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Identification of temporal and spatial signatures of broadband shock-associated noise

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Abstract

Broadband shock-associated noise (BBSAN) is a particular high-frequency noise that is generated in imperfectly expanded jets. BBSAN results from the interaction of turbulent structures and the series of expansion and compression waves which appears downstream of the convergent nozzle exit of moderately under-expanded jets. This paper focuses on the impact of the pressure waves generated by BBSAN from a large eddy simulation of a non-screeching supersonic round jet in the near-field. The flow is under-expanded and is characterized by a high Reynolds number \(\mathrm{Re}_\mathrm{j} = 1.25\times 10^6\) and a transonic Mach number \(M_\mathrm{j}=1.15\). It is shown that BBSAN propagates upstream outside the jet and enters the supersonic region leaving a characteristic pattern in the physical plane. This pattern, also called signature, travels upstream through the shock-cell system with a group velocity between the acoustic speed \(U_{\mathrm{c}}-a_\infty \) and the sound speed \(a_\infty \) in the frequency–wavenumber domain \((U_\mathrm{c}\) is the convective jet velocity). To investigate these characteristic patterns, the pressure signals in the jet and the near-field are decomposed into waves traveling downstream (\(p^+\)) and waves traveling upstream (\(p^-\)). A novel study based on a wavelet technique is finally applied on such signals in order to extract the BBSAN signatures generated by the most energetic events of the supersonic jet.

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Acknowledgements

The authors are thankful to L. Gefen from Università Degli Study di Roma, UniRomaTre for his help with the wavelet post-processing. This work was granted access to the HPC resources of CINES under the allocation 2016-[x20162a6074] made by GENCI. Moreover, It was supported by the Marie Curie Initial Training Networks (ITN) AeroTraNet 2 of the European Community’s Seventh Framework Programme (FP7) under Contract No. PITN-GA-2012-317142.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sherbrooke University, Sherbrooke, QC, J1K 2R1, Canada

    C. Pérez Arroyo & S. Moreau

  2. CERFACS, 42 Avenue Gaspard Coriolis, 31057, Toulouse, France

    G. Daviller

  3. ONERA, 2 Avenue Edouard Belin, 31000, Toulouse, France

    G. Puigt

  4. IMFT, CNRS, Université de Toulouse, Allée du Professeur Camille Soula, 31400, Toulouse, France

    C. Airiau

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  1. C. Pérez Arroyo
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  2. G. Daviller
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  4. C. Airiau
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Communicated by D. Zeidan and H. D. Ng.

Appendix: Validation of the simulation against experimental results

Appendix: Validation of the simulation against experimental results

This appendix presents a comparison between the numerical results obtained from the large eddy simulation of the non-screeching under-expanded supersonic single jet and the experimental results from LMFA [46] and VKI [61].

1.1 Jet aerodynamics

The averaged Mach number profile on the axis for the LES and the experimental results are shown in Fig. 21. The LES shows good agreement for shock-cell spacing in the first three shock cells. However, further downstream, there is a shift between the experimental and the numerical Mach number profiles. Nonetheless, the shock-cell spacing is only reduced by about 5%. Even though the amplitudes are higher than in the experimental results, they follow the same decay and they capture the end of the potential core at the same position.

Fig. 21
figure 21

Mach number profile at AXIS. LMFA experiment [46] (notched nozzle), LES

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The turbulence levels of the velocity components on the lip-line are shown in Fig. 22. As no turbulence injection is used in the LES, the initial turbulence levels at \(x/D=0\) are equal to zero. However, they reach the same levels of rms as in the experiments after one radius. The overshoot observed within the first two diameters can be explained by a rapid transition to turbulence. Due to this, the amplitude decays about 20% relative to the experiments at \(x/D=10\). Nonetheless, the rms values are high enough to be considered as turbulent flow as it can be deduced by the vorticity contours in Fig. 6.

Fig. 22
figure 22

Turbulence levels of the axial (\(U_\mathrm{rms}\)) and radial (\(V_\mathrm{rms}\)) component of velocity at LIPLINE (\(r/D=0.5\)). \(U_\mathrm{rms}\) and \(V_\mathrm{rms}\) from LMFA experiment [46]. \(U_\mathrm{rms}\) and \(V_\mathrm{rms}\) from LES

Full size image

The turbulence length-scale \(L_{uu}\) computed from the auto-correlations \(R_{uu}\) along the lip-line is illustrated in Fig. 23. The integration of \(R_{uu}\) is calculated up to the value 0.1. The length-scale obtained has the same growth rate, but shifted 1.5 diameters in the axial direction. This displacement is probably due to the inlet steady profiles at the exit of the nozzle, where the laminar vortex pairing triggering transition to turbulence occurs in a more abrupt fashion as seen in Fig. 22.

Fig. 23
figure 23

Axial turbulence length-scale at LIPLINE(\(r/D=0.5\)). LMFA experiment [46], LES

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1.2 Far-field acoustics

The sound pressure level (SPL) in the far-field at \(r/D=53\) from the nozzle exit plane is compared against experimental results from LMFA [46] (notched nozzle) and VKI [61]. The pressure fluctuations from the LES were propagated to the far-field using the Ffowcs Williams and Hawking’s analogy [48] and averaged over 20 azimuthal probes in order to artificially increase the convective time of the signal. The comparisons are shown in Fig. 24 for two different angles computed from the jet direction.

Fig. 24
figure 24

Far-field sound pressure level at \(r/D=53\) from the nozzle exit at a\(60^\circ \) and b\(120^\circ \) with respect to the jet direction. The vertical dashed line represents the cut-off \(\mathrm{St}\) for the LES. LMFA experiment [46], experiment VKI [61], and LES

Full size image

At \(60^\circ \) (Fig. 24a), the LES pressure spectra are dissipated by the cut-off Strouhal number of the mesh above \(\mathrm{St}\approx 2\). Good agreement is obtained with the experimental results between \(0.5<\mathrm{St}<2\). Below \(\mathrm{St}=0.5\), the amplitude differs because of the lack of convergence of the large structures and the differences in turbulence values and length-scales.

The BBSAN peak from the LES captured at \(120^\circ \) (Fig. 24b) has the correct amplitude (up to \(\mathrm{St}\approx 2\)), but it is shifted in frequency with respect to the experimental SPL due to the fact that the shock-cell length captured is 5% smaller than the experimental one. The screech is not captured in the numerical simulations. The initial conditions used for this LES and the fact that the interior of the nozzle is not modeled [12, 14, 62, 63] are probably the key reasons for not obtaining screech because the feedback loop cannot take place in the development of the boundary layer inside the nozzle.

Overall, the acoustic results show good agreement with experiments without screech, despite the slight differences in turbulence levels and turbulence length-scales. This shows that the phenomenon generating shock-cell noise is well represented in the simulation and can be studied without discussing the impact of screech on the flow. Moreover, from [46], it can be seen that the acoustic and aerodynamic results for the case at \(M_\mathrm{j}=1.15\) are the ones the least impacted by screech.

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Pérez Arroyo, C., Daviller, G., Puigt, G. et al. Identification of temporal and spatial signatures of broadband shock-associated noise. Shock Waves 29, 117–134 (2019). https://doi.org/10.1007/s00193-018-0806-4

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