3.1. Experimental measurements of periodic separation

A series of time-resolved schlieren snapshots is shown in figure 7. The snapshots were acquired with the $L_e/D_e = 0.7$ mixing-duct length operated at $M_j = 0.90$ with both streams unheated – an operating condition at which the nozzle produced a powerful howling. A close-up view of the flow at and just downstream of the nozzle exit is shown. The dark shape on the left-hand side of each snapshot at $x \lt 0$ is the silhouette of the final nozzle. The high spatial resolution of these snapshots is evident upon inspection of the axes against which they are plotted: the snapshots span less than 0.1 $D_e$ in the transverse and axial directions. Within all the snapshots, a horizontal, dashed line located at the lipline of the nozzle ( $y = 0.5D_e$ ) is shown for visual reference. The images were acquired at a nominal sampling rate of 270 kHz (the ‘actual’ sampling rate reported by the camera’s control software was about 270.968 kHz). A segment from the schlieren recording is provided as Supplementary Movie 1, which shows a larger field of view than the still snapshots printed in the figure. The still snapshots are intended to recreate and emphasise the key oscillations visible in the movie. Each snapshot is assigned a number that is printed above it, with an increase in the number representing a forward passage of time.

Figure 7. One cycle of separation/reattachment shown by down-sampled series of time-resolved schlieren images (every fourth image) with $L_e/D_e$ = 0.7, $M_j$ = 0.90, unheated. Flow is left to right. A segment of the schlieren recording is available for playback (see supplementary movie 1).

In snapshot 1 of figure 7, a bright white region is visible just below the nozzle lipline. This is the jet/ambient shear layer where transverse density gradients were present because the jet was unheated. The approximate location of the jet/ambient shear layer at the nozzle-exit plane is indicated with a rightward-pointing arrow in each snapshot. By comparing the relative position of the arrow and the dashed line across the snapshots, the shear layer’s oscillation in time is visible. Snapshots 1–6 show the shear layer moving away from the nozzle lip, while snapshots 7–10 show the shear layer returning to the nozzle lip. This is clear evidence that, when the howling occurred, the boundary layer just upstream of the nozzle’s exit periodically separated and reattached. At $M_j = 0.80$ and $M_j = 1.00$ , similar visualisations were obtained with an unheated and a heated core stream, which are not shown for brevity. These operating conditions did not produce any howling, and no periodic separation was visible in the visualisations.

A spectral proper orthogonal decomposition (SPOD) of the fluctuations in the schlieren images was conducted using the open-source MATLAB code documented by Towne, Schmidt & Colonius (Reference Towne, Schmidt and Colonius2018) (available at https://github.com/SpectralPOD/spod_matlab). Welch’s method of spectral estimation (Welch Reference Welch1967) was used with $2048$ -image, Hanning-windowed blocks with 50 % overlap across $8192$ total images (corresponding to a duration of approximately 0.03 s). Figure 8(a) shows all seven SPOD eigenvalue spectra obtained from analysing the schlieren video acquired with the core stream unheated. At the nominal frequency of the measured, fundamental acoustic tone (indicated by the vertical, dotted line), the leading-order SPOD mode’s eigenvalue spectrum contains a peak which indicates organised, high-amplitude oscillations at that frequency. As will be shown below in figure 9, the SPOD mode corresponding to this peak in the spectrum was inspected and confirmed to represent the periodic separation visible in the schlieren recording. As expected, harmonics of the fundamental tone are present as well.

Figure 8. SPOD eigenvalue spectra obtained from schlieren with $L_e/D_e = 0.7$ , $M_j = 0.90$ : (a,c) unheated and (b,d) heated core stream ( $T_{t1} \approx 533$ K, $TTR_1 \approx 1.85$ ). (c,d) Show same spectra as (a,b) over limited frequency range. Nominal, measured frequency of fundamental acoustic tones plotted as vertical, dotted lines.

Figure 9. Leading-order SPOD modes obtained from measured schlieren with $L_e/D_e = 0.7$ , $M_j = 0.90$ with (a) flow unheated and (b,c) heated core stream ( $T_{t1} \approx 533$ K, $TTR_1 \approx 1.85$ ). In each panel the frequency of the SPOD mode is shown in the top, left-hand corner and lipline of final nozzle shown as a white, dashed line. Normalised real part shown for arbitrary phase. Flow is left to right.

Figure 8(b) shows the eigenvalue spectra obtained by analysing a similar schlieren recording acquired with the core stream heated to $T_{t1} \approx 533$ K ( $TTR_1 \approx 1.85$ ). A segment from this schlieren recording is provided as Supplementary Movie 2. The separation frequency evident from the schlieren and the nominal frequency of the fundamental tone from acoustic measurements (indicated by a vertical dotted line) match with the core stream heated as well. The eigenvalue spectra are quite similar to those with the core stream unheated; however, a high-frequency feature is indicated by a downward-pointing arrow at a relatively high frequency. This is the signature of unsteady flow features in the wake of the core-nozzle lip as they exit the final nozzle. This conclusion was drawn by inspecting the leading-order SPOD mode at this frequency, which will be shown below in figure 9. Again, the instability in the wake of the core-nozzle lip is not discussed further because it is of too high a frequency to cause the howling studied here.

Close-up views of the fundamental peaks in the SPOD eigenvalue spectra are shown in figures 8(c) and 8(d), emphasising the excellent agreement between the nominal frequencies of the acoustic tones and the periodic separation. The peaks in the leading-order eigenvalue spectra (the top-most spectra in figures 8 c and 8 d) are more than two orders of magnitude greater than the spectra appearing beneath them. This indicates that the fluctuations in the video at the frequency of the fundamental acoustic tone are dominated by an orderly flow feature: the signature of the periodic separation. The excellent agreement between the acoustic tone’s frequency and the periodic separation’s frequency suggests a feedback phenomenon involving the flow instability responsible for the periodic separation caused the observed howling. Again, the nature of this flow instability is made clear in the following two subsections.

For completeness, salient leading-order SPOD modes associated with the schlieren recordings are shown in figure 9. In each of the three panels, the lipline of the final nozzle is shown as a horizontal, white dashed line and the corresponding frequency is printed on the top, left-hand corner. The SPOD mode shown in figure 9(a) corresponds to the fundamental frequency present in the schlieren with unheated flow. Alternating dark and bright regions can be seen on the far left near $x = 0$ and just below the lipline of the final nozzle (see red arrow). This is the signature of the periodic separation at the nozzle exit. Figure 9(b) shows a similar signature of the periodic separation, but also reveals a signature of the core-jet instability waves when the core stream is heated and there is substantial difference between the two streams’ jet velocities. Because similar phenomena were observed with the flow unheated, this instability is believed to occur as a passive response of the core jet to the fluctuations associated with the howling. The core-jet instability is not discussed further in this paper. For completeness, figure 9(c) shows the high-frequency signature of what is believed to be vortex shedding from the core-nozzle lip. This vortex shedding and the noise it produces were studied by Ramsey et al. (Reference Ramsey, Larisch and Ahuja2025). The vortex-shedding frequency is an order of magnitude higher than the howling frequency and is thus not needed to explain the observed howling.

3.2. Evidence that the boundary-layer separation is shock induced

RANS simulation results are shown in figure 10. In figure 10(a), a typical local-Mach-number distribution is shown at and just upstream of the final-nozzle exit for the $L_e/D_e$ = 0.7 mixing-duct length operated at $M_j$ = 0.90, unheated, which produced a powerful howling in the experiments. The white space in the centre, top portion of this image is the cross-section of the final nozzle, with the flow from left to right as indicated by the streamlines. A small-radius bend existed along the final-nozzle wall between its converging and final straight sections and, from this Mach-number distribution, it is evident that there was localised supersonic flow in this region. This is very similar to the locally supersonic flow found by prior researchers studying internally mixed nozzles with a small-radius, convex bend in the nozzle wall (e.g. see Garrison et al. Reference Garrison, Lyrintzis, Blaisdell and Dalton2005; Wright, Blaisdell & Lyrintzis Reference Wright, Blaisdell and Lyrintzis2006; Kube-McDowell, Lyrintzis & Blaisdell Reference Kube-McDowell, Lyrintzis and Blaisdell2010). The supersonic flow was necessarily over-expanded because the stagnation-to-ambient pressure ratio was subcritical. Therefore, to eventually reach the ambient pressure outside the nozzle, the supersonic region was terminated by a shock wave. The right half of the ‘Mach = 1 contour’ called out in the image indicates the location of a shock. The left half of this contour simply represents the locus of physical points where gas flowing through the supersonic region reached sonic speed. The inlaid view emphasises the lambda-shock structure, indicating that shock-induced boundary-layer separation occurred. The upstream leg of the lambda-shock structure is not made visible by the Mach = 1 contour because it was an oblique shock that turned the flow away from the wall but did not bring the flow back down to subsonic speeds.

Figure 10. RANS results for $L_e/D_e$ = 0.7, unheated: (a) map of local Mach number at $M_j$ = 0.90 and (b) maximum local Mach number inside nozzle across a range of $M_j$ .

A series of RANS simulations were conducted at several $M_j$ between 0.70 and 0.93, using the same $L_e/D_e = 0.7$ mixing-duct length and with unheated flow. The maximum local Mach number inside the nozzle was extracted from each simulation, and the results are summarised in figure 10(b). The specified $M_j$ is listed on the horizontal axis and the resulting maximum local Mach number is indicated on the vertical axis. A horizontal ‘sonic line’ is plotted at unity Mach number for reference. The RANS simulations suggested that a region of supersonic flow exists inside the nozzle above $M_j$ between 0.75 and 0.78. A linear interpolation suggested $M_j = 0.76$ and is indicated by the solid, vertical line: the ‘RANS sonic threshold.’ If shock-induced boundary-layer separation were necessary for the howling to occur, the $M_j$ above which howling ensued in experiments should be above the RANS sonic threshold. Experimental data (to be discussed in § 3.4) indicated that howling may occur for $M_j$ as low as 0.81. This is indicated in figure 10(b) by the shaded region to the right of the vertical, dotted line. These results show that, when howling occurred in the experiments, the computations indicate a region of supersonic flow exists inside the nozzle. Supersonic flow inside the nozzle (and thus a shock inside the nozzle) was evidently necessary for the howling to occur.

One may wonder why the howling does not occur precisely at the RANS sonic threshold. This is understood as follows. A shock’s presence inside the nozzle is a necessary but not sufficient condition for shock-induced boundary-layer separation. The shock must not only be present but also strong enough (i.e. it must bring about a sufficiently large adverse pressure gradient) to separate the boundary layer upon its interaction with it. This is consistent with our observations that the lowest $M_j$ that produced howling in experiments ( $M_j$ = 0.81, as noted in the previous paragraph) is higher than the RANS sonic threshold. A variety of indicators for the onset of flow instability were used for sensible judgement of the RANS results, suggesting that flow instability occurs for $M_j \geqslant 0.85$ . Therefore, we cautiously note that the RANS simulations themselves indicated a flow instability did not occur precisely at the RANS sonic threshold but rather at higher $M_j$ . Grid-refinement studies were not performed but would likely fine tune this finding. Detecting the onset of flow instability using RANS is imprecise, and we avoid weighing such indicators too heavily.

3.3. Explaining why the boundary-layer separation is periodic

HRLES results are presented here for $L_e/D_e = 0.7$ , $M_j = 0.90$ with the flow unheated. In figure 11, the mean-velocity distribution along the jet’s centreline obtained from the HRLES at $M_j = 0.90$ is shown to have close agreement with the ensemble-averaged particle-image velocimetry (PIV) measurements first reported by Ramsey et al. (Reference Ramsey, Mayo, Karon, Funk and Ahuja2022b ). Witze (Reference Witze1974) provided a model for a single, round jet (approximately what the nozzle produces at unheated operating conditions with both streams at the same pressure ratio) that was included in the figure to emphasise the shortening of the potential core of the jet that occurred along with the howling. This is a typical effect of jet excitation (e.g. see Ahuja, Lepicovsky & Burrin (Reference Ahuja, Lepicovsky and Burrin1982)). The agreement between the HRLES results and experimental data inspired confidence that the jet excitation present in the experiments was captured by the simulation.

Figure 11. Comparison of centreline mean-velocity distribution from HRLES simulation with experimental PIV data (Figure modified from Ramsey et al. (Reference Ramsey, Mayo, Karon, Funk and Ahuja2022b ).)

In figure 12, a series of snapshots of local Mach number from the HRLES simulation are shown. The snapshots show a close-up view at and just upstream of the nozzle exit, with white space in the snapshots corresponding to the nozzle’s cross-section. An animation of these simulation results is included as Supplementary Movie 3. From these snapshots of the HRLES results, it is evident that the periodic separation measured outside the nozzle in § 3.1 was the downstream signature of a periodic SWBLI occurring just inside the nozzle. The periodic SWBLI is understood as follows.

Figure 12. Snapshots from HRLES simulation showing instantaneous distributions of local Mach number. $L_e/D_e$ = 0.7, $M_j$ = 0.90, unheated.

Shock-induced boundary-layer separation occurred in the vicinity of the small-radius bend in the nozzle wall and, once separation occurred, the flow was no longer required to make as sharp a turn. Supersonic flow inside the nozzle then ceased to exist and the shock vanished. The shock that induced boundary-layer separation was no longer present and the boundary layer reattached to the nozzle wall. The flow accelerated again to locally supersonic speeds in the vicinity of the small-radius, convex bend. The shock was again strong enough to separate the boundary layer, and the cycle continued periodically. In Appendix A, it is shown via experiments that the howling can be suppressed by simply adding an appropriate boundary-layer trip just upstream of the small-radius, convex bend. These results are not discussed here to avoid detracting from the study of the howling. An axial velocity spectrum was computed from the HRLES using the fluctuations at a single point along the lipline of the final nozzle and indicated that the separation occurred at a frequency of 7747.7 $\pm$ 125 Hz. The frequency of oscillations found in the HRLES was greater than the typical frequency of measured flow oscillations at the same operating condition (approximately $6$ kHz, as shown in § 3.1). An explanation for this is forthcoming.

A look at pressure fluctuations in the HRLES reveals that a particular acoustic mode of the nozzle’s mixing duct was energised when the SWBLI took place. A time history of pressures inside the straight cylindrical portion of the nozzle’s mixing duct at $x = -0.37D_e$ was extracted from the HRLES at a sampling rate of about 110 kHz along a radial line running from one side of the duct to the other. Considering pressures along a single line rather than the entire cross-stream plane at a given axial position was justified because the orderly oscillations in the simulation appeared axisymmetric. The pressure fluctuations were extremely orderly, as seen in the segment of the time history in figure 13(a). This shows the pressure fluctuations over 10 $\tau$ , where $\tau$ is the period of the dominant oscillations. The pressure fluctuations were most powerful near the centreline of the duct. The pressure fluctuations at the duct’s wall were out of phase with and weaker than those at the duct’s centreline. Further, there was a radial position where the pressure fluctuations were nearly zero at all times: approximately a ‘pressure node.’ As will be clear shortly, this was the signature of a particular higher-order acoustic mode of the mixing duct.

Figure 13. Pressure fluctuations inside the nozzle’s mixing duct at $x=-0.37D_e=-0.39D_1$ in HRLES: (a) time history, (b) power spectral density on duct centreline and (c) radial profile of pressure-fluctuation magnitude in HRLES at $f =$ 7747.7 $\pm$ 125 Hz compared with $(0,1)$ acoustic mode of a round duct with uniform flow. $L_e/D_e$ = 0.7, $M_j$ = 0.90, unheated.

The power spectral density at the duct’s centreline estimated using single block of 440, Hanning-windowed time steps is shown in figure 13(b). A dominant 7747.7 $\pm$ 125 Hz discrete frequency was present. This is the same frequency as the periodic separation in the simulation. The discrete Fourier transform (in time) of the extracted pressure fluctuations along the entire radial line was taken and the magnitude of this Fourier transform at 7747.7 Hz is shown in figure 13(c). As seen in the time history, the fluctuations are strongest near the centre of the duct at $r=0$ and exhibit an approximate pressure node between the centreline at $r=0$ and the mixing duct’s wall at $r=D_2/2$ . Alongside the curve obtained from the HRLES, a result from classical duct acoustics is shown: the pressure-fluctuation magnitude of a round duct’s $(0,1)$ acoustic mode in the presence of uniform flow. Aside from discrepancies at larger $r$ at and near the mixing-duct wall (which could be due to refraction of sound by the boundary layer near the wall), there is excellent agreement. This shows that an excitation of the mixing duct’s $(0,1)$ acoustic mode accompanied the periodic SWBLI at the final nozzle’s exit. More fundamentally, this suggests that the SWBLI and associated oscillations were capable of exciting higher-order acoustic modes of the nozzle’s mixing duct. As will be evident shortly, experiments suggest that the SWBLI at the final nozzle’s exit predominantly coupled with the $(2,0)$ acoustic mode of the nozzle’s mixing duct at its cut-on frequency – not the $(0,1)$ mode found in the HRLES.

Additional work would be required to rigorously show that the excited higher-order acoustic mode was indeed a natural acoustic mode in the simulation – although this is likely the case. As explained earlier in § 1.3, natural acoustic modes have zero axial group velocity and do not transport energy along the duct. Given the HRLES time history, showing this condition is met would require a numerical evaluation of the time-averaged acoustic energy flux through a cross-stream plane at a given axial position inside the nozzle’s mixing duct. If this time-averaged acoustic energy flux was found to be zero, then it could safely be stated that the higher-order mode excited in the simulation was a natural acoustic mode. Short of determining the group velocity, the axial phase velocity of the waves appearing in the HRLES could be compared with values for the duct’s relevant natural acoustic modes obtained by theoretical means. However, we carefully recall that, at no point during the set up of the simulations (e.g. grid refinement) were there checks that the acoustic duct modes inside the nozzle were properly resolved. Although the SWBLI consistently appeared in our simulations, the frequency of oscillation exhibited some sensitivity to the grid refinement, indicating that further computational study is required. We prefer to exercise caution and recommend that, prior to making any stronger statements regarding the HRLES and carrying out acoustic energy-flux calculations, a detailed (and computationally expensive) HRLES campaign be conducted to ensure that the acoustics of the nozzle interior are properly resolved. This substantial additional work is not pursued here, because the HRLES has served its purpose of revealing that a SWBLI was responsible for the experimentally observed periodic separation and that this SWBLI could excite the higher-order acoustic modes of the nozzle’s mixing duct.

3.4. The fundamental acoustic tone’s frequency and amplitude

Turning back to experimental data, figure 14 shows the dependence of the fundamental acoustic tone’s frequency (figure 14 a) and amplitude (figure 14 b) on $M_j$ across the range $0.80 \lt M_j \lt 1.00$ in steps as small as 0.01. Within this figure, data acquired using three different mixing-duct lengths ( $L_e/D_e$ = 0.7, 2.0 and 3.0) are shown but measurements acquired using the $L_e/D_e$ = 0.7 mixing-duct length are discussed first.

In acquiring the data for the $L_e/D_e = 0.7$ mixing-duct length, acoustic measurements were made on four different days over the span of several months. On each of these days, the facility was set up as consistently as possible and measurements were made across jet Mach numbers from 0.80 to 1.00. For readability of the figure, the resulting ranges of fundamental-tone frequency and amplitude are shown (as opposed to every data point) for $M_j$ at which multiple measurements were made and the fundamental tone’s frequency fit comfortably with the dominant trend in the data. Due to the onset of the howling occurring at slightly different $M_j$ in measurements made on different days, only single data points appear at $M_j = 0.81$ and $M_j = 0.82$ . As was alluded to earlier, the howling was found to occur at $M_j$ as low as 0.81. Beginning at $M_j$ = 0.83, the $L_e/D_e$ = 0.7 data are shown as a narrow, shaded envelope running horizontally across the plot, indicating that the frequency of the fundamental tone was largely insensitive to $M_j$ . For $0.94 \leq M_j \leq 0.97$ , however, there are three data markers that appear above this envelope, indicating that the fundamental tone sometimes occurred at a higher frequency. All three of these higher-frequency data markers correspond to measurements made on the same day, and it is not presently clear what caused this. Only speculation is possible, which we avoid here.

The frequency of periodic separation in the HRLES simulations (7747.7 $\pm$ 125 Hz) was somewhat close to the frequency of these higher-frequency data markers, as shown by the data at $M_j = 0.90$ labelled ‘HRLES simulation ( $L_e/D_e$ = 0.7).’ As $M_j$ approached unity, the howling ceased. Measurements were made at $M_j$ = 0.99 on some days, but no tones were visible in spectra measured at this operating condition. The curves which run nearly horizontally across the plot show the cut-on frequencies of the mixing duct’s higher-order acoustic modes that fell close to the measured frequencies. These cut-on frequencies will be discussed shortly.

In figure 14(b), the grey, shaded envelope shows that there was substantial scatter in the fundamental tone’s amplitude on the different occasions it was measured, despite care taken to ensure the experimental set-up was as consistent as possible. Due to this scatter, not much may be said about the fundamental tone’s amplitude. It does appear, however, that the highest-amplitude tones were measured at $0.86 \leq M_j \leq 0.88$ . The scatter is particularly large at $M_j = 0.90$ . Further statements about this variation in tone amplitude would require an additional experimental campaign of many repeated experiments. Because we do not study the tone’s amplitude in great detail here, these additional experiments are not pursued. However, we can say that this day-to-day variation was larger than experiments repeated in close succession on the same day. A relatively small difference of $2$ dB has been observed between running the jet at $M_j = 0.90$ , then an operating condition that did not produce any howling and then $M_j= 0.90$ again without shutting off the flow. We focus on the tone’s frequency in pursuit of an understanding of the observed howling.

Figure 14. Measured fundamental acoustic tone with unheated core-stream (a) frequency and (b) sound pressure level (SPL) for several $M_j$ for three different mixing-duct lengths. ( $\theta$ = 90 $^{\circ }$ , lossless, fully corrected, projected to $R$ = 84.7 $D_e$ , and $\Delta f$ = 2.5 Hz.)

Returning to figure 14(a), increasing the nozzle’s mixing-duct length reduced the fundamental tone’s frequency slightly, and an explanation for this will be put forth in § 4. It is noted with caution (due to the large scatter in the fundamental tone’s amplitude mentioned in the previous paragraph) that the longer mixing-duct lengths tended to produce lower-amplitude fundamental tones, as seen in figure 14(b). Of course, this was not always the case. For example, at $M_j$ = 0.91 the $L_e/D_e$ = 3.0 mixing-duct length produced a tone that was of higher amplitude than the $L_e/D_e$ = 0.7 mixing-duct length produced during any of the many measurements made. With the measurements corresponding to the two longer mixing-duct lengths made on only one occasion, it is reasonable to expect that, had the measurements been repeated several times, the longer mixing-duct lengths would have produced a scatter in the fundamental tone’s amplitude similar to what is shown by the grey, shaded envelope corresponding to the $L_e/D_e$ = 0.7 mixing-duct length. Therefore, not much may be said about the role of the mixing-duct length on the fundamental tone’s amplitude.

In figure 14(a), the curves show the cut-on frequencies of the nozzle’s round mixing duct that were nearest the measured tone frequencies: those of the $(2,0)$ and $(0,1)$ modes. The procedure used to calculate these cut-on frequencies is outlined in Appendix B. There are no cut-on frequencies that lie between those of the $(2,0)$ and $(0,1)$ modes, and the cut-on frequencies were nearly constant across the range of $M_j$ plotted. This is because, although the cut-on frequencies do depend on $M_j$ , as $M_j$ increases from 0.80 to 1.00, the Mach number of the flow inside the mixing duct only increases slightly from 0.433 to 0.454 (according to quasi-one-dimensional, isentropic-flow calculations). As explained earlier in § 1.3, the cut-on frequencies are natural frequencies of the mixing duct at which aeroacoustic feedback phenomena are particularly likely to occur, provided a suitable, sound-generating flow instability exists at a nearby frequency. In the present case, this flow instability is the SWBLI near the final-nozzle exit – although the specific sound-generation mechanisms are not studied in detail in this paper. There is good agreement between the calculated cut-on frequencies and the measured tone frequencies, suggesting that the natural acoustic modes of the nozzle’s round mixing duct indeed coupled with the SWBLI in a feedback phenomenon that led to the observed howling. A coupling to the $(2,0)$ natural acoustic mode appears to predominate coupling with the $(0,1)$ natural acoustic mode at a slightly higher frequency. This may occur because the cut-on frequency of the $(2,0)$ mode was closer to the characteristic frequency of the SWBLI (i.e. the frequency it would tend to occur at in the absence of acoustic coupling). A substantial research effort would be required to rigorously support this, as the characteristic frequency of the SWBLI is not presently known.

The reader may recall that the signature of the mixing duct’s $(0,1)$ acoustic mode was found in the HRLES. A detailed HRLES simulation campaign would be required to explain this discrepancy, and this is beyond the scope of this paper. However, we again emphasise that the HRLES has served its purpose of showing that the SWBLI was the root cause for the periodic separation observed in the experiments and that the SWBLI can excite a higher-order acoustic mode of the nozzle’s mixing duct. In what follows, we focus on the $(2,0)$ acoustic mode because its cut-on frequency was in close agreement with the majority of the measured fundamental-tone frequencies in experimental data in figure 14(a). Although the measured frequencies were consistently higher than the calculated cut-on frequencies, an explanation for this is put forth in § 4.

We briefly note that natural acoustic modes of longitudinal nature (i.e. involving waves travelling along the axis of the mixing duct) cannot explain the measured trends because the fundamental tone’s frequency was relatively insensitive to $L_e$ .

3.5. Azimuthal order of the oscillations

As shown in § 3.1, the fundamental tones are attended by a periodic boundary-layer separation at the final-nozzle exit and an excitation of instability waves in the final jet at the same frequency. In § 3.3, we showed that this separation was caused by a periodic SWBLI inside the nozzle which excited an acoustic mode of the nozzle’s mixing duct. Basic calculations provided in § 3.4 indicated that the predominantly observed howling frequency was close to the cut-on frequency of an acoustic mode of the nozzle’s mixing duct. These observations alone strongly suggest that the howling was the result of a feedback phenomenon between the SWBLI and a natural acoustic mode of the nozzle’s mixing duct. Again, this coupling seems to predominantly involve the $(2,0)$ acoustic mode of the nozzle’s mixing duct. Before proceeding to more-sophisticated models for estimating the cut-on frequencies of the mixing duct’s higher-order acoustic modes, additional experimental data are provided. These data further support that a coupling between the SWBLI and the mixing duct’s $(2,0)$ acoustic mode took place when the howling occurred.

The feedback phenomenon studied here was, as explained several times now, caused by a coupling between the SWBLI and a natural acoustic mode of the nozzle’s mixing duct. In the event of such a coupling, the SWBLI that excited the natural acoustic mode will also be influenced by the natural acoustic mode. Both should exhibit the same azimuthal order. The instability waves excited in the final jet as a result of this coupling should exhibit this same azimuthal order, too. Measurements and accompanying theory appearing in § 4.5 of Ramsey (Reference Ramsey2024) provide extensive evidence that this is the case. For completeness, a select few velocity-field measurements obtained via stereoscopic PIV appearing in that reference are presented here. A description of the measurement system is foregone here but may be found in § 4.5 of (Ramsey Reference Ramsey2024), along with careful checks of the accuracy of the measured data.

Figure 15 shows a baseline PIV dataset obtained using the $L_e/D_e = 0.7$ configuration of the nozzle at an operating condition that did not produce any howling: $M_j = 0.70$ , unheated flow. These data were obtained in a cross-stream plane at $x = 2D_e$ downstream of the final nozzle’s exit. The reader should imagine the jet axis pointed out of the page when inspecting these data. Figure 15(a) shows the axial mean velocity, $U$ . This measured mean velocity was normalised by the ideally expanded core-jet velocity ( $U_{j1}$ ), which was nominally the same as that of the bypass stream at this operating condition. Figure 15(b) shows the axial turbulence intensity, which was computed as the standard deviation of axial velocity fluctuations ( $\sigma _x$ ) normalised by the ideally expanded core-jet velocity. These data were obtained from an ensemble of $N$ = 750 independent snapshots of the velocity field, which yielded suitable convergence of the mean and standard deviation of the velocity. Both the mean velocity and turbulence intensities seen are typical of a turbulent, round jet and compare well with the well-accepted published data (e.g. see Ahuja et al. (Reference Ahuja, Lepicovsky and Burrin1982); Bridges & Wernet (Reference Bridges and Wernet2011)).

Figure 15. Velocity-field measurements in a cross-stream plane at $x/D_e$ = 2 with $M_j=0.70$ , unheated: (a) axial mean and (b) axial turbulence intensity. $N = 750$ snapshots averaged. $L_e/D_e = 0.7$ .

Figure 16. Velocity-field measurements in a cross-stream plane at $x/D_e$ = 2 with $M_j = 0.90$ , unheated: (a) axial mean and (b) axial turbulence intensity. $N = 750$ snapshots averaged. $L_e/D_e = 0.7$ .

Figure 17. Velocity-field measurements in a cross-stream plane at $x/D_e$ = 1 with $M_j = 0.90$ , unheated: (a) axial mean and (b) axial turbulence intensity. $N = 750$ snapshots averaged. $L_e/D_e = 0.7$ .

Figure 16 shows measurements made using the same nozzle configuration at an operating condition that produced the howling: $M_j = 0.90$ , unheated flow. Figure 16(a) reveals a distorted mean flow, and figure 16(b) shows that the axial turbulence intensity is higher in the presence of the howling. This is expected of an excited jet. What is not expected is that there are four localised regions of relatively high turbulence intensity in the jet’s shear layer at 90-degree intervals from one another. This pattern is yet clearer in data obtained closer to the final-nozzle exit at $x = D_e$ in figure 17. If the instability waves excited in the final jet were axisymmetric ( $m=0$ ), the turbulence intensity would also appear nominally axisymmetric. Had these instability waves been flapping (one variety of $m=1$ instability waves that do not spin about the jet axis), there would be two regions of locally high turbulence intensity appearing on opposing sides of the jet. Although perhaps not obvious at first, these patterns in the measured turbulence intensity are the signature of $m = 2$ instability waves that remain preferentially oriented, rather than spinning about the jet axis. These waves might be suitably called ‘pinching’ instability waves due to their appearance in a cross-stream plane. These experimental data thus support that the feedback phenomenon involved a coupling between the SWBLI and the $(2,0)$ acoustic mode of the nozzle’s mixing duct. This is shown more rigorously in § 4.5 of Ramsey (Reference Ramsey2024) through comparisons with results of linear stability theory, where it is also shown that the same patterns discussed here can be found in measured PIV data obtained on five different days over the span of one month without modifying the experimental set-up. Other data appearing in that reference indicate the excited instability waves have $m = 2$ azimuthal order, too.

4.1. Further experimental data

The frequency of the fundamental tone was shown to be largely insensitive to $M_j$ and $L_e$ above in figure 14. However, the fundamental tone’s frequency was sensitive to $T_{t1}$ , as shown here in figure 18, where the total temperatures reported are averages across entire microphone recording durations. Figures 18(a) and 18(b) correspond to two different tests conducted using the $L_e/D_e = 0.7$ mixing-duct length, which are denoted ‘test 1’ and ‘test 2’. The ‘test 1’ data shown in figure 18(a) were measured as follows. The nozzle was operated at $M_j$ = 0.90, and then the core stream’s burner was lit. Subsequently, $T_{t1}$ rose to about 600 K ( $TTR_1 \approx 2$ ) over approximately 30 minutes. A total of 45 typical, 45-second microphone recordings were acquired during this period. For approximately 30 minutes, $T_{t1}$ was held around 600 K and a total of 15 45-second microphone recordings were acquired. Then, the burner was shut off and the jet was continuously run while the facility cooled. During this cooling period, a total of 34 shorter, 20-second microphone recordings were acquired with hopes of capturing the temperature dependence with a finer resolution in time. The ‘test 2’ data shown in figure 18(b) were measured on a different day, and a similar procedure was conducted with 20-second recordings acquired throughout. During test 2, $T_{t1}$ was only allowed to rise to about 525 K ( $TTR_1 \approx$ 1.77) and 31 recordings were obtained during this time. The burner was shut off immediately at that temperature (a practical decision made to reduce the test’s duration), and another 23 recordings were obtained as the temperature decreased.

Figure 18. Measured fundamental acoustic tone’s frequency as a function of core-stream total temperature ( $L_e/D_e = 0.7$ ): (a) test 1 and (b) test 2. ( $\theta$ = 90 $^{\circ }$ , $\Delta f$ = 2.5 Hz.)

By comparing the test 1 data in figure 18(a) and test 2 data in figure 18(b), it can be seen that the fundamental tone’s frequency had a repeatable dependence on $T_{t1}$ . This will be made clearer in the following figures that show both datasets on the same plot (e.g. figure 20). The frequency of the fundamental tone increased with $T_{t1}$ , varying more than 500 Hz as $T_{t1}$ increased from 300 to 600 K. A minor difference between temperature-rising and temperature-falling measurements may be seen in the test 1 data in figure 18(a) around $T_{t1} \approx 500$ K, with temperature-rising and temperature-falling measurements differing by approximately 100 Hz. This is believed to be due to residual heat given off by the nozzle hardware during the temperature-falling measurements. Due to the relatively long duration of test 1, the core nozzle was likely heat soaked before the temperature-falling measurements began. Due to the heat given off by the heat-soaked core nozzle, some of the air flowing through the nozzle was likely at a higher temperature when it reached the mixing duct than the upstream total temperature measurements suggested, resulting in a higher-frequency fundamental tone.

If the cut-on frequency of the mixing duct’s $(2,0)$ acoustic mode truly set the frequency of the fundamental acoustic tone, then calculations of the cut-on frequency corresponding to heated core operating conditions should reveal the same trend as shown in figure 18. With the core stream heated, the core and bypass streams have different temperatures and velocities in the mixing duct and a uniform-flow assumption cannot be made to calculate cut-on frequencies as was done in the previous section.

4.2. A dual-stream vortex-sheet model

An inviscid vortex-sheet model illustrated in figure 19 approximates the mean flow in the mixing duct as two, uniform-velocity regions separated by an infinitely thin shear layer located at $r = R_1$ ( $r$ is the radial coordinate), which was selected to be the inner radius of the core nozzle ( $R_1 = 20.3$ mm). This dual-stream flow is confined within a round, rigid duct with radius $R_2$ which was set to the radius of the mixing duct ( $R_2= 26.3$ mm). The details of this model problem are not new, and have been used in the past by Ahuja et al. (Reference Ahuja, Massey, Fleming, Tam and III1992) and Viswanathan, Morris & Chen (Reference Viswanathan, Morris and Chen1994) to analyse ducted supersonic jets, for example. The dispersion relation for this model problem is given in Appendix C for completeness.

Figure 19. Dual-stream vortex-sheet model: (a) axial view and (b) cross-sectional view.

With the model problem selected, the flow properties for each stream remain to be prescribed. The Mach number inside the nozzle’s mixing duct was calculated using isentropic and quasi-one-dimensional compressible-flow equations. With $M_j = 0.90$ and the area consumed by the core-nozzle lip subtracted from the area of the mixing duct, the mixing-duct Mach number was found to be $M = 0.449$ . The total pressure and the specific heats of both streams are the same such that, even in the heated core operating conditions, a quasi-one-dimensional analysis suggests that the two streams have the same $M$ inside the mixing duct. With the stagnation temperatures and pressures prescribed, all needed flow properties were known.

For a given set of flow conditions, the cut-on frequency of the mixing duct’s $(2,0)$ acoustic mode was then obtained by searching for neutral-wave solutions of the dispersion relation (i.e. frequency and wavenumber were real-valued) using a zero-searching algorithm implemented in MATLAB. The calculated cut-on frequency of the mixing duct’s $(2,0)$ acoustic mode is shown in figure 20 as a function of $T_{t1}$ along with all measured data from figure 18. The increase in frequency with increasing $T_{t1}$ seen in the measurements is captured well by the calculations, inspiring even greater confidence in our understanding of the mechanism responsible for the howling. However, a minor discrepancy remains. The calculated $(2,0)$ mode’s cut-on frequency is consistently lower than the measured tone frequencies. This discrepancy was also seen in the comparisons of calculated cut-on frequencies and measured tone frequencies at unheated operating conditions shown earlier in figure 14. We propose that the $(2,0)$ mode’s cut-on frequencies have consistently fallen below the measured tone frequencies due to neglecting the wake of the core nozzle in the model. In what follows, this wake is accounted for.

Figure 20. Cut-on frequency of $(2,0)$ acoustic mode of mixing duct calculated using dual-stream vortex-sheet model ( $M = 0.449$ , $T_{t2}$ = 293 K) compared with measured data from figure 18 (data from test 1 and test 2 both plotted). (Measurements made at $\theta$ = 90 $^{\circ }$ with $\Delta f$ = 2.5 Hz.)

4.3. Accounting for the core nozzle’s wake

To account for the core-nozzle wake, a different vortex-sheet model was used: a ‘tri-stream vortex-sheet model’ as illustrated in figure 21. In this model, a third, annular stream was included between the core and bypass streams to model the wake of the core nozzle. The velocity in the wake region was set to zero unless otherwise noted, and the static temperature of the wake was taken to be equal to $(T_{1}+T_{2})/2$ , where $T_1$ and $T_2$ are the static temperatures of the core and bypass stream inside the mixing duct (in absolute units) calculated using isentropic-flow equations. This latter assumption is justified by noting that the wake will be heated to some degree when the core stream was heated due to the simple fact that the core-nozzle hardware itself was at an elevated temperature. The static pressure inside the mixing duct was found as a function of the mixing-duct Mach number $M$ and the stagnation pressure, and the fluid properties of the wake were then known. The wake region’s inner radius $R_1$ and outer radius $R_\alpha$ were not constrained to match the core nozzle’s inner and outer radii. Due to the presence of boundary layers on the inner and outer surfaces of the core nozzle and the possibility of instabilities in the wake (e.g. vortex shedding from the core-nozzle lip), we allowed for the wake region to be thicker than the core-nozzle lip itself. The value of $M$ used for the tri-stream vortex-sheet model was taken to be that used in the dual-stream vortex-sheet model. This is left unjustified for now, but will be revisited at the end of this section.

Figure 21. Tri-stream vortex-sheet model: (a) axial view and (b) cross-sectional view.

To convey the wake dimensions in a concise manner, additional notation was needed. The thickness of the wake is defined as $\delta \equiv R_{\alpha } - R_{1}$ and alternatively expressed as $\delta = \delta _0 + \delta _- + \delta _+$ , where $\delta _0$ is the true thickness of the core-nozzle lip and $\delta _-$ and $\delta _+$ are the thicknesses of the wake region on the inside and outside of the core-nozzle lip, respectively. A so-called ‘wake-skew ratio’ defined as $\beta \equiv \delta _-/\delta _+$ is used to indicate a radial offset of the wake’s centre from the true centre of the core-nozzle lip. For example, if $\beta \gt 1$ , the wake was centred at a smaller radius than the centre of the core-nozzle lip. If $\beta = 1$ , the centre of the wake was aligned with the centre of the core-nozzle lip. In figures reporting calculations made using the tri-stream vortex-sheet model, the extent of the wake region will be shown graphically to eliminate any possible confusion. The dispersion relation for the tri-stream vortex-sheet model was derived as part of the present work and is given in Appendix D in its general form (i.e. the velocity in the wake region was not assumed zero for the derivation). As done for the dual-stream vortex-sheet model, the cut-on frequencies were found as neutral-wave solutions of the dispersion relation (i.e. frequency and wavenumber were real valued) using a zero-searching algorithm implemented in MATLAB.

The sensitivity of tri-stream vortex-sheet model calculations of the $(2,0)$ mode’s cut-on frequency to $\delta$ with $\beta = 4$ is shown in figure 22. The diagram in figure 22(a) illustrates the radial extent of the wake for three different wake thicknesses. In figure 22(b), increased $\delta$ (with $\beta = 4$ ) is shown to increase the $(2,0)$ mode’s cut-on frequency in a uniform manner across the entire range of $T_{t1}$ plotted. The sensitivity of the calculations to $\beta$ (with $\delta / \delta _0 = 8$ ) is shown in figure 23. The diagram in figure 23(a) illustrates the extent of the wake region for four different values of $\beta$ . Figure 23(b) shows that increased $\beta$ decreases the $(2,0)$ mode’s cut-on frequency, with $\beta$ having no influence when $T_{t1} = T_{t2}$ and having an increasingly strong effect as $T_{t1}$ is increased above $T_{t2}$ . This may be understood as follows.

For a given $\delta$ , increasing $\beta$ shifts the wake radially inward. This shift of the wake results in a smaller cross-sectional area being consumed by the core stream and wake, and the cross-sectional area consumed by the bypass stream is increased. With a heated core stream, this reduces the cross-sectional area consumed by the hotter core stream and wake, and increases the cross-sectional area consumed by the cooler bypass stream. This change can be shown to decrease the $(2,0)$ mode’s cut-on frequency even with zero mean flow, and thus the sensitivity to $\beta$ is the result of a modified sound-speed distribution inside the duct.

Figure 22. Effect of wake thickness for a given wake-skew ratio ( $\beta = 4$ ): (a) illustration of wake’s radial extent relative to core nozzle and mixing-duct wall and (b) $(2,0)$ mode’s cut-on frequency as a function of $T_{t1}$ . ( $M = 0.449$ , $T_{t2}$ = 293 K.)

Figure 23. Effect of different wake-skew ratios, $\beta$ , with $\delta /\delta _0 = 8$ : (a) illustration of wake’s radial extent relative to core nozzle and mixing-duct wall and (b) $(2,0)$ mode’s cut-on frequency as a function of $T_{t1}$ . ( $M = 0.449$ , $T_{t2}$ = 293 K.)

By taking $\delta /\delta _0 = 8$ and $\beta = 4$ , the calculated $(2,0)$ mode’s cut-on frequency was in excellent agreement with the measured tone’s frequency, as shown in figure 24. It is noted that the calculated cut-on frequency still fell below the measured frequency at lower $T_{t1}$ , but making the already-thick wake any thicker to improve the agreement seemed physically unjustifiable. We also avoided further complicating the model (e.g. by making $\delta$ and $\beta$ dependent upon $T_{t1}$ ) to improve this discrepancy.

Figure 24. Cut-on frequency of $(2,0)$ acoustic mode of mixing duct calculated using both vortex-sheet models ( $M = 0.449, T_{t2} = 293$ K) compared with data from figure 18 (data from test 1 and test 2 both plotted). (Acoustic measurements made at $\theta$ = 90 $^{\circ }$ and have $\Delta f$ = 2.5 Hz.)

The curious reader may wonder how the pressure profiles of the $(2,0)$ acoustic modes (i.e. the pressure eigenfunctions) compare with those of classical acoustic duct modes in the presence of uniform flow. Only minor changes are observed at the cut-on frequencies, as shown via example pressure eigenfunctions shown in Appendix E.

4.4. Comparing the modelled and computed wake

Admittedly, $\delta$ and $\beta$ were tuned until the calculations matched well with the experimental data, rendering the calculations an empirical fit of sorts. The thorough reader may be critical of this, and may wonder how the geometry of the modelled wake and true wake compare. We provide a comparison in figure 25 between the mean flow obtained via HRLES and the radial extent of the modelled wake for the inviscid vortex-sheet model. Figure 25(a) shows the local-Mach-number distribution throughout the nozzle at the $M_j = 0.90$ , unheated operating condition. The white space corresponds to the solid cross-sectional area of the nozzle, and the colour indicates the computed local Mach number. The darker colour appearing downstream of the core-nozzle lip (see $r/D_1\approx 0.5$ , $x/D_1\approx -0.7$ ) indicates the location of the most severe velocity deficit due to the wake of the core nozzle. Notably, the centre of the wake is directed slightly radially inward inside the nozzle’s mixing duct. The two solid black lines indicate the radial extent of the wake assumed in the inviscid vortex-sheet model with $\delta /\delta _0=8$ and $\beta = 4$ . A more quantitative comparison is offered in figure 25(b), where radial profiles of the mean local Mach number are compared with the inviscid profile that agreed best with the measured data. The wake assumed for the inviscid flow model is unsurprisingly thicker than the wake visible in the HRLES mean Mach-number profile; however, the computed wake is nearly centred in the wake assumed for the inviscid profile. We believe that, although thicker than the computed wake visible in the mean Mach-number profiles, the assumed wake for the inviscid flow model is reasonable.

Figure 25. Local Mach number inside nozzle obtained from HRLES and extent of wake assumed for inviscid flow model with $\delta /\delta _0 = 8$ and $\beta = 4$ : (a) local Mach-number distribution and (b) radial profiles at three axial positions. Here, $L_e/D_e = 0.7$ , $M_j = 0.90$ , unheated.

No assertion is made that the modelled wake should have the exact same geometry (thickness, say) as the true wake. After all, the simplicity of the vortex-sheet model used (e.g. the mean-velocity profile is assumed to be piecewise uniform) means that the wake used in the model should be thought of as an effective wake at best. For example, the boundary layer along the mixing duct’s wall, following Mason (Reference Mason1969) and Swinbanks (Reference Swinbanks1975), could slightly increase the cut-on frequencies of the duct. This offers one additional, possible reason why the calculated cut-on frequencies tended to be lower than the measured frequencies. Accounting for two realistic flow features (the core-nozzle wake and the boundary layer along the mixing-duct wall) would require yet more sophisticated modelling and is not pursued here. Further, the minor non-uniformity seen in the computed mean flow outside the wake is not accounted for. Cut-on frequency estimates that use the computed mean flow directly are also possible, but require more-complicated techniques not pursued here. Short of further research, the inviscid flow model used to capture the wake provides a reasonable explanation for why the calculated cut-on frequencies tended to be lower than the measured frequencies.

4.5. Mixing-length dependence of the howling frequency

As a final comparison of vortex-sheet model calculations and measurements, we briefly consider the measured dependence of the fundamental tone’s frequency on the mixing-duct length when the core stream was unheated (shown in figure 14 a). These data showed that increasing the mixing-duct length from $L_e/D_e$ = 0.7 to 3.0 reduced the fundamental tone’s frequency from between 6200 and 6300 Hz to approximately 5900 Hz. A similar mixing-duct-length dependence was also seen in the measurements published by Ramsey et al. (Reference Ramsey, Mayo, Karon, Funk and Ahuja2022b ), where increasing $L_e/D_e$ from 0.7 to 3.0 reduced the fundamental tone’s frequency from approximately 6100 to approximately 5700 Hz. Overall, experiments have suggested that increasing the mixing-duct length may reduce the fundamental tone’s frequency by between 300 and 400 Hz.

One might consider the boundary layer along the mixing duct’s wall as a possible explanation for the reduction in the howling frequency as the mixing duct was made longer. As the mixing duct’s length is increased, the maximum thickness of the boundary layer along the mixing duct’s wall will be greater. Thus, again following Mason (Reference Mason1969) and Swinbanks (Reference Swinbanks1975), a thickening of the boundary layer associated with increasing the mixing duct’s length would tend to increase the cut-on frequencies of the duct’s higher-order acoustic modes. This is opposite the measured trend.

Considering the wake of the core-nozzle lip offers a suitable explanation for this measured trend. As the mixing-duct length of the nozzle is increased, it is expected that the core nozzle’s wake is weaker (the wake has a reduced velocity deficit, say) when it reaches the downstream end of the mixing duct near where the periodic SWBLI occurred. A wake-strength parameter is used in what follows, and is defined as $\sigma \equiv 1 - U_w/U$ , where $U_w$ is the mean velocity in the wake and $U$ is the mean velocity of both the core and bypass streams used in the tri-stream vortex-sheet model (these are equal because only cases with unheated flow are considered in what follows). As a reminder, the wake velocity was set to zero in all preceding analyses. Examples of three mean-velocity profiles with three different $\sigma$ ( $\delta /\delta _0 = 8$ , $\beta = 4$ ) are shown in figure 26(a) for clarity. Figure 26(b) shows that, as the wake strength was decreased from 1 to 0, the cut-on frequency of the $(2,0)$ mode of the mixing duct decreased by about 350 Hz – commensurate with the frequency reduction observed experimentally when $L_e/D_e$ was increased from 0.7 to 3.0. Therefore, it seems that the dependence of the fundamental tone’s frequency on the mixing-duct length could have been due to the wake of the core nozzle becoming weaker (at the downstream end of the mixing duct, where the SWBLI occurred) as the mixing-duct length was increased.

Figure 26. Effect of wake strength on cut-on frequency of $(2,0)$ acoustic mode of mixing duct with core stream unheated: (a) example mean-velocity profiles for three different wake strengths and (b) calculated cut-on frequency as a function of wake strength. ( $M = 0.449, T_{t1} = T_{t2} = 293$ K, $\beta = 4$ , and $\delta /\delta _0 = 8$ .)

4.6. Sensitivity of vortex-sheet model results to Mach number

As noted earlier, the mixing-duct Mach number $M$ used in the tri-stream vortex-sheet model calculations was simply taken to be equal to that used in the dual-stream vortex-sheet model calculations. This assumption reduced undue complexity in our analysis, although $M$ might be expected to vary depending on the characteristics of the wake prescribed inside the mixing duct. For example, introducing a zero-velocity wake downstream of the core-nozzle lip reduces the effective flow area inside the mixing duct. Depending on the extent of the wake at the final-nozzle exit, the presence of this wake may change $M$ . In what follows, we avoid these complications and simply show that the tri-stream vortex-sheet model calculations are insensitive to changes ( $\pm$ 5 %) in $M$ .

Figure 27(a) shows that the calculations made using the dual-stream model change in a minor but appreciable way with $M$ – but, in this case, the procedure used to calculate $M$ was sound. More importantly, the tri-stream model calculations (using $\delta /\delta _0 = 8$ , $\beta = 4$ ) are extremely insensitive to the Mach number used as shown on the right in figure 27(b). This indicates that, while $M$ could be made to depend on $\beta$ , $\delta$ and $L_e$ , complicating the analysis in this way is not likely to yield much improvement. Even if the value of $M$ used for the analysis was not exactly correct, inserting a zero-velocity annulus between the core and bypass streams as a model for the wake of the core nozzle increased the calculated cut-on frequencies – providing reasonable explanations for why the cut-on frequencies of the dual-stream model fell below the measured ones and why increasing the nozzle’s mixing-duct length decreased the fundamental tone’s frequency.

Figure 27. Sensitivity of cut-on frequency calculations to mixing-duct Mach number $M$ : (a) dual-stream model (b) tri-stream model with $\beta = 4$ and $\delta /\delta _0 = 8$ . ( $T_{t2}$ = 293 K.)