonumber\\ &\quad +\dfrac{m_2\omega}{4}\sin(2(\theta-\varTheta_{\mu 2}))-\dfrac{ \varGamma \tan(2\chi) }{ 2 \omega^2 A^2 } +\dfrac{1}{A}\zeta_{\chi} . \end{align}

Preferential angle equation

(5.13)\begin{align} \dot{\theta} &= \dfrac{U_{\varTheta\,{eff}}}{R}-\dfrac{b_2}{4}\dfrac{\sin(2(\theta-\varTheta_{\beta 2}))}{\cos(2\chi)}\nonumber\\ &\quad -\dfrac{m_2\omega}{4}\cos(2(\theta-\varTheta_{\mu 2}))\tan(2\chi) + \dfrac{1}{A \cos(2\chi)}\zeta_{\theta}. \end{align}

Phase equation

(5.14)\begin{equation} \dot{\varphi} = \dfrac{b_2}{4}\sin(2(\theta-\varTheta_{\beta 2}))\tan(2\chi) + \dfrac{m_2\omega}{4}\dfrac{\cos(2(\theta-\varTheta_{\mu 2}))}{\cos(2\chi)} - \dfrac{\tan(2\chi)}{A}\zeta_{\theta} + \dfrac{1}{A}\zeta_{\varphi} . \end{equation}

The brackets around $\langle A \rangle$ were dropped for convenience: from now on, $A$ refers to the average amplitude over an azimuthal slice. Some terms need to be defined: $\varGamma$ is the intensity of the noise $\varXi (\varTheta,t)$ projected on $\mathrm {e}^{\mathrm {i}\varTheta }$ (Faure-Beaulieu & Noiray Reference Faure-Beaulieu and Noiray2020). The terms $\zeta _A, \zeta _\chi, \zeta _\theta$ and $\zeta _\varphi$ are white noises of intensities $\varGamma /2\omega ^2$. From the Fourier decompositions (5.10a,b) of the asymmetries of the shear layer response, only the coefficients $b_2$ and $m_2$ are still present in the slow equations, consistent with Noiray et al. (Reference Noiray, Bothien and Schuermans2011) and Faure-Beaulieu et al. (Reference Faure-Beaulieu, Indlekofer, Dawson and Noiray2021).

5.5. Effect of the mean flow on the aeroacoustic gain

In the slow equations, the effect of the swirl on the time evolution of the aeroacoustic mode explicitly appears in the term $U_{\varTheta \,{eff}}$, which is the effective azimuthal velocity (cf. Appendix A) and also implicitly in the terms $\beta _0$, $\alpha$, $\kappa$, $b_2$ and $m_2$, which come from spatial integrals of quantities directly related to the mean flow. To keep the model as simple as possible, only the effect of the mean flow on the aeroacoustic gain $\beta _0$ will be modelled. Section 4 showed that the splitting between the growth rates of the counterswirl and the co-swirl shear layer modes increases linearly with the swirl velocity. Therefore, it is reasonable to assume that the difference between the effective linear aeroacoustic growth rates for $\chi ={\rm \pi} /4$ (CCW spinning) and $\chi =-{\rm \pi} /4$ (CW spinning) also has this linear dependency. We propose a simple phenomenological model for the temporal evolution of the instantaneous aeroacoustic gain

(5.15)\begin{equation} \beta_0 = \beta_{ref} - \beta_s U_{\varTheta\,{eff}}\dfrac{\chi}{{\rm \pi}/4}, \end{equation}

where $\beta _{ref}$ is the gain in absence of mean flow and $\beta _s$ is a positive coefficient to account for the growth rate splitting: when $\chi$ and $U_{\varTheta \,{eff}}$ have the same sign, the gain decreases (co-swirl mode), and when they have opposite signs, it increases. With this model, the aeroacoustic growth rate difference between co-swirl and counterswirl waves is $\Delta \beta _0=-2\beta _s U_{\varTheta \,{eff}}$. This difference is assumed to be due only to the split $\varDelta \sigma$ of the growth rates of the counterspinning hydrodynamic fluctuations, giving $\Delta \beta _0=\Delta \sigma$. It follows from § 4 that

(5.16)\begin{equation} -\beta_s \times 2U_{\varTheta \,{eff}} =-8.9/5.2\times f/c \times 2U_{\varTheta \,{SL}}. \end{equation}

As explained in Appendix A, $U_{\varTheta \,{eff}}$ is approximately 10 times smaller than $U_{\varTheta \,{SL}}$, the typical azimuthal mean velocity in the shear layer. This gives values of $\beta _s$ of the order of $30\ {\rm m}^{-1}$.

5.6. Effect of the instability on the mean flow

As shown in Reference Faure-Beaulieu, Xiong, Pedergnana and NoirayPart 1, the coherent motion of the shear layer can lead to the emergence of a mean swirl. Therefore, the global azimuthal component $U_{\varTheta \,{eff}}$ is decomposed into two parts: the imposed swirl $U_i$, constant over a given experiment, and the self-induced swirl $U_s$, which slowly fluctuates over time depending on the instability's dynamics. In Reference Faure-Beaulieu, Xiong, Pedergnana and NoirayPart 1, we showed that the self-induced mean swirl appears only when the hydrodynamic fluctuations are sufficiently spinning and have large amplitude. Thus, $U_s$ is a function of $\chi$ and $A$. The perturbation analysis of Reference Faure-Beaulieu, Xiong, Pedergnana and NoirayPart 1 gave us an equation for the self-induced azimuthal velocity $\bar {u}_\varTheta$ when a non-swirling base flow $\bar {\boldsymbol {u}}_b$ was forced by an incompressible spinning helical shear layer wave $\tilde {\boldsymbol {u}}$

(5.17)\begin{align} &\bar{u}_{br}\dfrac{\partial \bar{u}_\varTheta}{\partial r} + \bar{u}_{bx}\dfrac{\partial \bar{u}_\varTheta}{\partial x} + \dfrac{\bar{u}_\varTheta \bar{u}_{br}}{r} -(\nu+\nu_t) \left( \dfrac{1}{r}\dfrac{\partial }{\partial r}\left(r\dfrac{\partial \bar{u}_\varTheta}{\partial r}\right) + \dfrac{\partial^2 \bar{u}_\varTheta}{\partial x^2} - \dfrac{\bar{u}_\varTheta}{r^2}\right) \nonumber\\ &\quad =- \dfrac{\partial \overline{\tilde{u}_r \tilde{u}_\varTheta}}{\partial r} - \dfrac{\overline{\partial \tilde{u}_x \tilde{u}_\varTheta}}{\partial x} - 2\dfrac{\overline{\tilde{u}_r \tilde{u}_\varTheta}}{r} , \end{align}

where $\nu _t$ is a turbulent viscosity. All the terms are of comparable order. The convective terms of the left-hand side (like $\bar {u}_{bx}\partial \bar {u}_\varTheta /\partial x$) are of the same order as the coherent Reynolds stresses on the right-hand side, giving the estimate $\bar {u}_\varTheta \sim \tilde{\boldsymbol{u}}^2/U_x$. Thus, the amplitude of the self-induced swirl is a quadratic function of the acoustic amplitude. Additionally, Reference Faure-Beaulieu, Xiong, Pedergnana and NoirayPart 1 showed that the self-induced swirl spins against the wave direction. For a CCW spinning mode, the following phenomenological model is adopted for the self-induced swirl

(5.18)\begin{equation} U_s=-\upsilon A^2, \end{equation}

with $\upsilon$ a positive constant. For a CW spinning mode, the sign is changed: $U_s=+\upsilon A^2$. For a mixed or a standing mode, the wave is decomposed into the sum of counterspinning waves of amplitudes $A_{CCW}$ and $A_{CW}$ and the ansatz (3.1) is rewritten as

(5.19)\begin{equation} p_a=A_{CCW}\cos(\omega t +\varphi-(\varTheta-\theta)) + A_{CW}\cos(\omega t +\varphi+(\varTheta-\theta)), \end{equation}

where $A_{CCW}=(\cos (\chi )+\sin (\chi ))A/2$ and $A_{CW}=(\cos (\chi )-\sin (\chi ))A/2$. For a mixed mode, the self-induced swirl is assumed to be the sum of the contributions of both spinning components

(5.20)\begin{equation} U_s =-\upsilon A_{CCW}^2 + \upsilon A_{CW}^2, \end{equation}

which simplifies to

(5.21)\begin{equation} U_s(A,\chi) =-\upsilon A^2 \sin(2\chi). \end{equation}

This corresponds to the induced swirl in steady conditions. However, when the mode state changes, it is expected that the flow needs a time delay to adapt. From the Navier–Stokes equations, one can write $\partial u_\varTheta /\partial t \sim \boldsymbol {u}\boldsymbol{\cdot}\boldsymbol {\nabla } u_\varTheta$, which gives a characteristic response time $\tau _d=R/U_x\sim 2$ ms for $U_x=60\ {\rm m}\ {\rm s}^{-1}$. This is certainly an under-estimation because the mean velocities in the cavity are smaller, of the order of $10\ {\rm m}\ {\rm s}^{-1}$ (which gives $\tau _d=6$ ms). An additional time-domain equation is introduced to account for the delayed response of the swirl

(5.22)\begin{equation} \dfrac{\partial U_{\varTheta\,{eff}}}{\partial t} + \dfrac{1}{\tau_d}(U_{\varTheta\,{eff}} - (U_i + U_s(A,\chi)))=0.\end{equation}

A rough estimate of $\upsilon$ can be obtained, starting from $\bar {u}_\varTheta \sim \tilde {\boldsymbol {u}}^2/U_x$. Locally, $\bar {u}_\varTheta$ is typically 10 times larger than the averaged swirl $U_{\varTheta \,{eff}}$ (see Appendix), and, from the experiments, one knows that acoustic amplitudes $A\sim 5$ kPa are typically associated with hydrodynamic velocity fluctuations $\tilde {u}\sim 50\ {\rm m}\ {\rm s}^{-1}$. Therefore, $U_{\varTheta \,{eff}}\sim (10^{-5}/U_x)A^2$, which gives $\upsilon \sim 10^{-9}\ {\rm Pa}^{-2}\ {\rm m}\ {\rm s}^{-1}$.

5.7. Time-domain simulations

The system of four coupled Langevin equations (5.11)–(5.14) and the self-induced swirl equation (5.22) are numerically solved together with the stochastic Runge–Kutta method of order 1. The values of the parameters were adjusted to best reproduce the dynamics of the state variables observed in the experiments. For most parameters, the choice of the calibrated value is constrained, because beyond a certain range, some key features of the state variables’ dynamics are lost. All the parameters were calibrated to the case $\dot {m}_s=0$, except those related to interactions with the swirl. Certain parameters can be unambiguously found from the experimental time traces: this is the case of the reactive asymmetry $m_2$, which is proportional to the beating frequency, easily obtained from the experiments. This also sets a maximum value for the growth rate $(\beta _0-\alpha )/2$, as Faure-Beaulieu et al. (Reference Faure-Beaulieu, Indlekofer, Dawson and Noiray2021) showed theoretically that the beating is suppressed when the ratio $(\beta _0-\alpha )/\omega$ is greater than a certain threshold. When no swirl is imposed, the experimental p.d.f. of $\chi$ has a preference for the CCW spinning direction, which can be explained by a misalignment of the preferential directions of the resistive and reactive components of the shear layer response to acoustic perturbation, i.e. $\varTheta _{\beta 2}$ and $\varTheta _{\mu 2}$. These are calibrated in the model to reproduce the preferential values of $\theta$ in the experiments and the asymmetry of the experimental p.d.f. of $\chi$. The coefficient of resistive asymmetry $b_2$ has a direct impact on the amplitude of the oscillations of $\chi$: the larger $b_2$, the smaller the amplitude. When $b_2$ is too large, the p.d.f. of $\chi$ becomes unimodal instead of bimodal. The saturation parameter $\kappa$ is chosen to match the global amplitude $A$ of the experiments. The noise amplitude $\varGamma$ modifies the intermittency of the dynamics: when $\varGamma =0$, the system only shows regular asymmetric beating with a statistical preference for the CCW side. A sufficiently high noise is necessary to reproduce the random transitions between regimes of beating and mixed mode. The values for the additional parameters $\beta _s$, $\upsilon$ and $\tau _d$, which control the interactions with the swirl, will be discussed later.

Figure 9 shows the simulated time traces. Each column is associated with a different imposed swirl $U_i$, with increasing values of from left to right: $U_i=0$, 0.03, 0.07 and $0.2\ {\rm m}\ {\rm s}^{-1}$. These values seem small, but it has to be kept in mind that $U_i$ is an effective velocity corresponding to an averaging over a section of the cavity (see Appendix A). We also stress the fact that the values chosen for $U_i$ are not equispaced, unlike the values of $\dot {m}_s$ in the experiments, increased by constant steps of $3\ {\rm g}\ {\rm s}^{-1}$. This non-constant stepping in the simulations was necessary to have a good match with the experiments. It is attributed to the fact that the penetration of the tangential jets of the swirler and the swirl intensity that is imparted to the cavity flow do not linearly depend on $\dot {m}_s$. Figure 10 shows a comparison between the p.d.f.s of the experimental state variables (row 1) and the ones from the simulations (row 2).The different shades of colour used in figure 9 are the same as in figure 10: darker colours corresponds to higher $U_i$.

Figure 9. Evolution of the state variables $A$, $\chi$, $\theta$, $\varphi$ from simulations, for increasing values of the imposed swirl: (a) $U_i=0$, (b) $U_i=0.03\ {\rm m}\ {\rm s}^{-1}$, (c) $U_i=0.07\ {\rm m}\ {\rm s}^{-1}$, (d) $U_i=0.2\ {\rm m}\ {\rm s}^{-1}$ from left to right. Other parameters are common to all the simulations: $\omega =2{\rm \pi} \times 793\ {\rm rad}\ {\rm s}^{-1}$, $\beta _{ref}=320\ {\rm s}^{-1}$, $\alpha =280\ {\rm s}^{-1}$, $\kappa =4\times 10^{-6}\ {\rm Pa}^{-2}\ {\rm s}^{-1}$, $b_2=2.8\ {\rm s}^{-1}$, $\varTheta _{\beta 2}=2{\rm \pi} /3$ rad, $m_2=0.012$, $\varTheta _{\mu 2}=\varTheta _{\beta 2}-{\rm \pi} /2-{\rm \pi} /8$, $\varGamma = 1\times 10^{14}\ {\rm Pa}^2\ {\rm s}^{-3}$, $\beta _s=27.7\ {\rm m}^{-1}$, $\upsilon =7.61\times 10^{-9}\ {\rm Pa}^{-2}\ {\rm m}\ {\rm s}^{-1}$, $\tau _d=0.007$ s.

Figure 10. The p.d.f.s of the state variables from experiments (row 1) and numerical simulations (row 2).