Spatiotemporal Koopman decomposition of second mode instability from a hypersonic schlieren video

Data-driven modal analysis methods provide a powerful way to decompose data into a sum of modes. The spatiotemporal Koopman decomposition (STKD) enables the computation of modes defined by global frequencies and growth rates in various spatial dimensions and time. The method is an extension of the dynamic mode decomposition (DMD) and higher-order dynamic mode decomposition (HODMD) that represents the data as a sum of standing and traveling, possibly growing or decaying, waves. In this paper, the STKD with HODMD is applied to schlieren video highlighting second mode instability waves traveling down the length of a 3-degree half-angle cone and a 7-degree half-angle cone, both at a freestream Mach number of 6. The HODMD is able to compute dominant modes and frequencies that align with those from associated experimental measurements of unsteady pressure fluctuations, and whose mode shapes clearly show the intensifying wavepacket structure of the waves. The STKD algorithm is used to compute streamwise wavenumbers, spatial growth rates, and wave speeds. The spatial growth rates from the STKD and the magnitudes of the HODMD mode shapes are used to compute the N-factor for waves of several frequencies. Overall, the STKD with HODMD is shown to be a useful tool for extracting spatiotemporal disturbance growth from a schlieren video.