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Spectral Proper Orthogonal Decomposition-based Bispectral Analysis of Nonlinear Flows

  • Author(s): Zhao, Xueqi
  • Advisor(s): Schmidt, Oliver
  • et al.

In this paper, we combine the spectral proper orthogonal decomposition (SPOD) and bispectral analysis to identify the dominant quadratic interaction triads in nonlinear flows, and also extract the corresponding bispectral modes involved in the quadratic interactions. The main idea of this approach is using n ranks of SPOD modes to reconstruct the Fourier realizations of flow data, calculating the bispectrum with the definition of integral measure bispectrum, and then taking the SPOD modes as the flow structures involved in quadratic interactions. This research is an alternative to the bispectral mode decomposition method (BMD). This SPOD-based bispectral analysis method is more time-saving compared with BMD method. Since the SPOD-based bispectral analysis method combines the SPOD and bispectral analysis, it can retain the properties of both SPOD and bispectral analysis. The SPOD inherits the optimality of POD, it can provide a hierarchy of modes in terms of energy, and it is distinct from the spatial POD in the sense of giving spatial-temporal-coherent modes. The bispectral analysis has been widely used for identifying quadratic interactions in the area of statistical analysis. These properties of the SPOD-based bispectral analysis approach can give us advantages to construct the reduced-order model of nonlinear flows which can reproduce the nonlinear dynamics in a more computationally efficient way. Also, the results given by this approach can help us interpreting the energy transfer mechanisms in nonlinear flow. To show the application of this approach, we first apply the approach on several artificial test datasets to check the feasibility and denoising ability of this approach. We find that the SPOD-based method has a good capability of resisting the effect of white noise. Then, we apply it to the data of two real fluid mechanics datasets, the direct numerical simulation (DNS) data of a cylinder flow at Re=500 and the particle image velocity (PIV) data of a Mach 0.6 open cavity flow. We compare the results of the SPOD-based approach with the results of bispectral mode decomposition (BMD). The outcomes given by these two approaches are very similar, which means that the SPOD method can be an alternative to the BMD method for important triads identification and flow structures extraction. We also show that in some cases, more than 1 ranks of SPOD modes are needed for more accurate representation in terms of both the flow energy and bispectrum value.

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