Velocity data were obtained in the turbulent wake of a circular cylinder with an orthogonal array of sixteen X-wires, eight in the (x, y)-plane and eight in the (x, z)-plane. By applying the phase-plane technique to these data, three types of critical points (where the velocity is zero and the streamline slope is indeterminate) were identified. Of these, foci and saddle points occurred most frequently, although a significant number of nodes was also found. Flow topology and properties associated with these points were obtained in each plane. Saddle-point regions associated with spanwise vortices provide the dominant contribution to the Reynolds shear stress and larger contributions to the normal stresses than focal regions. The topology was found to be in close agreement with that obtained from other methods of detecting features of the organized motion. The inter-relationship between critical points simultaneously identified in the two planes can provide some insight into the three-dimensionality of the organized motion. Foci in the (x, z)-plane correspond, with relatively high probability and almost negligible streamwise separation, to saddle points in the (x, y)-plane and are interpreted in terms of ribs aligned with the diverging separatrix between consecutive spanwise vortex rolls. Foci in the (x, z)-plane which correspond, with relatively weak probability, to foci in the (x, y)-plane seem consistent with a distortion of the vortex rolls in the (y, z)-plane.
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