Global instability of low-density jets

The global stability of laminar axisymmetric low-density jets is investigated in the low Mach number approximation. The linear modal dynamics is found to be characterised by two features: a stable arc branch of eigenmodes and an isolated eigenmode. Both features are studied in detail, revealing that, whereas the former is highly sensitive to numerical domain size and its existence can be linked to spurious feedback from the outflow boundary, the latter is the physical eigenmode that is responsible for the appearance of self-sustained oscillations in low-density jets observed in experiments at low Mach numbers. In contrast to previous local spatio-temporal stability analyses, the present global analysis permits, for the first time, the determination of the critical conditions for the onset of global instability, as well the frequency of the associated oscillations, without additional hypotheses, yielding predictions in fair agreement with previous experimental observations. It is shown that under the conditions of those experiments, viscosity variation with composition, as well as buoyancy, only have a small effect on the onset of instability

Resolvent modelling of jet noise: the need for forcing models

The singular value decomposition of the mean-flow-based resolvent operator, or resolvent analysis, has proven to provide essential insights into the dynamics of various turbulent flows. In this study, we perform a resolvent analysis of a compressible turbulent jet, where the optimisation domain of the response modes is located in the acoustic field, excluding the hydrodynamic region, in order to promote acoustically efficient modes. We examine the properties of the acoustic resolvent and assess its potential for jet-noise modelling, focusing on the subsonic regime. We compare resolvent modes with SPOD modes educed from LES data. Resolvent forcing modes, consistent with previous studies, are found to contain supersonic waves associated with Mach wave radiation in the response modes. This differs from the standard resolvent in which hydrodynamic instabilities dominate. Acoustic resolvent response modes generally have better alignment with acoustic SPOD modes than standard resolvent response modes. For the optimal mode, the angle of the acoustic beam is close to that found in SPOD modes for moderate frequencies. However, there is no significant separation between the singular values of the leading and sub-optimal modes. Some suboptimal modes are furthermore shown to contain irrelevant structure for jet noise. Thus, even though it contains essential acoustic features absent from the standard resolvent approach, the SVD of the acoustic resolvent alone is insufficient to educe a low-rank model for jet noise. But because it identifies the prevailing mechanisms of jet noise, it provides valuable guidelines in the search of a forcing model (Karban $\textit{et al.}$ 2022, An empirical model of noise sources in subsonic jets. arXiv preprint arXiv:2210.01866).Comment: 24 pages, 20 figure

Ambiguity in mean-flow-based linear analysis

Abstract</jats:p

A Parabolised Stability Equation based Broadband Shock-Associated Noise Model

International audienceWavepacket models have been used extensively to predict the noise produced from turbulent subsonic and supersonic jets. Such wavepackets, which represent the organised structures of the flow, are solutions to the linearised Navier-Stokes equations. Using a kinematic two-point model, Wong et al. [1] have indicated the importance of incorporating coherence decay in modelling broadband shock-associated noise (BBSAN) in supersonic jets. In this work, we aim to improve the model by using solutions from linear parabolised stability equations (PSE) to model the wavepacket part of the BBSAN source. The two-point coherence of the wavepackets is obtained from large-eddy simulation (LES) data of a M j = 1.5 fully-expanded isothermal supersonic jet [2]. The aim is to build a dynamic sound-source model for BBSAN that would improve on the simplified line-source model proposed by Wong et al. [3]. We find that a frequency dependent coherence decay length scale is important in order to suppress the higher-order harmonic peaks [4] and to obtain the correct BBSAN peak shape. Moderate agreement up to St = 1 was found between the current noise predictions and those from experimental data. I. Nomenclature ω = wavepacket frequency θ = azimuthal coordinate c s n = amplitude coefficient of the shock cells G = Green's function k s = shock-cell wavenumber k h = hydrodynamic wavenumber L = longitudinal extent of wavepacket L c = coherence length of wavepacket m = azimuthal mode number M j = ideally-expanded Mach number r = radial coordinate u s = shock cell velocity fluctuation u t = wavepacket fluctuationŝ u * ω = velocity fluctuations at a frequency ω x = axial coordinat

Acoustic resolvent analysis of turbulent jets

Abstract We perform a resolvent analysis of a compressible turbulent jet, where the optimisation domain of the response modes is located in the acoustic field, excluding the hydrodynamic region, in order to promote acoustically efficient modes. We examine the properties of the acoustic resolvent and assess its potential for jet-noise modelling, focusing on the subsonic regime. Resolvent forcing modes, consistent with previous studies, are found to contain supersonic waves associated with Mach wave radiation in the response modes. This differs from the standard resolvent in which hydrodynamic instabilities dominate. We compare resolvent modes with SPOD modes educed from LES data. Acoustic resolvent response modes generally have better alignment with acoustic SPOD modes than standard resolvent response modes. For the optimal mode, the angle of the acoustic beam is close to that found in SPOD modes for moderate frequencies. However, there is no significant separation between the singular values of the leading and sub-optimal modes. Some suboptimal modes are furthermore shown to contain irrelevant structure for jet noise. Thus, even though it contains essential acoustic features absent from the standard resolvent approach, the SVD of the acoustic resolvent alone is insufficient to educe a low-rank model for jet noise. But because it identifies the prevailing mechanisms of jet noise, it provides valuable guidelines in the search of a forcing model (Karban et al. in J Fluid Mech 965:18, 2023). Graphical abstract </jats:sec

Self-similar mechanisms in wall turbulence studied using resolvent analysis

International audienceSelf-similarity of wall-attached coherent structures in a turbulent channel at $Re_\tau =543$ is explored by means of resolvent analysis. In this modelling framework, coherent structures are understood to arise as a response of the linearised mean-flow operator to generalised frequency-dependent Reynolds stresses, considered to act as an endogenous forcing. We assess the self-similarity of both the wall-attached flow structures and the associated forcing. The former are educed from direct numerical simulation data by finding the flow field correlated with the wall shear, whereas the latter is identified using a frequency space version of extended proper orthogonal decomposition (Borée, Exp. Fluids , vol. 35, issue 2, 2003, pp. 188–192). The forcing structures identified are compared to those obtained using the resolvent-based estimation introduced by Towne et al. ( J. Fluid Mech. , vol. 883, 2020, A17). The analysis reveals self-similarity of both wall-attached structures – in quantitative agreement with Townsend's hypothesis of self-similar attached eddies – and the underlying forcing, at least in certain components

An empirical model of noise sources in subsonic jets

Modelling the noise emitted by turbulent jets is made difficult by their acoustic inefficiency: only a tiny fraction of the near-field turbulent kinetic energy is propagated to the far field as acoustic waves. As a result, jet-noise models must accurately capture this small, acoustically efficient component hidden among comparatively inefficient fluctuations. In this paper, we identify this acoustically efficient near-field source from large-eddy simulation data and use it to inform a predictive model. Our approach uses the resolvent framework, in which the source takes the form of nonlinear fluctuation terms that act as a forcing on the linearised Navier–Stokes equations. First, we identify the forcing that, when acted on by the resolvent operator, produces the leading spectral proper orthogonal decomposition modes in the acoustic field for a Mach 0.4 jet. Second, the radiating components of this forcing are isolated by retaining only portions with a supersonic phase speed. This component makes up less than 0.05 % of the total forcing energy but generates most of the acoustic response, especially at peak (downstream) radiation angles. Finally, we propose an empirical model for the identified acoustically efficient forcing components. The model is tested at other Mach numbers and flight-stream conditions and predicts noise within 2 dB accuracy for a range of frequencies, downstream angles and flight conditions