Demonstration and verification of exact DMD analysis applied to double-pulsed schlieren image of supersonic impinging jet

The exact dynamic mode decomposition (DMD) was applied to the nonsequential image dataset obtained by the double-pulsed schlieren measurement of a supersonic impinging jet, and the effect of the dataset length on the obtained spatial modes and estimated frequencies of the aeroacoustic fields was investigated. The Mach number of the jet was 2.0, the Reynolds number based on the diameter of the nozzle exit was \(1.0\times 10^6\) and the distance between the nozzle exit and the flat plate was four times the nozzle diameter long. The DMD modes extract the characteristic pattern and its frequency that relate to the aeroacoustic fields. The estimated frequencies of DMD modes were compared with the acoustic spectra measured using microphones. The estimated frequency of the DMD mode that has the largest amplitude approximately coincides with that of the highest peak in the acoustic spectra regardless of the dataset length. However, the variation in the estimated frequencies of the high-order DMD modes increases when the dataset length is short. Although the estimated frequencies of the second and third DMD modes did not match the peak frequencies of the acoustic spectra, the estimation accuracy of the frequency of the modes can be improved by recalculating the frequency based on the wavelength of the corresponding spatial mode. The order of the amplitude of DMD modes did not agree with the order of the peak magnitude in the acoustic spectra, except for the first mode. This is because the schlieren method visualizes the density gradient resulting in emphasizing the high-frequency fluctuations. This mismatch was mitigated by correcting the acoustic spectrum considering the first derivative of the acoustic spectrum. Therefore, the verification of the estimation accuracy considering the data characteristics is important when the exact DMD analysis is applied to the noisy experimental data.