Frequency shifts during whistling occurs as a transition between two phase synchronised limit cycles via a state of intermittency or abruptly

To investigate the dynamical states in the aeroacoustic system having a confined flow through the double orifices, we increase the $Re$ by changing the inlet airflow rate. The $Re$ is computed based on the bulk flow velocity $\bar{v}$ of the airflow across the orifice and the diameter $d$ of the orifice. Figure 2a shows the changes in the value of RMS of the acoustic pressure fluctuations $p^{\prime}_{\mathrm{rms}}$ as $Re$ is increased. As the $Re$ increases from 4000 to 12700, we observe a rise and fall in the value of $p^{\prime}_{\mathrm{rms}}$ reaching successive maxima followed by minima.

Figure 2b shows the corresponding changes in the dominant frequency (whistling frequency) of $p^{\prime}$ with $Re$. We obtain the dominant frequency from the magnitude spectrum of the acoustic pressure fluctuations, computed using the fast Fourier transform (FFT). The resolution of the amplitude spectrum considered here is 0.2 Hz. We observe a switch in dominant frequencies from 461 to 411, 535 to 443, 535 to 411, and 470 to 445 Hz when the value of $Re$ crosses the values $5150$, $7500$, $8300$, and $12000$, respectively; please refer to the intersection regime of R1-2, R2-3, R3-4, and R4-5 in figure 2b. A detailed view of the amplitude spectrum during the frequency shifts is represented in the waterfall plots of figure 2e-h.

The change in the value of $p^{\prime}_{\mathrm{rms}}$ begins from a local minimum at $Re=4000\pm 101$ (figure 2a), where the value of $p^{\prime}_{\mathrm{rms}}$ is approximately equal to 0.7 Pa. Upon increasing the $Re$, $p^{\prime}_{\mathrm{rms}}$ gradually increases to reach a local maximum at the value of $Re$ equal to $5150\pm 110$ (marked as i in figure 2a). We observe the state of LCO (figure 2c-i) at this local maximum ($p^{\prime}_{\mathrm{rms}}=9.5$ Pa). With further increase in $Re$, the value of $p^{\prime}_{\mathrm{rms}}$ decreases to a second local minimum ($p^{\prime}_{\mathrm{rms}}=1.7$ Pa, marked as ii in figure 2a). This transition from a local maximum to a local minimum is accompanied by the shift in the dominant frequency of $p^{\prime}$ (figure 2e) from 461 to 411 Hz. We observe that this frequency switching occurs via the state of intermittency (figure 2c-ii), which has bursts of periodic oscillation amidst the epochs of aperiodicity. The value of $p^{\prime}_{\mathrm{rms}}$ further increases, with increasing $Re$, to a subsequent higher local maximum ($p^{\prime}_{\mathrm{rms}}=46.6$ Pa, marked as iii in figure 2a) when the value of $Re$ equals $7500\pm 129$. The dynamical state corresponding to this local maximum is an LCO, as shown in figure 2c-iii.

Upon further increasing the Reynolds number ($Re>7450$), we note a sudden dip in the value of $p^{\prime}_{\mathrm{rms}}$ from 46 to 8.5 Pa. This decrease is accompanied by a shift in dominant frequency from 443 to 535 Hz (figure 2f and the interface of the region R2 & R3 in figure 2b). We again note that the switch in the dominant frequency occurs via the state of intermittency (refer to figure 8a of A). With a further increment of $Re$, we observe a similar trend of rise in $p^{\prime}_{\mathrm{rms}}$, reaching a local maximum and subsequently decreasing to a minimum with a change in the dominant frequency of $p^{\prime}$ fluctuations from 535 to 411 Hz. During this frequency shift from 535 to 411 Hz, we found that the transition occurs via the intermittency state (refer to figure 8c of A). Thus, when the value of the Reynolds number is increased from $4000$ to $9200$, we have shifts in whistling frequency between the regions R1-R2, R2-R3, and R3-R4 (figure 2b). During these frequency shifts, we observe that the dynamical state is intermittency.

Further, an increase in the $Re$ beyond $8600$ causes $p^{\prime}_{\mathrm{rms}}$ to rise again to reach a higher $p^{\prime}_{\mathrm{rms}}$ of 170 Pa at $Re\approx 12000\pm 165$; here we observe the dynamical state of LCO. With the continued increase in the value of $Re$, we observe an abrupt jump to another state of LCO having a lower amplitude with the value of $p^{\prime}_{\mathrm{rms}}=$ 60 Pa (refer to subfigures iv & v of figure 2d). This abrupt jump is accompanied by a slight frequency shift from 445 to 470 Hz.

Thus, from figure 2, the aeroacoustic system under consideration in our study, which has the flow through double orifices, exhibits the state of intermittency during the frequency shifts for the control parameter $Re$ in the range of 4000 to 8600. However, for $Re$ values greater than 8600, we observe an abrupt jump from one LCO to another LCO during the frequency shift. Motivated by these findings, we utilize a visualization technique based on the theory of recurrence to characterize the dynamical states of intermittency and LCO.

Recurrence is a measure attributed to the time constancy of a dynamical system [40]. A trajectory is said to be recurring at a location in phase space if it revisits the neighbourhood of the considered location after a specific time interval. To visualize the recurrence of the system in its phase space trajectory, a recurrence plot (RP) is built based on the recurrence matrix. In this study, we use delay embedding [41] to compute the time-delayed vectors for the signals obtained during the experiment to build the phase space trajectory. The uniform delay embedding with time delay $\tau$ and embedding dimension $D$ is used to rebuild the trajectory of the phase space. The value of the $\tau$ is computed from the function of average mutual information (AMI) [42]; The value of the $\tau$ at the first minimum of AMI is considered as the optimum $\tau$. The optimum dimension $D$ is computed using the false nearest neighbourhood (FNN) method [43]. Here, the delayed vector can be written as,

To compute the recurrence matrix, initially, we compute the distance between the location $i$ and all the other locations of the trajectory in phase portrait. Further, we choose a distance threshold $\epsilon$ and consider only those points to compute the recurrence matrix whose Euclidean distance is lesser than the threshold $\epsilon$. The threshold $\epsilon$ can be chosen such that the number of neighbouring locations is a small part of the total span of the attractor or choose a constant number such that each point has a fixed number of neighbours [44]. The equation for calculating the recurrence matric $R_{ij}$ is provided as,

Where $\Theta$ is a Heaviside step function. If the Euclidian distances $\left\|\mathbf{X}_{i}-\mathbf{X}_{j}\right\|$ is less than the threshold $\epsilon$, then the element of the matrix $R_{ij}$ is equal to one, else $R_{ij}$ is equal to zero. Entry 1 in the recurrence matrix corresponds to a recurrent state, which implies that the trajectory is revisiting its neighbourhood.

Figure 3 represent the plots for the recurrence matrix obtained for the acoustic $p^{\prime}$ oscillations for the state of intermittency at $Re\approx 5136$ and the state of LCO at $Re\approx 12077$. For the intermittency state, the phase space is reconstructed with an embedding dimension of $D=7$ and an optimum delay of $\tau_{opt}=0.7$ ms; For LCO $D$ is 5 and $\tau_{opt}$ is 0.5 ms. The recurrence matrix is obtained based on choosing a fixed value for the threshold $\epsilon=\lambda/5$, where $\lambda$ is the highest span between the pairs of locations of the trajectory in phase portrait [45]. For the state of intermittency, the recurrence plot is seen to have perforated black patches among white regions (cf. figure 3a). The occurrence of black patches in RP is due to the regime of low-magnitude aperiodic oscillations of the intermittency state. The white patches correspond to the periodic bursts of the intermittency state. Similar observations are made for the intermittency states at $Re\approx 7496$ and $Re\approx 8398$ (refer to figure 9 of B). During the states of LCO, we observe equidistant diagonal lines in the RP (cf. figure 3c). The time period of the LCO can be computed using the distance between the diagonal lines. The diagonal lines are also evident in the recurrence plots for the states of LCO observed at $Re\approx 4789$, 7288, 8120, and 12216 (cf. figure 9 of B).

Further, the information from the topology of the recurrence plot can be quantified using the recurrence quantification measure. To obtain and compare the quantifiable measure across various values of $Re$, we fix the threshold $\epsilon$ to a specific value. Here we choose the threshold $\epsilon$ to be of the size of the attractor corresponding to the aperiodic state of the intermittency during frequency shift.

We make use of the measure recurrence rate ($RR$) to investigate the variation in the dynamical state of the system with Reynolds number $Re$. The $RR$ is the measure of the density of the points that recur in the RP. The equation for $RR$ is given as,

where $N$ is the overall number of points in the trajectory. $RR$ is the average number of black dots ($R_{ij}=1$) in the recurrence plots [46].

We present the variation of the recurrence rate, $RR$, during the shift in whistling frequency in figure 4. The first column represents the variation of $p^{\prime}_{\mathrm{rms}}$ with $Re$ (cf. figure 4(a-d)-i). The second column denotes a shift in whistling frequency with $Re$ (cf. figure 4(a-d)-ii). The corresponding variations in $RR$ during the frequency shift are shown in the third column (cf. figure 4(a-d)iii). The plots corresponding to transitions via the state of intermittency are grouped in the orange background, and the abrupt transition from one LCO to another LCO is in the green background. We note that as the $Re$ varies, the curves of $RR$ increase and then decrease (cf. figure 4(a-c)iii). This observation can be anticipated as the number of black dots in the RP rises as the system approaches the state of intermittency as the pairwise separation length, during aperiodic epochs, now rarely cross the threshold $\epsilon$. Thus the curve of the recurrence measure $RR$ rises during the frequency shift. This variation in $RR$ confirms the presence of the intermittency state while the system transits from one whistling frequency to another. Whereas, during the abrupt transition from one LCO to another LCO in the $Re$ range of 11500 to 12500, we observe that the variation in $RR$ is negligible as both the states are of LCO and the pairwise distance regularly crosses the threshold $\epsilon$ (cf. figure 4ciii).

The co-existence of the acoustic and hydrodynamic subsystems in a confined flow gives rise to the synchronisation between the two. Thus, the synchronisation strength between the acoustic and hydrodynamics is an important criterion for understanding the mechanism through which intermittency (during the frequency shifts) occurs. We investigate the coupled dynamics between the acoustics and hydrodynamics of the current system using the theory of synchronisation. In this study, we cast acoustic ($p^{\prime}$) and hydrodynamic ($v^{\prime}$) fluctuations as two different subsystems. In the following subsection, we plot joint recurrence matrices to understand the level of synchronisation between acoustic and hydrodynamic subsystems during the LCO and the intermittency states.

The joint recurrence matrix (JRM) can be considered as an extension of the recurrence matrix to investigate the coupled dynamics of the two subsystems. A joint recurrence matrix helps visualize the recurrence of the trajectories, in the phase space, of the two subsystems at the same time [47, 48, 49].

The JRM for two subsystems having the time-delayed vectors $\mathbf{X}$ and $\mathbf{Y}$ is calculated by computing the element-wise product of the individual recurrence matrices ($R^{X}$, $R^{Y}$). The equation for JRM can be written as,

If the trajectories $X$ and $Y$ of the two subsystems recur simultaneously, then $JRM_{ij}=1$ else, $JRM_{ij}=0$.

Figure 5 represents the joint recurrence plots (JRP) of the phase trajectories of $p^{\prime}$ and $v^{\prime}$ corresponding to the intermittency state at $Re\approx 5136$ and LCO at $Re\approx 12077$ (cf. figure 5a and c). The corresponding time signal of the acoustic pressure $p^{\prime}$ and velocity $v^{\prime}$ fluctuations of the states of intermittency and LCO are shown in the subfigures b and d of figure 5. The $\epsilon$ is selected such that the recurrence rate (RR) for the individual recurrence matrix remains fixed, which is 0.1. A simultaneous recurrence of $p^{\prime}$ and $v^{\prime}$ would manifest as a black dot in the JRP. The black dots are spread in an irregular pattern during the desynchronised states of intermittency due to the aperiodic nature of the two subsystems. These black dots are sparsely distributed during the desynchronised state as a result of the fewer occurrences of the recurrences in $p^{\prime}$ and $v^{\prime}$, at the same time.

During the states of intermittency, we note the sparsely spaced irregular black patches due to the aperiodic epochs and the discontinuous diagonal lines due to the periodic epochs of $p^{\prime}$ and $v^{\prime}$ (cf. figure 5a). During the state of LCO, we observe that most of the area in JRP is filled with diagonal lines due to the periodic nature of the oscillations, as the periodic oscillations have more simultaneous recurrence than the aperiodic state. Note that the diagonal lines are more pronounced during LCO (cf. figure 5c), implying a higher correlation between $p^{\prime}$ and $v^{\prime}$.

We also present the JRP for the states of intermittency at $Re\approx 7496$ and $Re\approx 8398$ in the subfigures c and e of C. The occurrence of diagonal lines in the JRP corresponding to the states of LCO at $Re\approx$ 4789, 7288, 8120, and 12216 also depict the higher correlation between $p^{\prime}$ and $v^{\prime}$. Thus, from the JRP, we observe that there are transitions from one synchronised state of LCO to the next synchronised state of LCO via a state of intermittency. We see that the strength of synchronisation is high during the states of LCO and is low during the states of intermittency.

We now quantify the topology of the joint recurrence plots using the RQA measures recurrence rate $RR_{J}$ and determinism $DET_{J}$. The $RR$ is defined in equation 3. Determinism $DET$ quantifies the periodic dynamics of the system and is given as,

where $F(l)$ is the distribution frequency of the span of the diagonal lines in the recurrence plot. $DET$ represents the fraction of recurrence points in the RP that forms the diagonal lines. Determinism $DET_{J}$ in JRP measures phase synchronisation between the periodic signals.

The RQA measures $RR_{J}$ and $DET_{J}$ give us information on the variation in the synchronisation strength as the Reynolds number of the system varies. Figure 6 represents the changes of $DET_{J}$ and $RR_{J}$ with $Re$. The first column represents the variation of $p^{\prime}_{\mathrm{rms}}$ with $Re$ (cf. figure 6(a-d)-i). The second column denotes a shift in whistling frequency with $Re$ (cf. figure 6(a-d)-ii). The corresponding variations in $DET_{J}$ and $RR_{J}$ during the frequency shift are shown in the third and fourth columns, respectively (cf. figure 6(a-d)iii & iv). The plots corresponding to transitions via the state of intermittency are grouped in the orange background, and the abrupt transition from one LCO to another LCO is in the blue background. We note that as the $Re$ varies, the curves of $DET_{J}$ and $RR_{J}$ decrease and then rise, which indicates that the strength of the synchronisation reduces during the whistling frequency shift that occurs via the state of intermittency (cf. figure 6(a-c)iii & iv). There is an overall decrease in the values of $DET_{J}$ and $RR_{J}$ during the transition from high amplitude LCO to low amplitude LCO, which indicates that the synchronisation strength reduces and is manifested as the reduction in the amplitude of the LCO. Please note that the variation in these recurrence measures depends on how we define the recurrence threshold $\epsilon$ while computing the recurrence matrix. For a fixed $\epsilon$ (refer to Section 3.2) we observe that the measure RR rises as the system exhibits the state of intermittency during the frequency shift. Whereas for the varying threshold, the $\epsilon$ selected such that the recurrence rate $RR$ remains comstant, we observe a dip in the value of the measure $RR_{J}$ as the system exhibits the state of intermittency during the frequency shift.

We further make use of the probability of recurrence to identify the type of synchrony that persists between $p^{\prime}$ and $v^{\prime}$. Probability of recurrence quantifies the probability with which a state vector of the trajectory recurs after a time lag $\tau$ [50], and is given as,

The recurrence property of the signal is also associated with the phase of the signal. A recurrence of the signal is equivalent to an increment in phase by $2\pi$ [50]. The synchronisation of the two coupled subsystems implies the locking of their phases and frequencies. This locking of phase leads to the simultaneous appearance of the apexes of $P(\tau)$ of two signals in the plots of probability of recurrence. If the frequencies of the two signals are locked, and their amplitude remains uncorrelated, then the state is called a phase synchronised (PS) state. The PS state manifests as the simultaneous occurrence of the peaks, but with unequal heights, in the plots of probability of recurrence with the variation in the time lag. On the contrary, if a functional relationship exists between the subsystems, the apexes of $P(\tau)$ for the subsystems occur simultaneously, and also, the magnitude of the peaks are matched; this state is referred to as generalized synchronisation (GS). For the generalized synchronisation state, the RP and hence the probability of recurrence plots are identical [51].

In figure 7, we represent the plots for variation in the probability of recurrence with a time lag $\tau$, corresponding to the state of LCO and intermittency. In order to study the coupled behaviour of the two subsystems $p^{\prime}$ and $v^{\prime}$, we have overlapped the $P(\tau)$ functions for $p^{\prime}$ and $v^{\prime}$. We observe several peaks of $P(\tau)$ at regular intervals as the time lag $\tau$ increases, denoting the existence of a very high probability of recurrence for $p^{\prime}$ and $v^{\prime}$ during LCO (cf. figure 7a). Further, the peaks of $P(\tau)$ of the two subsystems occur simultaneously, implying that the trajectories of $p^{\prime}$ and $v^{\prime}$ are phase locked. Hence, we observe the state of phase synchronisation (PS) during LCO; note that the magnitude of the peaks of $P(\tau)$ for $p^{\prime}$ and $v^{\prime}$ are not matching, implying phase synchronisation.

During the periodic bursts of the state of intermittency, the peaks of $P(\tau)$ of the two subsystems occur simultaneously, implying that the trajectories of $p^{\prime}$ and $v^{\prime}$ are phase-locked. However, the amplitudes are mismatched, implying that the state is of phase synchronisation and not generalized synchronisation (cf. figure 7b). During the aperiodic epochs, we have a very low $P(\tau)$ of $p^{\prime}$ and $v^{\prime}$, and there is no correlation among them, implying the desynchronised state (cf. figure 7c). Thus it is evident from figure 7b-c that during the state of intermittency, we have the state of phase synchronisation during periodic epochs and a desynchronised state during the aperiodic regime of $p^{\prime}$ and $v^{\prime}$.

Thus, utilising the theory of synchronisation between acoustic and hydrodynamic fields of the aeroacoustic system, we observe a transition from one phase synchronised state (LCO) to another phase synchronised state (LCO) state as the $Re$ is increased. The shift between the phase-synchronised states happens via the intermittency state. The periodic epochs of the intermittency state have phase-synchronised periodic oscillations of pressure and hydrodynamic fluctuations. At the same time, the aperiodic epochs have desynchronised aperiodic oscillations of $p^{\prime}$ and $v^{\prime}$.