3.1. Ruling out structural vibration effect

At the start of this investigation structural vibrations were encountered affecting the noise spectra. Such effects were suspected when the frequency of a spectral peak would not change upon variation of $\mathit{Ma}_{j}$ , and confirmed when the amplitude of the spectral peak changed upon addition of stiffeners to the plate assembly. It was imperative to suppress these effects, first in order to confirm that the resonant tones were not due to structural vibration and then to enable a study of their fluid dynamic characteristics. The approach taken to resolve this issue was to progressively add stiffeners to the plate and its mounting structure until the noise spectra did not change any more.

Figure 2. SPL spectra for plate $L20$ ( $L=20.32~\text{cm}$ ) at four jet Mach numbers ( $\mathit{Ma}_{j}$ ): (a) for different rigidity of plate mounting structure, (b) for plates of different material. Pairs of traces are staggered successively by 15 dB, ordinate pertains to the pair at the bottom.

At first, the $L30$ plate ( $L=30.48~\text{cm}$ , see § 2) was mounted on a pair of posts placed symmetrically on its major axis. This pair, marked ‘3’ in figure 1(a), can be seen underneath the plate between two other pairs, marked 4 and 5. (In each pair the nearer post is marked and the two in a given pair are placed symmetrically about the $x{-}z$ plane.) Pairs 4 and 5 were added later as described in the following. The plate was firmly bolted to the 3.81 cm diameter posts of pair 3 and thought to be rigid enough for the experiment. However, a peak around 500 Hz appeared consistently in the noise spectra. This appeared to be the same frequency as that at which the plate would ‘sing’ upon an impact. It was estimated to be the first flexural mode about the major axis. There were also other peaks in the spectra suspected to be related to structural vibration.

Figure 3. SPL spectra for varying $z$ -location (in cm) of plate $L20$ ; $x_{TE}=21.6$ , $\mathit{Ma}_{j}=0.96$ .

At this point the pair of posts marked 4 was added. The noise spectra were checked as the structural changes were made in steps. Supports marked 8 and 9 were then firmly anchored to the vibration-isolated floor of the test chamber. This configuration, referred to as ‘mount 1’, took out most of the suspicious spectral peaks. These tests were done with the $L30$ as well as the $L20$ plate. Figure 2(a) shows pairs of SPL spectra taken at four jet Mach numbers for the $L20$ case. Comparison is made between ‘mount 1’ and the final configuration (‘mount 4’) that included another pair of posts (marked ‘5’ in figure 1 a) together with additional stiffeners on the under sides of the plate as well as both breadboards 6 and 7. ‘Mount 1’ still involved some structural modes as evident from the spectral peak around 2150 Hz, especially visible at higher values of $\mathit{Ma}_{j}$ . This went away when the stiffeners were added. These additions were made to the support mechanism in steps (from ‘mount 2’, to ‘3’, to ‘4’) until the spectra did not change any more. As a final proof another set of data was taken by replacing the $L20$ aluminium plate with a plate of the same dimensions but made of a different material (Bakelite). Figure 2(b) shows spectral pairs for the two plates at four values of $\mathit{Ma}_{j}$ . If any of the spectral peaks were due to a certain structural mode the frequency would decrease substantially with the Bakelite plate (by a factor of 2 from rough estimates). The four pairs of spectra are virtually identical, providing confidence that the observed peaks must be fluid dynamic in origin. (While the data compared in figure 2(a,b) were obtained in back-to-back experiments, after major changes in the setup and reinstallation, the relative amplitudes of the spectral peaks were noted to vary somewhat. This was likely to be due to slight differences in the exact configuration.)

3.2. Parametric dependence

Parametric dependence of the resonance was explored. Even though detailed data were taken with various plates, only sample results highlighting the key observations are presented in the following. Figure 3 shows SPL spectra for varying $z$ -location of plate $L20$ in a ‘waterfall’ pattern. The traces are staggered by one ordinate division with the scale pertaining to the trace at the bottom. The axial location of the plate’s trailing edge and leading edge were held at $x_{TE}=21.6$ and $x_{LE}=1.27$ , respectively. The $z$ -location (in cm) for each spectral trace is indicated in the figure. Each spectral trace is characterized by multiple peaks. A peak at about 1200 Hz becomes dominant around $z=-3.81$ . This is accompanied by a clearly audible tone (audible to bare ears from the control room of the test chamber). With varying $z$ -location, the frequency of the tone varies slightly; however, the amplitude changes significantly. The amplitude variations with $z$ for four different plates, all with $x_{TE}=21.6$ , are shown in figure 4. Overall sound pressure level (OASPL) data are shown. OASPL is dominated by the tone when it is strong and, therefore, was enough to provide an idea about the intensity of the resonance. For plates $L30$ and $L20$ , it is apparent that the amplitude is large within a range of $z$ . The tone occurs within that range and disappears when the plate is either too close to or too far from the jet. (Note that with the given $x_{TE}$ the leading edge of plate $L30$ touched the under surface of the nozzle when $z=-3.57$ , and thus it could not be moved any closer to the jet.) For the $L10$ case, there is a similar trend; however, for both $L10$ and $L5$ cases other noise components come into play when they are placed too close to the jet, possibly due to the impact of the flow also on the LE.

Figure 4. Overall sound pressure level for varying $z$ -location of different plates; $x_{TE}=21.6$ , $\mathit{Ma}_{j}=0.96$ .

Sample schlieren flow visualization images are shown in figure 5 for plate $L20$ . The data pertain to $x_{TE}=21.6$ for four $z$ -locations of the plate with $\mathit{Ma}_{j}=0.96$ . As stated before, the pictures represent ‘instantaneous’ snapshots with $1~{\rm\mu}\text{s}$ exposure time. For $z=-1.27$ and $-2.54$ (plate close to the jet in (a) and (b)) the resonant tone is either absent or weak (see amplitude data in figure 4) and the jet basically hugs the plate surface. At $z=-3.43$ , however, there is a clear tone at 1190 Hz and this is accompanied by an undulating flapping motion of the jet column (figure 5 c). A rolled-up vortical structure is apparent just past the trailing edge. For $z=-5.59$ , on the other hand, the tone has ceased and the path of the jet has become relatively straight (figure 5 d).

Figure 5. Schlieren images of flow field for varying $z$ -location of plate $L20$ ; $x_{TE}=21.6$ , $\mathit{Ma}_{j}=0.96$ . (a)  $z=-1.27$ , (b)  $z=-2.54$ , (c)  $z=-3.43$ and (d)  $z=-5.59$ .

Another set of schlieren data is shown in figure 6 illustrating the effect of jet Mach number. Data for four values of $\mathit{Ma}_{j}$ are shown for the $L30$ plate with $x_{TE}=21.6$ , $z=-3.81$ . At the lowest $\mathit{Ma}_{j}$ of 0.76 (figure 6 a) there is no audible tone and the path of the jet is straight. When the tone occurs, at $\mathit{Ma}_{j}=0.86$ in figure 6(b) or $\mathit{Ma}_{j}=0.99$ in figure 6(c), it is evident that the jet goes through a flapping motion similar to that seen in figure 5(c). (SPL spectra with variation of $\mathit{Ma}_{j}$ are shown later.) At $\mathit{Ma}_{j}=1.06$ (figure 6 d), on the other hand, the tone has ceased and the path of the jet has become straight. A shock structure in the jet plume has also appeared at the latter $\mathit{Ma}_{j}$ .

Figure 6. Schlieren images of flow field for plate $L30$ at four $\mathit{Ma}_{j}$ ; $x_{TE}=21.6$ , $z=-3.81$ . (a)  $\mathit{Ma}_{j}=0.76$ , (b)  $\mathit{Ma}_{j}=0.86$ , (c)  $\mathit{Ma}_{j}=0.99$ and (d)  $\mathit{Ma}_{j}=1.06$ .

Before proceeding with data on detailed parametric variation, the ‘directivity’ of the noise may be examined from the spectra data shown in figure 7. SPL spectra for all four measurement locations exhibit similar peaks at identical frequencies. The resonance is likely to involve a feedback mechanism, as discussed later, and involves sharp peak(s). These are heard everywhere and the frequencies can be measured from any of the spectra. It is also noted that the tone amplitude is larger in the upstream direction (similar to the screech phenomenon with imperfectly expanded jets). Data for several cases indicate that the tone is heard the loudest around ${\it\theta}=60^{\circ }$ . In this paper, we concern ourselves with the mechanism of frequency selection for the tone(s) and hence for brevity all subsequent data are presented only for the ${\it\theta}=60^{\circ }$ location.

Figure 7. SPL spectra for plate $L30$ with $x_{TE}=21.6$ and $z=-3.89$ ; $\mathit{Ma}_{j}=0.96$ . Data are for all four polar locations, as indicated; successive curves are staggered by 10 dB for easy identification.

Figure 8. SPL spectra for plate $L30$ with varying $x_{TE}$ ; $\mathit{Ma}_{j}=0.96$ . Plate $z$ -location varied approximately as $z=1.37+0.12x_{TE}$ . Ordinate pertains to the trace at the bottom, others staggered successively by 10 dB. Successive traces are for 1.27 cm increment in  $x_{TE}$ .

Detailed spectral data for parametric variation were obtained. Data for the $L30$ plate, with increments of 1.27 cm in trailing edge location $x_{TE}$ , are shown in figure 8. The values of $x_{TE}$ are indicated for alternate traces; the $z$ -location is changed as indicated in the figure caption in order to produce the resonance with approximately the maximum amplitude. (The equation for $z$ quoted in the figure caption was fitted after the experiment. Since each change in $z$ had to be done manually by shutting down the flow, the changes were done in coarse steps. The values are within $\pm 0.25~\text{cm}$ of the prediction from the equation.) With increasing distance a sharp spike around 1400 Hz ensues at $x_{TE}=21.6$ . The frequency of the tone decreases monotonically with further increase in $x_{TE}$ . Note that in each spectrum, besides the dominant peak there are other conspicuous peaks. The characteristics of these spectral peaks and the variation of their frequencies with $x_{TE}$ are analyzed and discussed in the following.

From figures 5–8, one might get the impression that the phenomenon under consideration might be akin to the classical edgetone phenomenon (Brown Reference Brown1937; Powell Reference Powell1961). Here, the resonance might be occurring due to sound pulses produced during the passage of the vortices over the plate’s trailing edge. The sound pulses travel back to the nozzle lip to produce nascent vortices completing the feedback loop. This notion, however, conflicts with the following data. Figure 9 shows spectral data where the trailing edge of the plate is held fixed ( $x_{TE}=21.6$ , $z=-3.94$ ) while the length $L$ is varied in increments of 1.27 cm. If the resonance were due simply to a feedback between the plate’s TE and the nozzle exit, one might expect the resonant frequency to remain unchanged. An inspection of figure 9 should convince one that this is not the case. Even though some of the spectral peaks appear to remain unchanged for smaller values of $L$ there are significant changes at larger $L$ ; a vertical dashed line is drawn at 1.64 kHz to make this clear. (The experiment was started with plates of fixed lengths. For $L=15.2$ , 20.3 and 30.5 cm the frequencies of the main peak around the dashed line were 1.64, 1.58 and 1.36 kHz, respectively. This was the anomaly that confronted us at the beginning of the investigation. Only after varying $L$ systematically do the underlying trends became clear.) The frequency variations are further discussed shortly.

Figure 9. SPL spectra for fixed TE location of plate ( $x_{TE}=21.6$ and $z=-3.94$ ) but varying $L$ (hence varying LE location, $x_{LE}$ );  $\mathit{Ma}_{j}=0.96$ . Traces are staggered and successive traces are for 1.27 cm increment in  $L$ .

It is apparent that the leading edge location also comes into play in determining the various spectral peaks. This is further examined in figure 10 where the leading edge is held fixed ( $x_{LE}=0$ , $z=-4.57$ ) while $L$ is varied, again in increments of 1.27 cm. Large changes in the spectral shape also take place with variation of  $L$ . Thus, LE location alone also does not determine the frequency.

Figure 10. SPL spectra for fixed LE location of plate ( $x_{LE}=0$ , $z=-4.57$ ) but varying $L$ (hence varying $x_{TE}$ );  $\mathit{Ma}_{j}=0.96$ . Traces are staggered and successive traces are for 1.27 cm increment in $L$ .

Figure 11. SPL spectra for varying $\mathit{Ma}_{j}$ with plate $L30$ ; $x_{TE}=21.6$ , $z=-3.81$ . Traces are staggered in the same manner as in previous figures.

While the data shown in figures 7–10 pertain to a fixed jet Mach number $\mathit{Ma}_{j}=0.96$ , the effect of $\mathit{Ma}_{j}$ for the $L30$ plate with fixed position is shown in figure 11. One finds that a sharp spike accompanied by an audible tone ensues with increasing Mach number from around $\mathit{Ma}_{j}=0.77$ , for the given configuration. The resonance persists throughout the transonic regime but abruptly stops around $\mathit{Ma}_{j}=1.06$ ; see schlieren images in figure 6.

The spectral data are now analyzed as follows. Each spectrum is scanned for the highest three peaks. Even though in some cases the smaller peaks may be at the noise level, a look at the characteristics of these peaks reveals some orderly trends. First, for a clarification of the method, the amplitudes of the peaks for the case corresponding to figure 9, are shown in figure 12. The circular, diamond and triangular symbols represent the spectral peaks in order of diminishing amplitudes. OASPL values are also shown in the figure by the square (solid) symbols. The undulations in the amplitudes of the largest peak (circular symbols) are apparently due to a ‘staging’ behaviour, as will become clear with the frequency data shown next.

Figure 12. Amplitudes of the three highest peaks in the spectra of figure 9; varying $L$ with fixed $x_{TE}=21.6$ , $z=-3.94$ ; $\mathit{Ma}_{j}=0.96$ .

Figure 13. Frequencies of the three highest peaks in the spectra of figure 9. Varying $L$ with fixed $x_{TE}=21.6$ , $z=-3.94$ ; $\mathit{Ma}_{j}=0.96$ . f1, f2 and f3 denote three peaks in order of deceasing amplitude. Lines for equation (3.1) (see text).

Variations in the frequencies of the three highest spectral peaks, corresponding to the case of figures 9 and 12 (fixed TE, varying $L$ ), are shown in figure 13. At small values of $L$ , the highest peak occurs around 800 Hz. The frequency decreases with increasing $L$ until at around $L=17.8$ the highest peak shifts to a higher frequency band. With further increase in $L$ , it shifts to a yet higher frequency, followed by some oscillations between the upper two bands. Overall, there appear to be three frequency bands starting at approximately 800, 1200 and 1600 Hz, for the case under consideration. Ignoring some data scatter, it is clear that there is an underlying pattern and all frequency data fall in these distinct bands. Peaks representing each band are always there in the spectra and one of them becomes dominant depending on geometric configuration. It is noted that generally the frequencies of the three peaks are not multiples or submultiples of one another. On occasions, they seem to be multiples but this could be a coincidence. Also shown in figure 13 are curves denoted ‘Equation (3.1)’ that will be described shortly.

Figure 14. Frequencies of the three highest peaks in the spectra of figure 10; fixed LE of plate ( $x_{LE}=0$ and $z=-4.57$ ) but varying $L$ (hence varying $x_{TE}$ ).

Similar frequency variations for the three highest spectral peaks corresponding to the data of figure 10 (fixed LE, varying $L$ ) are shown in figure 14. A similar observation can be made as for figure 13. Again, the frequency data are not random and fall in one of the multiple bands. The same appears true for jet Mach number variation with a fixed plate, as seen from the frequency data in figure 15 corresponding to the spectral traces of figure 11. Referring back to figure 8 (varying $x_{TE}$ with fixed $L$ ), a similar analysis yields the frequency data shown in figure 16. The frequencies of the highest peaks again cluster in certain bands. Finally, similar frequency data for the effect of $z$ -variation with the $L20$ plate are shown in figure 17. These data correspond to the spectral data of figure 3. A similar observation can be made as for figures 13–16.

Figure 15. Frequencies of the three highest peaks in the spectra of figure 11; varying $\mathit{Ma}_{j}$ for plate $L30$ , $x_{TE}=21.6$ , $z=-3.81$ .

Figure 16. Frequencies of the three highest peaks in the spectra of figure 8; varying $x_{TE}$ for plate $L30$ .

Figure 17. Frequencies of the three highest peaks in the spectra for varying $z$ with plate $L20$ ; $x_{TE}=21.6$ , $\mathit{Ma}_{j}=0.96$ .

3.3. Possible mechanism

Possible aerodynamics and acoustic feedback involved in the phenomenon are considered with the sketches in figure 18. The fact that both the LE and TE of the plate influence the frequency of the spectral peaks led us to first consider the following feedback path. In figure 18(a), the likely distribution of the vortices is sketched. Inspection of the schlieren images (figures 5 c, 6 b,c) suggests that there may be multiple vortices over the length of the plate at a given instant. It appears that in most cases there are two vortices in the bottom shear layer over the length of the plate at a given instant. Let us assume that there are $n$ such vortices. Each vortex produces a sound pressure pulse upon its passage past the TE. Let us also assume that the feedback takes place via a path underneath the plate. Then, using the notation in figure 18(a), the total period of the feedback process can be written as the sum of vortex passage time over the distance $x_{TE}/n$ and the acoustic feedback time over the distance $L+s$ . Assuming a vortex convection velocity equal to $0.5c\mathit{Ma}_{j}$ , where $c$ is the speed of sound in the ambient, the following equation can be written for the frequency:

Note that the thickness of the plate has been neglected in determining the paths and also that $L=x_{TE}-x_{LE}$ . The predictions from (3.1) have been shown in figures 13–17. In figure 13, predictions for $n=1$ , 2 and infinity are shown. To avoid clutter, the prediction only for $n=2$ is shown in the subsequent figures. It is apparent that the prediction ( $n=2$ ) approximates the band of frequency data at the bottom of each figure. Before considering the sketch in figure 18(b) let us note the following. The fact that the path around the leading edge might be playing a role is further demonstrated by the following exercise. When a gap between the plate’s LE and the underside of the nozzle (about 0.33 cm) is closed by sound absorbing foam material (figure 1 d), the resonance is eliminated or weakened significantly. This is evident from spectral data shown in figure 19(a). With addition of the foam material the amplitudes of the spectral peaks get significantly reduced, from the dashed curve for the ‘no foam’ case to the solid curve for the ‘foam’ case.

Figure 18. Schematics of flow and acoustic diffraction. (a) Likely distribution of vortices in the flow, (b) possible acoustic waves and diffraction from the LE.

Figure 19. Effect of LE modification; plate $L30$ , $x_{TE}=21.6$ , $\mathit{Ma}_{j}=0.96$ . (a) Effect of placing sound absorbing material near the LE (figure 1 d), $z=-3.89$ ; (b) effect of closing the gap between the LE and the underside of the nozzle, $z=-3.56$ .

It is clear that the feedback hypothesis considered above reasonably well explains the frequencies in the lowest band in all of figures 13–17. However, there were observations that conflicted with the assumed feedback via a path underneath the plate. It was found that if the gap between the underside of the nozzle and the plate’s LE was closed by touching the two surfaces (without the sound absorbing material but leaving a hard junction) the spectral shape remained practically unchanged. This is shown in figure 19(b). That is, a small gap or no gap made no difference in the spectra as long as the junction involved hard surfaces. Insertion of a metal strip in the gap (instead of the foam material) also made no difference in the spectra. However, as seen in figure 19(a), the peaks diminished significantly with the foam material. These observations suggested that the feedback may not be via a path under the plate and led to the following thought.

It is possible that diffraction from the LE creates a ‘new’ source of sound. This is schematically shown in figure 18(b). The sound originating from the TE, due to the passage of the vortices, may travel along a path over the surface (dashed wave fronts). The wave fronts are shown as if there were no flow to cause any distortion. They are also shown relatively packed for clarity (in reality they may be spaced wider, i.e. of longer wavelengths). The waves are diffracted at the LE whence new waves (solid fronts) are radiated in all directions. The new source due to diffraction at the LE is nearer to the nozzle lip and thus may have a stronger influence on the feedback in the following sense. The waves from the TE will also reach the nozzle lip but they have to travel through regions of flow and thus get distorted before impacting on the lip. The LE, on the other hand, is near the lip and in a region where there is no direct flow. Thus, the waves from the LE remain spatially coherent and undistorted when impacting on the nozzle lip. This might explain a stronger influence of the LE in the feedback process even though the original source of the sound waves is at the TE. The part of the dashed waves reaching the LE to get diffracted travels through regions where there is no significant flow (hugging the plate surface and partly through the boundary layer over the plate) and thus are not significantly distorted. With this scenario, it should be clear that the lengths of the feedback paths (TE to LE travelled by the dashed waves and LE to nozzle lip travelled by the solid waves, and passage over the distance $x_{TE}$ by the vortices) remain the same as before, yielding the same equation (3.1) for the frequency. It is plausible that a sharp but hard junction near the leading edge, when the gap is closed, would also produce a similar sound source via diffraction, to that sketched in figure 18(b). It is not clear how much change one would expect in the latter source strength while going from a small gap to no gap. However, the radiation to the aft (towards the nozzle lip) must not have changed significantly since the two spectra in figure 19(b) are virtually identical. In summary it appears that the diffraction scenario provides a clearer rationale for (3.1).

Figure 20. Focused schlieren images near the TE of plate $L30$ ; $\mathit{Ma}_{j}=0.96$ . Plate location ( $x_{TE}=21.6$ , $z=-3.81$ ) yielded resonant tone at 1375 Hz. Unsteady flow pattern at three phases within a tone period: (a)  ${\it\varphi}=0^{\circ }$ , (b)  ${\it\varphi}=120^{\circ }$ and (c)  ${\it\varphi}=240^{\circ }$ (arbitrary reference for phase ${\it\varphi}$ ).

A few earlier works on trailing-edge noise are worth noting here. Yu & Tam (Reference Yu and Tam1978) and Olsen & Boldman (Reference Olsen and Boldman1979) reported experimental results on the interaction of a rectangular (slot) jet with a flat plate. Both, however, involved wall jets (i.e. plate touching the lower lip of the nozzle), a configuration which did not produce a resonance in the present experiment. Yu & Tam presented shadowgraph data that exhibited large-scale turbulent motion similar to that seen in figure 5(a,b). Their data did not seem to exhibit the violent flapping motion seen in figures 5(c), 6(c) or figure 20 below. Their (averaged) spectral data were also smooth and did not display any peak. In an earlier paper, Tam & Yu (Reference Tam and Yu1975) also presented an analysis addressing sound generation by the interaction of the large-scale turbulent structures with the plate. Based on the analysis they noted, ‘… the noise directivity is strongly influenced by diffraction of sound at the leading edge of the plate’. For further comments on their analysis, as well as others, the reader may refer to Crighton (Reference Crighton and Hubbard1991), but none of the analyses or experiments addressed a resonant condition and the associated frequencies. Olsen & Boldman (Reference Olsen and Boldman1979), on the other hand, presented narrow-band spectral data; some did exhibit sharp spikes that must have been associated with discrete tones. Unfortunately, the physical configuration of the plate relative to the flow was not completely clear. The plate used was large (several metres long by 2.43 m wide) but only 0.32 cm thick. We have seen how structural resonance can come into play and it remains unclear if the tones reported were due to structural modes or fluid dynamic in origin. A few other past works involving jet–airframe interaction noise are also pertinent. Miller (Reference Miller1983) presented experimental results and considered diffraction from the TE as well as the LE of the wing to explain spectral content and directivity of radiated noise. The data presented by Mengle (Reference Mengle2011), in connection with the effect of engine stand-off distance from the wing (‘gulley height’), are worth noting. The noise spectra involved multiple peaks that were smeared due to the one-third-octave format of presentation but appeared to have similarity with some of the spectral shapes presented in this paper. Note that both of the latter references pertain to ‘under-the-wing’ engine configurations.

Referring back to the discussions of figures 13–17, it should be clear that (3.1) (with $n=2$ ) predicts the ‘fundamental’ or the lowest band of frequencies for every parametric variation considered. However, not everything is explained and these weaknesses are discussed in the next paragraph. Here, it should be mentioned that various other possibilities were considered as possible mechanisms for the phenomenon. For example, vortex shedding from the entrained flow over the plate’s leading edge was suspected as a source of unsteadiness. However, the exercise described with figure 19(b) ruled this out. Recall that by touching the LE with the under-surface of the nozzle the possibility of such vortex shedding was precluded, yet the resonant tone remained unaffected. Another possibility was considered based on interference of sound waves from the TE and LE (figure 18 b). This could produce interference patterns and the feedback could be most effective when a high-amplitude ‘crest’ of the interference coincided with the nozzle lip. However, such an interference pattern may be complex and distorted since, as indicated before, the sound waves travel through regions of varying flow speed. Thus, the latter possibility also seemed unlikely.

The upper frequency bands in figures 13–17 may be considered as ‘stages’ and clearly strong resonance can also occur at those frequencies depending on parameters. In many resonance phenomena, e.g. edgetone or tones from impinging jets (Rockwell & Naudascher Reference Rockwell and Naudascher1979), such a staging behaviour is observed. The various stages are harmonically related in the sense that, depending on the stage, a different but integral number of vortices occurs between the nozzle and the wedge (or ground in case of an impinging jet; the frequencies from one stage to another are often not exact multiples of each other). In the present case, if one assumes more than two vortices in the bottom shear layer ( $n>2$ ) the predicted frequency will be higher. This, however, is not enough to explain the frequencies in the upper stages. Predictions for $n=1$ and infinity are also shown in figure 13. First, it is clear that the curve for $n=1$ does not agree with any of the data. It is not clear why the ‘fundamental tone’ of the phenomenon is predicted with $n=2$ ; one may speculate that the nozzle aspect ratio has an influence in determining this factor. Second, one notes that in the limiting case of an infinite number of vortices the first term in parentheses on the right-hand side of (3.1) becomes zero. This still predicts frequencies lower than the uppermost band in all cases, as illustrated by the curve for $n=\infty$ (‘inf’) in figure 13. In essence, the mechanism of the tones in the upper stages of figures 13–17 is not fully understood. It is not pursued further at the present time and in the following we examine the unsteady flow characteristics for a specific case of the resonance.

3.4. Unsteady flow field under the resonance

In this subsection data for plate $L30$ with $x_{TE}=21.6$ and $z=-3.81$ at $\mathit{Ma}_{j}=0.96$ , yielding a tone at 1375 Hz, are discussed. Schlieren pictures of the flow field obtained at different phases within the tone period are obtained by using the microphone signal as a reference for triggering data acquisition. Such pictures at three phases are shown in figure 20 as examples. The axial motion of the unsteady flow structure with advancing phase can be seen clearly. At phase zero in figure 20(a), chosen arbitrarily, a trough in the unsteady flow structure is just leaving the trailing edge. One-third of the way into the period ( ${\it\varphi}=120^{\circ }$ in figure 20 b) a ‘trough’ occurs downstream of the TE. Two-thirds of the way ( ${\it\varphi}=240^{\circ }$ in figure 20 c), the trough is farther away while another trough upstream has approached the TE. The cycle then repeats itself.

Variations of the tone phase with streamwise distance, measured by the pressure transducer with the microphone signal as reference, are shown in figure 21(a). These data are taken at the centre span ( $y=0$ ) for four values of $z$ , as indicated in the legend. For negative $z$ -locations (closer to the plate) the data are obtained starting from the exit of the nozzle. For positive $z$ -locations the data could not be obtained all the way to the nozzle exit due to hardware constraints. It can be seen that the phase variations for either the upper or the lower pair of traverses are similar but there is a consistent difference between the two pairs. Average phase velocities of $0.56U_{j}$ and $0.52U_{j}$ are estimated from the upper and lower pairs of curves, respectively.

Figure 21. Streamwise variation of phase and amplitude of the tone fundamental; $L30$ plate at $x_{TE}=21.6$ , $z=-3.81$ producing tone at 1375 Hz. Data for four transverse ( $z$ ) locations ( $y=0$ ). (a) Phase variation, (b) amplitude (r.m.s. of pressure fluctuation) variation. Data obtained by spectral analysis of signal from pressure transducer relative to microphone signal.

Amplitude variations corresponding to the data of figure 21(a) are shown in figure 21(b). The transducer essentially sensed instantaneous total pressure. As stated in § 2, there was some uncertainty in the calibration and the absolute values of the amplitudes should be considered as qualitative. The amplitude data display the characteristics of an instability wave: initially growing, saturating to a peak and then decaying. The phase and amplitude variations in the transverse ( $z$ ) direction, measured just downstream of the TE of the plate, are shown in figure 22. The phase variation exhibits a 180° jump across the centreplane of the jet. The amplitude drops at this location, grows and then decays in either direction of  $z$ . These data are consistent with a flapping motion of the jet column and the asymmetric vorticity distribution sketched in figure 18(a).

Figure 22. Phase and amplitude variation for the resonance case of figure 21 in the transverse ( $z$ ) direction, downstream of the plate’s trailing edge ( $x=22.9$ , $y=0$ ).

The schlieren pictures showed the structure of the flow field on the $x{-}z$ cross-sectional plane. Phase-averaged flow field data were acquired to examine the corresponding unsteady characteristics on the cross-stream ( $y{-}z$ ) plane. These data were acquired by traversing the pressure transducer on the $y{-}z$ plane downstream of the plate’s trailing edge ( $x=22.9$ ), for 19 phases within the period. Data for four phases are shown in figure 23(b–e), as examples. Corresponding time-averaged data are shown in figure 23(a). (The reference ${\it\varphi}=0$ for these data and the pictures in figure 20 were assigned arbitrarily and there is no correspondence.) Estimated values of Mach number normalized by the jet Mach number $\mathit{Ma}_{j}$ are plotted. The jet cross-section is seen to go through contortions and the unsteady motion is not simply an up-and-down flapping of the jet column against the plate.

Figure 23. Cross-sectional distribution of $\mathit{Ma}/\mathit{Ma}_{j}$ for plate $L30$ at $x=22.9$ for the resonant condition of figures 20–22. Time-averaged data shown in (a); (b–e) show phase-averaged data at four phases: (b)  ${\it\varphi}=0^{\circ }$ ; (c)  ${\it\varphi}=80^{\circ }$ ; (d)  ${\it\varphi}=180^{\circ }$ (e)  ${\it\varphi}=260^{\circ }$ . Coordinates are in cm.

The three-dimensional nature of the flow field is further illustrated by time-averaged Mach number contours measured at a far downstream location. These data, shown in figure 24 for the measurement location of $x=64.5$ ( $x/D=12$ ) were acquired with standard Pitot probes. The distribution in figure 24(a) represents the same plate configuration as for figures 20–23 (plate $L30$ with $x_{TE}=21.6$ ), yielding the resonant condition. It is clear that an axis switch has taken place. The major axis of the jet was horizontal at the exit of the nozzle but by the measurement station it has become vertical. Corresponding data with the plate drawn back are shown in figure 24(b). In this case there was no resonance and the data demonstrate that there was no axis switching. The jet cross-section gradually became round in this case with increase in  $x$ .

Figure 24. Time-averaged $\mathit{Ma}/\mathit{Ma}_{j}$ contours at $x=64.5$ for a resonant and a non-resonant condition with plate $L30$ : (a)  $x_{TE}=21.6$ , $z=-3.81$ , $\mathit{Ma}_{j}=0.96$ (resonance at 1375 Hz); (b)  $x_{TE}=7.6$ , $z=-6.35$ , $\mathit{Ma}_{j}=0.96$ (plate drawn back, no resonance). Coordinates are in cm.

The data in figure 24(a) involve a resonance at $f_{p}\approx 1375~\text{Hz}$ , corresponding to a Strouhal number ( $f_{p}D/U_{j}$ ) of about 0.225, a value not too far from the ‘preferred mode’ for an equivalent axisymmetric jet. The resonance imposes a periodic forcing of the jet. Such forcing causes the rectangular jet to go through the axis switching due to the dynamics of the unsteady azimuthal vortical structures (Hussain & Husain Reference Hussain and Husain1989; Zaman Reference Zaman1996). A periodic forcing causes the sheet of azimuthal vorticity emanating from the nozzle to roll up into discrete vortex ‘rings’ (with initial shape determined by the geometry of the nozzle). The non-axisymmetric vortex rings go through self-induced contortions that lead to the axis switching. The key is the roll-up of the azimuthal vorticity into the non-axisymmetric rings. This may sometimes occur naturally, e.g. at low Reynolds numbers when the efflux boundary layer is laminar and susceptible to naturally occurring disturbances. Periodic perturbations (e.g. by artificial excitation or naturally occurring phenomena such as screech), also yield the same effect. Without such periodic perturbation the axis switching is seldom observed at higher jet Reynolds number and Mach number, as is the case in figure 24(b). Thus, the axis switching is likely to be a result rather than the cause of the resonance in the present case. All resonant tones with the jet–surface configuration under study are expected to result in such axis switching as long as the frequency corresponds to a Strouhal number not too far from the value noted above.

The unsteady flow field data presented in this subsection illustrate the effect of the resonance on the flow field. Clearly, the assumption of two-dimensionality should be viewed with caution in any analysis of the flow. Furthermore, any numerical simulation must capture the unsteady effects, for otherwise calculation of even the mean flow field (e.g. captured in figure 24 a) will not be possible.