Issue 
Acta Acust.
Volume 6, 2022
Topical Issue  Aeroacoustics: state of art and future trends



Article Number  36  
Number of page(s)  11  
DOI  https://doi.org/10.1051/aacus/2022033  
Published online  26 August 2022 
Scientific Article
Presence and properties of acoustic peaks near the nozzle of impinging rocket jets
Univ Lyon, École Centrale de Lyon, INSA Lyon, Université Claude Bernard Lyon I, CNRS, Laboratoire de Mécanique des Fluides et d’Acoustique, UMR 5509, 69134 Écully, France
^{*} Corresponding author: mathieu.vare@eclyon.fr
Received:
25
March
2022
Accepted:
8
August
2022
The presence and properties of acoustic peaks near the nozzle of impinging rocket jets have been investigated. Four jets at a Mach number of 3.1 impinging on a plate at a distance L = 15r_{0}, 20r_{0}, 25r_{0} and 30r_{0} from the nozzle, where r_{0} is the nozzle radius, have been computed using largeeddy simulations. In all cases, upstreamtravelling pressure waves are generated by the jet impingement on the plate, with amplitudes decreasing with the nozzletoplate distance. The nearnozzle pressure spectra contain peaks, at frequencies not varying much with this distance. For L ≥ 20r_{0}, the spectra are dominated by a lowfrequency peak, whereas two additional highfrequency peaks emerge for L = 15r_{0}. The lowfrequency peak is associated with the azimuthal mode n_{θ} = 0, whereas the two other ones are due to strong components for modes n_{θ} ≥ 1. As for nearnozzle tones for free and impinging jets at lower Mach numbers, the peak frequencies fall close to the frequency bands of the upstreampropagating guided jet waves, showing a link between the peaks and the latter waves. Regarding the peak levels, they do not change significantly with the nozzletoplate distance for the lowfrequency peak, but they decrease by 1.5 to 18 dB as the distance increases for the other peaks. Finally, for L ≥ 20r_{0}, the nearnozzle peak frequency is close to that of the strongest shearlayer structures, indicating a connexion between the upstream noise and these structures. For L = 15r_{0}, a shockleakage mechanism of a nearplate shock is found to generate the upstream noise.
Key words: Jet noise / Impinging rocket jet / Acoustic peaks
© The Author(s), published by EDP Sciences, 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
During the liftoff of a space launcher, the hot supersonic gases of the engines impinge on the launch area, which generates intense sound waves. These waves propagate to the launcher fairing, where they exert strong acoustic loads likely to damage the payload. The sound fields near the rocket structure thus require a specific attention from the aerospace industry, as highlighted in the review by Lubert et al. [1]. In order to characterize them, the impingement of the gases exhausted by the engines on the launchpad can be modeled as a jet impinging on a flat plate. Such a simplified setup has been studied experimentally and numerically for highsubsonic jets [2–8]. Intense tones were reported in the pressure spectra close to the jet nozzle. Similar tones were later found for impinging supersonic jets at Mach numbers lower than 2 in experiments [9–14] and simulations [15–18]. They are generated by a feedback loop establishing between the nozzle and the plate, formed by downstreampropagating Kelvin–Helmholtz instability waves and upstreampropagating guided jet waves. More recently, tones of weaker amplitude were also noticed in the vicinity of the nozzle of free ideally expanded jets at Mach numbers between 0.5 and 2 [19–21]. Their frequencies lie in the allowable frequency ranges of the upstreampropagating guided jet waves, highlighting a close link with these waves.
For impinging rocket jets at Mach numbers higher than 3, the presence of such peaks close to the nozzle is not obvious. It was not mentioned in references [22–24] reporting simulations of jets at Mach numbers from 3.1 to 3.66 impinging on a plate at a distance L between 30 and 40 nozzle radii r_{0} from the nozzle. In a recent work by the authors dealing with a similar configuration [25], a broad peak was, however, noticed at a low frequency in the pressure spectra near the nozzle. Its frequency is located in the allowable bands of the upstreampropagating guided jet waves, which suggests the establishment of a feedback phenomenon. Nevertheless, the peak is large and has a low amplitude, which questions the existence of such a feedback. In addition, it is unclear if the absence of marked peaks in the nearnozzle pressure fields of impinging rocket jets is due to the high Mach number of the jets or to the large nozzletoplate distance considered.
In the present paper, the presence of acoustic peaks in the nearnozzle pressure fields of rocket jets impinging on a flat plate is examined by performing LargeEddy Simulations (LES) of four overexpanded jets. The jets are at an exit Mach number M_{e} of 3.1 and a Reynolds number Re_{D} of 2 × 10^{5}. They impinge on a plate located at a distance of L = 15r_{0}, 20r_{0}, 25r_{0} or 30r_{0} from the nozzle. The nozzletoplate distances are shorter than those of previous works [22–25] in order to discuss the possible establishment of feedback mechanisms between the nozzle and the plate. The first objective is to study the effects of the nozzletoplate distance on the sound waves propagating in the upstream direction. For that, the jet flow and acoustic fields are described. The nearnozzle pressure spectra are notably scrutinized in order to detect the emergence of possible peaks. The second objective is to analyze the characteristics of the nearnozzle peaks. To this end, the azimuthal structure of the sound field in the nearnozzle region is revealed. The variations of the frequencies and amplitudes of the peaks with the nozzletoplate distance are described. The peak frequencies are also compared with the frequencies bands allowed for the upstreampropagating guided jet waves for the present jets according to a vortexsheet model. Finally, the last objective is to investigate the noise generation mechanisms of the impinging rocket jets. For this, properties of the flow fields close to the impingement area, including velocity spectra and motions of nearplate shocks, are presented.
This paper is organized as follows. The jet parameters and numerical methods used in the LES are documented in Section 2. The results of the simulations are presented in Section 3. Finally, concluding remarks are given in Section 4.
2 Parameters
2.1 Jet parameters
The exhaust parameters of the present jets are identical to those of the jets considered in a recent work [25]. The geometric parameters are described using cylindrical coordinates (r, θ, z). The jets have an exhaust Mach number M_{e} = u_{e}/c_{e} of 3.1 and a Reynolds number Re_{D} = u_{e}D/ν_{e} of 2 × 10^{5}, where u_{e} is the velocity, c_{e} is the sound speed, D = 2r_{0} is the nozzle diameter and ν_{e} is the kinematic viscosity at the jet nozzle exit. The exhaust temperature T_{e} and pressure p_{e} are equal to 738 K and 0.63p_{0}, where p_{0} = 10^{5} Pa is the ambient pressure. The jets impinge on a plate located at a distance L = 15r_{0}, 20r_{0}, 25r_{0} or 30r_{0} from the nozzle exit. The distance of L = 30r_{0} corresponds to that for the impinging jets in Varé and Bogey [25], whereas the other distances are shorter in order to discuss the presence of feedback phenomena. The jets are generated by a straight pipe nozzle of length 2r_{0}, at the inlet of which Blasius boundary layer velocity profiles with a thickness of 0.15r_{0} are imposed. The boundary layer transition from a fully laminar to a disturbed state is triggered by adding vortical disturbances noncorrelated in the azimuthal direction in the pipe at z = −r_{0} [26]. The mean velocity profiles obtained at the nozzleexit z = 0 for the different jets are close to each other and very similar to those for the jets in Varé and Bogey [25]. The radial profiles of axial turbulence intensity 〈u′_{z} u′_{z}〉^{1/2}/u_{e} at z = 0 are plotted in Figure 1 for the four impinging jets and for the corresponding free jet [25]. In all cases, they reach a peak at r ≈ 0.9r_{0}. The peak value ranges between 1% and 1.6% of the exit velocity, indicating weakly disturbed nozzleexit flow conditions. It is minimum for the free jet. For the impinging jets, it increases as the nozzletoplate distance is reduced, which suggests that the jet shear layers are disturbed by stronger upstreampropagating sound waves for shorter distances.
Figure 1 Nozzleexit profiles of axial turbulence intensity ${{\langle u\mathrm{\prime}}_{z}u\mathrm{\prime}}_{z}{\rangle}^{1/2}/{u}_{e}$ for L = 15r_{0}, L = 20r_{0}, L = 25r_{0}, L = 30r_{0}, and for the free jet [25]. 
2.2 Numerical methods
The numerical setup is identical to that used in previous works by the authors [8, 25]. In the simulations, the unsteady compressible Navier–Stokes equations are solved in cylindrical coordinates using an OpenMP based inhouse solver. The time integration is performed using a sixstage Runge–Kutta algorithm and the spatial derivatives are evaluated with elevenpoint lowdispersion finitedifference schemes [27] ensuring high accuracy down to five points per wavelength. At the end of each time step, gridtogrid oscillations are removed by applying a selective filtering [28]. This filter also dissipates kinetic turbulent energy near the grid cutoff frequency, acting as a subgridscale model [29]. Solid and adiabatic wall conditions are imposed at the nozzle and plate walls. In order to handle shock waves, a damping procedure using a dilatationbased shock detector and a secondorder filter are used to remove Gibbs oscillations in the vicinity of shocks [28]. For L ≤ 25r_{0}, to ensure the stability of the computation, this procedure is applied twice in the nearwall region for r ≤ 2r_{0} and L–2r_{0} ≤ z ≤ L. For the same reason, a standard fourthorder filter is also applied in this area for the cases L = 15r_{0} and L = 20r_{0}. The radiation boundary conditions of Tam and Dong [30] are implemented at the radial and lateral boundaries of the computational domain. They are combined with sponge zones using grid stretching and Laplacian filtering to prevent significant spurious reflections. The singularity on the jet axis is removed by applying the method of Mohseni and Colonius [31].
2.3. Computational parameters
In the four simulations, the numbers of points in the radial and azimuthal directions are equal to N_{r} = 501 and N_{θ} = 256, respectively. In the axial direction, the number of points N_{z} is equal to 1291 for L = 15r_{0}, 1531 for L = 20r_{0}, 1752 for L = 25r_{0} and 1910 for L = 30r_{0} yielding a total number of points between 170 and 250 millions. The grids extend out to r = 15r_{0} in the radial direction and down to z = L in the axial direction. The radial mesh grid used is identical to that in the LES of Varé and Bogey [25]. It notably enables a cutoff Strouhal number St = fD/u_{e} of 1.62 for an acoustic wave discretized with five points per wavelength, where f is the frequency, in the jet nearacoustic field. The variations of the axial mesh spacing Δz are plotted in Figure 2. In all cases, the value of Δz is minimum and equal to 0.0144r_{0} at the nozzle exit and reaches its maximum value at z = L – 10r_{0}. This value is equal to 0.0173r_{0} for L = 15r_{0}, 0.019r_{0} for L = 20r_{0}, 0.0205r_{0} for L = 25r_{0} and 0.022r_{0} for L = 30r_{0}. The mesh spacing is constant between z = L – 10r_{0} and z = L – 5r_{0}, and finally decreases down to Δz = 0.0144r_{0} at z = L.
Figure 2 Variations of axial mesh spacing Δz for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 
The radial variations of the mesh spacing at z = L on the plate in the wallnormal direction Δz^{+}, in wall units, are represented in Figure 3 to discuss the discretization of the wall jet created by the jet impingement. This spacing is computed using the relation Δz^{+} = u_{τ} Δz/ν, where u_{τ} is the mean friction velocity, Δz is the axial mesh spacing on the plate and ν is the kinematic viscosity. For all jets, it reaches a maximum value at a radial distance between r = 0.5r_{0} and r = r_{0}. This value is equal to 23 for L = 15r_{0} and decreases as the nozzletoplate distance is larger down to 15 for L = 30r_{0}. Farther from the axis, the dimensionless mesh spacing on the plate decreases for the four jets. It remains, however, much higher than 1, indicating that the turbulent boundary layer on the plate is not well resolved [32–35]. Nevertheless, the sound radiation of the wall jet is expected to be weak compared with that produced by the jet flow structures, as the wall jet velocity is significantly lower than the jet exhaust velocity.
Figure 3 Variations of the axial mesh spacing Δz^{+} on the plate, in wall units, for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 
The results in the present paper are obtained for a simulation time of 1000r_{0}/u_{e}. During the computations, density, the velocity components, pressure and their azimuthal Fourier coefficients up to the mode n_{θ} = 4 are recorded on the same twodimensional sections and at the same sampling frequencies as in Varé and Bogey [25]. These sections notably include the nozzleexit plane at z = 0 and four planes at the azimuthal angles θ = 0, 90, 180 and 270 degrees. The spectra presented below are calculated from these recordings and they are averaged in the azimuthal direction when possible. On the whole, 200,000 time steps were performed for each jet, consuming a total of 130,000 CPU hours, using 32 core nodes of Intel 6142 Skylake with a clock frequency of 2.6 GHz.
3 Results
3.1 Snapshots of the flow and acoustic fields
Fields of temperature and pressure fluctuations obtained inside and outside of the flow, respectively, are represented in Figure 4. In the temperature fields, diamond patterns typical of shock cells are present in the four jets downstream of the nozzle exit. The cells are progressively dampened by the turbulent mixing for z ≥ 10r_{0}. Wall jets are created by the impingement of the flow on the plate. Zones of high temperature are also found in the impingement area. In this region, the temperature decreases as the nozzletoplate increases, indicating weaker jetplate interactions for larger L.
Figure 4 Snapshots in the (z,r) plane of temperature fluctuations in the flow and of pressure fluctuations outside for (a) L = 15r_{0}, (b) L = 20r_{0}, (c) L = 25r_{0} and (d) L = 30r_{0}. The red arrows show the wavefronts of two consecutive upstreampropagating pressure waves for the nozzletoplate distances L = 15r_{0} and 25r_{0}. The color scales range from 0 to 780K for temperature, from red to white, and from −2000 to 2000 Pa for pressure, from black to white. 
In the pressure fields, waves originating from the impingement zone can be observed in the upstream direction. Their levels are typically of 3000 Pa for L = 15r_{0} but are lower as the nozzletoplate distance increases. The wavefronts of two consecutive upstreamtravelling sound waves are shown using red arrows for the nozzletoplate distances L = 15r_{0} and L = 25r_{0} in Figures 4a–4c to examine the frequency content of the upstream radiated noise. They are closer to each other for L = 15r_{0} than for L = 25r_{0}, indicating stronger highfrequency components in the first case. For all jets, inclined wavefronts are also seen to travel downstream. They are particularly noticeable for z ≥ 15r_{0} for L = 25r_{0} and L = 30r_{0} in Figures 4c and 4d. They are characteristic of Mach wave radiation, as noticed in previous simulations of jets at Mach numbers higher than 2 [36–39]. These waves are generated by turbulent coherent structures convected at a supersonic velocity.
3.2 Mean flow fields
The variations of the mean axial centerline velocity are represented in Figure 5a. The profiles obtained for the four jets are similar, suggesting that the nozzletoplate distance has little effects on the mean velocity fields. Downstream of the nozzle, the axial velocity oscillates strongly, due to the presence of 3 to 6 shock cells depending on the nozzletoplate distance. In particular, the plate is located in the third cell for L = 15r_{0}, at the end of the fourth cell for L = 20r_{0}, in the sixth cell for L = 25r_{0} and past the sixth cell for L = 30r_{0}. The amplitude of the oscillations decreases with the axial distance, because of the weakening of the cells by the turbulent mixing. The centerline velocity finally drops down to 0 at z = L. For the corresponding free jet, the end of the potential core, reached at the location z_{c} for which 〈u_{z}〉/u_{e} = 0.9, is found at z_{c} = 15.7r_{0} in Varé and Bogey [25]. Hence, the jet impinges on the plate before the end of the potential core for L = 15r_{0} and after the latter in the three other cases.
Figure 5 Variations of (a) the mean axial centerline velocity 〈u_{z}〉/u_{e} and (b) the axial turbulence intensity ${{\langle u\mathrm{\prime}}_{z}u\mathrm{\prime}}_{z}{\rangle}^{1/2}/{u}_{e}$ at r = r_{0} for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 
The variations of the axial turbulence intensity in the shear layer at r = r_{0} are plotted in Figure 5b. For all jets, the turbulence intensity remains below 1% from the nozzle exit down to z = 2.5r_{0}. Farther downstream, it increases sharply up to a peak value located around z ≈ 10r_{0}, equal to about 18% for all nozzletoplate distances. Farther downstream, the rms values of the velocity fluctuations decrease with the axial distance and reach a second peak value close to the plate, before falling abruptly down to 0 at z = L.
3.3 Pressure spectra
The pressure spectra computed near the nozzle at z = 0 and r = 2r_{0} are displayed as a function of the Strouhal number in Figure 6, along with the spectrum obtained for the free jet. For that jet, a peak centered around St = 0.06 is visible. It is linked to the broadband shockassociated noise (BBSAN) components produced by the interactions between the turbulent structures of the jet mixing layers and the shock cells [25]. For the impinging jets, the sound pressure levels are considerably higher than those for the free jet, by 5 to 22 dB depending on the nozzletoplate distance and the frequency. In that case, the noise generated by the jet impingement on the plate predominates strongly. For the three jets with L ≥ 20r_{0}, the spectra all reach a peak at a Strouhal number of 0.04, with an amplitude which does not vary significantly with the nozzletoplate distance. In contrast, for St ≥ 0.1, the sound levels increase as the plate is closer to the jet exit, and rise by approximately 3 dB for L = 25r_{0} and by 7 dB for L = 20r_{0} with respect to those for L = 30r_{0}. For L = 15r_{0}, the sound pressure spectrum is clearly different from the the spectra for the three larger nozzletoplate distances. In particular, the highfrequency components for St ≥ 0.1 are stronger by about 3 dB with respect to the case L = 20r_{0}, which is consistent with the pressure snapshots of Section 3.1. More importantly, the spectrum exhibits two additional peaks at St = 0.09 and 0.22.
Figure 6 Sound pressure levels (SPL) at z = 0 and r = 2r_{0} as a function of the Strouhal number St for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0} and the free jet [25]. 
The variations of the overall sound pressure levels (OASPL) obtained at z = 0 and r = 2r_{0} are represented as a function of the nozzletoplate distance in Figure 7. As expected, the levels are highest for L = 15r_{0} and decrease with L, from 158.8 dB for L = 15r_{0} down to 152.5 dB for L = 30r_{0}. Moreover, the levels for the largest nozzletoplate distance are 9.4 dB higher than those for the free jet, for which the value of L can be considered as infinite, showing that the impingement noise is significantly stronger than the mixing noise emitted by the free jet.
Figure 7 Overall sound pressure levels (OASPL) at z = 0 and r = 2r_{0} as a function of the nozzletoplate distance. 
3.4 Properties of the nearnozzle peaks
To examine the azimuthal structure of the nearnozzle pressure fields, the contributions of the first four azimuthal modes to the pressure spectra at z = 0 and r = 2r_{0} are shown in Figure 8 for L = 15r_{0} and L = 25r_{0}. The results for the two other cases are not presented here because they are similar to those for L = 25r_{0}. The azimuthal decompositions for the two nozzletoplate distances in Figure 8 display similarities. Peaks can be observed at comparable frequencies. They are found at St ≈ 0.04, 0.09 and 0.17 for n_{θ} = 0, at St ≈ 0.08 and 0.16 for n_{θ} = 1, at St = 0.2 for n_{θ} = 2–3 and at St = 0.3 for n_{θ} = 4. For both jets, the axisymmetric mode is dominant for St ≤ 0.06. As the frequency increases, the contributions of modes n_{θ} ≥ 1 become more significant. As a result, the levels for n_{θ} = 1 dominate for Strouhal numbers between 0.06 and 0.2, and the contributions of higher azimuthal modes to the pressure fields are notable for St ≥ 0.2 for n_{θ} = 2 and for St ≥ 0.3 for n_{θ} = 3–4. However, discrepancies are noted between the two nozzletoplate distances. For L = 15r_{0}, the peak at St = 0.09 for n_{θ} = 0 emerges 5 dB above that for L = 25r_{0}. The sound levels associated with the modes n_{θ} ≥ 1 for L = 15r_{0} are also stronger by about 5 dB than those for L = 25r_{0} over the whole frequency range. In particular, the mode n_{θ} = 1 contributes strongly to the levels at St = 0.09 and the contributions of the modes n_{θ} = 1 and 2 are significant at St = 0.22, causing the emergence of two peaks in the full spectrum.
Figure 8 Sound pressure spectra at z = 0 and r = 2r_{0} for (a) L = 15r_{0} and (b) L = 25r_{0}: full spectra and modes n_{θ} = 0, n_{θ} = 1, n_{θ} = 2, n_{θ} = 3 and n_{θ} = 4. 
The variations of the Strouhal numbers of the nearnozzle peaks for the first two azimuthal modes are represented in Figure 9 as a function of the nozzletoplate distance. The allowable frequency ranges of the upstreampropagating guided jet waves obtained using a vortexsheet model [40] are also displayed. These waves are organized in azimuthal and radial modes n_{θ} and n_{r}. For n_{θ} = 0 in Figure 9a, the Strouhal number of the dominant peak does not vary much with the nozzletoplate distance. It is located very close to the upper band of the first radial mode of the guided jet waves, with n_{r} = 1, indicating a link with these waves. The secondary peaks are found at frequencies slightly higher than those of the bands for the second, third and fourth radial modes of the guided jet waves, with n_{r} = 2 to 4. The discrepancies observed between the LES and the vortexsheet model may be due to the assumption of an infinitely thin shear layer in the model, leading to an underestimation of the frequencies of the guided jet waves and [17]. For n_{θ} = 1 in Figure 9b, the peak frequencies slightly vary with the nozzletoplate distance. They lie above the frequency bands of the first and second radial modes for the dominant and secondary peaks.
Figure 9 Strouhal numbers of the peaks in the nearnozzle pressure spectra as a function of the nozzletoplate distance for (a) n_{θ} = 0 and (b) n_{θ} = 1; • dominant and ∘ secondary peaks. The frequency bands of the upstreampropagating guided jet waves are in grey. 
Finally, the variations of the amplitudes of the peaks with the nozzletoplate distance are plotted in Figure 10 for the first two azimuthal modes. For n_{θ} = 0 in Figure 10a, the level of the dominant peak, associated with the guided jet mode n_{r} = 1, is close to 162 dB in all cases. On the contrary, for the secondary peak associated with n_{r} = 2, the peak amplitude decreases continuously with the nozzletoplate distance from 160 dB for L = 15r_{0} down to 142 dB for L = 30r_{0}. For n_{θ} = 1 in Figure 10b, the peak amplitude also decreases strongly with the distance. It is equal to 159 dB for L = 15r_{0} but only to 154 dB for L = 30r_{0}. Therefore, the levels of the highfrequency peaks strongly depend on the nozzletoplate distance.
Figure 10 Levels of the peaks in the nearnozzle pressure spectra as a function of the nozzletoplate distance for (a) n_{θ} = 0 and (b) n_{θ} = 1; • dominant and ∘ secondary peaks. 
3.5 Velocity spectra
To link the sound spectra with the jet flow structures, the power spectral densities of the fluctuations of axial velocity obtained in the shear layer at r = r_{0} between the nozzle exit and the plate are represented in Figure 11 for the four jettoplate distances. The frequencies of the peaks in the nearnozzle spectrum for L = 15r_{0} are indicated by dashed lines. In all cases, the levels are very weak from the nozzleexit plane down to z = 10r_{0}. These low levels can be explained by the fact that the shear layer is not centered on r = r_{0} because of the overexpansion of the jets. Farther downstream, down to z = L – 3r_{0}, spots of significant energy are observed in the spectra for Strouhal numbers lower than 0.15. These lowfrequency spots are related to largescale vortical structures convected downstream. The frequencies of the peak components decrease with the axial distance. For example, for L = 30r_{0} in Figure 11c, they vary from St = 0.1 at z = 11r_{0} down to St = 0.075 at z = 25r_{0}. In all cases, close to the plate, the levels are especially strong. They are highest in a spot around z = L – r_{0}, due to the flow impingement. High levels are found in these spots for Strouhal numbers below 0.3 for L = 15r_{0}, 0.2 for L = 20r_{0} and 0.11 for L = 30r_{0}. Given that the jet structures are deviated in the wall jet a few radii upstream of the plate, these components can be expected to be linked to shocks or compressions near the plate rather than to the flow vortical structures.
Figure 11 Power spectral densities of the fluctuations of axial velocity u′_{z} at r = r_{0} between z = 0 and L for (a) L = 15r_{0}, (b) L = 20r_{0}, (c) L = 25r_{0} and (d) L = 30r_{0}. The color scale is the same in the three cases and spreads over 3 dB, from white to black. The black dashed lines indicate the frequencies St = 0.04, 0.09 and 0.22. 
Outside of the impingement area, the highest levels are found at z_{max} = 11.7r_{0} for L = 15r_{0}, 16.5r_{0} for L = 20r_{0} and L = 30r_{0} and 20.1r_{0} for L = 25r_{0}. To highlight the frequency contents of the most energetic flow structures, the velocity spectra obtained at these positions are shown in Figure 12. They reach their maximum value at a Strouhal number depending on the nozzletoplate distance. This Strouhal number is equal to St = 0.07 for L = 15r_{0} and to St = 0.03 – 0.04 for L ≥ 20r_{0}. The frequencies of the strongest flow structures are lower for the three largest nozzletoplate distances, because the jet potential core closes several nozzle radii upstream of the plate in these cases. For L ≥ 20r_{0}, the peak frequency in the nearnozzle pressure spectra are close to those in the velocity spectra, suggesting a link between the most energetic flow structures and the nearnozzle pressure fields. However, for L = 15r_{0}, the peak frequency in the velocity spectrum is different from those in the nearnozzle sound spectra. In this case, the nearnozzle acoustic components may be not related to the strongest flow structures but to phenomena occuring in the impact zone such as shock leakage as discussed in what follows.
Figure 12 Power spectral densities of the fluctuations of axial velocity ${u}_{z\mathrm{\prime}}$ at r = r_{0} and z = z_{max} for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 
3.6 Sound generation near the plate
In order to examine the noise generation in the region of jet impingement, the fields of the axial gradient of density and of vorticity near the plate are represented in Figure 13 for L = 15r_{0} and L = 25r_{0}. In the vorticity fields, in both cases, the turbulent structures of the shear layers impinge on the plate, creating a wall jet. Near the jet axis, for L = 15r_{0} in Figure 13a, the level of vorticity is quite low, as the nozzletoplate distance is smaller than the size of the potential core for the free jet, whereas for L = 25r_{0} in Figure 13b, vortical structures are found as the mixing layers have merged. In the density fields, for L = 15r_{0} in Figure 13a, a normal shock is visible in the potential core just upstream of the plate at z ≈ 14.5r_{0}, resulting from an intense impact of the jet flow on the plate. A similar nearplate shock was observed for jets at Mach numbers lower than 2 impinging on normal [11, 18, 41, 42] or inclined [43, 44] plates. The shock extends across the mixing layers, where it is curved in the upstream direction at r = r_{0}. The distortion of the shock tip is due to the interactions between the shock and the coherent structures of the shear layers. A part of the shock leaks out of the flow, which creates a circular sound wave visible around z ≈ 13.5r_{0} and r_{0} ≤ r ≤ 3r_{0}. This sound generation mechanism, also involved in the production of screech noise [45–47], is referred to as shock leakage. It is shown here to be an efficient acoustic source in highsupersonic impinging jets in certain cases, as suggested by previous works [18, 41]. However, for L = 25r_{0} in Figure 13b, no clear shock structure is present in the density field. The shockleakage mechanism seems to be absent, which may explain the lower sound levels compared with the case L = 15r_{0}. The noise radiated in the upstream direction is produced by the impingement of vortical structures on the plate in that case.
Figure 13 Fields of density gradient ∂ρ/∂z (black) and vorticity (red) near the plate for (a) L = 15r_{0} and (b) L = 25r_{0}. The color scales range from 0 to 4ρ_{0}/r_{0} for the density gradient, from white to black, and from 0 to 7.5u_{e}/r_{0} for the vorticity, from white to red. 
To visualize the shockleakage mechanism for the case L = 15r_{0}, the density gradient and vorticity fields near the plate are represented in Figures 14a–14e at times t = 0 to 10r_{0}/u_{e} in increment of 2.5r_{0}/u_{e}. For t = 0 in Figure 14a, a nearplate shock is found on the axis around z = 14.5r_{0}. It extends across the mixing layer, where its tip is tilted in the upstream direction. Out of the mixing layer, a short line of strong density gradient is observed close to the shock tip, at z ≈ 14.2r_{0} and between r = r_{0} and r = 2r_{0}. This line is related to a pressure wave produced by the leakage of the shock out of the shear layer. For t = 2.5r_{0}/u_{e} in Figure 14b, the normal shock moves downstream as it interacts with a vortical structure impinging on the plate. Its tip is slightly inclined towards the plate. Outside of the flow, the wave generated at t = 0 propagates up to z ≈ 13.7r_{0}, which is consistent with the distance of 0.5r_{0} travelled by a sound wave between two snapshots. A second weak wavefront is also seen at z ≈ 14r_{0} and r ≈ 1.5r_{0}, close to a coherent structure impinging on the plate. For t = 5r_{0}/u_{e} in Figure 14c, the shock tip moves upstream and the sound waves produced previously propagate towards the nozzle. The shock tip is more and more curved in the upstream direction for t = 7.5r_{0}/u_{e} and 10r_{0}/u_{e} in Figures 14d and 14e. For t = 7.5r_{0}/u_{e} in Figure 14d, it extends in the radial direction up to r = 1.5r_{0}. Finally, for t = 10r_{0}/u_{e} in Figure 14e, it leaks out of the shear layer, which creates a new sound wave outside the flow at z ≈ 13.5r_{0}.
Figure 14 Fields of density gradient ∂ρ/∂z (black) and vorticity (red) near the plate for L = 15r_{0} at (a) t = 0, (b) t = 2.5r_{0}/u_{e}, (c) t = 5r_{0}/u_{e}, (d) t = 7.5r_{0}/u_{e} and (e) t = 10r_{0}/u_{e}. The color scales range from 0 to 4ρ_{0}/r_{0} for the density gradient, from white to black, and from 0 to 7.5u_{e}/r_{0} for the vorticity, from white to red. 
The motions of the nearplate shock for L = 15r_{0} are studied using the methodology proposed by Gojon and Bogey [18] to investigate the oscillations of nearwall Mach disks for underexpanded impinging jets. In practice, the position of the shock z_{s} is estimated by detecting the maximum of the gradient of axial velocity. The power spectral densities of the time variations of the position of the shock are then computed. The results obtained for r = 0 and r = 0.85r_{0} are represented in Figure 15. The frequencies of the peaks in the nearnozzle pressure spectra for the two first azimuthal modes are also plotted. On the jet axis at r = 0, where only the axisymmetric mode is present, two peaks are observed in the spectrum at St = 0.05 and 0.09. The frequency of the second peak is the same as that of the second nearnozzle peak for n_{θ} = 0. Closer to the mixing layer, the spectrum for r = 0.85r_{0} exhibits three peaks at St = 0.03, 0.09 and 0.16. The second and third peak frequencies are similar to those of the second nearnozzle peaks for n_{θ} = 0 and n_{θ} = 1, respectively. This result suggests that the highfrequency peaks in the nearnozzle pressure spectrum are linked to the motions of the nearplate shock.
Figure 15 Power spectral densities of the axial oscillations of the nearplate shock for L = 15r_{0} at r = 0 and r = 0.85r_{0}, and peak frequencies in the nearnozzle pressure spectra for n_{θ} = 0 and n_{θ} = 1. 
4 Conclusion
In this paper, the presence and properties of acoustic peaks near the nozzle of impinging rocket jets have been investigated using largeeddy simulations for nozzletoplate distances varying from 15 to 30 nozzle radii. For all distances, peaks are found to emerge weakly in the nearnozzle pressure spectra. The weak emergence of the peaks for all nozzletoplate distances indicates that no intense feedback mechanism establishes between the nozzle and the plate, even for L = 15r_{0}. The absence of strong resonance is likely due to the jet high Mach number, as the growth rate of the Kelvin–Helmholtz waves, forming the downstream part of possible feedback loops, decreases as the jet Mach number increases [48, 49]. In further works, this growth rate could be investigated. Regarding the peak frequencies, they do not vary much with the nozzletoplate distance. Similarly to the frequencies of the tones obtained in the nearnozzle pressure fields of jets at lower Mach numbers, they are located inside or near the allowable frequency ranges of the upstreampropagating guided jet waves, indicating a link with these waves for rocket jets. As for the sound levels, they increase as the plate is closer to the nozzle exit. For the shortest nozzletoplate distance, a normal shock is found near the plate, creating additional sound waves by a shockleakage mechanism. This shock is found to oscillate at frequencies close to those of peaks in the nearnozzle pressure spectra.
Conflict of interest
The authors declare no conflict of interest.
Acknowledgments
This work was financed by the IRICE IJES project RA0014963 (Installed Jet Effect Simulator, FEDERFSE RhôneAlpes). 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, members of FLMSN (Fédération Lyonnaise de Modélisation et Sciences Numériques), partner of EQUIPEX EQUIP@MESO, and to the resources of IDRIS (Institut du Développement et des Ressources en Informatique Scientifique) under the allocation 20212a0204 made by GENCI (Grand Equipement National de Calcul Intensif). It was performed within the framework of the LABEX CeLyA (ANR10LABX0060) of Université de Lyon, within the program Investissements d’Avenir (ANR16IDEX0005) operated by the French National Research Agency (ANR).
References
 C.P. Lubert, K.L. Gee, S. Tsutsumi: Supersonic jet noise from launch vehicles: 50 years since nasa sp8072. Journal of the Acoustical Society of America 151, 2 (2022) 752–791. [CrossRef] [PubMed] [Google Scholar]
 A. Powell: On edge tones and associated phenomena. Acta Acustica United with Acustica 3, 4 (1953) 233–243. [Google Scholar]
 G. Neuwerth: Acoustic feedback of a subsonic and supersonic free jet which impinges on an obstacle. NASA Technical Translation No. F15719, 1974. [Google Scholar]
 J.S. Preisser: Fluctuating surface pressure and acoustic radiation for subsonic normal jet impingement. NASA Technical Paper 1361, 1979. [Google Scholar]
 C.M. Ho, N.S. Nosseir: Dynamics of an impinging jet. Part 1. The feedback phenomenon. Journal of Fluid Mechanics 105 (1981) 119–142. [CrossRef] [Google Scholar]
 N.S. Nosseir, C.M. Ho: Dynamics of an impinging jet. Part 2. The noise generation, Journal of Fluid Mechanics 116 (1982) 379–391. [CrossRef] [Google Scholar]
 V. Jaunet, M. Mancinelli, P. Jordan, A. Towne, D.M. EdgingtonMitchell, G. Lehnasch, S. Girard: Dynamics of round jet impingement. AIAA Paper 2019–2769, 2019. [Google Scholar]
 M. Varé, C. Bogey: Generation of acoustic tones in round jets at a Mach number of 0.9 impinging on a plate with and without a hole. Journal of Fluid Mechanics 936 (2022) A16. [CrossRef] [Google Scholar]
 T.D. Norum: Supersonic rectangular jet impingement noise experiments. AIAA Journal 29, 7 (1991) 1051–1057. [CrossRef] [Google Scholar]
 A. Krothapalli, E. Rajkuperan, F. Alvi, L. Lourenco: Flow field and noise characteristics of a supersonic impinging jet. Journal of Fluid Mechanics 392 (1999) 155–181. [CrossRef] [Google Scholar]
 B. Henderson, J. Bridges, M. Wernet: An experimental study of the oscillatory flow structure of toneproducing supersonic impinging jets. Journal of Fluid Mechanics 542 (2005) 115–137. [CrossRef] [Google Scholar]
 A. Risborg, J. Soria: Highspeed optical measurements of an underexpanded supersonic jet impinging on an inclined plate, in: 28th International Congress on HighSpeed Imaging and Photonics, Vol. 7126, International Society for Optics and Photonics, 2009, p. 71261F. [Google Scholar]
 N.A. Buchmann, D.M. Mitchell, K.M. Ingvorsen, D.R. Honnery, J. Soria: High spatial resolution imaging of a supersonic underexpanded jet impinging on a flat plate, in: 6th Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion, 2011. [Google Scholar]
 D.M. Mitchell, D.R. Honnery, J. Soria: The visualization of the acoustic feedback loop in impinging underexpanded supersonic jet flows using ultrahigh frame rate schlieren. Journal of Visualization 15, 4 (2012) 333–341. [CrossRef] [Google Scholar]
 A. Dauptain, L.Y. Gicquel, S. Moreau: Largeeddy simulation of supersonic impinging jets. AIAA Journal 50, 7 (2012) 1560–1574. [CrossRef] [Google Scholar]
 R. Gojon, C. Bogey, O. Marsden: Investigation of tone generation in ideally expanded supersonic planar impinging jets using largeeddy simulation. Journal of Fluid Mechanics 808 (2016) 90–115. [CrossRef] [Google Scholar]
 C. Bogey, R. Gojon: Feedback loop and upwindpropagating waves in ideally expanded supersonic impinging round jets. Journal of Fluid Mechanics 823 (2017) 562–591. [CrossRef] [Google Scholar]
 R. Gojon, C. Bogey: Flow structure oscillations and tone production in underexpanded impinging round jets. AIAA Journal 55, 6 (2017) 1792–1805. [CrossRef] [Google Scholar]
 A. Towne, A.V.G. Cavalieri, P. Jordan, T. Colonius, O. Schmidt, V. Jaunet, G.A. Brès: Acoustic resonance in the potential core of subsonic jets. Journal of Fluid Mechanics 825 (2017) 1113–1152. [CrossRef] [Google Scholar]
 O.T. Schmidt, A. Towne, T. Colonius, A.V.G. Cavalieri, P. Jordan, G.A. Brès: Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability. Journal of Fluid Mechanics 825 (2017) 1153–1181. [CrossRef] [Google Scholar]
 C. Bogey: Acoustic tones in the nearnozzle region of jets: characteristics and variations between Mach numbers 0.5 and 2. Journal of Fluid Mechanics 921 (2021) A3. [CrossRef] [Google Scholar]
 S. Kawai, S. Tsutsumi, R. Takaki, K. Fujii: Computational aeroacoustic analysis of overexpanded supersonic jet impingement on a flat plate with/without hole, in: ASME/JSME 2007 5th Joint Fluids Engineering Conference, American Society of Mechanical Engineers, 2007, pp. 1163–1167. [Google Scholar]
 S. Tsutsumi, R. Takaki, H. Ikaida, K. Terashima: Numerical aeroacoustics analysis of a scaled solid jet impinging on flat plate with exhaust hole, in: 30th International Symposium on Space Technology and Science, 2015. [Google Scholar]
 J. Troyes, F. Vuillot, A. Langenais, H. Lambaré: Coupled CFDCAA simulation of the noise generated by a hot supersonic jet impinging on a flat plate with exhaust hole. AIAA Paper 2019–2752, 2019. [Google Scholar]
 M. Varé, C. Bogey: Flow and acoustic fields of rocket jets impinging on a perforated plate, AIAA Journal 60 (2022) 1–14. [Google Scholar]
 C. Bogey, O. Marsden, C. Bailly: Largeeddy simulation of the flow and acoustic fields of a Reynolds number 10^{5} subsonic jet with tripped exit boundary layers. Physics of Fluids 23, 3 (2011) 035104. [CrossRef] [Google Scholar]
 C. Bogey, C. Bailly: A family of low dispersive and low dissipative explicit schemes for flow and noise computations. Journal of Computational Physics 194, 1 (2004) 194–214. [CrossRef] [Google Scholar]
 C. Bogey, N. De Cacqueray, C. Bailly: A shockcapturing methodology based on adaptative spatial filtering for highorder nonlinear computations. Journal of Computational Physics 228, 5 (2009) 1447–1465. [CrossRef] [Google Scholar]
 D. Fauconnier, C. Bogey, E. Dick: On the performance of relaxation filtering for largeeddy simulation. Journal of Turbulence 14, 1 (2013) 22–49. [CrossRef] [Google Scholar]
 C.K.W. Tam, Z. Dong: Radiation and outflow boundary conditions for direct computation of acoustic and flow disturbances in a non uniform mean flow. Journal of Computational Acoustics 4, 02 (1996) 175–201. [CrossRef] [Google Scholar]
 K. Mohseni, T. Colonius: Numerical treatment of polar coordinate singularities. Journal of Computational Physics 157, 2 (2000) 787–795. [CrossRef] [Google Scholar]
 S. Viazzo, A. Dejoan, R. Schiestel: Spectral features of the wallpressure fluctuations in turbulent wall flows with and without perturbations using les. International Journal of Heat and Fluid Flow 22, 1 (2001) 39–52. [CrossRef] [Google Scholar]
 P. Schlatter, Q. Li, G. Brethouwer, A.V. Johansson, D.S. Henningson: Simulations of spatially evolving turbulent boundary layers up to Re_{θ}=4300. International Journal of Heat and Fluid Flow 31, 3 (2010) 251–261. [CrossRef] [Google Scholar]
 X. Gloerfelt, J. Berland: Turbulent boundarylayer noise: direct radiation at Mach number 0.5. Journal of Fluid Mechanics 723 (2013) 318–351. [CrossRef] [Google Scholar]
 F. Kremer, C. Bogey: Largeeddy simulation of turbulent channel flow using relaxation filtering: Resolution requirement and Reynolds number effects. Computers & Fluids 116 (2015) 17–28. [CrossRef] [Google Scholar]
 N. De Cacqueray, C. Bogey, C. Bailly: Investigation of a highMachnumber overexpanded jet using largeeddy simulation. AIAA Journal 49, 10 (2011) 2171–2182. [CrossRef] [Google Scholar]
 A. Langenais, F. Vuillot, J. Troyes, C. Bailly: Accurate simulation of the noise generated by a hot supersonic jet including turbulence tripping and nonlinear acoustic propagation. Physics of Fluids 31, 1 (2019) 016105. [CrossRef] [Google Scholar]
 T. Nonomura, K. Fujii: Overexpansion effects on characteristics of Mach waves from a supersonic cold jet. AIAA Journal 49, 10 (2011) 2282–2294. [CrossRef] [Google Scholar]
 P. Pineau, C. Bogey: Steepened Mach waves near supersonic jets: study of azimuthal structure and generation process using conditional averages. Journal of Fluid Mechanics 880 (2019) 594–619. [CrossRef] [Google Scholar]
 C.K.W. Tam, K.K. Ahuja: Theoretical model of discrete tone generation by impinging jets. Journal of Fluid Mechanics 214 (1990) 67–87. [CrossRef] [Google Scholar]
 B. Henderson: The connection between sound production and jet structure of the supersonic impinging jet. Journal of the Acoustical Society of America 111, 2 (2002) 735–747. [CrossRef] [PubMed] [Google Scholar]
 G. Sinibaldi, L. Marino, G.P. Romano: Sound source mechanisms in underexpanded impinging jets. Experiments in Fluids 56, 5 (2015) 105. [CrossRef] [Google Scholar]
 T. Nonomura, H. Honda, Y. Nagata, M. Yamamoto, S. Morizawa, S. Obayashi, K. Fujii: Plateangle effects on acoustic waves from supersonic jets impinging on inclined plates. AIAA Journal 54, 3 (2016) 816–827. [CrossRef] [Google Scholar]
 C. Brehm, J.A. Housman, C.C. Kiris: Noise generation mechanisms for a supersonic jet impinging on an inclined plate. Journal of Fluid Mechanics 797 (2016) 802–850. [CrossRef] [Google Scholar]
 T. Suzuki, S.K. Lele: Shock leakage through an unsteady vortexladen mixing layer: application to jet screech. Journal of Fluid Mechanics 490 (2003) 139–167. [CrossRef] [Google Scholar]
 J. Berland, C. Bogey, C. Bailly: Numerical study of screech generation in a planar supersonic jet. Physics of Fluids 19, 7 (2007) 075105. [CrossRef] [Google Scholar]
 D. EdgingtonMitchell, J. Weightman, S. Lock, R. Kirby, V. Nair, J. Soria, D. Honnery: The generation of screech tones by shock leakage. Journal of Fluid Mechanics 908 (2021) A46. [CrossRef] [Google Scholar]
 A. Michalke: Survey on jet instability theory. Progress in Aerospace Sciences 21 (1984) 159–199. [CrossRef] [Google Scholar]
 P.J. Morris: The instability of high speed jets. International Journal of Aeroacoustics 9, 1–2 (2010) 1–50. [CrossRef] [Google Scholar]
Cite this article as: Varé M. & Bogey C. 2022. Presence and properties of acoustic peaks near the nozzle of impinging rocket jets. Acta Acustica, 6, 36.
All Figures
Figure 1 Nozzleexit profiles of axial turbulence intensity ${{\langle u\mathrm{\prime}}_{z}u\mathrm{\prime}}_{z}{\rangle}^{1/2}/{u}_{e}$ for L = 15r_{0}, L = 20r_{0}, L = 25r_{0}, L = 30r_{0}, and for the free jet [25]. 

In the text 
Figure 2 Variations of axial mesh spacing Δz for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 

In the text 
Figure 3 Variations of the axial mesh spacing Δz^{+} on the plate, in wall units, for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 

In the text 
Figure 4 Snapshots in the (z,r) plane of temperature fluctuations in the flow and of pressure fluctuations outside for (a) L = 15r_{0}, (b) L = 20r_{0}, (c) L = 25r_{0} and (d) L = 30r_{0}. The red arrows show the wavefronts of two consecutive upstreampropagating pressure waves for the nozzletoplate distances L = 15r_{0} and 25r_{0}. The color scales range from 0 to 780K for temperature, from red to white, and from −2000 to 2000 Pa for pressure, from black to white. 

In the text 
Figure 5 Variations of (a) the mean axial centerline velocity 〈u_{z}〉/u_{e} and (b) the axial turbulence intensity ${{\langle u\mathrm{\prime}}_{z}u\mathrm{\prime}}_{z}{\rangle}^{1/2}/{u}_{e}$ at r = r_{0} for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 

In the text 
Figure 6 Sound pressure levels (SPL) at z = 0 and r = 2r_{0} as a function of the Strouhal number St for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0} and the free jet [25]. 

In the text 
Figure 7 Overall sound pressure levels (OASPL) at z = 0 and r = 2r_{0} as a function of the nozzletoplate distance. 

In the text 
Figure 8 Sound pressure spectra at z = 0 and r = 2r_{0} for (a) L = 15r_{0} and (b) L = 25r_{0}: full spectra and modes n_{θ} = 0, n_{θ} = 1, n_{θ} = 2, n_{θ} = 3 and n_{θ} = 4. 

In the text 
Figure 9 Strouhal numbers of the peaks in the nearnozzle pressure spectra as a function of the nozzletoplate distance for (a) n_{θ} = 0 and (b) n_{θ} = 1; • dominant and ∘ secondary peaks. The frequency bands of the upstreampropagating guided jet waves are in grey. 

In the text 
Figure 10 Levels of the peaks in the nearnozzle pressure spectra as a function of the nozzletoplate distance for (a) n_{θ} = 0 and (b) n_{θ} = 1; • dominant and ∘ secondary peaks. 

In the text 
Figure 11 Power spectral densities of the fluctuations of axial velocity u′_{z} at r = r_{0} between z = 0 and L for (a) L = 15r_{0}, (b) L = 20r_{0}, (c) L = 25r_{0} and (d) L = 30r_{0}. The color scale is the same in the three cases and spreads over 3 dB, from white to black. The black dashed lines indicate the frequencies St = 0.04, 0.09 and 0.22. 

In the text 
Figure 12 Power spectral densities of the fluctuations of axial velocity ${u}_{z\mathrm{\prime}}$ at r = r_{0} and z = z_{max} for L = 15r_{0}, L = 20r_{0}, L = 25r_{0} and L = 30r_{0}. 

In the text 
Figure 13 Fields of density gradient ∂ρ/∂z (black) and vorticity (red) near the plate for (a) L = 15r_{0} and (b) L = 25r_{0}. The color scales range from 0 to 4ρ_{0}/r_{0} for the density gradient, from white to black, and from 0 to 7.5u_{e}/r_{0} for the vorticity, from white to red. 

In the text 
Figure 14 Fields of density gradient ∂ρ/∂z (black) and vorticity (red) near the plate for L = 15r_{0} at (a) t = 0, (b) t = 2.5r_{0}/u_{e}, (c) t = 5r_{0}/u_{e}, (d) t = 7.5r_{0}/u_{e} and (e) t = 10r_{0}/u_{e}. The color scales range from 0 to 4ρ_{0}/r_{0} for the density gradient, from white to black, and from 0 to 7.5u_{e}/r_{0} for the vorticity, from white to red. 

In the text 
Figure 15 Power spectral densities of the axial oscillations of the nearplate shock for L = 15r_{0} at r = 0 and r = 0.85r_{0}, and peak frequencies in the nearnozzle pressure spectra for n_{θ} = 0 and n_{θ} = 1. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.