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Automated topology classification method for instantaneous velocity fields

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Abstract

Topological concepts provide highly comprehensible representations of the main features of a flow with a limited number of elements. This paper presents an automated classification method of instantaneous velocity fields based on the analysis of their critical points distribution and feature flow fields. It uses the fact that topological changes of a velocity field are continuous in time to extract large scale periodic phenomena from insufficiently time-resolved datasets. This method is applied to two test-cases : an analytical flow field and PIV planes acquired downstream a wall-mounted cube.

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Notes

  1. Let O be a critical point of a 2-D/2-C velocity field (U(x,y), V(x,y)), its jacobian matrix J is defined as \({\left(\begin{array}{*{20}l} {\frac{\partial U} {\partial x}} & {\frac{\partial U}{\partial y}}\\ \\ {\frac{\partial V}{\partial x}} & {{\frac{\partial V}{\partial y}}}\\ \end{array}\right)_O}.\)

  2. Due to the linear interpolation, if any given velocity field u a is compared to u b  = 0, every streamline of f is aligned with \({\varvec{\zeta}.}\)

  3. The Matlab cmdscale command is used.

  4. Let p = −trJ and q = detJ, where J is the jacobian matrix of the velocity field taken at the critical point. p and q are continuous functions as the critical point moves downstream. In some cases, this leads to a configuration where we no longer have q > p 2/4 (i.e. a focus), but q < p 2/4 instead (i.e. a node).

  5. With the notation defined in the next paragraph, Fig. 11a, b, respectively, represent pairings between A2–A4 fields and A12–A4 fields.

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Authors and Affiliations

  1. PSA Peugeot Citroën, Direction de la Recherche et de l’Innovation Automobile, Route de Gisy, 78943, Vélizy-Villacoublay Cedex, France

    S. Depardon & J. J. Lasserre

  2. Laboratoire d’Etudes Aérodynamiques, Téléport 2, 1 Av. Clément Ader, BP 40109, 86961, Futuroscope Chasseneuil, France

    S. Depardon, L. E. Brizzi & J. Borée

Authors
  1. S. Depardon
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  2. J. J. Lasserre
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  3. L. E. Brizzi
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  4. J. Borée
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Correspondence to S. Depardon.

Appendix: Multi-dimensional scaling (MDS)

Appendix: Multi-dimensional scaling (MDS)

In this appendix is shown a basic application of multi dimensional scaling [MDS, Lavin et al. (1998); Shepard (1962)]. Let d be a 3 × 3 matrix corresponding to the “field-to-field” distance between three velocity fields:

$$ d = \left(\begin{array}{lll} 0 & 4 & 9 \\ 4 & 0 & 6 \\ 9 & 6 & 0 \end{array}\right) $$
(8)

For example, for Nb = 3:

The results can be represented on a 2-D map (Fig. 16a), where:

  • Each point corresponds to an instantaneous velocity field

  • The euclidean distance δ n,p on this map matches their “field-to-field” distance d n,p . In this case δ n,p = d n,p .

In such representations, the important parameter is the distance between each point and the others, and not their coordinates or the origin of the frame of reference.

Fig. 16
figure 16

MDS 2-D (a) and 1-D (b) representation of d

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The results can also be represented on a 1-D axis (Fig. 16b). In that case, d ij ≠ δ ij . Multi dimensional scaling [MDS, Lavin et al. (1998); Shepard (1962)] is needed to compute a configuration where the euclidean distance between every pair of objects in this low order representation matches their real “field-to-field” distance i.e. to minimize \({\epsilon=\sqrt{{{\sum_{n,p}\left(\delta_{n,p} - d_{n,p}\right)^2}\over {\sum_{n,p}d_{n,p}^2}}}}.\) In this example, the 1-D representation yields ε = 0.07 (Fig. 16b). For any Nb, an exact representation requires a dimension of Nb−1, which is not applicable for a few hundreds datasets. Therefore, methods such as MDS is necessary to provide understandable representation on 2-D or 3-D maps.

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Depardon, S., Lasserre, J.J., Brizzi, L.E. et al. Automated topology classification method for instantaneous velocity fields. Exp Fluids 42, 697–710 (2007). https://doi.org/10.1007/s00348-007-0277-3

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