Inlet conditions for wave packet models in turbulent jets based on eigenmode decomposition of large eddy simulation data

Abstract

This paper makes contributions towards reduced-order models of wave packets in supersonic, turbulent jets. Wave packets are large-scale turbulent structures that are correlated and advected over distances that are large compared to the integral scales of turbulence, i.e., many jet diameters at the lowest frequencies. They are thought to be responsible for the peak noise radiated at shallow angles to the jet axis. Linear wave packet models based on the Parabolized Stability Equations (PSE) have been shown in the past to be in excellent agreement with statistical structures educed from experimental pressure and velocity data in subsonic jets. Here, we extend these models to supersonic jets and validate them using a Large Eddy Simulation (LES) database for an isothermal and a moderately heated Mach 1.5 turbulent jets. For supersonic jets, inlet conditions for PSE models are ambiguous, as a parallel flow stability analysis shows several unstable modes at the inlet cross section. We develop a bi-orthogonal decomposition and project the LES data onto the relevant families of instability waves. These serve as inlet conditions, including the amplitude and shape functions, for PSE solutions which are then favorably compared to the near-field pressure fields educed from LES. C 2013 AIP Publishing LLC. [http://dx

Wavepacket models for supersonic jet noise

“…As the PSE mode adopts the correct near-field acoustic behaviour downstream (see figure 6), it exhibits the velocity oscillations associated with this linear eigenmode in the core since a single axial wavenumber is being imposed by the ansatz. Examples of such acoustic modes were presented by Rodríguez et al (2013) for the jets under consideration here. PSE is unable to model the larger set of modes required to represent the full physics in this region.…”

“…Rodríguez et al (2013) present an alternate method of determining the amplitudes of the PSE wavepackets based on an adjoint-based projection of the LES fluctuation data near the nozzle exit plane onto the parallel-flow linear stability modes used to initiate the PSE. Figure 6 depicts visual comparisons of the linear PSE solutions with the first POD eigenfunctions for some of the Fourier modes under consideration.…”

Amplitude scaling of turbulent-jet wavepackets

“…Hence, strictly speaking, one cannot think of the procedure above as a projection as often done for stability eigenfunctions. 15,16 However, we are here interested in the determination of the amplitude of a single spatial function, and thus the approach can be used as method to estimate wavepacket amplitudes considering a jet slice at x o . Notice that in the upstream region both the SPOD mode and the PSE solution have similar growth rates, and because of this the amplitude parameter A is expected to depend only weakly on the choice of x o .…”

“…This can be done in an ad hoc way using a limited number of available measurements, or in a more theoretically-consistent manner by obtaining the amplitude of the Kelvin-Helmholtz mode near the nozzle exit by a projection using the adjoint mode. 16 Each combination of Strouhal number St and azimuthal mode m leads a separate free amplitude. How such amplitudes scale with St and m remains an open question.…”

A second golden age of aeroacoustics?

“…The parabolized stability equations (PSEs) provide an efficient way to compute unsteady perturbations about base flows, which organize into wavepackets for subsonic jets [145,146]. While this method has proved successful for subsonic jet noise [147], low-frequency noise from hot supersonic jets remains difficult to predict [148]. With increased computational power, it is also possible to directly compute global modes of supersonic jet noise [149] which do not rely upon the assumption of a streamwise slowly varying base flow.…”