Vector Field Topology in the Context of Separation and Attachment of Flows

The interaction of a flow with the objects embedded in it mostly results in attachment or detachment (also separation) of the flow to respectively from the object. Separation and attachment usually appears at isolated points or lines. In steady flows, separation and attachment are related to singularities and separatrices, i.e. vector field topology, of the wall-shear stress field [SGH06]. Consequently, a number of flow visualization techniques employing the concepts of vector field topology for the detection of separation and attachment have been proposed. First, Helman and Hesselink extracted separation and attachment lines (short: feature lines) as separatrices of the wall-shear stress field [HH90]. Later, Kenwright et al. [KHL99] presented a marching method that used the “separatrices” of the linear vector field given in triangle cells with special vector configurations to construct segments of piecewise defined feature lines. Only recently attracting singularities on surfaces have been used for illustrating vortices connected to object boundaries [WTS∗07]. These vortices are also connected to separation as they drag the flow away from the surface. Unfortunately, the mentioned methods using separatrices lead to incorrect results if applied to instantaneous vector fields, i.e. snapshots of unsteady flows [SJGH08]. Thus, the discussion of separatrices on surfaces as feature lines applies only to autonomous vector fields throughout the rest of the paper.