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.

If the square matrix A is real, its eigenvalues are either real or come in complex conjugate pairs. c In such case, A is called diagonalizable or non-defective. If A is defective, we have A = V JV −1 with J being the canonical Jordan form [31,33].

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 quasiparallel predictions were successful at some frequencies and azimuthal wavenumbers, but failed at others.

The purpose of this paper is to characterize and model waves that are observed within the potential core of subsonic jets and relate them to previously observed tones in the near-nozzle region. The waves are detected in data from a large-eddy simulation of a Mach 0.9 isothermal jet and modelled using parallel and weakly non-parallel linear modal analysis of the Euler equations linearized about the turbulent mean flow, as well as simplified models based on a cylindrical vortex sheet and the acoustic modes of a cylindrical soft duct. In addition to the Kelvin–Helmholtz instability waves, three types of waves with negative phase velocities are identified in the potential core: upstream- and downstream-propagating duct-like acoustic modes that experience the shear layer as a pressure-release surface and are therefore radially confined to the potential core, and upstream-propagating acoustic modes that represent a weak coupling between the jet core and the free stream. The slow streamwise contraction of the potential core imposes a frequency-dependent end condition on the waves that is modelled as the turning points of a weakly non-parallel approximation of the waves. These turning points provide a mechanism by which the upstream- and downstream-travelling waves can interact and exchange energy through reflection and transmission processes. Paired with a second end condition provided by the nozzle, this leads to the possibility of resonance in limited frequency bands that are bound by two saddle points in the complex wavenumber plane. The predicted frequencies closely match the observed tones detected outside of the jet. The vortex-sheet model is then used to systematically explore the Mach number and temperature ratio dependence of the phenomenon. For isothermal jets, the model suggests that resonance is likely to occur in a narrow range of Mach number,$0.82<M<1$.

To investigate the effects of the nozzle-exit conditions on jet flow and sound fields, large-eddy simulations of an isothermal Mach 0.9 jet issued from a convergent-straight nozzle are performed at a diameter-based Reynolds number of $1\times 10^{6}$. The simulations feature near-wall adaptive mesh refinement, synthetic turbulence and wall modelling inside the nozzle. This leads to fully turbulent nozzle-exit boundary layers and results in significant improvements for the flow field and sound predictions compared with those obtained from the typical approach based on laminar flow in the nozzle. The far-field pressure spectra for the turbulent jet match companion experimental measurements, which use a boundary-layer trip to ensure a turbulent nozzle-exit boundary layer to within 0.5 dB for all relevant angles and frequencies. By contrast, the initially laminar jet results in greater high-frequency noise. For both initially laminar and turbulent jets, decomposition of the radiated noise into azimuthal Fourier modes is performed, and the results show similar azimuthal characteristics for the two jets. The axisymmetric mode is the dominant source of sound at the peak radiation angles and frequencies. The first three azimuthal modes recover more than 97 % of the total acoustic energy at these angles and more than 65 % (i.e. error less than 2 dB) for all angles. For the main azimuthal modes, linear stability analysis of the near-nozzle mean-velocity profiles is conducted in both jets. The analysis suggests that the differences in radiated noise between the initially laminar and turbulent jets are related to the differences in growth rate of the Kelvin–Helmholtz mode in the near-nozzle region.