2. Experiments and analysis

The experimental set-up consists of a plenum chamber with a length of 420 mm and a diameter of 300 mm, as illustrated in figure 1. Connected to the plenum chamber is a square duct with a cross-section of 75 mm $\times$ 75 mm and a length of 750 mm. At the end of the duct are two orifice plates with a thickness of 15 mm and a gap of 50 mm between the two plates. The orifice is circular with a diameter of 25 mm generating a cavity flow between the orifice plates. A screw compressor provides the air supply, and airflow is regulated by an Alicat 500 MCR mass flow controller, which has an uncertainty range of $\pm$ 0.8 $\%$ of the reading and 0.20 $\%$ of full-scale flow measurements.

The flow is forced harmonically using an Ahuja 16 $\Omega$ impedance AU-60 speaker. The speaker is located 150 mm from the end of the settling chamber. The harmonic signal from the MATLAB signal generator is fed to the power amplifier (Crown XLI 800), and the gain is adjusted to a value of one unit for all experiments. The speaker is connected in parallel to the amplifier. The damping in the system was measured without flow prior to each experiment to ensure that it remained within $32.6 \pm {3}\,\rm s^{-1}$ by estimating the exponential decay rate of acoustic waves in the experimental chamber.

Acoustic measurements $p'$ are obtained using a PCB-378C10 microphone (with a sensitivity of $1 \rm mV\,(kPa^{-1})^{-1}$ and a $\pm$ 0.8 $\%$ uncertainty) placed behind the first orifice plate (745 mm from the settling chamber). The location of the microphone is sufficiently distant from a pressure node and the speaker to ensure reliable data acquisition (DAQ). An analogue to digital card (NI-6143,16 bits) is used to acquire the signals from the piezoelectric transducer at a sampling rate of 20 kHz for 5 s.

The fast Fourier transform technique is utilized to estimate the frequency components of the dynamic pressure ( $p'$ ) time series. We sample the datasets at a frequency of 20 kHz to avoid any spectral leakage and accurately capture the limit-cycle signal of the unforced system the frequency of which was estimated to be 150.6 Hz. The sampling frequency is high enough to satisfy the Nyquist criterion for frequencies in a reasonable neighbourhood of this natural frequency (Hearn & Metcalfe Reference Hearn and Metcalfe1995). We collected 1 00 000 data points at each forcing condition, equivalent to around 753 oscillation cycles at the natural frequency. The fast Fourier transform algorithm employed a bin size of 0.2 Hz for frequency resolution, leading to 65 537 bins.

Figure 1.

The experimental set-up consists of a plenum chamber connected to a duct with two orifice plates at its end. It includes a speaker for forcing, an amplifier, a microphone, a signal conditioner and a mass flow controller, all integrated and managed through a DAQ system installed on a personal computer (PC). All dimensions are in millimetres.

Various synchronization states are discerned through the behaviour of the instantaneous phase difference ( $\Delta \phi$ = $\phi _{r} -\phi _{f}$ ) between the response signal ( $p^{\prime}_{r}$ ) and the forcing signal $p^{\prime}_{f}$ . The instantaneous phases are obtained by applying the Hilbert transform to the respective signals. The Hilbert transform calculates the phase angle of each complex-valued signal using the arctangent function.

In the phase-locked state, the instantaneous phases and frequencies of the forcing and response signals satisfy $\Delta \phi =0$ and $\Delta \omega =0$ (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2001). However, in the phase-trapped state, the relative phase between the forcing signal and the response signal exhibits a bounded oscillatory pattern, leading to the synchronization of the mean frequencies of the signals without achieving instantaneous phase lock-in; i.e. $\Delta \phi \neq 0$ and $\Delta \omega =0$ (see figure 2 c). Phase-drifting (or phase-slipping) refers to the continuous increase or decrease over time of the unwrapped relative phase between the forcing and response signals. When the forcing frequency is less than the natural frequency, the phase difference between the response and the forcing signals will increase over time. This happens because the system’s natural oscillation frequency is faster than the forcing frequency (see figure 2 a). Similarly, when the forcing frequency is more than the natural frequency, the phase difference between the response and the forcing signals will increase over time as can be seen in figure 2(b).

We perform a transverse intersection of the trajectory of evolution of the response signal using techniques of dynamical systems theory and analyse the resulting intercepts to gain insights into the system’s topology and geometry. This method, known as a one-sided Poincaré map, reveals unique characteristics of different nonlinear responses. To illustrate, a limit cycle, represented by a closed trajectory, appears as a single blob on the map. Detailed information on obtaining the phase trajectory and one-sided Poincaré maps from the experimental data is presented in Appendix A.1.

Figure 2.

(a) Phase-drifting ( $f_f/f_n\gt 1$ ), (b) phase-drifting ( $f_f/f_n\lt 1$ ), (c) phase-trapping and (d) lock-in.

4. Modal power variation

To study modal power variation in the various regimes identified, we first calculate the spectral power of four key components: the natural mode power ( $P_n$ ); the forced mode power ( $P_f$ ) (and its harmonics or subharmonics where relevant); cumulative power ( $P_n$ + $P_f$ ); the total signal power ( $P_t$ ), normalized by the power of unforced oscillations ( $P_{n,u}$ ). These powers are represented as functions of the forcing power, normalized by the power required at lock-in $(\sigma _f/\sigma _{f,loc})^2$ . A process similar to bandpass filtering is performed around each frequency to obtain the modal power. This procedure involves integrating the power spectral density around each mode to isolate their contributions. The total power is obtained by integrating the power spectral density across the entire bandwidth. We ensure that the calculated total power matches (within a tolerance of 0.5 $\%$ ) the power of the mean squared fluctuations, as required by Parseval’s theorem (Porat Reference Porat2008).

The spectral power of the oscillations as a function of the forcing power for the system in the synchronous quenching regime ( $f_f=141.94$ Hz) and asynchronous quenching regime ( $f_f=129.8$ Hz) on the left-hand side of the Arnold tongue ( $f_f/f_n \lt 1$ ) are presented in figures 9(a) and 9(b), respectively. For low forcing amplitudes corresponding to phase-drifting dynamics, a linear decrease in the power of the natural mode ( $P_n$ ) and a simultaneous linear increase in the power of the forced mode ( $P_f$ ) is observed in both regimes. However, the total signal power is lower than the power of the unforced system.

The linear variations indicate that with each increment in forcing amplitude, a fixed portion of the power is withdrawn from the natural mode. The influence of external forcing on the natural mode suggests the presence of nonlinear interactions, although the power transfer out of the natural mode is linear. Nonlinear mechanisms that can lead to a reduction in $P_n$ are ones that diminish the natural mode’s ability to extract energy from the base flow, enhance dissipation or transfer energy from the natural mode to its sidebands.

The power increment in forcing frequency remains linear for all forcing amplitudes even beyond lock-in. A similar linear increase in $P_f$ is observed by Sreenivasan et al. (Reference Sreenivasan, Raghu and Kyle1989) and Kyle & Sreenivasan (Reference Kyle and Sreenivasan1993) in both self-excited and convectively unstable jets. Despite the linear increase in $P_f$ , the combined power ( $P_n$ + $P_f$ ) decreases linearly for small forcing amplitudes because $P_n$ diminishes more rapidly than $P_f$ grows. This decrease is especially pronounced in the asynchronous quenching regime.

For forcing amplitudes close to lock-in in the synchronous quenching regime, while $P_f$ increases linearly, $P_n$ decreases in a highly nonlinear and abrupt manner. This deviation from linearity indicates the activation of a second mechanism that enables additional power transfer from the natural mode on approaching the saddle-node bifurcation.

For forcing amplitudes close to lock-in in the asynchronous quenching regime, while $P_f$ increases linearly, $P_n$ starts decreasing much more as the dynamics transitions to phase-trapping. Finally, there is an abrupt drop in modal power $P_n$ as system enters lock-in.

After lock-in, cumulative power as well the total power is dictated by $P_f$ and its harmonics as the modal power in the natural mode is suppressed completely at lock-in in both the regimes. In the asynchronous quenching regime, the reduction in total power is up to $74.3\,\%$ at lock-in, whereas in the synchronous quenching regime, the reduction is up to $56.3\,\%$ . Increasing the forcing amplitude beyond lock-in increases the total signal power; however, this power remains well below the total power of the unforced system, especially in the asynchronous quenching regime.

Figure 9.

Spectral power of natural frequency ( $P_ n$ ), forcing frequency ( $P_ f$ ), $P_ n$ + $P_ f$ and total signal power ( $P_ t$ ) normalized by unforced system power ( $P_{n,u}$ ). These powers are represented as functions of the forcing power and normalized by the lock-in amplitude $(\sigma _f/\sigma _{f,loc})^2$ for (a) synchronous quenching ( $f_f/f_n$ = 0.94) and (b) asynchronous quenching ( $f_f/f_n$ = 0.86) on the left arm of the Arnold tongue.

Figure 10.

Spectral power of natural frequency ( $P_ n$ ), forcing frequency ( $P_ f$ ), $P_ n$ + $P_ f$ and total signal power ( $P_ t$ ) normalized by unforced system power ( $P_{n,u}$ ). These powers are represented as functions of the forcing power and normalized by the lock-in amplitude $(\sigma _f/\sigma _{f,loc})^2$ for (a) synchronous quenching ( $f_f/f_n$ = 1.02) and (b) asynchronous quenching ( $f_f/f_n$ = 1.17) on the right arm of the Arnold tongue.

In figure 10, spectral power for forcing performed on the right-hand side of the Arnold tongue ( $f_f/f_n \gt 1$ ) are shown for the synchronous (figure 10 a) and asynchronous (figure 10 b) regimes. For low forcing amplitudes corresponding to phase-drifting dynamics, $P_f$ increases linearly and $P_n$ appears nearly invariant in the synchronous quenching regime with negligible variation. This near invariance hints at the negligible influence of forcing on dynamics at the natural mode. As a consequence, there is negligible transfer of energy away from the natural mode which leads to an increase in the total signal power to values above the power of the unforced system.

A linear decrease in $P_n$ is more apparent in the asynchronous quenching regime for low forcing amplitudes corresponding to during phase-drifting dynamics leading to a near constant value of total signal power for low forcing amplitudes. In other words, energy supplied externally is dissipated more efficiently in the asynchronous quenching regime.

For forcing amplitudes close to lock-in, there is an abrupt large decrease in $P_n$ in the synchronous regime as is observed for $f_f/f_n \lt 1$ . However, there is a more pronounced nonlinearity in the variation of the $P_f$ for these forcing amplitudes. This indicates that dynamics at the natural frequency is able to influence the dynamics at the forcing frequency in this range of amplitudes which was not the case for $f_f/f_n \lt 1$ .

The dynamics in the asynchronous quenching regime is further complicated by the presence of two clear abrupt drops in $P_n$ corresponding to transitions to phase-trapping dynamics and lock-in. Transition to phase-trapping leads a much larger drop in $P_n$ in this case than was observed for $f_f/f_n\lt 1$ . There is a nonlinearity in the variation of $P_f$ close to lock-in as was observed in the synchronous quenching regime.

The abrupt drops in $P_n$ bring the total signal power to values lower than that observed for unforced system resulting in amplitude suppression at lock-in. On increasing the forcing amplitude beyond lock-in, one can observe a rapid rise in the total power for both synchronous and asynchronous quenching regimes, to values well beyond that observed for the unforced system. This overshoot or amplification was absent for forcing on the left-hand side of the Arnold tongue ( $f_f/f_n \lt 1$ ) where the increase in total power after lock-in is much more gradual.

In the case of synchronous quenching, the amplification is termed as resonant amplification; for asynchronous quenching, it is called non-resonant amplification. For resonant amplification, the shoot-up after lock-in happens at nearly the same levels as the levels at which the modal power at the forcing frequency increases. However, for non-resonant amplification, the shoot-up after lock-in is much more drastic and happens at much higher levels than the levels at which the modal power at the forcing frequency increases. This feature is particularly distinct from the variation of modal power on the left-hand side of the Arnold tongue where the increase in total power was always proportional to the variation of the modal power at the forcing frequency after lock-in. This amplification, possibly due to the asymmetry in the vortex distribution as discussed in the previous section, is responsible for the observed asymmetry of the Arnold map on either side of the natural frequency.

The reduction in total power at lock-in within the synchronous quenching regime reaches up to 38 % for cases where $f_f/f_n \gt 1$ , which is comparatively smaller than the reduction observed for $f_f/f_n \lt 1$ (57.3 %). Similarly, in the asynchronous quenching regime, the total power reduction at lock-in reaches 44 % when $f_f/f_n \gt 1$ , again demonstrating a comparatively smaller reduction than in cases where $f_f/f_n \lt 1$ (78.3 %).

In figure 11, the power transfer across all significant modes is shown for forcing on either side of the Arnold tongue in the $1:2$ lock-in regime. When external harmonic forcing is applied to the system near the subharmonic frequency, $P_n$ decreases with a concurrent rise in $P_f$ as forcing amplitude increases. In contrast to synchronous and asynchronous quenching regimes, a rise in the power of the second harmonic of the forced mode ( $P_{2f}$ ) is also observed with an increase in the forcing amplitude during phase-drifting dynamics before lock-in. Another significant observation is the presence of subharmonics at $f_f/2$ and $f_f/3$ corresponding to $T^3$ quasiperiodicity. The modal power at $f_f/3$ increases linearly until lock-in when it abruptly disappears as period-two oscillations set in.

Resonant amplification of the total signal power is also observable for forcing amplitudes beyond lock-in for forcing on the right-hand side of the Arnold tongue centred at the subharmonic of the natural frequency (see figure 11 b). The amplification is resonant as forcing results in the excitation of the first harmonic ( $2f_f$ ) of the forcing frequency which is located closely to the right of the natural frequency ( $f_n$ ). Forcing on the left-hand side of this Arnold tongue does not result in amplification (see figure 11 a). This indicates that amplification is a phenomenon dictated by the forcing frequency or its harmonics being larger than the natural frequency.

Figure 11.

Spectral power of natural frequency ( $P_ n$ ), forcing frequency ( $P_ f$ ), $P_ n$ + $P_ f$ , subharmonics ( $P_ {f/2}, P_ {f/3}$ ) and total signal power ( $P_ t$ ) normalized by unforced system power ( $P_{n,u}$ ). These powers are represented as functions of the forcing power and normalized by the lock-in amplitude $(\sigma _f/\sigma _{f,loc})^2$ for (a) subharmonic lock-in without resonant amplification ( $f_f=73.5\, \textrm{Hz}$ ), (b) subharmonic lock-in with resonant amplification ( $f_f=76.2\, \textrm{Hz}$ ).

Figure 12.

Spectral power of natural frequency ( $P_ n$ ), forcing frequency ( $P_ f$ ), $P_ n$ + $P_ f$ , subharmonic ( $P_ {n/2}$ ) and total signal power ( $P_ t$ ) normalized by unforced system power ( $P_{n,u}$ ). These powers are represented as functions of the forcing power and normalized by the lock-in amplitude $(\sigma _f/\sigma _{f,loc})^2$ for de-lock-in regime ( $f_f=98.5\, \textrm{Hz}$ ).

In the de-lock-in regime, the primary difference from the $1:2$ lock-in regime for forcing amplitudes until lock-in is the absence of three-frequency dynamics (no modal power at $f_f/3$ ) and a much smaller increase in the energy content of the harmonic $2f_f$ (figure 12). At higher amplitudes beyond lock-in, de-lock-in can be observed as an increase in the power of the natural mode. Some energy also gets transferred to the subharmonic ( $f_n/2$ ) of the natural frequency resulting in complex dynamics. Resonant amplification of total power is observed due to the excitation of the harmonic $2f_f\gt f_n$ of the forcing frequency as described for the $1:2$ lock-in regime.

The reduction in total power at lock-in is up to 30.3 % for the $1:2$ lock-in forcing regime and up to 28.3 % for the de-lock-in regime. The decrease in overall power at lock-in is less compared with asynchronous and synchronous quenching regimes, suggesting that these regimes are unsuitable for optimal control. In the de-lock-in regime, there is the possibility of de-lock-in at higher amplitude ratios with complex dynamics, which makes it risky for control purposes. In the $1:2$ lock-in regime, there is the presence of a three-frequency regime with subharmonics which complicates control action. In addition, there is also a possibility of resonant amplification.

In summary, to suppress self-sustained oscillations in the cavity flow by external forcing, we need the forcing frequency ( $f_f$ ) to be sufficiently detuned from the natural frequency ( $f_n$ ) to result in asynchronous quenching. Secondly, we have to ensure that $f_f/f_n \lt 1$ so that amplification is avoided. And finally, we have to ensure that the detuning is not so large as to end up in other $p:q$ lock-in regimes or de-lock-in regimes. These conditions pose a strict constraint on the optimal frequency range for effective control action.

In open-loop control, asynchronous quenching emerges as a promising approach for suppressing limit-cycle oscillations. However, a significant limitation of this method is that it necessitates prior knowledge of $f_n$ , which can be difficult to determine in systems where even brief excursions into a high-amplitude limit-cycle regime may result in catastrophic mechanical or thermal damage. Furthermore, this challenge is compounded by the potential for $f_n$ to drift due to variations in operating conditions or system parameters.

For robust control, it is desirable to develop a method for suppressing limit-cycle oscillations that is less dependent on the detuning between forced and natural modes. Specifically, a strategy is needed that facilitates both asynchronous and synchronous quenching while avoiding any amplification of the forcing signal.

5. Phenomenological model

Consider a double orifice plate configuration in a duct of length $L$ that forms a cavity of length $L_c$ . Flow enters the cavity, forming vortices as the free shear layer develops from the first orifice. These vortices advect downstream with the flow and impinge on the second orifice location, leading to pressure fluctuations of dipole type (Matveev Reference Matveev2005). These pressure fluctuations travel upstream with the speed of sound to the first orifice location, exciting the free shear layer, which, under appropriate flow conditions, also excites the longitudinal acoustic mode of the duct, leading to self-sustained oscillations or a limit cycle. In what follows, a simplified mathematical model is presented to capture this mechanism. The model is inspired by the work of Matveev & Culick (Reference Matveev and Culick2003) and incorporates elements from van der Pol and Duffing oscillators to capture the observed dynamics.

Certain assumptions are incorporated to facilitate a clear mathematical representation, which is reasonable for the flow field under consideration. Specifically, one-dimensional acoustic theory is employed for the duct (treated as open–open) due to the dominance of longitudinal acoustic modes. Additionally, a low Mach number assumption is made so that the effects of mean flow (treated as a constant inside the cavity) in the wave equation can be neglected. Further, the cavity length $L_c$ is considered small compared with the acoustic wavelength (compact source assumption). The effects of turbulence or flow unsteadiness on vortex advection are ignored, and only the acoustic fluctuations are considered when describing vortex dynamics. Similarly, the only source term in the wave equation arises from the circulation generated by the vortices. Finally, the effects of viscosity and heat transfer on the wave equation are neglected. The momentum and energy equations for the flow may then be written as

(5.1) \begin{equation} {\frac {\partial u^{\prime }}{\partial t}}+{\frac {1}{\rho _{0}}}{\frac {\partial p^{\prime }}{\partial x}}=u\Gamma (t)\delta (x-x_c) \end{equation}

and

(5.2) \begin{equation} {\frac {\partial p^{\prime }}{\partial t}}+\gamma {p_{0}}{\frac {\partial u^{\prime }}{\partial x}}=0 \end{equation}

where $p^{\prime}$ , $u^{\prime}$ are acoustic pressure and velocity fluctuations, $p_{0}$ , $\rho _{0}$ are mean pressure and density, $\gamma$ is the ratio of specific heat and $\Gamma$ is the circulation strength of the vortex. The motivation for the forcing term comes from the expression of the lift generated by a circulation distribution (this is different from the form described in Matveev & Culick Reference Matveev and Culick2003) and the parameter $x_c=L-Lc$ . Circulation is assumed to be purely time-dependent as it is obtained by spatially integrating the velocity field over a closed contour. Subtracting the time derivative of (5.2) from that of (5.1), the resulting expression can be written as

(5.3) \begin{equation} \begin{aligned} & \frac {\partial ^{2} p^{\prime }}{\partial t^{2}}-c_{0}^{2} \frac {\partial p^{\prime }}{\partial x^{2}}=-\Gamma \frac {\partial u}{\partial x} \delta \!\left (x-x_{c}\right )\!. \\ \end{aligned} \end{equation}

We employ the Galerkin projection method (Zinn & Lores Reference Zinn and Lores1971) to spatially average the wave equation to obtain ordinary differential equations for the fundamental acoustic mode. The acoustic variables ( $u^{\prime}$ and $p^{\prime}$ ) are expanded using basis functions that meet the boundary conditions of the system as follows:

(5.4) \begin{align} p^{\prime }(x,t) &=\gamma p_{0} \eta (t)\sin (kx),\\[-9pt]\nonumber \end{align}

(5.5) \begin{align} u^{\prime }(x,t) &=\frac {\dot {\eta }(t)}{k}\cos (kx),\\[-9pt]\nonumber \end{align}

where $k=\pi /L$ . Substituting (5.4), (5.5) in (5.3), and spatially integrating over the length of the duct, we obtain second-order ordinary differential equations for the time coefficient $\eta$ of the acoustic mode,

(5.6) \begin{equation} \begin{aligned} \ddot {\eta }(t)+\omega ^{2}\eta (t)=\kappa \dot {\eta }(t)\Gamma. \end{aligned} \end{equation}

Damping can be modelled, taking inspiration from the van der Pol oscillator to account for linear and nonlinear damping effects as was done in Noiray & Schuermans (Reference Noiray and Schuermans2013) and Guan et al. (Reference Guan, Gupta, Kashinath and Li2019a ). To model the sudden jump phenomenon observed in experiments in the transition region of $1:1$ to $1:2$ lock-in, an additional cubic stiffness term is added as in the Duffing oscillator. Kashinath et al. (Reference Kashinath, Li and Juniper2018) and Guan et al. (Reference Guan, He, Murugesan, Li, Liu and Li2019c ) have previously employed this strategy to capture the jump phenomena in a ducted flame near $3:4$ lock-in regime.

Figure 13.

Schematic representation of the vortex shedding and impingement from double orifice cavity set-up and its active control by harmonic acoustic perturbations.

The final oscillator equation for acoustics then becomes

(5.7) \begin{equation} \begin{aligned} \ddot {\eta }+\eta (\omega ^{2}-\beta \eta ^{2})+\dot {\eta }[{\rm \mu} (1-\zeta \eta ^{2})-\kappa {\Gamma }]=0 .\\ \end{aligned} \end{equation}

A value of ${\rm \mu} =0.2$ is selected, which generates an unforced self-excited waveform consistent with experimental observations. The value of $\zeta =0.95$ is chosen to capture the asymmetry of the Arnold tongue. This approach is different from conventional modelling employed for van der Pol oscillators. The parameter $\kappa =({2\pi }/{L^2})\sin ^2(({\pi }/{L})x_c)$ represents the coupling factor for vortex–acoustic interaction, which is obtained as 1.25. The cubic-stiffness parameter $\beta$ is chosen as 0.1 by matching the frequency ( $f_f/f_n=0.60$ ) at which the amplitude-response curve peaks in the experiments.

In (5.7) $\Gamma$ , which represents circulation, appears as a driving (or damping term) that needs to be modelled. When a vortex begins to form, its circulation is zero. As the vortex develops, $\Gamma$ grows. The vortex detaches once the circulation achieves a critical value and gets advected away from the source location. Therefore, the rate of change of circulation at the leading edge of the cavity at the first orifice can be expressed as follows:

(5.8) \begin{equation} \begin{aligned} \frac {{\rm d} \Gamma }{{\rm d} t} &= \frac {1}{2}\left (u_{0}+u^{\prime }_{x_c}\right )^{2}-\alpha u_{0}\left (u_{0}+u^{\prime }_{x_c}\right ). \\ \end{aligned} \end{equation}

In this equation, the first term represents the growth rate of circulation at the step as defined in Matveev & Culick (Reference Matveev and Culick2003), and the second term represents the rate at which circulation is lost as the shear layer advects downstream. The coefficient $\alpha$ characterizes the fraction of the mean velocity ( $u_{0}$ ) at which the shear layer advects away from the first orifice. Experimental studies suggest that $\alpha =0.45$ can be appropriate for the present study (Dotson, Koshigoe & Pace Reference Dotson, Koshigoe and Pace1997). Using Galerkin expansion and substituting the value of $u^{\prime }$ from (5.5) into (5.8), we obtain the rate of change of circulation in terms of the acoustic modes as

(5.9) \begin{equation} \begin{aligned} \frac {{\rm d} \Gamma }{{\rm d} t}& =a \dot {\eta }^{2} + b \dot {\eta } - c \\ \end{aligned} \end{equation}

where $a = ({0.5}/{k})\cos ^2( {(\pi }/{L})x_c)$ , $b = ({(1-\alpha )u_0L}/{\pi })\cos (({\pi }/{L})x_c)$ and $c = (0.5- \alpha )u^2_0$ . The values of $a$ , $b$ and $c$ are obtained as 0.56, 0.45 and 0.675 on substituting values from experiments.

The parameters $a$ , $b$ and $c$ , which dictate vortex dynamics can be directly related to the physical parameters $u_0$ , $L$ and $L_c$ . Parameter $a$ captures the nonlinear contribution of acoustic velocity fluctuations to the circulation growth. The value of $a$ determines the lock-in amplitude. This parameter $a$ can also broaden the lock-in range, allowing synchronization over a wider frequency band. Parameter $b$ represents the rate at which circulation is lost as the shear layer advects downstream. Experimental results presented in the previous sections demonstrate that all harmonics, including the even harmonics, are present in the response signals, which conventional van der Pol equation cannot reproduce (only odd harmonics are captured by van der Pol equation). This is typically remedied by introducing additional term $\dot {\eta }^2$ in the oscillator equation as is done in Rayleigh–van der Pol oscillators. In our system of equations $\dot {\eta }^2$ term appears indirectly through circulation in (5.7), and parameter $b$ dictates the relative strength of the even harmonics. This approach, where the Rayleigh term appears as an integrated effect, is necessary to observe the nonlinear phase drift present during de-lock-in. Parameter $c$ which incorporates the effect of mean velocity and decay due to vortex advection, acts as a bifurcation parameter in the unforced oscillator. On varying $c$ (flow rate), the system transitions from near-zero amplitudes to self-sustained limit cycle oscillations in the absence of forcing.

Equations (5.7) and (5.9) constitute a system of coupled oscillator equations, hereafter referred to as ROPM. The fast equation (5.7) representing acoustics contains features of van der Pol and Duffing oscillators. The slow equation (5.9) represents the dynamics of vortex generation and advection at the location of the first orifice plate. For harmonically forced dynamics, we add a sinusoidal forcing term to the right-hand side of (5.7) to obtain

(5.10) \begin{equation} \begin{aligned} \ddot {\eta }+\eta (\omega ^{2}-\beta \eta ^{2} )+\dot {\eta } [{\rm \mu} (1-\zeta \eta ^{2} )-k{\Gamma } ]=\sigma _f \sin (2\pi f_f t ) \end{aligned} \end{equation}

where $\sigma _f$ denotes the non-dimensional amplitude and $f_f$ denotes the forcing frequency.

Understanding the sensitivity of model parameters which cannot be explicitly determined and the sequential procedure of tuning them is of utmost importance in reproducing experimental results. The parameter ${\rm \mu}$ , which governs both linear and nonlinear damping and determines the unforced system behaviour has to be within the range of 0.1–0.3 so that qualitative behaviour of the model remains the same (Thompson & Stewart Reference Thompson and Stewart2002). For instance, values of ${\rm \mu}$ above this range results in steepened limit cycles.

After adjusting the value of ${\rm \mu}$ , we adjust the cubic stiffness parameter $\beta$ according to the frequency at which the maximum amplitude in the response curve is observed during the experiments. The value of $\beta$ , if maintained within a low range (−0.05 $\leqslant$ $\beta$ $\leqslant$ 0.2), results in the properties of a soft cubic spring.

Then, the value of $\zeta =0.95$ is chosen to match the asymmetry of the Arnold tongue seen in experiments. When $\zeta \leqslant 1.5$ , it correctly captures the asymmetry observed in experiments. The value of $\zeta$ above 1.5 results in the steepening of the right arm of the Arnold tongue and affects the saturation behaviour observed in the synchronization map.

Here ROPM fundamentally differs from existing models utilized for modelling coupled vortex–acoustic interaction. First, it builds upon Matveev’s model by introducing terms that account for vortex advection. Second, the model avoids discontinuities associated with vortex shedding and impingement present in Matveev’s model. Finally, the model provides a phenomenological pathway based on experimental observations to arrive at a model that contains elements of van der Pol and Duffing oscillators.

The system of coupled oscillators was solved using multistep variable-order algorithm (Shampine & Reichelt Reference Shampine and Reichelt1997). When the forcing term is set to zero, the model gives a solution equivalent to a unforced limit-cycle state with frequency $f_n$ observed in experiments. The forcing amplitude ( $\sigma _f$ ) and forcing frequency $f_f$ are systematically varied to study the impact of forcing on the limit cycle solution.

5.1. Comparison of model results with experiments

Synchronous quenching is observed when the oscillator (ROPM) is forced near the natural frequency, i.e. $f_f/f_n = 0.98$ . As the forcing amplitude increases, phase-drifting is observed before lock-in, similar to what is observed in the experiments (figure 3). In figure 14, it is evident that the torus suddenly collapses to a blob on the Poincaré map, indicating saddle-node bifurcation. The model correctly captures side-bands obtained in experiments mainly due to the ability of the model to capture second harmonics. The model also exhibits a suppression-type behaviour and not a locking-type behaviour as the natural frequency in the pressure spectrum is not pushed (or pulled) towards the forcing frequency. This observation is in agreement with experiments.

In figure 15, we present a comparison of four indicators, similar to those shown in figure 14, for our phenomenological model forced far from the natural frequency ( $f_f/f_n = 0.86$ ) where asynchronous quenching was observed in experiments. As the forcing amplitude ( $\sigma _f$ ) increases, we observe a decrease in the amplitude of the natural mode without any concurrent pulling of the natural frequency towards the forcing frequency ( $f_f$ ). For low forcing amplitudes, phase-drifting is present which transitions to phase-trapping prior to lock-in. The torus grows during the phase-drifting dynamics and shrinks in size once dynamics enters phase-trapping (figure 15 e). Hence, we have a Neimark–Sacker bifurcation in the opposite direction leading to torus death, leading the dynamics from $T^2$ quasiperiodicity to $1:1$ lock-in as the forcing amplitude increases.

Figure 14.

(a) Pressure time series, (b) frequency spectrum, (c) phase difference plots (d) and one-sided Poincaré maps for various system dynamics on varying the forcing amplitude in synchronous quenching regime ( $f_f/f_n=0.98$ ), obtained from ROPM.

Figure 15.

(a) Pressure time series, (b) frequency spectrum, (c) phase difference plots, (d) one-sided Poincaré maps for various system dynamics on varying the forcing amplitude in asynchronous quenching regime ( $f_f/f_n=0.86$ ) and (e) torus growth during phase-drifting and subsequently shrinking during phase-trapping, obtained from ROPM.

Figure 16.

(a) Pressure time series, (b) frequency spectrum, (c) phase difference plots (d) and one-sided Poincaré maps for various system dynamics on varying the forcing amplitude in de-lock-in regime ( $f_f/f_n=0.65$ ), obtained from ROPM.

As we reduce the forcing frequency towards $1:2$ lock regime, a window of de-lock-in appears. As the amplitude of forcing ( $\sigma _f=0.089$ ) increases from the unforced state, we can observe quasiperiodic behaviour and the appearance of a torus is evident in the Poincaré map (figure 16 d ii). Phase-drifting is observed prior to lock-in. As we further increase the amplitude, we can observe a sudden collapse of the torus to a blob, leading the system to lock-in via saddle-node bifurcation. On further increasing the forcing amplitude after lock-in ( $\sigma _f=0.65$ ), we can observe that phase delocks via nonlinear phase-drifting, which matches the observations in experiments. The route to de-lock-in involves period-doubling bifurcation from the lock-in state indicated by the presence of a frequency component at $f_n/2$ , similar to that observed in experiments.

The ROPM also correctly captures the subharmonic ( $1:2$ ) lock-in phenomena observed in the experiments. We forced the system slightly above and below the natural subharmonic frequency. In figure 17, the natural mode is forced at $f_f/f_n=0.48$ , and the forcing amplitude is progressively increased until lock-in is observed. When the forcing amplitude is small ( $\sigma _f=0.18$ ), we can observe various peaks at $f_n$ and $f_f$ that lead to phase-drifting and formation of a $T^2$ torus. When forcing strength is increased further, a three frequency regime corresponding to a higher-order $T^3$ torus is observed as in experiments. This transition from $T^2$ to $T^3$ quasiperiodicity happens when the modal strength at $2f_f$ exceeds the strength at $f_n$ . At lock-in, dynamics exhibits period-two oscillations as indicated by the signal and two blobs in the Poincaré map (figure 17(d iv).

Figure 17.

(a) Pressure time series, (b) frequency spectrum, (c) phase difference plots (d) and one-sided Poincaré maps for various system dynamics on varying the forcing amplitude in $1:2$ lock-in regime ( $f_f/f_n=0.49$ ), obtained from ROPM.

5.1.1. Synchronization map obtained from ROPM

The synchronization map obtained from the ROPM is compared with the one obtained in the experiment in figure 18.

Figure 18.

Comparison of synchronization map (Arnold tongue) obtained from the experiments, discrete map (Britto & Mariappan Reference Britto and Mariappan2019) and ROPM. Zoomed inset shows a comparison of lock-in boundary in the de-lock-in region for experiments (red dashed line) and model (blue dashed line).

The model correctly predicts the onset of the transition from the saddle-node bifurcation region to the torus-death bifurcation regimes of the torus. The model also predicts asymmetry in the Arnold tongue and the saturation of the right-hand branch for $f_f/f_n\gt 1$ .

The jump in forcing amplitude at $f_f /f_n$ = 0.6 is captured with the addition of the cubic stiffness parameter $\beta$ in the model. However, the model tends to overestimate the lock-in amplitude at the jump location when compared with experimental findings. The presence of second subharmonics in the unforced state was found to be crucial in capturing essential dynamics of $1:2$ lock-in and even predicting comparable trends for lock-in amplitude as in experiments (figure 18). This is a striking improvement over other existing models available to study vortex–acoustic interaction.

Recently, (Britto & Mariappan Reference Britto and Mariappan2019) used a discrete map equation approach to model vortex shedding; the results of this discrete map are also presented in figure 18. The discrete model correctly predicts the map in the synchronous quenching regime; however, the model fails to predict the experimental observations in the region of asynchronous quenching. This is because the discrete map equation (Britto & Mariappan Reference Britto and Mariappan2019) contains only a single equation to generate a synchronization map and cannot account for the complexity in dynamics far away from the forcing frequency. The presented oscillator model, however, offers great flexibility in solving synchronization problems as key parameters, which are physically motivated, can be adjusted to match the experimental results.

5.1.2. Modal power variation obtained from ROPM

The spectral power variation at the dominant modes on the left-hand ( $f_f/f_n=0.96$ Hz) side of the Arnold tongue obtained from ROPM in the synchronous and asynchronous quenching regimes is presented in figures 19(a) and 19(b), respectively. There is a linear reduction in power of natural mode for low forcing amplitudes in both regimes. Close to lock-in there is a single drastic in the synchronous quenching regime, whereas an additional drop in $P_n$ is observed in the asynchronous quenching regime due to the transition to phase-trapping dynamics prior to lock-in. Power remains below the unforced system’s power for forcing amplitudes larger than that required for lock-in.

Figure 19.

Spectral power of natural frequency ( $P_ n$ ), forcing frequency ( $P_ f$ ), $P_ n$ + $P_ f$ and total signal power ( $P_ t$ ) normalized by unforced system power ( $P_{n,u}$ ). These powers are represented as functions of the forcing power and normalized by the lock-in amplitude $(\sigma _f/\sigma _{f,loc})^2$ for synchronous quenching (a) left-hand side ( $f_f/f_n=0.96$ ) and (b) left-hand side ( $f_f/f_n=0.86$ ), for asynchronous quenching (c) right-hand side ( $f_f/f_n=1.02$ ) of the Arnold tongue and (d) right-hand side ( $f_f/f_n=1.17$ ) of the Arnold tongue obtained from ROPM.

The spectral power variation at the dominant modes on the right ( $f_f/f_n=1.02$ Hz) side of the Arnold tongue obtained from ROPM in the synchronous and asynchronous quenching regimes is presented in figures 19(c) and 19(d), respectively. Amplification (both resonant and non-resonant) is observed in experiments after lock-in, and the total signal power goes beyond that of the unforced system’s power even for low forcing amplitudes, as observed in experiments. The model also correctly captures the mild nonlinearity in $P_f$ near lock-in.

Figure 20.

Spectral power of natural frequency ( $P_ n$ ), forcing frequency ( $P_ f$ ), $P_ n$ + $P_ f$ , subharmonics ( $P_ {f/2}$ , $P_ {f/3}$ ) and total signal power ( $P_ t$ ) normalized by unforced system power ( $P_{n,u}$ ). These powers are represented as functions of the forcing power and normalized by the lock-in amplitude $(\sigma _f/\sigma _{f,loc})^2$ for (a) de-lock-in ( $f_f/f_n=0.65$ ) and (b) subharmonic regime ( $f_f/f_n=0.48$ ) obtained from ROPM.

Figure 20 shows the power transfer across all modes for the sub- $1:2$ lock-in and de-lock-in regimes. For low forcing amplitudes, there is minimal reduction of $P_n$ and the total power remains nearly invariant. The variation of $P_{f_f}$ is linear. However, there is an increase in the spectral power ( $P_{2f}$ ) as the forcing amplitude increases in a nonlinear manner.

In the $1:2$ lock-in regime, there is a three-frequency regime indicated by the spectral contents of $P_{f_f/2}$ and $P_{f_f/3}$ . At lock-in, the spectral content of $P_{f_f/3}$ drops to zero and we have period-two oscillations. In the de-lock-in regime, at high forcing amplitudes beyond lock-in, de-lock-in can be observed as a slight increase in the power of the natural mode can be observed (figure 20 a). Resonant amplification is observed in the de-lock-in regime and on the right arm of the Arnold tongue of the $1:2$ lock-in regime. All these observations match the trends seen in experiments.

To summarize, we have shown that the model presented can capture essential features of synchronization phenomena at least in cavity flow systems involving vortex–acoustic interaction.