3.2.1. SPOD spectra on unaltered images ( $M_{\!j}=0.9$ )

In this subsection, we will first examine the SPOD analysis when performed in the original SAFS images, without any pre-processing. The SPOD spectra from the analysis of schlieren images are compared with the near-field acoustic tones in figure 5. The corresponding mode shapes and their direction of propagation are subsequently exhibited in figure 6. Within the discussion, we identify the modal structure related to the peaks by using the subscript $(m,n)$ , where $m$ is the azimuthal wavenumber in (1.1), $n$ is the ordered root number (the $n{\rm th}$ root of the dispersion relation according to Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017)). The mode shape is inferred from careful examination of the entire context encompassing mode shape, location of the mode in the dispersion diagram and nearest dispersion line from the vortex sheet theory.

The two circular nozzles examined (SMC000 $2''$ and Conical $1.5''$ ) have different outlet diameters to ensure that any frequencies found are not an artefact of the camera sampling or the optical train; the flow phenomena should scale such that the physical frequencies collapse to an approximately constant Strouhal number. We also note that the relative amplitudes of the SPOD spectral features and high-ranking modes are highly dependent on the spatial extents of the visibility window. Therefore, for the two nozzles of different diameters, the SPOD processing window was chosen such that the same spatial extent for both jets was considered: $(0,-1.34)\lt (x/D,y/D)\lt (4.38,1.34)$ .

Figure 6.

(a,b) Spectra from SPOD for $M_{\!j}=0.9$ and eigenmodes corresponding to selected frequencies at the (c,d) first, (e, f) second and (g,h) third (smaller) peak. Arrows indicate the propagation direction of the waves observed. Centred red-white-blue colourmap used to plot the modes in panels (c–f) is scaled to 30 % of the maximum absolute value contained in the field. Peaks marked with $\times$ are camera artefacts. See supplementary movie 2 for cyclic movies corresponding to the modes shown.

An examination of the SPOD spectra presented in figure 5 shows that both circular nozzles contain a series of prominent and somewhat broad peaks. A linear frequency scale is chosen to facilitate comparison between peak frequencies. Note that certain single-frequency peaks in the SPOD spectra, marked with an $\times$ in all spectra shown, are not physical, and are believed to be sampling artefacts related to the camera pixel electronics and jitters in the laser intensity, and will henceforth be excluded from the discussion in all SPOD results that follow.

The detectable SPOD peaks in figure 5(a) for the SMC000 nozzle appear to match the first two peaks observed in the acoustic measurements performed in the anechoic SHJAR facility at corresponding conditions (Zaman et al. Reference Zaman, Bridges, Upadhyay and Brown2024); the tone series has also been shown to persist in the far-field in the forward flight arc. Figure 5(b) shows a similar match for the first few peaks between the SPOD and near-field spectra for the Conical $1.5''$ nozzle. We note that microphone measurements for this nozzle were acquired in the CW17 (non-anechoic) facility and contain some contamination due to reflections from the walls and equipment in the room. Therefore, the results in figure 5(b) are presented only to provide a reference for the general behaviour of the near-field acoustics for the Conical $1.5''$ nozzle with respect to the peaks observed in SPOD, and the presence of other potentially spurious peaks should not be over-analysed.

Although the match in SPOD and NF microphone peaks seems to be fairly good for both cases in figure 5, we note that a small peak located at $\textit{St}_{(0,2)}$ in figure 5(a) lies between the two acoustic peaks detected by the microphone. As will be discussed later in this section, this peak corresponds to the $(0,2)$ axisymmetric mode, gleaned from its SPOD mode shape. However, this shows that the match is not excellent and that there is more to the story. This mismatch will be elaborated further when we discuss the upstream/downstream decomposition presented in § 3.2.3.

The acoustic peaks in the near-field spectra of figure 5, outside of the jet, are identified as sawtooth patterns where the peaks have a slow rise and sharp fall with increasing frequency; whereas the SPOD peaks have a sharp rise and slow fall. This seems to be consistent with the behaviour for sound generated by a source inside of the jet, as documented in the calculations of Bogey (Reference Bogey2025). The sharp rise inside the jet (i.e. in the SPOD peak) would be explained by the sudden cut-on behaviour for these first two modes, as the cut-on marks the frequency at which trapped acoustic waves become propagative. Although figure 5 shows only the first two peaks with a sawtooth behaviour, it is further clarified in § 3.2.3 that this also occurs for the subsequent peaks.

In figure 6, we present a snapshot of the real part at an arbitrary phase of the mode shapes corresponding to the relevant peaks observed in the SPOD spectra for the two round nozzles. The spectra for the first three SPOD modes are presented once again in figure 6(a,b) to facilitate discussion. The logarithmic frequency axis is chosen here to highlight the relative prominence of the spectral peaks with respect to the broadband hump related to the turbulent shear layer. The first two spectral peaks for both the nozzles are observed at $\textit{St}_{(0,1)}\approx 0.42$ and $\textit{St}_{(1,1)}\approx 0.65$ , respectively, and the broadband humps peak at $\textit{St}\sim 1.2$ . For the SMC000 nozzle, several more peaks are observed in the SPOD spectrum compared with the conical nozzle.

The SPOD mode shapes related to the first few peaks shown in figure 6(c–h) are most noteworthy. The mode shape for the $\textit{St}_{(0,1)}\approx 0.42$ peak shown in figure 6(c,d) is an axisymmetric $(0,1)$ acoustic wave propagating upstream and trapped within the jet potential core, as indicated by the yellow arrows in the viewgraphs. The direction of propagation can be gleaned from the complex-valued coefficients of the SPOD mode, and is visually evident when the cycling movie is advanced in time for that specific frequency by multiplying the SPOD mode by $e^{2\pi if t}$ . Corresponding movies to the plots presented in figure 6 are provided as supplementary movie 2. The black arrows in the same figures present the downstream-advecting shear layer structures, which are comparatively much longer in wavelength and can only be distinguished in the movies. However, the upstream-travelling trapped waves are salient in our experimental data and easily distinguishable in the mode shapes presented. The (0,1) axisymmetric mode is also shown as a wavenumber decomposition in figure 7(a). The wavenumber decomposition of the SPOD modes follows the analysis presented by Edgington-Mitchell et al. (Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022), allowing the phase velocity direction to be gleaned from the sign of the wavenumber $k$ , where negative $k$ waves have upstream phase velocity. As seen in figure 7(a), a peak is observed at $kD\sim -16$ for this SPOD mode.

Further examining the SPOD modes presented in figure 6(e, f) corresponding to the second peak at $\textit{St}_{(1,1)}\approx 0.65$ , we see that a helical ( $m=1$ ) upstream phase velocity trapped wave is also observed for both nozzles, which can be gleaned by the opposite-polarity intensity zones across the jet axis. Its wavenumber decomposition, shown in figure 7(b), shows a broader set of negative wavenumbers.

Finally, the second axisymmetric mode at $\textit{St}_{(0,2)}\approx 0.82$ is visible in both nozzles, as evidenced in figure 6(g,h), though with a significantly weaker signature when compared with the downstream-convecting shear layer modes, which are also present in the image. Once again, the wavenumber decomposition separates the upstream phase velocity trapped waves from the downstream Kelvin–Helmholtz waves, as shown in figure 7(c). The family of shear layer waves related to Kelvin–Helmholtz instability are present throughout the SPOD spectrum over all frequencies comprising the broadband hump, and correspond to the majority of the modes observed in our decomposition. They are, however, more well understood, and are omitted from the discussion to retain the focus of our study on the trapped waves and guided jet modes.

These observations are, to the knowledge of the authors, the first imaging-based fully time-resolved documented experimental visualisation of the trapped waves predicted by Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) in free subsonic jets. As previously discussed, the first two spectral peaks for the waves revealed by SPOD match the acoustic tones measured by Zaman et al. (Reference Zaman, Bridges, Upadhyay and Brown2024) for the same nozzles, thereby establishing, for the first time experimentally, a connection between trapped waves in the jet potential core and acoustic emission in the forward direction in the far-field.

Figure 7.

Wavenumber decomposition of the SPOD modes shown in figure 6 for the SMC000 nozzle.

Figure 7. Long description

Panel A: A heatmap graph shows acoustic spectra at a frequency of 2363 Hz and a Strouhal number of 0.415. The horizontal axis is labeled kD, ranging from -60 to 60. The vertical axis is labeled y/D, ranging from -1.0 to 1.0. The heatmap displays a concentration of acoustic energy around the center. Panel B: A heatmap graph shows acoustic spectra at a frequency of 3926 Hz and a Strouhal number of 0.689. The horizontal axis is labeled kD, ranging from -60 to 60. The vertical axis is labeled y/D, ranging from -1.0 to 1.0. The heatmap displays a more dispersed pattern of acoustic energy with distinct regions of higher intensity. Panel C: A heatmap graph shows acoustic spectra at a frequency of 4688 Hz and a Strouhal number of 0.824. The horizontal axis is labeled kD, ranging from -60 to 60. The vertical axis is labeled y/D, ranging from -1.0 to 1.0. The heatmap displays a concentrated region of high acoustic energy near the center with additional regions of lower intensity.

3.2.2. Empirical dispersion diagrams at $M_{\!j}=0.9$

To ascertain that the upstream phase velocity waves are indeed propagated in the correct direction and are not an artefact of the temporal aliasing due to the limited sampling rate of our camera, we also construct the dispersion diagrams shown in figure 8 for the SMC000 $2''$ and Conical $1.5''$ nozzles operating at $M_{\!j}=0.9$ . To build these diagrams, a long rectangular window at the core of the nozzle stream (shown in figure 4 a) is defined and the pixel intensity values are averaged in the $y$ direction; forming a row of intensity values for each frame acquired. The averaging in the $y$ direction is done to improve the resilience of this approach to the high-energy turbulent signal, in an effort to improve the signal-to-noise ratio of the dispersion diagrams. The window is deliberately selected off-centre to highlight the helical modes that would cancel out with a symmetric $y$ window. The rows are then arranged in an $x$ – $t$ diagram as shown in figure 4(e, f), producing a wave diagram. The dispersion diagram $k$ – $\omega$ can be then obtained by taking the two-dimensional (2-D) Fourier transform of the wave diagram. A total of 508 space-time blocks (fast Fourier transform (FFT) images) were averaged using the Welch method to obtain better estimates of the underlying dispersion diagram, as shown in figure 8. A logarithmic grey scale spanning 40 dB (in power spectral density) is used to highlight some of the weaker features related to the higher order guided and trapped waves herein discussed.

Figure 8.

Welch-averaged dispersion diagrams for the pixel row at the jet core for the SMC000 $2''$ and Conical $1.5''$ nozzles at $M_{\!j}=0.9$ ( $M_a=0.835$ ) using 32 $x$ – $t$ blocks and 2048 images/block plotted along with vortex sheet model predictions. Colour scale represents pixel amplitude squared (power per bin) on a logarithmic scale spanning 40 dB. Least dispersive waves (inflection points) from model marked with a star.

In the Fourier plane, there will be four quadrants that represent the wave propagation directions based on phase velocity. Defining phase velocity as $v_{ph}=\omega /k$ , the top-right quadrant plotted in figure 8 represents downstream phase velocity waves (positive $k$ and positive $\omega$ ). The bottom-left quadrant also has downstream phase velocity waves (negative $k$ and negative $\omega$ ), equivalent to the top-right quadrant since the underlying data are real-valued. The top-left and bottom-right quadrants represent upstream phase velocity waves ( $k$ and $\omega$ have opposite signs). Presenting the four quadrants provides the opportunity to show annotations in one of the quadrants and still present the unannotated data in the opposite quadrant.

Highlighted in figure 8 as a black dashed line, we see the downstream propagation at the jet speed $M_{\!j}=0.9$ . Most hydrodynamic waves present in the contours propagate at this speed and, therefore, this is where most of the energy in the diagram is present. Note that the bright band with the same inclination repeats cyclically in the horizontal (spatial frequency) direction throughout the diagram, showing as diagonal stripes in the dispersion plane. This occurs because a wave travelling downstream at $M_{\!j}=0.9$ with the temporal Nyquist frequency has a wavenumber much lower than the spatial Nyquist frequency (see table 1 for values). More specifically, an unaliased high-wavenumber spatial wave ( $k\lt k_{ny}$ ) that moves at the speed $u_{\!j}$ will cause an oscillation of pixel intensity at a high temporal frequency that is aliased ( $\textit{St}\gt St_{ny}$ ), such that it will wrap around the dispersion diagram and show first as a negative temporal frequency, then as a positive temporal frequency at the incorrect frequency (but correct wavenumber), and so on, resulting in a striped dispersion diagram that separates the non-aliased waves (close to the origin) from the aliased waves (the repeated stripes above the positive and negative Nyquist limits).

Perhaps the more interesting feature that can be gleaned from both dispersion diagrams in figure 8 is the group of highly dispersive upstream phase velocity waves that show with very good contrast in the top-left quadrant before the aliased waves dominate the chart (i.e. for $-25\lesssim kD\lt 0, St\gt 0$ ). There are between 6 and 8 groups for the SMC000 $2''$ nozzle (depending on how the different regions are clustered), and 6 groups of upstream phase velocity waves distinguishable for the Conical $1.5''$ nozzle, as presented in figure 8. The upstream-travelling wave speeds approximately asymptote to the upstream-propagating acoustic speed (shown as a green dashed line in figure 8).

Figure 9.

Magnified view of the empirical dispersion diagram for the SMC000 nozzle (from figure 8 a) for $M_{\!j}=0.9$ , $M_a=0.835$ , $T_{\!j}/T_\infty =0.861$ , and its comparison with the vortex sheet theory. Yellow dots from upstream SPOD analysis in § 3.2.3.

Figure 9. Long description

A line graph depicting the empirical dispersion diagram for the SMC000 nozzle. The horizontal axis is labeled k D and ranges from -60 to 0. The vertical axis is labeled S t and ranges from 0 to 2. The graph includes multiple lines representing different modes and inflection points. The black dashed line represents u j, the green dashed line represents u j - a j, and the red dashed line represents -a infinity. The blue dashed line represents (0, n) modes, and the magenta dashed line represents (1, n) modes. Yellow dots indicate upstream SPOD modes, and blue and magenta stars indicate (0, n) and (1, n) inflection points, respectively. Yellow dotted lines represent NF tones. The graph shows various trends and intersections of these lines, indicating the relationship between different modes and inflection points.

A closer view of the dispersion relationship showing only the top-left quadrant where upstream-travelling waves are is highlighted in figure 9 for the SMC000 nozzle. We can note that the bright regions containing waves in the dispersion diagram match closely with the predictions for the vortex sheet mode shown as dashed blue and magenta lines calculated based on (3.15) of Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017). Also note the prediction for the vortex sheet and the soft duct (not shown) models match almost exactly up to the inflection point of the vortex sheet curve (marked with a blue/magenta star); so we choose to omit the duct mode from our charts to minimise clutter and ensure the experimental result of this manuscript is presented without being exceedingly overlapped by prediction curves.

As discussed by Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) and also evident in the magnified view of figure 9, none of the waves identified by SPOD (where peak $k$ is identified and displayed as yellow dots in figure 9) transfer energy upstream faster than the upstream-propagating sound wave inside the potential core because their group velocity (the slope of the dispersion curve) is smaller in magnitude than the upstream sound wave. For the longer waves (small $|kD|$ , waves to the right of the minima), the slope of the dispersion curve is positive – meaning the wave’s phase velocity points upstream, but energy propagates downstream. These longer waves, while observed in the dispersion diagram, are not detected by SPOD.

As seen in figure 9, the general shape of the empirical dispersion curves from our experiments is well matched by the prediction from the vortex sheet model, but the quantitative positions of the solutions are only an excellent match for the first axisymmetric $(0,1)$ and helical $(1,1)$ modes. The remaining modes show an increasing mismatch, owing to the simplicity of the vortex sheet assumptions such as perturbation linearity, flow parallelism, infinitely thin shear layer and zero viscosity. Nonetheless, under the additional experimental context of the evidence provided herein, it appears to be clear that the vortex sheet dispersion relationships are sufficiently close for a first-order analytical model prediction.

3.2.3. Upstream/downstream phase velocity decomposition at $M_{\!j}=0.9$

Now that a discussion on the empirical dispersion relations has been elaborated, a decomposition of the waves based on their phase velocity direction will further elucidate the role of the various waves observed in our images. To accomplish this decomposition, we perform two completely separate SPOD analyses, one for the upstream phase velocity waves and another for the downstream phase velocity waves. This is done by performing the following steps as a modification to the original SPOD processing code:

Figure 10.

Results of the SPOD performed only on upstream or downstream phase velocity waves for SMC000 at $M_{\!j}=0.9$ , $M_a=0.835$ , $T_{\!j}/T_\infty =0.861$ . (a) SPOD spectra for Mode 1 versus NF microphones. SPOD peaks marked with $\times$ are camera artefacts. Shaded red areas highlight regions between stationary points for first three jet modes to facilitate grouping. (b) Upstream SPOD mode shapes for upstream peaks. (c) Downstream SPOD mode shapes for downstream peaks. (d) Upstream SPOD mode shapes matching downstream peak frequencies.

1. (i) collect a block with $N_{\textit{fft}}=2048$ images, yielding a three-dimensional (3-D) matrix with $(n_x\times n_y\times N_{\textit{fft}})=512\times 1024\times 2048$ ;

2. (ii) perform a 2-D FFT row-by-row, in the $n_x\times N_{\textit{fft}}$ directions. This yields $n_y$ dispersion diagrams like the ones presented in figure 8, one for each row of the images;

3. (iii) for upstream waves, set the FFT coefficients corresponding to quadrants 1 and 3 to zero. For downstream waves, set quadrants 2 and 4 to zero;

4. (iv) reconstruct the block time series by performing a row-by-row inverse 2-D FFT;

5. (v) perform one-dimensional FFT only in the $N_{\textit{fft}}$ (time) direction for all pixels in the image;

6. (vi) construct the SPOD block matrix (columns $\hat {q}$ in equation (20) of Schmidt & Colonius (Reference Schmidt and Colonius2020));

7. (vii) continue the SPOD algorithm.

This analysis was performed for the SMC000 nozzle. It isolates the upstream phase velocity waves from the downstream phase velocity waves prior to performing SPOD. Therefore, the two independent SPOD analyses yield one SPOD spectrum each, which are presented as black and blue lines in figure 10(a), only for the first mode. Unfortunately, it is not possible to isolate group velocity directions with this technique, as the slopes of the 2-D FFT peaks cannot be identified reliably. Note that this wavenumber decomposition analysis does not change the underlying physics of this problem, nor does it change the interactions between waves with opposing group velocities. This analysis will effectively serve to separate the K–H instability waves (downstream phase velocity) from the trapped/guided jet waves (upstream phase velocity), so we can more easily observe the weaker upstream phase velocity waves.

The spectrum in figure 10(a) effectively isolates the upstream phase velocity waves (black line) from the downstream phase velocity waves (blue line). We also add the near-field microphone spectrum as a transparent red line and the position of the zero group velocity waves at $\textit{St}_{\textit{min}}$ and $\textit{St}_{\textit{max}}$ corresponding to the trough and peak of the vortex sheet dispersion curve for comparison. The reference lines are bound by a red window to facilitate distinction between the different jet modes, and lines corresponding to modes $(1,2)$ and higher order modes are suppressed to avoid clutter, as there is overlap between $\textit{St}_{\textit{min}}$ and $\textit{St}_{\textit{max}}$ for these modes.

First, we note that the downstream SPOD spectrum shows a broadband hump that corresponds to the K–H instability waves, which is continued as aliased waves in the folded SPOD spectrum of the upstream waves (i.e. the black and blue lines connect at the Nyquist limit because the broadband hump continues as negative frequencies and shows as upstream waves). This means that we can effectively identify that the modes in the upstream SPOD spectrum at Strouhal numbers greater than ${\sim} 1.2$ are also K–H structures propagating in the wrong direction due to aliasing.

Now that the broadband K–H hump feature in the spectrum is addressed, we can focus our attention on the individual peaks in the downstream SPOD spectrum (blue), marked with (1d, 2d, …). These peaks also correspond to K–H structures, but these structures have increased energy in the SPOD spectrum. The shape of the structures marked in these peaks is shown in figure 10(c), all of them propagating with downstream phase velocity. Not only do these waves have increasing wavenumber with temporal frequency (as expected), but also their mode shape clearly alternates according to the mode series previously identified. For example, peaks 1d, 3d and 5d correspond to axisymmetric modes since their structures have the same polarity on both sides of the shear layer, whereas peaks 2d and 4d correspond to helical modes, as the structures change polarity across the jet centreline.

Most notably, however, is that the downstream SPOD peak series is an excellent match with the near-field microphone peaks (shown in transparent red lines). In contrast, this is not the case for the upstream peak series (1u, 2u, …), which loosely matches for the first microphone peak at 1u, but increasingly becomes misaligned for the subsequent peaks. However, the upstream phase velocity peak series clearly shows a series of trapped wave mode shapes matching the series of axisymmetric and helical jet modes, as evidenced in figure 10(b).

Leveraging our observations of the $k{-}St$ diagrams presented in figure 9, we observe three effects that compound to produce the upstream SPOD peaks at a frequency misaligned with the near-field acoustics. First, the soft-duct cut-on frequency $\textit{St}_{\textit{min}}$ causes the trapped waves for a given mode to start very close to the SPOD peak frequency, which appears to have significant energy in the dispersion relation of figure 8. Second, SPOD isolates temporally coherent waves by forcing them to be single-frequency, meaning the cut-on frequency $\textit{St}_{\textit{min}}$ will contain more wavenumbers at the same frequency because it corresponds to a zero group velocity point, compounding their energy. Third, our SAFS naturally highlights high-wavenumber waves, as the intensity signal is proportional to the density gradient $\partial \rho /\partial x$ , which for a sinusoidal wave $\rho ^\prime =\rho _0^\prime \textrm{e}^{\textrm{i}(kx-\omega t)}$ will already produce an intensity signal proportional to the wavenumber $k$ . Most importantly, we observe the acoustic peaks occur at frequencies that are increasingly far from both zero group velocity temporal frequencies $\textit{St}_{\textit{min}}$ and $\textit{St}_{\textit{max}}$ for the higher order modes of the jet. Although the shape of the upstream spectrum and the cut-on behaviour inside of the jet at $\textit{St}_{\textit{min}}$ agrees with what is observed by Bogey (Reference Bogey2025) in the simplified problem set-up of a monopole source inside a laminar jet and by Bogey (Reference Bogey2021) on a LES of a round jet, the near-field peak (which coincides with the far-field peaks in the forward-flight directions) does not match either $\textit{St}_{\textit{min}}$ or $\textit{St}_{\textit{max}}$ , suggesting $\textit{St}_{\textit{max}}$ is simply a cut-off frequency for the supported waves, not the frequency where far-field propagative acoustic waves are produced.

The observation that the K–H wave peaks match the NF microphone spectral peaks is also notable. If the K–H waves have a broadband receptivity but peaks exist in the spectrum, then the forcing amplitude must be higher at the peak frequencies.

Finally, we highlight that the shape of the upstream phase velocity waves at the mode shapes with frequencies matching the near-field acoustic peaks, presented in figure 10(d), show only waves related to the trapped wave (duct) mode. It is evident the waves in figure 10(d), 1d–3d, are trapped waves because the wavenumber/frequency of these waves places them on region $\unicode {x2460}$ of the dispersion diagram in figure 1, which is further corroborated when looking at all waves detected in the upstream SPOD analysis, which are plotted as yellow dots in the magnified dispersion chart of figure 8 (wavenumber identification from SPOD modes is detailed in Appendix A). We detect no guided jet waves (region $\unicode {x2462}$ of the dispersion diagram) in the SPOD decomposition up to mode 6, even though they should exist according to the vortex sheet dispersion relationship. One possibility for the absence of guided waves in our data is that our schlieren apparatus, which highlights higher wavenumber waves, is not sensitive enough to detect these low-wavenumber waves as effectively.

3.3.1. SPOD spectra on unaltered images

Similar SPOD processing was applied to the full-frame images acquired at $M_{\!j}=0.983$ (no upstream/downstream decomposition), corresponding to an acoustic Mach number of $M_a=0.9$ for the unheated jet. The raw images for the $M_{\!j}=0.983$ case look qualitatively very similar to the images presented in figure 4 and are omitted for brevity.

Figure 11.

Direct comparison between SPOD spectra and near-field microphone spectra for the circular nozzle cases at $M_{\!j}=0.983$ . Peaks marked with $\times$ are camera artefacts.

First, we examine the SPOD spectra compared directly with the near-field microphone spectra for $M_{\!j}=0.983$ , as displayed in figure 11. As is evident in figure 11, the SPOD peaks are completely misaligned from the near-field microphone spectra in both nozzles. The spectrum of figure 11(a) for the SMC000 nozzle clearly shows that there are no acoustic peaks in the near-field corresponding to the two low-frequency tones related to the trapped waves observed in SPOD. What is gleaned from the comparison in figure 11, therefore, is that the trapped waves at $M_{\!j}=0.983$ have acoustic propagation that is confined to the jet potential core as duct-like modes and do not efficiently transmit outside of the shear layer. As previously discussed in § 3.2.3, trapped waves for a given mode shape are most prominent near the zero group velocity region (with cut-on near $\textit{St}_{\textit{min}}$ ). As expected, SPOD highlights these waves trapped in the jet core which belong to the same family as the radiating waves, but oscillate at a lower temporal frequency. The frequency misalignment between trapped and radiated waves for a given mode is greatly exaggerated because of the positioning of the dispersion lines, as will be further evidenced in § 3.3.2.

Figure 12.

(a,b) Spectra from SPOD for $M_{\!j}=0.983$ and most energetic eigenmodes corresponding to selected frequencies at the (c,d) first, (e, f) second and (g,h) third peak. Plots produced similarly to figure 6. Peaks marked with $\times$ are camera artefacts. SPOD mode shape movies are provided as supplementary movie 3.

A summary of the SPOD analysis for the $M_{\!j}=0.983$ cases is presented in figure 12. The SPOD spectra shown in figure 12(a,b) for the SMC000 $2''$ and Conical $1.5''$ nozzle show a broadband hump, related to the shear layer instability, and two very tall peaks at a frequency significantly lower than the shear layer group. The sharper peaks correspond to the trapped wave modes shown in figure 12(c–g) and are now shifted to a significantly lower frequency when compared with the $M_{\!j}=0.9$ case. Movies for the modes shown in figure 12 are also provided as supplementary movie 3.

The Conical $1.5''$ nozzle SPOD spectrum now contains two plateau-shaped peaks, whereas the SMC000 $2''$ nozzle spectrum also contains an additional smaller peak indicated by ‘g’ in figure 12(a). The first peak (panels c,d) for both nozzles, with the tallest value located at $\textit{St}=0.153$ but spanning $0.15\lt St_{(0,1)}\lt 0.19$ , is the most energetic feature in the images and comprises a group of upstream-propagating axisymmetric trapped waves with a very short wavelength $\lambda$ when compared with $M_{\!j}=0.9$ , as displayed in figure 12(c). The second peak/plateau (panels e, f), located in the range $0.23\lt St_{(1,1)}\lt 0.31$ , is constituted by a family of upstream-propagating helical waves. Note the approximate ratio of the centre bands $\textit{St}_{(1,1)}/St_{(0,1)}\approx 0.27/0.17 = 1.59$ is very close to the $5/3$ ratio observed by Zaman et al. (Reference Zaman, Bridges, Upadhyay and Brown2024), but the physical frequency of the peaks is completely mismatched from the acoustics as previously discussed.

Specifically, for the SMC000 $2''$ nozzle, another small peak is present in the spectrum corresponding to the mode shape shown in figure 12, matching a $(0,2)$ axisymmetric mode shape. This peak is absent in the SPOD spectrum for the Conical $1.5''$ nozzle when performing the computation with all pixels in the image. It is likely that the energy of the shear layer structures in other regions of the image overwhelm the signal intensity and, hence, the norm used for SPOD highlights the shear layer structures, of the kind shown in figure 12(h), instead of the trapped waves.

Examining the waves comprising the SPOD peaks in more detail, we can observe all mode shapes correspond to trapped waves. The trapped waves for the first axisymmetric $(0,1)$ and helical $(1,1)$ modes are displayed in detail in figure 13(a) for the SMC000 $2''$ (first two columns) the waves belonging to the $(0,2)$ axisymmetric mode for the SMC000 nozzle are relatively weaker and are omitted here. The family of trapped waves comprises several wavepackets that begin and end at different downstream locations within the jet. Further examination of figure 13(a) shows that the higher frequency portion of the trapped wave band tends to be confined further downstream of the jet, whereas the lower frequency portion of the band is trapped further upstream, eventually reaching the lip of the nozzle. These trapped waves appear to match the R3 duct mode described by Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017), where the wave reflection is terminated by an evanescent wave in both upstream and downstream sides of the wave packet, meaning the entire wave packet stands between two streamwise locations. The streamwise location of the trapped wave consistently moves downstream as frequency is increased, until the conditions for its existence disappear or it exits the limited viewing window of our SAFS system. Finally, notice that some of the helical modes in figure 13(a) (i.e. $\textit{St}=0.229$ , $\textit{St}=0.236$ for SMC000) are not completely symmetric around the nozzle axis, though it is unclear whether this is a feature of the SPOD processing.

Turning our attention to figure 13(b), which shows the wavenumber decomposition of the SPOD modes shown in figure 13(a), we see that a peak is observed at negative wavenumbers for all cases. The peak slowly trends towards lower $|k|$ with higher $\textit{St}$ for both modes. The peaks are also broadband, which is consistent with the wavepacket shape not extending over the entire jet length, following the Fourier uncertainty relation. For the higher frequency mode shapes for the $(0,1)$ mode of the SMC000 nozzle in figure 13(a) ( $\textit{St}=0.178$ , $\textit{St}=0.194$ ), two different wave packets of different wavelengths are superposed, which show as two separate peaks in the corresponding charts of figure 13(b). This makes sense as the empirical dispersion diagram (see figure 14) displays U-shaped curves, meaning that the same temporal frequency will potentially have two different wavenumbers, and therefore two distinct phase and group velocities. The fact they both appear at certain frequencies in the same SPOD mode means the two different wavenumbers are correlated, providing some support that a feedback between these two opposing group velocity waves may exist in this jet.

Figure 13.

(a) SPOD waves for SMC000 $2''$ nozzle at $M_{\!j}=0.983$ . Last row is eigenvalue-weighted reconstruction for a sample time snapshot. A movie corresponding to this figure is provided as supplementary movie 4. (b) Wavenumber decomposition of corresponding SPOD modes.

Figure 13. Long description

Panel A: This panel contains multiple scatter plots showing SPOD waves for SMC000 nozzle configurations. Each subplot represents different frequencies and Strouhal numbers. The x-axis is labeled x/D and the y-axis is labeled y/D. The last row shows eigenvalue-weighted reconstructions for a sample time snapshot. Panel B: This panel contains heatmaps showing the wavenumber decomposition of corresponding SPOD modes. Each subplot represents different frequencies and Strouhal numbers. The x-axis is labeled kD and the y-axis is labeled y/D.

The last row in figure 13(a) presents the sum of all the trapped waves within its corresponding peak/plateau frequency band, weighed by their corresponding SPOD eigenvalue. It represents the reconstruction of the first SPOD mode based on the band-limited SPOD frequencies composing a given trapped wave shape. Evidently, the reconstruction shape varies over time as the waves propagate upstream and the wavepackets interact; and here we display a representative snapshot of this time series. It becomes evident that despite each frequency containing a wave packet limited to a given streamwise extent, the wave packets combine to a full wave train along the majority of the axial extent of the jet probed (i.e. in the range $0\lt x/D\lt 4$ ). We believe the waves would have extended further downstream if the SAFS optics were not imposing excessive blur in the raw images for the $x/D\gt 4$ locations.

3.3.2. Empirical dispersion diagrams at $M_{\!j}=0.983$

Figure 14 presents the empirical dispersion diagrams generated in the same manner as described in § 3.2.2. The general observations are very similar: a large fraction of the waves observed are downstream-convecting at the jet speed $u_{\!j}$ , following the black dashed line, with similar aliasing artefacts as observed for the $M_{\!j}=0.9$ case. The upstream-travelling group of waves are shown in the top-left and bottom-right quadrants of the $k{-}\omega$ spectrum as six visible groups of waves for both SMC000 $2''$ and Conical $1.5''$ nozzles, though some of the groups are very weak in intensity. Note that when we increase the Mach number of the jet to $M_{\!j}=0.983$ , the quantity of wavenumbers with near-zero group velocity becomes significantly higher and waves with very small wavelength (high $|k|$ ) begin to exist. Since these high-wavenumber waves have larger streamwise density gradients (Settles Reference Settles2001), they become more prominent to our schlieren system.

Figure 14.

Welch-averaged dispersion diagrams produced with the same procedure as figure 8 for $M_{\!j}=0.983$ ( $M_a=0.9$ ).

Figure 14. Long description

Panel A: The graph for the SMC000 2-inch nozzle shows a wavenumber decomposition of SPOD modes. The x-axis is labeled kD and ranges from -50 to 50. The y-axis is labeled St and ranges from -2.0 to 2.0. Various modes and tones are marked with different colors and line styles. Panel B: The graph for the Conical 1.5-inch nozzle shows a similar wavenumber decomposition. The x-axis is labeled kD and ranges from -50 to 50. The y-axis is labeled St and ranges from -2.0 to 2.0. Different modes and tones are also marked with various colors and line styles. Both graphs display peaks and trends that indicate the behavior of the modes at different wavenumbers.

Note in figure 14, the upstream propagating trapped waves that are responsible for the peaks in the SPOD spectra are the most prominent near zero group velocity regions, with the cut-on near $\textit{St}_{\textit{min}}$ for each mode. Due to the shape of the dispersion curves at $M_{\!j}=0.983$ , the strongest trapped waves for each mode occur at temporal frequencies that are much lower and are, therefore, fairly separated from the frequencies of the radiated waves, as shown by the yellow dotted lines.

A more magnified view of the empirical dispersion diagrams for the SMC000 nozzle is provided in figure 15 to facilitate the comparison with the vortex sheet dispersion curves. We can observe the bright regions on our dispersion diagram form bands that match well with the theoretical dispersion curves, but the bands themselves have significant thickness and energy is not concentrated in a narrow line as the theoretical relation would predict. This is despite our choice of processing window excluding the shear layer (see yellow dashed rectangle in figure 4 a). As the bands have thickness in some cases comparable to the distance between two prediction lines, we can state that the real jet is only approximated by the vortex sheet.

We can also see in figure 15 that the guided jet waves (group $\unicode {x2462}$ in figure 1) are not conclusively detected in our observations. Although one could initially ascribe this absence to our schlieren apparatus preferring high-wavenumber waves, it is evident in figure 15 that waves with $|kD|\gtrsim 5$ are detectable and waves with $|kD|\gtrsim 10$ are well above the noise floor. Therefore, the guided waves for the higher-order modes (modes above ( $1,2$ )) should be as visible in figure 1. A very faint signature may be argued to exist around the curve for modes ( $1,2$ ) and ( $0,3$ ), but at this point, we cannot confirm their existence with the data we captured.

3.3.3. Upstream/downstream phase velocity decomposition at $M_{\!j}=0.983$

Figure 15.

Magnified view of the empirical dispersion diagram for the SMC000 nozzle for $M_{\!j}=0.983$ , $M_a=0.9$ , $T_{\!j}/T_\infty =0.833$ , and its comparison with the vortex sheet theory.

Figure 16.

Results of the SPOD performed only on upstream or downstream phase velocity waves for SMC000 at nominal $M_{\!j}=0.983$ , $M_a=0.9$ , $T_{\!j}/T_\infty =0.838$ . SPOD spectra for Mode 1 versus NF microphones. SPOD peaks marked with $\times$ are camera artefacts.

Figure 16. Long description

A line graph titled ‘Upstream/downstream SPOD and mic spectra’ with the horizontal axis labeled ‘f (kHz)’ ranging from 0 to 20 and the vertical axis labeled with a logarithmic scale ranging from 10 to 1000. The graph includes four data lines: Upstream SPOD in black, Downstream SPOD in blue, NF mics in red, and vertical dashed lines indicating St_min and St_max at M_j = 0.982. The Upstream SPOD line shows a peak around 1 kHz and decreases thereafter. The Downstream SPOD line increases steadily up to around 10 kHz. The NF mics line shows multiple peaks and troughs. An annotation indicates aliasing at the Nyquist limit.

The upstream/downstream phase velocity decomposition discussed in § 3.2.3 was also performed for the SMC000 at $M_{\!j}=0.983$ case, as displayed in figure 16. In this condition, the decomposition was not necessarily required because the upstream peaks shown in the black traces of figure 16(a) already clearly separate from the downstream waves shown in blue.

However, we show this result not only for completeness, but also because we initially expected that the SPOD analysis of the downstream-only (positive $k$ ) waves would reveal new peaks closely matching the NF microphone spectra in a similar manner to the $M_{\!j}=0.9$ condition because the upstream phase velocity waves were separated; which turned out not to be the case. Although a large number of peaks is indeed present around the K–H spectral hump in the blue trace of figure 16, none of them seem to have significant prominence over the variability that could be ascribed to spectral noise. Furthermore, some peaks match the acoustic tones seen in the near-field, but peaks in between the NF tones can also be identified in the downstream SPOD spectrum. This lack of prominence of the K–H peaks and excessive number of intermediate local peaks indicates this jet Mach number behaved differently with respect to the trapped waves; demonstrating a much lower degree of coupling of the K–H spectrum than the $M_{\!j}=0.9$ case.

Note the lines for the zero group velocity $\textit{St}_{\textit{min}}$ from the vortex sheet prediction are also provided in figure 16 as red dashed lines for the first three modes. We do not provide the mode shapes, as they match the mode shapes shown in figure 12. We see that the $\textit{St}_{\textit{min}}$ lines at $M_{\!j}=0.982$ match the positions where the upstream SPOD peaks find their cut-on frequency, whereas the $\textit{St}_{\textit{max}}$ lines appear to be the cut-off of the near-field peaks, similar to what was observed for $M_{\!j}=0.9$ . Finally, note that the Mach number used to create the $\textit{St}_{\textit{min}}$ , $\textit{St}_{\textit{max}}$ lines was adjusted from the nominal $M_{\!j}=0.983$ because the vortex sheet prediction lines are very sensitive to Mach number close to $M_{\!j}=1$ , and our NPR uncertainty of $\pm 0.003$ results in a jet Mach number uncertainty of $\pm 0.0014$ , which is sufficient to justify the mismatch that we observed against the prediction curves at $M_{\!j}=0.983$ .