Gradient-based Approach for Determination of Oscillating Flow Fields in PIV

Abstract

Oscillating velocity vector fields are often observed in fluid flow fields. A flow field caused by the Karman vortices is a typical example of an oscillating velocity vector field. In particle image velocimetry (PIV), a velocity vector field is determined from an image sequence representing a fluid flow field visualised by particles (Adrian, 1991; Raffel et al., 2007). When we focus on an oscillating velocity vector field, we can expect that the oscillatory characteristic of the field is useful for determining a high accurate velocity vector field. Thus, utilising the oscillatory characteristic as an additional constraint is an interesting topic for the PIV approaches, such as the matching-based approach and the gradient-based one. Thematching-based approach utilises a pattern-matching procedure, which calculates a crosscorrelation function between two image templates on the two successive image frames. The peak position of the obtained two-dimensional cross-correlation function provides a displacement vector for a brightness pattern during one frame, that is, a velocity vector. Under the low density of particles, it is possible to track a particular particle during two or more successive image frames by the pattern-matching procedure (Hassan et al., 1992). When the density of particles is in the middle range, the pattern-matching procedure for a particle distribution function is utilised to detect its velocity vector (Willert and Gharib, 1991). The gradient-based approach first estimates spatio-temporal gradients on an image brightness function, and then derives the basic constraint equation consisting of the gradients and two velocity components (Horn and Schunck, 1981). The basic constraint equation is derived from the correspondence of a moving image brightness pattern during a short time period. Next, it organises the error function consisting of the basic constraint equation and additional one(s) modelling the characteristics of a fluid flow field. Finally, minimising the error function by an optimisation method provides the two velocity components. There are several problems in the matching-based approach and the gradient-based one. With the gradient-based approach, it is difficult to determine high speed flow fields. In contrast to this, while the matching-based approach can determine such the high speed flow fields, the sub-pixel accuracy of the approach is generally unreliable. The gradient-based approach 4