2.1. Jet nozzle and velocity measurements

The nozzle has a round outlet whose exit diameter is $D = 6$ mm. Its internal walls are defined by a fifth-order polynomial with a $34:1$ area contraction between the settling chamber and the nozzle outlet. The streamwise, transverse (radial) and cross-stream coordinates are denoted by $x$, $y$ and $z$, respectively, as shown in figure 1. The nozzle outlet can be moved along the transverse ($y$) direction, enabling the jet position to be varied relative to the standing acoustic waveform in the enclosure. However, the nozzle outlet always protrudes $1.67D$ into the enclosure itself so as to minimize wall effects. Helium gas is discharged from the nozzle into the enclosure air (at temperature 293 K), creating a jet whose density ratio is $S \equiv \rho _j / \rho _\infty = 0.14$, well below the upper limit for global instability (Monkewitz et al. Reference Monkewitz, Bechert, Barsikow and Lehmann1990; Kyle & Sreenivasan Reference Kyle and Sreenivasan1993). The helium is supplied from a compressed gas cylinder and is metered with a mass flow controller (Alicat MCR-500). Figure 2(a) shows radial profiles of the normalized time-averaged streamwise velocity, $\bar {u}/\bar {u}_{max}$, for a range of Reynolds numbers, $700 \leqslant Re \equiv U_j \rho _j D/\mu _j \leqslant 3990$, where $U_j$ is the jet centreline velocity, and $\mu _j$ is the jet dynamic viscosity. The data are acquired with a hot-wire anemometer positioned just beyond the nozzle outlet ($x/D \approx 0.1$); the measurement procedures are described further below. It can be seen in figure 2(a) that the velocity profile is top-hat with thin shear layers. This aids the detection of global instability by keeping the critical $Re$ low, thus limiting the ability of convective modes to amplify inherent disturbances (Hallberg et al. Reference Hallberg, Srinivasan, Gorse and Strykowski2007). Figure 2(b) shows that the velocity fluctuations in the jet core are weak ($u^{\prime }_{rms}/\bar {u} < 0.35\,\%$), while figure 2(c) shows that the transverse curvature – defined as $D$ normalized by the initial momentum thickness $\theta _0$ – scales linearly with $Re^{1/2}$. This linear scaling indicates that the initial shear layers are laminar, which further aids the detection of global instability.

Figure 2. Characterization of the jet base flow: (a) normalized time-averaged streamwise velocity, and (b) its local fluctuations, both as functions of the radial position. Also shown is (c) the transverse curvature as a function of the square root of the Reynolds number. The data are acquired with a hot-wire anemometer positioned just beyond the nozzle outlet ($x/D \approx 0.1$).

The hydrodynamic response of the jet to acoustic perturbations is characterized with hot-wire anemometry (HWA) and particle image velocimetry (PIV). HWA is used when long time traces are needed to identify the topology of the jet attractor in its reconstructed phase space, whereas PIV is used when spatiotemporal information is needed to better understand the flow symmetry. The HWA system consists of a constant-temperature bridge operated at an overheat ratio of 1.8 and connected to a single-normal probe (Dantec 55P11) equipped with a platinum-plated tungsten wire (diameter $5\,\mathrm {\mu }{\rm m}$, length 1.25 mm). Its frequency response is around $10^4$ Hz, well above the natural global frequency of the jet (§ 4: $f_n \sim 10^3$ Hz). Previous studies have shown that when an open shear flow becomes globally unstable, a large section of its domain becomes temporally synchronized such that its dynamics can be resolved with only local measurements taken at a single spatial location (Huerre & Monkewitz Reference Huerre and Monkewitz1990; Broze & Hussain Reference Broze and Hussain1994). In both the unforced (§ 4) and forced (§ 5) jet experiments, the HWA probe is positioned on the jet centreline, $1.5D$ downstream of the nozzle outlet, i.e. at $(x/D,y/D)=(1.5,0)$, with uncertainty $\pm 0.017D$ in each axis. This sampling location is chosen because it is far enough downstream for a nonlinear global mode to emerge and interact with the applied forcing (Hallberg & Strykowski Reference Hallberg and Strykowski2008), but it is not so far downstream as to be outside the potential core, where fluctuations in helium concentration could contaminate the HWA data (Li & Juniper Reference Li and Juniper2013a,Reference Li and Juniperc). Furthermore, sampling the HWA data in the potential core is preferred because it provides direct measurements of the wavemaker (Lesshafft et al. Reference Lesshafft, Huerre, Sagaut and Terracol2006; Qadri, Chandler & Juniper Reference Qadri, Chandler and Juniper2018), which is the flow region that prescribes the intrinsic dynamics of the global mode (Huerre & Monkewitz Reference Huerre and Monkewitz1990; Chomaz Reference Chomaz2005). The HWA probe wire itself is oriented parallel to the $z$ axis (see figure 1) so that it senses simultaneously both streamwise ($x$) and transverse ($y$) velocity components. This probe orientation enables both axisymmetric ($m=0$) and transverse flapping ($m=\pm 1$) oscillations in the jet to be resolved. For each test run, the HWA voltage is digitized at 25 600 Hz for 8 s on a 16-bit data acquisition system, generating a time trace of the local $x$–$y$ velocity, $\gamma (t)$. Here the conventional symbol for velocity, $u(t)$, is not used because it is reserved for the $x$ velocity component measured with PIV (see below) and because the HWA probe is sensitive to both the $x$ and $y$ velocity components. The $\gamma (t)$ data are examined via: (i) the power spectral density (PSD), as computed with the algorithm of Welch (Reference Welch1967); (ii) the instantaneous phase difference, as computed with the Hilbert transform; and (iii) the phase space, as reconstructed via nonlinear time series analysis based on the time-delay embedding theorem of Takens (Reference Takens1981) (Appendix A).

Planar stereoscopic time-resolved PIV measurements are performed to quantify the spatiotemporal evolution of the jet velocity under different forcing conditions. The PIV system consists of a high-speed dual-cavity Nd:YLF laser (Litron LDY303HE), a set of sheet-forming optics (Thorlabs), and two high-speed cameras (Photron FASTCAM SA1.1). Both cameras are fitted with Scheimpflug adapters and 180 mm lenses, with their axes aligned at 12.5$^\circ$ to the measurement plane normal. The measurement plane itself is illuminated with a laser sheet (thickness 1 mm) positioned in the $x$–$y$ plane. Oil-droplet seeding is introduced into both the jet flow and the enclosure air via a Laskin nozzle. Image pairs are recorded at 5442 Hz, which is more than five times $f_n$; the image separation time is $\delta t = 12\,\mathrm {\mu }{\rm s}$. The image resolution is $768$ (height) $\times$ $512$ (width) pixels at a depth of 12 bits, spread over a field-of-view measuring $4.44D \times 2.96D$. The image pairs are processed with the software DaVis 8.2.2 from LaVision. First, the image pairs are preprocessed with two filters: (i) a sliding background subtraction filter of width 32 pixels; and (ii) a particle intensity normalization filter of width 8 pixels. Next, the image pairs are processed with a multipass cross-correlation algorithm. The initial interrogation area contains $64 \times 64$ pixels, with a 50 % window overlap for the first two passes. The final interrogation area contains $32 \times 32$ pixels, which corresponds to $1.11 \times 1.11$ mm$^2$ in physical space; the window overlap for the last pass is 75 %, producing a final vector spacing of $0.046D$. The velocity components in the streamwise ($x$), transverse ($y$) and cross-stream ($z$) directions are denoted by $u$, $v$ and $w$, respectively, with the magnitude of the total velocity vector given by $V \equiv \sqrt {u^2 + v^2 + w^2}$. In each test run, 4000 image pairs are recorded, resulting in the same number of instantaneous vector fields. This represents over 650 cycles of the natural jet oscillation at $f_n$, which is sufficient for statistical convergence.

2.2. Acoustic pressure field

As mentioned earlier, the jet discharges into a rectangular enclosure, in which planar acoustic standing waves are established to perturb the jet. These waves arise from resonance between the natural acoustic modes of the enclosure and the pressure oscillations generated by a pair of loudspeakers (Monacor KU-516) mounted at opposite ends of the enclosure (figure 1). Each loudspeaker is fitted with its own acoustic resonance tube and is driven by a sinusoidal voltage signal sent from a function generator (Aim–TTi TGA1244) via a power amplifier (Crown CE1000). The frequency of the signal (i.e. the forcing frequency $f_f$) is varied around the natural global frequency of the jet ($0.8 \leq f_f/f_n \leq 1.2$) in order to explore the $1:1$ Arnold tongue and its synchronization phenomena. The enclosure has width $W=0.22$ m ($z$ direction), height $H=0.59$ m ($x$ direction), and variable length $L=0.96$–$1.35$ m ($y$ direction) so that the acoustic resonance frequency can be tuned carefully to match $f_f$. The ratio of the enclosure width to the jet diameter ($W/D=36.7$) is large enough to keep confinement effects negligible in the near field, $0 \leqslant x/D \leqslant 4$. At every value of $f_f$, the enclosure contains three full standing waves. The wavelength of these transverse acoustic waves is much larger than the jet diameter ($\lambda /D > 53.5$), implying that the jet can be regarded as acoustically compact in the near field. The acoustic pressure fluctuations ($p^\prime$) in the enclosure are measured with four microphones (B&K Type 4939-A-011: $\pm 2.07 \times 10^{-5}$ Pa uncertainty, $4.13$ mV Pa$^{-1}$ sensitivity) mounted flush along the transverse wall (figure 1). The multiple microphone method is used to reconstruct the acoustic field in the enclosure from estimates of the incident ($p_i$) and reflected ($p_r$) planar acoustic waves (Seybert & Ross Reference Seybert and Ross1977; Jang & Ih. Reference Jang and Ih1998). The stationarity of the acoustic mode is verified by ensuring that the standing-wave ratio is sufficiently close to zero: $-0.01 < (|p_i|-|p_r|)/(|p_i|+|p_r|) < 0.01$. The jet is positioned at five discrete locations along the acoustic standing wave, producing a variety of forcing conditions: pure axial forcing (position $\mathbb {A}$ in figure 3, a pressure antinode), pure transverse forcing (position $\mathbb {E}$, a pressure node), and combinations of axial and transverse forcing (positions $\mathbb {B}$, $\mathbb {C}$ and $\mathbb {D}$, between a pressure antinode and a pressure node).

Figure 3. Decomposition of the acoustic pressure field in the jet enclosure. The modal amplitude is shown as a function of $m$ at five different locations in a planar acoustic standing wave: a pressure antinode (position $\mathbb {A}$), a pressure node (position $\mathbb {E}$), and between a pressure antinode and node (positions $\mathbb {B}$, $\mathbb {C}$ and $\mathbb {D}$). At the pressure node (position $\mathbb {E}$), the superposition of the $m=\pm 1$ helical modes leads to a transverse flapping motion. In the experiments, three full standing waves are present in the jet enclosure, but for illustration purposes, only one is shown here.

As discussed in § 1.3, placing a jet in an acoustic standing wave can preferentially excite certain hydrodynamic modes, depending on the exact nozzle location. Following O'Connor et al. (Reference O'Connor, Acharya and Lieuwen2015), we use the Jacobi–Anger expansion to decompose the planar acoustic perturbations in the jet enclosure into a superposition of nozzle-centric modes:

(2.1)\begin{align} \textrm{e}^{\textrm{i}ky} + \textrm{e}^{-\textrm{i}ky} &= \sum_{m ={-}\infty}^{\infty} C_m(kr)\, \textrm{e}^{\textrm{i} m \theta} +\sum_{m ={-}\infty}^{\infty} C_m(kr)\,\textrm{e}^{-\textrm{i} m \theta} \end{align}

(2.2)\begin{align} &= \sum_{m ={-}\infty}^{\infty} \textrm{i}^{m}\,{\rm J}_m(kr)\,\textrm{e}^{\textrm{i} m \theta} +\sum_{m ={-}\infty}^{\infty} (-\textrm{i})^{m}\,{\rm J}_m(kr)\,\textrm{e}^{-\textrm{i} m \theta}, \end{align}

where $k$ and $m$ are, respectively, the wavenumbers in the transverse ($y$) and azimuthal ($\theta$) directions, $r$ is the nozzle-centric radial coordinate, $C_m$ is the modal amplitude, and ${\rm J}_m$ is the Bessel function of the first kind. For an acoustically compact nozzle ($kr \ll 1$), a Taylor series expansion to leading order gives ${\rm J}_m(kr) = (kr)^m/2^m{m!}$. Figure 3 shows that the acoustic disturbances generated at the pressure antinode (position $\mathbb {A}$) are dominated by the $m=0$ mode, with no contribution from the $m=\pm 1$ modes (see inset), indicating pure axial forcing. Moving away from the pressure antinode (i.e. to positions $\mathbb {B} \rightarrow \mathbb {D}$), the $m=0$ mode gradually weakens, while the $m=\pm 1$ modes gradually strengthen. These variations, however, are not symmetric in space: the $m=0$ mode remains significantly stronger than the $m=\pm 1$ modes, even at positions $\mathbb {C}$ and $\mathbb {D}$. It is not until arriving fully at the pressure node (position $\mathbb {E}$) that the acoustic disturbances become dominated by the $m=\pm 1$ modes, with no contribution from the $m=0$ mode, indicating pure transverse forcing. Crucially, for a given amplitude of the acoustic standing wave, the axial forcing ($m=0$) produced at the pressure antinode (position $\mathbb {A}$) is more than an order of magnitude stronger than the transverse forcing ($m=\pm 1$) produced at the pressure node (position $\mathbb {E}$). Therefore, to provide representative comparisons between axial and transverse forcing, we adjust the amplitude of the acoustic standing wave such that the modal amplitude is kept constant across all five nozzle positions.

5.1. Synchronization maps

To provide an overview of the amplitude response and lock-in boundaries, we show in figure 5 the synchronization maps for the five jet positions examined in figure 3. These maps contain surfaces of $\eta _\gamma$ in a parameter space defined by $a$ and $f_f/f_n$. The red and blue regions correspond, respectively, to amplification and attenuation of the self-excited jet oscillations. For each value of $f_f/f_n$, the minimum value of $a$ required for lock-in, $a_{loc}$, is indicated by circular markers. As noted in § 1.2, the region above these markers is known as the $1:1$ Arnold tongue (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). The filled (black) markers denote a saddle-node bifurcation along the phase-locking route to lock-in, whereas the hollow (white) markers denote an inverse Neimark–Sacker bifurcation along the suppression route to lock-in; these two routes will be examined in §§ 5.3.1 and 5.3.2. In this study, we define lock-in as a completely synchronous state in which the instantaneous phase difference between the forcing signal and the forced self-excited system is constant in time; this state is known as phase locking in the synchronization literature and can be reached via either the phase-locking route or the suppression route (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). Prior to lock-in ($a < a_{loc}$), the jet is either partially synchronized (phase trapping) or desynchronized (phase drifting), depending on its proximity to the Arnold tongue; these intermediate quasiperiodic states will also be examined in §§ 5.3.1 and 5.3.2.

Figure 5. Synchronization maps from the forced jet experiments. The surface colours denote the normalized response amplitude ($\eta _\gamma \equiv \gamma ^{\prime \ast }_{rms}/\gamma ^\prime _{rms}$), shown in a parameter space defined by the forcing amplitude ($a$) and the normalized forcing frequency ($f_f/f_n$). The minimum value of $a$ required for lock-in, $a_{loc}$, is indicated by circular markers, which form the lower boundaries of the $1:1$ Arnold tongue. The filled (black) markers denote a saddle-node bifurcation along the phase-locking route to lock-in, whereas the hollow (white) markers denote an inverse Neimark–Sacker bifurcation along the suppression route to lock-in.

The synchronization maps in figure 5 reveal that $a_{loc}$ increases as $f_f/f_n$ deviates from 1, producing a classic $\vee$-shaped Arnold tongue centred on the natural mode ($f_f/f_n = 1$). The shape and position of the Arnold tongue remain similar across all five jet positions, indicating that the jet always locks in at the same set of forcing amplitudes regardless of the symmetry of the imposed perturbations, i.e. regardless of whether the forcing is purely axial (position $\mathbb {A}$), purely transverse (position $\mathbb {E}$), or a mix of both (positions $\mathbb {B}$, $\mathbb {C}$ and $\mathbb {D}$).

An examination of the $\eta _\gamma$ surfaces shows that when the jet is forced purely axially (figure 5, position $\mathbb {A}$), its oscillations within the Arnold tongue are amplified when $f_f/f_n < 1$ (red regions) but attenuated when $f_f/f_n > 1$ (blue regions). This is consistent with the experiments by Li & Juniper (Reference Li and Juniper2013a), who found the same asymmetry in a similar jet subjected to pure axial forcing from a loudspeaker mounted upstream of the nozzle outlet. In figure 5, the asymmetry in $\eta _\gamma$ also appears at all the jet positions between the pressure antinode and node (positions $\mathbb {B}$, $\mathbb {C}$ and $\mathbb {D}$). This is not surprising given that the acoustic perturbations at these intermediate jet positions are still dominated by the axial mode (figure 3), just as they are at the pressure antinode itself (figure 5, position $\mathbb {A}$). As § 5.4 will show, the asymmetry in $\eta _\gamma$ is caused by non-resonant amplification of the forcing signal when $f_f/f_n < 1$.

When the jet is forced purely transversely (figure 5, position $\mathbb {E}$), its oscillations become suppressed across the full range of $f_f/f_n$. Crucially, this suppression occurs not just inside the Arnold tongue but also outside it, demonstrating the robustness of this control technique (i.e. axisymmetry breaking) to variations in both $f_f$ and $a$. The degree of suppression is generally high: the oscillation amplitude can be reduced to less than 10–15 % of that of the unforced jet in most of the parameter space (figure 5, position $\mathbb {E}$). As § 5.4 will show, this suppression of the self-excited jet oscillations occurs via asynchronous and synchronous quenching, both without any amplification of the forcing signal.

We have shown from a synchronization perspective that breaking the axisymmetry of a globally unstable jet via transverse forcing can lead to a more effective means of open-loop flow control than preserving the axisymmetry via axial forcing. In particular, although the shape and position of the Arnold tongue ($a_{loc}$) are relatively insensitive to the symmetry of the imposed perturbations, the jet oscillations themselves are suppressed more readily with transverse forcing than with axial forcing. Compared with pure axial forcing (position $\mathbb {A}$) and axially dominated forcing (positions $\mathbb {B}$, $\mathbb {C}$ and $\mathbb {D}$), pure transverse forcing (position $\mathbb {E}$) can produce larger reductions in $\eta _\gamma$ with fewer limitations on the required values of $f_f$ and $a$. By contrast, when $f_f/f_n < 1$, axial forcing, even when combined with transverse forcing, can cause $\eta _\gamma$ to exceed unity. This amplification occurs because the axisymmetric global mode ($m=0$) is preferentially receptive to perturbations matching its own spatial structure, a phenomenon that will be examined further in § 5.4.

The synchronization maps for the model are shown in figure 6. As before, the surface colours denote the normalized response amplitude ($\eta _x \equiv x_{rms}^{\prime \ast }/x_{rms}^\prime$), and the circular markers denote the onset of lock-in via a saddle-node bifurcation (filled black markers) or an inverse Neimark–Sacker bifurcation (hollow white markers). The forcing amplitude is defined as $b \equiv (b_1^2 + b_2^2)^{1/2}$, with the minimum value of $b$ required for lock-in denoted by $b_{loc}$. Comparing figures 5 and 6, we find many similarities but also some key differences between the jet and the model. In both systems, the minimum forcing amplitude required for lock-in ($a_{loc}$, $b_{loc}$) increases with detuning regardless of the symmetry of the imposed perturbations, resulting in a familiar $\vee$-shaped Arnold tongue centred on the natural mode ($f_f/f_n=1$, $\omega _f/\omega _n=1$). Furthermore, when forced purely transversely (figures 5 and 6, position $\mathbb {E}$), the self-excited oscillations in both systems become suppressed readily across a wide range of forcing frequencies and amplitudes, highlighting again the effectiveness of axisymmetry breaking as a quenching strategy. However, a notable feature of the jet that could not be captured by the model is the asymmetry in $\eta _\gamma$ about $f_f/f_n = 1$, which occurs whenever the imposed perturbations are axially dominated, i.e. at positions $\mathbb {A}$–$\mathbb {D}$ in figure 5. This discrepancy was also noted by Li & Juniper (Reference Li and Juniper2013a), but its origin has yet to be fully established. Nevertheless, it is encouraging to see how such a simple model can still accurately – albeit qualitatively – capture the extensive amplitude suppression observed in the purely transversely forced jet, including its insensitivity to both $f_f$ and $a$ (figures 5 and 6, position $\mathbb {E}$).

Figure 6. The same as for figure 5 but from the forced VDP model ((3.1) and (3.2)).

5.2. Amplitude suppression and symmetry breaking of the wavemaker region

The extensive amplitude suppression observed in the purely transversely forced jet (figure 5) was established based on local measurements made with a spatially fixed HWA probe. To verify that the observed suppression is not simply an artefact of the nonlinear global mode shifting away from the HWA probe, we show in figures 7(a–c) the spatial distribution of the total velocity fluctuation (i.e. the root-mean-square of $V^{\prime }$, or $V^{\prime }_{rms}$), as measured with stereoscopic time-resolved PIV, for both axial and transverse forcing as well as for no forcing at all. Our focus is on the potential core, rather than on the regions downstream, because that is where the wavemaker of the global mode is known to reside (Lesshafft et al. Reference Lesshafft, Huerre, Sagaut and Terracol2006; Qadri et al. Reference Qadri, Chandler and Juniper2018). We demarcate the potential core with a black line representing the regions where $\bar {V}$ exceeds 95 % of its value at the nozzle outlet centreline. In figures 7(d–f), we show the normalized PSD of $V^{\prime }$ along the jet centreline ($y/D=0$) at different axial stations.

Figure 7. Spatial distribution of the jet dynamics, as measured with stereoscopic time-resolved PIV: (a–c) the root-mean-square of the total velocity fluctuation $V^\prime _{rms}$; (d–f) the normalized PSD of $V^{\prime }$ along the jet centreline ($y/D=0$); (g–i) the dominant DMD mode with contours of the out-of-plane vorticity fluctuation; and (j) the normalized histogram of the phase shift $\varTheta$ between the left and right shear layers. Three representative conditions are shown: (left) unforced, (middle) pure axial forcing at $a_{loc}$, and (right) pure transverse forcing at $a_{loc}$. With either axial or transverse forcing, the forcing frequency is fixed at $f_f/f_n = 1.09$.

When the jet is unforced (figure 7a), $V_{rms}^{\prime }$ increases from around zero at the nozzle outlet ($x/D \approx 0$) to a local maximum within the potential core ($x/D \approx 1.5$, coinciding with the HWA probe location; § 2.1), before eventually decreasing downstream towards the tip. The corresponding PSD (figure 7d) shows that these self-excited oscillations occur at the fundamental frequency of the natural global mode ($f_n$). Farther downstream ($x/D > 2.3$), the subharmonic ($f_n/2$) overtakes the fundamental ($f_n$) to become the new dominant mode. This coincides with regions of large $V_{rms}^{\prime }$ outside the potential core. Such strong $f_n/2$ oscillations in the downstream region are known to arise from period doubling associated with vortex pairing, possibly aided by disturbance amplification from local convective modes (Kyle & Sreenivasan Reference Kyle and Sreenivasan1993). From these observations, we can conclude that the potential core of the jet is dominated by a global hydrodynamic mode at its natural fundamental frequency ($f_n$).

When the jet is forced purely axially at $a_{loc}$ and $f_f/f_n = 1.09$ (figure 7b), $V_{rms}^{\prime }$ becomes lower than that of the unforced jet, particularly in the potential core, which is now noticeably shorter. This amplitude reduction for $f_f > f_n$ is consistent with the $\eta _\gamma$ data (figure 5) extracted from our local HWA measurements. The corresponding PSD (figure 7e) shows that, as is the case without forcing, most of the potential core oscillates at the fundamental frequency of the dominant mode, which is now at $f_f$ (not $f_n$) because the jet has locked into the forcing. For a similar reason, the regions downstream oscillate at the subharmonic of the forced mode ($f_f/2$), rather than at the subharmonic of the natural global mode ($f_n/2$).

When the jet is forced purely transversely at $a_{loc}$ and $f_f/f_n = 1.09$ (figure 7c), $V_{rms}^{\prime }$ becomes even lower than that of the purely axially forced jet (figure 7b), with most of the decrease occurring in the potential core. Also decreasing is the streamwise extent of the potential core itself, although most of its spatial domain ($x/D < 2$) remains dominated by the $f_f$ mode, with the regions downstream dominated by the $f_f/2$ mode (figure 7f). Crucially, the fact that regardless of whether the forcing is axial (figure 7b) or transverse (figure 7c), the decrease in $V_{rms}^{\prime }$ occurs spatially throughout most of the potential core, where the wavemaker is known to reside (Lesshafft et al. Reference Lesshafft, Huerre, Sagaut and Terracol2006; Qadri et al. Reference Qadri, Chandler and Juniper2018), confirms that our single-point HWA measurements are indeed a reliable indicator of the jet response.

To assess the symmetry of the vortical structures in the jet, we show in figures 7(g–i) the dominant spatial modes extracted via dynamic mode decomposition (DMD; Schmid Reference Schmid2010; Rowley & Dawson Reference Rowley and Dawson2017) of the in-plane velocity field ($u,v$) measured with PIV. These DMD modes are dominant in the sense that their amplitudes are the highest in their respective spectra; they correspond to the $f_n$ mode in the unforced jet (figure 7g) and to the $f_f$ mode in both the purely axially forced jet (figure 7h) and the purely transversely forced jet (figure 7i). Superimposed over the DMD modes are contours of the out-of-plane vorticity fluctuation. In all three cases (figures 7g–i), the jet dynamics are periodic, either naturally or due to lock-in, with each DMD mode featuring a train of counter-rotating vortical structures advecting along the shear layers on both sides of the jet centreline.

In both the unforced jet (figure 7g) and the purely axially forced jet (figure 7h), the vortical structures are antisymmetric about the jet centreline, which is consistent with the presence of an $m=0$ global mode. To quantify the phase shift between the left and right shear layers ($\varTheta$), we apply the Hilbert transform to instantaneous streamwise profiles of vorticity extracted at $y/D = \pm 0.5$, and then plot the data in figure 7(j) as a normalized histogram of $\varTheta$. We find that the histogram peaks at $\varTheta \approx 0$ for both the unforced and axially forced jets, confirming that the left and right shear layers are rolling up in phase with each other. This is the classic behaviour of an axisymmetric jet dominated by an $m=0$ global mode.

In the purely transversely forced jet (figure 7i), however, the vortical structures are no longer antisymmetric about the jet centreline. Instead, they roll up and advect out-of-phase by ${\rm \pi} /2$ (figure 7j), producing a transverse flapping motion that breaks the axisymmetry of the $m=0$ global mode. Put together, these findings provide compelling evidence that axial forcing preserves the inherent axisymmetry of the jet, whereas transverse forcing breaks that axisymmetry by inducing the staggered roll-up of vortical structures in the shear layers. The latter effect will prove to be crucial in achieving asynchronous and synchronous quenching of the natural global mode, without amplification of the forced mode.

5.3. Bifurcation routes to lock-in

Having examined the spatial characteristics of the forced jet response, we now turn to identifying the bifurcations responsible for lock-in. We first consider the purely axially forced jet (§ 5.3.1, position $\mathbb {A}$) so as to establish a calibrated baseline against the axial forcing experiments of Li & Juniper (Reference Li and Juniper2013a,Reference Li and Juniperc). We then compare this to the purely transversely forced jet (§ 5.3.2, position $\mathbb {E}$) so as to determine the effects of axisymmetry breaking. In the interest of brevity, we do not show results for combined axial–transverse forcing (positions $\mathbb {B}$, $\mathbb {C}$ and $\mathbb {D}$) because they are qualitatively similar to those for pure axial forcing (position $\mathbb {A}$). This is expected because all four of these jet positions (positions $\mathbb {A}$, $\mathbb {B}$, $\mathbb {C}$ and $\mathbb {D}$) are dominated by axial perturbations ($m=0$), with little to no contribution from transverse perturbations ($m=\pm 1$), as was shown in figure 3. For each type of forcing, we consider two representative values of $f_f$, one close to $f_n$ and one far from $f_n$. This is so that we can investigate the full range of synchronization dynamics around the $1:1$ Arnold tongue.

5.3.1. Axial forcing: the jet at position $\mathbb {A}$

Figure 8 shows the synchronization dynamics of the jet when forced axially at a frequency close to its natural global frequency: $f_f/f_n = 1.04$. We consider the time trace, the PSD, the instantaneous phase difference between the forcing signal and the forced self-excited jet, $\Delta \phi _{f,\gamma }$, and the one-sided Poincaré map, all extracted from the $\gamma (t)$ signal. For comparison, we show in figure 9 the same indicators but for the axially forced model. In both systems, we increase the forcing amplitude incrementally until reaching the onset of lock-in, which occurs at $A \equiv a/a_{loc} = 1$ for the jet and at $B \equiv b/b_{loc} = 1$ for the model.

Figure 8. Synchronization dynamics of the jet when forced axially (position $\mathbb {A}$) at a frequency close to its natural global frequency: $f_f/f_n=1.04$. The (a) time trace, (b) PSD, (c) temporal evolution of $\Delta \phi _{f,\gamma }$, and (d) one-sided Poincaré map are shown for nine forcing amplitudes, including the unforced case ($A=0$). Lock-in occurs via the phase-locking route, which involves a Neimark–Sacker bifurcation at $A = 0 \rightarrow 0.25$ followed by a saddle-node bifurcation at $A = 0.93 \rightarrow 1$. The unforced state is shown in green, the quasiperiodic states are shown in blue, and the onset of lock-in is shown in red.

Figure 9. The same as in figure 8 but for the axially forced model at $\omega _f/\omega _n=1.04$.

When unforced (figure 8, $A = 0$), the jet is hydrodynamically self-excited, exhibiting a nonlinear global mode at $f_n$ with progressively weaker components at the subharmonic and superharmonics (both odd and even; figures 8a,b). In the absence of forcing, $\Delta \phi _{f,\gamma }$ is undefined, so we plot in the bottom row of figure 8(c) the instantaneous phase evolution of the unforced jet, $-\phi _{\gamma }$, which is seen to drift continuously in time without phase slips. This behaviour is consistent with the absence of pronounced amplitude modulation in the time trace (figure 8a) (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). In the Poincaré map (figure 8d), the intercepts of the phase trajectory are concentrated at a single spot, indicating the presence of a closed repetitive orbit in the phase portrait, a signature feature of a period-1 limit-cycle attractor.

When forced at intermediate amplitudes (figure 8, $A = 0.25$–$0.93$), the jet becomes quasiperiodic. This is evidenced by the appearance of amplitude modulation in the time trace (figure 8a) and by the emergence of sharp peaks at both the forced ($f_f$) and natural ($f_n^\ast$) modes in the PSD (figure 8b). The values of $f_f$ and $f_n^\ast$ are incommensurable and, through the action of nonlinear resonant three-wave interactions (Schmid, Henningson & Jankowski Reference Schmid, Henningson and Jankowski2002), they generate a train of spectral peaks at linear combinations of $f_f$ and $f_n^\ast$. Quasiperiodicity is also demonstrated by $\Delta \phi _{f,\gamma }$ drifting unboundedly in time (figure 8c) and by the formation of a closed ring of trajectory intercepts in the Poincaré map (figure 8d). Taken together, these observations indicate that the jet has transitioned from a period-1 limit-cycle attractor to an ergodic two-dimensional torus attractor $\mathbb {T}^2$ via a Neimark–Sacker bifurcation at $A = 0 \rightarrow 0.25$ (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003; Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). Several other features are worth noting in this quasiperiodic regime. As $A$ increases, the natural mode at $f_n^\ast$ gradually shifts towards the forced mode at $f_f$, whose value is dictated by the external forcing and is thus fixed. This temporal adjustment process, known as frequency pulling in the synchronization literature, is a hallmark feature of the phase-locking route to lock-in (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003; Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). As for the phase difference (figure 8c), frequency pulling causes the time-averaged slope of $\Delta \phi _{f,\gamma }$, or $\langle {\Delta \dot {\phi }_{f,\gamma }}\rangle$, to approach zero gradually as $A$ increases. Physically, this implies that the time scale of the natural mode is adjusting to match that of the forced mode. This process occurs without significant changes in the oscillation amplitude (figure 8a). Consequently, lock-in along the phase-locking route occurs via adjustments in the frequency rather than in the amplitude, which is physically consistent with this route arising only when $f_f$ is sufficiently close to $f_n$. Another notable feature is that at relatively high forcing amplitudes (figure 8c, $A = 0.74$–$0.93$), $\Delta \phi _{f,\gamma }$ drifts increasingly nonlinearly, spending long epochs at nearly constant values and then dropping abruptly by one complete cycle of $2{\rm \pi}$ (see insets of figure 8c). In synchronization theory, the sudden loss or gain of an integer number of complete cycles is known as phase slipping and is another defining feature of the phase-locking route (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). As $A$ increases towards the value required for lock-in ($A=1$), the time between successive phase slips increases. Physically, this implies that the jet is spending more and more time oscillating at $f_f$, with fewer phase slips. Meanwhile, the Poincaré map shows that the phase trajectory continues to wrap around the ergodic $\mathbb {T}^2$ torus attractor, whose scale grows monotonically as $A$ increases (figure 8d). This confirms that the jet is still oscillating quasiperiodically at two incommensurable frequencies.

When forced at a critical amplitude (figure 8, $A = 1$), the jet transitions from $\mathbb {T}^2$ quasiperiodicity to $1:1$ lock-in: it now oscillates at only $f_f$ and its harmonics, with no sign of the original natural mode at $f_n^\ast$ (figure 8b). As a result, the time trace is no longer modulated (figure 8a), and $\Delta \phi _{f,\gamma }$ no longer drifts or slips but instead remains constant in time (figure 8c), indicating phase locking (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). As expected, this transition from $\mathbb {T}^2$ quasiperiodicity to $1:1$ lock-in sees the ring structure in the Poincaré map replaced by a single cluster of trajectory intercepts, indicative of a period-1 orbit (figure 8d). Crucially, the collapse of the ring structure occurs suddenly rather than gradually (figure 8d, $A = 0.93 \rightarrow 1$). According to synchronization analyses of the forced VDP oscillator (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009), the suddenness of this collapse indicates that the $\mathbb {T}^2$ attractor is still alive but has become resonant through a saddle-node bifurcation. Consequently, a pair of periodic orbits – one stable orbit and one saddle orbit – is born on the resonant $\mathbb {T}^2$ surface, with lock-in occurring when the phase trajectory snaps on to the stable orbit. This synchronization mechanism, involving a saddle-node bifurcation and frequency pulling, is intrinsic to the phase-locking route to lock-in (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009).

As mentioned earlier, figure 9 shows the same four indicators as in figure 8 but for the axially forced model. We find that the model can reproduce qualitatively the main synchronization phenomena exhibited by the jet. These include: (i) a Neimark–Sacker bifurcation from a period-1 limit cycle to $\mathbb {T}^2$ quasiperiodicity when axial forcing is first applied; (ii) amplitude modulation of the time trace, along with phase drifting and slipping during $\mathbb {T}^2$ quasiperiodicity; (iii) a gradual pulling of $f_n^\ast$ towards $f_f$ as the forcing amplitude increases, causing $\langle {\Delta \dot {\phi }_{f,\gamma }}\rangle$ to approach zero gradually; and (iv) a saddle-node bifurcation from $\mathbb {T}^2$ quasiperiodicity to $1:1$ lock-in at a critically high forcing amplitude. However, the model cannot reproduce the train of spectral peaks observed at and around the subharmonic and superharmonics of $f_n$. This is because the VDP kernel used in the model contains only cubic nonlinearity, thus generating only odd harmonics (Nayfeh & Mook Reference Nayfeh and Mook1995). Although incorporating other forms of nonlinearity may improve the comparison, doing so is not a priority here. Our primary aim is to use the simplest possible model to capture phenomenologically the most salient synchronization phenomena of the jet, such as its Arnold tongue, bifurcation routes to lock-in, phase dynamics, and oscillation quenching mechanisms.

Next we increase $f_f$ so that it is far from $f_n$ (figure 10, $f_f/f_n = 1.17$). We find several important differences relative to the earlier case where $f_f$ is close to $f_n$ (figure 8, $f_f/f_n = 1.04$). In particular, lock-in now occurs not through a gradual pulling of $f_n^\ast$ towards $f_f$, but through a gradual reduction in the amplitude of the natural mode. This fundamental change in the synchronization mechanism can be seen in the PSD (figure 10b), where the frequency ($f_n^\ast$) of the spectral peak corresponding to the natural mode remains constant as $A$ increases. The difference can also be seen in the temporal evolution of the phase difference (figure 10c), where $\langle {\Delta \dot {\phi }_{f,\gamma }}\rangle$ remains constant as $A$ increases, until it eventually snaps to zero at the onset of phase trapping ($A=0.78$) (Aronson, Ermentrout & Kopell Reference Aronson, Ermentrout and Kopell1990; Thévenin et al. Reference Thévenin, Romanelli, Vallet, Brunel and Erneux2011).

Figure 10. Synchronization dynamics of the jet when forced axially (position $\mathbb {A}$) at a frequency far from its natural global frequency: $f_f/f_n=1.17$. The (a) time trace, (b) PSD, (c) temporal evolution of $\Delta \phi _{f,\gamma }$, and (d) one-sided Poincaré map are shown for nine forcing amplitudes, including the unforced case ($A=0$). Lock-in occurs via the suppression route, which involves a Neimark–Sacker bifurcation at $A = 0 \rightarrow 0.18$ followed by an inverse Neimark–Sacker bifurcation at $A = 0.91 \rightarrow 1$. The unforced state is shown in green, the quasiperiodic states are shown in blue, and the onset of lock-in is shown in red.

As $A$ increases from the onset of phase trapping, the ring structure in the Poincaré map shrinks gradually (figure 10d, $A = 0.78 \rightarrow 1$), indicating that the underlying $\mathbb {T}^2$ attractor is collapsing gradually to its death via an inverse Neimark–Sacker bifurcation (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). This gradual collapse stands in stark contrast to the sudden collapse seen in figure 8(d), where the detuning is small. According to synchronization analyses of the forced VDP oscillator (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009), this difference is consistent with the jet locking into the forcing via the suppression route, rather than the phase-locking route. In other words, instead of adjusting its phase dynamics (i.e. advancement or retardation of its temporal rhythm) to match that of the forcing, the jet experiences a suppression of its natural amplitude dynamics (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003). Because in this example $f_f/f_n$ is sufficiently larger than 1, the suppression of the natural mode occurs without resonant amplification of the forced mode, resulting in the oscillation amplitude at lock-in (figure 10a, $A=1$) being lower than that in the unforced state (figure 10a, $A=0$). In synchronization theory, a suppression of the natural mode by non-resonant forcing is often referred to as asynchronous quenching (Bogoliubov & Mitropolsky Reference Bogoliubov and Mitropolsky1961; Minorsky Reference Minorsky1967, Reference Minorsky1974), which will be examined further in § 5.4.

For comparison, we show in figure 11 the same four indicators as in figure 10 but for the axially forced model. As is the case when $f_f$ is close to $f_n$ (figures 8 and 9), we find that the model can reproduce qualitatively the main synchronization phenomena exhibited by the jet. Crucially, these include features specific to the suppression route to lock-in, such as: (i) a gradual reduction in the amplitude of the natural mode as $A$ increases, without a simultaneous pulling of $f_n^\ast$ towards $f_f$; (ii) a constant value of $\langle {\Delta \dot {\phi }_{f,\gamma }}\rangle$ followed by an abrupt snap to zero at the onset of phase trapping; and (iii) a gradual reduction in the size of the ring structure in the Poincaré map, indicating an inverse Neimark–Sacker bifurcation from $\mathbb {T}^2$ quasiperiodicity to $1:1$ lock-in at a critically high forcing amplitude.

Figure 11. The same as in figure 10 but for the axially forced model at $\omega _f/\omega _n=1.17$.

In summary, we have investigated the forced synchronization of a prototypical hydrodynamic oscillator – a globally unstable low-density axisymmetric jet – located at the pressure antinode of a planar acoustic standing wave, where axial (symmetric, $m=0$) perturbations were produced at different amplitudes and frequencies. We found that around the $1:1$ Arnold tongue, the jet exhibits a complex range of synchronization phenomena, such as: (i) phase drifting, slipping and trapping during quasiperiodicity on a $\mathbb {T}^2$ attractor at intermediate $A$ values; (ii) phase locking on a periodic orbit at critically high $A$ values; and (iii) two bifurcation routes to lock-in – the phase-locking route when the detuning is small, and the suppression route when the detuning is large. We then showed that these synchronization phenomena could be captured qualitatively by the simple VDP model proposed in § 3. These findings are broadly consistent with the experiments conducted by Li & Juniper (Reference Li and Juniper2013a,Reference Li and Juniperc) on a similar hydrodynamically self-excited jet forced axially from upstream with a loudspeaker mounted in the nozzle plenum. This further validates the present experimental methodology and data analysis, paving the way for an exploration of the effects of axisymmetry breaking via transverse forcing.

5.3.2. Transverse forcing: the jet at position $\mathbb {E}$

Figure 12 shows the synchronization dynamics of the jet when forced transversely at a frequency close to its natural global frequency: $f_f/f_n = 1.04$. Comparing this with figure 8, we find that switching from axial to transverse forcing at small detuning leads to a dramatic change in the jet response. In particular, although the jet still transitions from unforced periodicity to $\mathbb {T}^2$ quasiperiodicity via a Neimark–Sacker bifurcation when forcing is first applied (figure 12, $A = 0 \rightarrow 0.14$), the subsequent transition to $1:1$ lock-in no longer occurs via a saddle-node bifurcation, as would be expected from figure 8 and from weakly nonlinear analyses of the classic (single-mode) VDP model when $f_f$ is close to $f_n$ (Pikovsky et al. Reference Pikovsky, Rosenblum and Kurths2003; Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). Instead, we find in figure 12 that as $A$ increases in the quasiperiodic regime, the frequency ($f_n^\ast$) of the natural mode remains constant but its amplitude decreases gradually, indicating asynchronous quenching without frequency pulling. In turn, this causes $\langle {\Delta \dot {\phi }_{f,\gamma }}\rangle$ to remain constant with increasing $A$, until it eventually snaps to zero at the onset of phase trapping (figure 12c, $A=0.92$). By tracking the ring structure in the Poincaré map (figure 12d), we find a gradual shrinkage of the $\mathbb {T}^2$ attractor en route to lock-in, providing evidence of an inverse Neimark–Sacker bifurcation (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). Put together, these observations reveal that when its axisymmetry is broken via transverse forcing, the jet locks in via the suppression route, even when the detuning is small. This is strikingly different from the phase-locking route taken by the axially forced jet (figure 8). An intrinsic feature of the suppression route is a marked reduction in the oscillation amplitude en route to lock-in (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009). Here, such a reduction is indeed observed, both in the time trace (figure 12a) and in the synchronization map (figure 5). Figure 13 shows that the main synchronization phenomena exhibited by the transversely forced jet – such as the amplitude reduction, phase trapping, and suppression route to lock-in via an inverse Neimark–Sacker bifurcation – are well captured by the transversely forced model.

Figure 12. Synchronization dynamics of the jet when forced transversely (position $\mathbb {E}$) at a frequency close to its natural global frequency: $f_f/f_n=1.04$. The (a) time trace, (b) PSD, (c) temporal evolution of $\Delta \phi _{f,\gamma }$, and (d) one-sided Poincaré map are shown for nine forcing amplitudes, including the unforced case ($A=0$). Lock-in occurs via the suppression route, which involves a Neimark–Sacker bifurcation at $A = 0 \rightarrow 0.14$ followed by an inverse Neimark–Sacker bifurcation at $A = 0.92 \rightarrow 1$. The unforced state is shown in green, the quasiperiodic states are shown in blue, and the onset of lock-in is shown in red.

Figure 13. The same as in figure 12 but for the transversely forced model at $\omega _f/\omega _n=1.04$.

Next we increase $f_f$ so that it is far from $f_n$ (figure 14, $f_f/f_n = 1.17$). We find that the jet responds qualitatively similarly to when $f_f$ is close to $f_n$ (figure 12, $f_f/f_n = 1.04$). Specifically, lock-in still occurs via the suppression route: (i) the natural mode becomes quenched without its frequency $f_n^\ast$ being pulled towards $f_f$; (ii) phase trapping occurs just before the onset of lock-in; and (iii) the $\mathbb {T}^2$ attractor shrinks gradually to its death via an inverse Neimark–Sacker bifurcation. For comparison, we show the corresponding model response in figure 15. As is the case for small detuning (figures 12 and 13), we find that the model can reproduce qualitatively the main synchronization phenomena exhibited by the jet, including the substantial reduction in the oscillation amplitude en route to lock-in (figure 15a).

Figure 14. Synchronization dynamics of the jet when forced transversely (position $\mathbb {E}$) at a frequency far from its natural global frequency: $f_f/f_n=1.17$. The (a) time trace, (b) PSD, (c) temporal evolution of $\Delta \phi _{f,\gamma }$, and (d) one-sided Poincaré map are shown for nine forcing amplitudes, including the unforced case ($A=0$). Lock-in occurs via the suppression route, which involves a Neimark–Sacker bifurcation at $A = 0 \rightarrow 0.49$ followed by an inverse Neimark–Sacker bifurcation at $A = 0.95 \rightarrow 1$. The unforced state is shown in green, the quasiperiodic states are shown in blue, and the onset of lock-in is shown in red.

Figure 15. The same as in figure 14 but for the transversely forced model at $\omega _f/\omega _n=1.17$.

5.4. Asynchronous quenching

In § 5.3.1, we showed that when forced axially, the jet locks in via the phase-locking route when the detuning is small, but via the suppression route when the detuning is large. In § 5.3.2, we showed that when forced transversely, the jet locks in exclusively via the suppression route, regardless of whether the detuning is small or large. According to synchronization theory based on a self-excited oscillator with a single natural mode (Balanov et al. Reference Balanov, Janson, Postnov and Sosnovtseva2009), only the suppression route can cause asynchronous quenching of the natural mode without resonant amplification of the forced mode. The findings of §§ 5.3.1 and 5.3.2 thus suggest that transverse forcing would be more effective than axial forcing in suppressing the global axisymmetric oscillations of a low-density jet. Below, we provide definitive evidence of this and show that, like most other synchronization phenomena observed in the jet, asynchronous quenching can be qualitatively reproduced by the coupled oscillator model of § 3.

Figure 16 shows the spectral power of the natural mode ($P_n^{\prime \ast }$), of the forced mode ($P_f^{\prime \ast }$), and of the total $\gamma ^\prime$ signal ($P_t^{\prime \ast }$), all as functions of the forcing power ($A^2$) for both axial and transverse forcing. We consider two representative non-resonant values of $f_f$, one below $f_n$ (figure 16a,c, $f_f/f_n = 0.82$) and one above $f_n$ (figure 16b,d, $f_f/f_n = 1.17$). Both $f_f$ values are sufficiently far from $f_n$ for lock-in to occur via the suppression route (figure 5). We compute the spectral power by integrating the PSD around each respective mode ($\pm 5$ Hz) and then normalizing the result by the total power of the unforced system ($P_t^{\prime }$). We compute the total power by integrating the PSD across the entire bandwidth, with verification that it obeys Parseval's theorem, i.e. that it is equal (within $0.22\,\%$) to the mean squared fluctuation.

Figure 16. Evidence of asynchronous quenching in the jet: spectral power as a function of the forcing power for (a,b) axial forcing, and (c,d) transverse forcing. For both types of forcing, lock-in occurs via the suppression route. Two representative values of $f_f$ are shown: (a,c) one below $f_n$, and (b,d) one above $f_n$.

5.4.1. Axial forcing: the jet at position $\mathbb {A}$

Figures 16(a,b) show that, regardless of whether $f_f/f_n < 1$ or $f_f/f_n>1$, the axially forced jet undergoes a suppression of its natural mode en route to lock-in (grey shading). This is demonstrated by $P_n^{\prime \ast }$ decaying to zero as $A^2$ increases to 1 and beyond, providing clear evidence of asynchronous quenching (Bogoliubov & Mitropolsky Reference Bogoliubov and Mitropolsky1961; Minorsky Reference Minorsky1967, Reference Minorsky1974). Meanwhile, $P_f^{\prime \ast }$ increases, but in two different ways. When $f_f/f_n > 1$ (figure 16b), $P_f^{\prime \ast }$ increases linearly with $A^2$ up to and beyond lock-in. When $f_f/f_n < 1$ (figure 16a), however, $P_f^{\prime \ast }$ increases linearly at first but then nonlinearly at higher $A^2$ values, jumping sharply at the onset of lock-in ($A^2 = 1$). This accelerated growth in $P_f^{\prime \ast }$ is indicative of an amplification of the forced mode via nonlinear interactions with the natural mode (Odajima et al. Reference Odajima, Nishida and Hatta1974; Abel et al. Reference Abel, Ahnert and Bergweiler2009). Because here the forcing is non-resonant (figure 16a, $f_f/f_n = 0.82$), this phenomenon is referred to as non-resonant amplification (Nayfeh & Mook Reference Nayfeh and Mook1995). Such an amplification of the forced mode causes $P_t^{\prime \ast }$ to rise above that of the unforced system ($P_t^{\prime \ast }/P_t^{\prime } > 1$). Consequently, if the control objective is to suppress the global axisymmetric oscillations of a low-density jet using axial forcing, then it is not simply enough for $f_f$ to be sufficiently far from $f_n$ to activate the suppression route; one must also ensure that $f_f/f_n > 1$, as this enables asynchronous quenching to occur without resonant or non-resonant amplification en route to lock-in (figure 16b). If $f_f$ is somehow allowed to fall below $f_n$, then asynchronous quenching still occurs, but its effect is counteracted by non-resonant amplification of the forced mode, resulting in the total power increasing above that of the unforced system as lock-in is approached (figure 16a). We believe that this is the first time that asynchronous quenching has been observed alongside non-resonant amplification in a forced self-excited hydrodynamic system. As § 5.4.2 will confirm, this discovery provides a compelling incentive to consider axisymmetry breaking, via transverse forcing, as an alternative method of suppressing a nonlinear global mode.

Figures 17(a,b) show the spectral power of the axially forced model at the two forcing frequencies shown in figures 16(a,b): $\omega _f/\omega _n = 0.82$ and $1.17$. For both values of $\omega _f/\omega _n$, the model can capture qualitatively the asynchronous quenching observed in the jet: $P_n^{\prime \ast }$ decays to zero as $B^2$ increases to 1 and beyond. However, the model cannot capture the non-resonant amplification of the forced mode, regardless of whether $\omega _f/\omega _n < 1$ or $\omega _f/\omega _n> 1$. Instead, it exhibits a proportional relationship between $P_f^{\prime \ast }$ and $B^2$, resulting in $P_t^{\prime \ast }$ remaining well below that of the unforced state en route to and at the onset of lock-in ($B^2 = 1$). The inability of the model to capture non-resonant amplification when the forcing frequency is below the natural frequency is why the asymmetry about $f_f/f_n = 1$ in the $\eta _{\gamma }$ data from the jet (figure 5) is absent in the $\eta _{x}$ data from the model (figure 6). Nevertheless, in both the jet and the model, continuing to increase the forcing amplitude (or forcing power) after lock-in eventually causes the total amplitude to rise above that of the unforced state (figure 5).

Figure 17. The same as in figure 16 but for the axially or transversely forced model.

5.5. Synchronous quenching

To examine the amplitude response at small detuning, we show in figure 18 the spectral power of the jet and of the model for a range of forcing frequencies, both close to and far from the natural frequency. The forcing is either axial or transverse, but always at a relatively low amplitude (see the caption of figure 18).

Figure 18. Evidence of synchronous quenching in (a,c) the jet, and (b,d) the model: spectral power as a function of the normalized forcing frequency for (a,b) axial forcing, and (c,d) transverse forcing. The forcing amplitude is (a) $a = 0.998$ Pa, (b) $b = 0.019$, (c) $a = 1.320$ Pa, and (d) $b = 0.049$.

In the axially forced jet (figure 18a), we find that as $f_f/f_n$ approaches 1 from either the low side or the high side, $P^{\prime \ast }_n$ decays to zero whilst $P^{\prime \ast }_f$ grows to a maximum. This indicates that synchronous quenching of the natural mode is occurring alongside resonant amplification of the forced mode (Keen & Fletcher Reference Keen and Fletcher1969; Odajima et al. Reference Odajima, Nishida and Hatta1974; Abel et al. Reference Abel, Ahnert and Bergweiler2009; Mondal et al. Reference Mondal, Pawar and Sujith2019). Here the degree of resonant amplification is strong enough to enable $P^{\prime \ast }_f$ to counterbalance the quenching of $P^{\prime \ast }_n$, resulting in $P^{\prime \ast }_t$ exceeding $P^{\prime }_t$ just below $f_f/f_n = 1$. In the axially forced model (figure 18b), we find the same qualitative behaviour involving the simultaneous occurrence of synchronous quenching and resonant amplification when $f_f/f_n$ is close to 1. Similar behaviour has been observed recently in self-excited thermoacoustic systems subjected to similar resonant forcing (Guan et al. Reference Guan, Gupta, Kashinath and Li2019a,Reference Guan, Gupta, Wan and Lib; Mondal et al. Reference Mondal, Pawar and Sujith2019). However, we believe that this is the first time that synchronous quenching has been observed alongside resonant amplification in a self-excited hydrodynamic system, reinforcing the universality of such synchronization phenomena in physically disparate systems.

In the transversely forced jet (figure 18c), we find markedly different behaviour: although synchronous quenching of the natural mode still occurs ($P^{\prime \ast }_n \rightarrow 0$), the forced mode is not amplified to a degree sufficient to cause $P^{\prime \ast }_t$ to exceed $P^{\prime }_t$, implying the absence of resonant amplification. The same qualitative behaviour is observed in the transversely forced model (figure 18d). This demonstrates that synchronous quenching of the natural mode occurring without resonant amplification of the forced mode is the primary mechanism by which transverse forcing is able to reduce the overall response amplitude when $f_f$ is close to $f_n$.