Transition to chaos in an acoustically driven cavity flow

We consider the unsteady regimes of an acoustically driven jet that forces a recirculating flow through successive reflections on the walls of a square cavity. The specific questions addressed are whether the system can sustain states of low-dimensional chaos when the acoustic intensity driving the jet is increased, and, if so, what are the pathway to it and the underlying physical mechanisms. We adopt two complementary approaches, both based on data extracted from numerical simulations: (i) We first characterize successive bifurcations through the analysis of leading frequencies. Two successive phases in the evolution of the system are singled out in this way, both leading to potentially chaotic states. The two phases are separated by a drastic simplification of the dynamics that immediately follows the emergence of intermittency. The second phase also features a second intermediate state where the dynamics is simplified due to frequency locking. (ii) Nonlinear time series analysis enables us to reconstruct the attractor of the underlying dynamical system and to calculate its correlation dimension and leading Lyapunov exponent. Both these quantities bring confirmation that the state preceding the dynamic simplification that initiates the second phase is chaotic. Poincaré maps further reveal that this chaotic state in fact results from a dynamic instability of the system between two nonchaotic states respectively observed at slightly lower and slightly higher acoustic forcing.

Figure 1

Schematic representation of the cavity including the axis of the acoustic beam used to generate a circulating flow pattern. The acoustic beam enters the cavity by an opening in the center of one of its faces with an angle of $\pi /4$ and reflects on three of the vertical walls before leaving the cavity through the opening where it entered. The acoustic beam axis remains in the horizontal midheight plane ($x,y$) and is composed of four rectilinear parts. $O$ is the origin of the reference frame used thereafter. This corresponds to the experimental setup used by Cambonie et al. [26]. The plane transducer is 28.5 mm in diameter and operates at 2 MHz.

Figure 2

Normalized acoustic forcing $I$ generated by the acoustic beam in the horizontal midheight plane ($z/H=0$), computed using the Rayleigh integral. The acoustic beam enters at ($x/H=0$, $y/H=-0.571875$), the center of a vertical wall, with a $\pi /4$ angle and is reflected on the other vertical walls while slowly diverging and dissipating (which explains the progressive loss of intensity), before leaving the cavity at its inlet location.

Figure 3

Normalized time-averaged velocity fields and rms fluctuations for various acoustic forcing. [(a), (d)] In-plane velocity magnitude $|\overline{{\mathit{u}}_h}|/A^{*}$ and segments of 2D streamlines in the horizontal midheight plane ($z/H=0$). [(a), (b)] Numerical results for $A=1.5$ and $A=2.4$. Dashed lines represent the acoustic beam axis. [(c), (d)] Experimental Particle Image Velocimetry (PIV) measurements for $P=1$ W and $P=2$ W. The velocity fields are normalized using respectively $A^{*}=1.5\times {10}^6$ and $A^{*}=2.4\times {10}^6$ for the sake of the comparison with the numerical results. Dashed lines represent the expected position of the acoustic beam. [(e), (f)] In-plane velocity magnitude $|\overline{{\mathit{u}}_v}|/A^{*}$ and segments of 2D streamlines in the vertical plane ($y=0$) for $A=1.5$ and $A=2.4$. [(g), (h)] Normalized rms fluctuations $|{\mathit{u}}_{h,\text{rms}}|/A^{*}$ in the horizontal midheight plane for $A=1.5$ and $A=2.4$. $S_1$ is the point where velocity time series are recorded for subsequent spectral and nonlinear dynamics analyses. Arrows on this point indicate the transverse direction. [(i), (j)] Normalized experimental rms fluctuations $|{\mathit{u}}_{h,\text{rms}}|/A^{*}$ in the horizontal midheight plane for $P=1$ W and $P=2$ W.

Figure 4

Normalized velocity fields for a steady configuration ($A=1$). (a) In-plane velocity magnitude $|{\mathit{u}}_h|/A^{*}$ and segments of 2D streamlines in the horizontal midheight plane ($z=0$). Dashed lines represent the acoustic beam axis. (b) In-plane velocity magnitude $|{\mathit{u}}_v|/A^{*}$ and segments of 2D streamlines in the vertical plane ($y=0$).

Figure 5

Evolution of the cavity flow energy with increasing acoustic forcing $A$. Blue circles: time-averaged energy $E_a$ in the cavity, from Eq. (6). Red squares: fluctuation energy $E_f$ in the cavity, from Eq. (7).

Figure 6

Temporal evolution of the jet transverse velocity fluctuation ($u_t^{\prime }$) for increasing acoustic forcing $A$. The transverse velocity fluctuation $u_t^{\prime }$ is the fluctuating part of the velocity component perpendicular to the acoustic axis, in the horizontal midheight plane at point $S_1$ [see point $S_1$ and associated arrows in Figs. 3 and 3]. The velocity is normalized by the standard deviation of the velocity fluctuations $\sigma \left(u_t^{\prime }\right)$. Note that the durations observed on these plots correspond to several hours in physical time ($5\times {10}^3$ to ${10}^5$ s). It is worth stressing that such long time series are indispensable to adequately represent the system's dynamics. However, experimentally obtaining data over such long periods of time would be extremely difficult.

Figure 7

Power spectral density $E$ as a function of the acoustic forcing $A$ and the frequency $f$. $E$ is computed using fast Fourier transform of the time series of $u_t^{\prime }$ following Welch's method [39]. This picture singles out the appearance of frequencies associated with bifurcations (Hopf and period doubling) and highlights potentially chaotic regimes (at $A=2.3$ and $A>3.2$). Dashed lines identify particular frequencies evolving with $A$. Note that the observed range of frequency values is consistent with the very low frequencies experimentally observed by Cambonie et al. [26] (of the order of $5\times {10}^{-3}\mathrm{Hz}$, or $f=128$).

Figure 8

(a) Recurrence map of $u_t^{\prime }$ for acoustic forcing $A=2.3$. The regions with diagonal patterns correspond to time intervals for which the system is quasiperiodic. White horizontal and vertical bands correspond to intermittencies, for which the dynamical system's evolution loses quasiperiodicity. For the sake of simplicity, only a portion of the signal is represented here, where only 11 of the 25 intervals of intermittent behavior ($I1$ to $I11$) from the full measured signal are visible. [(b), (c)] Enlargements of the recurrence map. $T_a$ and $T_b$ are the periods associated with the characteristic frequencies $f_a$ and $f_b$ on Fig. 7.

Figure 9

Recurrence map of $u_t^{\prime }$ for an acoustic forcing $A=2.2$. The diagonal patterns indicating a quasiperiodic behavior are clearly visible on this figure.

Figure 10

Evolution of characteristic frequencies with the acoustic forcing $A$. Blue squares: normalized frequency $f_c$. Purple triangles: normalized frequency $f_d^{\prime }$. Red circles: ratio $f_d^{\prime }/f_c$ (right axis). The peak frequencies are obtained from local maxima in the power spectral densities, and hence bear an uncertainty $\mathrm{\Delta }f/f_{max}=1/\left(t_f{f}_{max}\right)<4.5\times {10}^{-4}$ linked to the time-series duration $t_f$. The frequency $f_c$ appears to evolve smoothly, with the exception of the cases between $A=3.1$ and $A=3.13$, where it is abnormally high. The fact that the ratio $f_d^{\prime }/f_c$ is an integer for these cases suggests that this abnormal behavior is due to frequency locking.

Figure 11

False neighbors number $N_f$ and correlation sums $C_2$ for $A=2.3$ from velocity time series taken at different positions in the cavity. (a) Illustration of the positions investigated in the cavity midheight plane. The dashed line is the axis of the acoustic beam. (b) The fall to zero of the number of false neighbors indicates a global embedding dimension of $D_e=3$. (c) The power law exponent of $C_2$ in the inertial range gives the correlation dimension $D_2$. The spatial homogeneity of $D_e$ and $D_2$ in this case suggests that the whole cavity is governed by the same dynamical system.

Figure 12

(a) Example of mutual information computed from $u_t^{\prime }$ for $A=2.2$. The first minimum is highlighted by the vertical dashed line and gives an optimal timescale $\mathrm{\Delta }{t}^{*}\approx 0.003$. (b) Evolution of the optimal time scale $\mathrm{\Delta }{t}^{*}$ with the acoustic forcing $A$. From left to right, the vertical dotted lines delimit the regions identified on Fig. 7: the steady region (green), the first sequence (blue), the second sequence (orange), and the regimes with an attractor of higher dimensions (red).

Figure 13

Evolution of the attractors represented in three-dimensional phase spaces $\left[s\left(t\right),s(t-\mathrm{\Delta }{t}^{*}),s(t-2\mathrm{\Delta }{t}^{*})\right]$ for increasing acoustic forcing $A$. Each vertical bar represents an acoustic forcing for which nonlinear properties were computed from time series of the velocity. The first sequence (bottom attractors) and second sequence (top attractors) leading up to chaos are clearly visible. Gaps in this evolution represent regions where the dynamics fundamentally changes (as reflected by the dimensions of the attractors).

Figure 14

Evolution of the number of false neighbors $N_f$ for increasing phase-space dimension $N$ for an acoustic forcing $A=2.2$. Different critical divergence ratios $r$ are tested and lead to different values for the embedding dimension $D_e$.

Figure 15

Correlation sum $C_2$ computed from $u_t^{\prime }$ for an acoustic forcing $A=2.3$. The correlation sum is computed for different embedding dimensions $D_e$ in the range from $D_e=1$ to 10 (represented by different colours) to ensure that it is not computed on a folded attractor. A fairly constant slope is observed in the scaling (central) range and indicates a fractal dimension of $D_2=2.35$. For small scales ($\epsilon <7\times {10}^{-7}$), the correlation sum is known to be dominated by noise [40]. For large scales ($\epsilon >5\times {10}^{-6}$), the self-similarity is broken by the finite extension of the attractor.

Figure 16

Variations of the embedding and correlation dimensions ($D_e$ and $D_2$) with the acoustic forcing $A$. The first and second sequences are clearly visible as consecutive increases of the dimensions separated by a sudden drop.

Figure 17

(a) Comparison of the Poincaré section of the attractors for $A=2.2$, $A=2.3$, and $A=2.4$. $S_s$ and $S_p$ are the coordinates of the intersections between the Poincaré section and the attractors. The signal is normalized using $\sigma$, the standard deviation of the time series for $A=2.2$. The Poincaré section plane is obtained using POD to ensure an optimal perpendicularity with the attractor trajectories. For $A=2.3$, this representation singles out the two unstable orbits (surrounded by dashed boxes), where the system remains $80\%$ of the time. One of these orbits is represented in panel (c), in the three-dimensional phase space. The less dense outer region for $A=2.3$ matches the region occupied by the attractor at $A=2.2$. For $A=2.4$, the attractor is represented with its transient part (purple points) that travels on the same Poincaré section as for $A=2.3$, before stabilizing (four purple circles, with two of them being juxtaposed). The final attractor for $A=2.4$ is represented in the three-dimensional phase space in panel (b). By symmetry, another attractor is supposed to exist for $A=2.4$, as indicated by white circles.

Figure 18

Time evolution of $u_z^{\prime }$ for an acoustic forcing $A=2.3$, normalized using the standard deviation of the time series $\sigma$. Vertical dotted lines delimit the regions of intermittencies.

Figure 19

Poincaré sections for a high-dimensional system ($A=3.2$). The blue section is obtained from a local quantity (transverse velocity at the jet). The red section is obtained from a global quantity (integral of the vertical vorticity in one quarter of the horizontal plane at midheight of the lower half of the cavity). For both signals, high frequencies have been filtered out at $f=1000$ using a Gaussian filter. A structure close to a 2D torus is clearly visible on the red section.