Aaron Towne

Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet

This note contains a correction to equation (16) in AIAA Paper 2015-2217: ”Stochastic and nonlinear forcing of wavepackets in a Mach 0.9 jet,” as well as some additional clarification of the empirical resolvent mode (ERM) method described in the same paper. The notation and terminaology used in this note follows that of the paper. Before turning attention to ERM, it’s useful to write proper orthogonal decomposition (POD) as an optimization problem in a form similar to that used later for ERM.

Super- and multi-directive acoustic radiation by linear global modes of a turbulent jet

The mean flow stability of a Mach 0.9 turbulent jet is investigated by means of global linear theory with a focus on acoustic effects. A novel class of resonant acoustic modes that are trapped within the potential core, and whose eigenvalues appear as discrete branches in the global stability spectrum, is studied in detail. A dispersion relation is reconstructed from the global modes, and shown to accurately predict energy bands observed in the PSD of a high-fidelity LES. Similarly, the acoustic far-field radiation patterns of the trapped modes are compared to the LES. A favorable agreement between the global mode waveforms and coherent structures educed from the LES is found for both the trapped acoustic wave component inside the core and the far-field radiation.

Trapped acoustic waves in the potential core of subsonic jets

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and that have been previously detected as tones in the near-nozzle region. Using three models (the linearized Euler equations, a cylindrical vortex sheet, and a cylindrical duct with pressure release boundary conditions), we show that these waves can be described by linear modes of the jet and correspond to acoustic waves that are trapped within the potential core. At certain frequencies, these trapped waves resonate due to repeated reflection between end conditions provided by the nozzle and the streamwise contraction of the potential core. Our models accurately capture numerous aspects the potential core waves that are extracted from large-eddy-simulation data of a Mach 0.9 isothermal jet. Furthermore, the vortex sheet model indicates that this behavior is possible for only a limited range of Mach numbers that is consistent with previous experimental observations.

Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic tools in turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg-Landau equation and a turbulent jet.

One-way spatial integration of hyperbolic equations

In this paper, we develop and demonstrate a method for constructing well-posed one-way approximations of linear hyperbolic systems. We use a semi-discrete approach that allows the method to be applied to a wider class of problems than existing methods based on analytical factorization of idealized dispersion relations. After establishing the existence of an exact one-way equation for systems whose coefficients do not vary along the axis of integration, efficient approximations of the one-way operator are constructed by generalizing techniques previously used to create nonreflecting boundary conditions. When physically justified, the method can be applied to systems with slowly varying coefficients in the direction of integration. To demonstrate the accuracy and computational efficiency of the approach, the method is applied to model problems in acoustics and fluid dynamics via the linearized Euler equations; in particular we consider the scattering of sound waves from a vortex and the evolution of hydrodynamic wavepackets in a spatially evolving jet. The latter problem shows the potential of the method to offer a systematic, convergent alternative to ad hoc regularizations such as the parabolized stability equations.

Large eddy simulation for jet noise: azimuthal decomposition and intermittency of the radiated sound

To improve understanding and modeling of jet-noise source mechanisms, extensive experimental and numerical databases are generated for an isothermal Mach 0.9 turbulent jet at Reynolds number Re = 10. The large eddy simulations (LES) feature localized adaptive mesh refinement, synthetic turbulence and wall modeling inside the nozzle to match the fully turbulent nozzle-exit boundary layers in the experiments. Long LES databases are collected for two grids with different mesh resolutions in the jet plume. Comparisons with the experimental measurements show good agreement for the flow and sound predictions, with the far-field noise spectra matching microphone data to within 0.5 dB for most relevant angles and frequencies. Preliminary results on the radiated noise azimuthal decomposition and temporal intermittency are also discussed. The azimuthal analysis shows that the axisymmetric mode is dominant at the peak radiation angles and that the first 3 Fourier azimuthal modes of the LES data recover more than 97% of the total acoustic energy at these angles. The temporal analysis highlights the presence of recurring intermittency in the radiated sound for the low-frequency range and main downstream angles. At these frequencies and angles, temporally-localized bursts of noise can reach levels up to 3 or 4 dB higher (or lower) than the long-time average.

Tonal dynamics and sound in subsonic turbulent jets

Acoustic waves trapped in the potential core of subsonic turbulent jets have recently been observed and explained by Towne et al. We show that these waves also radiate outside the jet, primarily into the upstream arc. We provide an experimental identification of the Mach-number dependence of the phenomenon, which indicates that the modes are active even when evanescent, probably due to turbulent forcing. Finally, we show that for Mach numbers lower than about 0.8, the strong tonal dynamics and sound radiation (up to 170dB) that occur when a sharp edge is placed close to the jet are related to a resonance mechanism involving convective hydrodynamic wavepackets and a ‘slow’, upstreampropagating, trapped acoustic mode. A Helmholtz scaling of the resonance at higher Mach number suggests involvement of the ‘fast’ trapped modes in the range 0.8 ≤ M ≤ 1.

Improved Parabolization of the Euler Equations

Abstract : We present a new method for stability and modal analysis of shear flows and their acoustic radiation. The Euler equations are modified and solved as a spatial initial value problem in which initial perturbations are specified at the ow inlet and propagated downstream by integration of the equations. The modified equations, which we call one-way Euler equations, differ from the usual Euler equations in that they do not support upstream acoustic waves. It is necessary to remove these modes from the Euler operator because, if retained, they cause instability in the spatial marching procedure. These modes are removed using a two-step process. First, the upstream modes are partially decoupled from the down- stream modes using a linear similarity transformation. Second, the error in the first step is eliminated using a convergent recursive filtering technique. A previous spatial marching method called the parabolized stability equations uses numerical damping to stabilize the march, but this has the unintended consequence of heavily damping the downstream acoustic waves. Therefore, the one-way Euler equation could be used to obtain improved accuracy over the parabolized stability equations as a low-order model for noise simulation of mixing layers and jets.

Continued development of the one-way Euler equations: application to jets

An efficient method for calculating linearized disturbances to shear flows that accurately captures their acoustic radiation was recently introduced (Towne & Colonius, AIAA Paper 2013-2171, 2013). The linearized Euler equations are modified such that all upstream propagating acoustic modes are removed from the operator. The resulting equations, called one-way Euler equations, can be stably and efficiently solved in the frequency domain as a spatial initial value problem in which initial perturbations are specified at the flow inlet and propagated downstream by integration of the equations. In this paper, we continue the development of this method with the aim of using it to model wavepackets and their acoustic radiation in turbulent jets. Before turning attention to jets, two dimensional mixing layer noise results computed using the one-way Euler equations are shown to be in excellent agreement with a direct solution of the full Euler equations. The one-way Euler operator is then shown to accurately represent all downstream modes that exist in supersonic and subsonic parallel jets, while properly eliminating the upstream acoustic modes. Finally, the method is applied to a turbulent Mach 0.5 jet mean flow obtained from experimental measurements. The near-field one-way Euler results are similar to those obtained using a previous spatial marching technique called the parabolized stability equations. However, the one-way Euler solutions also include the acoustic fields. With further development, the results suggest that the one-way Euler equation could be used to obtain improved accuracy over the parabolized stability equations as a low-order jet noise model.

Spectral analysis of jet turbulence

Informed by LES data and resolvent analysis of the mean flow, we examine the structure of turbulence in jets in the subsonic, transonic, and supersonic regimes. Spectral (frequency-space) proper orthogonal decomposition is used to extract energy spectra and decompose the flow into energy-ranked coherent structures. The educed structures are generally well predicted by the resolvent analysis. Over a range of low frequencies and the first few azimuthal mode numbers, these jets exhibit a low-rank response characterized by Kelvin-Helmholtz (KH) type wavepackets associated with the annular shear layer up to the end of the potential core and that are excited by forcing in the very-near-nozzle shear layer. These modes too the have been experimentally observed before and predicted by quasi-parallel stability theory and other approximations--they comprise a considerable portion of the total turbulent energy. At still lower frequencies, particularly for the axisymmetric mode, and again at high frequencies for all azimuthal wavenumbers, the response is not low rank, but consists of a family of similarly amplified modes. These modes, which are primarily active downstream of the potential core, are associated with the Orr mechanism. They occur also as sub-dominant modes in the range of frequencies dominated by the KH response. Our global analysis helps tie together previous observations based on local spatial stability theory, and explains why quasi-parallel predictions were successful at some frequencies and azimuthal wavenumbers, but failed at others.