4.2.1. Tones in the jet potential core

In order to shed light on the waves in the jets, a space–time Fourier transform has been applied to the pressure fluctuations inside the jet potential core for azimuthal modes $n_{\theta }=0$ and 1, as in previous work on free, impinging and screeching jets (Bogey & Gojon Reference Bogey and Gojon2017; Towne et al. Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017, Reference Towne, Schmidt and Brès2019; Gojon et al. Reference Gojon, Bogey and Mihaescu2018), but also for modes $n_{\theta }=2$ to 8. The pressure fluctuations are taken at a fixed radial position, depending on the azimuthal mode, from the nozzle exit at $z=0$ down to $z=0.7z_c$. The latter position allows us to reduce the contributions of the pressure disturbances of aerodynamic nature, particularly significant around the end of the potential core, while permitting a substantial spatial extent in the axial direction.

The spectra obtained for the jets with tripped boundary layers for $n_{\theta }=0$ to 5 are represented in figure 12(a–f) as a function of wavenumber and Strouhal number. For positive wavenumbers, strong components lie near the line $k=\omega /(0.75u_j)$. They correspond to the footprints left in the jet potential core by the shear-layer turbulent structures convected by the flow. For negative wavenumbers, high levels appear along bands near the dispersion curves predicted for the guided jet waves using the vortex-sheet model. This is clearly visible for $n_{\theta }=0$ and 1, as in Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) also for a Mach number 0.9 jet, as well as for higher azimuthal modes. Differences can be noted between the bands and the dispersion curves. They are less important in the spectra for the untripped jets with thinner boundary layers, not shown for brevity. Therefore, they can be attributed to the assumption of an infinitely thin shear layer in the vortex-sheet model. In agreement with previous studies (Tam & Ahuja Reference Tam and Ahuja1990; Bogey & Gojon Reference Bogey and Gojon2017), the bands are slightly above the dispersion curves far from the line $k=-\omega /c_0$. Near the line, the opposite trend is observed for $n_{\theta }=0$ to 2 in figure 12(a–c). In this zone, the bands have larger portions with a negative slope than the dispersion curves, yielding waves with a negative group velocity over wider frequency ranges. This is especially true for $n_{\theta }=3$ to 5 in figure 12(d–f), for which upstream-propagating waves are found near $k=-\omega /c_0$ for all radial modes, contrary to the model prediction. In that case, they are restricted to Strouhal numbers around that of the limit points $L$ given by the model. Moreover, the bands are thicker as the azimuthal mode number increases. In the bands, the energy is rather evenly distributed and no peak can be detected, unlike the results for impinging and screeching jets (Bogey & Gojon Reference Bogey and Gojon2017; Gojon et al. Reference Gojon, Bogey and Mihaescu2018) for instance. However, the energy levels are, overall, highest between the Strouhal numbers of the two stationary points $S_{min}$ and $S_{max}$, as pointed out by Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017). In most cases, the maximum levels even appear mainly located between $S_{min}$ and $S_{max}$ where the waves propagate in the downstream direction.

Figure 12. Frequency–wavenumber spectra of pressure fluctuations in the potential core of the jet at ${M}=0.90$ with tripped boundary layers at (a) $r=0$ for $n_{\theta }=0$, (b) $r=0.2r_0$ for $n_{\theta }=1$, (c) $r=0.3r_0$ for $n_{\theta }=2$, (d) $r=0.3r_0$ for $n_{\theta }=3$, (e) $r=0.4r_0$ for $n_{\theta }=4$ and (f) $r=0.4r_0$ for $n_{\theta }=5$ as a function of $(kD,{St}_D)$; dispersion curves of the guided jet waves (black lines), points $L$ (black circles), $S_{max}$ (red circles) and $S_{min}$ (blue circles); $k=-\omega /c_0$ (black dashed lines), $k=\omega /(0.75u_j)$ (black dash-dotted lines). The grey-scale levels spread over 25 dB.

Spectra of pressure fluctuations obtained for $n_{\theta }=0$ to 5 in the potential core of the jets with tripped boundary layers and with untripped boundary layers with $\delta _{BL}=0.025r_0$ are plotted in figure 13(a–f) as a function of ${St}_D$. To be consistent with the frequency–wavenumber spectra of figure 12, they are computed at the same radial positions, by averaging between $z=0$ and $z=0.7 z_c$. The allowable frequency ranges predicted using the vortex-sheet model for the upstream-propagating guided jet waves are displayed using the same colour code as in figures 7 and 9. To avoid an overlapping of the dark-grey bands, in which upstream-propagating but also downstream-propagating waves can exist, only the bands for the first four radial guided jet modes are represented for $n_{\theta }=0$. In the same way, only the bands for $n_r=1-3$ and $n_r=1-2$ are shown for $n_{\theta }=1,2$ and $n_{\theta }=3-5$, respectively. For all azimuthal modes, despite the presence of strong broadband aerodynamic components, visible in the spectra of figure 12 for positive wavenumbers, large acoustic peaks emerge in the spectra. They lie exclusively within the dark-grey bands, indicating that they are closely linked to the presence of $v_g^+$ guided jet waves. These results are in line with the findings of Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017), who demonstrated the possibility of resonant interactions between the former waves and $v_g^-$ guided waves in high subsonic jets between the frequencies of the stationary points $S_{min}$ and $S_{max}$ for $n_{\theta }=0$ and 1. In the present work, these interactions are found to be possible for higher azimuthal modes. On the basis of the frequency–wavenumber spectra of figure 12, two types of resonances can occur (Towne et al. Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017). The first one involves $v_g^+$ and $v_g^-$ duct-like waves located on both sides of points $S_{min}$ and the second one happens between $v_g^+$ duct-like and $v_g^-$ free-stream waves around points $S_{max}$.

Figure 13. Sound pressure levels in the potential core of the jets at ${M}=0.90$ with tripped boundary layers (black lines) and with untripped boundary layers with $\delta _{BL}=0.025r_0$ (red lines) at (a) $r=0$ for $n_{\theta }=0$, (b) $r=0.2r_0$ for $n_{\theta }=1$, (c) $r=0.3r_0$ for $n_{\theta }=2$, (d) $r=0.3r_0$ for $n_{\theta }=3$, (e) $r=0.4r_0$ for $n_{\theta }=4$ and (f) $r=0.4r_0$ for $n_{\theta }=5$ as a function of ${St}_D$; allowable ranges for the upstream-propagating guided jet waves with (dark grey shading) and without (light grey shading) downstream-propagating guided waves for (a) $n_r=1-4$, (b,c) $n_r=1-3$ and (d–f) $n_r=1-2$.

4.2.2. Tones in the jet near-nozzle region

To identify which of the guided jet waves have a significant radial support outside of the jets, a space–time Fourier transform has been applied to the pressure fluctuations at $r=1.1r_0$, between $z=0$ and $z=0.7z_c$ as previously in the potential core, for $n_{\theta }=0$ to 8. The spectra for $n_{\theta }=0$ to 5 are represented in figure 14(a–f) as a function of $k$ and ${St}_D$ for $k\leq 0$ only. Strong components of aerodynamic nature are observed near $k=0$ for low Strouhal numbers due to the proximity of the shear layer. In spite of this, spots of high levels are found for all azimuthal modes in the vicinity of the dispersion curves predicted for the guided jet waves using the vortex-sheet model. With respect to the elongated bands obtained in the in-core spectra of figure 12, the spots do extend along the curves but are restricted to very limited parts. Their levels are highest close to the line $k=-\omega /c_0$, rapidly decrease farther from it and are negligible to the left of the local maximum point. Therefore, the waves located between the limit points $L$ and the stationary points ${S}_{max}$ of the dispersion curves, i.e. the so-called free-stream waves in Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017), are detected just outside of the jets, whereas the other, duct-like, waves are not. The frequency ranges of the free-stream waves also appear to be narrower at a higher azimuthal mode. These results are in agreement with the variations of the eigenfunction magnitude of the guided jet waves outside of the shear layer in figures 5(b) and 6(c), and with the merging of points $L$ and ${S}_{max}$ as the azimuthal mode number increases in figures 7 and 9 for ${M}=0.90$. Given their negative group velocities, the free-stream waves propagate in the upstream direction, and can be expected to mark the pressure spectra in the near-nozzle region.

Figure 14. Frequency–wavenumber spectra of pressure fluctuations of the jet at ${M}=0.90$ with tripped boundary layers at $r=1.1r_0$ for (a) $n_{\theta }=0$, (b) $n_{\theta }=1$, (c) $n_{\theta }=2$, (d) $n_{\theta }=3$, (e) $n_{\theta }=4$ and (f) $n_{\theta }=5$ as a function of $(kD,{St}_D)$; dispersion curves of the guided jet waves (black lines), points $L$ (black circles), $S_{max}$ (red circles) and $S_{min}$ (blue circles); $k=-\omega /c_0$ (black dashed lines). The grey-scale levels spread over 25 dB. Only $k\leq 0$ is shown.

The spectra of pressure fluctuations computed near the nozzle exit at $z=0$ and $r=1.5r_0$ for the jets with tripped boundary layers and with untripped boundary layers with $\delta _{BL}=0.025r_0$ are represented in figure 15(a,b) as a function of ${St}_D$. The contributions of the first six azimuthal modes are also depicted. Tonal peaks emerge in the full spectra as well as for the azimuthal modes. They are very similar to those found in the near-nozzle spectra reported in Suzuki & Colonius (Reference Suzuki and Colonius2006), Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) and Brès et al. (Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018) for jets at the same Mach number as the present jets but at Reynolds numbers $Re_D\simeq 10^6$. They are stronger and narrower for the jet with untripped boundary layers than for the other one. However, the peak frequencies are almost identical in the two cases. For a given azimuthal mode, the first peak falls very near the dash-dotted line indicating the Strouhal number of point ${S}_{max}$, or $L$ in the absence of ${S}_{max}$, obtained on the dispersion curves for the radial guided jet mode $n_r=1$ using the vortex-sheet model. This supports that the peaks are due to the upstream-propagating waves highlighted in figure 14. Regarding the full spectra, the first, second and third peaks at ${St}_D\simeq 0.4$, $0.6$ and 1 coincide with the first peaks of modes $n_{\theta }=0$, 1 and 2 in red, blue and green, respectively. The fourth peak corresponds to the first peak of mode $n_{\theta }=3$ in magenta, enhanced by the second peak of mode $n_{\theta }=1$ in blue. The higher peaks also consist of combinations of peaks of different modes, for instance modes $n_{\theta }=2$ and 4 for the fifth peak and modes $n_{\theta }=1$, 3 and 5 for the sixth peak. The complex structure of the peaks can be explained by the great resemblance, and quasi superposition in some instances, of the dispersion curves for different azimuthal modes, discussed in § 3.2 based on figure 8. This issue was mentioned by Suzuki & Colonius (Reference Suzuki and Colonius2006) who remarked that the frequency of the first peak of mode $n_{\theta }=2$ is nearly the same as that of the second peak of mode $n_{\theta }=0$ in their experimental spectra.

Figure 15. Sound pressure levels obtained at $z=0$ and $r=1.5r_0$ for the jets at ${M}=0.90$ (a) with tripped boundary layers and (b) with untripped boundary layers with $\delta _{BL}=0.025r_0$ as a function of ${St}_D$: full spectra (black lines) and $n_{\theta }=$ 0 (red lines), 1 (blue lines), 2 (green lines), 3 (magenta lines), 4 (yellow lines) and 5 (light blue lines). Dash-dotted lines: Strouhal numbers at points $S_{max}$ or $L$ on the dispersion curves for the guided jet modes ($n_{\theta }$, $n_r=1$) using the same colours as for the solid lines.

The spectra calculated at $z=0$ and $r=1.5r_0$ for the jets with tripped boundary layers and untripped boundary layers with $\delta _{BL}=0.025r_0$ for modes $n_{\theta }=0$ to 5 are plotted in figure 16(a–f) as a function of $St_D$. The allowable ranges for the upstream-propagating guided jet waves according to the vortex-sheet model are represented in grey as in figure 13. The frequency ranges of the free-stream waves between points $L$ and $S_{max}$ on the dispersion curves are also indicated by oblique hatching, when possible. Compared with the peaks obtained in the potential core in figure 13, the near-nozzle peaks are narrower and exhibit a sharper decrease on the right side of the stationary points ${S}_{max}$. Moreover, instead of fully filling the dark-grey bands where $v_g^-$ and $v_g^+$ guided jet waves can both exist, they appear limited to the hatched bands. This trend is clearly observed for modes $n_{\theta }\geq 1$ in figure 16(b–f), for which points $L$ and $S_{max}$ are very close or superimposed on the dispersion curves, which gives rise to very tonal peaks. The present results are in agreement with the eigenfunctions of figure 5(b) and the transfer functions of 6(c). They show that the near-nozzle peaks are mainly related to the free-stream guided jet waves. In particular, it appears that among the resonant interactions possibly occurring between $v_g^-$ and $v_g^+$ guided waves in the jet potential core, only those involving $v_g^-$ free-stream waves can contribute significantly to the near-nozzle pressure field. Given that the peak levels are higher at point $S_{max}$ than at point $L$ when the two points are sufficiently distinct from each other, as for mode ($n_{\theta }=0$, $n_r=2$) in figure 16(a) and for mode ($n_{\theta }=1$, $n_r=1$) in figure 16(b), this may be especially true for the waves resonating around the stationary point ${S}_{max}$.

Figure 16. Sound pressure levels obtained at $z=0$ and $r=1.5r_0$ for the jets at ${M}=0.90$ with tripped boundary layers (black lines) and with untripped boundary layers with $\delta _{BL}=0.025r_0$ (red lines) for (a) $n_{\theta }=0$, (b) $n_{\theta }=1$, (c) $n_{\theta }=2$, (d) $n_{\theta }=3$, (e) $n_{\theta }=4$ and (f) $n_{\theta }=5$ as a function of ${St}_D$; allowable ranges for the upstream-propagating guided jet waves with (dark grey shading) and without (light grey shading) downstream-propagating guided waves and between the ${St}_D$ at points $L$ and $S_{max}$ (hatched) for (a) $n_r=1-4$, (b–c) $n_r=1-3$ and (d–f) $n_r=1-2$, ${St}_D$ at points $L$ on the dispersion curves (black dash-dotted lines).

The Strouhal numbers of the peaks obtained for the six jets at ${M}=0.90$ in the spectra at $z=0$ and $r=1.5r_0$ for $n_{\theta }=0$, 1 and 2 are represented in figure 17(a–c) as a function of boundary-layer thickness $\delta _{BL}$ at the nozzle-pipe inlet. The allowable bands predicted for the upstream-propagating guided jet waves using the vortex-sheet model, as well as the points $L$, $S_{max}$ and $S_{min}$ on the dispersion curves given by the model, are also displayed. Over the wide range of boundary-layer thicknesses considered, the peak Strouhal numbers do not vary appreciably despite the ratio of 16 between the largest and the smallest values of $\delta _{BL}$. This is in line with the experimental results of Zaman & Fagan (Reference Zaman and Fagan2019) for two jets with boundary-layer thicknesses differing by a factor of 3. In the same way, the peak frequencies are very similar for tripped and untripped boundary layers, as was the case for the two initially laminar and turbulent jets computed by Brès et al. (Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018). For all jets, even for the one with $\delta _{BL}=0.4r_0$ for which the vortex-sheet model is a very rough approximation, the peaks are located between the points $L$ and $S_{max}$, or near the point $L$ when $S_{max}$ does not exist. These results provide further evidence about the links between the near-nozzle peaks and the free-stream guided jet waves. In the zones between points $L$ and $S_{max}$, the peaks are closer to the second point than to the first one for the first two radial modes for $n_{\theta }=0$ in figure 17(a). This again supports the possible contributions of waves resonating around the stationary point $S_{max}$ to the near-nozzle pressure fields.

Figure 17. Peak Strouhal numbers in the spectra of pressure fluctuations at $z=0$ and $r=1.5r_0$ for the jets at ${M}=0.90$ with tripped (filled black circles) and untripped (black circles) boundary layers for (a) $n_{\theta }=0$, (b) $n_{\theta }=1$ and (c) $n_{\theta }=2$ as a function of $\delta _{BL}/r_0$; allowable ranges for the upstream-propagating guided jet waves with (dark grey shading) and without (light grey shading) downstream-propagating guided waves, ${St}_D$ at points $L$ (black dash-dotted lines), $S_{max}$ (red lines) and $S_{min}$ (blue lines) on the dispersion curves.

The sensitivity of the near-nozzle acoustic peaks to the jet initial conditions is examined by comparing some properties of the near-nozzle peaks obtained for the six jets at ${M}=0.90$. Only the peaks associated with the guided jet modes $n_r=1$ for $n_{\theta }=0$, 1 and 2 are considered. The peak intensities are depicted in figure 18(a). They are lower for the jet with tripped boundary layers than for the jets with untripped ones, and for the latter jets, they grow with the boundary-layer thickness except for $\delta _{BL}\geq 0.2r_0$. Thus, overall, the more noise generated by the jets (Bogey & Bailly Reference Bogey and Bailly2010; Bogey Reference Bogey2018), the higher the levels of the near-nozzle acoustic peaks.

Figure 18. Near-nozzle peaks for the jets at ${M}=0.90$ with tripped (filled circles) and untripped (circles) boundary layers for the radial modes $n_r=1$ of the guided jet waves for $n_{\theta }=0$ (red), $n_{\theta }=1$ (blue) and $n_{\theta }=2$ (green): (a) peak levels, (b) ratio between the peak levels and the minimum values for higher ${St}_D$ and (c) peak widths as a function of $\delta _{BL}/r_0$. Dash-dotted lines: ${\rm \Delta} {St}_D$ between points $S_{max}$ and $L$ using the above colours for $n_{\theta }$.

In order to quantify the degree of emergence of the peaks, the ratios between the peak levels and the first minimum values reached for a higher frequency are plotted in figure 18(b). As for the peak intensities, they are minimum for the jet with tripped boundary layers and are higher as the boundary layer is thicker for the initially fully laminar jets. Therefore, the near-nozzle peaks are more prominent for the jets with mixing layers containing stronger large-scale coherent structures, and inversely weaker fine-scale turbulent structures, yielding a weaker broadband noise in the upstream direction.

Finally, the peak widths at half of maximum are given in figure 18(c). They increase with the boundary-layer thickness. This trend can be attributed to the effects of the shear-layer thickness on the dispersion curves of the guided jet waves near the line $k=-\omega /c_0$ (Tam & Ahuja Reference Tam and Ahuja1990; Bogey & Gojon Reference Bogey and Gojon2017). Indeed, as mentioned in § 4.2.1 and illustrated by the frequency–wavenumber spectra of figure 12, the free-stream guided jet waves are obtained over wider frequency ranges for a thicker shear layer. In that case, the band-pass filtering of the upstream-propagating waves by the guided jet modes has a larger width. For comparison, the widths estimated as the frequency differences between points $L$ and $S_{max}$ on the dispersion curves using the vortex-sheet model for modes ($n_{\theta }=1$, $n_r=1$) and ($n_{\theta }=2$, $n_r=1$) are shown in figure 18(c). A fairly good agreement is found with the peak widths for $n_{\theta }=1$. For $n_{\theta }=2$, the peak width is underestimated by the model, which is expected due to the discrepancies between the numerical and theoretical dispersion curves near $k=-\omega /c_0$ in figure 12(c).

4.2.3. Tones in the jet acoustic far field

The LES near-field fluctuations obtained for the jet at ${M}=0.90$ with tripped boundary layers have been propagated to the far field using an in-house OpenMP-based solver of the isentropic linearized Euler equations in cylindrical coordinates based on the same numerical methods as the LES. Two calculations are performed as in previous studies (Bogey & Sabatini Reference Bogey and Sabatini2019; Bogey Reference Bogey2021). They are carried out from the velocity and pressure fluctuations recorded during time $T=3000r_0/u_j$ at $r=15r_0$ and at $z=-1.5r_0$ and $z=L_z=40r_0$ at a sampling frequency corresponding to a Strouhal number of ${St}_D=12.8$. They allow us to compute the pressure waves radiated at a distance of $150 r_0$ from the nozzle exit, where far-field acoustic conditions are expected to apply (Ahuja, Tester & Tanna Reference Ahuja, Tester and Tanna1987; Viswanathan Reference Viswanathan2010), between the angles $\phi =15^{\circ }$ and $165^{\circ }$ relative to the jet direction. Grids containing up to $1.6 \times 10^9$ points with a uniform mesh spacing of $0.075 r_0$ in the axial and radial directions and $N_{\theta }=256$ points in the azimuth are used. This mesh spacing, leading to ${St}_D=5.9$ for an acoustic wave discretized by five points per wavelength, is identical to that in the LES near pressure field. The sound pressure spectra thus determined at six angles $\phi$ between $30^{\circ }$ and $165^{\circ }$ are represented in figure 19(a–f) as a function of ${St}_D$. The contributions of modes $n_{\theta }=0$ to 8 are also shown. Because of the different shapes of the spectra (Mollo-Christensen, Kolpin & Martucelli Reference Mollo-Christensen, Kolpin and Martucelli1964; Tam Reference Tam1998), the level axis ranges from 74 up to 116 dB/${St}_D$ in figure 19(a) for $\phi =30^{\circ }$, but only up to 104 dB/${St}_D$ in figure 19(b) for $\phi =60^{\circ }$ and to 98 dB/${St}_D$ in figure 19(c–f) for $\phi \geq 90^{\circ }$.

Figure 19. Sound pressure levels obtained for the jet at ${M}=0.90$ with tripped boundary layers at $150r_0$ from the nozzle exit for (a) $\phi =30^{\circ }$, (b) $\phi =60^{\circ }$, (c) $\phi =90^{\circ }$, (d) $\phi =135^{\circ }$, (e) $\phi =150^{\circ }$ and (f) $\phi =165^{\circ }$ as a function of ${St}_D$: full spectra (black lines) and $n_{\theta }=$ 0 (red lines), 1 (blue lines), 2 (green lines), 3 (magenta lines), 4 (yellow lines), 5 (light blue lines), 6, 7 and 8 (grey lines). Dash-dotted lines: Strouhal numbers at points $S_{max}$ or $L$ on the dispersion curves for the guided jet modes ($n_{\theta }$, $n_r=1$) using the same colours as for the solid lines; measurements of Bridges & Brown (Reference Bridges and Brown2005) for an isothermal jet at ${M}=0.9$ and ${Re}_D=10^6$ (black triangles).

In the downstream and sideline directions, in figure 19(a–c), the pressure spectra exhibit the characteristics typically found in the far field of subsonic jets. The axisymmetric mode is dominant in the downstream direction and is soon overwhelmed by modes $n_{\theta }=1$ and 2 as the radiation angle increases (Juvé, Sunyach & Comte-Bellot Reference Juvé, Sunyach and Comte-Bellot1979; Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012; Brès et al. Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018). More importantly given the focus of the present work, there are peaks neither in the spectra for the full pressure signals, nor in those for the different azimuthal modes. This is obvious in figure 19(c) for the angle $\phi =90^{\circ }$, for instance. In particular, there is no trace of the undulations noticed around ${St}_D=1$ in the spectra of Zaman & Fagan (Reference Zaman and Fagan2019) at an angle of $60^{\circ }$ for jets at Mach numbers close to 1. This supports the hypothesis of the authors that these undulations are due to unwanted reflections by some uncovered surfaces in their experiments.

In the upstream direction, as suggested by the experiments of Jaunet et al. (Reference Jaunet, Jordan, Cavalieri, Towne, Colonius, Schmidt and Brès2016), peaks appear in the full spectra at high radiation angles. They are barely detectable at $\phi = 135^{\circ }$ in figure 19(d), clearly visible at $\phi = 150^{\circ }$ in figure 19(e) and predominant at $\phi = 165^{\circ }$ in figure 19(f). Their frequencies and azimuthal structures at the latter angle are very close, if not identical, to those of the near-nozzle peaks in figure 15(a). In summary, the $i$th peak in the full spectrum corresponds to the first peak of the azimuthal mode $n_{\theta }=i-1$. The latter peak is located near the Strouhal number of the point ${S}_{max}$, or $L$ when ${S}_{max}$ is lacking, obtained on the dispersion curve for the guided jet mode ($n_{\theta }=i-1$, $n_r=1$) using the vortex-sheet model. Therefore, the free-stream guided jet waves contribute significantly to the far-field noise for very large radiation angles. It can be noted that the prominence and tonal shape of the peaks vary with the angle. Thus, the most apparent peaks are the peaks for modes $n_{\theta }= 4-8$ at $\phi = 135^{\circ }$, for $n_{\theta }= 2-5$ at $\phi = 150^{\circ }$ and for $n_{\theta }= 0-3$ at $\phi = 165^{\circ }$. Finally, the emergence of peaks in the spectra results in stronger noise components at $\phi = 165^{\circ }$ than at $150^{\circ }$. This trend is similar to that observed for supersonic jets generating screech tones in the upstream direction.

To better describe the noise variations with the radiation angle, the overall sound pressure levels computed at $150 r_0$ from the nozzle exit are plotted in figure 20 as a function of $\phi$. They decrease monotonically with the angle between $\phi = 25^{\circ }$ and $150^{\circ }$, in agreement with the experimental data of the literature (Bridges & Brown Reference Bridges and Brown2005; Bogey et al. Reference Bogey, Barré, Fleury, Bailly and Juvé2007), and then are nearly constant between $\phi = 150^{\circ }$ and $165^{\circ }$. The effects of the emergence of peaks in the spectra for large radiation angles are more visible on the sound levels associated with the different azimuthal modes. For $\phi \leq 135^{\circ }$, in line with previous studies (Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012; Brès et al. Reference Brès, Jordan, Jaunet, Le Rallic, Cavalieri, Towne, Lele, Colonius and Schmidt2018), the stronger modes are the axisymmetric mode for $\phi \leq 45^{\circ }$ and modes $n_{\theta }=1$ and 2 for $\phi \geq 45^{\circ }$. In addition, for all modes, the levels decrease between $\phi =60^{\circ }$ and $135^{\circ }$. For $\phi \geq 135^{\circ }$, more surprisingly, the contributions of modes $n_{\theta }=0$ and 1 sharply grow with the radiation angle. As a consequence, at $\phi = 165^{\circ }$, the first helical mode predominates, closely followed by the axisymmetric mode. For larger angles, given the tendencies obtained between $\phi =150^{\circ }$ and $165^{\circ }$, one can expect a further increase of the sound levels for modes $n_{\theta }=0$ and 1 and the predominance of mode $n_{\theta }=0$ near $\phi = 180^{\circ }$. This could be checked in future studies.

Figure 20. Overall sound pressure levels obtained for the jet at ${M}=0.90$ with tripped boundary layers at $150r_0$ from the nozzle exit as a function of the angle $\phi$: full spectra (black lines) and $n_{\theta }=$ 0 (red lines), 1 (blue lines), 2 (green lines), 3 (magenta lines), 4 (yellow lines), 5 (light blue lines), 6 and 7 (grey lines); measurements of Bridges & Brown (Reference Bridges and Brown2005) for an isothermal jet at ${M}=0.9$ and ${Re}_D=10^6$ (black triangles).

4.3.1. Tones in the jet potential core

A space–time Fourier transform has been applied to the pressure fluctuations inside the jet potential core for modes $n_{\theta }=0$ to 8. The fluctuations are located between $z=0$ and $0.7z_c$ at radial positions depending on the azimuthal mode as in § 4.2.1. The spectra for the jets with tripped boundary layers at ${M}=0.75$, $1.10$ and $2$ for $n_{\theta }=0$ to 2 are represented in figure 21(a–i) as a function of wavenumber and Strouhal number. As for the tripped jet at ${M}=0.90$ in figure 12, strong aerodynamic components are found along the line $k=\omega /(0.75u_j)$. Bands of high energy are also observed near the dispersion curves of the guided jet modes predicted by the vortex-sheet model, for negative wavenumbers but also for positive wavenumbers in the supersonic cases, as expected. Similar results were obtained by Towne et al. (Reference Towne, Schmidt and Brès2019) for jets at ${M}=0.70$, 0.80, 0.9 and 1.50 for $n_{\theta }=0$. The agreement between the bands and the dispersion curves is good in figure 21(a–c) for ${M}=0.75$, fair in figure 21(d–f) for ${M}=1.10$ and rather poor in figure 21(g–i) for ${M}=2$. This may be due to the fact that the dispersion curves are determined from the linearized equation of motion of a compressible flow by assuming disturbances of small amplitudes, which is not necessarily true for supersonic jet velocities. For all Mach numbers, the energy is fairly well distributed along the bands. This is the case in particular for ${M}=0.75$, in figure 21(a–c), where the levels are significant from the limit point $L$ of the dispersion curves on $k=-\omega /c_0$ to the inflexion point $I$ and far beyond that point. For ${M}=2$, in figure 21(g–i), the levels are, however, much stronger for positive wavenumbers than for negative ones. Focusing on the region near the line $k=-\omega /c_0$, the bands all exhibit a portion with negative slopes between the limit point on the line and the local maximum point. This indicates the presence of upstream-propagating guided jet waves for all modes, including those for which such waves should not exist according to the vortex-sheet model. This is the case, for instance, in figure 21(f) for the mode ($n_{\theta }=2$, $n_r=1$) at ${M}=1.10$. This discrepancy, also noticed for the tripped jet at ${M}=0.90$ for $n_{\theta }\geq 3$ in previous section, is most likely due to the infinitely thin shear layer in the vortex-sheet model.

Figure 21. Frequency–wavenumber spectra of pressure fluctuations in the potential cores of the jets with tripped boundary layers at (a–c) ${M}=0.75$, (d–f) ${M}=1.10$ and (g–i) ${M}=2$, at (a,d,g) $r=0$ for $n_{\theta }=0$, (b,e,h) $r=0.2r_0$ for $n_{\theta }=1$ and (c,f,i) $r=0.3r_0$ for $n_{\theta }=2$, as a function of $(kD,{St}_D)$; dispersion curves of the guided jet waves (black lines), points $L$ (black circles), $S_{max}$ (red circles), $S_{min}$ (blue circles) and $I$ (green circles); $k=-\omega /c_0$ (black dashed lines), $k=\omega /(0.75u_j)$ (black dash-dotted lines). The grey-scale levels spread over 25 dB.

The pressure spectra calculated in the potential core of the six jets with tripped boundary layers for $n_{\theta }=0$ and 1, by averaging between $z=0$ and $0.7 z_c$ at $r=0$ and $r=0.2r_0$, respectively, are depicted in figure 22(a,b) as a function of $St_D$. For the two modes and for modes $n_{\theta }\geq 2$, not shown for brevity, the spectra are smooth for ${M}=0.60$ but peaks can be seen for higher Mach numbers. The peaks clearly appear for ${M}=0.75$ at $St_D\geq 1.5$, dramatically emerge for ${M}=0.90$ and are hardly visible for supersonic Mach numbers. This trend is in good agreement with the statement of Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) that the acoustic tones in the jet potential core, attributed to resonating trapped waves, should reach their strongest prominence between ${M}\simeq 0.82$ and 1. For ${M}=0.90$, as illustrated in figure 13, the peaks lie within the frequency bands of the $v_g^+$ duct-like waves. For ${M}=0.75$, the peaks are close to the Strouhal numbers of the inflexion points $I$ on the dispersion curves. For supersonic Mach numbers, they are near the frequencies of the stationary points $S_{max}$, around which interactions are possible between $v_g^+$ and $v_g^-$ guided jet waves. Finally, it can be remarked that in the frequency–wavenumber spectra, the dominant components are those associated with aerodynamic fluctuations for ${M}=0.75$ in figure 21(a–c) and with the guided jet waves with positive wavenumbers for ${M}=2$ in figure 21(g–i). The strengthening of these two components, in comparison with those related to the guided jet waves with negative wavenumbers, may be one reason for the weakening of the peaks in the spectra as the Mach number deviates from ${M}=0.90$.

Figure 22. Sound pressure levels obtained in the potential core of the jets with tripped boundary layers at (a) $r=0$ for $n_{\theta }=0$ and (b) $r=0.2r_0$ for $n_{\theta }=1$ as a function of ${St}_D$ for ${M}=$ 0.60 (black lines), $0.75$ (red lines), $0.90$ (blue lines), $1.10$ (black dashed lines), $1.30$ (red dashed lines) and $2$ (blue dashed lines).

4.3.2. Tones in the jet near-nozzle region

To get information on the pressure waves just outside of the jet flow, a space–time Fourier transform has been applied to the pressure fluctuations at $r=1.1r_0$ between $z=0$ and $0.7z_c$ for $n_{\theta }=0$ to 8, as in § 4.2.2. The spectra for the jets with tripped boundary layers at ${M}=0.75$, $1.10$ and 2 for $n_{\theta }=0$ to 2 are presented in figure 23(a–i) as a function of $k$ and ${St}_D$ for $k\leq 0$. In the region with $k\geq 0$, not shown for clarity, the spectra are dominated by very strong aerodynamic components as the in-core spectra of figure 21. However, contrary to the latter, they do not reveal spots of notable energy near the dispersion curves of the guided jet modes provided by the vortex-sheet model for supersonic Mach numbers. For $k\leq 0$, despite the wide dark patch due to aerodynamic disturbances at low Strouhal numbers, high levels are found close to the dispersion curves. As for the jet at ${M}=0.90$ in figure 14, the levels are significant only in the vicinity of $k=-\omega /c_0$. For ${M}=0.75$, in figure 23(a–c), they are negligible on the left side of the inflexion point $I$ of the dispersion curves. For ${M}=1.10$, in figure 23(d–f), they quickly decrease to the left of the local maximum point, corresponding to the stationary points ${S}_{max}$ of the dispersion curves. For ${M}=2$, in figure 23(g–i), a similar trend is observed for the first radial modes. For modes $n_r\geq 2$, however, the decay on the left side of the local maximum point is less rapid. This may be caused by the fact that for the jet at ${M}=2$, represented in figure 10(e) between $z=0$ and $12r_0$, some of the points considered from $z=0$ down to $z=0.7z_c=16.4r_0$ at $r=1.1r_0$ to compute the spectra lie inside the jet flow. Therefore, only the guided jet waves located approximately between the limit points $L$ on $k=-\omega /c_0$ and points $I$ in the subsonic case, and between points $L$ and ${S}_{max}$ in the supersonic cases, clearly extend out of the jets. These waves, with a negative group velocity, are able to propagate in the upstream direction up to the near-nozzle region. These results are consistent with the eigenfunction magnitudes obtained for the guided jet waves at $r=1.5r_0$ using the vortex-sheet model, displayed as a function of the Strouhal number in figure 6(b,d,e) for ${M}=0.75$, $1.10$ and 2.

Figure 23. Frequency–wavenumber spectra of pressure fluctuations of the jets with tripped boundary layers at (a–c) ${M}=0.75$, (d–f) ${M}=1.10$ and (g–i) ${M}=2$, at $r=1.1r_0$ for (a,d,g) $n_{\theta }=0$, (b,e,h) $n_{\theta }=1$ and (c,f,i) $n_{\theta }=2$, as a function of $(kD,{St}_D)$; dispersion curves of the guided jet waves (black lines), points $L$ (black circles), $S_{max}$ (red circles), $S_{min}$ (blue circles) and $I$ (green circles); $k=-\omega /c_0$ (black dashed lines). The grey-scale levels spread over 25 dB. Only $k\leq 0$ is shown.

The spectra of pressure fluctuations obtained at $z=0$ and $r=1.5r_0$ for the jets with untripped boundary layers with $\delta _{BL}=0.2r_0$ are presented in figure 24(a) as a function of the Mach number, using a logarithmic scale. They are normalized by their respective peak values, yielding a maximum value of 1 for each jet velocity. The contributions of modes $n_{\theta }=0$ and $1$ to the spectra are highlighted in figure 24(b,c). A zoom between ${M}=0.75$ and 0.85 is provided in Appendix A, based on the results for the jets at Mach numbers increasing in increments of ${\rm \Delta} {M}=0.01$ in figure 1. Well-organized peaks are visible in the spectrograms. For ${M}\leq 0.65$, the dominant peaks are at ${St}_D \simeq 0.70$ for both $n_{\theta }=0$ and $1$. As discussed in Appendix B, they happen at the vortex-pairing frequency and may result from the establishment of a feedback loop between the Kelvin–Helmholtz instability waves and the upstream-propagating sound waves generated by the first stage of vortex pairings in the initially laminar shear layers. For higher Mach numbers, the peaks form continuous bands, with the first two bands associated with the axisymmetric and the first helical modes, respectively. The bands have central Strouhal numbers decreasing with the Mach number, and emerge hardly from the background noise below ${M}= 0.75$ but quite distinctly above. They are similar to those measured near the nozzle lips of free jets by Jaunet et al. (Reference Jaunet, Jordan, Cavalieri, Towne, Colonius, Schmidt and Brès2016) and Zaman & Fagan (Reference Zaman and Fagan2019) for $0.6\lesssim {M} \leq 1$, and resemble the allowable frequency bands obtained for the upstream-propagating guided jet waves in figure 7. They persist for supersonic Mach numbers up to ${M}=2$, without any discontinuity or energy jump from one band to another. In particular, they do not turn into screech-tone bands, as it was the case for the non-ideally expanded jets of Zaman & Fagan (Reference Zaman and Fagan2019).

Figure 24. Power spectral densities of (a) pressure fluctuations at $z=0$ and $r=1.5r_0$, normalized by their peak values, and contributions of modes (b) $n_{\theta }=0$ and (c) $n_{\theta }=1$ for the jets with untripped boundary layers with $\delta _{BL}=0.2r_0$ as a function of $({M},{St}_D)$. The grey scale ranges logarithmically from $10^{-2}$ to $10$.

The spectra of the pressure fluctuations at $z=0$ and $r=1.5r_0$ for the six jets with tripped boundary layers at ${M}=0.60$, $0.75$, $0.90$, $1.10$, $1.30$ and $2$ are represented in figure 25(a–f) as a function of ${St}_D$, along with the contributions of modes $n_{\theta }=0$ to 8. The Strouhal numbers of specific points on the dispersion curves predicted by the vortex-sheet model for the guided jet modes ($n_{\theta }$, $n_r=1$) are also indicated by dash-dotted lines. For a given mode, the reported point is the inflexion point $I$ when the dispersion curve has no portion with a positive slope. Otherwise, the point is the stationary point $S_{max}$, or the limit point $L$ on the line $k=-\omega /c_0$ when there is no local maximum on the curve. The first case happens for all modes for ${M}=0.60$ in figure 25(a), and the second one for all modes for ${M}\geq 0.90$ in figure 25(c–f). For ${M}=0.75$ in figure 25(b), both cases occur and the dash-dotted lines are associated with points $I$ for $n_{\theta }=0-3$, $S_{max}$ for $n_{\theta }=4$ and $L$ for $n_{\theta }=5-8$. According to figure 6, these lines reveal the cutoff Strouhal numbers of the band-pass filtering of the sound waves propagating in the upstream direction outside of the jets by the guided jet modes.

Figure 25. Sound pressure levels obtained at $z=0$ and $r=1.5r_0$ for the jets with tripped boundary layers at (a) ${M}=0.60$, (b) ${M}=0.75$, (c) ${M}=0.90$, (d) ${M}=1.10$, (e) ${M}=1.30$ and (f) ${M}=2$ as a function of ${St}_D$: full spectra (black lines) and $n_{\theta }=$ 0 (red lines), 1 (blue lines), 2 (green lines), 3 (magenta lines), 4 (yellow lines), 5 (light blue lines), 6, 7 and 8 (grey lines); ${St}_D$ at points $I$ (dashed lines) and $S_{max}$ or $L$ (dash-dotted lines) on the dispersion curves for the guided jet modes ($n_{\theta }$, $n_r=1$) using the same colours as for the solid lines.

In figure 25, peaks emerge in the spectra for the full pressure signals, strongly for ${M} \geq 0.75$ but hardly for ${M}=0.60$. This is in agreement with the experimental results of Suzuki & Colonius (Reference Suzuki and Colonius2006) and Zaman & Fagan (Reference Zaman and Fagan2019). They correspond to peaks in the spectra for the modes $n_{\theta }=0$ to $n_{\theta }^{max}$, with $n_{\theta }^{max}$ depending on the Mach number and being equal to 8 for ${M}=0.75$ and to 1 for ${M}=2$, for instance. For a given azimuthal mode, the first peak falls very near the associated dash-dotted line. The second peak is much weaker than the first one for subsonic Mach numbers in figure 25(a–c), but is significant for supersonic ones for $n_{\theta }=0$ and 1 in figure 25(d–f). Coming back to the first peak for each $n_{\theta }$, it is tonal and well above the background noise when the dash-dotted line is plotted for a point $S_{max}$ or $L$. For a point $I$, in contrast, the peak can be more or less broadband and prominent. For ${M}=0.75$, in figure 25(b), the peak is very narrow for mode $n_{\theta }=3$ in magenta, and broadens for a lower azimuthal mode. The peaks are still larger and less pronounced for ${M}=0.60$ in figure 25(a) than for ${M}=0.75$. These results demonstrate the close links between the near-nozzle peaks and the upstream-propagating guided jet modes over the wide range of Mach numbers considered in this study. Moreover, they suggest that the tonal character of the peaks is related to the shape of the transfer function of the band-pass filtering associated with the latter modes outside of the jet, illustrated in figure 6. Indeed, the peaks are tonal for a sharp cutoff, but are gradually broader for a smoother one. This can explain the different shapes of the peaks for Mach numbers above and below ${M}=0.75$.

As was done in figure 16(a–c) for jets at ${M}=0.90$, the pressure spectra obtained at $z=0$ and $r=1.5r_0$ for $n_{\theta }=0$, 1 and 2 for the jets at ${M}=0.75$ with tripped boundary layers and with untripped boundary layers with $\delta _{BL}=0.2r_0$ are represented in figure 26(a–c). The allowable ranges for the upstream-propagating guided jet waves according to the vortex-sheet model are displayed in grey using the same colour code as in figures 7 and 9. Among these waves, those located between points $L$ and $I$ on the dispersion curves are specifically highlighted with greenish grey. For the two jets, and especially for the untripped one, peaks appear in the spectra, despite the fact that downstream-propagating guided jet waves cannot exist at ${M}=0.75$ for $n_{\theta }=0$, 1 and 2. This result, along with those reported in Appendix A for the jets at Mach numbers varying from ${M}=0.75$ up to ${M}=0.85$ in increments of ${\rm \Delta} {M}=0.01$, may cast doubt on the importance of the acoustic resonance possibly occurring in the jet potential core in the generation of the near-nozzle peaks. In figure 26, overall, the peaks lie in the greenish–grey bands. They strongly decrease on the right side of the bands, all the more rapidly that the peak is at a higher frequency. This is clearly observed, for instance, for the untripped jet in figure 26(b) for $n_{\theta }=1$. These trends are in agreement with the transfer functions of figure 6(b), revealing that along the dispersion curves of the guided jet modes, the decay of the wave magnitude outside of the jet is maximum around the inflexion point $I$, and that the decay rate at that point increases with the radial mode number. They provide additional evidence of the importance of the steepness of the band-pass filter associated with the latter modes on the near-nozzle peak shape. Regarding the Mach number effects for ${M}\leq 0.75$, the decrease of the spectra around point $I$ is slower for ${M}=0.60$ in figure 25(a) than for ${M}=0.75$ in figure 25(b). Again, this can be explained by the lower decay rates in the transfer functions for ${M}=0.60$ in figure 6(a) than for ${M}=0.75$ in figure 6(b).

Figure 26. Sound pressure levels obtained at $z=0$ and $r=1.5r_0$ for the jets at ${M}=0.75$ with tripped boundary layers (black lines) and with untripped boundary layers with $\delta _{BL}=0.2r_0$ (red lines) for (a) $n_{\theta }=0$, (b) $n_{\theta }=1$ and (c) $n_{\theta }=2$ as a function of ${St}_D$; the allowable ranges for the upstream-propagating guided jet waves are coloured in greenish-grey between the Strouhal numbers at points $L$ and $I$ on the dispersion curves of the modes and in light grey otherwise.

To examine the near-nozzle peak properties for a supersonic Mach number, the pressure spectra obtained at $z=0$ and $r=1.5r_0$ for $n_{\theta }=0$, 1 and 2 for the jets at ${M}=1.30$ with tripped boundary layers and with untripped boundary layers with $\delta _{BL}=0.2r_0$ are gathered in figure 27(a–c). The frequency ranges for the upstream-propagating guided jet waves according to the vortex-sheet model are displayed in grey. Downstream-propagating waves are permitted in the bands. The Strouhal numbers at the points $L$ of the dispersion curves of the waves on $k=-\omega /c_0$ are indicated by dash-dotted lines. They allow us to roughly locate the frequencies of the upstream-propagating guided jet waves when these waves are not predicted by the vortex-sheet model, refer to the frequency–wavenumber spectra in figure 23(g–i). In figure 27, the peaks are typically one order of magnitude higher for $n_{\theta }=0$ and 1 than for $n_{\theta }=2$. They are all near the grey bands, when available, or the dash-dotted lines, otherwise, despite the relatively poor agreement between the dispersion curves given by the model and the simulations in figure 21(d–i) for the jets at ${M}=1.10$ and 2. Therefore, for supersonic jets, the near-nozzle peaks can also be related to the upstream-propagating guided jet modes. In this case, interactions are possible between free-stream upstream-propagating and duct-like downstream-propagating waves close to the cutoff frequencies of the modes, as suggested by figure 23(d–i).

Figure 27. Sound pressure levels obtained at $z=0$ and $r=1.5r_0$ for the jets at ${M}=1.30$ with tripped boundary layers (black lines) and with untripped boundary layers with $\delta _{BL}=0.2r_0$ (red lines) for (a) $n_{\theta }=0$, (b) $n_{\theta }=1$ and (c) $n_{\theta }=2$ as a function of ${St}_D$; allowable ranges for the upstream-propagating guided jet waves (dark grey shading); Strouhal numbers at points $L$ on the dispersion curves (black dash-dotted lines).

The Strouhal numbers of the first three peaks associated with the guided jet modes in the near-nozzle spectra are represented as a function of the Mach number in figure 28(a–c) for the tripped jets for $n_{\theta }=0$, 1 and 2 and in figure 29(a,b) for the untripped jets for $n_{\theta }=0$ and 1. For the latter jets, the Strouhal numbers of the peaks obtained for ${M}\leq 0.65$ close to the vortex-pairing frequency, as documented in Appendix B, are also shown. The allowable frequency bands for the upstream-propagating guided jet waves, as well as the points $L$, $S_{max}$, $S_{min}$, $I$ and $I^{\prime }$ on the dispersion curves, defined in § 3 based on the vortex-sheet model, are displayed. For both tripped and untripped jets, for all azimuthal modes, the circles remarkably follow the variations of the guided jet modes over the entire Mach number range. They are found between points $L$ and $I$ for ${M}\leq 0.75$ and points $L$ and $S_{max}$ for ${M}\geq 0.80$. Thus, they all lie within the frequency ranges over which free-stream upstream-propagating guided jet waves are possible according to figure 6. This further supports that the presence of the near-nozzle peaks is mainly due to a filtering of the upstream-travelling sound waves by the guided jet modes, the amplitude of the waves with frequencies in specific ranges being preserved while that of the other waves decreases because of their evanescent nature.

Figure 28. Mach number variations of $\circ$ the Strouhal numbers of the first three peaks in the spectra of pressure fluctuations at $z=0$ and $r=1.5r_0$ for the jets with tripped boundary layers for (a) $n_{\theta }=0$, (b) $n_{\theta }=1$ and (c) $n_{\theta }=2$; allowable ranges for the upstream-propagating guided jet waves with (dark grey shading) and without (light grey shading) downstream-propagating guided waves, points $L$ (black lines), $S_{max}$ (red lines), $S_{min}$ (blue lines) and $I$ (green lines).

Figure 29. Mach number variations of the peak Strouhal numbers in the spectra of pressure fluctuations at $z=0$ and $r=1.5r_0$ for the jets with untripped boundary layers with $\delta _{BL}=0.2r_0$ for (a) $n_{\theta }=0$ and (b) $n_{\theta }=1$: peaks associated with the first three guided jet modes (black circles) and peaks at the vortex-pairing frequencies (black diamonds); allowable ranges for the upstream-propagating guided jet waves with (dark grey shading) and without (light grey shading) downstream-propagating guided waves, points $L$ (black lines), $S_{max}$ (red lines), $S_{min}$ (blue lines), $I$ (green lines) and $I^{\prime }$ (green dashed lines).

The intensities, degrees of emergence and full widths at half-maximum of the spectral peaks at $z=0$ and $r=1.5r_0$ are represented as a function of the Mach number in figure 30 for the tripped jets and in figure 31 for the untripped jets. For the tripped jets, the peaks are those associated with the first radial guided jet modes $n_r=1$ for $n_{\theta }=0$, 1 and 2, while for the untripped jets, modes $n_r=1$ and 2 are both considered for $n_{\theta }=0$ and 1. For the latter jets, zoomed views between ${M}=0.75$ and 0.85 are also available in Appendix A.

Figure 30. Near-nozzle peaks associated with the radial modes $n_r=1$ of the guided jet waves for $n_{\theta }=0$ (red circles), $n_{\theta }=1$ (blue circles) and $n_{\theta }=2$ (green circles) for the jets with tripped boundary layers: (a) peak levels, (b) ratios between the peak levels and the minimum values for higher ${St}_D$ and (c) peak widths as a function of ${M}$; ——–${M}^8$, – – – ${M}^3$; dash-dotted lines: ${\rm \Delta} {St}_D$ between points $I$ and $L$ or between $S_{max}$ and $L$ using the same colours for $n_{\theta }$ as for the circles.

Figure 31. Near-nozzle peaks associated with the guided jet modes (a–c) $n_r=1$ and (d–f) $n_r=2$ (circles) and at the vortex-pairing frequencies (diamonds) for $n_{\theta }=0$ (red) and $n_{\theta }=1$ (blue) for the jets with untripped boundary layers with $\delta _{BL}=0.2r_0$: (a,d) peak levels, (b,e) ratios between the peak levels and the minimum values for higher ${St}_D$ and (c,f) peak widths as a function of ${M}$; ——–${M}^8$, – – – ${M}^3$. Dash-dotted lines: ${\rm \Delta} {St}_D$ between points $I$ and $L$ or between $S_{max}$ and $L$ using the same colours for $n_{\theta }$ as for the symbols.

In figures 30(a) and 31(a,d), the peak levels increase roughly as ${M}^8$ for ${M}\leq 1$ and as ${M}^3$ for ${M}\geq 1$, following the typical scaling laws of aerodynamic noise for subsonic jets (Lighthill Reference Lighthill1952) and supersonic jets (Ffowcs Williams Reference Ffowcs Williams1963). This indicates, unsurprisingly, that the pressure waves propagating up to the near-nozzle region are acoustic waves generated by the jets. In most cases, there are no significant deviations from the ${M}^8$ law between ${M}=0.5$ and 1, suggesting that the acoustic resonances which can occur in the jet potential core for $0.80 \leq {M}\leq 1$ have a rather limited effect on the near-nozzle peak intensities. Given the 3–5 dB excess observed between the red circles and the trend line around ${M}= 0.85$ in figure 31(a), this may, however, not be true for the untripped jets for the peaks associated with the first radial axisymmetric guided jet mode. In figure 31(a), the levels of the peaks obtained for the untripped jets at low Mach numbers at the vortex-pairing frequency ${St}_D \simeq 0.70$ are also represented. For the axisymmetric mode, they strongly increase as the Mach number decreases. A tone can even be observed at ${St}_D \simeq 0.70$ in the near-nozzle spectrum for the jet at ${M} = 0.50$, provided in figure 35(a) of Appendix B. This may result from a feedback loop establishing between the Kelvin–Helmholtz instability waves and the sound waves generated by the vortex pairings.

Regarding the peak emergence, it is difficult to identify a clear trend in figure 30(b) for the tripped jets. For the untripped jets, however, the peak emergence gradually increases from ${M}=0.50$ up to ${M}\simeq 0.85$ and then decreases with the Mach number for ${M}\geq 0.85$ in figure 31(b,e). The increase may be linked to the steepening of the filters induced by the guided jet modes at their upper cutoff frequencies, shown in figure 6(a–c), providing lower noise levels just above these frequencies. As for the decrease, it may be due to the fact that at a higher Mach number the bands of the filters are closer to each other, leading to smaller bands without upstream-propagating guided jet waves as illustrated in the spectra of figure 23(f,i).

Finally, in figures 30(c) and 31(c,f), the peak widths decrease significantly between ${M}=0.50$ and 1, and then very slightly for ${M}\geq 1$. These variations can be explained by the narrowing of the allowable frequency bands for the free-stream guided jet modes as the Mach number increases. To demonstrate this, the band widths, estimated as the frequency differences between points $L$ and $I$ or points $L$ and $S_{max}$ on the dispersion curves obtained using the vortex-sheet model, are plotted. They are in fairly good agreement with the peak widths nearly up to ${M}= 1$ in all cases. Above ${M}= 1$, except for the mode ($n_{\theta }=0$, $n_r=2$) in figure 31(f), they are much smaller than the peak widths, which is not surprising considering the considerable discrepancies observed between the dispersion curves given by the model and the simulations near $k=-\omega /c_0$ in figure 23(d–i) for supersonic Mach numbers.