3.1.1. Forced response to transverse forcing

Figures 3(a–e) show spatial maps of $V^\prime _{\textit{rms}}$ for the jet when forced transversely at five different amplitudes. The black line marks the potential core boundary, defined as the region where the mean velocity $\overline {V}$ exceeds $95\,\%$ of its value at the nozzle outlet centreline.

Figure 3.

Spatial maps of (a–e) the velocity fluctuations and (f–j) their spectra for a globally unstable jet forced transversely at $f_{\!f}/f_n=1.09$ . Root mean square velocity fluctuations $V_{\textit{rms}}^\prime$ for five forcing amplitudes: (a) unforced, (b) low-amplitude forcing, (c) moderate-amplitude forcing, (d) critical amplitude at lock-in, and (e) above the lock-in threshold. (f–j) The PSDs of the centreline velocity fluctuations ( $V_{c}^\prime$ ) along $x/D$ for the corresponding forcing conditions. The unforced jet (a, f) exhibits strong periodic oscillations at a well-defined natural frequency ( $f_n$ ). Below lock-in, asynchronous oscillations occur with spectral broadening around $f_{\!f} /f_n = 1$ in the PSD in (g,h), and the oscillation amplitudes gradually decrease with $A_{\!f}$ in (b,c). At lock-in and beyond, the jet is synchronised to $f_{\!f}$ in (i,j), with a substantial drop in its oscillation amplitude in (d,e).

The unforced jet exhibits low-amplitude oscillations at the nozzle outlet ( $x/D \approx 0$ ). With downstream development, the jet amplitude $V^\prime _{\textit{rms}}$ grows and reaches a local peak within the potential core at $1 \lt x/D \lt 2$ , as seen in figure 3(a). The corresponding PSD (figure 3 f) shows the fundamental frequency of the natural global mode ( $f_n$ ) as the dominant spectral feature, evidenced by a sharp peak at $f_n$ . Farther downstream ( $x/D \gt 2.3$ ), a dominant spectral peak emerges at $f_n/2$ . This subharmonic dominance coincides with large $V_{\textit{rms}}^\prime$ values outside the potential core, typically associated with vortex pairing (Kyle & Sreenivasan Reference Kyle and Sreenivasan1993).

When forced below the lock-in threshold, $V_{\textit{rms}}^\prime$ within the potential core decreases below the unforced levels, particularly in the region $1 \lt x/D \lt 2$ (figures 3 b,c). The corresponding PSDs (figures 3 g,h) reveal the emergence of a new spectral peak at $f_{\!f}$ , while the peak at $f_n$ broadens relative to the unforced case. These observations suggest the presence of several weaker spectral peaks surrounding $f_n$ . The coexistence of two incommensurate frequencies ( $f_n$ and $f_{\!f}$ ) and additional frequencies around $f_{\!f}$ indicates that the global mode exhibits quasi-periodic rather than purely periodic dynamics.

At lock-in (figure 3 d), $V_{\textit{rms}}^\prime$ within the potential core takes on substantially lower values. Further increases in the forcing amplitude above the lock-in threshold produce even greater reductions in $V_{\textit{rms}}^\prime$ without resonant amplification (figure 3 e). In the corresponding PSDs (figures 3 i, j), the spectral peak at $f_{\!f}$ becomes sharp and dominant, with complete suppression of the peaks at $f_n$ , indicating synchronisation of the global mode to the transverse forcing. The suppression of the global mode in the potential core of this synchronised jet – as illustrated by the transition from dark to light shading in figures 3(a,e) – indicates asynchronous quenching, consistent with the findings of our previous work (Kushwaha et al. Reference Kushwaha, Worth, Dawson, Gupta and Li2022). In the downstream region ( $x/D \gt 2$ ), the subharmonics of the forcing frequency ( $f_{\!f}/2$ ) dominate the PSD, attributable to vortex pairing under synchronised conditions.

3.1.2. Forced response to axial forcing

Figure 4 shows the hydrodynamic response of the jet under axial forcing applied at $f_{\!f}/f_n=1.09$ for five different amplitudes leading up to lock-in and beyond. As before, the colour maps depict the spatial distribution of $V_{\textit{rms}}^\prime$ , with the black line delineating the potential core boundary. Comparing figures 3 and 4 reveals both qualitative similarities and notable differences in the forced response between the axially and transversely forced jets. Below lock-in, the axially forced jet exhibits dynamics that initially resembles the transversely forced jet. Both forcing configurations produce: (i) a transition from limit-cycle periodicity to quasi-periodicity, evidenced by the emergence of a spectral peak at $f_{\!f}$ accompanied by additional peaks in the vicinity of $f_n$ ; and (ii) a modest reduction in $V'_{\textit{rms}}$ within the potential core, as indicated by changes in shading intensities in figures 4(b,c). These similarities suggest that both forcing types can disrupt the natural global mode and induce complex dynamical states at low forcing amplitudes. However, notable differences emerge as the forcing amplitude rises above the lock-in threshold. Unlike the transversely forced jet, the axially forced jet does not exhibit significant suppression of its amplitude at synchronisation. Instead, at forcing amplitudes exceeding the lock-in threshold, $V'_{\textit{rms}}$ recovers to levels comparable to the unforced state. This behaviour stands in sharp contrast to the continued attenuation of the oscillation amplitude observed under transverse forcing. This asymmetric response is consistent with the findings of Kushwaha et al. (Reference Kushwaha, Worth, Dawson, Gupta and Li2022), who demonstrated that transverse forcing leads to suppression across a wide range of forcing frequencies: $0.8\lt f_{\!f}/f_n\lt 1.2$ . In contrast, axial forcing exhibits a distinctly asymmetric response about $f_{\!f}/f_n = 1$ , wherein suppression occurs when $f_{\!f}/f_n \gt 1$ , while amplification occurs when $f_{\!f}/f_n \lt 1$ , as illustrated in figure 5(a).

Figure 4.

The quantities shown are the same as those in figure 3 but for the axially forced jet. When synchronised with $f_{\!f}$ , the axially forced jet maintains substantial oscillation amplitudes throughout the lock-in regime, contrasting sharply with the oscillation suppression arising from transverse forcing.

Figure 5.

(a,b) Root mean square of the total velocity fluctuations ( $V^\prime _{\textit{rms}}$ ), and (c,d) the normalised PSD of $V^\prime$ along the centreline of the jet when it is forced purely axially, at (a,c) $f_{\!f}/f_n=0.82$ and (b,d) $f_{\!f}/f_n=1.09$ . (e) The normalised PSD of the velocity fluctuations acquired from HWA experiments when the jet is forced purely axially across a range of frequencies ( $0.8\lt f_{\!f}/f_n\lt 1.18$ ) at lock-in.

The contrasting amplitude responses observed under axial forcing are closely linked to the distinct spectral signatures in the downstream region ( $x/D \gt 2$ ). When the jet is axially forced above its natural frequency ( $f_{\!f}/f_n = 1.09$ ), its PSD reveals a pronounced spectral peak at the subharmonic frequency ( $f_{\!f}/2$ ), as shown in figure 5(d). This subharmonic emergence signifies large-scale vortex interactions and marks the onset of vortex pairing, a key nonlinear phenomenon discussed further in § 3.3.3. Conversely, when forced below its natural frequency ( $f_{\!f}/f_n = 0.82$ ), the jet oscillates strictly at $f_{\!f}$ , with no discernible $f_{\!f}/2$ subharmonic component in the PSD (figure 5 c). The absence of this subharmonic signature implies that successive vortices advect downstream without merging, thereby preserving their individual coherence. This lack of nonlinear energy transfer from the fundamental mode to the subharmonic components contributes directly to the observed amplification of the oscillation amplitude under synchronised conditions. These spectral differences are further substantiated by HWA measurements taken at $x/D \approx 1.5$ , as shown in figure 5(e). These local PSD measurements corroborate the global trends: subharmonic content remains absent when $f_{\!f}/f_n \lt 1$ , while pronounced subharmonic peaks appear when $f_{\!f}/f_n \gt 1$ . Collectively, these observations underscore the central role of local vortex interactions in shaping the global response of the jet to axial forcing. The connection between vortex advection and global-mode modulation will be examined further in § 3.3.

3.3.1. In response to transverse forcing

Figure 7 shows maps of the normalised phase-averaged vorticity ( $\omega _z^*$ ) in the $x$ – $y$ plane at six time instants within one natural oscillation cycle of the unforced self-excited jet. To quantitatively characterise these dynamics, we first identify coherent vortical structures using the $\lambda _2 \lt 0$ criterion, which distinguishes rotational cores from shear-dominated regions (Jeong & Hussain Reference Jeong and Hussain1995). Following the identification of vortical structures near the nozzle exit at $t/T = 0$ , we track their temporal evolution via successive phase snapshots by identifying regions of $\lambda _2 \lt 0$ that exhibit partial spatial overlap with those detected in the preceding phase instant. The streamwise and transverse positions of vortex cores are quantified by computing the $x$ -centroid and $y$ -centroid coordinates of the $\lambda _2\lt 0$ regions, as per Saffman (Reference Saffman1992):

(3.1) \begin{align} x_v^{*} = \frac { \iint _S 2\pi x^{*}y^{*}\omega _{z}^{*} \,{\rm d}x^{*}\, {\rm d}y^{*}}{ \iint _S 2\pi y^{*}\omega _{z}^{*} \,{\rm d}x^{*}\, {\rm d}y^{*}}, \quad y_v^{*} = \frac { \iint _S 2\pi x^{*} y^{*} \omega _{z}^{*} \,{\rm d}x^{*}\, {\rm d}y^{*} }{ \iint _S 2\pi x\omega _{z}^{*} \,{\rm d}x^{*}\, {\rm d}y^{*} }, \end{align}

where $^{*}$ represents non-dimensionalisation, with $x_v^{*} =x/D$ and $y_v^{*}=y/D$ denoting the non-dimensional $x$ -centroid and $y$ -centroid, respectively, and $\omega _{z}^{*} = \omega _{z}D/u_e$ denoting the non-dimensional $z$ -component of vorticity.

Figure 7.

Maps of the normalised iso-vorticity $\omega ^* = \omega D/U_e$ in the $x{-}y$ plane corresponding to six phases of the natural oscillation cycle of the unforced jet. When unforced, the vortical structures on both sides of the jet centreline evolve and advect in phase with each other.

To further quantify the vortex dynamics, we compute the circulation of individual vortical structures during their downstream advection, and the total circulation on either side of the jet centreline for various forcing amplitudes. This enables us to assess the influence of the forcing amplitude on the vortex growth characteristics, and determine the pinch-off time, which is defined as the moment when vortical structures cease entraining vorticity from the surrounding shear layers. The methodology for estimating the vortex ring and total circulation follows the approach established by Gharib et al. (Reference Gharib, Rambod and Shariff1998), with specific modifications made to exclude contributions from the preceding oscillation cycles. To isolate the circulation of individual vortices, we establish upper and lower spatial bounds relative to the vortex core centroid at each time instant, with circulation calculations performed for vortices tracked starting from $t/T = 0$ and progressing through successive time steps. The circulations for left-hand-side (LHS) and right-hand-side (RHS) vortices are computed separately by excluding positive and negative vorticity regions, respectively. The total circulation is estimated via a domain integration approach wherein the lower spatial bound remains fixed at $x/D = 0$ , while the upper bound translates in accordance with the instantaneous vortex structure centroid position. The non-dimensional circulation ( $\varGamma ^*$ ) is computed by integrating the vorticity magnitude over regions satisfying $|\omega _z^*| \geqslant 1$ at each time step, as expressed mathematically in (2.2). This framework enables systematic tracking of vortex formation, advection and interaction processes throughout an oscillation cycle, providing quantitative insight into the symmetry-breaking phenomena observed within the phase-averaged vorticity fields.

In the unforced state (figure 7), the jet exhibits clear axisymmetric behaviour: spatially growing instabilities within the laminar shear layers undergo synchronous roll-up near the nozzle exit ( $x/D \approx 0$ ), forming symmetric vortex cores at $t/T=0$ (figure 7). These vortical structures evolve into coherent axisymmetric vortex ring structures whose LHS and RHS components remain synchronised, occupying identical axial positions while evolving along congruent downstream trajectories (figure 8 a). This spatial and temporal alignment indicates that both LHS and RHS vortices advect in phase with a common convective velocity. As these vortical structures advect downstream, they continuously entrain and accumulate vorticity from the trailing shear layers, resulting in a steady increase in both individual vortex circulations ( $\varGamma _{vr}$ ) and the total circulation ( $\varGamma _{\textit{tot}}$ ) on both sides of the jet centreline (figure 8 f). These advecting vortices undergo pinch-off within the streamwise region $1 \lt x/D \lt 2$ where their self-induced velocity exceeds the mean shear-layer velocity. This pinch-off process is reflected in the circulation plateau at $\varGamma _{vr}^* \approx 1$ at $t/T \approx 1$ for both LHS and RHS vortices (figure 8 f), indicating cessation of vorticity entrainment. In the far field ( $x/D \gt 2$ ), successive vortical structures undergo pairing and merging, inducing structural deformation and coherence degradation through nonlinear vortical interactions (figure 7). This downstream interaction correlates with the emergence of the $f_n/2$ subharmonic in the far field, consistent with the spectral characteristics observed in figure 3.

Figure 8.

Temporal evolution of vortical structures under transverse forcing at $f_{\!f}/f_n = 1.09$ for five forcing amplitudes. (a–e) Axial positions of vortex cores as functions of normalised time $t/T$ for LHS (blue) and RHS (red) vortical structures. At lock-in (d) and beyond (e), vortical structures on both sides of the jet centreline exhibit phase differences $90^\circ$ and $120^\circ$ , respectively. (f–j) Vortex ring circulation $\varGamma ^*_{vr}$ and total circulation $\varGamma ^*_{\textit{tot}}$ as functions of $t/T$ for the corresponding forcing conditions. The vortex ring circulation initially grows with $t/T$ before saturating at $\varGamma ^*_{vr}\approx 1$ across all forcing amplitudes, indicating that the vortical structures reach a saturation threshold where additional vorticity entrainment from the shear layers ceases despite continued increases in the forcing amplitude.

Figure 9.

The quantities shown are the same as those in figure 7 but for the quasi-periodic state when forced transversely at two different amplitudes: (a) low amplitude, $A_{\!f} =0.56$ ; and (b) moderate amplitude, $A_{\!f} =0.72$ . While the initial shear layers on both sides of the jet centreline roll up in phase with each other, the vortical structures become out of phase as they evolve and advect downstream.

Figure 10.

The quantities shown are the same as those in figure 7 but at a lock-in state when forced transversely (a) at the critical lock-in amplitude $A_{\!f} =1$ and (b) above the lock-in amplitude $A_{\!f} =1.45$ . Under lock-in conditions, vortical structures on both sides of the jet centreline evolve and advect out of phase with each other, producing symmetry-breaking phenomena that underlie the suppression mechanism characteristic of transverse forcing.

When transversely forced at amplitudes below the lock-in threshold, the jet exhibits quasi-periodicity characterised by complex temporal dynamics (Kushwaha et al. Reference Kushwaha, Worth, Dawson, Gupta and Li2022). In this state, the initial shear layers near the nozzle exit roll up axisymmetrically at $t/T = 0$ , preserving in-phase vortex formation on both sides of the jet centreline. This symmetric roll-up is evident in figures 9(a,b) and quantitatively corroborated by figures 8(b,c), revealing that during early convection ( $t/T \lt 1$ ), the LHS and RHS vortex cores follow nearly congruent axial trajectories. However, as the vortical structures advect progressively downstream beyond $t/T \gt 1$ , systematic phase divergence develops between the cores. This asymmetric evolution manifests through increasingly distinct axial trajectories for the respective vortex cores, as seen in figure 8(b). This implies differential convective velocities between the two sides for any given time instant. The resulting asymmetric evolution leads to preferential circulation accumulation on one side of the centreline, while the individual vortex circulation strength at pinch-off remains approximately symmetric across the jet centreline (figure 8 g). In the far field ( $x/D \gt 2$ ), vortical structures undergo straining and deformation processes that promote structural fragmentation (figure 9 a). This degradation is evidenced by the bifurcation of the LHS vortex into two distinct segments near $x/D \approx 2.5$ at $t/T = 2$ (figure 8 b). At moderate forcing amplitudes (figures 9(b) and 8(c,h)), this premature splitting event initiates a cascade of effects: divergent convective velocities emerge between the separated vortical segments, culminating in the degradation of overall vortical coherence. This phenomenon is attributed to the intensified transverse velocity fluctuations accompanying the increased forcing amplitude. When forced transversely at the critical amplitude required for synchronisation, the jet undergoes profound symmetry breaking that fundamentally restructures the spatial distribution of vortical structures. The initial shear-layer roll-up processes become temporally phase-shifted, with the LHS and RHS shear layers no longer rolling up synchronously (figure 10 a). This asymmetric roll-up results in vortex cores consistently occupying distinct axial positions throughout the forcing cycle, leading to a persistent spatial separation between opposing vortical structures (figure 8 d). Consequently, a consistent $90^\circ$ phase lag develops between opposing vortices, producing a staggered arrangement of vortical structures in the near field (figure 10 a). This spatiotemporal offset manifests as an unequal total circulation distribution on either side of the jet centreline, with one side exhibiting preferential circulation accumulation relative to the other (figure 8 i). When the forcing amplitude exceeds the lock-in threshold, the asymmetric evolution intensifies markedly: the axial separation between vortex cores further increases (figures 10(b) and 8(e)), causing the phase difference between vortical structures to rise from $90^\circ$ to approximately $120^\circ$ . This staggered configuration induces additional geometric distortion of the vortex ring structure, producing pronounced tilting of the vortical elements relative to the jet centreline, and amplifying the departure from axisymmetric flow organisation. Despite the pronounced out-of-phase roll-up dynamics of the shear layers, a remarkable feature emerges: the peak circulation strength of the individual vortical structures remains invariant across all forcing amplitudes (figures 8 i, j). Each vortical structure exhibits linear circulation growth during the vorticity entrainment phase, followed by saturation at $\varGamma _{vr}^* \approx 1$ . The temporal offset inherent to asymmetric vortex formation results in staggered pinch-off times, but both sides ultimately achieve an equivalent circulation strength at their respective saturation points.

3.3.2. Symmetry breaking in a transversely forced jet

Having demonstrated that transverse forcing fundamentally disrupts the inherent axisymmetry of the $m=0$ global mode through systematic phase shifting of opposing shear layers, we next provide a quantitative metric to assess the symmetry-breaking phenomenon. First, we extract instantaneous streamwise vorticity profiles at $y/D =\pm 0.5$ , and apply the Hilbert transform to these profiles. We then construct normalised histograms of the phase shift $\varTheta$ between the LHS and RHS vortices, as shown in figure 11. For the unforced jet, the histogram peaks sharply at $\varTheta =0$ , confirming the in-phase roll-up of the axisymmetric $m=0$ global mode. However, figure 11 also shows that transverse forcing induces systematic phase divergence, which intensifies with increasing forcing amplitudes. Specifically, when forced below the critical amplitudes for which the jet exhibits quasi-periodicity, both the LHS and RHS vortical structures formed from the in-phase roll up of the initial shear layers gradually become out of phase as they advect downstream. This is evidenced by a broadening of the phase difference histogram around small but finite values of $\varTheta$ in figure 11, with $\varTheta$ peaking at approximately $\pi /4$ . At lock-in, the histogram peaks at approximately $\varTheta =\pi /2$ , corroborating the temporal offset between opposing shear-layer roll-up events. When the forcing amplitude exceeds the lock-in threshold, the phase difference further increases to $2\pi /3$ , producing pronounced staggered vortical arrangements. These progressive phase shifts demonstrate that transverse forcing generates a transverse flapping motion that systematically breaks the $m=0$ axisymmetry by inducing temporally offset, spatially staggered roll-up of vortical structures in the opposing shear layers.

Figure 11.

Normalised histogram of the phase shift $\varTheta$ between the left- and right-hand shear layers of a jet forced transversely at $f_{\!f} /f_n = 1.09$ for four different amplitudes: (i) unforced where $\varTheta$ peaks at 0, (ii) moderate forcing during quasi-periodicity where $\varTheta$ peaks at $\pi /4$ , (iii) lock-in onset where $\varTheta$ peaks at $\pi /2$ , and (iv) beyond lock-in where $\varTheta$ peaks at $2\pi /3$ . Progressive increases in phase shift values $\varTheta$ indicate that transverse forcing breaks the $m=0$ axisymmetry.

While modal interactions between the natural $m=0$ mode and the excited $m=\pm 1$ modes can lead to oscillation suppression, mean-flow modification represents another plausible suppression mechanism. Specifically, transverse forcing can act as a counter to the natural axisymmetric mean flow by inducing strong in-phase transverse velocity fluctuations around the jet centreline. This modified mean flow can nonlinearly interact with the $m=0$ global mode, weakening the axisymmetric oscillations, and producing a drastic amplitude reduction. This stabilisation mechanism through mean-flow modification is analogous to that demonstrated by Skene, Qadri & Schmid (Reference Skene, Qadri and Schmid2020), where counter-rotating forcing stabilised swirling jets through analogous base-flow alterations.

3.3.3. In response to axial forcing

Figure 12 shows the phase-averaged vorticity distribution in the $x{-}y$ plane, akin to figure 8, but focusing on the axially forced jet. As with the unforced case, figure 12(a) shows that the initial shear layers on both sides of the jet centreline roll up axisymmetrically, forming coherent vortical structures that remain in phase throughout near-field advection. These synchronously advecting vortical structures produce nearly equal total circulations on either side of the jet centreline, as quantitatively corroborated by figures 12(a) and 12( f).

Figure 12.

The quantities shown are the same as those in figure 8 but for the axially forced jet. At lock-in (d) and beyond (e), both the LHS and RHS vortical structures on either side of the jet centreline occupy identical axial positions during downstream advection, confirming in-phase evolution that preserves the axisymmetric flow topology.

In the quasi-periodic regime, the spatial organisation of vortical structures in the near field of the jet exhibits striking similarities to configurations observed under transverse forcing (cf. figures 10(a,b) and 13(a,b)). The axial trajectories of both LHS and RHS vortex cores remain nearly aligned throughout the initial convection period $t/T \lt 1$ , as demonstrated in figures 12(b,c). This indicates that the in-phase roll-up of the initial shear layers between opposing sides of the jet centreline remains preserved during the early stages of vortical formation (figures 13 a,b). However, fundamental distinctions emerge during downstream evolution when $t/T \gt 1$ . The axially forced jet exhibits no evidence of vortex fragmentation or splitting as the vortical structures advect downstream, contrasting markedly with bifurcation behaviour characteristic of transverse forcing. Notably, the phase disparity that develops between advecting vortical structures on opposing sides of the jet centreline is substantially higher for the axially forced jet than for the transversely forced jet, as evidenced in figures 12(b,c). This enhanced phase disparity generates differential convective velocities between the LHS and RHS vortical structures, establishing an asymmetric vorticity flux distribution that preferentially accumulates circulation on the side characterised by higher convective velocity. Consequently, the total circulation strength $\varGamma _{\textit{tot}}^*$ becomes markedly unequal between the two sides of the jet centreline, as quantitatively demonstrated in figures 12(g,h). The progressive downstream advection of vortical structures at differential convective velocities amplifies this asymmetric evolution, ultimately producing a heightened degree of non-axisymmetric flow organisation during the quasi-periodic state (figures 13 a,b). This non-axisymmetric distribution of vortical structures within the near field exhibits direct correlation with amplitude reduction in the oscillation response, demonstrating a consistent mechanism that operates irrespective of the specific forcing symmetry, whether axial or transverse.

Figure 13.

The quantities shown are the same as those in figure 7 but for the quasi-periodic state when forced axially at two different forcing amplitudes: (a) low amplitude, $A_{\!f} =0.31$ ; and (b) moderate amplitude, $A_{\!f} =0.57$ . While the initial shear layers on both sides of the jet centreline roll up in phase with each other, the vortical structures become out of phase as they evolve and advect downstream.

When the forcing amplitude reaches or exceeds the lock-in threshold – the critical forcing amplitude at which the jet achieves synchronisation to $f_{\!f}$ – the spatial organisation of vortical structures undergoes transition from the asymmetric to the natural axisymmetric configuration characteristic of the unforced jet, as visualised in figures 14(a,b). This axisymmetric spatial realignment emerges as a consequence of phase-coherent shear-layer roll-up processes, wherein successive vortices formed in the near field of the jet maintain consistent phase relationships throughout their downstream evolution. This transition to axisymmetry arrangement is reflected in figures 12(d,e), which show that LHS and RHS vortex cores consistently occupy identical axial positions throughout their downstream advection process. This spatial alignment ensures that both LHS and RHS vortical structures advect with equal convective velocities, thereby establishing a uniform vorticity flux distribution across the jet centreline (figures 12 i, j). The circulation strength $\varGamma _{vr}^*$ for both LHS and RHS vortical structures exhibits simultaneous, in-phase growth from $t/T=0$ , subsequently reaching a concurrent plateau at maximum circulation strength $\varGamma _{vr}^*=1$ at approximately $t/T\approx 1$ , as evidenced in figures 12(i, j). This synchronous temporal evolution of vortical structures on both sides of the jet centreline closely resembles the behaviour observed in the unforced jet configuration, indicating that axial forcing at or above the lock-in amplitude effectively preserves the $m=0$ azimuthal symmetry of the natural jet while imposing temporal periodicity.

Figure 14.

The quantities shown are the same as those in figure 7 but at a lock-in state when forced axially (a) at critical lock-in amplitude $A_{\!f} =1$ and (b) above lock-in amplitude $A_{\!f} =1.68$ . Under lock-in conditions, vortical structures on both sides of the jet centreline evolve and advect in phase, preserving the axisymmetric topology of the unforced jet. This in-phase evolution maintains structural symmetry and prevents the symmetry-breaking dynamics observed under transverse forcing, thereby sustaining strong oscillation amplitudes as the jet transitions through lock-in.

Figure 15.

The quantities shown are the same as those in figure 7 but for the axially forced jet at lock-in when forced at (a) $f_{\!f}/f_n=1.09$ and (b) $f_{\!f}/f/n=0.8$ . When $f_{\!f}/f_n\gt 1$ as in (a), vortical structures remain more compact and undergo vortex merging as they advect downstream. Conversely, when $f_{\!f}/f_n\lt 1$ as in (b), the coherent vortical structures exhibit increased spatial growth and persistence without undergoing vortex merging as they advect downstream.

Figure 16.

The quantities shown are the same as those in figure 15 but for the transversely forced jet.

We now examine the development of vortical structures in the near field under lock-in conditions for both axially and transversely forced jets at two distinct frequency ratios: $f_{\!f}/f_n\lt 1$ and $f_{\!f}/f_n\gt 1$ . Figures 15 and 16 reveal that the streamwise separation between successive vortices exhibits pronounced dependence on the frequency ratio for both forcing symmetries. Vortex spacing increases substantially when $f_{\!f}/f_n \lt 1$ (figures 15(b) and 16(b)) compared to conditions when $f_{\!f}/f_n \gt 1$ (figures 15(a) and 16(a)), reflecting the wavelength characteristics of the induced perturbations. Lower forcing frequencies ( $f_{\!f}/f_n \lt 1$ ) generate longer wavelengths and correspondingly increased inter-vortex spacing, which inhibits interactions with downstream structures formed during previous cycles and hinders vortex merging, as evidenced in figures 15(b) and 16(b). Under these conditions, the vortical structures on both sides of the jet continue to grow while maintaining structural coherence during downstream advection. Conversely, when forced at $f_{\!f}/f_n \gt 1$ , successive vortices exhibit reduced streamwise separation, facilitating interactions with downstream structures and promoting vortex merging. The advecting vortical structures consequently undergo rapid loss of structural coherence, and experience diffusive spreading during the merging process, as evidenced in figures 15(a) and 16(a). Although the spatiotemporal dynamics of vortical structures exhibit similar trends for both forcing orientations, a fundamental asymmetry emerges in the amplitude response of axially forced jets. Specifically, axial forcing produces oscillation amplitude enhancement when $f_{\!f} \lt f_n$ , while generating amplitude attenuation when $f_{\!f} \gt f_n$ , as demonstrated in figure 5. This asymmetric behaviour about $f_{\!f}/f_n = 1$ can be attributed to the interplay between the jet response time scale ( $t_r \propto 1/f$ ) and power transfer mechanisms involving the fundamental mode, subharmonic mode and applied forcing (Davitian et al. Reference Davitian, Getsinger, Hendrickson and Karagozian2010a ). For $f_{\!f} \lt f_n$ , the extended response time ( $t_{r,f} \gt t_{r,n}$ ) provides greater temporal duration for forcing input to act upon the jet, resulting in enhanced power transfer from the external forcing to the global mode. This increased energy transfer manifests as higher oscillation amplitudes. Conversely, at $f_{\!f} \gt f_n$ , the reduced interaction time ( $t_{r,f} \lt t_{r,n}$ ) limits power transfer efficiency, leading to diminished oscillation amplitudes. Furthermore, when subjected to axial forcing at $f_{\!f} \gt f_n$ , energy redistribution occurs within the flow field through nonlinear interactions between successive vortices, particularly vortex pairing phenomena. This process facilitates energy transfer from the dominant global mode to subharmonic modes, contributing to amplitude reduction. The absence of such energy redistribution mechanisms when $f_{\!f} \lt f_n$ results in oscillation amplification. In contrast to axial forcing, transversely forced jets exhibit no such disparity in their amplitude response, regardless of whether $f_{\!f}/f_n \lt 1$ or $f_{\!f}/f_n \gt 1$ . This fundamental difference stems from the topological modifications imposed by transverse forcing on the naturally occurring vortical structures. Transverse forcing fundamentally reorganises the spatial distribution of vortical structures into a staggered arrangement within the near field, redistributing energy from the dominant global mode to alternative modes through systematic breaking of the $m = 0$ axisymmetric character of the global mode, as detailed in § 3.4.

3.4.1. Periodic states: unforced and forced synchronisation

Figures 17(a–c) show the normalised SPOD eigenvalue spectra of the vorticity field as functions of $f_{\!f}/f_n$ , providing insights into the relative distribution of modal energy between the leading SPOD mode (mode 1) and the subsequent modes of lower energy. A substantial separation between the eigenvalues of mode 1 and succeeding modes indicates that the leading mode possesses significantly greater energy and exerts dominance over the prevalent coherent structures within the flow field. The presence of a prominent peak in the eigenvalue spectra at the fundamental frequency $f_n$ and $f_{\!f}$ for unforced and forced configurations, respectively (as seen in figures 17 a–c), suggests that most of the energy is contained within these dominant modal frequencies.

Figure 17.

The SPOD of the vorticity field under periodic oscillations for three forcing conditions: (a,d) unforced jet, (b,e) axially forced jet at lock-in, and (c, f,g) transversely forced jet at lock-in. Shown are (a–c) SPOD eigenvalue spectra normalised by the total flow energy, (d–f) the leading SPOD modes with the highest eigenvalue at $f_n$ (unforced) and at $f_{\!f}$ (forced cases), and (g) the second SPOD mode for the transversely forced jet at $f_{\!f}$ .

Figures 17(d–g) show the SPOD modes of the vorticity field at the most energetic frequency for three different forcing conditions: unforced, axially forced and transversely forced. The jet exhibits periodic oscillations for all three cases, due to natural self-excitation or lock-in. Figures 17(d) and 17(e) show the leading SPOD mode at $f_n$ and $f_{\!f}$ for the unforced and axially forced jets, respectively. These dominant spatial modes manifest as a series of counter-rotating vortical structures forming coherent wavepackets that advect along the shear layers on both sides of the jet centreline. Notably, these localised wavepackets appear particularly energetic at $1 \lt x/D \lt 2$ , which coincides with the region of maximum velocity fluctuations. This region of maximum amplitude is downstream of the wavemaker region due to convective amplification of upstream disturbances (Lesshafft et al. Reference Lesshafft, Huerre, Sagaut and Terracol2006; Coenen et al. Reference Coenen, Lesshafft, Garnaud and Sevilla2017; Qadri et al. Reference Qadri, Chandler and Juniper2018); the region of maximum amplitude also differs from the region of maximum receptivity to forcing. Both the unforced jet (figure 17 d) and the purely axially forced jet (figure 17 e) exhibit vortical wavepackets with an antisymmetric pattern about the jet centreline, characteristic of an axisymmetric jet dominated by a global mode with azimuthal wavenumber $m = 0$ . Their corresponding SPOD spectra (figures 17 a,b) reveal a significant difference in the eigenvalues between the two leading SPOD modes (modes 1 and 2) at the dominant frequency. Physically, this implies that the majority of the energy supplied by the axial forcing to the jet is concentrated in the leading SPOD mode, which shares the same characteristic features as that of the unforced jet exhibiting $m=0$ oscillations. Put together, this finding provides compelling evidence that axial forcing not only preserves the inherent axisymmetry of the $m=0$ global mode but also reinforces it.

The SPOD spectra of the transversely forced jet (figure 17 c) reveal a marked reduction in eigenvalue separation between modes 1 and 2, suggesting departure from low-rank modal behaviour. This reduction indicates that the dynamics of the transversely forced jet undergoing periodic oscillations cannot be adequately represented by the leading mode alone, requiring contributions from multiple modes for complete characterisation. In figures 17( f,g), we show the first two leading SPOD modes (modes 1 and 2) at $f_{\!f}$ for the transversely forced jet. In contrast to unforced or axially forced configurations, these modes reveal that the vortical wavepackets are no longer antisymmetric about the jet centreline; instead, they exhibit a phase relationship of $\pi /2$ . The more energetic mode (mode 1) is localised upstream, while the subdominant mode (mode 2) is localised in the downstream region. Collectively, these findings indicate that transverse forcing redistributes the power associated with the $m=0$ axisymmetric mode to asymmetric modes, weakening the global mode and breaking its inherent axisymmetry. This modal energy redistribution provides the fundamental mechanism for the substantial amplitude suppression observed in transversely forced configurations, as the coherent energy concentration characteristic of the natural global mode becomes dispersed across multiple modal structures.

3.4.2. Quasi-periodic states

Figures 18(a,b) show the eigenvalue spectra for the vorticity field during quasi-periodic oscillations under both axial and transverse forcing conditions. As before, the substantial eigenvalue separation between the leading mode and subsequent modes, observed in both axially forced (figure 18 a) and transversely forced (figure 18 b) configurations, highlights the dominant role of mode 1 in governing the spatiotemporal dynamics of the flow field during this dynamical state. In addition to the prominent peaks at $f_{\!f}$ and $f_n$ , the eigenvalue spectra also reveal an additional peak at $f_{\!f}-f_n$ for the transversely forced case, which corresponds to the beating frequency. However, the amplitude at $f_{\!f}-f_n$ is approximately one order of magnitude lower than that of the primary spectral peaks at $f_{\!f}$ and $f_n$ , indicating that its energetic contribution to the overall jet dynamics remains comparatively limited.

Figure 18.

The SPOD of the vorticity field under quasi-periodic conditions for (a,c,d) the axial forcing and (b,e, f) the transverse forcing. Shown are (a,b) SPOD eigenvalue spectra normalised by the total flow energy, (c–f) the leading SPOD modes at $f_{\!f}$ and $f_n$ for the axially forced cases (c,d) and the transversely forced cases (e, f).

Figures 18(c–f) show the dominant SPOD mode at $f_n$ and $f_{\!f}$ for both the axially forced and transversely forced jets, representing the two most dominant frequencies in the SPOD spectra. For the axially forced jet, the spatial mode structures at both frequencies exhibit qualitatively similar characteristics, as visualised in figures 18(c,d). Specifically, the vortical wavepackets on both sides of the jet centreline maintain neither axisymmetric nor antisymmetric arrangements; instead, they demonstrate out-of-phase advection relative to each other. Crucially, the vortex strengths of both LHS and RHS wavepackets exhibit comparable magnitudes, indicating that the axial forcing maintains a balanced energy distribution between the two shear layers despite the phase asymmetry.

For the transversely forced jet (figures 18 e, f), the spatial SPOD mode reveals both similarities and differences compared to the axially forced case. Both LHS and RHS vortical wavepackets exhibit a staggered arrangement, indicating persistent out-of-phase behaviour analogous to the axially forced configuration. However, a notable disparity emerges in the vortex strengths between the LHS and RHS wavepackets, with this asymmetry persisting across both spatial mode structures extracted at $f_n$ (figure 18 e) and $f_{\!f}$ (figure 18 f). Physically, this asymmetry implies that the transverse forcing preferentially excites the shear layers on the side of the jet centreline relative to the other, constituting a fundamental departure from the inherent axisymmetry of the natural mode structure. This preferential excitation mechanism provides physical insight into how jets exhibiting identical dynamical states can be characterised by fundamentally different modal structures, resulting in distinct flow field dynamics. The SPOD mode shape corresponding to $f_{\!f}-f_n$ (not shown here for brevity) reveals that the spatial structure associated with this beating frequency is predominantly localised in the far-field region at $x/D \gt 2$ , well outside the potential core. This region is characterised by vortex pairing and merging events.

These findings provide insight into the underlying physical mechanisms through which different external forcing symmetries (i.e. axial and transverse) govern the flow dynamics in fundamentally different ways. While both forcing types can produce quasi-periodic states with comparable temporal characteristics, the spatial organisation of coherent structures reflects the specific symmetry properties imposed by each forcing configuration, driving the energy distribution and evolutionary pathways of the resulting flow field.