4.1. Aeroacoustic instability characteristics in dual-impinging jets

Figure 3 summarises the far-field tonal characteristics of rectangular and circular DIJs across the tested parameter space (M = 1.0–1.2, h/D = 2–3), and highlights a representative high Mach case to illustrate the contrast in spectral organisation between the two nozzle geometries. For both nozzle types, the spectra at h/D = 3, M = 1.2 exhibit several discrete peaks, but their alignment with the classical single-loop feedback prediction differs between the rectangular and circular configurations. As shown in figure 3(a), the rectangular DIJs exhibit multiple acoustic peaks after the twin jets impinge on the flat plate. Here, the Strouhal number St was defined as follows:

(4.1) \begin{equation} St=\frac{\textit{fD}}{U} ,\end{equation}

where U is the ideally expanded jet-exit velocity defined in § 2.1. The fundamental frequency and its harmonics, calculated using Powell’s formula, are indicated by dashed lines in figure 3 for harmonic orders ranging from n = 1 to n = 5. The convective velocity U _c was estimated based on an empirical model (Gojon, Bogey & Marsden Reference Gojon, Bogey and Marsden2015)

(4.2) \begin{equation} U_{c}=0.65U-\left(0.65U-0.5U_{e}\right)\frac{1}{1+h/D} .\end{equation}

Figure 3.

Far-field tonal characteristics of rectangular and circular DIJs. (a) Representative far-field sound pressure level (SPL) at h/D = 3 and M = 1.2, obtained from the reference microphone at $\Theta$ = 40°. The black dashed lines denote the fundamental feedback frequency f ₁ and its harmonics with phase lag p = 0. The blue bands indicate the corresponding harmonic ranges obtained when the phase-lag term is varied within −0.2 ≤ p ≤ 0, consistent with the shocklet-related time-delay interpretation of p and the commonly reported bounds of p in impingement literature. (b) Sensitivity of the harmonic alignment to the p, quantified by the mean deviation between the measured peak Strouhal numbers and the nearest Powell-predicted harmonic frequencies. (c) Peak Strouhal numbers of the measured far-field tones over the investigated operating matrix, extracted from the same reference microphone at $\Theta$ = 40°. Rectangular and circular symbols denote the respective nozzle types, with larger markers highlighting the dominant peak at each Mach number. (d) Far-field tonal intensity of representative peaks as a function of observer angle θ, measured by the microphone array at 120D in the plane normal to the impingement plate.

Figure 3. Long description

Panel A: Two line graphs depict the far-field acoustic spectra for rectangular and circular direct impinging jets at h/D = 3 and M = 1.2. The x-axis represents the Strouhal number (St) and the y-axis represents the SPL in decibels (dB). The rectangular jet spectrum shows peaks at various Strouhal numbers, labeled as n = 1 to n = 5. The circular jet spectrum also shows peaks at similar Strouhal numbers, with an additional annotation indicating a phase lag. Panel B: A line graph shows the sensitivity of harmonic alignment to the phase-lag term (p). The x-axis represents the phase-lag term (p) and the y-axis represents the mean error in percent. The graph shows a decreasing trend in mean error as the phase-lag term increases from -0.2 to 0. Panel C: Two scatter plots depict the peak Strouhal numbers of measured far-field tones over the investigated operating matrix. The x-axis represents the Strouhal number (St) and the y-axis represents the Mach number (M). Rectangular and circular symbols denote the respective nozzle types, with larger markers highlighting the dominant peak at each Mach number. Panel D: A line graph shows the far-field tonal intensity of representative peaks as a function of observer angle (θ). The x-axis represents the observer angle (θ) in degrees and the y-axis represents the SPL in decibels (dB). The graph compares the tonal intensity for rectangular and circular nozzles, with annotations indicating specific harmonic modes.

To estimate the fundamental frequency f ₁ in (1.1), the ambient speed of sound c _a was determined from the environmental conditions and is approximately 340 m/s. Here, U _e is the jet-exit velocity. Powell’s feedback formulation further includes a phase-lag term p to account for non-instantaneous hydrodynamic–acoustic interactions. Recent studies link p to a finite shocklet-related time delay in supersonic impingement, with reported values spanning p = 0 (Gojon et al. Reference Gojon, Bogey and Marsden2016) to −0.4 (Krothapalli et al. Reference Krothapalli, Rajkuperan, Alvi and Lourenco1999). In the present weakly supersonic DIJ conditions (M = 1.2), we adopt p = 0 as a first-order reference and assess the robustness of the harmonic alignment by considering a modest negative range (p = −0.2) based on the shocklet-delay magnitude reported by Weightman et al. (Reference Weightman, Amili, Honnery, Soria and Edgington-Mitchell2017b ). In figure 3(a), the black dashed lines denote f ₁ and its harmonics predicted with p = 0. The blue bands indicate the corresponding frequency bands obtained by varying the phase-lag term over p ∈ [−0.2, 0]. To make the choice of p more quantitative, figure 3(b) quantifies the sensitivity of the harmonic alignment to p using the mean deviation between the measured peak Strouhal numbers and the nearest Powell-predicted harmonic frequencies. The minimum deviation occurs at p = 0, which supports its use as the baseline setting in the present configuration. Importantly, the strongest rectangular-DIJ peak at St = 0.42 remains outside any single integer-harmonic band over the range p ∈ [−0.2, 0], indicating that its non-harmonic character is robust against reasonable variations of the phase-lag term. For the rectangular DIJs, two prominent spectral peaks fall within the vicinity of the third and fourth harmonic bands (around n = 3.17, St = 0.37, and n = 4.01, St = 0.46), indicating that part of the response is compatible with the classical feedback scaling. The strongest peak at St = 0.42 (n = 3.63) lies between the third and fourth harmonics outside any single integer-harmonic band. This deviation is consistent with a geometry-conditioned modification of the effective feedback loop, whose wall-pressure manifestation is examined in §§ 4.2–4.4 using co-resonance analysis and modal decomposition. A similar sensitivity of impingement-tone behaviour to nozzle geometry was reported for non-circular impinging jets by Balakrishnan & Srinivasan (Reference Balakrishnan and Srinivasan2019). Tam & Norum (Reference Tam and Norum1992) attributed such peaks to symmetric and antisymmetric feedback instability modes. In contrast, the circular nozzle exhibits a single dominant spectral peak at n = 3.92, corresponding to the fourth harmonic of the fundamental feedback frequency f ₁, located at St = 0.45, a feature widely reported in literature (Arthurs & Ziada Reference Arthurs and Ziada2012; Stahl et al. Reference Stahl, Gaitonde, Bhargav and Alvi2022). Figure 3(c) compares the acoustic peak spectra of the rectangular and circular DIJs under various conditions. Rectangular and circular symbols represent the respective nozzle types, with larger markers denoting the dominant peak at each Mach number. Feedback-loop frequencies are indicated as lines. In rectangular DIJs, dominant spectral peaks are frequently observed between adjacent harmonics, whereas in circular DIJs the dominant tones track the harmonic lines more closely. The observer-angle dependence of the principal tones is further summarised in figure 3(d). In the far field at 120D, the rectangular nozzle does not introduce a pronounced redistribution of tonal directivity in the plane normal to the impingement plate. The common harmonic-related peak exhibits a broadly similar angular trend for the circular and rectangular DIJs, and the additional rectangular-specific peak near n = 3.63 does not produce a qualitatively different directivity pattern within the microphone-array plane shown in figure 1(a).

To capture the unsteady loading on the impingement surface, PSP was applied to the region of interest. Figure 4 presents the time-averaged, instantaneous and root mean square (r.m.s.) results for both rectangular and circular DIJs. At an impinging distance of h/D = 3, each nozzle forms a high-energy impingement zone reflecting the nozzle-exit shape, with an interference band between the two nozzles at the centre (figure 4 a). Over the monitored region (7D × 5D), the surface-averaged mean pressure yields a total average pressure of 123.7 kPa for the circular DIJs and 113.5 kPa for the rectangular DIJs. The lower mean level for the rectangular case is consistent with the more rapid decay of rectangular jets associated with flapping along the short axis (Karnam et al. Reference Karnam, Baier and Gutmark2020), and corresponds to an 8.25 % reduction in the average impingement pressure. Projecting the nozzle shapes onto the obstacle reveals that the pressure fluctuations are more pronounced at the centre and edges (figure 4 b), leading to larger r.m.s. values (figure 4 c). For rectangular jets, two corner vortices are also observed near the diagonal corners of the projection, a classical feature of rectangular jets (Zaman Reference Zaman1996). The r.m.s. data along the centrelines of the x- and y-axes (figure 4 d) indicate that the rectangular nozzle exhibits a broader pressure influence along the major axis (y-axis). Nevertheless, in the twin-jet interference region (−0.1 < y/D< 0.1, −2 < x/D< 2), the coupling range and intensity of the rectangular twin jets decreases. This anisotropic near-wall imprint is consistent with prior studies of supersonic rectangular and twin rectangular jets. In particular, the non-axisymmetric shock/shear-layer organisation of rectangular jets can retain a strong directional preference over the early downstream region (Mazharmanesh et al. Reference Mazharmanesh, Nogueira, Weightman and Edgington-Mitchell2025), while in twin rectangular jets the coupling state has a marked influence on the near-field pressure fluctuations (Samimy et al. Reference Samimy, Katterle, Leahy, Webb, Yarlagadda and Hiler2024). The wall-pressure footprints therefore remain compatible with the expected free-jet and twin-jet behaviour of rectangular nozzles before impingement.

Figure 4.

(a) Time-averaged and (b) transient pressure field from PSP (upper: instantaneous total pressure contours; lower: instantaneous fluctuating pressure contours). (c) The r.m.s. of pressure field and (d) r.m.s. profiles along y-axis (left) and x-axis (right).

Figure 5.

(a) Pressure-gradient amplitude indicator of the time-averaged PSP pressure field, and (b) PSP pressure spectrum at y/D = 1.75 along the x-axis for rectangular (left) and circular (right) DIJs. Since the plate coordinates are normalised by D, S _g is reported in units of kPa/D.

Although the rectangular DIJs reduce the mean impingement pressure, they reorganise the near-wall-pressure field and introduce stronger spatial gradients on parts of the plate. To quantify this spatial non-uniformity, figure 5(a) shows the distribution of a pressure-gradient amplitude indicator over the impingement plate, and figure 5(b) presents pressure spectra at selected plate locations. In this study, S _g is used to characterise the spatial variation of the time-averaged PSP pressure field, while the pressure spectra at y/D = 1.75 are used to characterise the streamwise frequency content. The gradient indicator S _g of the time-averaged pressure p(i, j) at a given point was calculated as follows:

(4.3) \begin{equation} S_{g}=\left| \frac{p(i+1,j)-p(i-1,j)}{2{\unicode[Arial]{x0394}} x}\right| +\left| \frac{p(i,j+1)-p(i,j-1)}{2{\unicode[Arial]{x0394}} y}\right| .\end{equation}

Here, Δx and Δy denote the grid spacings in the x- and y-directions, respectively. This definition applies a centred-difference approximation in each direction and combines their absolute magnitudes to highlight regions of strong near-wall-pressure variation. In figure 5(a), elevated values of S _g appear mainly along the outer regions of the nozzle projection. Compared with the circular nozzle, for which the pressure field spreads more uniformly outward, the rectangular nozzle exhibits stronger directional preference and greater non-uniformity in the near-wall-pressure variation, in conformity with the conclusions of Prasad (Reference Prasad2024). In the frequency domain, circular DIJs exhibited consistent spectral characteristics along the x-axis, corresponding to the annular-symmetric modal features (Davis et al. Reference Davis, Edstrand, Alvi, Cattafesta, Yorita and Asai2015). Conversely, the rectangular DIJs show a more complex behaviour: at the centreline of the upper jet (y/D = 1.75), adjacent measurement points may exhibit significantly different spectra (figure 5 b). At x/D = 0, dominant fluctuations occur at n = 3.17 and n = 4.01. In the regions −0.5 < x/D < −0.1 and 0.1 < x/D< 0.5, multiple dominant frequencies coexist, including n = 3.17, 3.63 and 4.01. Among these, the strongest peak appears at n = 3.63, consistent with far-field spectral observations (figure 3). With increasing distance from the jet centre, in the ranges −0.9 < x/D < −0.5 and 0.5 < x/D< 0.9, harmonic peaks diminish and non-harmonic peaks weaken, indicating reduced spectral coherence. This suggests that, while circular DIJs show a strong correlation between wall pressure and far-field acoustic fluctuations over the entire nozzle projection range, rectangular DIJs still exhibit such correlations, but with varying dominant frequencies depending on jet–wall interaction zones. Multiple dominant peaks are observed across the plate, suggesting that more than one coherent structure may be relevant, depending on frequency and location.

4.2. Aeroacoustic correlation analysis for far- and near-field pressures

Given the consistency between the PSP-resolved near-field pressure fluctuations and the far-field microphone signal in both frequency and relative intensity, we employed COP analysis (Noda et al. Reference Noda, Nakakita, Wakahara and Kameda2018) to quantify near- to far-field spectral correlation and to identify the regions on the plate that are most strongly linked to the radiated tone. The implementation of the COP method consists of three main steps, as illustrated in figure 6(a). In step I, a fast Fourier transform (FFT) is applied to convert the time-domain pressure signals into the frequency domain, allowing identification of peaks at matching frequencies. Then, in step II, the auto-spectral densities R _xx and R _yy for the near- and far-field signals, respectively, as well as their cross-spectral density R _xy , are computed to obtain the squared coherence Coh²(f)

(4.4) \begin{equation} \mathrm{Coh}^{2}\left(f\right)=\frac{\left| R_{\textit{xy}}(f)\right| ^{2}}{R_{\textit{xx}}(f)R_{\textit{yy}}(f)} .\end{equation}

A higher Coh²(f) value indicates a stronger correlation between near-field pressure fluctuations and far-field acoustic pressure at the corresponding frequency, implying a greater contribution of the surface pressure to the radiated sound. Finally, the COP, Cop(f) is calculated as the product of Coh²(f) and the r.m.s. amplitude of near-field pressure fluctuations at each frequency (step III)

(4.5) \begin{equation} \mathrm{Cop}(f)=\mathrm{Coh}^{2}(f)\times p'_{\textit{rms}}(f) .\end{equation}

Figure 6.

(a) Schematics of COP analysis, and (b) COP analysis for aerodynamic noise sources of circular DIJs at St = 0.45. (c) Sensitivity of the COP source distribution to the analysis bandwidth, shown for f _a ± 5 and f _a ± 15 Hz around the target frequency.

Figure 6. Long description

Panel A: The schematic of COP analysis includes three steps. Step I shows FFT with microphone and PSP data. Step II displays the coherence formula. Step III presents the COP formula using RMS data. Panel B: The COP analysis for aerodynamic noise sources of circular DIJs at St = 0.45 includes three heat maps. The RMS heat map shows pressure distribution in kPa. The Coh squared heat map shows coherence values. The Cop heat map shows COP values in kPa. Panel C: The sensitivity of the COP source distribution to the analysis bandwidth is shown for fa 5 and fa 15 Hz around the target frequency. Both heat maps display pressure distribution in kPa.

The resulting COP spectrum not only reflects the coherence between surface-pressure fluctuations and far-field acoustic output but also quantifies the effective source strength, thus enabling spatial mapping of potential acoustic sources on the solid surface. This approach has recently been adopted for the joint analysis of PSP data and far-field acoustic measurements (Peng et al. Reference Peng, Gu, Li and Liu2018; Matsui et al. Reference Matsui, Kasai, Sugioka, Asai and Nonomura2023). As an example, figure 6(b) shows the COP results for circular DIJs at n ≈ 4 (St = 0.45), demonstrating the method’s capability to extract localised aerodynamic noise sources on the plate. The corresponding bandwidth-sensitivity check in figure 6(c) confirms that the identified source regions are robust to modest variations in the analysis bandwidth. In this study, both the PSP and microphone datasets used for COP analysis consist of 20 000 samples, corresponding to 1 s of continuous PSP snapshots captured by high-speed camera. A frequency resolution of 1 Hz is used to ensure reliable r.m.s. calculations, and a rectangular window with 50 % overlap is applied in the spectral estimation. To define the analysis band around each target frequency f _a , we benchmark our selection against representative COP implementations in the literature. Nakakita & Ura (Reference Nakakita and Ura2020) employed a 40 Hz band centred on the dominant tone (f _a ± 10 Hz) in an unsteady-PSP/microphone correlation framework. More recently, Sijtsma et al. (Reference Sijtsma, Dreux, Delescluse, Leneveu and Marjono2024) adopted a one-third-octave band centred at 1000 Hz in a generalised COP formulation using grouped sensors. Guided by these established practices and considering the frequency-alignment uncertainty inherent to fast PSP, we select a 20 Hz band (f _a ± 10 Hz) as a consistent and conservative choice for the present tonal-dominated DIJ conditions. A sensitivity check using slightly narrower and wider bands, f _a ± 5 and f _a ± 15 Hz, yielded very similar COP source distributions, while mainly affecting the absolute COP amplitude rather than the dominant spatial structure, as shown in figure 6(c). For circular DIJs, the dominant acoustic sources are primarily distributed in three regions: the jet core, the wall-jet region and the interference zone between the two nozzles (Zhang et al. Reference Zhang, Wang, Wei, Fan, Peng and Liu2025), corresponding respectively to direct jet impingement, enhanced pressure fluctuations due to wall-jet development and jet–jet interaction along the centreline.

Figure 7.

The COP analysis for sound source of rectangular DIJs at (a) St = 0.24, (b) St = 0.37, (c) St = 0. 42 and (d) St = 0.46.

As for the rectangular DIJs, we extracted acoustic peaks at frequencies around n = 2, 3, 3.6 and 4, corresponding to St = 0.24, 0.37, 0.42 and 0.46, respectively, as shown in figure 7. Overall, compared with the strong coherence between impingement pressure and aerodynamic noise exhibited by circular DIJs at the acoustic peak frequency (St = 0.45), the rectangular cases show a modestly lower Coh²(f) than the circular cases at the selected peaks. This difference reflects the altered footprint and azimuthal organisation imposed by the rectangular exits. Additionally, the elongated outlet shape of rectangular DIJs may cause a broader spatial distribution of the impingement region, further weakening the spatial and temporal correlation between near-field and far-field pressure signals. Since the acoustic peak at n = 2 is not sufficiently prominent, the resulting Cop(f) distribution is not discussed further. For n = 3 (figure 7 b) and n = 4 (figure 7 d), the COP analysis reveals aerodynamic noise source structures similar to those of circular DIJs, two vertically symmetric annular distributions, indicating both structural similarity and harmonic base frequency alignment between the two nozzle types. A detailed comparison of COP results on the impingement plate shows that, for rectangular DIJs, dominant pressure fluctuations remain concentrated in the jet core and wall-jet regions, while the fluctuation intensity in the interference region between the two nozzles is significantly weakened. Moreover, for rectangular DIJs, the originally circular annular structures become elliptical, a direct consequence of the rectangular nozzle-exit shape, which also reduces the spatial extent of the interference region. The reduced coverage and diminished pressure fluctuation amplitude in the interference region are consistent with the r.m.s. distribution of pressure fluctuations shown in figure 4(d). At n = 3.7 (figure 7 c), the rectangular DIJs exhibit a distinct jet–plate interaction mechanism, giving rise to a Cop(f) distribution that corresponds to the strongest impingement tone observed. Unlike the classical annular-symmetric pattern, this aerodynamic noise source structure exhibits a twin quadrupole-like shape, with pronounced pressure fluctuations appearing on either side of the centreline along the y-axis of each nozzle. This result confirms the absence of peak pressure fluctuations near x/D = 0 for mode n = 3.7, as observed in figure 5(b). The overall source distribution is aligned with the edge of the nozzle projection area on the plate, with a discontinuity appearing along the far end of the long axis. At this stage, we regard this feature as a plate-pressure footprint consistent with geometry-conditioned rectangular-jet organisation. In this sense, the rectangular nozzle introduces a much clearer directional organisation on the impingement plate than in the observer plane sampled by the far-field microphone array, indicating that its primary geometric effect is to redistribute the near-wall-pressure field rather than to strongly alter the far-field directivity within the measured vertical plane at 120D. The COP analysis establishes a connection between the pressure fluctuations on the impingement plate and the far-field acoustic pressure fluctuations, both in the spatial and frequency domains. It maps the time-domain far-field signal measured by the microphone onto frequency-correlated acoustic-source structures on the impingement surface. Building upon this foundation, the subsequent analysis examines the dynamic evolution of the pressure fluctuations measured by PSP, with particular attention to the aeroacoustic instabilities present in rectangular DIJs.

4.3. Modal dominance variations in rectangular DIJs

To shed light on the characteristics of coherent structures induced by high-speed rectangular DIJs, we employed CS-SPOD to analyse the pressure fluctuation field on the impingement plate. Here, we employ the cyclostationary extension of SPOD to extract the phase-dependent impinging structure of the plate-pressure field. The present PSP signals exhibit clear periodic modulation at the fundamental loop frequency and its harmonics (see cyclic-coherence evidence in Appendix B), motivating the use of CS-SPOD, which extracts modes that maximise correlated energy across the frequency set $\Omega$ _γ = {…, γ–2α ₀, γ – α ₀, γ, γ + α ₀, γ + 2α ₀, …}. This use is consistent with the original formulation of Heidt & Colonius (Reference Heidt and Colonius2024). In the present analysis, α ₀ is taken as the fundamental cycle frequency associated with the dominant loop tone identified in § 4.1. We therefore analyse the wall-pressure statistics using CS-SPOD with α ₀ as the cycle frequency to quantify how this fundamental modulation organises the spatial structures and their harmonic companions on the plate. Each record was divided into blocks of 520 samples, corresponding to t _end = 40T ₀ = 40/α ₀, to provide sufficient cycles for stable second-order estimates. We use 67 % overlap with a Hamming window to reduce cyclic leakage and estimator variance, following standard cyclostationary practice (Antoni Reference Antoni2009). The resulting 100 blocks provide the statistical convergence required for the subsequent CS-SPOD eigen-decomposition.

Figure 8.

(a) The CS-SPOD energy spectrum of circular DIJs, and (b) CS-SPOD eigenvalues at γ _St = 0, normalised by the total energy at that frequency set.

Figure 9.

Real component of the CS-SPOD modes of circular DIJs (a) at four dominant modes with γ _St = 0, and (b) with four different frequency deviation γ _St of mode 1.

Figure 9. Long description

Panel A: Four heat maps showing the real component of CS-SPOD modes of circular DIJs at four dominant modes with St = 0. Each heat map is labeled as Mode 1, Mode 2, Mode 3, and Mode 4. The x-axis and y-axis are labeled as x/D and y/D, respectively. The color scale ranges from -1 to 1, with darker colors indicating lower values and lighter colors indicating higher values. Panel B: Four heat maps showing the real component of CS-SPOD modes of circular DIJs with four different frequency deviations St of mode 1. Each heat map is labeled with different values of St: 0.003, 0.02, 0.04, and 0.06. The x-axis and y-axis are labeled as x/D and y/D, respectively. The color scale ranges from -1 to 1, with darker colors indicating lower values and lighter colors indicating higher values.

Taking the circular DIJs as an example, we present their CS-SPOD energy spectrum and modal-energy distribution (figure 8) as a benchmark. Under the condition of h/D = 3 and M = 1.2, it is known that circular DIJs exhibit a strong pressure fluctuation peak at the fourth harmonic (n = 4), and consequently, a strong CS-SPOD energy peak is observed only at γ _St = 0 (figure 8 a). This indicates that the frequency set composed of the fundamental α ₀ and its harmonics nα ₀, $\Omega$ _{γSt = 0} = {α ₀, 2α ₀, 3α ₀, …}, dominates the energy content in the CS-SPOD decomposition. Here, γ _St denotes the non-dimensional frequency deviation γ from the fundamental and its harmonics, defined as γ _St = γD/U. In this study, the frequency resolution of γ _St is (α ₀/40) × D/U ≈ 3 × 10⁻³. Within the frequency set $\Omega$ _{γSt = 0}, the first two modal energies (λ _j/Σλ _j) account for 42.8 % and 17.6 % of the total energy, respectively. We then present the normalised real components of CS-SPOD modes for various mode numbers and frequency deviations, as shown in figure 9. Within $\Omega$ _{γSt = 0}, the coherent structures identified become increasingly fragmented as the mode number increases (figure 9 a). Notably, the first CS-SPOD mode at $\Omega$ _{γSt = 0} exhibits an annular-symmetric structure that closely resembles the acoustic source pattern obtained from COP analysis (figure 6 b). The leading CS-SPOD mode at $\Omega$ _{γSt = 0} yields a surface-pressure footprint that is spatially consistent with the high-COP regions extracted at the same tone frequency. This agreement provides a useful cross-check between two independent analyses of the same PSP–microphone dataset. While CS-SPOD captures the most spatially coherent and energetic organisations of the plate-pressure field at a given frequency, COP analysis identifies the effective source regions contributing to far-field radiation. The spatial consistency supports the use of the first CS-SPOD mode as a compact representation of the dominant wall-pressure organisation at the fundamental cycle frequency, which we use as a baseline for comparing rectangular and circular configurations. It is found that the influence of frequency deviation γ _St on mode identification is significantly greater than that of mode number. Since the circular DIJs exhibit a single-frequency acoustic peak (n = 4) under the current conditions, larger values of γ _St correspond to broadband aerodynamic noise components that deviate from the dominant tonal feature, manifesting as spatially incoherent structures on the impingement plate (figure 9 b). Based on these observations, the subsequent CS-SPOD analysis focuses on the first-order mode within each characteristic frequency set, as it consistently captures the most spatially coherent and energetically dominant structures associated with the acoustic sources of interest.

Figure 10.

(a) The CS-SPOD energy spectrum of rectangular DIJs, and (b) CS-SPOD eigenvalues at γ _St = 0 and 0.03, normalised by the total energy at that frequency set.

In figure 10, we present the CS-SPOD energy spectrum and modal eigenvalues of rectangular DIJs at two selected frequency deviations, γ _St = 0 and γ _St = 0.03. In contrast to the circular DIJs, the rectangular configuration exhibits three distinct energy peaks in the CS-SPOD spectrum: one at γ _St = 0 and two symmetric peaks at |γ _St | = 0.03. These peaks align well with the far-field acoustic peaks shown in figure 3, indicating that, in addition to the fundamental frequency set $\Omega$ _{γSt = 0}, the rectangular DIJs also exhibit a more energetic frequency set. Considering the symmetry of the spectrum, we focus our analysis on the positive branch $\Omega$ _{γSt = 0.03}. Comparing the energy contribution of the first two modes (figure 10 b), we observe that, for $\Omega$ _{γSt = 0}, the first and second modes account for 29.4 % and 17.1 % of the total energy, respectively. These values are significantly lower than those of the circular DIJs, suggesting reduced mode dominance and potentially limiting the clarity of coherent structure identification. In contrast, for the frequency set $\Omega$ _{γSt = 0.03}, the first mode dominates with an energy fraction of 47.3 %, while the second mode only accounts for 11.6 %, indicating a much stronger and clearer leading coherent structure at this offset frequency. To verify the convergence of CS-SPOD eigenvalues for the model problem, we further examine the distribution and relative errors of the first 50 modes derived from pressure fluctuations on the impingement plate (figure 11). Here, the frequency deviation is fixed at γ _St = 0.03 to evaluate the effect of signal duration, t _end , on the resolution of CS-SPOD modal energy. As shown in figure 11(a), the modal-energy spectrum becomes stable when t _end reaches 40T ₀, indicating that this duration is sufficient to achieve convergence of the CS-SPOD eigenvalues. Using the modal-energy distribution at t _end = 80T ₀ as a reference, we quantify the relative errors of each mode at t _end = 5T ₀, 10T ₀, 20T ₀ and 40T ₀. As illustrated in figure 11(b), the relative errors of the first two modes (red and blue dots) and the average of the first five modes (white dots) are all below 5 % for t _end = 40T ₀, confirming the statistical convergence of the CS-SPOD results. To further assess the uncertainty associated with the modal-energy fractions at the dominant off-harmonic peak, we applied bootstrap and jackknife resampling to the analysis blocks (Efron & Stein Reference Efron and Stein1981; Benedict & Gould Reference Benedict and Gould1996), and the resulting 95 % confidence intervals are shown in figure 11(c). The two resampling procedures yield closely similar uncertainty ranges. In particular, the confidence interval of the first modal-energy fraction remains clearly separated from those of the lower modes, which supports the statistical significance of its dominant contribution despite the finite ensemble size and overlap of the analysis blocks.

Figure 11.

(a) The CS-SPOD modes’ energy of rectangular DIJs for multiple signal durations t _end , and (b) relative error for different CS-SPOD modes with different t _end, showing the convergence of CS-SPOD eigenvalues of the model problem. (c) Bootstrap and jackknife 95 % confidence intervals of the modal-energy fractions for the first five modes at γ _St = 0.03, showing the uncertainty associated with the finite ensemble of overlapping blocks.

Figure 11. Long description

Panel A: A line graph shows the CS-SPOD modes’ energy for rectangular dual-impinging jets over multiple signal durations. The x-axis represents the mode number, and the y-axis represents the normalized energy. The graph includes multiple lines indicating different signal durations, with annotations showing specific durations like 80T0, 40T0, 20T0, and 10T0. Panel B: A line graph illustrates the relative error for different CS-SPOD modes at various signal durations. The x-axis represents the signal duration, and the y-axis represents the error percentage. The graph includes lines for different modes, labeled as Mode 1, Mode 2, and Mode 3, along with a line for the mean error. Panel C: A box plot displays the 95 percent confidence intervals of the modal-energy fractions for the first five modes at St = 0.03. The x-axis represents the mode number, and the y-axis represents the normalized energy. The plot includes two sets of box plots, one for Bootstrap and one for Jackknife, with annotations indicating the different methods.

Figure 12.

(a) Evolution of annular-symmetric modes in rectangular DIJs at $\Omega$ _{γSt = 0}, and (b) normalised modal strength on centreline (x/D = 0), showing the convection velocities on the impingement plate.

Figure 12 presents the first-order CS-SPOD mode and the normalised modal strength along the centreline (x/D = 0) for the frequency set $\Omega$ _{γSt = 0}. At this frequency set, the dominant mode exhibits a classical annular-symmetric structure expanding around the nozzle exit. The jet column undergoes necking, converging into a central jet core, which then rapidly expands outward and re-forms into new cores downstream. In this study, two distinct modal peaks are consistently identified on the impingement plate in the outer region of the jet core (y/D < −1.75), as shown in figure 12(b), indicating the presence of a relatively stable double-layered annular structure. A representative convection velocity C _v of the annular-symmetric inner annular structure at $\Omega$ _{γSt = 0} along the radial direction was estimated from the phase-dependent shift of the peak in the normalised modal strength. Over the interval from 0 to π, the peak position varies approximately linearly with phase, while the interval from π to 2π exhibits the same trend with opposite sign. We therefore determined C _v from a linear fit of the peak position against phase over the first half-cycle, and used the second half-cycle as a symmetry check. The fitted slope gives C _v = 0.34D/0.25T = 0.57U ± 0.02U (95 % confidence interval). The 95 % confidence interval was obtained from the uncertainty of the fitted slope in the linear regression of peak position against phase. Additional estimates based on different phase intervals yielded closely similar values, indicating that the inferred convection velocity is not sensitive to the specific phase increment used. This representative value agrees well with the empirical prediction U _c = 0.60U for circular nozzles (equation (4.2)). The link between this radial propagation velocity and the axial convection velocity lies in the flow-acoustic resonance observed under lock-in conditions (Wei et al. Reference Wei, Zhang, Fan, Wang, Peng and Liu2024). Specifically, at characteristic frequencies where strong flow-acoustic coupling is present, pressure disturbances propagating along the impingement surface tend to synchronise with the convection of coherent structures in the jet shear layer, resulting in comparable velocities. This phenomenon has been confirmed by both numerical simulations (Gojon & Bogey Reference Gojon and Bogey2018) and experimental measurements (Davis et al. Reference Davis, Edstrand, Alvi, Cattafesta, Yorita and Asai2015). For comparison, we also present in figure 13 the real part of the first-order CS-SPOD mode for the circular DIJs at the same frequency set $\Omega$ _{γSt = 0}, from which the phase-dependent dynamics of the lower annular shear layer was extracted to compute C _v . The downstream propagation velocity along the y-direction was found to be 0.60U, in close agreement with the empirical value.

Figure 13.

(a) Evolution of annular-symmetric modes in circular DIJs at $\Omega$ _{γSt = 0}, and (b) phase-dependent dynamics of the lower annular structure’s shear layer.

Figure 13. Long description

Panel A: This panel contains four heatmap plots showing the evolution of annular-symmetric modes in circular dual-impinging jets at Strouhal number 0. Each heatmap is labeled with a different angle (theta = π/4, π/2, 3π/4, π) and shows the normalized variable Φ^P/||Φ^P||_max. The x and y axes are labeled x/D and y/D, respectively, where D is the diameter. The color scale ranges from -1 to 1, indicating the intensity of the modes. Panel B: This panel features a scatter plot with a fitting curve. The x-axis is labeled Phase and ranges from 0 to π, while the y-axis is labeled y-position and ranges from 0 to 0.6. The scatter plot shows experimental data points with error bars, and the fitting curve represents the modeled relationship between phase and y-position.

This agreement indicates that the annular-symmetric response of the circular DIJs follows the canonical convective timing of the classical single-loop feedback branch. It also provides a useful reference for examining whether the rectangular-specific peak at St = 0.42 can be interpreted as a shifted harmonic caused by a different convection velocity. Figure 14 therefore evaluates the sensitivity of the Powell-predicted harmonic locations to the assumed value of U _c /U. Within the propagation-velocity range inferred from the rectangular annular-symmetric wall-pressure branch, C _v /U = 0.57 ± 0.02, the fourth harmonic remains above the measured St = 0.42 peak. Even at the lower bound of this range, U _c /U = 0.55, the distance between St = 0.42 and the nearest integer harmonic remains approximately 0.022. Exact alignment with the fourth harmonic would require U _c /U = 0.506, which lies below the range inferred from the measured annular-symmetric branch. Thus, changing U _c /U within the present mode-based reference range does not move the St = 0.42 peak into natural alignment with an integer harmonic. This supports the interpretation that the peak is not simply a shifted harmonic of the annular-symmetric feedback branch. The circular case therefore provides a useful baseline for judging whether the rectangular-specific peak reflects a modified harmonic response or the appearance of a different modal organisation. This interpretation is consistent with previous supersonic impinging-jet studies showing that symmetric and antisymmetric oscillation modes can occupy distinct tonal branches under resonant conditions. Notably, the first-order mode for the circular DIJs also exhibits a clear annular-symmetric structure under this frequency set. This observation suggests that, under ideal conditions, once a closed acoustic feedback loop is established (originating at the nozzle exit and reflecting off the impingement plate), jets exhibiting annular-symmetric impingement structures tend to lock onto the fundamental frequency and its integer harmonics. Consequently, both rectangular and circular DIJs that produce similar annular-symmetric impingement structures and coherent modal propagation paths can exhibit pronounced frequency responses at integer multiples in the $\Omega$ _{γSt = 0} set.

Figure 14.

Sensitivity of the harmonic alignment of the rectangular DIJ peak at St = 0.42 to the convection velocity U _c /U. (a) Powell-predicted harmonic Strouhal numbers for n = 1–4, evaluated at p = 0. The brown horizontal line denotes the measured rectangular-DIJ peak at St = 0.42. The red shaded band indicates the propagation-velocity range inferred from the annular-symmetric wall-pressure branch in rectangular DIJs, C _v /U = 0.57 ± 0.02. The blue vertical line denotes U _c /U = 0.60, which is close to the circular-DIJ value and the empirical estimate from equation (4.2). (b) Minimum distance between St = 0.42 and the nearest integer harmonic. Matching St = 0.42 to the nearest harmonic requires U _c /U = 0.506, while at U _c /U = 0.55, the lower bound of the rectangular-DIJ range, the remaining distance is approximately 0.022.

By comparing the common peak intensity at n = 4 for rectangular and circular DIJs in figure 3(a), it is evident that the peak amplitude is significantly attenuated for the rectangular nozzles. Given that both nozzle types exhibit nearly identical mean impingement intensity and convection velocity along the y-direction, this attenuation can be attributed to the reduced energy contribution within the frequency set $\Omega$ _{γSt = 0} and a weakened interference region. As shown in figure 10(a), the frequency set $\Omega$ _{γSt = 0} carries a lower proportion of total modal energy in the case of rectangular DIJs. This suggests that the annular-symmetric feedback loop, which dominates the acoustic resonance mechanism in circular DIJs, is not the primary contributor in the rectangular configuration. To illustrate this, a schematic of the simplified annular modes for both rectangular and circular DIJs is provided in figure 15(a). In rectangular configuration, the two nozzles are identically oriented, with their major axes parallel to the plate y-axis and their centres aligned along the same axis. This configuration alters the resulting annular-symmetric structure from a circular to an elliptical shape. Furthermore, the apparent transport speed of this annular-symmetric footprint along the plate x-direction is reduced to approximately 0.47U, which spatially confines the effective interference region. Despite this, both the rectangular and circular DIJs exhibit phase-dependent variations of normalised modal strength under the nozzle projection area on the impingement plate, which follow a regular sinusoidal trend, as shown in figure 15(b). This reflects a similar periodic jet behaviour in both configurations. However, the first-order mode energy contribution under $\Omega$ _{γSt = 0} is significantly lower in rectangular DIJs, which compromises the coherence and integrity of the annular structure. As a result, the interference between the two jets is weakened. This observation is supported by the distribution of the acoustic source structures on the impingement plate, as shown in figure 7, where the source intensity in the interference region is notably weaker for rectangular DIJs. Consequently, despite exhibiting a similar resonance mechanism, the lower modal energy and diminished interference strength in rectangular DIJs result in a weaker acoustic peak at the shared characteristic frequency (n = 4).

Figure 15.

(a) Schematics of simplified annular-symmetric modes for rectangular and circular DIJs, and (b) phase-dependent variations of mean modal strength under the nozzle projection area on the impingement plate.

At γ _St = 0.03, the evolution of modal structures in rectangular DIJs reveals deviations in the dominant acoustic peak frequency, suggesting a distinct flow–acoustic coupling mechanism compared with the circular configuration (figure 13). At the strongest peak frequency (St = 0.42), the modal structure on the impingement plate exhibits a clear transition between rectangular and annular patterns (figure 16 a). Drawing upon the spatial mode characteristics reported in studies of free jets (Rodríguez, Jotkar & Gennaro Reference Rodríguez, Jotkar and Gennaro2018; Rodríguez et al. Reference Rodríguez, Stavropoulos, Nogueira, Edgington-Mitchell and Jordan2023; Mazharmanesh et al. Reference Mazharmanesh, Nogueira, Weightman and Edgington-Mitchell2025), we refer to the corresponding leading wall-pressure organisation at this frequency set as a flapping-antisymmetric mode with four stages. In rectangular DIJs, this mode is associated with a geometry-conditioned antisymmetric organisation of the jet response. This interpretation is also compatible with recent free-jet results showing that supersonic rectangular jets can support a flapping-type branch when the downstream shock/shear-layer organisation departs from the nozzle-exit symmetry (Mazharmanesh et al. Reference Mazharmanesh, Nogueira, Weightman and Edgington-Mitchell2025). The coupled impingement process then selects and amplifies its plate-pressure signature. The stage labels I–IV provide a compact description of the phase-dependent surface-pressure morphology of the leading CS-SPOD mode, and are used here for clarity of presentation. Unlike the annular-symmetric modes that radiate uniformly outward, this mode exhibits a spatially antisymmetric distribution with distinct dynamical stages. Figure 16(b) shows the mean modal strength extracted from the nozzle projection region on the impingement plate, capturing the essential features of this flapping-antisymmetric mode. At the zero-phase point (stage I, stage corresponding to the COP-identified source footprint), the coherent structure displays a bi-antisymmetric distribution with a central nodal line along the streamwise direction, and its spatial distribution is consistent with the COP distribution discussed in § 4.2. Energy concentrates along the two longer edges of the rectangular footprint, while the shorter edges exhibit localised instabilities. This configuration likely reflects the influence of shear-layer perturbations originating from the short axis, as observed in prior studies (Karnam et al. Reference Karnam, Baier, Gutmark, Jeun, Wu and Lele2021). As the phase advances (stage II, transition stage), the surface-pressure footprint evolves from a bi-antisymmetric rectangular pattern toward an annular-like distribution, accompanied by a gradual weakening of the long-edge-dominated features. By phase = π/2, the rectangular signature has nearly vanished, giving way to a hollow, annular-like footprint. In the subsequent phase interval (stage III, recovery stage), a central high-amplitude patch reappears in the plate-pressure field, and the overall footprint regains a more rectangular character. In the final stage (stage IV, perturbation stage), the plate-pressure footprint shows renewed localisation near the short-axis edges, consistent with the re-emergence of antisymmetric flapping signatures on the surface. During this final stage, the mean modal strength remains relatively unchanged, yielding a configuration that persists over a finite phase interval. This four-stage evolution recurs twice within a single CS-SPOD cycle, with the 180°–360° phase interval mirroring the 0°–180° development, thereby forming a complete cycle of the flapping-antisymmetric mode. The modal behaviour described above is consistent with a geometry-conditioned antisymmetric organisation of the rectangular jets. This interpretation is supported by three observations from the present data. First, the COP footprint at St = 0.42 is concentrated near the nozzle-edge regions rather than in the centre interference region, as shown in figure 7(c). Second, the dominant frequencies vary across the plate, as shown in figure 5(b), which indicates that the strongest non-harmonic response is confined to specific jet–wall interaction zones. Third, the leading mode at $\Omega$ _{γSt = 0.03} is much more dominant than that at $\Omega$ _{γSt = 0}, as shown in figure 10(b). Notably, while bi-axial flapping modes reported in free rectangular jets are characterised by transverse anti-symmetry and streamwise periodicity (Mazharmanesh et al. Reference Mazharmanesh, Nogueira, Weightman and Edgington-Mitchell2025), the impingement configuration introduces additional constraints and mode interactions. These results suggest that the dominant antisymmetric branch is favoured by the rectangular geometry, while the coupled impingement process selects and strengthens its plate-pressure signature. In this study, the term flapping-antisymmetric mode refers to the leading wall-pressure organisation associated with this frequency set under impinging and coupled conditions.

Figure 16.

(a) Evolution of flapping-antisymmetric mode at $\Omega$ _{γSt = 0.03}, (b) phase-dependent variations of mean modal strength and (c) schematic of simplified flapping modes for rectangular DIJs.

4.4. Co-resonance amplification in rectangular DIJs

To quantify the influence of twin-jet coupling on the tonal response of rectangular DIJs, we employ the co-resonance framework of Stahl et al. (Reference Stahl, Gaitonde, Bhargav and Alvi2022). In the DIJs, in addition to the feedback loops formed individually by each jet, the acoustic propagation paths between the two jets establish a broader global feedback loop (G_DIJ ) (figure 17 a). The interaction between the global DIJ loop (G_DIJ ) and the single-jet loop (S_DIJ = f ₁) provides a kinematic route for constructive or destructive interference, depending on their phase alignment. Following Stahl’s framework (2022), the co-resonance factor R_DIJ (0–1) is defined to quantify the degree of phase matching between the single-jet and global loops

(4.6) \begin{align}& N=n\frac{S_{\textit{DIJ}}}{G_{\textit{DIJ}}}=n\boldsymbol{\cdot }f_{1}\boldsymbol{\cdot }\left(\frac{2h}{U_{c}}+\frac{2\sqrt{h^{2}+S^{2}}}{c_{a}}\right)\! ,\end{align}

(4.7) \begin{align}&\qquad\quad R_{\textit{DIJ}}=1-2\left| N-\text{round}(N)\right|\! , \end{align}

where n denotes the integer coupling state associated with the two-loop phase relation, and S/D and h/D denote the nozzle spacing and impingement distance, respectively. A larger R_DIJ indicates a stronger potential for constructive two-loop phasing at a given frequency, whereas a smaller value suggests a tendency toward cancellation. As shown in figure 17(b), R_DIJ varies quasi-periodically with the Strouhal number St, highlighting the frequency-selective nature of two-loop phasing. Peaks observed at frequencies with high R_DIJ are therefore consistent with constructive coupling. For the rectangular DIJs, this interpretation is especially relevant to the non-harmonic peak at St = 0.42. As shown in §§ 4.2 and 4.3, the modal identity of this peak is inferred from the COP footprint and the CS-SPOD wall-pressure mode, rather than from the co-resonance model itself. The role of R_DIJ is therefore to assess whether this observed flapping-antisymmetric response is favoured by the phase relation between the single-jet and global dual-jet loops. The peak near St = 0.42 falls close to a strong co-resonant phase-matching condition, which is consistent with its large amplitude in the dual-jet configuration. Stahl et al. (Reference Stahl, Gaitonde, Bhargav and Alvi2022) noted that the integer index can, in principle, take any positive value, while practical interpretations often emphasise integer-multiple coupling as a phase-delay-consistent reference (Jeun, Wu & Lele Reference Jeun, Wu and Lele2021). In the present rectangular DIJs, the flapping-antisymmetric mode identified suggests that geometry-conditioned mode selection can shift the dominant peak away from the classical integer-harmonic sequence. We therefore interpret RDIJ as a complementary phase-consistency measure for the observed peaks. The framework is not used as a stand-alone predictor of modal existence, but as a test of whether an observed peak is favoured or suppressed by dual-jet phase coupling.

Figure 17.

(a) Schematic of co-resonance feedback mechanism and (b) the co-resonance factor R_DIJ for DIJs.

Figure 17. Long description

Panel A: A schematic diagram of the co-resonance feedback mechanism. It shows an upstream process and a downstream process with dual-nozzles and an impinging plate. The diagram includes labels for the upstream process, downstream process, dual-nozzles, impinging plate, and various arrows indicating the direction of processes and feedback loops. Panel B: A line graph showing the co-resonance factor RDIJ for dual-impinging jets. The x-axis is labeled St and the y-axis is labeled RDIJ. The graph highlights regions of strong instabilities and shows lines representing constructive and destructive interference.

Figure 18.

Acoustic spectrum for SIJs and DIJs of (a) rectangular and (b) circular nozzles to show the two-nozzle coupling effect.

Figure 19.

Schematic diagram illustrating the main aeroacoustic instabilities induced by rectangular dual-jets’ interference effect.

Figure 19. Long description

Panel A: The initial nozzle modulation effect shows the transition from a circular to a rectangular nozzle. The mean pressure and coefficient of pressure distributions are displayed for both configurations. Panel B: The dual-jets interference effect is depicted with different values of n and R DIJ. Constructive interference at n equals 3.7 and R DIJ equals 0.92 enhances instabilities with a SPL increase of 13.4 dB. At n equals 4 and R DIJ equals 0.39, the interference has minimal impact on the SPL, showing a 3.3 dB increase. Destructive interference at n equals 2 and R DIJ equals 0.19 weakens instabilities with a SPL decrease of 7.9 dB. Panel C: The downstream wall blockage effect illustrates different modes: flapping-antisymmetric mode, annular-symmetric mode, and weakened mode. The weakened mode shows suppressed peaks in the dual-jets.

We compared the acoustic spectra of SIJ (black line) and DIJs (red line) for both rectangular and circular nozzles to illustrate the coupling effects between the two nozzles (figure 18). The black dashed lines indicate the integer-multiple relationship of the classic feedback-loop frequencies. At n = 2, both rectangular and circular DIJs show a noticeable reduction in tonal amplitude, with values of 7.9 and 7.6 dB, respectively. The co-resonance factor at this frequency is the same for both cases, with R_DIJ = 0.19, indicating that the coupling between the jets suppresses the acoustic peak at this position. At n = 4, R_DIJ = 0.39, suggesting that the interference between the twin jets has minimal impact on the tonal amplitude, with respective increases of 3.3 and 0.6 dB for the circular and rectangular DIJs. When comparing the SIJ, it is evident that the rectangular SIJ exhibits a weaker tonal amplitude near n = 4, implying a lower degree of mode amplification in each rectangular nozzle. The rectangular-specific peak near n = 3.6 coincides with a high R_DIJ , indicating favourable two-jet phasing for this mode. The strong tonal intensity near n = 3.6 is therefore interpreted as the result of a rectangular-geometry-favoured antisymmetric branch that is further amplified by favourable dual-jet phasing, with a total increase of 13.4 dB. The key findings regarding the role of dual-jet interference in the observed tones are (figure 19): (i) the rectangular nozzle geometry favours an antisymmetric hydrodynamic organisation that is absent from the circular baseline. In the dual-nozzle configuration, jet–plate feedback and favourable two-jet phasing strengthen this branch, so that its plate-pressure signature becomes a prominent tonal response near n = 3.6. The resulting peak reflects both geometry-conditioned modal preference and constructive co-resonant coupling. The high value of R_DIJ therefore rationalises why this branch is strongly amplified in the dual-jet configuration, even though the branch itself is selected by the rectangular-nozzle geometry rather than predicted uniquely by the two-loop model. The convection-velocity estimates and the sensitivity check in figure 14 are most directly relevant to the annular-symmetric branch at $\Omega$ _{γSt = 0}, for which both the circular and rectangular DIJs exhibit comparable representative propagation speeds. By contrast, the strongest rectangular-specific peak at St = 0.42 is associated with the flapping-antisymmetric mode at $\Omega$ _{γSt = 0.03}, whose phase-dependent wall-pressure organisation does not admit a single monotonic transport path comparable to that of the annular mode. We therefore do not interpret the St = 0.42 peak as arising from a modified convection velocity within the classical harmonic branch. Instead, it is more consistently understood as the manifestation of a different modal branch, selected by the rectangular geometry and strengthened by coupled impingement and favourable two-loop phasing. (ii) For the annular-symmetric mode-dominated acoustic peaks, when the value of R_DIJ is small, the interference effect of the dual jets is limited or, due to destructive interference, can even suppress aeroacoustic instability. In this case, the dominant modes formed by the upper and lower nozzles remain relatively independent, and their energy contributions are lower. (iii) The coupling effects from the dominant modes of the dual jets transition from strong to weak as follows: rectangular nozzle’s flapping-antisymmetric mode, circular nozzle’s annular-symmetric mode and rectangular nozzle’s annular-symmetric mode.

This interpretation is based on the measured wall-pressure modes and the acoustic spectra, rather than on a separate reconstruction of the free-jet development upstream of the plate. The present results also suggest a broader qualitative interpretation of the rectangular-nozzle effect beyond the specific operating point used for the detailed modal analysis. Within the parameter range examined in figure 3(c), the rectangular DIJs more frequently exhibit dominant peaks that deviate from the classical integer-harmonic feedback lines, whereas the circular DIJs remain more closely aligned with the harmonic prediction. In the present framework, these off-harmonic peaks are associated with the flapping-antisymmetric branch identified in § 4.3, which modifies the classical feedback organisation and gives rise to a rectangular-specific tonal response, most prominently near n = 3.6. This tendency should not be interpreted as a universal frequency band detached from operating conditions. Rather, it reflects the modal-selection outcome produced jointly by nozzle geometry, impingement distance and twin-jet coupling. The roles of S/D and h/D can be interpreted directly through the co-resonance model. In this formulation, both parameters enter the phase-matching quantity N, and therefore affect the co-resonance factor R _DIJ . Changes in h/D alter not only the local jet–plate interaction time but also the relative weighting of the downstream convective path and the upstream acoustic return path. Changes in S/D modify the global dual-jet propagation path and the phase relation between the single-jet and global loops. As a result, variations in either parameter are expected to shift the degree of phase matching for the flapping-antisymmetric branch, and hence its relative prominence, rather than merely translating the acoustic peaks in a kinematic manner. This interpretation is consistent with previous studies showing that interaction strength in dual impingement reorganises both the near-wall-pressure field and the acoustic response (Bhargav et al. Reference Bhargav, Mehta, Kumar and S. Alvi2022; Mehta et al. Reference Mehta, Natarajan, Sellappan, Gustavsson and Kumar2022). The sensitivity to nozzle AR is expected to follow the same geometric logic. The nozzle AR of 2 : 1 is a representative rectangular-jet configuration, and the associated flapping-antisymmetric response is consistent with the broader tendency of non-axisymmetric jets to favour geometry-conditioned flapping-type branches. As the nozzle departs further from axisymmetry, the asymmetry between the major- and minor-axis shear-layer development is expected to become more influential, which would favour a stronger departure from the circular-like annular response (Gojon et al. Reference Gojon, Gutmark and Mihaescu2019; Mazharmanesh et al. Reference Mazharmanesh, Nogueira, Weightman and Edgington-Mitchell2025). In this sense, higher-AR rectangular jets may be expected to exhibit a more pronounced redistribution of modal dominance. The planar jet may be viewed as an idealised limiting case of this trend, for which flapping-type organisation is especially prominent (Thomas & Goldschmidt Reference Thomas and Goldschmidt1986; Matsubara et al. Reference Matsubara, Alfredsson and Segalini2020). These results therefore suggest that the observed rectangular-specific tone is not an isolated anomaly of the chosen configuration, but part of a broader geometry-conditioned shift in modal preference in high-speed DIJs.