Symmetry breaking of azimuthal waves: Slow-flow dynamics on the Bloch sphere

Depending on the reflectional and rotational symmetries of annular combustors for aeroengines and gas turbines, self-sustained azimuthal thermoacoustic eigenmodes can be standing, spinning, or a mix of these two types of waves. These thermoacoustic limit cycles are unwanted because the resulting intense acoustic fields induce high-cycle fatigue of the combustor components. This paper presents a new theoretical framework for describing, in an idealized annular combustor, the dynamics of the slow-flow variables, which define the state of an eigenmode, i.e., if the latter is standing, spinning or mixed. The acoustic pressure is expressed as a hypercomplex field and this ansatz is inserted into a one-dimensional wave equation that describes the thermoacoustics of a thin annulus. Slow-flow averaging of this wave equation is performed by adapting the classic Krylov-Bogoliubov method to the quaternion field to derive a system of coupled first-order differential equations for the four slow-flow variables, i.e., the amplitude, the nature angle, the preferential direction, and the temporal phase of the azimuthal thermoacoustic mode. The state of the mode can be conveniently depicted by using the first three slow-flow variables as spherical coordinates for a Bloch sphere representation. Stochastic forcing from the turbulence in annular combustors is also accounted for. This new analytical model describes both rotational and reflectional explicit symmetry-breaking bifurcations induced by the nonuniform distribution of thermoacoustic sources along the annulus circumference and by the presence of a mean swirl.

Figure 1

Simplified geometry of a thin annular combustor. $\sigma$ is a poloidal cross-section on which the wave equation is averaged.

Figure 2

Representation of the mode state on the Bloch sphere. The markers correspond to the three cases displayed in Fig. 3.

Figure 3

Representation of the acoustic pressure in the chamber for three different states of the first azimuthal mode ($n=1$), each of them being described by a point on the Bloch sphere: (a) standing mode with $\chi =0$ and $\theta =\pi /6$; (b) counterclockwise spinning mode with $\chi =\pi /4$; (c) counterclockwise mixed mode with $\chi =\pi /6$ and $\theta =\pi /6$.

Figure 4

Stream lines describing the evolution of the state of the linearly unstable azimuthal-thermoacoustic-mode on the Bloch sphere, for a uniform distribution of the thermoacoustic feedback, with no mean swirl. Parameters: ${\beta }_0=160\mathrm{rad}{\mathrm{s}}^{-1}$, $\alpha =100\mathrm{rad}{\mathrm{s}}^{-1}$, $\kappa =0.5\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{Pa}}^{-2}$, ${\omega }_n=1167\mathrm{rad}{\mathrm{s}}^{-1}$. Symbols: $\circ$, attractors; $*$, repellers; $+$, saddles. The color scale is given in Fig. 5. (a) plane $\chi =0$ and (b) plane $\theta =0\mathrm{mod}\pi$.

Figure 5

Color bar used for the contours of the streamline plots throughout the paper. The values correspond to the norm of the stream vectors normalized by $A_0({\beta }_0-\alpha )$ with $A_0$ the amplitude of the stable spinning modes in the symmetric configuration. This bar is represented in logarithmic scale to better visualize the state change speed near equilibrium points.

Figure 6

Stream lines in the plane $\theta =0mod\pi$ for the system Eq. (84). Parameters: ${\beta }_0=160\mathrm{rad}{\mathrm{s}}^{-1}$, $\alpha =100\mathrm{rad}{\mathrm{s}}^{-1}$, $\kappa =0.5\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{Pa}}^{-2}$, ${\omega }_n=1167\mathrm{rad}{\mathrm{s}}^{-1}$. ${\mathrm{\Gamma }}_{\text{thr}}=2.3.{10}^{1}0{\mathrm{Pa}}^2{\mathrm{s}}^{-3}$. Symbols: $\circ$, attractors; $+$, saddles. The color scale is given in Fig. 5. (a) $\mathrm{\Gamma }=1.6.{10}^{1}0{\mathrm{Pa}}^2{\mathrm{s}}^{-3}<{\mathrm{\Gamma }}_{\mathrm{thr}}$ and (b) $\mathrm{\Gamma }=2.5.{10}^{1}0{\mathrm{Pa}}^2{\mathrm{s}}^{-3}>{\mathrm{\Gamma }}_{\mathrm{thr}}$.

Figure 7

Stream lines describing the evolution of the state of the linearly unstable azimuthal-thermoacoustic-mode on the Bloch Sphere, for a nonuniform distribution of the thermoacoustic feedback: $c_{2n}=0.25<C_{2n}=0.375$. ${\beta }_0=160\mathrm{rad}{\mathrm{s}}^{-1}$, $\alpha =100\mathrm{rad}{\mathrm{s}}^{-1}$, $\kappa =0.5\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{Pa}}^{-2}$, ${\omega }_n=1167\mathrm{rad}{\mathrm{s}}^{-1}$. Symbols: $\circ$, attractors; $*$, repellers; $+$, saddles. The color scale is given in Fig. 5. (a) plane $\chi =0$, (b) plane $\theta =0$ mod $\pi$, (c) plane inclinated of ${\chi }_{\mathrm{eq}}$, showing the two attractors $\chi ={\chi }_{\mathrm{eq}}$ and $\chi =-{\chi }_{\mathrm{eq}}$. (d) Plane z=cst. including the points $\{\chi ={\chi }_{\mathrm{eq}},\theta =0\}$ and $\{\chi ={\chi }_{\mathrm{eq}},\theta =\pi \}$, corresponding to the same state.

Figure 8

Stream lines for a large azimuthal asymmetry: $c_{2n}=0.5>C_{2n}=0.375$. ${\beta }_0=160\mathrm{rad}{\mathrm{s}}^{-1}$, $\alpha =100\mathrm{rad}{\mathrm{s}}^{-1}$, $\kappa =0.5\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{Pa}}^{-2}$, ${\omega }_n=1167\mathrm{rad}{\mathrm{s}}^{-1}$. Symbols: $\circ$, attractors; $*$, repellers; $+$, saddles. The color scale is given in Fig. 5. (a) plane $\chi =0$ and (b) plane $\theta =0$ mod $\pi$.

Figure 9

Streamlines for $M=0.005$. $M_1=0.0107>M$. The green symbols are the equilibrium points: $\circ$, attractors; $*$, repellers; $+$, saddles. Parameters: ${\beta }_0=160\mathrm{rad}{\mathrm{s}}^{-1}$, $\alpha =155\mathrm{rad}{\mathrm{s}}^{-1}$, $\kappa =0.5\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{Pa}}^{-2}$, ${\omega }_n=1167\mathrm{rad}{\mathrm{s}}^{-1}$. The maximum value of the color scale is not the same as shown in Fig. 5. (a) plane $\chi =0$ and (b) plane $\theta =0$ mod $\pi$.

Figure 10

Streamlines in the plane $\theta =0$ for different values of $M$. $M_1=0.0107$, $M_2=0.0313$. Symbols: $\circ$, attractors; $*$, repellers; $+$, saddles. The parameters are the same as described in the caption of Fig. 9. The same color scale is used for the the four first plots. (a) $M=0$, (b) $M=0.005<M_1$, (c) $M=M_1$, (d) $M=0.02M_1<M<M_2$, (e) $M=M_2,\mathrm{and}\left(\mathrm{f}\right)M=0.1>M_2$.

Figure 11

Streamlines for $M=0.005$, for the model without convective terms in the thermoacoustic source term. The green symbols are the equilibrium points: $\circ$, attractors; $*$, repellers; $+$, saddles. Parameters: ${\beta }_0=160\mathrm{rad}{\mathrm{s}}^{-1}$, $\alpha =155\mathrm{rad}{\mathrm{s}}^{-1}$, $\kappa =0.5\mathrm{rad}{\mathrm{s}}^{-1}{\mathrm{Pa}}^{-2}$, ${\omega }_n=1167\mathrm{rad}{\mathrm{s}}^{-1}$. (a) plane $\chi =0$ and (b) plane $\theta =0$ mod $\pi$.