1 Introduction

The need to reduce fuel consumption and emissions has been an important driver for the development of ultra-high bypass ratio (UHBR) aeroengines (Hugues 2011). Moreover, since a larger diameter engine also produces less jet noise at constant thrust, the size of aeroengines never stopped increasing. This has reduced the distance between the jet and the wing, since the ground clearance must be preserved. We have now reached the point where significant jet-wing installation effects are impacting both the aerodynamic performance (Otter et al. 2019) and noise generation (Lawrence and Self 2015). The approach phase is particularly crucial as the fully-deployed flap has its trailing edge close to the upper shear layer of the aeroengine exhaust jet. This paper is considering jet-wing installation effects corresponding to the aeroacoustic interaction of a jet with the trailing edge of a flat plate.

Large Reynolds number jets contain a broad range of turbulent scales, associated with a broadband noise that can be strongly reinforced by the presence of the wing and flap. Different installation effects can be distinguished (Lawrence et al. 2011), depending on the position of the flap: in the non-linear (vortical) part of the hydrodynamic field, in the linear (potential) hydrodynamic near-field, or in the acoustic field. According to Ref. Arndt et al. (1997), the dividing line between the hydrodynamic and acoustic fields lies around \(k \, r = 2\), with \(k = \omega / c_0\) the wavenumber of the pressure fluctuation calculated from the angular frequency and speed of sound, and r is the radial distance from the center of mixing layer (roughly aligned with the nozzle lip line). Thus, the relative position of the flap with respect to the jet will have important implications for the scattering of noise since the decay of the pressure fluctuations, the phase velocity of their trace on the wall and their spatial coherence are quite different in the hydrodynamic and acoustic regions, especially at low Mach numbers.

In addition to the broadband noise, experiments conducted at different scales have revealed the presence of multiple anharmonic tones when the flap is placed in the hydrodynamic region, both in the near-field pressure over the flap surface and in the acoustic far-field (Lawrence and Self 2015). The tones are the result of a resonant loop, involving a primary noise generation mechanism and a feedback path. It is commonly understood that the primary noise generation is associated with the interaction of the convectively unstable Kelvin–Helmholtz vortical wave with the flap trailing edge (Jordan et al. 2018). Concerning the feedback from the trailing edge to the nozzle lip, a scenario involving free-stream sound waves propagating outside the jet column was initially proposed (Lawrence and Self 2015; Umeda et al. 1987), but more recent work suggested that another set of upstream-propagating trapped waves, developing in the jet column, close the loop (Towne et al. 2017; Schmidt et al. 2017; Jordan et al. 2018). The occurrence of such resonant tones has been explored experimentally in this work.

Noise reduction techniques have been thoroughly investigated in literature for isolated jets (Casalino et al. 2008) using for example tabs or chevrons (Zaman et al. 2011), asymmetrical (Upadhyay et al. 2017; Bridges 2012) or beveled nozzles (Bridges et al. 2015; Viswanathan et al. 2006). These techniques have shown promising noise reduction potential for isolated jets, some of them have found their way to production engines and have effectively reduced aircraft noise during take-off, when the jet constitutes a significant contribution to the noise footprint. But it remains unclear how these noise mitigation technologies affect the noise produced by the jet when it interacts with the airframe, given the different noise mechanisms that are involved.

Some mitigation techniques were proven to be effective in installed conditions. Chevrons, which were initially developed for jet noise reduction irrespectively of installation effects, were shown to reduce the noise of installed jets as well, albeit with smaller gains compared to isolated conditions (Kamliya Jawahar et al. 2021; Mengle et al. 2006; Wang et al. 2021). Furthermore, the efficiency of chevrons depends on the strength of the jet-wing interaction (Bastos et al. 2017). For weak interactions, the wing is essentially an acoustic scattering surface and the noise reduction that is obtained over a broad frequency range is similar to that observed for free jets. But once the wing interacts with the hydrodynamic region of the jet (strong interaction), it plays also the role of an equivalent source and the benefit of chevrons becomes almost negligible at low frequencies (Bastos et al. 2017; Kamliya Jawahar et al. 2021), while the noise regeneration at high frequency might increase even more than for isolated jets (Bastos et al. 2017). Likewise, zigzag vortex generators (Proenca and Lawrence 2022a) implemented inside a nozzle showed up to 3 dB noise reduction on the overall sound pressure level for installed configurations, while a noise regeneration is observed at high frequencies due to the increased wing-reflected jet-mixing noise.

Besides, some nozzle alterations that reduce the noise of isolated jets don’t bring any benefit once installed. This was shown for lobed nozzles (Lyu and Dowling 2019), for example. Relatively few studies reported a mitigation of installed jet noise by a modification of the nozzle outlet shape. As an example, oval nozzles were reported to be noisier than round nozzles, except when the ellipse minor axis is aligned with the wing trailing edge (Proenca and Lawrence 2022b). The noise mitigation was explained to be related to a decrease of the surface of the wing that is wetted by the jet flow.

Recent efforts have been directed at the modification of the trailing edge of the surface interacting with the jet instead. Porous media inserted in the trailing edge can yield up to 10 dB noise reduction when it is located in the hydrodynamic region of the jet (Rego et al. 2021; Jente 2023; Kamliya Jawahar et al. 2023; Kamliya Jawahar and Azarpeyvand 2021). Acoustic liners on the flap (Rego et al. 2022) showed also noise reductions in the order of 7 dB and 10 dB on the far-field spectra, at the resonant frequency of single and double degree-of-freedom liners, respectively. A flexible trailing edge (Mancinelli et al. 2022) yields also a decrease of the installed jet noise by about 2 dB on the overall sound pressure level. However, the airworthiness and certification potential of those structural alterations of the flap remain open issues.

The present paper investigates global shape modifications of the nozzles and the corresponding produced noise in isolated and installed configurations. Both beveled and rectangular nozzles, with different aspect ratios, are considered. Section 2 first introduces the main parameters of the problem, the corresponding lab-scale facility and the nozzle designs, as well as the commissioning of the setup. The fundamental installed jet noise mechanisms are explored in Sect. 3 by varying the axial and radial distance of the plate relative to the jet. The mitigation of the installed jet noise by rectangular and beveled nozzles is explored in Sect. 4. A final discussion of the results is given in Sect. 5.

2 Experimental Campaign

The installed jet configuration is defined by the parameters shown in Fig. 1. The nozzle outlet inner diameter is D, the Mach number is \(M_a=U_j/c_0\), where \(U_j\) is the jet velocity and \(c_0\) is the reference speed of sound. A solid surface with chord C has its trailing edge at a distance L from the nozzle outlet plane and a crosswise distance H from the jet centre-line. The relative positioning of the plate, defined by the main parameters H/D and L/D, has been varied to investigate different flow and/or acoustic interactions.

Fig. 1
figure 1

Sketch of relevant parameters in a jet installed configuration and corresponding investigated plate positions

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2.1 Experimental Facility

The investigation has been carried out in the JAFAAR facility (Jet Aeroacoustic Facility for Aeronautical and Aerospace Research) of the von Kármán Institute for Fluid Dynamics, shown in Fig. 2. The facility is operated with a 40 bar pressure network that allows a maximum mass flow rate of 3 kg/s. The flow is quietened by means of a silencer that also minimizes the noise produced by the pressure lines, in which up to 4 bar of stagnation pressure can be reached. Further details on the design of the silencer are provided in Ref. Guariglia et al. (2018). The jet discharges in an anechoic room with dimensions 4 \(\times\) 3 \(\times\) 4 m\(^3\) and with a cut-off frequency of about 200 Hz. Grids (shown at the bottom of Fig. 2a) are located within the upstream pipe to homogenize the flow and decrease the free stream turbulence intensity. A duct contraction with an area ratio equal to 36, based on a 7th order polynomial, reduces the risk of separation while favouring boundary layer relaminarization (Schram 2003). The operation of the jet flow is monitored measuring the stagnation pressure and temperature before the nozzle contraction. The outlet jet velocity \(U_j\) is computed through isentropic relations.

Fig. 2
figure 2

a Implemented nozzle and b anechoic room and microphone antenna

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Different nozzle heads manufactured by rapid prototyping (presented in Sect. 2.2) are fitted to the end of the contraction. The outlet diameter of the baseline circular nozzle is \(D=0.05\) m. Squeezed between the contraction and the nozzle head, an exchangeable tripping ring ensures a turbulent boundary layer at the nozzle outlet. Further information relative to the tripping geometry and its corresponding effect on the radiated noise are available in the “Appendix B”.

The solid surface figuring the airframe is a flat plate of thickness 10 mm, with a slanted angle of 30\(^\circ\) and a remaining blunt trailing edge of 1 mm thickness. The plate has a chord of \(C/D=3\) (\(C=0.15\) m) and a span of 18D (\(D=0.9\) m). It has been verified in previous studies (Kucukcoskun et al. 2013) that this span length is large enough to make the scattering by the plate side-edges negligible compared with the primary source mechanisms. The plate is mounted on a carriage seen in Fig. 2a and recessed from the support to minimize the possible acoustic interactions. This 3-axis carriage system allows a plate positioning in the range of \(0.62 \le H/D \le 1\) and \(1.8 \le L/D \le 2.6\). The corresponding positions are illustrated in Fig. 1. The intermediate plate position \((H/D, L/D) = (0.62, 2.0)\) , corresponding to \((H, L) = (31\,\textrm{mm}, 100\,\textrm{mm})\), is taken as reference for the results presented in Sect. 4 and shown in grey in Fig. 1. For that reference position, the plate leading edge is located 1D (50 mm) upstream of the nozzle lip.

A microphone antenna, shown in Fig. 2b, is equipped with 19 Bruel and Kjaer 1/4 inch microphones with a spatial resolution of 5\(^\circ\), covering polar angles between 30\(^\circ\) and 115\(^\circ\). The antenna is fixed on a motorized carriage that allows an azimuthal rotation of the polar arc around the nozzle axis, covering a range of 90\(^\circ\) from the plate mid-span plane to the plate plane. The microphones are located at 20D (1 m) from the jet center. The definition of the \((\theta , \phi )\) angles are illustrated in Fig. 3a and the directivity mesh of the considered observers is shown in Fig. 3b together with the angle bounds covered by the antenna. A low pass filter of 100 kHz and an acquisition frequency of 250 kHz are used in order to reach \(St=fD/U_j=10\) for \(M_a=0.9\). For such large frequencies, the microphone measurements are corrected by the calibration curves provided by the manufacturer, assuming a zero-angle incidence of the incoming acoustic waves. The antenna structure is covered with melamine foam in order to minimize possible acoustic reflections and make the support as transparent acoustically as possible, as verified in “Appendix  A”. The nozzle body and plate supporting structure have also been wrapped in acoustic absorbing foam to minimize acoustic reflections.

Fig. 3
figure 3

a Definition of angular positions of the listener in the rear and side planes and b radiation map covered by the microphone antenna

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The microphone signals have been recorded for 10 s and corrected using a Bruel &Kjaer pistonphone type 4231 calibrator. The power spectral density (PSD) is obtained using the welch method, using windows of \(2^{11}\) points, multiplied by the Hanning function and a 50% overlap. The spectra presented in the result sections are by default normalised at 1 m distance, corresponding to the microphone setup default diameter. The following noise metrics have been calculated:

  • Sound Pressure Level:

    $$\begin{aligned} \text {SPL}[\text {dB}/St] = 10 \, \log _{10} \left( \frac{\text {PSD}(p')[\text {Pa}^2/\text {Hz}]}{p'_\text {ref}}\frac{U_j}{D}\right) , \end{aligned}$$

    with \(p'\) being the pressure time signal and \(p'_\text {ref}=20~ \upmu\)Pa. When indicated in the figures, a 1/3 octave band filtering may be applied to the PSD (Pierce 2019).

  • Sound Power Level:

    $$\begin{aligned} \text {PWL}[\text {dB}/St] = 10 \, \log _{10} \left( \frac{\int _S I\ \text {d}S}{P_\text {ref}}\right) , \end{aligned}$$

    where the surface of integration S is the area covered by the microphone antenna as illustrated in Fig. 3b, \(P_\text {ref}=10^{-12}\) W/m\(^2\) and

    $$\begin{aligned} I = \frac{\text {PSD}(p')[Pa^2/Hz]}{\rho _0 c_0}\frac{U_j}{D} \end{aligned}$$

    with \(\rho _0=1.225\) kg/m\(^3\), \(c_0=340\) m/s, being the density and speed of sound for air at 15\(^\circ\)C.

  • OverAll Sound Pressure/Power Level are obtained by integration across the frequency axis of the SPL/PWL, and denoted OASPL[dB] and OAPWL[dB] respectively.

  • \(\Delta\)SPL, \(\Delta\)PWL, \(\Delta\)OASPL and \(\Delta\)OAPWL are the difference between two spectra or overall values in dB.

2.2 Modified Nozzle Geometries

As mentioned in Sect. 1, the noise produced by installed jets depends on the plate location relative to the jet shear layer. The most direct way to achieve noise reduction is to have the plate trailing edge outside of the hydrodynamic region of the jet, or at least to reduce the span of the plate that is ‘wetted’ by this region. For a given distance between the jet axis and the plate, and fixed cross-section of the nozzle outlet, the two following approaches have been attempted:

  • Using beveled nozzles: the starting location of the shear layer on the plate side is pushed further downstream. As a result, depending on the side plate radial position H or downstream location L, the plate trailing edge can be located out of the shear layer or in an early development stage of the jet shear layer. In the present study, an arbitrary axial distance of 0.5D on the top nozzle lip has been used to design the beveled nozzles, resulting in a bevel angle of 26.57\(^\circ\). As the nozzle lip is extended in the parallel outlet section, a deviation of the jet plume, as observed in Kamliya Jawahar and Azarpeyvand (2022) and Shur et al. (2007) is not expected, preserving possibly a similar jet thrust.

  • Using rectangular nozzles: reducing the jet dimension perpendicular to the plate, allows to increase the distance between the jet shear layer and the plate. The design of the rectangular nozzles has been performed using super-ellipses, ensuring the same cross-sectional area and then the same mass flow. From the round nozzle to the rect-19 nozzle shown in Fig. 4, the major axis of 50 mm is kept similar, reducing then the minor axis, as reported in Table 1, and the radius of curvature at the corners. Going towards rect-15, the same radius of curvature of 2.7 mm as for rect-19 is used, the major axis is increased and the minor axis is adjusted to keep the same cross-section. The sizes of the major and minor axes and the corresponding aspect ratio are shown in Table 1. A 7th order polynomial is used in the streamwise direction to transition from the round to rectangular shape, following the same principles as for the main nozzle contraction.

Table 1 Tested nozzles
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Additional nozzles have also been tested, shown in Fig. 4. The radius of curvature of 2.7 mm of rect-19 has been reduced to the minimum allowable by the manufacturing process for rect-19r with a resulting radius of curvature of 0.2 mm. To preserve the same cross-sectional area at the nozzle outlet, the minor axis of rect-19r is reduced by 0.4 mm compared to rect-19, resulting in a slightly different aspect ratio of 1.27:1. rect-19b has the same shape as rect-19 adding a similar beleved angle than for round-b. Finally, the nozzle round-r is a mix of the round and rect-19 nozzles.

Fig. 4
figure 4

Tested nozzle heads

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2.3 Free Jet Noise Measurements

The noise emissions by the circular jet have been measured in absence of the plate, for Mach numbers ranging from 0.1 to 0.6 in order to verify the consistency of the measurements with literature. The Mach number scaling is applied in Fig. 5a, using different processing techniques and compared to the Mach number scaling law of \(M^{7.5}\) as reported in Bailly et al. (2016) and Juvé et al. (1980) for practical jet measurements or the theoretical scaling \(M^8\) as in Lighthill (1952). The use of the root-mean-square of the entire time signal is being polluted by the lowest frequencies, having large amplitudes but related to the anechoicity of the measurement room. When using a fixed integration band \(0.1 \le St \le 10\) or a single \(St=1\) number value, the scaling with the Mach number is seen to compare well with the theoretical value of \(M^{7.5}\) for Mach numbers above 0.3. At the low Mach number of 0.1, the acoustic results are in the background noise of the anechoic room and the microphone sensitivity. The corresponding spectra for Mach numbers above 0.3 are plotted in Fig. 5b with a \(M^{7.5}\) scaling of the sound intensity. A good collapse of the spectra over the frequencies considered is observed. This confirms that our jet flow can also be considered as free of transitional effects or spurious monopolar/dipolar sources. A similar scaling for installed jets has been addressed in literature (Kamliya Jawahar et al. 2023).

Fig. 5
figure 5

a Mach scaling law for an observer located at \((\theta , \phi )=(90, 180)\) for the round nozzle using different processing techniques: (blue) rms of the pressure signal, (red) spectral integration over \(0.1 \le St \le 10\) and (black) specific value of the spectrum at \(St=1\). (plain) \(M_a^{7.5}\) and (dash) \(M_a^8\) scaling of the sound intensity. b SPL using a \(M_a^{7.5}\) scaling of the sound intensity for a jet \(M_a\) of (plain) 0.3, (dash) 0.4, (dot) 0.5 and (dash-dot) 0.6

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Fig. 6
figure 6

Third-octave band SPL spectra of the noise radiated from round jets at \(M_a=0.6\) scaled for an observer located at 1 m distance and measured in different facilities: (plain) SJET model based on NASA SHJAR data, (dash) Doak Lab. from the University of Southampton and (dash-dot) JAFAAR Lab. from the von Kármán Institute. a \((\theta , \phi )=(118, 180)\), b \((\theta , \phi )=(96, 180)\) and c \((\theta , \phi )=(50, 180)\)

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A comparison of the present round nozzle results, in isolated configuration, are compared with results obtained in other facilities in Fig. 6. The nozzle diameter of 50 mm and the acoustic Mach number of \(M_a=0.6\) are similar for all cases. The first dataset is provided by the NASA sJet empirical jet noise prediction model (Khavaran and Bridges 2009a) built over a large set of experimental data collected at the NASA SHJAR facility (Khavaran and Bridges 2009b), as similarly used in Gryazev et al. (2022). It considers a far-field array of microphones located on an arc of 100D. The second experimental dataset is obtained from the Doak Laboratory of the University of Southampton, on a line microphone array located at 41.6D from the jet axis. All results are scaled to a 1 m receiver location and the operating conditions are corrected to match the International Standard Atmosphere (ISA) norm. In the case of the present measurements, for which the discharge room temperature and humidity are not available, values of 15\(^{\circ }\)C and 50% have been assumed respectively. As the jet noise sources are aligned and distributed along the jet axis, the microphone should be located far enough (at least 50D according to Koch et al. 2003) to consider the jet sources as a point source. To have the different antenna configurations comparable, including those located geometrically close to the source, a proper source point location has to be determined to correct the corresponding radiation angles. In the present case, a source point location on the jet axis and 7D downstream of the jet outlet has been selected, based on the documented source location results for unheated subsonic jets (Viswanathan 2007). As seen in Fig. 6, the present results compare well with the data of other facilities with a maximum deviation of about 1 dB at low frequencies, and for all radiation angles. The main discrepancies are observed for the rearward angle, possibly due to the repositioning of the point source and the facility-to-facility corrections.

3 Installed Jet Noise

The insertion of the flat plate in the hydrodynamic field of the jet is causing a significant noise increase, as shown in Fig. 7, where the radial distance between the jet axis and the plate has been varied for a fixed streamwise separation (a), and varying the streamwise separation for a fixed radial distance (b). Consistently with previous literature (Lawrence et al. 2011; Lawrence and Self 2015), different installation effects can be distinguished at low and high frequencies. At low frequencies, the plate interacts with the hydrodynamic pressure field of the plate, adding a broadband dipolar source corresponding to the reaction force exerted by the plate on the surrounding fluid, as explained by Curle (1955). Taking \(k r = 2\) as the limit between the hydrodynamic and the acoustic fields (Arndt et al. 1997), the hydrodynamic interaction should prevail up to Strouhal values equal to 1.3, 2.1 and 4.4, for \(H/D = 0.9\), 0.75 and 0.62, respectively. The results in Fig. 7 show indeed a much smaller increase of the noise emissions induced by the plate above those St values. At high frequencies, the plate is indeed immersed in the acoustic field of the jet, causing only a modest increase of the noise levels due to acoustic scattering (in this instance acoustic reflections since the microphones are located on the jet side, with respect to the plate). Acoustic reflections by a solid surface are generally more pronounced when the dimensions of the body make it acoustically non-compact. Taking the quarter-wavelength criterion for chordwise non-compactness (\(C > \lambda /4\)), we find that significant acoustic reflections should be observed for \(St>0.14\). This justifies the increase of a couple of decibels that is indeed observed in the high-frequency range.

Fig. 7
figure 7

Influence of the plate position on the SPL for an observer located at \((\theta , \phi )=(90, 180)\) for a \(L/D=2.6\) and varying radial distances of (dash) \(H/D=0.62\), (dot) \(H/D=0.75\), (dash-dot) \(H/D=0.9\) and b \(H/D=0.62\) and varying axial distances of (dash) \(L/D=2.6\), (dot) \(L/D=2.2\), (dash-dot) \(L/D=1.8\). (plain) Isolated jet

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Besides those effects, the presence of the plate is causing the appearance of tones, emerging above the broadband noise by more than 10 dB in some cases. This is again consistent with previous observations, e.g. made in Ref. Lawrence and Self (2015). Comparing Fig. 7a and b indicates that the frequency of the tones is dictated by the streamwise distance L/D between the jet outlet and the plate trailing-edge, consistently with the advection-propagation feedback mechanism that was introduced above. A change of the lateral distance H/D alters the maximum frequency covered by the tonal components: a deeper penetration of the plate in the hydrodynamic field of the jet is seen to induce tones extending to higher frequencies, the low-frequency tones being less affected.

In terms of directivity, the installation effects are seen in Fig. 8 to concern mostly the azimuthal angles normal to the plate, with a more reduced impact at grazing azimuthal angles.

Fig. 8
figure 8

Directivity of the variation of OASPL, integrated over \(0.3 \le St \le 10.0\), between isolated and installed jet configurations for varying H/D and L/D distances

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Having characterized the jet installation effects for the circular nozzle outlet, we will now examine how different nozzle outlet shapes can possibly lead to an attenuation of these adverse effects.

4 Mitigation of the Adverse Installation Effects with Non-circular Nozzles

4.1 \(\Delta \text {OASPL}\) directivity maps

The directivity maps shown in Fig. 9 provide a summary of the positive and negative effects brought by the modified nozzles shown in Fig. 4 above, respecting the same graphical layout. It displays the difference between the OASPL (integrated over the interval \(0.3 \le St \le 10.0\)) measured for the installed non-circular nozzles and the OASPL measured for the circular nozzle (round in Fig. 4) taken as reference.

Fig. 9
figure 9

Directivity of the variation of OASPL, integrated over \(0.3 \le St \le 10.0\), between round and modified nozzles for installed jet configurations at \(H/D=0.62\) and \(L/D=2.0\)

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4.1.1 Circular Versus Rectangular Nozzles

It appears that going from the round nozzle to the rect-22 nozzle induces larger noise emissions over the whole directivity map. As the same observation was made in absence of the plate, it can be concluded that the rect-22 nozzle is noisier than the baseline irrespectively of installation effects. It had been also observed by Ahuja (1990) that rectangular nozzles with aspect ratio \(AR=3.6\) can be slightly noisier than circular ones in subsonic conditions, especially for the radiation angles perpendicular to the long edge of the nozzle. Similar conclusions were obtained by Upadhyay et al. (2017) for a rectangular nozzle outlet with \(AR=4\), operated at \(M_a=0.9\). Despite its small aspect ratio of 1.12:1, the rect-22 nozzle features some corners that are likely to induce streamwise vortical structures.

These streamwise vortical structures, inducing an enhanced mixing of the jet, were already reported in Ref. Massey and Ahuja (2002) to increase the high-frequency noise emissions but to reduce the dominant noise emissions at lower frequencies that are associated with the large-scale wavepacket-like structures, radiating preferentially along the downstream radiation angles. Such noise reduction at the downstream radiation angles is however not visible for the rect-22 nozzle, which only exhibits a fairly uniform noise increase, suggesting that this aspect ratio is not sufficient to affect the large-scale coherent structures of the jet and accordingly reduce the low-frequency downstream emissions. In contrast, directivity maps obtained for the nozzles rect-19 (\(AR=1.26\)), rect-17 (\(AR=1.59\)) and rect-15 (\(AR=2\)) show the expected noise decrease at downstream polar radiation angles, but only for the azimuthal angles that are approaching the direction of the plate. Perpendicularly to the plate, the trend is reversed: increasing the aspect ratio decreases noise on the jet sideline, but increases it at downstream angles. It should be noted that these trends are mainly induced by the installation effects, the directivity of rectangular nozzles having been shown in former investigations to be very similar to that of circular nozzles, even for aspects ratios larger than tested in this work (Bridges 2012; Tam and Zaman 2000).

To explain the noise reduction, it should be first reminded that the long edge of the rectangular nozzle is parallel to the plate so that increasing the AR is also augmenting the distance between the shear layer and the plate, potentially beyond the \(k \, r =2\) limit for some high frequencies. (This will be examined in the spectra below.) The hydrodynamic interaction being associated with a dipolar source carried by the plate and radiating along its main normal, it seems sensible that increasing the AR is reducing the noise emitted along the azimuthal-polar angles where this dipole was dominant.

Note that this reasoning rests on the assumption that the long and short axes of the jet haven’t switched, as it is often observed for rectangular jets. However, based on the literature (Zaman 1996), it is expected that axis switching, if it would take place at all, would occur much more downstream than the location where the plate trailing edge is located in our experiments.

Comparing the contour plots of the rect-19 and rect-19r nozzles, it would seem that the sharper corners of the rect-19r nozzle are bringing a significant noise reduction at large azimuthal angles. This impression will be revised in Sect. 4.2 below, where we calculate the differences between the spectra.

4.1.2 Circular Versus Flattened Nozzle

Following the previous argumentation, one is tempted to reduce the hydrodynamic installation effects by flattening the contour of the nozzle on the side of the plate, in order to increase the distance between the turbulent pressure near-field and the plate. The \(\Delta \text {OASPL}\) map shown in Fig. 9 for the nozzles round-r confirms the pertinence of this approach, the noise reduction perpendicularly to the plate being of the same order of magnitude as with the rect-15 nozzle, but without the penalty of noise increase at shallow azimuthal angles.

4.1.3 Circular and Rectangular Nozzles: Non-beveled Versus Beveled

As introduced, beveled nozzles offer a potential for noise reduction in absence of installation effects. The directivity maps shown in Fig. 9 for the nozzles round-b and rect-19b indicate that a beveled nozzle geometry attenuates also substantially the adverse installation effects. The maximum noise reduction reaches about 2 dB perpendicularly to the plate, with nearly no polar-azimuthal angles showing a noise increase.

4.2 Narrowband Spectra and Third-Octave \(\Delta \text {SPL}\) Spectra

As a complement to the above analysis of the \(\Delta \text {OASLP}\) maps, we will now analyse the point spectra extracted at two locations: \((\theta ,\phi ) = (90, 180)^\circ\) and \((90, 270)^\circ\), i.e. at directions normal and parallel to the plate.

The next figures permit to compare the noise spectra (in \(\text {dB}/St\)) in isolated and installed conditions (top), and to estimate more precisely the deviations (in \(\Delta \text {dB}\)) between the levels taking as reference the round nozzle in isolated and installed configurations (bottom). The \(\Delta \text {dB}\) spectra have been integrated into third-octave bands to enhance readability. These figures also allow to discriminate between the jet self-noise in isolated configuration and the installation effects.

In the next two subsections, we restrict the analysis to the nozzles that showed the best potential in the \(\Delta \text {OASPL}\) directivity maps: the rectangular ones rect-19 and rect-15, and the beveled ones round-b and rect-19b.

4.2.1 Circular Versus Rectangular Nozzles

The results shown in Fig. 10 for the isolated jets confirm that changing the outlet shape from circular to rectangular increases its noise emissions, especially at high frequencies. The penalty reaches about 1 dB and 2.5 dB for the rect-19 and rect-15 nozzles, respectively, at a St of 10. Note that the rect-15 nozzle also exhibits a penalty at the lowest frequencies, around \(St = 0.1\). This trend, previously observed for rectangular nozzles, is amplified with the AR according to Bridges (2012). The jet self-noise also decreases with the azimuthal angle, especially for high frequencies, with less noise being radiated for observers located on the wide side of the rectangular nozzle. This effect is limited for the present aspect ratio (maximum 2:1) but is supposed to increase with the aspect ratio according to Bridges (2012).

This noise penalty is similarly observed for \(St>2\) once the nozzles are installed, showing that this effect is related to the jet self-noise. This is however largely over-compensated in the installed configurations, for the highest frequency tones associated with the hydrodynamic feedback mechanism in particular. The attenuation of those peaks reaches about 2.5 dB and 5 dB for the rect-19 and rect-15 nozzles, respectively. But once again, the rect-15 nozzle comes also with a low-frequency broadband penalty for \(St < 0.4\), that is even amplified once the jet is installed.

The above observations hold for both radiation directions in Fig. 10a and b, the benefits being logically much less important at grazing angles from the plate, where the hydrodynamic feedback tones are not as marked. At the grazing angles, the rectangular nozzles are essentially noisier than the circular ones.

Fig. 10
figure 10

Influence of rectangular nozzles for a plate positioning of \(H/D=0.62\) and \(L/D=2.0\) for observers located at a \((\theta , \phi )=(90, 180)\) and b \((\theta , \phi )=(90, 270)\). (top) Narrowband SPL of (black) round, (blue) rect19 and (red) rect15 nozzles, and (bottom) delta SPL of the 1/3 octave band spectra of rectangular nozzles relative to the round nozzle. (plain) isolated case, (dash) installed configurations

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Still considering the rectangular nozzles, the effect of the radius of curvature of the corners is shown in Fig. 11, comparing the results obtained for the nozzles rect-19 and rect-19r. While in isolated conditions, the difference is marginal and within the measurement uncertainty, a meaningful reduction is obtained for the installed case in the band \(0.3< St < 2\), for the radiation angle normal to the plate. The peak of attenuation is obtained for the hydrodynamic feedback tone having the highest frequency (for the St just above 1). It seems unlikely for this attenuation to be due to the quite small recess (0.4 mm) of the rect-19r nozzle side edge parallel to the plate. An alternative explanation may involve the radius of curvature of the corners, the associated corner streamwise vortices, and the potential influence of the latter on the development of the large-scale wavepacket-like structures that interact with the plate. The fact that the difference between the rect-19 and rect-19r spectra is negligible at grazing radiation angles supports the idea of a mechanism involving the dipolar sources distributed on the plate, resulting from the aerodynamic interaction with the jet turbulent structures. In any case, detailed flow measurements would be necessary to elucidate this question, which are left to future investigations.

Fig. 11
figure 11

Influence of corner radius on the rectangular nozzles for a plate positioning of \(H/D=0.62\) and \(L/D=2.0\) for observers located at a \((\theta , \phi )=(90, 180)\) and b \((\theta , \phi )=(90, 270)\). (top) Narrowband SPL of (blue) rect-19 and (red) rect-19r nozzles, and (bottom) delta SPL of the 1/3 octave band spectra of the rect-19r nozzle relative to the rect-19 nozzle. (plain) isolated case, (dash) installed configurations

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4.2.2 Circular Versus Beveled Nozzles

The \(\Delta \text {OASPL}\) directivity maps of Fig. 9 presented the beveled nozzles as the most promising ones for the reduction of installed jet noise. The spectra in Fig. 12 confirm it. Firstly, the isolated beveled nozzles round-b and rect-19b produce nearly identical spectra in isolated conditions, without any significant additional mixing noise, for both normal and grazing radiation directions. As previously reported in a previous study by the same authors (Christophe et al. 2023), the jet self-noise is not influenced by the change of the beveled angle. This might be explained by the nozzle design, where the nozzle lip is extended in the parallel outlet section, compared to other studies (Kamliya Jawahar and Azarpeyvand 2022; Viswanathan 2005; Viswanathan et al. 2006), where the scarfing is performed in a convergent part of the nozzle. In such a case, a plume deviation was observed and a noise reduction was achieved in the azimuthal directions on the side of the longer lip. Secondly, once installed, the reduction of the hydrodynamic interaction noise is substantial and fairly broadband until \(St \simeq 3\), above which the interaction becomes of acoustic nature and much less affected by the nozzle shape. The broadband installation noise decrease in low and mid-frequencies is proportional to the nozzle beveled angle (Christophe et al. 2023). A slight penalty, reaching about 0.5 dB at \(St = 10\), is observed for the rectangular beveled nozzle rect-19b.

At the difference of the rectangular nozzles, the beveled ones have their downstream lip (the one nearer the plate) shifted by 0.5D downstream, which is seen to shift the frequencies of the hydrodynamic resonance peaks in addition to reducing their amplitude. The frequency shift has to be related to advection time of the turbulent structures from the shifted nozzle lip to the plate trailing edge. Postulating an explanation for the amplitude reduction is, however, more difficult. One possibility is the reduced amplitude of the spatially-growing wavepackets when they reach the plate trailing edge, due to the shortened distance from the nozzle lip. The azimuthal coherence of those wavepackets may also be altered by the non-axisymmetric nozzle outlet, making their interaction acoustically less efficient. Detailed flow diagnostics will again be necessary in order to verify these first assumptions.

Fig. 12
figure 12

Inluence of beveled nozzles for a plate positioning of \(H/D=0.62\) and \(L/D=2.0\) for observers located at a \((\theta , \phi )=(90, 180)\) and b \((\theta , \phi )=(90, 270)\). (top) Narrowband SPL of (black) round, (blue) rect19-b and (red) round-b nozzles, and (bottom) delta SPL of the 1/3 octave band spectra of beveled nozzles relative to the round nozzle. (plain) isolated case, (dash) installed configurations

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4.3 Effect of Lateral Separation H/D

The installation effects can be evaluated from the overall sound power level OAPWL, integrated over the range of angles covered by the directivity maps of Fig. 9. This allows to account for the noise performance of a nozzle for all observer positions in a single value. The evolution of the OAPWL for varying lateral distances between the nozzle and plate and for a couple of representative nozzle shapes is shown in Fig. 13. The results for the minimum distance \(H/D=0.62\) are consistent with the previous observations, with the maximum installation added noise (quantified by the difference between the solid and dash-dotted values) observed for the baseline circular nozzle. The minimum added noise is found for the round-b nozzle. The installed noise levels of the round-r and rect-17 nozzles lie between the baseline and beveled nozzles, the rectangular nozzle being slightly noisier in isolated conditions.

When H/D increases, the installation added noise reduces logically for all nozzles, but at a slower rate for the round-r and rect-17 nozzles than for the two other ones. For \(H/D=1\), it turns out that the OAPWL of the isolated round-r and rect-17 nozzles becomes larger than the installed OAPWL of the round and round-b nozzles.

Fig. 13
figure 13

OAPWL radiated by a jet at \(M_a=0.6\) in (dash) isolated configuration and (plain) installed configuration for varying radial distance of the plate H/D and a fixed axial distance of \(L/D=2.0\). (black) round, (blue) round-b, (red) round-r and (green) rect-17 nozzles

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These results are better understood by looking at the spectral distribution of the sound power levels, shown in Fig. 14 for the minimum and maximum values of H/D. It appears that the benefits brought by the nozzle variants are nearly cancelled when the lateral distance is large enough to prevent the occurrence of the hydrodynamic resonance tones. round-b is the only nozzle that reduces slightly the hydrodynamic broadband installation effect, while round-r and rect-17 only bring noise penalties at high frequencies, without any observable gain.

Fig. 14
figure 14

Influence of the nozzle geometry for a plate positioning of a \(H/D=0.62\) and b \(H/D=1.0\) for a fixed \(L/D=2.0\). (top) Narrowband PWL of (black) round, (blue) round-b and (red) round-r and (green) rect-17 nozzles, and (bottom) delta PWL of the 1/3 octave band spectra of modified nozzles relative to the round nozzle. (plain) Isolated and (dash) installed configurations

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5 Discussion

The previous results have shown that the noise reduction offered by the different nozzle shapes is mostly achieved through the attenuation of the hydrodynamic feedback tones that are observed in the range \(0.4<St<2\), for the baseline circular nozzle and baseline streamwise-radial separation between the nozzle axis and the plate. The attenuation of the feedback tones is more significant for the highest frequencies. When the lateral distance is large enough to avoid hydrodynamic feedback tones, a moderate mitigation of the low-frequency broadband hydrodynamic interaction effect could still be achieved by the round-b nozzle.

All the rectangular nozzles are louder than the baseline, when the aspect ratio exceeds a certain threshold between 1.26 and 2, at high (\(St>2\)) and low (\(St<0.3\)) frequencies. This can be attributed to the streamwise vortices emerging from the corners, enhancing mixing noise. But this mild penalty is largely compensated by the attenuation of the feedback tones in the intermediate frequency range.

Different conclusions may however be reached at full scale and accounting for the A-weighing of the spectra. Indeed, for a modern UHBR aeroengine and the same value of L/D as in this study, the feedback tones would be found at frequencies of the order of 100 Hz and below, and significantly attenuated in a dBA spectrum. In that case, the high-frequency noise increase of large AR nozzles may be more penalizing. The circular beveled nozzle would remain the best option. This exploratory study should then be complemented by further investigations where the bevel angle is varied, in order to further improve the noise reduction if possible. Since beveled nozzles can be accompanied by a mild reduction of thrust for subsonic jets (Viswanathan 2006), aeropropulsive efficiency should be included in a multi-objective optimization.