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

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Abstract

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.

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Acknowledgements

The present work was supported by the Japan Society for the Promotion of Science, KAKENHI Grants No. JP20H00278 and JP19KK0361, and by Japan Science and Technology Agency, FOREST Grant Number JPMJFR202C. Y. Ozawa was supported by the Japan Society for the Promotion of Science, KAKENHI Grants 19H00800. T. Nagata was supported by Japan Science and Technology Agency, CREST Grant Number JPMJCR1763.

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Authors and Affiliations

  1. Department of Aerospace Engineering, Graduate School of Engineering, Tohoku University, 6-6-01, Aramaki Aza Aoba, Aoba-ku, Sendai, Miyagi, 980-8579, Japan

    Kasumi Ohmizu, Yuta Ozawa, Takayuki Nagata, Taku Nonomura & Keisuke Asai

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  1. Kasumi Ohmizu
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  2. Yuta Ozawa
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  3. Takayuki Nagata
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  4. Taku Nonomura
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  5. Keisuke Asai
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Correspondence to Yuta Ozawa.

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Appendix

Appendix

1.1 Appendix: Effect of masked region on DMD spectra

Although the present study assumes that the acoustic waves propagate with the ambient sound speed, the schlieren image that is originally used for the DMD calculation includes the mainstream area of the jet with the different propagation speeds. This may induce the estimation error of the DMD calculation. Therefore, the exact DMD was performed on the schlieren images with the newly defined mask, which also masks the region of the mainstream area of the jet. Figure 14 shows a comparison of the DMD spectra with different masked regions and illustrates that the presence or absence of the jet region had almost no effect on the estimated frequencies of the DMD modes in the present dataset.

Fig. 14
figure 14

Comparison of the DMD spectra with different mask regions

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1.2 Appendix: Effect of intermittency and dataset length on DMD amplitude

The relation between the intermittency and the deviation in the DMD amplitude was investigated. Figure 15 shows the scalogram of the microphone data calculated by the wavelet analysis. It was confirmed that the amplitude of screech tone changes within the measurement time due to the intermittency of the screech tone.

Fig. 15
figure 15

Scalogram of the microphone data

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Figures 16 and 17 show the histograms of the DMD amplitude for each dataset length. Similar histograms were obtained for mode 1 and 2, respectively, regardless of the dataset length. However, only N1250_4 deviated from the others, where the amplitudes were swapped between the modes 1 and 2. This indicates that the intermittency of the screech tone is not related to the amplitude switching.

Fig. 16
figure 16

Histogram of the DMD amplitude of the first mode

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Fig. 17
figure 17

Histogram of the DMD amplitude of the second mode

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Fig. 18
figure 18

Effect of the dataset length in the estimated DMD amplitude

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Figure 18 shows the relation between the dataset length and the estimated DMD amplitude. Additional cases of N1000 and N500 were computed and the average of the estimated amplitudes for each dataset was plotted. The error bars show the variation in the amplitude. This figure illustrates that the shorter dataset length results in a larger variation in the DMD amplitude. In the present data, the variation of the estimated amplitudes is sufficiently small at \(N\ge 2500\), and the amplitudes can be calculated stably.

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Ohmizu, K., Ozawa, Y., Nagata, T. et al. Demonstration and verification of exact DMD analysis applied to double-pulsed schlieren image of supersonic impinging jet. J Vis 25, 929–943 (2022). https://doi.org/10.1007/s12650-022-00836-9

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