A topological evaluation procedure to assess the integrity of a PIV vector field

Particle image velocimetry (PIV) provides a field of discrete vectors to represent a continuum velocity field. Various methods have been adopted to evaluate the integrity of the discrete vectors. In contrast, the present communication provides a systematic technique whereby the integrity of the measured field can be assessed using basic topological principles. Starting with the recognition that PIV provides a vector field overlaid on a planar surface, the analyst can identify the holes (to be punched through the surface of a sphere) and the handles (to be added to the sphere's surface) that will represent the appropriate surface for the topological analysis. These operations define the a priori Euler characteristic (χ_A) for the subject PIV image. The experimental Euler characteristic (χ_E) will be known from the properties of the measured vector field: nodes, saddles, etc. A necessary condition for the integrity of the measured vector field is that χ_E   =   χ_A. The topological bases for the integrity evaluation, including the important constraint of ensuring a smooth collapsed sphere, are carefully explained and described with examples. (paper)