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Surface selections and topological constraint evaluations for flow field analyses

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Abstract

The isolated singular points (nodes, saddles) of a continuous vector field (e.g., velocity, shear stress, pressure gradient, vorticity, etc.) that are overlaid on a given surface must be compatible with the Euler characteristic of that surface, Xsurface. All surfaces can be fashioned from a sphere plus handles plus holes, and Xsurface=2−Σholes−2Σhandles=Σnodes−Σsaddles. This establishes an a priori constraint for the nodes and saddles that can be tested against the observed vector field to determine whether the experimental (or computational) observations are compatible with the known constraint. Numerous examples, including a clarification of, and a correction to, published results are given.

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Notes

  1. These references were kindly supplied by a reviewer.

  2. A formal definition of a sphere is provided in the Appendix.

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Acknowledgements

The author’s original experience with the motivating elements for this exposition were gained as an Alexander von Humboldt Fellow at the University of Karlsruhe. Professor W. Rodi served as the research host for this experience. Grateful appreciation is expressed to the AvH and Professor Rodi. MSU mathematics colleagues, Professors R. Miller and J. McCarthy, have provided substantial guidance and instruction for the topological elements presented herein. Useful discussions and clarifications were also gained from a 1995 interaction with Professors A.E. Perry and M.S. Chong at the University of Melbourne. Expert assistance with the preparation of the figures has been provided by M. Dusel and A. Butki.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Michigan State University, East Lansing, Michigan, 48824, USA

    John F. Foss

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  1. John F. Foss
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Correspondence to John F. Foss.

Appendix

Appendix

The following definition was provided to the author by Professor J.D. McCarthy. It is repeated here for mathematical completeness in relation to the centrally important understanding of a sphere in this communication.

An appropriate definition, in the context of the present communication, which is focused on three-dimensional Euclidean space, R3, can be expressed as follows. A sphere can be understood as that of a smooth two-dimensional submanifold S of R3 diffeomorphic to the standard two-dimensional sphere S2 in R3 (i.e., the set of all points in R3 at a distance of 1 from the origin (0, 0, 0) of R3):

  1. 1.

    A subset S of R3

  2. 2.

    With a smoothly varying two-dimensional tangent space at each point p of S

  3. 3.

    Admitting a smooth map F: S2→S with a smooth inverse map G: S→S2

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Foss, J.F. Surface selections and topological constraint evaluations for flow field analyses. Exp Fluids 37, 883–898 (2004). https://doi.org/10.1007/s00348-004-0877-0

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