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Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows

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Abstract

We present a Parseval tight wavelet frame for the representation and analysis of velocity vector fields of incompressible fluids. Our wavelets have closed form expressions in the frequency and spatial domains, are divergence free in the ideal, analytic sense, have a multi-resolution structure and fast transforms, and an intuitive correspondence to common flow phenomena. Our construction also allows for well defined directional selectivity, e.g. to model the behavior of divergence free vector fields in the vicinity of boundaries or to represent highly directional features like in a von Kármán vortex street. We demonstrate the practicality and efficiency of our construction by analyzing the representation of different divergence free vector fields in our wavelets.

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Acknowledgements

The author thanks Eugene Fiume for continuing support. Michal Jarzabek and Philipp Herholz helped with implementing the construction.

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  1. Otto-von-Guericke Universität Magdeburg, Magdeburg, Germany

    Christian Lessig

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  1. Christian Lessig
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Correspondence to Christian Lessig.

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Communicated by T. Richter

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Lessig, C. Divergence Free Polar Wavelets for the Analysis and Representation of Fluid Flows. J. Math. Fluid Mech. 21, 18 (2019). https://doi.org/10.1007/s00021-019-0408-7

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