3.1. Duct-like upstream-propagating waves and the guided jet mode

The first upstream-propagation mechanisms that we consider are the duct-like modes presented in figure 3. These are the same waves predicted theoretically by Tam & Hu (Reference Tam and Hu1989), suggested to govern subsonic jet resonance in Tam & Ahuja (Reference Tam and Ahuja1990), and recently observed experimentally for the first time in Li et al. (Reference Li, He, Zhang, Hao, Wu and Liu2024). The present data, however, provide some surprising detail regarding the nature of these waves. In still frames, the upstream-propagating waves look distinctly like the stationary shock waves formed in an underexpanded jet; however, as demonstrated in figure 2(a), these waves are actually propagating upstream. Though these waves are well predicted by linear models, here their amplitude is sufficiently strong that their interaction with the mean flow is nonlinear, more closely resembling a train of upstream-propagating shock waves rather than acoustic waves. While the waves associated with the $m = 0$ azimuthal mode evident in figures 3(a) and 3(d) bear a resemblance to the stationary shock cells for an underexpanded jet, figure 3(b,c) reveal that waves associated with an $m=1$ resonance trace a helical pattern through the core of the jet that has no analogue in steady supersonic flow.

As discussed in § 1, predictive models for resonance that assume that the upstream-propagating wave is the duct-like mode (or the equivalent guided jet mode) of figure 3 perform very well for subsonic jets and some ideally expanded (Varé & Bogey Reference Varé and Bogey2023) and underexpanded (Ferreira et al. Reference Ferreira, Fiore, Parisot-Dupuis and Gojon2023) supersonic jets. A visual inspection of the Fourier-decomposed pressure fields presented in Bogey & Gojon (Reference Bogey and Gojon2017) and Gojon & Bogey (Reference Gojon and Bogey2017) illustrate why: the pressure fields inside and outside the jet are almost uncorrelated for some underexpanded-jet cases, which indicates that the upstream-propagating wave cannot be the guided jet mode. In other cases, a waveform that is contiguous across the shear layer is evident, consistent with the structure of the guided jet mode. The majority of ideally expanded jet cases also exhibit this contiguous waveform. For the cases where the phase of the wave is uncorrelated across the shear layer, a different form of wave must be involved in the resonance. In the next subsection, we instead consider an upstream-propagating wave that exists entirely in the freestream.

Figure 4. A nonlinear acoustic wave propagating through the freestream exterior to the jet. (a–d) A time series with 200 $\unicode{x03BC}\textrm s$ between each frame; here, $\textit{NPR} = 3.0$ , $z/D_e = 3$ , $Re=6.5\times 10^5$ . Visualisations are a composite of $\partial \rho / \partial x$ (upper) and $\partial \rho / \partial y$ (lower); the two visualisations were taken separately and manually phase matched. The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure4/Figure4.ipynb.

Figure 5. Instantaneous fluctuating velocity field obtained from PIV data in Weightman et al. (Reference Weightman, Amili, Honnery, Edgington-Mitchell and Soria2016). Profiles are streamwise velocity, averaged over 20 vectors in the radial direction, corresponding to coloured rectangles superimposed on the contour map.

3.2. Freestream upstream-propagating waves

The second form of upstream-propagating wave that we consider is a nonlinear wave travelling in the freestream, an example of which is shown in figure 4. These images, which are a posteriori reconstructions based on images of both the $\partial \rho / \partial x$ and $\partial \rho / \partial y$ gradients, were obtained from the visualisation of a jet from a converging rectangular nozzle at $\textit{NPR} = 3.0$ , impinging on a cylindrical surface at height $z/D_e = 3.0$ . The sensitivity of the schlieren technique to the direction of wave propagation is highlighted here; sections of the wave are missing in each gradient-field image. The strong nonlinearity of this wave is evident in the sharpness of the wavefront visible in the schlieren data, indicative of something more resembling an inverse-sawtooth rather than sinusoidal wave. To more clearly define the shock-like nature of these waves, we reconsider particle image velocimetry (PIV) data presented in Weightman et al. (Reference Weightman, Amili, Honnery, Edgington-Mitchell and Soria2016). These data concern a jet from the converging axisymmetric nozzle operating at pressure ratio $\textit{NPR} = (p_0/p_{\infty})$ and impinging on the cylindrical surface at $z/D=3.5$ . A contour of instantaneous streamwise velocity fluctuation is presented in figure 5, along with two streamwise velocity profiles extracted by averaging across 20 vectors in the vertical direction. Sharp wavefronts are visible in the acoustic field of the jet on both sides, corresponding to the upstream-propagating wave that forms part of the resonance loop. Considering that the streamwise velocity change across the waves is $\Delta u \approx 40\,\textrm m\,\textrm s^{-1}$ , the Rankine–Hugoniot relations indicate that the wave is in fact a shock with Mach number $M_s \approx 1.07$ , which would correspond to an effective sound pressure level $\textit{SPL} \approx 175\,\textrm dB$ . The regions over which the velocity is calculated are not exactly normal to the wave, so the actual Mach number may be slightly higher. Though the curves in figure 5 do not seem to exhibit a step change in velocity, this is due to the well-known particle lag behind the shock wave (Melling Reference Melling1997); considering the PIV and schlieren together, it is clear that this wave is shock-like.

Figure 6. Upstream-propagating wave that is nonlinear from its inception, with $\textit{NPR} = 4.0$ , $z/D = 4.5$ , $Re=6.5\times 10^5$ . (a,c) $\partial \rho / \partial x$ schlieren visualisations of the first and last instants where the exemplar upstream-propagating wave is visible. (b) Schlieren visualisations of the upstream-propagating wave in the Lagrangian frame of reference. (d) Image intensity profiles across the upstream-propagating wave as a function of axial position. The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure6_7.

These upstream-propagating nonlinear freestream waves are visible in the data at a wide range of operating conditions. Figure 6 presents an example from the converging–diverging axisymmetric nozzle, operating close to its design point $\textit{NPR} = 4.0$ , and impinging on the cylindrical surface at distance $z/D = 4.5$ . Figure 6(a) represents an initial snapshot in time, where a wave that has just reached the nozzle plane is visible, as well as a newly created wave at $x/D \approx 3.5$ ; it is the latter wave that we will track in this image. Figure 6(c) represents the penultimate wave position considered in this figure, while figure 6(b) presents three snapshots of the wave as it propagates upstream. Figure 6(d) presents streamwise plots of light intensity, averaged across 40 px in the vertical direction, plotted as a function of distance from the nozzle. The shape of the wavefront, at least as captured in the schlieren visualisation, changes minimally during its upstream propagation; the peak amplitudes are similar, as are the half-widths.

A careful examination of figure 6(c) reveals the presence of a different phenomenon: a new diffuse wavefront at $x/D \approx 4.0$ , seemingly replacing the sharp nonlinear wave from figure 6(a) in the next period of the resonance cycle. Figure 7 presents a clearer exemplar of this less-shock-like wave; the initial wavefront at $x/D = 3.5$ in figure 7(a), presented in close-up in figure 7(b), appears to represent a diffuse collection of waves, rather than a single nonlinear wave. Tracking this collection of waves as it moves upstream reveals a relatively rapid coalescence into a single sharp wavefront, with a narrow peak of high amplitude appearing by $x/D \approx 2$ . During this coalescence, the half-width of the intensity peak shrinks dramatically, along with a concomitant increase in the peak amplitude. Strong acoustic waves undergoing nonlinear steepening during propagation is a well-known phenomenon (Gee et al. Reference Gee, Sparrow, James, Downing, Hobbs, Gabrielson and Atchley2008; Baars et al. Reference Baars, Tinney, Wochner and Hamilton2014), though it is difficult to ascertain from these data whether we are observing a classical nonlinear steepening, the coalescence of waves from distributed sources, or most likely a combination of both. It is perhaps worth emphasising at this point that figures 6 and 7 are obtained from the same sequence of images, and as is clear from the sequence in figure 6, switching between sharp and diffuse waveforms can occur over a single period of the resonance cycle. No clear link between the amplitude or shape of the downstream-propagating KH wavepacket and the sharpness of the upstream-propagating wave could be educed from the data.

Figure 7. Acoustic waves undergoing coalescence to form a single nonlinear wave, with $\textit{NPR} = 4.0$ , $z/D = 4.5$ , $Re=6.5\times 10^5$ . (a,c) $\partial \rho / \partial x$ schlieren visualisations of the first and last instants where the exemplar upstream-propagating wave is visible. (b) Schlieren visualisations of the upstream-propagating wave in the Lagrangian frame. (d) Image intensity profiles across the upstream-propagating wave as a function of axial position. The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure6_7.

If we continue with our consideration of an axisymmetric jet impinging on a cylinder, which has so far been able to produce both shock-like waves and coalescing waves, then a further increase in nozzle pressure ratio to $\textit{NPR} = 4.8$ will introduce yet another member of the upstream-propagating-wave family: a merging of two initially nonlinear waves, as depicted in figure 8. Here, we have two shock-like waves that appear to have originated from different sources in the impingement region or wall jet, that merge together into a single complex waveform. As described in detail in Willis et al. (Reference Willis, Cormack, Tinney and Hamilton2023), the merging of two nonlinear waves at grazing angle of incidence is well predicted by the Khokhlov–Zabolotskaya–Kuznetsov equation; even very weakly nonlinear acoustic waves, at grazing angles of incidence, can undergo a nonlinear merging resulting in the formation of a central Mach stem (Marchiano et al. Reference Marchiano, Coulouvrat, Baskar and Thomas2007). As demonstrated in figure 5, the upstream-propagating waves generated by an impinging jet can be shock-like from their inception, meaning that they easily surpass the criteria for the formation of a central Mach stem at these angles of incidence (Karzova et al. Reference Karzova, Khokhlova, Salze, Ollivier and Blanc-Benon2015).

Figure 8. Exemplar image sequence for nonlinear wave coalescence, images spaced by $33\,\unicode{x03BC}\textrm s$ , with $NPR = 4.8$ , $z/D = 4.5$ , $Re=8.6\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure8/Figure8.ipynb.

We have seen examples thus far of duct-like waves, freestream waves, coalescence of multiple weak waves, and merging of shock-like waves. At what impingement conditions do these occur? Regrettably, no simple parametric map appears to be possible, as a jet operating at a single condition can, over a short period of time, produce single waves, double waves that coalesce, and double waves that do not coalesce. Examples of single waves and distinct double waves are provided in figure 9; all four images presented here are taken from the same jet at $\textit{NPR} = 2.5$ striking a cylindrical surface at $z/D = 3.0$ . For other conditions, such as that presented in figure 7, both initially shock-like and initially diffuse upstream-propagating waves were observed. Shock-like waves were observed at both the low and high ends of the NPR range considered here, and for impingement over a range of distances from the nozzle. While a direct link between the nature of the upstream-propagating wave and the impingement conditions is appealing, at least in these data, nature appears uncooperative. Nonetheless, we have classified one key set of ingredients for resonance, in the form of the various different ways that the loop can be closed by upstream-propagating waves. We now turn our attention to the mechanisms by which these diverse waves are generated.

Figure 9. Examples of both single and double nonlinear waveforms generated at the same operating conditions: $\textit{NPR} = 2.50$ , $z/D = 3.0$ , $Re=5.9\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure9/Figure9.ipynb.

4.1. Vortex sound

We first address the generation mechanism associated with the arrival of vortices at the wall, though with the caveat that we will not present new data here, but only attempt to summarise existing research, and contextualise it in terms of new knowledge. The sound generated through the interaction of an isolated vortex ring with a solid wall has been studied in detail (Kontis et al. Reference Kontis, An, Zare-Behtash and Kounadis2008), and is generally well predicted by the theory of vortex sound (Howe Reference Howe2003). An axisymmetric vortex ring impinging upon a wall produces sound of a quadrupolar nature (Nakashima & Inoue Reference Nakashima and Inoue2008), with the quadrupole aligned such that the directions of maximum radiation are normal and tangential to the wall, centred on the vortex core. While the shear-layer vortices associated with the KH wavepacket are not identical to an isolated vortex ring, their pressure distribution is similar, with an $m=0$ wavepacket producing peak pressure fluctuations at the centreline of the jet (Tam & Hu Reference Tam and Hu1989). The closest analogy is a compressible vortex ring generated from a shock tube, which carries a trailing jet rather than being an isolated structure. The impact of such a structure on a solid wall produces a pressure maximum at the centreline of the vortex ring (Minota, Nishida & Lee Reference Minota, Nishida and Lee1997), behaviour that is also observed in the collision of two compressible vortices at a plane of symmetry (Minota, Nishida & Lee Reference Minota, Nishida and Lee1998), equivalent to the collision of a vortex ring with an inviscid wall. Thus it is reasonable to hypothesise that the mechanism of vortex sound will produce an effective source at the core of the jet.

The fact that the source is located within the core of the jet, in the context of recent work, explains why the upstream-propagating waves associated with this mechanism are either duct-like for subsonic jets (Tam & Ahuja Reference Tam and Ahuja1990; Varé & Bogey Reference Varé and Bogey2022a ) or associated with the radially extended guided jet mode for supersonic jets (Tam & Norum Reference Tam and Norum1992; Varé & Bogey Reference Varé and Bogey2023). Nogueira et al. (Reference Nogueira, Cavalieri, Martini, Towne, Jordan and Edgington-Mitchell2024) demonstrated that for a given range of frequencies, sound generated in the core of the jet will undergo reflection and transmission processes at the shear layer that will produce an upstream-propagating wave that is either duct-like (when total reflection occurs) or decays exponentially in the radial direction (the previously named guided jet mode). This is, in fact, the only mechanism by which acoustic waves generated in the core of the jet can propagate upstream. We thus have an explanation for why in some regimes resonance predictions based on the bands of existence of the guided jet mode produce exceptionally robust predictions and explanations of impinging jet resonance (Varé & Bogey Reference Varé and Bogey2023). Further, as mentioned in § 1, for these same conditions, predictions of sound based on Curle’s analogy match closely with the simulation data, which supports the argument that vortex sound is the relevant mechanism for these flows.

To summarise, we thus hypothesise that when the dominant mechanism of sound production is vortex sound, the upstream-propagating wave will be the so-called guided jet mode, i.e. a finite-frequency-band internal reflection and transmission of acoustic waves generated in the jet core near the wall. We note also that the collision of even subsonic compressible vortex rings can produce waves of sufficient amplitude and nonlinearity to be classified as shocks, rather than acoustic waves, as demonstrated in Minota et al. (Reference Minota, Nishida and Lee1998) and more recently in Bauer et al. (Reference Bauer, Thomas, Baker, Johnson, Williams, Langenderfer and Johnson2022). The impact of the vortex upon the impingement surface is thus a candidate for producing some of the shock-like waves observed in § 3. It is not immediately apparent whether a weak shock generated within the jet could only propagate upstream for the finite-frequency range predicted by the model of Nogueira et al. (Reference Nogueira, Cavalieri, Martini, Towne, Jordan and Edgington-Mitchell2024); the model is linear and thus assumes waves of infinitesimal amplitude. Nonetheless, though these waves are strong for acoustic waves, they are still very weak for shock waves, thus we assume that the frequency constraints suggested by the linear model should hold in at least a qualitative sense.

The direct interaction of the vortex with the wall is not the only mechanism by which acoustic waves may be generated, however, particularly when the incoming jet is supersonic. We turn now to other potential mechanisms.

4.2. Wall-jet shocklet

Figure 10. Image sequence showing full-field and magnified shadowgraph of a shocklet in the wall jet generating an upstream propagating wave, with $\textit{NPR} = 5.5$ , $z/D=3.5$ , $Re=7.8\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure10/Figure10.ipynb.

We commence our discussion of other mechanisms by revisiting the wall-jet shocklet phenomenon first discovered in Weightman et al. (Reference Weightman, Amili, Honnery, Soria and Edgington-Mitchell2017b ). Then, as now, the shocklet was challenging to visualise, as it is located within the highly turbulent wall jet, and exists for only tens of microseconds at a time. Nonetheless, in figure 10 we present some new shadowgraph visualisations of the shocklet, obtained by using a rectangular jet impinging on a cylindrical surface. The data here are consistent with the explanation in Weightman et al. (Reference Weightman, Amili, Honnery, Soria and Edgington-Mitchell2017b ): the large vortex associated with the KH wavepacket moves from the primary jet to the wall jet, and in doing so generates a transient shock structure in the wall jet. As the vortex convects further down the wall and removes the imposed velocity field, the shocklet moves upstream, generating a nonlinear compression wave. Figure 10(a) shows the shocklet forming as the vortex moves past, and figure 10(b) shows the retrograde tilting of the shocklet as the vortex moves away. The resultant nonlinear upstream-propagating wave is visible in figures 10(c) and 10(d).

Though the mechanism described here does not differ from that in Weightman et al. (Reference Weightman, Amili, Honnery, Soria and Edgington-Mitchell2017b ), the new data herein have the advantage that they span a much larger range of operating conditions and nozzle and surface geometries. The location of the shocklet, and the degree of motion that it experiences, are observed to be highly case dependent. A general trend is that the shocklet moves radially from the jet with increasing Mach number. It is, in fact, not visible in most datasets for lower pressure ( $\textit{NPR} \leq 3.0$ ), but this does not mean that it is not present; it is simply that the turbulent wall jet so obscures this region that its presence or lack thereof cannot be determined. There are also many cases where the shocklet is visible in the images, but is not observed to undergo the tilting motion responsible for the production of acoustic waves. With the caveat that the flow is complex, and the relative strength of waves cannot be rigorously determined from schlieren images, in the present dataset it appears that the wall-jet shocklet is consistently responsible for the upstream-propagating wave that closes resonance when $\textit{NPR} \geq 4.5$ and $z/D \leq 4.0$ . Given what is known of the mechanism underpinning the wall-jet shocklet, it is reasonable to propose that the likely condition for its existence is that the wall jet fluctuates between supersonic and subsonic with the passage of large-scale vortices. With some caution, we hypothesise that the wall-jet shocklet as a mechanism for sound production plays no role in resonance for subsonic and low-Mach supersonic jets, and an increasing role as jet Mach number increases. We also suggest that the shocklet might be observed for lower NPR when the jet impinges onto a curved surface, as the wall jet will maintain a higher mean velocity in these cases. However, in the present data we also find the shocklet producing the upstream-propagating wave in the case of impingement upon a flat plate.

Critically, the shocklet generates the upstream-propagating wave well outside the jet core. Thus the frequency-dependent nature of the guided jet mode is irrelevant here; the guided jet mode only governs sound waves that originate from within the core of the jet (Nogueira et al. Reference Nogueira, Cavalieri, Martini, Towne, Jordan and Edgington-Mitchell2024). The shocklet also generates waves that are sharp and nonlinear from their inception; at the moment of its birth, the wave is actually a shock, and only through diffraction does it decay to something that might be more appropriately described as a nonlinear acoustic wave, acknowledging that the difference between the two is only a matter of degree. This shocklet thus provides a possible generation mechanism for the initially sharp waves observed § 3.

4.3. Shock leakage

The wall-jet shocklet is not the only shock-related mechanism for the production of sound, however. As detailed in § 1, the wall-jet shocklet shares many properties with the shock-leakage phenomenon identified in screeching jets (Edgington-Mitchell et al. Reference Edgington-Mitchell, Weightman, Lock, Kirby, Nair, Soria and Honnery2021); both mechanisms involve the unsteady motion of a shock under the action of a transient velocity field induced by the passage of a vortex. The authors of this paper, as well as others (Gojon & Bogey Reference Gojon and Bogey2017), have for some time hypothesised that shock leakage should be active at the stand-off shock in supersonic jets, and indeed calculations based on the ray-tracing model of Shariff & Manning (Reference Shariff and Manning2013) using experimentally derived velocity fields support this hypothesis, though they have not yet been published. Indeed, such a mechanism was observed in the high-fidelity numerical simulations of Varé & Bogey (Reference Varé and Bogey2022b ), who recognised it as the same mechanism active in screeching jets. In their data, the phenomenon was observed to produce strong nonlinear upstream propagating waves at some conditions, but not at others. In the present dataset, we identify multiple conditions where upstream-propagating waves do indeed appear to be generated by a shock-leakage mechanism. Before presenting some of these examples, we must acknowledge a significant limitation in the nature of schlieren imaging, and thus place a caveat on the discussion to follow. As described in § 4.1, the collision of a vortex ring with a wall can produce not only acoustic waves, but a shock wave that travels retrograde to the motion of the vortex. If such a mechanism is active in these impinging jets (and it likely is for many operating conditions), then an upstream-propagating shock may well be generated within the core of the jet. The arrival of this wave at the stand-off shock will displace the stand-off shock, and this displacement may, from the perspective of a schlieren image, manifest in a similar manner to shock leakage. Given that the turbulent wall jet prevents direct observation of where the vortex impacts the wall, we cannot categorically state whether in all cases the motion of the stand-off shock that we observe is driven by the passage of downstream-propagating vortices, or the arrival of upstream-propagating shocks. There are, however, cases where the stand-off shock is sufficiently far from the wall that shock leakage can be observed before the vortex arrives at the wall. The model of Shariff & Manning (Reference Shariff and Manning2013), which dictates that the primary determinant of shock leakage is the strength of the imposed vorticity field, predicts that significant leakage will occur for the PIV data previously presented in Weightman et al. (Reference Weightman, Amili, Honnery, Soria and Edgington-Mitchell2017b , Reference Weightman, Amili, Honnery, Edgington-Mitchell and Soria2019). Which mechanism is dominant is beyond the scope of the present work, and it may even be that the two mechanisms are synchronised.

With these caveats in mind, we turn to our first example, presented in figure 11. In an attempt to render the phenomenon clearly, we combine three different imaging approaches here: schlieren of the $\partial \rho / \partial x$ gradient, shadowgraph, and an a posteriori correlation-based phase average of the shadowgraph. Note here that the schlieren and shadowgraph were not taken simultaneously, and have been manually phase matched. The generation of a single upstream-propagating wave, produced by a rectangular jet impinging on a cylinder, is shown in images separated by $20\,\unicode{x03BC}\textrm s$ ; the cyan arrow indicates the wave in question, and the magenta arrow indicates the vortex responsible for its production. Figure 11(a) shows the initial leakage of the wave, which occurs just after the passage of the large KH vortex, consistent with the phenomenon observed in screeching jets (Edgington-Mitchell et al. Reference Edgington-Mitchell, Weightman, Lock, Kirby, Nair, Soria and Honnery2021). Figure 11(b) presents two slightly different manifestations of the shock-leakage process; a single contiguous upstream wave is visible in the schlieren, whereas in the shadowgraph (and more clearly in the phase-averaged version), a discontinuity is evident between segments of the upstream-travelling wave. This particular manifestation, which is visible in the present data at many different operating conditions, is a form of shock leakage that has not been observed in screeching jets to the best of our knowledge. The resonance process in impinging jets is much stronger than that observed in screeching jets, with the higher-amplitude forcing producing KH vortices that grow much more rapidly. Given the relationship between vortex strength and shock leakage (Suzuki & Lele Reference Suzuki and Lele2003), this will result in a much larger rotation of the wavefront normal of the section of shock that ‘leaks’. We see the effects of this stronger rotation here: part of the shock has actually been rotated so strongly that it is directed back towards the core of the jet. The discontinuity between the two wave segments is divided into regions where the wave is freely propagating in near-quiescent fluid, and where it is propagating through the shear layer of the jet (thus moving more slowly in the laboratory-fixed reference frame). By figure 11(c), the inner portion of the wave has been transmitted into the jet itself, leaving only the outer portion to continue propagating upstream. Figure 11(d) shows the commencement of a new shock-leakage process on the lower side of the jet, as the wave on the upper side continues towards the nozzle.

Figure 11. Visualisation of the shock-leakage phenomenon. Left: $\partial \rho / \partial x$ schlieren. Centre: shadowgraph. Right: phase-averaged shadowgraph. Here, $\textit{NPR} = 3.4$ , $z/D = 3.0$ , $Re=6.0\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure11/Figure11.ipynb.

4.4. Dual mechanisms

With three distinct mechanisms of upstream-wave generation identified, all of which are capable of producing the sharp nonlinear waveforms observed in figure 2(b), we now seek to explain the various manifestations of double-wave phenomena: pairs of waves that either merge (figure 2 c) or remain separate (figure 9). All three mechanisms are linked to the action of the large vortical structures that comprise the downstream-propagating portion of the resonance loop. In the case of vortex sound, it is the interaction of the vortices themselves with the wall that produces the upstream wave, whereas for shock leakage, it is the interaction with the stand-off shock, and for the shocklet, with the wall jet. As all three mechanisms are linked to the same vortical structure, it is reasonable to assume that there may be situations where more than one of these mechanisms is active. The vortex-sound mechanism should always produce an upstream-propagating wave if the induced velocity field of the vortex is influenced by the wall; the degree to which the vortex interacts with the wall will be governed by how far from the wall the mean flow develops a significant non-streamwise velocity component, which in turn is highly dependent on jet Mach number and impingement height. The shock-leakage mechanism is known to depend primarily on the strength of the vorticity fluctuations in the shear layer, but in impinging jets, the structure of the stand-off shock, annular or otherwise, likely also plays a role. The exact requirements for the shocklet to generate upstream waves remain unclear, but seem to be linked to both the strength of the vortical structure and the speed of the wall jet.

Figure 12. Phase-averaged shadowgraph showing the two separate mechanisms generating upstream-propagating waves in a single period of the resonance cycle, with $\textit{NPR} = 5.5$ , $z/D=3.5$ , $Re=7.8\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure12/Figure12.ipynb.

Figure 12 shows an exemplar sequence of phase-averaged shadowgraph images, whereby two discrete upstream-propagating waves are generated as part of the impingement cycle. The first wave, produced by a shocklet in the wall jet, is visible in figure 12(a); note that this wave was not actually produced by the vortex visible near the impingement surface, but was produced by its predecessor. This phenomenon, whereby two closely spaced waves can be generated by separate vortices, is observed for a range of operating conditions. Figure 12(b) shows the genesis of the second wave, a shock-like wave that is either the result of a shock-leakage process or the nonlinear wave resulting from the vortex interacting with the wall. figures 12(c) and 12(d) show the waves propagating upstream, with slightly different directionality, and a significant space between them.

Having considered a case where dual mechanisms generate pairs of waves with significant spatial separation, we now consider the same geometry with only a slight increase in back pressure, and observe a phenomenon at the opposite extreme, where two mechanisms generate waves that merge. The shadowgraph images in figure 13 show two discrete waves generated at different points in the wall jet: one with an origin at $y/D \approx -2$ produced by a shocklet, and another at $y/D \approx -1$ that is likely the product of either another shocklet, or a vortex-sound mechanism. Tracking the wave in figure 13(b–g), the same merging process observed in figure 8 is evident. Here, we thus have our explanation for the merging phenomenon (Willis et al. Reference Willis, Cormack, Tinney and Hamilton2023) observed in those earlier figures: two separate waves are produced by different mechanisms, and in fact are associated with different vortices in a given resonance cycle. At some flow conditions, as in figure 12, these waves remain distinct, while at others, such as in figure 13, they merge into a single waveform of even higher amplitude.

Figure 13. Shadowgraph showing merging of two initially nonlinear waves from spatially distinct sources, with $\textit{NPR} = 6$ , $z/D = 3.5$ , $Re=8.5\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure13/Figure13.ipynb.

The previous two phenomena (merging and non-merging waves) were observed at different pressure ratios. This might give rise to some hope that synchronisation of the waves depends in a straightforward manner on the jet Mach number. Regrettably, the flow continues to resist such straightforward classification; figure 14 shows that even at a single operating condition, waves can be observed remaining distinct, as in figure 14(a,c), or merging, as in figure 14(b,d). It is reasonable to imagine that if there is some phase delay between the production of the two waves, there should be a specific frequency at which these waves merge. In practice, however, these are highly turbulent flows, and the stochastic nature of the turbulence produces sufficient jitter in the phase and amplitude of the KH wavepacket that there is in fact a range of frequencies where merging may or may not be observed for a given resonance cycle. While our discussion has focused on how the waves behave as they move back towards the nozzle (indicated with a cyan arrow), looking outside the coloured boxes in figures 14(a) and 14(b) reveals what appears to be an additional wave (indicated by the magenta arrow); there are two distinct upstream-propagating waves by the time we reach the nozzle. One of these waves (cyan) has resulted from the merging process of two waves as depicted in figure 14(d): the magenta wave appears to have a different origin, a different direction of propagation, and a clear spatial separation from the merged or unmerged wave pair. This wave will be the subject of our final proposed mechanism, which is, in the view of the authors, perhaps the most intriguing, but also the hardest to clearly visualise.

Figure 14. Demonstration that waves generated by two mechanisms can (a,c) remain separate, (b,d) merge, with $\textit{NPR} = 3.8$ , $z/D=3.0$ , $Re=6.7\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure14/Figure14.ipynb.

Figure 15. (a) Initial and (b) final shadowgraph images showing the production of a double nonlinear freestream wave. (c–g) A sequence of a posteriori phase-averaged shadowgraphs, with the two components of the wave described in the text indicated by cyan and magenta arrows. Here, $\textit{NPR}=2.5$ , $z/D=3.0$ , $Re=5.9\times 10^5$ . The directory including exemplar image sequences for this figure and the accompanying Jupyter notebook can be accessed at https://www-cambridge-org.demo.remotlog.com/S0022112025103832/JFM-Notebooks/files/Figure15/Figure15.ipynb.

To study this final mechanism, we consider a series of shadowgraph images for an axisymmetric jet impinging upon the hemispherical surface, presented in figure 15. Figures 15(a) and 15(b) represent the initial and final states that we will consider; figure 15(a) is the beginning of a shock-leakage process on the lower half ( $-y$ ) side of the jet, while figure 15(b) is the production of two discrete waves on the upper side ( $+y$ ) side of the jet. Figure 15(c–g) are an a posteriori phase average based on a region of interest that moves with the wave in question; this is thus not a true phase average, but a post hoc phase average that has been constructed to emphasise particular features. The motivation for so doing is that while some phenomena appear quite clear when animated, they are difficult to identify in a series of still frames; the raw videos without phase averaging are provided in supplementary movie 11. The images in figures 15(a) and 15(c) show that as the initial shock-leakage event on the $-y$ side of the jet takes place, part of the wave (labelled in magenta) is rotated sufficiently that it propagates back into the jet, rather than towards the nozzle. This portion of the wave travels towards the jet centreline above the stand-off shock, until it reaches a region where the retrograde flow exceeds the speed of sound, typically a nascent oblique shock, and merges with this shock. In some configurations, components of this wave are observed to exit the jet on the other side, though in many cases, the timing is such that no component of this transverse wave can cross the jet above the stand-off shock. Our focus here instead is the portion of this wave that moves below the stand-off shock, in the region where the flow is subsonic. That this wave exists at all is surprising; the wavefront normal is rotated almost at right angles to its undisturbed shape, as shown in figure 15(d). By figure 15(e), the arrival of a vortex on the $+y$ side of the jet has begun the motion of the shock (in cyan) that will shortly produce another upstream-propagating wave by way of shock leakage. The transverse wave is close but measurably lagging the new wave associated with shock leakage, still completing its transverse motion as the section of shock indicated in cyan begins to move upstream in figure 15(f). Finally, in figure 15(g), the two discrete waves are seen moving upstream; leading the way is the more recently produced wave from the $+y$ side, while close behind is a wave whose provenance was on the $-y$ side of the jet. The final directivity of these waves is quite different; the transverse wave is strongest in the radial direction, while the shock leakage process sends strong waves directly back upstream towards the nozzle.

In § 3, we identified five distinct families of upstream-propagating waves: duct-like, single nonlinear freestream, double nonlinear freestream, merging nonlinear waves, and coalescing diffuse waves. We have now proposed generation mechanisms for all of these bar the last. We also have not directly linked the mechanism described in Henderson et al. (Reference Henderson, Bridges and Wernet2005), i.e. the bulk motion of the wall jet acting as a mass displacement and thus sound source, to any of these waves. The nature of the present data precludes us from conclusively demonstrating a link between the two, but we nonetheless hypothesise that these diffuse waves could be produced by a large-scale displacement of the wall jet rather than the highly localised shock-associated sources discussed earlier. The data in Henderson et al. (Reference Henderson, Bridges and Wernet2005) suggest that the movement of the wall-jet boundary is subsonic relative to the ambient air, meaning that the resultant waves would not be shock-like. The displacement of the boundary is also going to be distributed over a non-compact region on the surface of the wall, meaning that the source of the wave will also be distributed. Thus while we cannot demonstrate it categorically, large displacements of the wall jet have the necessary characteristics to be responsible for the generation of these diffuse collections of weaker upstream-propagating waves, which then coalesce into a single nonlinear wave.

On a final note, we have observed a number of configurations where two strong waves are produced by the passage of a single vortical structure. This presents something of a challenge for the traditional model of resonance in such a flow: two upstream-propagating waves transferring energy to the jet shear layer near the nozzle should in turn produce two vortical structures, not one. Though interpretation is complicated by path-integration effects, in many cases where dual waves were observed, only one of the two was observed to interact with the shear layer near the nozzle, with either the other not reaching the nozzle at all, or only a weak component of the wave reaching the nozzle. There are a few select cases where some form of complicated amplitude modulation appeared to be taking place, which could suggest a more complex feedback process, but in the majority of cases, if two waves exist and did not merge, only one was involved in resonance.