3.1. Far-field acoustic results

Results obtained from the experimental far field acoustics tests are presented for the overexpanded conditions of NPR 2.6 and NPR 3. The Strouhal number ranges were computed using the isentropic exit velocity (U _{j, NPR 2.6} = 377.7 m s^–1, U _{j, NPR 3} = 399.9 m s^–1) and the nozzle equivalent diameter (St = fD _e /U _j ). Figure 2 summarises the acoustic intensity and directivity across two observation planes, the Symmetry plane, and the Twinjet plane as introduced in figure 1(b,c). For clarity, the results from the Symmetry plane are denoted with a solid line and those from the Twinjet plane are denoted using a dotted line. Additionally, in the OASPL plot values from the Symmetry plane are represented with hollow symbols and values from Twinjet plane are represented with solid symbols.

Figure 2. Far field acoustic results for NPR 2.6 (gold) and NPR 3 (blue). Staggered frequency spectra from the Symmetry plane (solid) and Twinjet plane (dotted) from azimuthal angles (a) $\xi =45^{\circ}$ , (b) $\xi =90^{\circ}$ . Frequency specific directivities for NPR 2.6 – (c) Symmetry plane, (d) Twinjet plane; NPR 3 – (f) Symmetry plane, (g) Twinjet plane. (e) The OASPL distributions from both planes.

Figures 2(a) and 2(b) reveal the frequency spectra obtained from two azimuthal angles. The spectra are offset by a value of 20dB for clear delineation across NPRs and observation planes. At NPR 2.6 (represented in gold), a prominent screech tone was observed at St ≈ 0.51 (10.4 kHz) with the harmonic identified at St ≈ 1.02 (20.8 kHz). At the upstream angle of $\xi =45^{\circ}$ , the SPL values from these frequencies were comparable in the Symmetry plane as shown in figure 2(a) while the primary screech tone was dominant in the Twinjet plane.

This trend was partly reproduced at $\xi =90^{\circ}$ , as seen in figure 2(b) accompanied by a significant increase in magnitudes of the primary screech tone (+4.3 dB) and the harmonic (+5.6 dB) when compared with the levels recorded at $\xi =45^{\circ}$ .

At NPR 3 (represented in blue), the primary screech tone was revealed at St ≈ 0.38 (8.15 kHz) accompanied by a harmonic at St ≈ 0.75 (16.25 kHz). The acoustic levels at $\xi =45^{\circ}$ were largely dominated by the primary screech on both observation planes as seen in figure 2(a). Conversely, in the jet normal direction at $\xi =90^{\circ}$ , the primary screech tones and first harmonic exhibited comparable SPL levels across both observation planes as seen in figure 2(b). Notably, a significant overlap between the broadband shock associated noise (BBSN) frequencies and the first harmonic frequency was observed that resulted in an intensity match with the primary tone in the Twinjet plane. This pattern of harmonic and BBSN frequency concurrence in twin-jet arrangements, previously noted in the research conducted by Jeun et al. (Reference Jeun, Karnam, Wu, Lele, Baier and Gutmark2022), Samimy et al. (Reference Samimy, Webb, Esfahani and Leahy2023) and Kuo, Cluts & Samimy (Reference Kuo, Cluts and Samimy2017), underscores a complex interplay between harmonic generation and shock–turbulence interactions that contribute to shock-associated noise production.

To deduce the directivity patterns, four distinct peak frequencies were identified across all directivity angles at each operating condition. These plots are presented as pairs visualising directivity patterns from the Symmetry plane and Twinjet plane in figures 2(c,d) and 2(f,g) for NPR 2.6 and NPR 3, respectively. The lines correspond to the large scale turbulence (LST) noise peak (gold), the primary screech tone (red), the first harmonic (black) and the BBSN peak (blue) relevant to each condition. At NPR 2.6, the LST frequency recorded at St ≈ 0.13 (2.8 kHz) exhibited peak noise levels at the far downstream angles ( $\xi =148^{\circ}$ , spectra; Appendix A) in both the Symmetry plane and Twinjet plane as seen in figures 2(c)and 2(d). This behaviour is well documented (Tam Reference Tam1995) and expected due to the growth and collapse of large scale turbulent structures that occurs towards the end of the jet potential core (Tam et al. Reference Tam, Viswanathan, Ahuja and Panda2008). The directivity pattern for the primary screech tone in the Symmetry plane resembles Norum’s (Norum Reference Norum1983) prediction for circular jets. The similarity between Norum’s data and the current result could likely be linked to the symmetric exit area of the square nozzles in the current study. In contrast, other non-circular nozzles such as rectangular jets show the emergence of screech tones and the associated harmonics from the shear layers that start from the longer nozzle walls housing the C-D section. The directivity pattern associated with screech tones consist of multiple propagation vectors as documented by Gutmark et al. (Reference Gutmark and Bicker1990), Gojon et al. (Reference Gojon, Gutmark and Mihaescu2019) and Malla et al. (Reference Malla and Gutmark2017). Additionally, findings that associate the production of these screech tones with shock turbulence interaction were deduced from the computational study by Shen & Tam (Reference Shen and Tam2002) and from the experimental studies of Panda (Reference Panda1999) and Edgington-Mitchell et al. (Reference Chen, Gojon and Mihaescu2021, 2022). The intricate directivity pattern in the Twinjet plane, coupled with that in the Symmetry plane suggests that current nozzle design demonstrates aspects of both symmetric and non-axisymmetric nozzles. The first harmonic exhibited pronounced directivity towards $\xi =90^{\circ}$ , especially noticeable in the Symmetry plane, whereas diminished noise levels at this frequency were recorded in the Twinjet plane. The BBSN peak frequency was recorded at St ≈ 1.12 (23 kHz) and showed identical directionality across both planes of observation.

At NPR 3, the LST frequency recorded at St ≈ 0.19 (4.1 kHz) and the BBSN peak frequency at St ≈ 0.92 (20 kHz) exhibited patterns consistent with those observed at NPR 2.6. The former frequency peaked towards the downstream angles ( $\xi =144^{\circ}$ , spectrum; Appendix A) and the latter demonstrated peak intensity in the jet normal direction with highest recorded values in the Twinjet plane as seen figures 2(f) and 2(g). Although the noise distribution of the primary screech tone trended similar to that observed at previous condition within the Symmetry plane, an azimuthal peak was observed at $\xi =140^{\circ}$ diverging from the value of $\xi =126^{\circ}$ for NPR 2.6. This shift suggests that the generation of large-scale coherent structures could be a prominent feature of the screech production mechanism at NPR 3. Additionally, the first harmonic exhibited a notable increase of ≈ 9.5 dB in the Twinjet plane compared with the Symmetry plane marking a reversal in the trend observed at NPR 2.6. This could be indicative of a significant change in jet behaviour at the higher pressure condition. As documented by Ahn et al. (Reference Ahn, Mihaescu, Karnam and Gutmark2023) and Karnam et al. (Reference Karnam, Ahn, Gutmark and Mihaescu2022) it is known that the jets undergo symmetric oscillations in the Twinjet plane at NPR 3. The observed variations in directivity for the screech peak and the harmonic at NPR 2.6, when compared with NPR 3, hint at a distinct mode of jet oscillation contrasting prior findings.

The OASPL presented in figure 2(e) encapsulates the cumulative impact across all relevant frequency ranges. For NPR 2.6 the noise levels remain consistent across both observation planes at the lower azimuthal angles and increase gradually towards the jet downstream. Peak noise levels in the Symmetry plane were recorded at 119.9 dB at the azimuthal angle of $\xi =140^{\circ}$ , while the noise peak in Twinjet plane was recorded at 119.7 dB at the azimuthal location of $\xi =144^{\circ}$ . At NPR 3, noise levels from the Twinjet plane clearly dominate the upstream and jet normal directions. This can be attributed to the first harmonic tone from the prior discussion on frequency directivity. However, the noise generated from the Symmetry plane is dominant in the downstream angles primarily due to the symmetric oscillation mode that the jets undergo at this condition. Peak noise levels of 126.1 dB ( $\xi =144^{\circ}$ ) and 124 dB ( $\xi =148^{\circ}$ ) were recorded in the Symmetry and Twinjet planes, respectively (additional spectra included in Appendix A).

3.2. Flow field: schlieren image analysis – primary screech frequency

High-speed schlieren images were analysed to identify spatial and temporal flow variations using spectral analyses. This section examines data acquired at the acquisition rate of 41 kHz (F ₁). Temporal decomposition of the image set was performed using the spectral proper orthogonal decomposition (SPOD) technique as detailed by Schmidt & Colonius (Reference Schmidt and Colonius2020). This was coupled with an implementation of the spatial decomposition technique outlined by Edgington-Mitchell et al. (Reference Chen, Gojon and Mihaescu2021, 2022) at the screech frequency to identify the corresponding bidirectional wave propagation.

Figure 3. Metrics derived from schlieren images (image intensity normalised with peak values), for NPR 2.6 (a–c), NPR 3 (d–f). Temporal average schlieren images (a,d). Logarithmic intensity standard deviation for Twinjet plane (b,e). Logarithmic intensity standard deviation for C-D plane (c,f).

An initial assessment of the jet structure was conducted by examining the average and statistical intensity distributions from schlieren images as depicted in figure 3. The logarithmic intensity standard deviation ( $\log _{10} \sigma _{i}$ , where $\sigma _{i}$ is the intensity standard deviation), computed using the fluctuating component of the light intensity at each spatial location, can be a useful technique to visualise the effect of periodic flow oscillations on the jet near field (Edgington-Mitchell Reference Edgington-Mitchell2019; Karnam et al. Reference Karnam, Saleem and Gutmark2023). At NPR 2.6, a clear shock train emerging from both nozzles was observed as seen in figure 3(a). The overexpanded nature of the flow led to the formation of a prominent shock system that extends up to X/h ≈ 5.2 from the nozzle exit. In contrast at NPR 3, the flow exits into a weaker oblique shock system due to the lower degree of overexpansion compared with the previous condition as seen in figure 3(d). The secondary shock system that is characteristic of sharp throat nozzles (Munday et al. Reference Munday, Gutmark, Liu and Kailasanath2011; Karnam et al. Reference Karnam, Saleem and Gutmark2023) and was previously overshadowed by the exit shock system emerges at this condition. The shock train extends from the nozzle exit to X/h ≈ 6.3. In comparison with NPR 2.6, the shock-expansion system at NPR 3 are fainter beyond X/h = 3 and lack spatial clarity due to the intense flow oscillations that alter the average location of these shocks.

The manifestation of the screech tones observed in the far field acoustics was captured in the standard deviation profiles depicted in the subsequent figures. Studies (Panda Reference Panda1999; Edgington-Mitchell Reference Edgington-Mitchell2019; Karnam et al. Reference Karnam, Saleem and Gutmark2023) have shown that screeching jets lead to the formation of standing waves in the jet near field resulting from the superposition of downstream travelling hydrodynamic waves along the jet shear layers and upstream travelling acoustic waves emitted by shock–turbulence interactions. At NPR 2.6, the standing wave pattern on each jet features distinct peaks along both shear layers in the Twinjet plane as seen in figure 3(b). The outer shear layers exhibited pronounced crests that extended radially outward with an upstream directivity. Furthermore, the jets exhibit mirror symmetry along the Symmetry plane, with notable alignment of crest locations on both shear layers hinting towards the emergence of coupled oscillations within the Twinjet plane. These features were found to extend along the upper and lower shear layers of the jets in the nozzle C-D plane as depicted in figure 3(c). This is a clear indication of coupled instability waves travelling along all shear layers of the jet and aligns with the findings noted in Karnam et al. (Reference Karnam, Saleem and Gutmark2023) for rectangular jets operating at the same NPR. A peculiar feature is the lack of standing waves along the inner shear layers at NPR 2.6 in the Twinjet plane. This can be indicative of the presence of destructive wave interference leading to diminished wave amplitudes along the inner shear layers.

The LES study of Ahn et al. (Reference Ahn, Mihaescu, Karnam and Gutmark2023) and experimental work of Karnam et al. (Reference Karnam, Ahn, Gutmark and Mihaescu2022) have previously documented the phase locked oscillations observed at NPR 3. The effect of this coupling dynamic is shown in figure 3(e). The internozzle region at the higher NPR features a prominent standing wave pattern that extends upstream of the nozzle exit. This is accompanied by a wave pattern that emerges along the outer shear layers. Conversely, no identifiable patterns emerged in the C-D plane of observation as illustrated in figure 3(f). This is a strong indication of the spatial instabilities being dominant in the Twinjet plane. Edgington-Mitchell et al. (Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022) and Li et al. (Reference Li, Wu, Liu, Zhang, Hao and He2023) demonstrated the use of temporally averaged schlieren images to compute the spatial wavenumber spectra of supersonic jet shock systems, allowing for the identification of wavenumbers linked to the most distinct shock cells ( $k=2\pi /\lambda _{S}$ where $\lambda _{S}$ is shock cell wavelength). A similar approach was implemented in the current study on the temporal average of the schlieren images and the axial velocity snapshots from the PIV data set as discussed in Appendix B. This led to the extraction of the wavenumbers corresponding to the first (k _S1 ) and second (k _S2 ) shock cells. The inverse correlation between the spatial wavelength of the shock cells and the wavenumber led to a lower k values for the same shock cells at the higher NPR with k _{S1, NPR 2.6} = 7.88, and k _{S1, NPR 3} = 6.309, respectively.

While the seminal research by Powell (Reference Powell1953a ,Reference Powell b ) indicated acoustic feedback as the primary screech closure mechanism, subsequent work by Shen & Tam (Reference Shen and Tam2002), Panda (Reference Panda1999), Mancinelli et al. (2019, 2021), Edgington-Mitchell et al. (Reference Chen, Gojon and Mihaescu2021a ,b, Reference Edgington-Mitchell, Beekman and Nogueira2022), Edgington-Mitchell et al. (Reference Edgington-Mitchell, Jaunet, Jordan, Towne, Soria and Honnery2018) and Nogueira et al. (Reference Nogueira, Jaunet, Mancinelli, Jordan and Edgington-Mitchell2022a ,Reference Nogueira, Jordan, Jaunet, Cavalieri, Towne and Edgington-Mitchell b ) have provided strong evidence that supports screech generation stems from triadic interactions between KH instability waves and the quasisteady shock structure of the jet. These interactions have been known to result in the generation of the three primary components: acoustic waves, shock leakage and the internal G-JM. These have been documented in circular (Shen & Tam Reference Shen and Tam2002; Edgington-Mitchell et al. Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022; Nogueira et al. Reference Nogueira, Jordan, Jaunet, Cavalieri, Towne and Edgington-Mitchell2022), rectangular (Gojon et al. Reference Gojon, Gutmark and Mihaescu2019; Karnam et al. Reference Karnam, Saleem and Gutmark2023; Li et al. Reference Li, Wu, Liu, Zhang, Hao and He2023) and elliptical (Edgington-Mitchell et al. Reference Edgington-Mitchell, Beekman and Nogueira2022) nozzles. The primary generation mechanism involves the downstream propagating KH instabilities interacting with the shock cell tips impinging on the jet shear layers leading to the generation of acoustic waves. In highly unstable jet states resonance in the internal neutral waves at the screech frequency and the interacting shocks can result in the generation of an upstream travelling wave mode that completes the excitation loop by propagating the flow instability towards the nozzle exit along the internal wall of the jet shear layer. Additionally, unstable perturbations in these jets can lead to shock impingement between consecutive vortices resulting in the leakage of internal pressure waves into the ambient. This was identified as the source of shock leakage by Manning & Lele (Reference Manning and Lele2000). In a recent study, Nogueira et al. (Reference Nogueira, Cavalieri, Martini, Towne, Jordan and Edgington-Mitchell2024) have formulated a model to predict G-JM waves resulting from these triadic interactions. The current study utilises spatial Fourier decomposition to identify the signatures of the various components of the triadic interaction as discussed in Appendix B. Throughout this discussion the interaction between the first shock cell and the KH instability waves is referred to as the optimal interaction (k _KH –k _S1 ) and the interaction with the second shock cell is referred to as the suboptimal interaction (k _KH –k _S2 ).

In the current study, the block size for SPOD was chosen relative to the acquisition frequency (N_F1 = 820, N_F2 = 2240) to achieve a frequency resolution of 50Hz, aligning with the frequency resolution of the acoustic measurements. To mitigate signal discontinuities between blocks, a Hamming window was employed, featuring a 75 % overlap among blocks. The principal instability frequencies and their associated harmonics within the Nyquist cutoff limits (Nq_F1 = 20.5 kHz, Nq_F2 = 56 kHz) were discerned from the SPOD analyses for both operational conditions. Subsequently, spatial analysis was performed on the energy distributions within the complex plane at each frequency, thereby revealing the directional wave propagation and the spatial wave interactions. The energy distribution associated with the primary mode at St ≈ 0.51 for NPR 2.6 is shown in figure 4(a) which clearly indicates that the jets undergo coupled oscillations in the Twinjet plane. The inner shear layers of the jets were found to be out of phase with each other by $\pi$ radians, a phenomenon that extends to the outer shear layers as well. Critically, this phase misalignment along the inner shear layers leads to destructive interference resulting in minimal energy levels along the symmetry line in the internozzle region. This is in agreement with the observations made in the intensity standard deviation in figure 3(b). Due to this interference, peak energy levels are concentrated along the outer shear layers, specifically from X/h = 1.5 to X/h = 4. Also observed was the upstream acoustic emission from the outer shear layer and the downstream travelling coherent structures along the jet shear layers. Contrary to the energy distribution in figure 4(a), the phase distribution does not exhibit mirror symmetry around the nozzle’s Symmetry plane. This behaviour is consistent with the results observed in twin circular nozzles in the experimental study by Knast et al. (Reference Knast, Bell, Wong, Leb, Soria, Honnery and Edgington-Mitchell2018) and the LES results of Ahn et al. (Reference Ahn, Lee and Mihaescu2021b ).

Figure 4. (a) Mode-1 SPOD result for NPR 2.6 at St ≈ 0.51. (b) Energy contours for normalised wavenumber (kh) versus radial distance (Z/h). White vertical line, acoustic wavenumber (k _±a ); yellow dashed line, KH instability wavenumber (k _KH ); magenta dashed line, first shock wavenumber (k _S1 ); cyan dashed line, second shock wavenumber (k _S2 ); red solid line, optimal interaction (k _KH –k _S1 ); magenta solid line, suboptimal interaction (k _KH –k _S2 ); white dotted line, nozzle lip line. (c) Reconstructed energy profile for the KH instability wave (k _KH ). (d) Reconstructed energy profile for upstream acoustic waves (k _–a ). (e) Reconstructed energy profile for G-JM (k _g ). View normal to Twinjet plane.

Figure 5. Amplitudes along inner and outer lip line for jet centred at Z/h = 1.05 from SPOD mode ( $\psi _{\mathbf{SPOD}}$ ) and spatial energy profiles: KH instability(k _KH ); acoustic (k _–a ); G-JM (k _g ). Dashed lines are shock cell locations.

However, in both cases the nozzles were purely converging circular nozzles with an internozzle spacing of s/h = 3 undergoing corotating helical coupling mode. The result of the spatial decomposition from the SPOD mode and the expected interaction wavenumbers are depicted in figure 4(b). As is convention, positive wavenumbers represent downstream propagating waves, and negative wavenumbers represent upstream propagating waves. The contour colours represent the energy intensity expressed in logarithmic scale of the spectral energy content.

While the convective velocity for supersonic flows is widely accepted (Tam & Tanna Reference Tam and Tanna1982; Anand et al. Reference Anand, Jodele, Shaw, Russell, Prisell, Lyrsell and Gutmark2021; Harper-Bourne and Fisher Reference Harper-Bourne and Fisher1973) as being ≈0.7U _j this value was observed to vary depending on the jet operating condition and nozzle geometry (Saleem et al. Reference Saleem, Karnam, Rodriguez, Liu and Gutmark2023). Following the methodology of Jeun et al. (Reference Jeun, Karnam, Wu, Lele, Baier and Gutmark2022), the wavenumber for the KH instability waves was identified by locating the peak energy signature in the downstream direction (k > 0). This value is indicated by the yellow dashed line. The red solid line marks the optimal interaction, and the magenta solid line marks the suboptimal interaction. Energy distribution across both the jets was found to be concentrated within the nozzle lip lines indicated by the dashed white lines. The energy peaks were found to be associated with the KH instability waves which propagate in the downstream direction. Conversely, notable upstream features appear close to the acoustic speed of sound (k _–a = –2.99) and represent the acoustic emission generated by the screech tones and the G-JM. The convergence of k _–a and the optimal interaction (k _KH –k _S1 = –3.67) highlights the influence of the first shock cell in the generation of the primary screech tone. The energy signature associated with the G-JM was identified as the secondary upstream lobe at k _g = –4.03. A defining characteristic of G-JM is the presence of high energy concentrations close to the jet shear layer that experience rapid drop off at larger radial locations outside the jet (Edgington-Mitchell et al. Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022). This trend was clearly observed at k _g , while the suboptimal interaction (k _KH – k _S2 = –10.74) lands in the duct-like region with energy distribution contained entirely within the jet shear layers.

These regions consist of waves with negative phase velocity that can appear at these wavenumbers (Towne et al. Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) that are energised by the suboptimal interaction but have a positive group velocity that prevents them from closing the screech resonance loop. Furthermore, the internozzle region showed minimal energy concentrations across both upstream and downstream domains reinforcing the hypothesis of destructive interference caused due to the phase offset. Figure 4(c–e) were generated by reconstructing specific wavenumbers as outlined in Appendix B. This reconstruction allows for the calculation of the phase velocities of the waves which are annotated on each contour. The waveforms associated with the KH instability waves are depicted in figure 4(c). As observed in the wavenumber spectra, these waves exhibit the highest energy concentrations among all other wave components. The contours illustrate both the phase and likely locations of the coherent structures responsible for generating these waves. Note that the phase offset between the inner and outer shear layer of each individual jet manifests at the nozzle exit and propagates downstream. These waves are characterised by the energy concentrations occurring close to the nozzle lip line permeating the jet hydrodynamic region. The phase disparity between the inner and outer shear layers leads to higher energy levels along the outer shear layers. Notably, the computed phase speed of 0.68U _j closely aligns with the initial estimate of 0.7U _j .

Figure 4(d) depicts the energy distribution linked to the acoustic resultant of the triadic interactions directed towards the jet upstream. The waves exhibit a diminishing energy profile moving downstream past X/h = 3, suggesting their origination point lies closer to the nozzle exit. It’s noteworthy that the phase disparity observed in these acoustic waves aligns with that identified in the SPOD analysis. The contour for the G-JM, depicted in figure 4(e), mirrors the phase pattern seen with KH instability waves, yet these waves propagate upstream, guided along the shear layers as indicated by the concentrated energy regions close to the lip lines. A distinct difference in energy levels was observed between the inner and outer shear layers. The outer layers experience no significant external interactions apart from shear driven mixing. However, the inner shear layers engage in consistent phase interactions along the axial direction up to X/h ≈ 5 resulting in increased wave amplitudes. The computed wave speed of 0.74a ₀ for the G-JM coincides with the observations made in previous studies (Shen & Tam Reference Shen and Tam2002; Edgington-Mitchell et al. Reference Edgington-Mitchell, Li, Liu, He, Wong, Mackenzie and Nogueira2022; Karnam et al. Reference Karnam, Saleem and Gutmark2023) where a ₀ is the ambient speed of sound.

A comparison of the wave amplitudes along the inner and outer lip lines from the jet centred on Z/h = 1.05 is illustrated in figure 5. The dotted lines mark the locations of the shock inflection points along the jet shear layer and represent the axial span of the first four shock cells. For the inner shear layer, the maximum amplitude identified from the SPOD contour ( $\psi _{\mathbf{SPOD}}$ ) was at X/h = 2.86, which coincided with the end of the third shock cell as seen in figure 5(a). The coincident peaks of $\psi _{\mathbf{SPOD}}$ and k _KH plots indicate that the coherent structures play a key role in screech generation. Additionally, the energy profiles for both levels gradually increase downstream, reflecting the growth of coherent structures. In contrast, the peak amplitude from the SPOD mode along the outer shear layer was identified at X/h = 2.06, ahead of the third shock cell accompanied by a rapid drop in SPOD energy levels as seen in figure 5(b).

This behaviour is driven by mixing between the ambient air and the outer shear layers. This trend is partially mimicked by the acoustic component that is generated through the interaction between KH instabilities and shock cells. However, the inner shear layers exhibit a more gradual decrease in energy levels due to the coupled interactions spanning the length of the jet. The surge in flow energy and the positioning of the peaks along the outer shear layer highlight the influence of the first shock cell in energising the flow leading to the generation of acoustic and flow instabilities. On the other hand, the amplitudes of the G-JM were more substantial along the inner shear layers driven by jet coupling. It is important to note that true levels for k _g are found in regions inwards of the lip line.

Spatial spectral decomposition was conducted on the dataset for the condition of NPR 3 with the corresponding results depicted in figure 6. Contrary to NPR 2.6, the inner shear layers at NPR 3 show pronounced phase locking with consistent phase match extending the length of the observable field of view as seen in SPOD results illustrated in figure 6(a). The outer shear layers, while being phase matched with each other, were found to be $\pi$ radians out of phase with respect to the inner shear layers. This phenomenon was attributed to the symmetric oscillations in the Twinjet plane (Ahn et al. Reference Ahn, Lee and Mihaescu2021b ; Karnam et al., Reference Karnam, Ahn, Gutmark and Mihaescu2022; Ahn et al. Reference Ahn, Mihaescu, Karnam and Gutmark2023). Additionally, a standing wave was observed in the internozzle region resulting from the superposition of the incident and reflected acoustic waves. The result of the spatial decomposition from the SPOD mode and the expected interaction wavenumbers are depicted in figure 6(b). The phase locked behaviour observed in the SPOD mode translated to energy distributions spanning the inner shear layers along both flow directions. Notably, the energy distribution tied to the KH instability waves (yellow dashed line) spans across the inner shear layers while also extending across the jets to the outer shear layers. Similarly, distributions associated with the upstream wavenumbers, acoustic emission (white solid lines) and the G-JM (k = –4.03), revealed continuous energy bands spanning across the inner shear layers. The optimal interaction (red line) emerges as the key driver behind the generation of the acoustic component and G-JM.

Figure 6. (a) Mode-1 SPOD result for NPR 3 at St ≈ 0.38. (b) Energy contours for normalised wavenumber (kh) versus radial distance (Z/h). White vertical line, acoustic wavenumber (k _±a ); yellow dashed line, KH instability wavenumber (k _KH ); magenta dashed line, first shock wavenumber (k _S1 ); cyan dashed line. second shock wavenumber (k _S2 ); red solid line, optimal interaction (k _KH –k _S1 ); magenta solid line, suboptimal interaction (k _KH –k _S2 ); white dotted line, nozzle lip line. (c) reconstructed energy profile for the KH instability wave (k _KH ). (d,c) Reconstructed energy profile for upstream acoustic waves (k _–a ).(e) Reconstructed energy profile for G-JM (k _g ). View normal to Twinjet plane.

The position of the optimal interaction line indicates that apart from the acoustic component, shock leakage could be an expected resultant. Figure 6(c–e) illustrate the reconstructed energy profiles at the k _kH , k _–a and k _g wavenumbers. For the KH (k _KH ) instability waves, adjacent shear layers initially display a phase difference smaller than $\pi$ radians contrasting the behaviour observed for NPR 2.6. As the waves propagate downstream, the phase offset converges towards $\pi$ radians. Additionally, the region between the jets exhibited higher energy concentration for NPR 3 as seen in figure 6(c) that can be attributed to the constructive interference due to the synchronised oscillation of the jets. The computed wave propagation speed of 0.74U _j closely aligns with the value of 0.71U _j observed from the LES results (Ahn et al. Reference Ahn, Mihaescu, Karnam and Gutmark2023). On the other hand, the acoustic component is characterised by a constant phase offset of $\pi$ radians across adjacent shear layers while preserving phase coherence across the inner shear layers as seen in figure 6(d). A closer examination revealed the relative dominance of the two components (k _KH , k _–a ) in the SPOD energy contour.

Figure 7. Wave amplitude along inner and outer lip line of the top jet from SPOD mode ( $\psi _{\mathbf{SPOD}}$ ) and spatial energy profiles: KH instability(k _KH ); acoustic (k _–a ); G-JM (k _g ). Dashed lines represent shock cell locations.

Closer to the nozzle exit the acoustic component’s influence is prominent marked by a constant phase difference across adjacent shear layers. Farther downstream at X/h > 4 the influence of the KH instability is more pronounced. This is corroborated by the energy profiles of the acoustic component with peak values closer to the nozzle exit. Along the outer shear layer, the influence of KH waves is stronger at downstream locations, while upstream acoustics influence regions closer to the nozzle exit. The G-JM is identifiable by the characteristic energy concentrations found close to the nozzle lip line in figure 6(b) at k _g = –4.02. Figure 6(e) depicts the reconstructed energy profiles for k _g . Mirroring the phase coherence followed by the other components, the G-JM originates as a resultant from the optimal interaction with coupled energy magnitudes increasing in amplitude towards the jet downstream. This trend indicates mode amplification driven by shear layer mixing. Characteristically, the energy profiles along the outer shear layers dissipate rapidly at outward radial locations.

The wave amplitude comparison extracted along the nozzle lip lines is depicted in figure 7. Along the inner lip line, the flow energy dynamics is predominantly driven by KH instabilities as evidenced by the correlation between k _KH and $\psi _{\mathbf{SPOD}}$ in figure 7(a). Peak amplitudes were identified between the third and fourth shock cell and at downstream location of X/h = 6.5. The persistent trend of high energy across the jet length illustrates the role played by the first shock cell on the triadic interactions. Conversely the acoustic component is the dominant contributor along the outer shear layer, particularly near the nozzle exit as depicted in figure 7(b). The amplitude trends of $\psi _{\mathbf{SPOD}}$ align closely with k _–a along the first three shock cells, beyond which a shift to k _KH was observed. The overall peak amplitude along the outer shear layer occurs at X/h = 5.8 for $\psi _{\mathbf{SPOD}}$ due phase inversion in relation to the inner shear layer. The trends in k _g amplitudes across both shear layer highlight jet coupling with higher values appearing along the inner shear layer. Across both shear layers the amplitudes of k _g broadly follow the trends exhibited by KH instabilities peaking at axial locations similar to k _KH . Higher G-JM amplitude along the inner shear layers can be attributed to the onset of mixing in the internozzle region. The energy trends along the inner and outer shear layers for NPR 3 oppose the results discussed for NPR 2.6 (figure 4) where energy concentrations were dominant along the outer shear layers. This emphasises the role of oscillation phase on jet development along inner shear layers where in phase oscillations drives stronger flow fluctuations.

3.3. Flow field: schlieren image analysis – screech harmonic frequency

To capture the flow dynamics at higher frequencies at the condition of NPR 2.6 the image acquisition rate was increased to 112 kHz (F ₂) thereby extending the Nyquist limit to 56 kHz. This led to a reduction in the field of view to 6 h × 2 h (length × width). A total of 4000 samples were collected at F₂ and an SPOD analysis was performed with a block size of N_F2 = 2240 samples per block with a 75 % overlap between blocks resulting in a frequency resolution of 50 Hz. The findings at the harmonic frequency of St ≈ 1.02 (20.8 kHz) are illustrated in figure 8. The energy distribution contour from the leading mode of the SPOD results at the harmonic frequency is illustrated in figure 8(a). Unlike the primary frequency tone, the inner shear layers displayed symmetric phase-energy distribution at the harmonic frequency. Significant interactions were observed in the internozzle region between X/h = 1 and X/h = 4 prior to shear layer mixing. The spatial decomposition of the dominant SPOD mode illustrated in figure 8(b) unveiled several noteworthy findings. The energy associated with the KH instabilities was highly localised in regions close to the shear layer of the jet at k _kH = 8.1 with a computed wave propagation speed u _p = 0.6U _j . Limited hydrodynamic interactions spanning the internozzle region were also observed in the upstream region of the spatial spectral contour.

Figure 8. Spatial energy distribution for NPR 2.6 at S t ≈ 1.02. (a) Energy distribution at mode 1 from SPOD results. (b) Contour of energy distribution for normalised wavenumber (kh) versus radial distance (Z/h). Line nomenclature see figure 7. (c) Isolated upstream components. (d) Isolated downstream components. (e) Energy levels obtained at centreline (Z/h = 0, -•-) and lip line (Z/h = 0.55, $ \cdot$ - -) from (b),(d) and (e). Vertical dashed lined indicate locations of shock inflection points. View normal to Twinjet plane.

The energy signature associated with the upstream acoustic waves (k _–a = –6.03) was found to be higher on the jet centred at Z/h = –1.55. Examination of the cyclic phase patterns of the SPOD revealed a staggered upstream wave propagation to be the cause of the non-uniform energy distribution. Critically, the triadic interaction between the second shock cell and the KH instability waves was identified as a key factor in harmonic production indicated by the position of the interaction wavenumber (magenta line). The distribution in the adjoining wavenumbers resembles the duct-like regions with energy levels fully contained within the jet shear layers. This indicates the absence of the G-JM with the primary closure mechanism being acoustic propagation. These observations are in partial agreement with the computational study of Martin et al. (2023) where instability generation of the second harmonic tone in an AR 4 rectangular jet was associated with triadic interactions of KH instability waves and shock cells.

However, in the study by Martin et al. (2023) a G-JM signature was identified as the closure mechanism for harmonic generation. But unlike the current case, the rectangular jet in their study exhibited an antisymmetric phase pattern at the harmonic frequency. In the current study, the symmetric oscillations at the harmonic frequency in twinjet configurations can lead to strong wave superposition. This can be a potential factor in altering the energy dynamics from the triadic distribution in favour of the acoustic waves making them the primary method of screech closure. The upstream and downstream flow components from the SPOD mode energy contour were isolated and are illustrated in figure 8(c,d). The staggered production of upstream component is clear from figure 8(c). Peak energy levels occur close to X/h = 3.35 and are driven towards the nozzle exit by continuous production of upstream waves along the shock tips. The non-uniformity in these features can be attributed to the presence of flow, hydrodynamic and acoustic components with the main contribution coming from the hydrodynamic part.

In the regions closer to the nozzle exit, the relative dominance of the acoustic component leads to a uniform phase distribution. On the other hand, the downstream propagating flow structures feature strong energy concentrations in regions close to the nozzle lip line. These areas are typically dominated by coherent flow structures seen in figure 8(d). Axial energy variations from these contours were extracted along the nozzle symmetry line (Z/h = 0) and the inner lip line (Z/h = 0.55) and are depicted in figure 8(e). Black plots indicate upstream component, red plots indicate the downstream component, and green plots indicate the overall SPOD mode energy. Values from the centreline are denoted by -•-, while those from the lip line are denoted by $ \cdot$ - -. The overall peak was identified in the SPOD mode energy along the nozzle lip line X/h = 3.05 depicted by the dashed green line. Notably, the intensity levels for the SPOD mode energy and the upstream components (black, -•-) were identical along the nozzle centreline indicating the dominance of acoustics in the internozzle region. Peak energy levels for the downstream components (red, $ \cdot$ - -) were identified along the nozzle lip line as expected due to the coherent structures traversing along the shear layers. This contributed to the SPOD mode energy in the shear layer mimicking the behaviour of the pure downstream components. The locations of the peak energy levels after the second shock cell reinforce the prior discussion attributing the suboptimal interaction as the primary driver for harmonic instability generation.

The temporal and spatial energy distributions for the condition of NPR 3 at St ≈ 0.75 are illustrated in figure 9. Figures 6 and 9 originate from the same dataset and share the larger field of view. The SPOD mode energy results depicted in figure 9(a) show identical symmetric phase energy distribution as observed in the first harmonic results for NPR 2.6. Shear layer phase locking observed at the primary screech frequency of St ≈ 0.38 is also evident in the first harmonic at NPR 3. Moreover, a phase deviation was noted in the outer shear layers relative to the inner ones, along with a noticeable disparity in energy levels between the inner and outer shear layers. The higher mixing experienced by the outer shear layers resulted in rapid reduction in energy levels along the axial direction. In contrast, the inner shear layers show an increase in overall energy along the axial direction peaking close to X/h = 4. Beyond this location the effect of flow mixing leads to rapid deterioration of the shear layer boundary definition. The spatial decomposition results illustrated in figure 9(b) show trends similar to those observed at NPR 2.6. Utilising the KH instability wavenumber at k _KH = 6.23 the phase velocity of the coherent structures in the downstream direction was computed to be 0.68U _j which is lower than the computed phase velocity at the primary screech frequency. Additionally, the triadic interaction between the KH instability wave and the second shock, k _KH – k _S2 , was identified to strongly influence the generation of the instability tone. This observation is critical as it emphasises the effect of optimal and suboptimal interaction on the flow.

Figure 9. Spatial energy distribution for NPR 3 at St ≈ 0.75. (a) Energy distribution at mode 1 from SPOD results. (b) Contour of energy distribution for normalised wavenumber (kh) versus radial distance (Z/h ). Line nomenclature see figure 7(c). (c) Isolated upstream components. (d) Isolated downstream components. View normal to twinjet plane.

Although the precise nature of the interaction that leads to the symmetric or antisymmetric phase behaviour is not fully understood, it is apparent that the suboptimal interaction consistently leads to phase symmetry across the inner shear layers. An interesting note was the absence of energy signatures associated with G-JM indicating that screech closure was once again driven primarily by the upstream propagation of the acoustic component. Isolated upstream and downstream SPOD contours are illustrated in figures 9(c) and 9(d), respectively. The majority of energy from flow structures was found to be concentrated in the coherent structures along the inner shear layers, primarily within the downstream flow component. Conversely, the acoustic component dominated the space between the nozzles and along the outer shear layers. A notable observation from the flow separation analysis was the detection of oblique waves, which originate from the outer shear layers of the jet and propagate upstream as seen in figure 9(c).

This phenomenon aligns with findings from the SPOD analysis of the LES data as reported in Ahn et al. (Reference Ahn, Mihaescu, Karnam and Gutmark2023). These oblique waves were found to originate near X/h = 6, exhibiting an angular directivity of 58° relative to the jet centreline (with the nozzle exit located at 0°). Considering the angular placement of far-field microphones, these waves are the likely cause of the SPL spikes observed in the 90° and 100° microphones shown in figure 2(g) at this frequency. In the following section results from PIV analysis will be discussed to understand the effect of the flow instabilities on critical flow metrics.

3.4. Flow field: PIV

Results were extracted from PIV images using a three strep postprocessing technique. This consisted of the cross-correlation computations, followed by a correlation accuracy check using vector peak ratio (Q), where vector groups with Q < 1.5 were discarded, as values approaching 1 could indicate random correlation peaks unrelated to particle locations (LaVison GmbH 2017). Finally, a local dynamic median filter was used to identify large erroneous vectors based on an root-mean-square–median difference criteria applied across both axial and radial vector components simultaneously. This filtering stage was repeated over two iterations to identify and recompute outliers with a local mean value. The final vector field consisted of 174 × 130 (length × width) vectors per camera resulting in a vector density of one vector every 0.63 mm. A traversable twin camera set-up was utilised to capture images resulting in a total field of view of 23.4h × 4.68h (accounting for image overlap and stitching) in the axial and radial directions, respectively.

Figure 10. Axial velocity contours: (a) NPR 2.6; (b) NPR 3. (c) Axial velocity values along geometric centreines for NPR 2.6 and NPR 3. Black dashed line represents the sonic line (M _local = 1). View normal to Twinjet plane.

The average normalised axial velocity profiles for the conditions of NPR 2.6 and NPR 3 are illustrated in figure 10. The formulation proposed by André, Castelain & Bailly (Reference André, Castelain and Bailly2014) was implemented to compute local Mach number from the mean velocity using

(3.1) \begin{align}M=\left\{\frac{{u^{2}}_{mean}}{\gamma RT_{t}-{u^{2}}_{mean}(\gamma -1)/2}\right\}^{1/2},\end{align}

where M is the local Mach number, u _mean is the mean velocity of the flow ( $u_{mean}=({u_{x}}^{2}+{u_{y}}^{2})^{0.5}$ ) and T _t is the total temperature of the plenum. Since the nozzle operates at ambient temperature, it can be assumed that the variation in the jet total temperature and the reservoir temperature are negligible. The sonic line (M _local = 1) was extracted from the local Mach number contours and is overlaid as a black dashed line in figure 10.

At NPR 2.6, the experimental results showed the formation of a Mach disc close to the nozzle exit X/h = 0.21 in both the jets as seen in figure 10(a). The termination of the supersonic core was identified at X/h = 6.9 from the nozzle exit marked by the sonic line. A constant reduction in the radial span of the jet supersonic regions was noted with the side closer to the symmetry line undergoing accelerated degradation compared with the side along the outer shear layers. Interactions between the jets along the symmetry line were minimal up to X/h = 3.6 beyond which a notable increase in axial velocity was observed. This axial position coincides with the strong antisymmetric coupling observed in the SPOD profiles in figure 4(a). and the minimal energy zone in the internozzle region from the intensity standard deviation in figure 3(b). Peak mixing velocity in the internozzle region was observed close to X/h = 15 beyond which the inner shear layers merged and were indistinguishable from each other. Subsequently, the jets coalesce and continue as a single jet in the observable domain accompanied by a steady inward deflection of the outer shear layers towards the nozzle symmetry line. For NPR 3, the supersonic jet core extends up to X/h = 7.4 from the nozzle exit, driven by the higher jet operating condition as seen in figure 10(b). No Mach disk formation was observed at this condition. Though pronounced oscillations induced by screech were observed at NPR 3, they did not alter the onset of mixing along the inner shear layers which started at X/h = 3.7, similar to NPR 2.6.

However, the rapid increase in the axial velocity along the nozzle symmetry plane indicated that the jets undergo more intense mixing at NPR 3. The degradation of the jet core along the inner shear layers was immediately apparent past X/h = 2.8 marked by the rapid thickening of the inner shear layers. This resulted in the jets demonstrating a strong deflection towards the symmetry line. The enhanced mixing along the symmetry line culminated in the complete merging of the inner shear layers by X/h = 14, beyond which the jets coalesce into a single column. The centreline axial velocities are presented in figure 10(c) and highlight the axial locations of the shock-expansion pairs for both operating conditions. The length of the potential core, typically defined as the axial location where the local axial velocity drops below 95 % of U _j , was located at X/h = 5.2 for NPR 2.6 and at X/h = 4.47 for NPR 3. Notably, the centreline velocities for all cases converge once shear layer merging begins at X/h = 15.

A clearer understanding of the flow dynamics can be achieved by analysing the turbulence intensities and Reynold’s stresses that highlight the isotropic and anisotropic turbulence features. Figures 11 and 12 illustrate these quantities as measured for NPR 2.6 and NPR 3, respectively. Also highlighted are the locations of the shock inflection points on the jet shear layers to help quantify the spatial span of the shock cells. At NPR 2.6, seven prominent shock cells were identified as seen in figure 11(a). The imperfect nature of flow expansion leads to constant reduction in the axial shock cell width driven by the pressure equalisation between the supersonic core and the ambient air.

Figure 11. Average axial velocity and turbulence quantities for NPR 2.6. (a) Normalised axial velocity. (b) Normalised TKE. (c) Normalised axial reynolds stresses (R _xx ). (d) Normalised radial reynolds stresses (R _yy ). Black dashed line indicates the sonic line. White dashed lined indicate the location of the shock inflection points.

The average shock cell width across the first three shocks was computed as 0.844h. The turbulence kinetic energy (TKE) distribution in figure 11(b) illustrates the distribution of isotropic turbulence intensity with clear peaks emerging in the outer shear layers of the jets and comparatively low levels along the nozzle symmetry line. This reduction is the consequence of the antisymmetric phase coupling observed in the inner shear layers in schlieren image analysis results discussed in § 3.2, figure 4(c). Additionally, the coupled nature of interactions occurring along the inner shear layers led to faster shear layer spreading as noted by the uneven growth across the inner and outer shear layers. The Reynolds stress profiles presented in figures 11(c) and 11(d) highlight the influence of the axial and radial velocity components on turbulence development. The axial stresses tend to be dominant driven by the larger fluctuation levels that occur along the streamwise direction. The readers must note that the intensity scale for the R _yy distributions has been modified to better reveal flow structures. The axial stresses (R _xx ) primarily influence turbulence production along the shear layers as indicated by the peak distributions observed in figure 11(c).

A notable increase in R _xx that indicates the onset of shear layer expansion was identified at X/h = 1.24, just aft of the first shock cell. This observation aligns with the findings from § 3.2 which emphasises the influence of triadic interactions between turbulent structures and the first shock cell being a critical component of instability closure. In comparison, the stress levels along the outer shear layers reach matching levels at X/h = 2.16, just aft of the second shock cell, close to the peak energy location observed in the SPOD profile in figure 5(b). While the radial stresses exhibit comparable patterns along the shear layers, their peak values were identified more internally relative to the shear layer boundaries. This can be attributed to radial fluctuations in flow velocity as illustrated in figure 11(d). A notable finding is the minimal turbulence levels within the supersonic core of the jets across all energy distributions. This is anticipated considering that the jet oscillations at NPR 2.6 lack the amplitude to notably disturb the shock train, which would induce rapid velocity changes.

Figure 12. Average axial velocity and turbulence quantities for NPR 3. (a) Normalised axial velocity. (b) Normalised TKE. (c) Normalised axial reynolds stresses (R _xx ). (d) Normalised radial reynolds stresses (R _yy ). Black dashed line indicates the sonic line. White dashed lined indicate the location of the shock inflection points.

For NPR 3, the turbulence and shear stress profiles from the experimental data are illustrated in figure 12. While the distribution of peak turbulence levels along the outer shear layers broadly resembles that of NPR 2.6, distinct differences are evident. The TKE distribution revealed a large increase in radial jet spreading driven by the symmetric phase locked oscillations of the inner shear layers in the Twinjet plane shown in figure 12(b). A prominent feature observed across TKE, R _xx and R _yy profiles was the distinct branching of flow energy along the inner shear layers and into the internozzle regions originating from X/h = 1.1 and reaching maximum radial span at X/h = 5.2, beyond the fourth shock cell. Turbulence peaks were identified aft of the first shock cell at X/h = 2.2 along the inner shear layer mirroring the findings from the SPOD analysis discussed in § 3.2. Notably, elevated TKE levels were observed in the trailing region of the supersonic core from X/h≈ 5 to X/h = 7.4 deviating from the trend observed for NPR 2.6. A closer examination of the R _xx and R _yy revealed some interesting findings. In figure 12(c), axial velocity fluctuations led to high shear stress concentrations along the inner shear layer spanning the first three shock cells between X/h = 1.1 and X/h = 4. Beyond this region the rapid onset of mixing led to dissipation of the shear stresses. On the other hand, the axial stresses along the outer shear layers showed gradual growth in the streamwise direction with peak values observed close to X/h = 7.5. These fluctuations were notably absent in regions inside the supersonic core. In contrast, shear stress peaks of R _yy driven by strong radial velocity fluctuations were identified inside the trailing edge of the supersonic core in the same axial range stated above as seen in figure 12(d). This suggests that the amplitude of flow oscillation plays a vital role in shaping the core turbulence distribution in highly screeching jets. Moreover, the symmetric coupling observed in twin jet set-ups can greatly influence the amplitudes of jet oscillations, as evidenced by the comparative results for NPR 2.6 and NPR 3.

3.5. Flow field: proper-orthogonal-decomposition-based selective reconstruction

To assess the impact of the jet oscillation amplitude on instability tone generation, a reconstruction study was implemented based on the snapshot method of proper orthogonal decomposition (POD) formulated by Sirovich (Reference Sirovich1987). Research by Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011) and Edgington-Mitchell et al. (Reference Edgington-Mitchell, Oberleithner, Honnery and Soria2014) has demonstrated that in flows with periodic instability generation, the first two POD modes fully capture the complete oscillation cycle in flows dominated by periodic instabilities. Additionally, the temporal coefficients associated with these modes are coupled with each other and produce a circular pattern when plotted in a correlated phase space.

This relationship allows for these modes to be utilised in flow reconstruction and identification of the dominant flow features. The current analysis employed the methodology from Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011) to compute the POD modes from the axial and radial velocity components. The energy distribution of the POD modes and the phase portraits computed from the radial velocity (v) components are presented in figure 13. Energy trends from the first fifteen modes are illustrated in figure 13(a). The plot depicts mode energy distribution from axial (u) and radial velocities with increasing mode value.

Figure 13. The POD mode energies and phase portraits. (a) Energy percentage of individual mode energies ( $\epsilon$ ) from experiment. Here $\blacktriangle$ ,axial velocity(u) components; •, radial velocity(v) components; NPR 2.6, gold; NPR 3, blue. Phase portrait of first two normalised POD temporal components extracted from radial velocity snapshots for (b) NPR 2.6 and (c) NPR 3. First axial velocity POD mode ( $\varphi _{{u_{x}}}$ ) for (d) NPR 2.6 and (e) NPR 3. The phase portraits are normalised using the $\mathrm{r}_{\mathrm{m}}=\sqrt{{a_{1}}^{2}+{a_{2}}^{2}}$ where $a_{1}$ and $a_{2}$ are the temporal coefficient for the respective POD mode.

The distribution of energy across the first two modes at NPR 2.6 and NPR 3 revealed significant insights into the dynamics of jet oscillations. At NPR 2.6, the first two axial modes accounted for 5.7 % of the total energy, while the radial modes contributed to 6.87 %. In contrast, at NPR 3, the energy contribution from the first two modes increased significantly to 17.5 % for axial modes and 23.4 % for radial modes. This increase in energy levels at NPR 3 is anticipated, considering the phase-locked oscillations that disturb the jets, resulting in substantial flow fluctuations compared with NPR 2.6. Notably, the energy proportion of the first two radial modes exceeds that of the axial modes. Despite axial velocity components typically harbouring more absolute energy than radial components, the jets undergo higher relative amplitude fluctuations in the radial direction, thus yielding a higher relative energy in these modes. The result of these rapid radial oscillations was the large concentrations of radial Reynold’s stresses observed in the interior regions of the jets. The temporal components associated with the first two radial POD modes were utilised to construct phase portraits as illustrated in figure 13(b,c). These contours represented as a probability density function distribution exhibited a pronounced circular pattern across both operating conditions. This pattern indicates that the first two modes are coupled and hence encapsulate the complete cyclic oscillation of the instability mode. As detailed in Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011) these modes can be utilised to comprehensively express the oscillating flow field through spatial reconstruction. Figures 13(d) and 13(e) show the energy distribution for the first axial POD mode for NPR 2.6 and NPR 3, respectively. Strong coupling effects along the inner shear layers are evident at both conditions. Additionally, the variations in the oscillation phase clearly align with the results observed in the SPOD analysis with the jets undergoing phase offset oscillations at NPR 2.6 and phase matched oscillations at NPR 3 along the inner shear layers. The triple decomposition technique formulated by Hussain & Reynolds (Reference Hussain and Reynolds1970) was utilised to isolate the coherent component from the flow using

(3.2) \begin{align}u\left(x,t\right)=\overline{u}\left(x\right)+u^{\prime\prime}(x,t)+u_{\boldsymbol{c}}(x,t),\end{align}

where u is the time varying velocity field $\overline{u}$ is the temporal average, u” is the fluctuating component and u _c is the coherent component. Alternatively, the coherent flow component can also be represented using the first two POD modes (Edgington-Mitchell et al. Reference Edgington-Mitchell, Oberleithner, Honnery and Soria2014). To identify the various states of fluctuation velocity reconstruction was performed using the first two POD modes from the axial and radial components. The velocity contours were passed through a cross-correlation based image filter to isolate images at a singular fluctuation phase to emulate phase locking (Alkislar et al. Reference Alkislar, Krothapalli and Lourenco2003) as discussed in Appendix B. These filtered velocity components were then utilised to compute the coherent velocity fluctuations and directional streamlines illustrated in figure 14.The colour bar indicates the range of this magnitude. The streamlines were computed using the axial and radial coherent velocity fluctuations. An algorithm was developed to overlay the coherent velocity magnitude computed using

(3.3) \begin{align}U_{c}=\sqrt{{u_{rec}}^{2}+{v_{rec}}^{2}}\end{align}

on top of the streamlines. In (3.3), u _rec and v _rec are the coherent velocity fluctuations in the axial and radial directions, respectively.

Figure 14. Streamlines of binned coherent velocities coloured with coherent velocity magnitude: (a) NPR 2.6; (b) NPR 3. Dashed lines indicate locations of shock inflection points. View normal to Twinjet plane.

For NPR 2.6, peak fluctuations manifested just beyond the first shock cell, accompanied by vortical formations as depicted by the streamlines in figure 14(a). The vortices remained within the bounds of the shear layers, avoiding extension into the internozzle area, due to the phase differences between oscillations along the inner shear layers. This manifested as minimal velocity distributions along the nozzle symmetry line. Similar vortical formations were observed along the outer shear layers, intensifying post the first shock cell and progressing downstream. These formations tend to shift towards the jet core, influenced by transverse velocity fluctuations, as indicated by the R _yy stress component previously discussed.

Conversely, NPR 3 experiences stronger coherent fluctuations, fuelled by synchronised oscillations characteristic of this operational state. Notably robust vortical areas were detected following the first shock cell, affirming insights from the SPOD analysis, as shown in figure 14(b). Additionally, the formations emerging along the inner shear layers gradually drift towards the nozzle symmetry line, propelled by significant transverse fluctuations. This movement and the resultant hydrodynamic oscillations likely contribute to the branched pattern seen in the turbulence and Reynolds stress distributions in figure 13. These formations exhibit swift dissipation after X/h = 4, correlating with the observed decline in turbulence within the TKE profiles. A key insight from these observations is that the interplay between the first shock cell and coherent structures initiates vortical formations pivotal for instability proliferation in screeching jets, with the amplitude of flow oscillations impacting their size and axial span.