Acoustic resonance in the potential core of subsonic jets

Abstract

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,$0.82<M<1$.

Analysis of Coherent Structures in an Under-Expanded Supersonic Impinging Jet Using Spectral Proper Orthogonal Decomposition (SPOD)

“…This decomposition method identifies energy-ranked modes where each oscillates at a single frequency, and has been overlooked since its inception despite its benefits over the common spatial form of POD [43] and dynamic mode decomposition [16]. There are few techniques in the literature for computing SPOD modes [13,44,45]. In this study, we used the procedure developed by Towne et al [13], as it was found to be computationally more efficient compared to others.…”

“…There are few techniques in the literature for computing SPOD modes [13,44,45]. In this study, we used the procedure developed by Towne et al [13], as it was found to be computationally more efficient compared to others. The details of the procedure can be found in the comprehensive paper by Towne et al [13].…”

“…In recent years, there has been considerable interest in data-driven techniques to reveal the characteristics of coherent structures, especially in jet flows [13][14][15][16][17][18]. In the recent series of large eddy simulation studies of ideally expanded and under-expanded supersonic slot and round impinging jets [5,8,19], the discrete Fourier transform of the near-field pressure has been used to obtain the frequencies of the acoustic tones.…”

“…Dynamic mode decomposition (DMD) is a more recent data-driven approach proposed by Schmid [16], where the flow field is decomposed into modes, with each mode having a growth/decay rate at a single frequency. A fundamental issue of POD, DMD, and Fourier decomposition is their weakness in identifying coherent structures when they have low energy or multiple frequencies [13]. More recently, spectral proper orthogonal decomposition (SPOD) was revitalised by Towne et al [13].…”

“…A fundamental issue of POD, DMD, and Fourier decomposition is their weakness in identifying coherent structures when they have low energy or multiple frequencies [13]. More recently, spectral proper orthogonal decomposition (SPOD) was revitalised by Towne et al [13]. SPOD is based on the original decomposition method of Lumley [22], where the flow is decomposed into energy-ranked spatial modes at discrete frequencies.…”