3.1. Self-pulsation – a state of globally unstable flow

Before we can understand the effects of forcing of gas jets on the spray from a gas-centred swirl coaxial (GCSC) injector, it is important to identify the operating conditions that produce the naturally pulsating flow originated from the global instability. The confinement of the shear layer (a recess region) and the velocity deficit (resulting from the finite separating lip) are important geometrical aspects of the atomizer that control the existence of the global instability (Juniper & Candel Reference Juniper and Candel2003; Juniper Reference Juniper2006; Matas Reference Matas2015; Matas et al. Reference Matas, Delon and Cartellier2018). Since all the experiments were performed using the same injector geometry, these geometrical parameters remain fixed. The only operating condition that is a relevant control parameter is then the momentum flux ratio (MFR), which is a combination of density ratio and velocity ratio. The relative influence of density and velocity ratios essentially dictates the transition to the local absolute instability (a precursor to the global mode) (Monkewitz Reference Monkewitz1988; Juniper & Candel Reference Juniper and Candel2003). Since the working fluids are the same throughout (water and air), the velocity ratio primarily defines the MFR. However, to be consistent with the literature on GCSC injectors, we prefer to choose MFR as a control parameter in the present study.

As the MFR is increased, the GCSC spray exhibits a number of non-pulsating flow regimes before transitioning to a self-oscillatory flow (Sivakumar & Kulkarni Reference Sivakumar and Kulkarni2011). To illustrate the qualitative difference between the non-pulsating and naturally pulsating sprays, we show time-resolved sequences of the spray morphologies in figure 2. The condition shown in figure 2(a) for $MFR=2.1$ is before the onset of self-pulsation. The near-orifice (local) region of the spray does not depict any temporal variation, and the spray breaks up with spatial evolution of the unstable shear layer. The spray formation at these lower MFRs is governed primarily by the convective instability. With an increase in the gas velocity, the shear instability (velocity gradient) at the liquid–gas interface becomes stronger and the pulsating flow starts appearing after a particular MFR. Consequently, figure 2(b), corresponding to $MFR=2.7$, shows the oscillatory axisymmetric (varicose) structure in the near orifice region (indicated by arrows), which gives rise to distinct pulses. This axisymmetric pulsating mode is similar to the mode observed in the case of a self-oscillating, low-density jet (Monkewitz et al. Reference Monkewitz, Lehmann, Barsikow and Bechert1989, Reference Monkewitz, Bechert, Barsikow and Lehmann1990; Hallberg & Strykowski Reference Hallberg and Strykowski2006; Hallberg et al. Reference Hallberg, Srinivasan, Gorse and Strykowski2007). The critical shear interaction between the gas and the annular liquid stream within confinement of the recess region is considered as the possible cause of the transition to the global instability. The change in the near-orifice spray structure is the manifestation of this transition in the instability mechanism. With this understanding, most of the studies were carried out by tracing the time-resolved measurement of spray width close to the injector exit (at $0.5D_o$).

Figure 2.

Sequence of spray images at 1 ms intervals for $u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$: (a) $MFR=2.1$ (non-pulsating);(b) $MFR=2.7$ (pulsating).

To demonstrate the transition to global instability, we further show the time series of the spray width fluctuations ($W'$) and its PSD at various MFRs in figure 3. From the time traces, one may notice that the lowest three MFRs (1.6–2.4) have a relatively smaller amplitude of fluctuations, indicating a convectively unstable spray. There is a sudden increase in the amplitude of fluctuations when $MFR \geq 2.7$, signifying the onset of global instability. As the momentum flux of the gas jet increases beyond a critical value, the axisymmetric instabilities of a gas jet (Crow & Champagne Reference Crow and Champagne1971) are crucial in the creation of the varicose structure of the instabilities (Sahoo & Gadgil Reference Sahoo and Gadgil2022). However, at very high MFR ($\geq$ 5.9), the gas momentum becomes high enough to cause the prompt breakup of the liquid film within the recess region, and dilutes the quality of the measurements.

Figure 3.

Dynamics of a GCSC injector for MFR ranging from non-pulsating to pulsating spray at ${u_l = 2.4\,{\rm m}\,{\rm s}^{-1}}$ measured at $0.5D_o$ from spray images: (a) time series, and (b) PSD. The $f_n$ appears after certain MFR, indicating bifurcation to a global mode.

For low MFR flows ($MFR=1.6\unicode{x2013}2.4$), the PSD of spray width fluctuations in figure 3(b) shows that there is a broadband noise in the spectrum at a frequency of around 400 Hz, without any clear peak. Such flows are globally stable. These flows may have a broadband spectrum or may not show any preferred mode, as observed here. They spatially amplify a disturbance wave at the extrinsic frequency of the perturbation. As the MFR is increased to 2.7, a sharp peak emerges in the PSD. This indicates that the global instability sets in the flow and it now behaves as a self-excited oscillator having its own intrinsic frequency. This transition is observed over a narrow range of MFR (Sahoo & Gadgil Reference Sahoo and Gadgil2021), and the frequency of the global mode is seen to vary linearly with the gas velocity (Matas et al. Reference Matas, Hong and Cartellier2014; Sahoo & Gadgil Reference Sahoo and Gadgil2021). Such bifurcation of states is classical to many hydrodynamic systems (Li & Juniper Reference Li and Juniper2013a; Shoji et al. Reference Shoji, Harris, Besnard, Schein and Karagozian2020). This transition happens when the MFR is sufficient that the small perturbations have the minimum inertia required to overcome the stabilizing effects of viscosity. The power for this self-oscillation is extracted continuously from the base flow through baroclinic torque established in the shear layer through opposing density and velocity gradients (Lesshafft & Huerre Reference Lesshafft and Huerre2007; Li & Juniper Reference Li and Juniper2013a). Along with the natural frequency, its harmonics may also be observed in the PSD of these self-pulsating sprays, which suggests that the fluctuations are not purely sinusoidal in time. At high MFRs ($MFR> 6$), the clear peak in PSD ceases to exist due to the early breakup of the bulk liquid, as explained above.

Based on the experimental evidence, we conclude that the present flow of gas–liquid shear layer with confinement and velocity deficit is likely to undergo a transition from convective to absolute instability under certain operating conditions, even though it is not proven rigorously here. The change in the PSD further substantiates the existence of a globally unstable mode with intrinsic frequency over a range of MFR values. This flow configuration thus represents a self-oscillatory system having startling similarities with other widely studied systems, such as low-density jets (Monkewitz & Sohn Reference Monkewitz and Sohn1988; Sreenivasan et al. Reference Sreenivasan, Raghu and Kyle1989; Li & Juniper Reference Li and Juniper2013a) and also transverse jets (Megerian et al. Reference Megerian, Davitian, Alves and Karagozian2007; Getsinger et al. Reference Getsinger, Gevorkyan, Smith and Karagozian2014).

It is important that in the present studies, the operating conditions should be such that they support globally unstable modes. At extremely high MFRs (> 6), the current method cannot reveal the detailed insight of the self-oscillatory nature of the spray, and also the other modes like swirling mode start interfering with the axisymmetric mode (Sahoo & Gadgil Reference Sahoo and Gadgil2022). Similarly, for higher liquid flow rates, the gas flow rates must be proportionately higher for pulsation to occur. In such a scenario, the acoustic forcing may be insufficient with respect to gas inertia to realize the lock-in phenomenon. Based on these criteria, the operating conditions are finalized for the present investigations and shown in table 1.

3.2. Spray response to the forced gas jet

We first demonstrate the distinct forced response of globally stable and globally unstable spray. An axisymmetric sinusoidal forcing is provided to the gas jet, which imparts perturbations to the shear layer. Here, we perform the experiments by sweeping frequencies at constant forcing amplitude to assess different response characteristics. Except for the results shown in § 3.3.2, the gas jet is excited at 117 dB in the rest of this study. This amplitude of forcing was found to be sufficient to alter the spray behaviour at various operating conditions.

The near-orifice spray structures shown in figure 4 show that the effect of acoustic forcing is more perceptible in the case of a non-pulsating spray at $MFR=2.1$. The formation of axisymmetric pulses or varicose mode of instability is clearly evident in figure 4(a) as a consequence of external forcing. This is definitely distinct in comparison with the spray seen in figure 2(a). We also note that the fluctuation amplitude in width in the case of an acoustically excited sheet is higher than in the case of an unforced sheet for a non-pulsating spray (not shown here). This effect may be attributed to the spatial amplification of the perturbation imposed by the acoustic forcing. For a pulsating spray (figure 4b), however, as the axisymmetric pulsating spray is already present as a global mode, the qualitative effect of the forcing is not noticeable from the spray images for $MFR=2.7$. Further, since the flow has an intrinsic frequency, the time trace of the spray width shows frequency modulations instead of amplification of perturbations.

Figure 4.

Sequence of spray images of a flow from the GCSC injector where gas is acoustically excited at frequency 500 Hz and 117 dB. The time interval between subsequent images is 1 ms for $u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$:(a) $MFR=2.1$ (non-pulsating); (b) $MFR=2.7$ (pulsating).

To evaluate different spectral characteristics in the presence of forcing, we plot the PSD of the temporal spray-width variation for the non-pulsating spray ($MFR = 2.1$) and the self-pulsating spray ($MFR=2.7$) by varying the forcing frequency from 400 to 1000 Hz; this is shown in figure 5. The spectrum of the forced condition (black) is superimposed on that of the unexcited condition (red) for each forcing frequency. Figure 5(a) shows that when the acoustic forcing is applied to the non-pulsating spray at a particular frequency, only the forcing frequency ($f_f$) appears in the spectrum. The harmonics of the forcing frequency may also be noticed. Although the forced response of the non-pulsating spray resembles that of the naturally pulsating spray (in the absence of forcing), there is a significant difference. The self-pulsating spray exhibits the intrinsic frequency that is unique for a given flow condition. However, the non-pulsating spray operating at a constant $MFR=2.1$ shows pulsating behaviour at a frequency equal to the forcing frequency. As the forcing frequency changes, the dominant frequency in the spectrum changes accordingly. The flow thus acts as an amplifier of instability with imposed frequency. Convectively unstable flows have been reported to act as an amplifier of external perturbations in a similar way (Sreenivasan et al. Reference Sreenivasan, Raghu and Kyle1989). It may also be observed from figure 5(a) that the energy content in the dominant frequency increases until 600 Hz, and reduces as the frequency of excitation increases beyond 600 Hz. A similar observation was also reported in our previous work (Sahoo et al. Reference Sahoo, Chauhan, Gadgil, Kumar and Kumar2021).

Figure 5.

The superposition of unforced spray spectra and frequency response of spray due to forcing of the gas jet at frequencies 400–1000 Hz from bottom to top for $u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$: (a) $MFR=2.1$ (before self-pulsation); (b) $MFR=2.7$ (during self-pulsation, $f_n = 410$ Hz). Before self-pulsation, only $f_f$ is observed in the spectrum, and in a self-pulsating spray, multiple frequency peaks, including both $f_n$ and $f_f$, are observed.

The spray at $MFR=2.7$, shown in figure 5(b), is an absolutely unstable flow with a certain natural frequency of self-oscillation ($f_n$). When the forcing is applied to such a flow, the spectrum shows multiple frequency peaks corresponding to the natural mode, the forcing frequency and their harmonics. It is indicative of the fact that multiple periodic modes can coexist in a forced self-oscillating system. Further, it should be noted in the context of figure 5(b) that the natural mode is always a more dominant one, and the energy content in the forcing frequency decreases as it goes away from the natural mode frequency.

When the forcing frequency and the natural frequency are incommensurate, the nonlinear interactions of both of the frequencies can be evidenced from the spectrum by noting several peaks that are present. Significant among those are the frequencies that follow the linear combination of both of these frequencies ($|{p}f_f \pm {q}f_n|$, where $p$ and $q$ are integers). Craik (Reference Craik1988) has called this a consequence of wave–triad interactions. Here, we show this for two cases in figure 6. Some easily identifiable frequencies are marked at their corresponding peaks in the spectrum. Note that the beat frequency $|f_f - f_n|$ is one of the very significant frequencies less than both natural and forcing frequencies. This establishes the quasi-periodic nature of this self-oscillatory system. It is also observed that the frequencies apart from the natural and forcing frequencies are more significant when $f_n < f_f$. A skewed response of the forcing self-oscillatory systems about the natural frequency, along with the quasi-periodic behaviour, is also observed in low-density jets (Li & Juniper Reference Li and Juniper2013a) and transverse jets (Shoji et al. Reference Shoji, Harris, Besnard, Schein and Karagozian2020). We emphasize the fact that the shadowgraphic measurements of this self-oscillatory multiphase system have a lot of noise due to the high Reynolds number of the gas jet ($\approx 2.8\times 10^4\unicode{x2013}5.6\times 10^4$). As a result, not many peaks are easily identifiable in the spectrum. However, the universal behaviour of forcing of self-oscillation systems is clearly demonstrated in liquid–gas multiphase flows through this study. Further, we focus on the frequencies close to the self-pulsation frequencies at different liquid and gas flow conditions.

Figure 6.

Quasi-periodicity of forced self-oscillating spray: $f_n$ and its harmonics, $f_f$, and their linear combinations, observed for $u_l = 2.8\,{\rm m}\,{\rm s}^{-1}$, $MFR=3.8$ (self-pulsation with $f_n = 590$ Hz) excited at 117 dB with (a) $f_f = 540$ Hz, (b) $f_f = 660$ Hz.

3.3. Phenomenon of frequency lock-in

3.3.1. Changing forcing frequencies near to the self-pulsation frequency

We examine the lock-in phenomenon of the pulsating spray by forcing the gas jet at frequencies close to the natural frequency of self-pulsation. The frequencies on either side of the natural mode are employed. The frequency response at these forcing conditions is shown in figure 7 for $MFR=4.2$ at $u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$ (case 3). The unforced self-pulsation condition shown in the bottommost PSD of figure 7 depicts a clear peak at 525 Hz along with its harmonics. The excitation frequency is changed approximately from $(f_n - 50)$ Hz to $(f_n + 50)$ Hz at intervals of 10 Hz such that $|1-f_f/f_n| < 0.09$. The forcing amplitude for all frequencies is maintained constant at 117 dB, for which the range of the frequencies responsible for lock-in is estimated. The notable observations related to the frequency lock-in are as follows.

1. (i) When $|f_f - f_n| > 20$ Hz, the natural mode $f_n$ shifts slightly towards $f_f$, called $f_{n}^*$. The spray responds at both frequencies $f_f$ as well as $f_{n}^*$. This is the frequency pulling behaviour of this self-oscillatory system. A similar observation is reported for a low-density jet by Li & Juniper (Reference Li and Juniper2013a).

2. (ii) When $|f_f - f_n| > 20$ Hz, and the forcing frequency is tending towards the natural frequency, the amplitude of the peak corresponding to $f_n$ reduces as $f_f$ comes closer to $f_n$ (as observed for $f_f = 480\unicode{x2013}500$ Hz and 540–560 Hz). This is a consequence of the fact that the acoustic mode pulls the power from the self-oscillatory mode when both the frequencies are close enough. As $f_f$ and $f_n$ become closer, the envelope around them widens, and the longer tail of the envelope shifts from right to left as the forcing frequency surpasses the natural frequency.

3. (iii) When $|f_f - f_n| < 20$ Hz ($f_f \approx 510\unicode{x2013}540$ Hz), merging of $f_n$ and $f_f$ is observed. This synchronization of frequencies is called lock-in (Li & Juniper Reference Li and Juniper2013a,Reference Li and Juniperb; Kashinath, Li & Juniper Reference Kashinath, Li and Juniper2018; Shoji et al. Reference Shoji, Harris, Besnard, Schein and Karagozian2020). During lock-in, the combined power of intrinsic oscillations and forcing results in the increase in amplitude of peaks in the PSD.

4. (iv) When the self-oscillating spray locks into the forcing, its power spectrum is nearly identical to an unforced spray. The PSD marks the existence of only forcing frequency and its harmonics, without any trace of the natural mode. This observation has important implications in the context of high-energy density combustors explained later, in § 3.5. To our knowledge, the lock-in behaviour has not been reported earlier in the case of a multiphase self-oscillatory system. It should be emphasized here that the frequency envelope where the spray locks into the forcing is $\pm$20 Hz for the forcing amplitude 117 dB. Li & Juniper (Reference Li and Juniper2013a) have shown in the case of a low-density jet that the range of frequency envelope widens for increasing values of forcing amplitude. One can predict a similar result in the present self-oscillatory system as the lock-in phenomenon here is analogous to that of a low-density jet.

Figure 7.

Superposition of the PSD of spray from a forced GCSC injector at forcing amplitude 117 dB and the forcing frequency near to the self-pulsation frequency ($f_n = 525$ Hz) for $u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$, $MFR=4.2$, on the unforced spray. The bottom subplot is without forcing. The forcing frequency (given to the right) is increased from bottom to top at 10 Hz increments. At this forcing amplitude, the lock-in is between 510 and 540 Hz, approximately, and other subplots demonstrate the quasi-periodicity of the forced conditions ($|f_n - f_f| > 20$ Hz).

Similar plots demonstrating the quasi-periodicity and lock-in are also shown for $u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$, $MFR=2.7$ (case 2, $f_n = 410$ Hz), $u_l = 2.8\,{\rm m}\,{\rm s}^{-1}$, $MFR=3.8$ (case 4, $f_n = 590$ Hz), and $u_l = 2.8\,{\rm m}\,{\rm s}^{-1}$, $MFR=5.8$ (case 5, $f_n = 705$ Hz) in figure 8 for a few forcing frequencies near to the natural frequency. Lock-in is observed between 390 and 430 Hz for case 2 (figure 8a), between 570 and 610 Hz for case 4 (figure 8b), and between 690 and 720 Hz for case 5 (figure 8c). Other characteristic features are the same as observed in case 3 (as shown in figure 7). Note that in case 3, the perturbation amplitude ${u'}/{u_g}$ is less than the perturbation amplitude in case 2, and more than the perturbation amplitude in cases 4 and 5. Therefore, the peaks corresponding to the excitation frequency just beyond lock-in are observed to have more distinct peaks of $f_f$ in case 2 compared to cases 4 and 5.

Figure 8.

Superposition of PSDs of the spray from a forced GCSC injector at 117 dB near to the self-pulsation frequency and unforced spray. The forcing frequency (given to the right) is increased from bottom to top at 10 Hz increments: (a) $u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$, $MFR=2.7$, $f_n = 410$; (b) $u_l = 2.8\,{\rm m}\,{\rm s}^{-1}$, $MFR=3.8$, $f_n = 590$; (c) $u_l = 2.8\,{\rm m}\,{\rm s}^{-1}$, $MFR=5.8$, $f_n = 705$. Lock-in is observed when $f_f$ is closer to $f_n$ ($|f_n - f_f| < 20$ Hz), and quasi-periodicity is observed when $f_f$ is away from $f_n$ ($|f_n - f_f| > 20$ Hz).

3.3.2. Changing forcing amplitude near to self-pulsation frequency

To assess the sensitivity of the lock-in phenomenon, we force the gas jet with external perturbation at different forcing amplitudes (dB levels). The flow condition chosen for demonstration is of case 3 ($u_l = 2.4\,{\rm m}\,{\rm s}^{-1}$, $MFR=4.2$), which has a global mode at 525 Hz as shown in figure 9(i). Initially, we look into the case when the forcing frequency is almost equal to the pulsation frequency ($f_f = 520$ Hz). It is shown in the previous subsection that the jet locks into the forcing at this frequency for the perturbation amplitude 117 dB. Here, the perturbation amplitude is varied from 110 to 122 dB. It may be observed from figure 9 that the spray responds only to the forcing frequency represented by the sharp peak at $f_f$ even at the lowest forcing amplitude, 110 dB. As the forcing amplitude is increased from figure 9(ii,iii), the spectral peak (at $f_f$) gets sharper, indicating a stronger lock-in of the flow at higher forcing amplitudes. This suggests that the spray locks into the forcing even at the lowest forcing amplitude (covered in the present work) if $f_f$ is close enough to $f_n$.

Figure 9.

The PSD of a self-pulsating spray (case 3) from a GCSC injector by forcing the gas jet near the self-pulsation frequency ($f_n = 525$ Hz) with increasing forcing amplitude for $f_f = 520$ Hz. Forcing magnitudes are given in the right bottom corner. Panel (i) is without forcing. A single frequency with increasing amplitude is observed.

To complement the above understanding on lock-in, we force the gas jet of the self-pulsating spray with increasing magnitudes of forcing amplitudes (110–122 dB) at constant frequencies that are slightly away from the natural frequency. For instance, figure 10(a) shows the frequency response of case 3 when the forcing frequency is $(f_n - 25)$ Hz. The response for the forcing frequency of $(f_n + 25)$ Hz is shown in figure 10(b) similarly. The following important observations can be made.

1. (i) When $f_f < f_n$ or $f_f > f_n$, the peak amplitude of $f_n$ decreases with increase in the forcing amplitude (for reference, the power content values are indicated alongside the peaks corresponding to $f_n$ in figure 10). The spray exhibits quasi-periodicity until the forcing amplitude of 117 dB. However, the dominant spectral peak at $f_n$ vanishes, and a single peak at $f_f$ is seen in the PSD, indicating lock-in when the forcing amplitude increases to 122 dB. It is already demonstrated in § 3.3.1 that the spray for case 3 does not lock into the forcing at 117 dB when $|f_f - f_n| > 20$ Hz; however, lock-in may be observed clearly for $f_f = 500$ or 550 Hz when a higher forcing amplitude 122 dB is applied. This observation reiterates the fact that the critical forcing amplitude responsible for lock-in increases as the forcing frequency deviates more from the natural mode. Although this is not verified over more forcing conditions due to the limitations of the acoustic driver unit, the observation is in line with the forced response of the other self-oscillatory systems (Li & Juniper Reference Li and Juniper2013a; Shoji et al. Reference Shoji, Harris, Besnard, Schein and Karagozian2020).

2. (ii) There are signs of a subtle asymmetry about $f_n$, though not obvious from figure 10, in the nature of the response of the spray to the external forcing. The spectral energy on the peak observed in figure 10(b(v)) is more than that in figure 10(a(v)) at the lock-in condition at 122 dB. This indicates a relatively strong lock-in when $f_f > f_n$. Also, at 110 dB, the spray is not seen to respond at $f_f = 500$ Hz (see figure 10a(ii)). A similar asymmetry (Shoji et al. Reference Shoji, Harris, Besnard, Schein and Karagozian2020) and a contrasting one (Li & Juniper Reference Li and Juniper2013a) have been reported earlier for other self-oscillatory systems. Based on the observed similarities between the spectral characteristics of the pulsating spray and the other self-oscillatory systems, it may be hypothesized that this spray locks in readily when $f_f > f_n$.

Figure 10.

The PSD of the self-pulsating spray from a GCSC injector by forcing the gas jet at a frequency far from the self-pulsation frequency ($f_n = 525$ Hz) with increasing forcing amplitude. Each subplot (i) is without forcing. Plots are for (a) $f_f = 500$ Hz, (b) $f_f = 550$ Hz. The amplitude values are mentioned in the right bottom corner. A value shown in parentheses beside $f_n$ is the magnitude of the peak. Lock-in is observed in subplot (v) of both cases.

3.4. Model and nonlinear time series analysis

The van der Pol (VDP) oscillator model is a well-known second-order differential equation that predicts many features of nonlinear dynamics associated with self-excited systems. In fluid dynamics, this model has been used to study the forced response of many hydrodynamically self-excited flows (Li & Juniper Reference Li and Juniper2013a,Reference Li and Juniperb; Kashinath et al. Reference Kashinath, Li and Juniper2018; Shoji et al. Reference Shoji, Harris, Besnard, Schein and Karagozian2020). Our experimental results establish the self-oscillatory nature of the GCSC spray and its response to the sinusoidal forcing depicting various features analogous to the other self-excited flows. We therefore extend the use of the VDP oscillator model to examine the response of the self-pulsating spray against the experimental observations. We used an external acoustic forcing sinusoidal in time to the spray with natural mode during the experiments. Hence the VDP oscillator model can be represented as

(3.1)\begin{equation} \ddot{z}-\epsilon(1-z^{2}) \dot{z}+\omega_{n}^{2} z=B \sin (\omega_{f} t). \end{equation}

Here, $z$ represents the observable that changes dynamically, having a natural mode at angular frequency $\omega _{n}$. The forcing is represented by the amplitude $B$ and angular frequency $\omega _{f}$. The feedback parameter involved in the nonlinear term is $\epsilon$, which essentially dictates the nonlinear interactions. When $|z| < 1$, the system tends to get linearly self-excited and amplify, and when $|z| > 1$, nonlinear self-limitation occurs and oscillations decay.

We use the natural frequency of a particular case along with the forcing frequencies to replicate the observations of the experiments through the model. The primary requirement of implementation of the VDP model is the correct estimation of the feedback parameter ($\epsilon$) and the forcing parameter ($B$). For $\epsilon$, we follow the understanding provided by Li & Juniper (Reference Li and Juniper2013a). The value of $\epsilon$ is chosen to be 40 based on the matching of the lock-in phenomenon extracted from this model to the experimentally observed lock-in at $f_f = 550$ Hz for case 3 at 120 dB of forcing amplitude. It should be noted that although the value of $\epsilon$ decides the degree of nonlinearity, it affects only the nature of bifurcation observed at higher-frequency deviations, without altering the essential features of lock-in and quasi-periodicity (Li & Juniper Reference Li and Juniper2013a). As will be demonstrated, the value $\epsilon = 40$ fairly predicts the important dynamical characteristics of the system. The value of $B$ is set at $2.8\times 10^{6}$ for forcing amplitude 120 dB. The rest of the values of $B$, corresponding to other forcing amplitudes, are determined by taking a value proportional to ${u'}/{u_g}$ with respect to the value at 120 dB. This method is repeated for other self-oscillating conditions. Equation (3.1) is solved numerically in Matlab using an ODE solver.

The important outcomes of the application of the VDP oscillator model to various cases of self-pulsating spray with external forcing are discussed here. Figure 11 shows the forced frequency response obtained from the VDP oscillator model for cases 2, 3 and 4. We use constant values of $\epsilon$ ($= 40$) and $B$ ($= 2 \times 10^6$), corresponding to 117 dB forcing amplitude, to replicate the effect of different forcing frequencies. The PSDs of the time series (in figure 11) obtained by solving (3.1) for various values of $f_f$ correspond to the cases shown in figures 8(a), 7 and 8(b), respectively. The lowermost PSD in all subplots of figure 11 represents the unforced condition ($B = 0$), and the presence of a global mode may be confirmed by the strong peak at $f_n$. In particular, we discuss other features with respect to figure 11(b). The spray is seen to lock into the forcing completely for rows (v)–(viii) where the PSD shows a clear peak at $f_f$ without any trace of the natural mode. The frequencies over which the lock-in is reproduced are same as those in figure 7. At other forcing frequencies, the spectrum shows $f_n$, $f_f$ and their linear combinations. This is the state of quasi-periodicity. It is to be noted here that the third harmonics are observed here as the VDP equation has a third-order nonlinearity.

Figure 11.

The PSD obtained from the VDP model taking $\epsilon = 40$, $B = 2\times 10^{6}$, which corresponds to 117 dB in the experiments for three different pulsating conditions forced at different frequencies, shown alongside each condition: (a) case 2 ($f_n = 410$ Hz), (b) case 3 ($f_n = 525$ Hz), (c) case 4 ($f_n = 590$ Hz). These spectra correspond to figures 8(a), 7 and 8(b), respectively, obtained from the experimental data. Subplot (i) in each case is without forcing. Lock-in may be seen approximately from subplots (v)–(viii) for all cases. The rest of the subplots demonstrate the quasi-periodicity.

Further, we also plot the spectra for case 3 with changing $B$ value in order to emulate the effect of forcing amplitude. It is evident that at $f_n = f_f$, the amplitude of the peak in the spectrum increases with increase in $B$ (not shown here). The plots for $f_f < f_n$ and $f_f > f_n$ are shown in figures 12(a) and 12(b), respectively. Note that the lock-in of frequencies is observed beyond $B = 2\times 10^{6}$, which corresponds to 117 dB of the acoustic forcing. This can be compared with the spectrum shown in figure 10. However, the asymmetry about $f_n$ observed in experimental measurements is not clearly observed from the model predictions. It may be a characteristic feature of the VDP model as the weak asymmetry was also observed in the case of low-density jets (Li & Juniper Reference Li and Juniper2013a) modelled as a VDP oscillator. Thus the VDP oscillator model not only shows its capability to predict the dynamics of lock-in and quasi-periodicity, but also extends its utility to a multiphase system in the class of self-excited flows. The typical bifurcations observed when a forced self-excited system transitions from self-oscillations to quasi-periodicity to lock-in can also be explored with tools like the Poincaré map. The noise in experimental data limits the construction of the Poincaré map. Hence we demonstrate the existence of these bifurcations with the help of time series data generated from the VDP model in the Appendix.

Figure 12.

The PSD obtained from the VDP oscillator by changing $B$ values (given above each plot) that correspond to different forcing amplitudes at $f_n = 525$ Hz ($\epsilon = 40$). These cases correspond to the experimental cases shown in figure 10: (a) $f_f$ (500 Hz) $< f_n$ (corresponds to figure 10a); (b) $f_f$ (550 Hz) $> f_n$ (corresponds to figure 10b). Each subplot (i) is without forcing ($B = 0$). Subplots (ii)–(v) are the quasi-periodic conditions, and lock-in corresponds to subplot (v) of both cases.