Mathieu Varé

Christophe Bogey

Generation of acoustic tones in round jets at a Mach number of 0.9 impinging on a plate with and without a hole

The generation of acoustic tones in four round jets at a Mach number of 0.9 impinging on a plate at a distance L = 6r 0 of the nozzle exit, where r 0 is the nozzle radius, has been investigated by large-eddy simulation. Three plates are perforated by a hole of diameter h = 2r 0 , 3r 0 and 4.4r 0 , centered on the jet axis, whereas the fourth plate has no hole, in order to study the effects of the hole on the jet flow and acoustic fields. In all cases, acoustic tones emerge in the jet near field, upstream but also downstream of the plate for the perforated plates. Their frequencies are similar for all jets and close to those predicted for aeroacoustic feedback loops establishing between the nozzle and the plate, involving flow disturbances convected downstream and waves propagating upstream at the ambient speed of sound. Their levels, however, decrease with the hole diameter, by a few dB for h 3r 0 but approximately by 40 dB for h = 4.4r 0 . The features of the feedback loops are identified by computing two-dimensional space-time correlations and frequency-wavenumber spectra of the pressure fluctuations in the jet mixing layers. These loops are found to be closed by free-stream upstream-propagating guided jet waves, produced by the impingement of vortical structures on the plate for h 3r 0 and by the scattering of the jet aerodynamic pressure fluctuations by the hole edges for h = 4.4r 0 . Finally, an acoustic analogy is used to show that the contributions of the pressure fluctuations on the plate to the radiated noise are dominant.

Introduction

It is well known that intense acoustic tones are generated by the impingement of jets on a plate. Such tones were observed experimentally for high subsonic jets by many researchers, including Powell (1953), Neuwerth (1974), Preisser (1979), Ho & Nosseir (1981) and Nosseir & Ho (1982). In these studies, a staging phenomenon of the main tone frequency was measured when the nozzle-to-plate distance increases, which has led Powell (1953) to explain the tone generation by a feedback mechanism between the turbulent structures convected downstream from the nozzle to the plate and the acoustic waves propagating upstream from the plate to the nozzle. Similar feedback loops were noticed for supersonic impinging jets as well. For ideally-expanded supersonic jets, the establishment of these loops has been studied experimentally by Norum (1991) and numerically by Gojon et al. (2016) and Bogey & Gojon (2017). For underexpanded jets, the loops have been visualised in the experimental works of Henderson et al. (2005), Risborg & Soria (2009), Buchmann et al. (2011) and Mitchell et al. (2012) and in the simulations of Dauptain et al. (2012) and Gojon & Bogey (2017a).

The upstream waves closing the feedback loop have also been analyzed in several studies. In their aeroacoustic feedback model, Ho & Nosseir (1981) assumed that they are free-stream acoustic waves propagating outside of the jet. However, they were later identified as guided † Email address for correspondence: mathieu.vare@ec-lyon.fr jet waves propagating mostly inside the jet, as suggested in the theoretical work of Tam & Ahuja (1990) for round jets and Tam & Norum (1992) for planar jets. These guided jet waves are defined by dispersion relations and classified in modes depending on their radial and azimuthal structures. In particular, their properties allow us to explain the frequencies and the axisymmetric or helical nature of the feedback tones, as highlighted in Gojon et al. (2016), Gojon & Bogey (2017a), Bogey & Gojon (2017) and Jaunet et al. (2019) for subsonic and supersonic impinging jets. They are involved in other resonance phenomena, for examples in screech generation mechanism, as studied by Gojon & Bogey (2017b), Edgington-Mitchell et al. (2018), Mancinelli et al. (2019), Edgington-Mitchell (2019), or in jet-plate interactions, as investigated by Jordan et al. (2018) and Tam & Chandramouli (2020). The guided jet waves also play a role in the generation of acoustic tones near the nozzle (Towne et al. 2017;Brès et al. 2018;Bogey 2021b) and in the upstream acoustic far field (Bogey 2021b) of free jets. Using a vortex-sheet model, Towne et al. (2017) have shown that within restricted frequency bands, both downstream and upstream-propagating guided jet waves can exist for Mach numbers between 0.82 and 1 and can interact with each other. The near-nozzle tones in both experiments and simulations are located in these bands. In a recent study, Bogey (2021a) investigated numerically the generation of these tones for jets at Mach numbers varying between 0.5 and 2. The tone properties, in terms of frequency, intensity, azimuthal structure, width and prominence, were detailed. They were shown not to strongly vary around Mach numbers of 0.82 and 1. In particular, over the whole Mach number range, the peak frequencies lie within the frequency bands of the free-stream upstream-propagating guided jet waves and the peak levels follow the typical scaling laws of jet noise. Very recently, Bogey (2021b) has shown using data from both experiments and simulations that the near-nozzle tones propagate in the far field in the upstream direction.

Jets impinging on a plate with a hole have been the subject of fewer studies than jets impinging on a flat plate. For subsonic jets, tones similar to those obtained for a nonperforated plate were first observed by Sondhauss (1854) and Rayleigh (1945) and they were referred to as hole tones. In the same way as without a hole, they are generated by a feedback mechanism between the hole edges and the nozzle. Indeed, the same staging phenomenon of the tone frequency as the nozzle-to-plate distance increases was revealed by several experiments, such as those of Chanaud & Powell (1965) and Vinoth & Rathakrishnan (2011). Hole tone generation has been studied over a wide range of Mach numbers. For instance, Langthjem & Nakano (2005) and Matsuura & Nakano (2012) simulated a jet at a Mach number of 0.03 impinging on a plate with a hole located one nozzle diameter from the jet exit. Meganathan & Vakili (2006) explored experimentally the effects of the jet velocity on the hole tone frequencies for Mach numbers varying from 0.2 up to 0.8. They noticed frequencies staging with the jet velocity, suggesting again a feedback mechanism. High subsonic and supersonic jets were considered in the experimental works of Umeda et al. (1988) and Umeda & Ishii (1993). Umeda et al. (1988) notably visualised the feedback loop between the hole edges and the nozzle for jets at Mach numbers of 0.94 and 1.54 using shadowgraphy. They also noted a tonal radiation at the feedback frequency downstream of the plate. More recently, jets impinging on a plate with a hole have been studied numerically for a Mach number of 3.7 by Kawai et al. (2007) and Tsutsumi et al. (2015) and for a Mach number of 3.1 by Troyes et al. (2019) and Varé & Bogey (2021), with the aim of modelling a rocket launch, during which the supersonic gas jets exhausted by the engines impinge on the launch area and are canalized in a trench dug under the rocket. No feedback loop was found for these high Mach numbers. Concerning the effects of the hole diameter on the feedback loop, Vinoth & Rathakrishnan (2011) compared the flow and pressure fields of two jets at a Mach number of 0.8 impinging on a flat plate and on a plate with a hole of same diameter as the nozzle. The two jets generate tones at similar frequencies, suggesting little effects of the hole on the feedback mechanism. In the same way, for a Mach number of 0.94, Umeda et al. (1988) remarked no significant changes in the frequencies of the tones for hole diameters varying from 1.8 up to 2.4 nozzle diameters. Despite the preceding studies, several questions remain about the influence of the size of the hole on the feedback properties. In particular, the effects of the hole diameter on the tone intensities need to be carefully described. As the hole diameter increases, these intensities are expected to decrease, due to weaker interactions between the jet turbulent structures and the plate. To the best of the authors knowledge, this hypothesis has not been confirmed yet. It is also unclear whether the broadband acoustic levels depend on the hole diameter. Furthermore, it is interesting to study a jet passing through a perforated plate with no impingement of the jet turbulent structures on the plate.

For such a jet, the interactions between the flow and the plate are weaker than for the cases with a small hole, which could lead to a less pronounced aeroacoustic resonance, and hence differences in the sound generation mechanisms. Finally, the tonal radiation downstream of the plate pointed out by Umeda et al. (1988) has not been examined thoroughly. The noise sources generating this radiation, which may be related to the feedback mechanism, have not been clearly identified. In the present work, the generation of hole tones is investigated by performing the large-eddy simulations of four subsonic round jets impinging on a plate with and without a hole. The jets are at a Mach number of 0.9 and a Reynolds number of 10 5 and are initially highly disturbed. They impinge on a plate located at a distance L = 6r 0 from the nozzle exit, where r 0 is the jet radius. Three of the plates have a hole of diameters h = 2r 0 , 3r 0 and 4.4r 0 , whereas the fourth plate has no hole. The nozzle-to-plate distance is the same as that in the experiments of Umeda et al. (1988), in which the establishment of a feedback loop between the nozzle and the plate was observed. The hole diameters are chosen to study noise generation mechanisms for different interactions between the jet and the plate. Indeed, for h = 2r 0 and 3r 0 , the jet impinges on the hole edges whereas for h = 4.4r 0 , it passes through the hole without being distorted. The first objective of this study is to investigate the influence of the hole and its size on the tone generation. For that purpose, the characteristics of the jet flow and acoustic fields will be detailed, and compared with those obtained for a free jet with the same exit conditions (Bogey (2018(Bogey ( , 2021a))). In particular, pressure spectra will be computed in order to highlight the tonal frequencies. The contributions of the first two azimuthal modes will also be examined in order to identify the oscillation modes of the jets.

The second aim of this work is to study the upstream part of the feedback loops establishing between the nozzle and the plate, which can be related, given previous works (Tam & Ahuja (1990);Gojon et al. (2016); Bogey & Gojon (2017)), to the free-stream upstream-travelling guided jet waves. The production of these waves at the plate, their upstream propagation, as well as their role in generating new instability waves in the jet mixing layers when hitting the nozzle lips, will be revealed by calculating two-dimensional space-time correlations and frequency-wavenumber spectra of the pressure fluctuations in the jet shear layers. Finally, the nature of the noise sources will be investigated by separating the sound components linked to the flow turbulence and those related to the forces exerted by the flow on the plate. To this end, Curle's analogy will be employed to evaluate the contributions of the variations of the pressure on the plate to the pressure field.

The paper is organized as follows. The jet parameters and numerical methods used in the LES are documented in section 2. In section 3, the results of the simulations are presented.

In particular, vorticity and pressure snapshots, mean and turbulent flow fields and pressure spectra are shown. The properties of the tones emerging in the spectra are analyzed and compared with the results of aeroacoustic feedback models. Two dimensional space-time correlations and frequency-wavenumber spectra of the pressure fluctuations in the mixing layer are provided. An acoustic analogy is then used to study the nature of the noise sources.

Finally, concluding remarks are given in section 4.

Parameters

Jet parameters

The jets computed in this work have a Mach number M = u j /c 0 of 0.9 and a Reynolds number Re D = u j D/ν of 10 5 , where u j is the jet velocity, D is the nozzle diameter and ν is the air kinematic viscosity. They exhaust from a cylindrical nozzle of radius r 0 , at ambient temperature and pressure T 0 = 293 K and p 0 = 10 5 Pa. They impinge on a plate located at L = 6r 0 downstream of the nozzle exit, with a width of e = 0.4r 0 . In one case, the plate has no hole whereas for three other ones it has a hole of diameter h = 2r 0 , 3r 0 , and 4.4r 0 .

The four cases are referred to as jetnohole, jetsmallhole, jetmediumhole and jetlargehole respectively. The nozzle-to-plate distance, the plate width and the hole diameter h = 4.4r 0 are the same as in the experiments of Umeda et al. (1988). At the nozzle inlet, a Blasius laminar boundary-layer profile with a boundary-layer thickness of δ = 0.15r 0 is imposed for the velocity. Vortical disturbances non-correlated in the azimuthal direction are added in the boundary layer at z = -r 0 to create velocity fluctuations at the nozzle exit, following a procedure described in Bogey et al. (2011b). Their amplitude is the same as in the free jet at M = 0.9 in Bogey (2018Bogey ( , 2021a) ) to reach a peak turbulence intensity of 9% at the nozzle exit. The profiles of mean and root-mean-square axial velocities thus obtained at the nozzle exit are represented in figure 1. In figure 1(a), the mean velocity profiles are very similar for the four jets and are close to the Blasius laminar boundary-layer profile imposed at the inlet.

As for the turbulent intensities in figure 1(b), they reach a peak of 9% for the four jets, as in the corresponding free jet.

Numerical and computational parameters

The numerical framework of this study is similar to that of recent simulations of free jets (Bogey (2021a)) and supersonic impinging jets (Gojon et al. (2016); Bogey & Gojon (2017)).

The unsteady compressible Navier-Stokes equations are solved in cylindrical coordinates (r, θ, z) using an OpenMP based in-house solver. The time integration is performed using a six-stage Runge-Kutta algorithm (Berland et al. (2007)) and the spatial derivatives are evaluated with an eleven-point low-dissipation and low-dispersion finite-difference scheme (Bogey & Bailly (2004)). At the end of each time step, a selective filtering is applied in order to remove grid-to-grid oscillations (Berland et al. (2007)). In these zones, they are combined with sponge zones using grid stretching and Laplacian filtering to avoid significant numerical reflections (Bogey & Bailly (2002)). The singularity at r = 0 is treated by applying the method of Mohseni & Colonius (2000). The first point close to the axis is thus located at r = ∆r/2, where ∆r is the radial mesh size near the jet axis. The effective azimuthal resolution near the origin of the polar coordinates is reduced down to 2π/16 in order to increase the admissible time step of the simulation (Bogey et al.

(2011a)).

The mesh grids used for the four simulations contain between 540 millions and 1.4 billions of points, as reported in table 1. They extend out to r = 15r 0 in the radial direction and down to z = 6r 0 and z = 40r 0 in the axial direction for the flat and perforated plates, respectively. The radial mesh spacing, represented in figure 2(a), is equal to ∆r = 0.014r 0 on the jet centerline and decreases down to ∆r = 0.0036r 0 at r = r 0 in the shear layers. It then increases to reach a maximum value of ∆r max = 0.075r 0 for r > 6.2r 0 , which allows a Strouhal number St = f D/u j of 5.9 to be obtained for an acoustic wave with five points per wavelength.

In the azimuthal direction, the grid is uniform and N θ = 1024 points are used. The axial mesh spacing ∆z, presented in figure 2(b), is minimal and is equal to ∆z = 0.0072r 0 at the nozzle exit. It increases and reaches ∆z = 0.012r 0 between z = 2r 0 and z = 4r 0 . Farther downstream, the axial mesh spacing is reduced down to ∆z = 0.0072r 0 near the plate at z = 6r 0 , as at the nozzle exit. Downstream of the plate, it raises up to ∆z = 0.05r 0 at z = 40r 0 .

The extremum values of the mesh spacings and the stretching rates in both axial and radial directions are the same as in the study of Bogey (2018), where a grid convergency study was performed for the free jet with the same ejection conditions. The simulation time after the transient period is equal to 500r 0 /u j for all jets. During the simulations, density, velocities and pressure along the jet axis at r = 0, along the lip line at r = r 0 , on the surface at r = 15r 0 , on the hole edges at r = h/2, and at z = -2r 0 , z = 0, on the faces of the plate at z = L and z = L + e, and at z = 40r 0 are recorded at a sampling frequency corresponding to a Strouhal number of St = 12.8. Density, velocity components and pressure at the azimuthal angles θ = 0, 90, 180 and 270 degrees are also saved at a frequency twice smaller than for the signals at constant r and z. In addition, the azimuthal Fourier coefficients of the density, pressure and velocity fields are computed up to the mode n θ = 4 for 0 r 15r 0 and 0 z 40r 0 .

The spectra presented in the next sections are calculated from these recordings and they are averaged in the azimuthal direction, when possible.

Results

Snapshots

Snapshots of the vorticity norm obtained down to z = 10r 0 for the impinging jets and the free jet are represented in figure 3. Additional videos movie1 and movie2 are also provided for jetnohole and jetmediumhole, respectively. For all jets, the mixing layers are highly disturbed near the nozzle exit. Farther downstream, they thicken with the axial distance because of the development of large-scale vortical structures. For jetnohole, jetsmallhole and jetmediumhole, in figures 3(a-c), these structures impinge on the plate. A wall jet is then created, as the entire jet flow, or only a part of it, is diverted in the radial direction. For the perforated plates, indeed, a significant part of the jet flow passes through the hole. The flow fields near the impingement region are dominated by large distorted structures, which are particularly visible between z = 4r 0 and z = 6r 0 . Downstream of the plate, coherent structures can also be seen in the flow. This is the case for instance around z = 8r 0 in figure 3(c). For jetlargehole, in figure 3(d), the mixing layers do not appear significantly affected by the plate. On the contrary, their development in the axial direction seems very similar to that for the free jet in figure 3(e).

Snapshots of the vorticity norm and of the pressure fluctuations obtained in a whole (z, r) section are given in figure 4. In figures 4(b-d), downstream of the plates with a hole, the shear layers develop down to the end of the potential cores, as for the free jet in figure 4(e).

In the pressure fields, for jetnohole, jetsmallhole and jetmediumhole, strong acoustic waves are generated by the impingement of the jet turbulent structures on the plate. Periodically separated wave fronts are observed, indicating a tonal radiation upstream of the plate, but also downstream for jetsmallhole and jetmediumhole. The amplitudes of the waves are of the order of 0.01p 0 , which is much higher than the amplitude obtained in the pressure field of the free jet, typically of 0.001p 0 . In the upstream direction, the waves are slightly stronger for jetsmallhole than for jetmediumhole, but also, more surprisingly, for jetsmallhole than for jetnohole. Moreover, for jetsmallhole and jetmediumhole, the amplitude of the pressure waves appear to be stronger in the upstream region than in the radial direction, which is not the case for jetnohole. Therefore, the acoustic radiations are visibly more directive in the two first cases with a perforated plate than for the no-hole case. For jetlargehole, in figure 4(d), the acoustic waves show a less organized pattern. An acoustic radiation originating from the hole can however be detected upstream and downstream of the plate. A phase jump of the pressure waves is noted across the plate, producing an acoustic pattern similar to that emitted by installed jets (Nogueira et al. 2017). The sound radiation seems to be caused by the scattering of the jet aerodynamic pressure fluctuations by the hole edges. The pressure waves thus generated are 4 times weaker than those produced by the other impinging jets.

Despite this, they are approximately twice as strong as the mixing-noise acoustic components radiated by the free jet in figure 4(e), namely low-frequency components in the downstream direction and high-frequency components for large radiation angles. For the latter jet, the sound radiation is very directive in the downstream direction, contrary to those of the jets impinging on a perforated plate.

Flow field

The mean density fields of the jets are represented in figure 5. In all cases, density is lowest in the shear layers. Furthermore, except for jetlargehole and the free jet, compression cells are visible inside the jets. For jetnohole, in figure 5(a), three compression cells are found.

The first one is located near z = 2r 0 , and the second one is between z = 3r 0 and z = 4r 0 .

The third, and most intense one, corresponds to the impingement area, and extends from z = 4.5r 0 down to the plate in the axial direction, and up to r = 1.8r 0 on the plate in the radial direction. Similar compression cells have been found using Schlieren imaging in the experiments of Neuwerth (1974) for a jet at a Mach number of 0.9 impinging on a plate Farther from the axis, it grows nearly linearly, as the wall jet approaches self similarity. This self-similarity is typical of wall jets, refer to the work of Poreh et al. (1967), Launder & Rodi (1983) or Van Hout et al. (2018), for instance. For jetsmallhole, the mass flow rate of the wall jet also grows continuously with the radial distance, but is lower than for jetnohole, which can be explained by the fact that, due to the hole in the plate, only a fraction of the jet flow is diverted in the wall jet. For jetmediumhole, the mass flow rate is still reduced compared with jetsmallhole, because of the larger hole diameter.

The profiles of centerline mean axial velocity in the impinging jets between the nozzle and the plate are presented in figure 7(a). The centerline velocity decreases down to zero at z = 6r 0 on the plate for jetnohole, but remains close to u j for the three other impinging jets.

Small oscillations are however visible between z = 2r 0 and z = 6r 0 for jetsmallhole and jetmediumhole. They can be linked to the compression cells observed in the density fields in these cases. Such oscillations do not appear for jetlargehole.

The variations of the jet shear-layer momentum thickness δ θ are represented in figure 7(b).

For jetlargehole, the shear-layer spreading is the same as for the free jet. The shear layers of the three other jets for which the mixing-layer turbulent structures impinge, fully or partially, on the plate, develop more rapidly. More precisely, they are thinner between z = 0 and z = 4.5r 0 , and much thicker for z 4.5r 0 , for jetnohole than for the two other jets. The difference near the plate can be attributed to the stronger wall jet in the case with a non-perforated plate.

The profiles of maximum radial velocity in the wall jets are plotted in figure 7(c). For jetnohole, the maximum velocity increases in the impingement region, where the jet flow is diverted into the radial direction, up to 0.87u j at r = 2r 0 . For r 2r 0 , the wall jet then spreads cylindrically, causing a decrease of the velocity as r -1 . For jetsmallhole and jetmediumhole, the velocities also first grow in the impingement zone and then decay farther from this zone. They are, however, lower as the hole diameter is larger, leading to peak values only of 0.48u j for jetsmallhole and 0.11u j for jetmediumhole.

The rms values of axial velocity fluctuations estimated at r = r 0 along the nozzle-lip line are displayed in figure 8(a) between z = 0 and z = 6r 0 . For the impinging jets, they are very similar to those in the free jet down to z = 2r 0 . For jetnohole, they fall down to zero on the plate. For jetsmallhole and jetmediumhole, they also decrease between z = 2r 0 and z = 4r 0 , but increase farther downstream. Finally, for jetlargehole, they grow continuously and are slightly higher than those for the free jet. r = 2r 0 and of 0.10u j on the hole edges at r = 1.5r 0 , respectively. The wall jets are therefore weaker in these two cases, as expected.

The spectra of the radial velocity fluctuations computed near the nozzle lip, at r = r 0 and z = 0.4r 0 , for the impinging and the free jets are represented in figure 9 as a function of the Strouhal number. The contributions of the two first azimuthal modes are also shown.

For all jets, a broadband hump is observed for both modes around a Strouhal number of St = 1.7, which is close to the frequency St θ = f δ θ /u j = 0.016 predicted for the Kelvin-Helmholtz instability waves using linear stability analysis (Michalke 1984). For the impinging jets, in addition, tones also emerge in the spectra, at similar frequencies but with levels depending on the presence and diameter of a hole in the plate. The tones are strong for jetnohole, jetsmallhole and jetmediumhole, but much weaker for jetlargehole. In the first case, in figures 9(a-c), very intense tones are found for the axisymmetric mode at St = 0.41 for jetnohole and St = 0.40 for the two other jets, and at their harmonic frequencies. A small peak also appears for n θ = 1 around St = 0.7. For both modes, the frequencies of the dominant tones are lower than those expected for the most-amplified Kelvin-Helmholtz instability waves. These results suggest that the tones are due to aeroacoustic feedback loops establishing between the nozzle and the plate, i.e. that the jet shear layers are forced by the acoustic waves propagating in the upstream direction closing the loops. These feedback loops and upstream-travelling waves will be investigated later. For jetlargehole, in figure 9(d), a peak can also be seen at St = 0.40 for n θ = 0. However, the peak has a much smaller amplitude than the dominant tones for the other impinging jets, and does not appear in the spectrum of the full velocity signal, indicating a weaker aeroacoustic resonance in this case. previous jets and they look like the isocontours obtained for the free jet in figure 11(e). Jet mixing noise components therefore seem to be most significant downstream. Concerning the acoustic levels, upstream of the plate, they are overall highest for jetnohole and they decrease as the hole is larger. Downstream of the plate, a similar trend is noticed, with highest levels for the smallest hole and lowest levels for the free jet.

Pressure spectra

The pressure spectra obtained for the four impinging jets near the nozzle and downstream of the plate, when possible, are plotted in figure 12 as a function of the Strouhal number, together with the spectra for the free jet.

Near the nozzle, at z = 0 and r = 1.5r 0 , in figure 12(a), several intense tones emerge for jetnohole, jetsmallhole and jetmediumhole at very similar frequencies regardless of the presence of a hole in the plate and its size. The dominant tone is located at a Strouhal number of St = 0.41 for jetnohole, and of St = 0.40 for jetsmallhole and jetmediumhole. In the three cases, it is 25 to 30 dB greater than the broadband noise level. Secondary tones, not harmonics of the dominant tones, appear at St = 0.69 for jetnohole, St = 0.64 for jetsmallhole and St = 0.60 for jetmediumhole. For jetlargehole, a tonal peak is also found at the Strouhal number of 0.40. However, the acoustic levels are much weaker than for the other impinging jets, and are reduced by 20 to 55 dB compared with jetnohole. They are even close to the levels estimated for the free jet for Strouhal numbers higher than 0.7, and only 10 to 15 dB higher for lower frequencies. Tonal peaks can also be noted in the spectra for the free jet, at frequencies comparable to those of the tones for the impinging jets, namely St = 0.41, 0.70, 1.0, 1.30 and 1.63 (Bogey (2021a)).

For the impinging jets, the tones can be assumed to be generated by feedback mechanisms between the nozzle and the plate (Powell (1953)), consisting of two steps. During the first step, coherent structures are convected in the shear layer to the plate, where their impingement produces acoustic waves. During the second step, these waves propagate upstream to the nozzle, where they excite the shear layer instability waves, creating new coherent structures and closing the feedback loop. A model was proposed by Ho & Nosseir (1981) to predict the frequencies of such a loop. It was built by considering the feedback period as the sum of two characteristic times. The first one is the time of convection of the flow structures from the nozzle to the plate and the second one is the time of propagation of the acoustic waves in the upstream direction at the ambient speed of sound c 0 . The feedback frequency f can thus be predicted by

f = Nu c L(1 + M c ) (3.1)
where N is an integer, u c is the mean convection velocity and M c = u c /c 0 is the convection Mach number. The integer N represents the order of the feedback mode and corresponds to the number of coherent structures between the nozzle and the plate. The velocity u c , computed for all the jets in this study, is close to the classical approximation of (2/3) × u j .

The feedback frequencies given by the model for N = 3 and N = 5 using this value for the (2021a). These waves will be examined in section 3.4.2. Interestingly, the feedback tones for the impinging jets and the peaks for the free jets emerge at very similar frequencies.

The feedback frequencies therefore appear to be mainly determined by the properties of the upstream-propagating guided jet waves. Regarding the tones intensities, the feedback loops produce strong tones for hole diameters h 3r 0 and much weaker tones for h = 4.4r 0 , when the flow structures do not impinge on a solid surface.

The pressure spectra computed at z = 12r 0 and r = 15r 0 downstream of the plate are represented in figure 12 four frequencies mentioned above. These results further support a link between the feedback loops and these modes. Finally, for n θ = 0, additional tones are observed, namely two tones at St = 0.80 and 1.20 for h 3r 0 , which are the harmonics of the first tone at St = 0.40 and a tone at St = 0.60 for h = 4.4r 0 .

The levels of the dominant tones at z = 0 are represented in figure 15 as a function of the hole diameter for the modes n θ = 0 and n θ = 1. Two radial positions, r = 1.5r 0 and 15r 0 , are considered, and only the tones emerging by more than 3 dB from the broadband levels are displayed. For n θ = 0 in figure 15(a), near the nozzle, the level of the dominant tone is strong for h 3r 0 , reaching a peak of 180 dB for h = 2r 0 . The amplitudes of the tone for jetsmallhole and jetmediumhole are slightly higher than for jetnohole. For h = 4.4r 0 , the tone level is drastically reduced by about 40 dB. This can be explained by the fact that for h 3r 0 the upstream sound waves are created by the impingement of the jet turbulent structures on the plate, whereas for h = 4.4r 0 , they are produced by the scattering of the jet pressure fluctuations by the plate. In the latter case, the tone level is still 15 dB higher than that for the free jet. Farther away from the jet axis, at r = 15r 0 , the tone levels are highest for jetnohole and decrease monotonously with the hole diameter, contrary to the levels of the near-nozzle tones. This is expected given the different directivities of the jets highlighted in figure 4. For n θ = 1, in figure 15(b), the level of the dominant near-nozzle tone also decreases continously with the hole diameter. It is reduced by 5 dB between h = 0 and 1.5r 0 , then by 15 dB for h = 4.4r 0 , tending towards the level for the free jet.

Investigation of the feedback loop

Visualisation of the feedback loop

In order to visualise the feedback loop, two-dimensional spatial correlations of the jet pressure fields are computed in a section (z, r). The pressure fluctuations p ′ at a reference point (z 1 , r 1 ) at time t are correlated with the fluctuations of pressure in the section (z, r) at time t + δt,

defining the dimensionless coefficient R R (r, z, δt) = p ′ (r 1 , z 1 , t)p ′ (r, z, t + δt) p ′2 (r 1 , z 1 , t) 1/2 p ′2 (r, z, t) 1/2 (3.2)
where δt is the time delay between the signals and . denotes time averaging. In this way, the structures and time variations of the waves correlated with the pressure fluctuations at the reference point are extracted. This method has notably been used to investigate noise generation in free jets at a Mach number of 0.9 in a recent work (Bogey ( 2019)).

In the present study, due to the presence of tones emerging by more than 20 dB in the sound spectra, mainly associated with the mode n θ = 0, the pressure signals are strongly correlated. Therefore, links between the pressure at a reference point and flow structures can not be clearly evidenced by computing correlations from the full signals or from the axisymmetric mode. However, the tones are much less emerging for the mode n θ = 1.

Correlations calculated for this mode are consequently shown in what follows.

The coefficients R are first evaluated for n θ = 1 for the four impinging jets for a reference point placed near the jet nozzle at z 1 = 0 and r 1 = 1.5r 0 , in order to specifically examine the characteristics of the upstream acoustic radiation. Similar results are obtained for jetnohole, jetsmallhole and jetmediumhole. The correlations for the latter jet are represented in figure 16, for time delays δt varying from -15r 0 /u j to 6r 0 /u j . They can also be seen in the supplementary video movie3. For δt = -15r 0 /u j and -12r 0 /u j , in figures 16(a,b), the correlation levels are highest around the shear layers of the jet, at z ≃ 2.5r 0 at δt = -15r 0 /u j and z ≃ 4r 0 at δt = -12r 0 /u j . They are linked to flow coherent structures convected in the jet flow direction, which constitute the downstream part of the feedback mechanism. For δt = -9r 0 /u j and -6r 0 /u j , in figures 16(c,d), the correlation levels increase and they are strongest near the hole, as the flow structures impinge on the hole edges. For δt = -3r 0 /u j , in figure 16(e), the correlation levels are greater than 0.5, which is significantly higher than the levels for the previous time delays. A curved region of strong positive correlations, in red, is aligned with a circle centered on the hole edge, at z = L and r = 1.5r 0 . It can be related to an acoustic wave produced by the jet impingement on the plate, which forms the upstream part of the feedback loop. The correlations are significant outside but also inside of the jet. The propagation of an upstream wave travelling at the ambient sound velocity in the jet column is thus highlighted, as noticed for supersonic impinging jets by Bogey & Gojon (2017). This wave is related to the guided jet waves studied later in section 3.4.2. For δt = 0, in figure 16(f), the correlation level is equal to 1 at the point (z 1 , r 1 ), as expected. Moreover, as for the previous time delay, the acoustic wavefront has a circular shape, and the correlations are strong both in the jet flow and near pressure fields. For later time delays δt = 3r 0 /u j and 6r 0 /u j , in figures 16(g,h), the correlation levels are lower than 0.2, except for two regions.

The first one is due to the upstream propagation of the acoustic waves. The second one is found around the shear layers at z = r 0 for δt = 3r 0 /u j and at z = 3r 0 for δt = 6r 0 /u j , revealing the generation and convection of a new coherent structure in the flow, and thus the downstream part of a new feedback cycle.

The correlations R obtained for jetlargehole are displayed in figure 17 and in the additional video movie4 for time delays δt varying from -15r 0 /u j to 6r 0 /u j . For δt = -15r 0 /u j , in figure 17(a), the correlations are weak, with levels lower than 0.3. They are stronger for δt = -12r 0 /u j , in figure 17(b), exhibiting a wavepacket structure with four lobes around the shear layer at z ≃ 3r 0 . Later, the wavepacket structure is convected down to the plate, as in figure 17(c) for δt = -9r 0 /u j for instance. It is then scattered by the hole edges for δt = -9r 0 /u j and -6r 0 /u j , in figures 17(c,d). The scattering of the aerodynamic pressure fluctuations generates acoustic waves, causing high correlation levels upstream and downstream of the plate for δt = -3r 0 /u j , aligned with a circle centered on the hole edge in figure 17(e). Upstream, in particular, the waves propagate up to the reference point in figure 17(f). As for jetmediumhole, non-negligible correlation levels are also found inside the jet, indicating the propagation of an upstream wave at the sound speed in the jet potential core. This wave belongs to the guided jet modes described in section 3.4.2. Afterward, the correlation levels finally decrease. Spots of significant correlations are nevertheless found Figure 17: Correlations R of p ′ (r = 1.5r 0 , z = 0, t) with p ′ (r, z, t + δt) for n θ = 1 for (a) δt = -15r 0 /u j , (b) δt = -12r 0 /u j , (c) δt = -9r 0 /u j , (d) δt = -6r 0 /u j , (e) δt = -3r 0 /u j , (f) δt = 0, (g) δt = 3r 0 /u j and (h) δt = 6r 0 /u j for jetlargehole, circle centered on (z = L, r = 2.2r 0 ) The color scale ranges between ±1, from blue to red.

near the shear layers at z ≃ r 0 for δt = 3r 0 /u j in figure 17(g) and z ≃ 4r 0 for δt = 6r 0 /u j in figure 17(h). They can be attributed to a flow coherent structure convected downstream, initiating a new feedback cycle, as mentioned above for jetmediumhole. Two-dimensional spatial correlations are also computed for n θ = 1 for a reference point located at z = 2L and r = 15r 0 to shed light on the sound generation downstream of the plate.

The results obtained for jetmediumhole and jetlargehole are represented in figures 18 and 19, respectively, for the time delays δt = -15r 0 /u j , -10r 0 /u j and -5r 0 /u j . For δt = -15r 0 /u j , weak correlations are found for jetmediumhole near the hole in figure 18(a). The correlations for jetlargehole in figure19(a) are stronger and reveal a wavepacket structure just outside of the jet shear layers upstream of the plate. Later, for δt = -10r 0 /u j and then for δt = -5r 0 /u j , curved regions of correlation levels higher than 0.6 and centered on the hole edge are observed for both jets in figures 18(b,c) and 19(b,c). Therefore, the sound waves radiated downstream of the plate are related to phenomena occurring close to the hole edge. These phenomena can be of different nature depending on the hole diameter and consist of the interactions of the hole edge with the vortical structures passing through the hole for jetmediumhole and with the jet aerodynamic pressure waves, surrounding the flow structures, for jetlargehole.

Guided jet modes

In order to discuss the nature of the upstream-travelling waves closing the feedback loops in the impinging jets, frequency-wavenumber spectra have been computed as recently in Bogey (2021a). They are calculated from the pressure fluctuations in the jet shear layer at r = r 0 between z = 0 and z = 4r 0 for jetnohole and jetsmallhole and between z = 0 and z = 10r 0 for the other jets for the two first azimuthal modes n θ = 0 and 1. The spectra obtained for jetmediumhole, jetlargehole and the free jet are represented in figure 20 as a function of k and St, for negative wavenumbers. The results for jetnohole and jetsmallhole are not shown but they look like those for jetmediumhole. The dispersion curves of the guided jet waves given by a vortex-sheet model (Tam & Ahuja (1990)) are also displayed using red lines. Finally, the tone frequencies emerging in the LES pressure spectra in section 3.3.2 are indicated by red triangles placed at k = 0. The dispersion curves start on the sonic line k = -ω/c 0 . Farther from this line, as the wavenumber increases, in absolute value, the frequency of the waves increases up to a local maximum, then it is reduced down to a local minimum and finally increases again. Three kinds of waves can be defined depending on their positions on the dispersion curves, as proposed by Towne et al. (2017). The waves between the sonic line and the local maximum have a negative group velocity v g = dω/dk and a support outside the jet. They can be referred to as upstreamtravelling free-stream waves. In the other cases, the waves are entirely confined in the jet.

They propagate downstream for the waves between the two local extremum frequencies on the curves and upstream otherwise. They will be denominated as downstream-and upstreampropagating duct-like waves, respectively. For jetmediumhole, in figures 20(a,d), lines of high energy extending over the whole range of wavenumbers are found at the Strouhal numbers of the tones in the LES sound spectra. Except for the tone at St = 1.20 for n θ = 0, they intersect the dispersion curves of the guided jet waves predicted by the vortex-sheet model, in regions where the waves are the upstream-travelling free-stream ones. Therefore, these waves are likely to close the feedback loops generating the tones. As for the tone at St = 1.20, it can be expected to result from nonlinear effects between the two tones at St = 0.40 and St = 0.80.

For jetlargehole, in figures 20(b,e), no lines are visible at the Strouhal numbers of the tones, contrary to jetmediumhole, which is most probably due to the quite weak tones in this case where the hole diameter is the largest. Curved bands are however clearly observed near the dispersion curves of the guided jet waves for the vortex-sheet model. They are very similar to those found for the free jet in figures 20(c,f). On these bands, the intensity is high near the line k = -ω/c 0 , decreases farther from it and is negligible for low wavenumbers. These variations of the intensity can be explained by the fact that the guided jet waves close to the line k = -ω/c 0 are free-stream waves, whereas those far from the line are duct-like waves (Tam & Ahuja 1990;Towne et al. 2017). Concerning the LES tones, their frequencies lie in the frequency bands of the free-stream upstream-propagating modes, highlighting again the role of these waves in the feedback loops. Moreover, as seen previously in figure 14, the tone frequencies do not vary much with the hole diameter. Despite different noise generation mechanisms, namely the scattering of the jet pressure for jetlargehole and vortical structures impinging on the plate for the other jets, they seem to be mainly determined by the frequency of the free-stream upstream-propagating guided jet waves. The tone frequencies are also close to those produced by jet-flap interactions at the same Mach number (Jordan et al. 2018), supporting again the idea that they are weakly influenced by the nature of the interactions between the jet and the plate.

To visualise both the downstream and the upstream parts of the feedback loops, the frequency-wavenumber spectrum obtained for the tonal case jetmediumhole for n θ = 0 is presented in figure 21 for negative but also positive wavenumbers. The lines found at the tone frequencies for k 0 extend for k 0. In the latter region, the strongest levels are close to the line k = ω/u c with u c = (2/3) × u j . Therefore, the downstream component of the feedback loops consists of flow structures convected in the jet mixing layers at the velocity u c , as expected. These structures develop at the tone frequencies, due to the forcing of the shear layers near the nozzle lips at these frequencies, evidenced in the velocity spectra of figure 9.

3.5. Noise source investigation using Curle's analogy

Curle's analogy

The noise radiated by flows interacting with rigid bodies can be investigated using acoustic analogies. In particular, an analogy, in which the noise generated by such flows consists of et al. (2005) to study the sound radiated by the flow around a cylinder at a Mach number of 0.12. In the first work, the authors showed that the pressure far-field predicted by the analogy is close to that obtained by a DNS computation. In an older study, Curle's analogy was employed by Preisser (1979) to correlate the far-field noise with the pressure on the plate for subsonic impinging jets.

For the present impinging jets, the term of Curle's analogy associated with the variations of pressure on the solid surfaces is considered to evaluate the acoustic field upstream and downstream of the plate. The amplitude of the pressure fluctuations due to this contribution is given by

p ′ (x, t) = 1 4π ∂ ∂ x i S 1 R P i y, t - R c 0 dS(y) (3.3)
where R = |x -y| is the distance between a source on the plate and the observer and P i = (pδ i j -τ i j )l j is the force per unit area exerted on the fluid by the plate in the direction x i , with τ i j the viscous stress tensor and l j the coordinates of the outward normal from the fluid. Here, the pressure normal to the plate is assumed to be dominant compared with the shear stresses, as in the work of Preisser (1979). Under this assumption and by considering that the observer is in the acoustic far field, the equation (3.3) can be simplified as

p ′ (x, t) = 1 4πc 0 ∂ ∂t S x i -y i R 2 p y, t - R c 0 l i dS(y) (3.4)
where the time derivative is evaluated using fourth-order centered finite differences. Equation (3.4) will be used to estimate the fluctuations of pressure both upstream and downstream of the plate. It may be noted that these fluctuations can also be evaluated using an analytical Green's function designed for a perforated plate, such as in the work of Howe (1979). This approach has not been carried out in the present work.

To examine the source terms in equation (3.4), the pressure spectra obtained on the plate at z = L and r = 3r 0 are displayed in figure 22 for the four impinging jets. The levels are highest for jetnohole and they decrease with the hole diameter. For jetnohole, jetsmallhole and jetmediumhole, tones are found at a Strouhal number of 0.4 and its harmonic frequencies, while for jetlargehole, a hump emerging by about 15 dB from the broadband levels is observed for frequencies lower than St = 0.7. In all cases, the spectra look like those near the nozzle in figure 12, suggesting that the spectral content of the sound waves computed using the analogy will be similar to that of the LES sound fields. The resemblance between the pressure spectra on the plate and those in the acoustic field has also been noted for supersonic impinging jets in the numerical study of Gojon & Bogey (2018).

Acoustic field upstream of the plate

The pressure fields obtained upstream of the plate using equation (3.4) for the four impinging jets are represented in figures 23(a-d). For all jets, periodic waves originating from the plate are observed. They look like the acoustic waves propagating in the LES pressure fields, given in figures 23(e-h), as expected. In the two cases, for r 5r 0 , the levels of fluctuating pressure are close and the wavefronts are located at similar positions. In particular, the sound waves radiated in the radial direction are well reproduced using the analogy for jetsmallhole and jetmediumhole, in figures 23(b,f) and 23(c,g). Therefore, the surfacic term of Curle's analogy is able to predict an acoustic radiation consistent with the LES results, verifying the validity of the analogy for the present study. Close to the jet flow, for r 5r 0 , some discrepancies, such as a phase shift, can be seen between the pressure waves obtained from the analogy and the LES. However, the comparison between the two pressure fields is not fully relevant here due to the fact that equation (3.4) is reliable only in the far field.

The acoustic spectra computed near the nozzle at z = 0 and r = 1.5r 0 using equation (3.4) are compared with the LES spectra in figure 24. For the four impinging jets, a good agreement is found. In particular, the effects of the hole and its size are well reproduced. For jetnohole, jetsmallhole and jetmediumhole, in figures 24(a-c), the spectra obtained from the analogy and the LES exhibit tones at the same frequencies, including the dominant tone at St = 0.40-0.41 and its harmonics, unsurprisingly given the frequency contents of the pressure spectra on the plate. However, the tone levels do not match perfectly. For instance, the dominant tones are 4 to 10 dB stronger in the LES than using the acoustic analogy. These discrepancies may be related to the noise radiation of aerodynamic phenomena such as the distortion of the vortical structures near the plate, not taken into account using the analogy. As for the broadband levels obtained using the analogy, for jetsmallhole and jetmediumhole, they are similar to those of the LES, with less than 3 dB of differences. For jetnohole, they are close to the LES levels for St > 1, but they are about 5 dB lower for St < 1. This result suggests that in this case, the contribution of the pressure fluctuations on the plate to the broadband noise is dominant at high frequencies. As mentioned in section 3.5.1, the reflected and scattered sound fields are related to the surfacic term of Curle's analogy, whereas the direct sound field consists of the jet mixing noise and of sound components linked to the flow distortion near the plate. With the aim of evaluating more precisely these two contributions to the pressure field, the sound levels estimated at z = 0 using Curle's analogy for jetsmallhole and jetlargehole are shown in figure 25, together with the levels obtained by the LES and for the free jet. The results for jetnohole and jetmediumhole, similar to those for jetsmallhole, are not represented for brevity. In all cases, the sound levels from the analogy and the LES decrease with the radial distance. For jetsmallhole, in figure 25(a), close to the nozzle, the results from the LES and the analogy exhibit discrepancies. Compared with the levels in the LES, for r 3r 0 , the levels predicted by the analogy are lower, with differences up to 9 dB, whereas for 3r 0 r 8r 0 , they are higher, with a maximum discrepancy of 4 dB. These differences may be due to the neglection of the aerodynamic noise sources in the analogy, especially those related to the impact of the jet mixing layers on the plate. Farther from the axis, for r 8r 0 , the levels given by the analogy are close to those from the LES, with less than 2 dB of differences. The present results highlight a significant contribution of the fluctuations of pressure on the plate to the noise radiated upstream. For jetlargehole, in figure 25(b), the levels obtained using the analogy are similar to those from the LES, showing less than 3 dB of differences. This good agreement suggests that the upstream-propagating sound waves are mainly produced by the scattering of the jet pressure fluctuations by the plate.

Acoustic field downstream of the plate

The pressure fields obtained downstream of the plate using Curle's analogy for the three jets impinging on a perforated plate are represented in figures 26(a-c). In all cases, periodic and almost circular pressure waves are found to originate from the hole in the plates. Their wavelengths and their magnitudes are similar to those of the sound waves propagating in the LES fields, shown in figure 26(d-f). Therefore, the variations of pressure on the plate by the jet flow generate the main part of the downstream acoustic radiation for z 20r 0 . However, the shapes of the wavefronts are slightly different, especially close to the jet flow.

These differences may be due to the refraction of sound waves by the flow, not predicted by the analogy.

The pressure spectra calculated at z = 12r 0 and r = 15r 0 using the analogy are plotted in figure 27. They are very similar to those obtained from the LES. For jetsmallhole and jetmediumhole, in figures 27(a,b), they display peaks at the dominant feedback frequency St = 0.40 and its harmonic frequencies. For the two first peaks for jetsmallhole and the first one for jetmediumhole, the levels are close to the LES levels, with differences lower than 4 dB. Furthermore, the broadband levels determined using the analogy and the LES do not differ by more than 3 dB. Therefore, the acoustic radiation downstream of the plate seems to be mainly produced by the pressure fluctuations on the plate due to the flow impingement.

Discrepancies can however be noted. Indeed, for the third and fourth tones for jetsmallhole and for the second and third tones for jetmediumhole, the levels evaluated using the analogy do not agree well with the LES results. Therefore, these tones may be due to noise generation mechanisms not taken into account by Curle's analogy, such as the interactions of turbulent structures just downstream of the plate. Finally, for jetlargehole, in figure 27(c), the spectrum predicted at z = 12r 0 and r = 15r 0 using the analogy is close to that of the LES, with a very good agreement for St < 0.6 and discrepancies lower than 3 dB for higher frequencies. The sound waves reaching the point at z = 12r 0 and r = 15r 0 thus appear to be mainly generated by the scattering of the jet pressure waves by the hole edge, and not by the turbulent jet developing past the hole.

To quantify more accurately the contribution of the scattered sound waves to the downstream pressure field, the sound levels obtained at r = 15r 0 for z 10r 0 using the analogy for jetsmallhole and jetlargehole are displayed in figure 28. They are compared with the levels for the LES and for the free jet. The results for jetmediumhole, not shown here, look like those for jetsmallhole. For jetsmallhole, in figure 28(a), the levels from the analogy do not vary much with the axial distance, and are approximately equal to 140 dB. They are close to those of the LES down to z = 22r 0 , with less than 2 dB of discrepancies. Farther downstream, the LES levels decrease down to 130 dB at z = 40r 0 . This reduction may be due to the refraction of the sound waves by the jet flow, neglected in the analogy. For jetlargehole, in figure 28(b), the levels in the LES are almost constant and close to 122 dB, while those for the analogy are lower and decrease with the axial distance. For z 15r 0 , the analogy provides levels lower by less than 3 dB than the LES levels, which suggests that most of the noise in this region is produced by the waves scattered by the plate. Farther downstream, the discrepancy raises up to 9 dB at z = 40r 0 . Therefore, far from the plate, the sound radiation does not seem to be substantially due to the interactions between the jet and the plate. In particular, for z 30r 0 , the LES levels are similar to those for the free jet, implying that the jet mixing noise is the main noise component for low radiation angles.

Conclusion

In this paper, the production of tones by a round jet at a Mach number of 0.9 impinging on a plate with and without a hole of varying diameter h has been investigated using compressible large-eddy simulations. For all plate geometries, tones emerge in the pressure spectra. Their frequencies are found to be nearly independent of the presence and of the size of the hole in the plate, in agreement with the frequencies expected for aeroacoustic feedback loops establishing between the nozzle lips and the plate, involving turbulent structures convected downstream by the jet flow and acoustic waves travelling upstream. The latter waves moreover appear to belong to the family of the free-stream upstream-propagating guided waves of the jet, allowing us to explain the azimuthal structure of the jet oscillation mode at each tone frequency. Regarding the tone intensities, they decrease when the hole in the plate is larger, weakly when the hole diameter increases from h = 2r 0 to h = 3r 0 , but very strongly between h = 3r 0 and h = 4.4r 0 . The reason for that is shown to be the different mechanisms producing the acoustic waves closing the feedback loops. Indeed, these waves are generated directly by the impingement of the jet flow structures on the plate for the two smallest holes, but by the scattering of the jet aerodynamic pressure fluctuations by the plate for the largest hole, leading to much weaker tones in that case. These results emphasize the importance of the nature of the interactions between the jet flow and the plate in producing strongly emerging acoustic tones. Finally, an acoustic analogy based on the variations of the pressure on the plate has also been used to study the nature of the dominant noise sources. It provides results in good agreement with the LES, showing the significant contribution of the pressure fluctuations on the plate to the pressure field. In the future, it could be interesting to examine whether the present phenomena observed for a high-subsonic jet will also be obtained for supersonic jets impinging on a perforated plate. In particular, this may not be the case for impinging rocket jets, for which no acoustic tones have been reported in the literature.

Figure 1 :

 1 Figure 1: Nozzle-exit radial profiles of (a) mean axial velocity u z and (b) axial turbulence intensity u ′ z u ′ z 1/2 /u j : jetnohole, jetsmallhole, jetmediumhole, jetlargehole and ---free jet.

Figure 2 :

 2 Figure 2: Variations of (a) radial and (b) axial mesh spacings; positions of the upstream and downstream faces of the plate.

Figure 3 :

 3 Figure 3: Snapshots in the (z, r) plane of the vorticity norm for (a) jetnohole, (b) jetsmallhole, (c) jetmediumhole, (d) jetlargehole and (e) the free jet. The color scales range from 0 up to 15u j /r 0 , from white to red.

Figure 4 :

 4 Figure 4: Snapshots in the (z, r) plane of the vorticity norm in the flow and of the fluctuating pressure outside for (a) jetnohole, (b) jetsmallhole, (c) jetmediumhole, (d) jetlargehole and (e) the free jet. The color scales range from 0 up to 7u j /r 0 for the vorticity norm, from white to red, and between (a-c) ±0.01p 0 , (d) ±0.0025p 0 and (e) ±0.001p 0 for the pressure, from blue to red.

Figure 5 :

 5 Figure 5: Mean density fields in the (z, r) plane for (a) jetnohole, (b) jetsmallhole, (c) jetmediumhole, (d) jetlargehole and (e) the free jet. The color scale ranges from 0.95ρ 0 up to 1.05ρ 0 , from blue to red.

Figure 6 :

 6 Figure 6: Variations of (a) the jet mass flow rate m in the axial direction and (b) the mass flow rate of the wall jet m W J in the radial direction for jetnohole, jetsmallhole, jetmediumhole, jetlargehole and ---the free jet (Bogey (2018)).

Figure 7 :Figure 8 :

 78 figure 8(b). For jetnohole, the turbulent intensity shows a local maximum at r = r 0 . It then increases up to 0.2u j at r = 4r 0 , and decreases farther from the jet axis. For jetsmallhole and jetmediumhole, the turbulent levels are lower, and reach maximum values of 0.15u j at

Figure 9 :Figure 10 :

 910 Figure 9: Power spectral densities of radial velocity fluctuations u ′ r obtained at r = r 0 and z = 0.4r 0 for (a) jetnohole, (b) jetsmallhole, (c) jetmediumhole, (d) jetlargehole and (e) the free jet; full signal, n θ = 0 and n θ = 1, ---St θ = f δ θ /u j = 0.016.

Figure 11 :

 11 Figure 11: Overall sound pressure levels (OASPL) in the (z, r) plane for (a) jetnohole, (b) jetsmallhole, (c) jetmediumhole, (d) jetlargehole and (e) the free jet. The levels are separated by increments of 3 dB.

Figure 12 :

 12 Figure 12: Sound pressure levels (SPL) at (a) z = 0 and r = 1.5r 0 and (b) z = 12r 0 and r = 15r 0 as a function of the Strouhal number St = f D/u j ; jetnohole, jetsmallhole, jetmediumhole, jetlargehole and ---free jet.

  convection velocity are equal to St = 0.42 and 0.69, respectively. The first value is in good agreement with the Strouhal numbers of 0.40 and 0.41 of the dominant tones emerging in the spectra. The second frequency value is close or equal to the Strouhal numbers of the tones at St = 0.60, 0.64 and 0.69 in the spectra of jetnohole, jetsmallhole and jetmediumhole. These results support the establishment of feedback loops at the tone frequencies. For the free jet, the tonal peaks in the spectra are due to the presence of guided jet waves, as documented by Towne et al. (2017) and Schmidt et al. (2017) and discussed very recently by Bogey

Figure 13 :

 13 Figure 13: Sound pressure levels (SPL) at z = 0 and r = 1.5r 0 for (a) jetnohole, (b) jetsmallhole, (c) jetmediumhole and (d) jetlargehole for the azimuthal modes n θ = 0 and n θ = 1; SPL for --n θ = 0 and --n θ = 1 for the free jet.

Figure 14 :

 14 Figure 14: Strouhal numbers of the tones as a function of the hole diameter for (a) n θ = 0 and (b) n θ = 1, frequencies obtained from equation (3.1) for increasing values of N.

Figure 15 :

 15 Figure 15: Dominant tone levels at z = 0 and • r = 1.5r 0 and △ r = 15r 0 as a function of the hole diameter for (a) n θ = 0 and (b) n θ = 1.

Figure 16 :

 16 Figure16: Correlations R of p ′ (r = 1.5r 0 , z = 0, t) with p ′ (r, z, t + δt) for n θ = 1 for (a) δt = -15r 0 /u j , (b) δt = -12r 0 /u j , (c) δt = -9r 0 /u j , (d) δt = -6r 0 /u j , (e) δt = -3r 0 /u j , (f) δt = 0, (g) δt = 3r 0 /u j and (h) δt = 6r 0 /u j for jetmediumhole; circle centered on (z = L, r = 1.5r 0 ). The color scale ranges between ±1, from blue to red.

Figure 18 :

 18 Figure 18: Correlations R of p ′ (r = 15r 0 , z = 12r 0 , t) with p ′ (r, z, t + δt) for n θ = 1 for (a) δt = -15r 0 /u j , (b) δt = -10r 0 /u j and (c) δt = -5r 0 /u j for jetmediumhole;circle centered on (z = L, r = 1.5r 0 ). The color scale ranges between ±1, from blue to red.

Figure 19 :

 19 Figure 19: Correlations R of p ′ (r = 15r 0 , z = 12r 0 , t) with p ′ (r, z, t + δt) for n θ = 1 for (a) δt = -15r 0 /u j , (b) δt = -10r 0 /u j and (c) δt = -5r 0 /u j for jetlargehole; circle centered on (z = L, r = 2.2r 0 ). The color scale ranges between ±1, from blue to red.

Figure 20 :

 20 Figure 20: Frequency-wavenumber spectra of the pressure fluctuations at r = r 0 for (a,d) jetmediumhole, (b,e) jetlargehole and (c,f) the free jet for (top) n θ = 0 and (bottom) n θ = 1; dispersion curves and * lower limits of the guided jet modes for a vortex-sheet model, ◭ LES tone frequencies and --k = -ω/c 0 . The greyscale levels spread over 25 dB.

Figure 21 :

 21 Figure 21: Frequency-wavenumber spectrum of the pressure fluctuations at r = r 0 for jetmediumhole for n θ = 0; dispersion curves and * lower limits of the guided jet modes for a vortex-sheet model, ◭ LES tone frequencies, --k = -ω/c 0 and --k = ω/u c . The greyscale levels spread over 25 dB.

Figure 22 :

 22 Figure 22: Pressure spectra on the plate at z = L and r = 3r 0 for jetnohole, jetsmallhole, jetmediumhole and jetlargehole.

Figure 23 :

 23 Figure 23: Pressure fields obtained upstream of the plate (top) using equation (3.4) and (bottom) from LES for (a,e) jetnohole, (b,f) jetsmallhole, (c,g) jetmediumhole and (d,h) jetlargehole. The color scales range between (a,b,c,e,f,g) ±0.01p 0 , and (d,h) ±0.0025p 0 , from blue to red.

Figure 24 :Figure 25 :

 2425 Figure 24: Sound pressure levels obtained at z = 0 and r = 1.5r 0 from LES and using equation (3.4), for (a) jetnohole, (b) jetsmallhole, (c) jetmediumhole and (d) jetlargehole, as a function of the Strouhal number.

Figure 26 :

 26 Figure 26: Pressure fields obtained downstream of the plate (top) using equation (3.4) and (bottom) from LES for (a,d) jetsmallhole, (b,e) jetmediumhole and (c,f) jetlargehole. The color scales range between (a,b,d,e) ±0.01p 0 , and (c,f) ±0.0025p 0 , from blue to red.

Figure 27 :

 27 Figure 27: Sound pressure levels obtained at z = 12r 0 and r = 15r 0 from LES and using equation (3.4) for (a) jetsmallhole, (b) jetmediumhole and (c) jetlargehole, as a function of the Strouhal number.

Figure 28 :

 28 Figure 28: Overall sound pressure levels obtained downstream of the plate at r = 15r 0 from LES and using equation (3.4), for (a) jetsmallhole and (b) jetlargehole, ---levels for jetfree.

Table 1 :

 1 Mesh parameters: numbers of points N r , N θ and N z in the radial, azimuthal and axial directions, and total number of points.

	This filter enables to dissipate kinetic

Acknowledgments

This work was financed by ArianeGroup and the DGA (Direction Générale de l'Armement) and by the IRICE IJES project RA0014963 (Installed Jet Effect Simulator, FEDER-FSE Rhône-Alpes). It was granted access to the HPC resources of PMCS2I (Pôle de Modélisation et de Calcul en Sciences de l'Ingénieur et de l'Information) of Ecole Centrale de Lyon, PSMN (Pôle Scientifique de Modélisation Numérique) of ENS de Lyon and P2CHPD (Pôle de Calcul Hautes Performances Dédiés) of Université Lyon I, members of FLMSN (Fédération Lyonnaise de Modélisation et Sciences Numériques), partner of EQUIPEX EQUIP@MESO, and to the resources of TGCC (Très Grand Centre de calcul du CEA) under the allocation 2020-2a0204 made by GENCI (Grand Equipement National de Calcul Intensif). It was performed within the framework of the Labex CeLyA of Université de Lyon, operated by the French National Research Agency (grant no. ANR-10-LABX-0060/ANR-16-IDEX-0005).

Declaration of interests

The authors report no conflict of interest. , C. 2018 Grid sensitivity of flow field and noise of high-Reynolds-number jets computed by largeeddy simulation. International Journal of Aeroacoustics 17 (4-5), 399-424.