The spatiotemporal dynamics of the coherent structures in an under-expanded supersonic impinging jet are studied using a spectral proper orthogonal decomposition technique. For this analysis, a large eddy simulation of an under-expanded supersonic impinging jet at a pressure ratio of 3.4 and a stand-off distance of 2 jet diameters at a Reynolds number of 50,000 is performed. The mean flow fields illustrate some striking features of this flow, such as an oblique shock, a stand-off shock, a Mach disk, and a recirculation bubble. The spectral proper orthogonal decomposition method is applied to time-resolved three-dimensional flow fields. The accumulative energy of modes within each azimuthal mode number reveals that the first three azimuthal modes contain most of the energy of the flow. The spectra of these azimuthal modes show that the flow exhibits a low-ranked behaviour with discrete frequencies at the optimal symmetric azimuthal mode while other two azimuthal modes have negligible contributions in this behaviour. Three peaks are observed in the spectra of the optimal symmetric azimuthal mode. The spatial fields of the streamwise velocity and pressure of these peaks show that the complex structures are consequences of the under-expansion, Mach disk, and the impingement. Strong hydrodynamic instabilities exist in the shear layer of the jet in the optimal azimuthal mode at each of these dominant frequencies. High-amplitude acoustic waves are also present in the near-field of the jet. These acoustic waves are strong at the nozzle lip, suggesting that a feedback loop linking these two processes exists for dominant frequencies in the optimal mode. High cross-spectrum density of near-field pressure fluctuations and streamwise velocity fluctuations near the nozzle lip at these frequencies confirms the hydro-acoustic coupling, which is necessary to close the feedback loop.
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