Resolvent-based jet noise models: a projection approach

Linear resolvent analysis has demonstrated encouraging results for modeling coherent structures in jets when compared against their data-deduced counterparts from high-fidelity large-eddy simulations (LES). However, leveraging resolvent modes for reconstructing statistics of the far acoustic field remains elusive. In this study, we use a LES database to produce an ensemble of realizations for the acoustic field that we project on to a limited set of n resolvent modes. The projections are done on a restricted acoustic output domain, r/D= [5,6], and allow for the LES realizations to be recast in the resolvent basis via a data-deduced, low-rank, n x n cross-spectral density matrix. We find substantial improvements to the acoustic field reconstructions with the addition of a RANS-derived eddy-viscosity model to the resolvent operator. The reconstructions quantitatively match the most energetic regions of the acoustic field across Strouhal numbers, St= [0−1], and azimuthal wavenumbers, m= [0,2], using only three resolvent modes. Finally, the characteristics of the resulting n x n covariance matrices are examined and suggest off-diagonal terms may be neglected for n ≤ 3. Results are presented for round, isothermal, Mach 1.5 and 0.9 jets

Modeling intermittent wavepackets and their radiated sound in a turbulent jet

We use data from a new, carefully validated, Large Eddy Simulation (LES) to investigate and model subsonic, turbulent, jet noise. Motivated by the observation that sound-source dynamics are dominated by instability waves (wavepackets), we examine mechanisms by which their intermittency can amplify their noise radiation. Two scenarios, both involving wavepacket evolution on time-dependent base flows, are investigated. In the first, we consider that the main effect of the changing base flow consists in different wavepacket ensembles seeing different steady mean fields, and having, accordingly, different acoustic efficiencies. In the second, the details of the base-flow time dependence also play a role in wavepacket sound production. Both short-time-averaged and slowly varying base flows are extracted from the LES data and used in conjunction with linearized wavepacket models, namely, the Parabolized Stability Equations (PSE), the One-Way Euler Equations (OWE), and the Linearized Euler Equations (LEE). All results support the hypothesized mechanism: wavepackets on time-varying base flows produce sound radiation that is enhanced by as much as 20dB in comparison to their long-time-averaged counterparts, and ensembles of wavepackets based on short-time-averaged base flows display similar amplification. This is not, however, sufficient to explain the sound levels observed in the LES and experiments. Further work is therefore necessary to incorporate two additional factors in the linear models, body forcing by turbulence and realistic inflow forcing, both of which have been identified as potentially important in producing the observed radiation efficiency

Evaluation of PSE as a Model for Supersonic Jet Using Transfer Functions

Parabolized Stability Equations (PSE) have been shown to model wavepackets and, consequently, the near field of turbulent jets with reasonable accuracy. Because of these capabilities, PSE is a promising reduced-order model to derive control laws that could be employed to reduce the sound generation of a jet. The purpose of this work is to apply PSE to obtain time-domain transfer functions that could estimate both the fluid-dynamic and the acoustic fields of a supersonic jet. The results of this model were compared to results obtained from a database of a well-validated large-eddy simulation of a supersonic jet. Based on the unsteady pressure data at a input position, the time-domain pressure field was estimated using transfer functions obtained using PSE and an empirical method based on the LES data. The prediction scheme employed is a single-input-single-output (SISO), linear model. The unsteady pressure predicted by PSE showed good agreement with the LES results, especially if the input position is outside the mixing layer. For this region, the prediction capabilities of PSE are comparable to those of empirical transfer functions. The agreement is good even for output points taken in the acoustic field, showing that it is possible to estimate the time-domain behaviour of Mach-wave radiation using transfer functions. This indicates that PSE could not only be used to predict the sound generation, but also to open up new potentialities to attenuate noise by means of closed-loop control of the flow. The exploration of the regions where the method displayed good agreement, presented in this work, can guide the positioning of sensors and actuators for experimental implementation of closed-loop control in a jet

Stability of Temporally Evolving Supersonic Boundary Layers over Micro-Cavities for Ultrasonic Absorptive Coatings

Ultrasonic absorptive coatings, consisting of regularly-spaced arrays of micro-cavities, have previously been shown to effectively damp second-mode instability for the purpose of delaying transition in hypersonic boundary layers. However, previous simulations and stability analysis have used approximate porous-wall boundary conditions. Here we investigate the feasibility of using direct numerical simulation to directly compute the hypersonic boundary layer including the micro-cavities. In order to keep the problem computationally tractable, we restrict our attention to the two-dimensional case (which is relevant since the second-mode is initially two dimensional), and we show that temporally evolving layers display qualitatively similar behavior to spatially developing boundary layer and instabilities. We validate the numerical method by comparing the simulation results to temporal linear stability analysis of the (frozen) velocity and temperature profiles from the direct numerical simulation. Two-dimensional linear simulations of the boundary layer on a flat plate and over a porous coating are performed, and it is shown that the presence of the cavities attenuates the instability waves, as expected from theory

Eddy viscosity for resolvent-based jet noise models

Response modes computed via linear resolvent analysis have shown promising results for qualitatively modeling both the hydrodynamic and acoustic fields in jets when compared to data-deduced modes from high-fidelity, large-eddy simulations (LES). For an improved quantitative prediction of the near- and far-field, the role of Reynolds stresses must also be considered. In this study, we propose a methodology to deduce an eddy-viscosity model that optimally captures the nonlinear forcing of resolvent modes. The methodology is based on the maximization of the projection between resolvent analysis and spectral proper orthogonal decomposition (SPOD) modes using a Lagrangian optimization framework. For a Mach 0.4 round, isothermal, turbulent jet, four methods are used to increase the projection coefficients: linear damping, spatially constant eddy-viscosity field, a turbulent kinetic energy derived viscosity field, and an optimized eddy-viscosity field. The resulting projection coefficients for the optimized eddy-viscosity field between SPOD and resolvent can be increased to over 90% for frequencies in the range St = 0.35−1 with significant improvements to St < 0.35. We find that the use of a frequency-independent turbulent kinetic energy turbulent viscosity model produces modes closely inline with optimal results, providing a preliminary eddy-viscosity resolvent model for jets

Stability of Temporally Evolving Supersonic Boundary Layers over Micro-Cavities for Ultrasonic Absorptive Coatings

Ultrasonic absorptive coatings, consisting of regularly-spaced arrays of micro-cavities, have previously been shown to effectively damp second-mode instability for the purpose of delaying transition in hypersonic boundary layers. However, previous simulations and stability analysis have used approximate porous-wall boundary conditions. Here we investigate the feasibility of using direct numerical simulation to directly compute the hypersonic boundary layer including the micro-cavities. In order to keep the problem computationally tractable, we restrict our attention to the two-dimensional case (which is relevant since the second-mode is initially two dimensional), and we show that temporally evolving layers display qualitatively similar behavior to spatially developing boundary layer and instabilities. We validate the numerical method by comparing the simulation results to temporal linear stability analysis of the (frozen) velocity and temperature profiles from the direct numerical simulation. Two-dimensional linear simulations of the boundary layer on a flat plate and over a porous coating are performed, and it is shown that the presence of the cavities attenuates the instability waves, as expected from theory

High-frequency wavepackets in turbulent jets

Wavepackets obtained as solutions of the flow equations linearised around the mean flow have been shown in recent work to yield good agreement with the amplitudes and phases of turbulent fluctuations in jets. Compelling agreement has been demonstrated up to Strouhal numbers, St ≈ 1. We extend the range of validity of wavepacket models to higher values, 1.0 < St < 4.0, by comparing Parabolised Stability Equation solutions with well resolved large-eddy simulation data. The initial growth rates of the high-frequency fluctuations continue to be well predicted, but saturation occurs earlier and agreement with simulation begins to deteriorate upstream of the end of the potential core of the jet. Results show that near-nozzle dynamics for a broad range of frequencies can be modelled using linearised models, which capture well the spatial growth of Kelvin-Helmholtz wavepackets for all the studied Strouhal numbers