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A characteristic dynamic mode decomposition

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Abstract

Temporal or spatial structures are readily extracted from complex data by modal decompositions like proper orthogonal decomposition (POD) or dynamic mode decomposition (DMD). Subspaces of such decompositions serve as reduced order models and define either spatial structures in time or temporal structures in space. On the contrary, convecting phenomena pose a major problem to those decompositions. A structure traveling with a certain group velocity will be perceived as a plethora of modes in time or space, respectively. This manifests itself for example in poorly decaying singular values when using a POD. The poor decay is counterintuitive, since a single structure is expected to be represented by a few modes. The intuition proves to be correct, and we show that in a properly chosen reference frame along the characteristics defined by the group velocity, a POD or DMD reduces moving structures to a few modes, as expected. Beyond serving as a reduced model, the resulting entity can be used to define a constant or minimally changing structure in turbulent flows. This can be interpreted as an empirical counterpart to exact coherent structures. We present the method and its application to a head vortex of a compressible starting jet.

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Notes

  1. Here we note that we do not make use of all data using this approach, specially if the dataset is periodic. This is unlike the method of Rowley and Marsden [22] which implements a pure shift. But as we are not willing to choose the method on economic grounds, there is not much to do about this for the moment.

  2. Transformation in time can be achieved by choosing the snapshots equidistantly in \(t'\) as defined above. Since abundant timesteps are available from the existing numerical simulation, this can be easily done by choosing the right snapshots. On the other hand, since the vortex head is expanding with time, via a more general approach proposed by Rowley et al. [23], also a scaling can be employed. We refer the reader to that article for further information.

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Acknowledgements

The authors would like to gratefully acknowledge Juan José Peña Fernández for providing simulated data for this study. Major parts of this work were performed during the sabbatical leave of the first author in 2015. He wishes to thank both his own university and La Sapienza, Rome, for this opportunity. The second author would like to appreciate Prof. Christoph Egbers (Brandenburg University of Technology Cottbus-Senftenberg), for the invaluable support during this research.

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Authors and Affiliations

  1. Technical University of Berlin, Berlin, Germany

    Jörn Sesterhenn

  2. Brandenburg University of Technology, Cottbus, Brandenburg, Germany

    Amir Shahirpour

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  1. Jörn Sesterhenn
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  2. Amir Shahirpour
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Correspondence to Jörn Sesterhenn.

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Communicated by Daniel J. Bodony.

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Sesterhenn, J., Shahirpour, A. A characteristic dynamic mode decomposition. Theor. Comput. Fluid Dyn. 33, 281–305 (2019). https://doi.org/10.1007/s00162-019-00494-y

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