Data Analysis from Velocity Vectors

In applications, the velocity information is often necessary but not sufficient and other quantities will be of interest as well. The velocity field obtained from PIV measurements can be used to estimate relevant quantities by means of differentiation and integration. The vorticity field is of special interest because, unlike the velocity field it is independent of the frame of reference. In particular, if it is resolved temporally, the vorticity field can be much more useful in the study of flow phenomena than the velocity field. This is particularly true in highly vortical flow such as turbulent shear layers, wake vortices and complex vortical flows. Integral quantities can also be obtained from the velocity data. The instantaneous velocity field obtained by PIV can be integrated, yielding either a single path integrated value or another field such as the stream function. Analogous to the vorticity field, the circulation obtained through path integration is also of special interest in the study of vortex dynamics, mainly because it is also independent of the reference frame. In the following section, data analysis for calculation of various derived quantities from PIV measurements are presented.

Velocity differentials

The differential terms are estimated from the velocity vectors obtained from PIV. Since PIV provides the velocity vector field sampled on a two dimensional evenly spaced grid specified as finite differencing can be employed to get the spatial derivatives. There are a number of finite difference schemes that can be used to obtain the derivatives. The truncation error associated with each operator is estimated by means of a Taylor series expansion. The actual uncertainty in differentiation is due to that in the uncertainty of the velocity estimate It can be obtained using standard error propagation methods assuming individual data to be independent of the other. There are two schemes that reduce the error associated with differentiation: Richardson extrapolation and least squares approach. The former minimizes the truncation error while the least squares approach reduces the effect of random error, i.e. the measurement uncertainty, These approaches are briefly discussed bellow.

Least squares estimate of the first derivative:

Here, the accuracy is of the order of and the associated uncertainty is

The derivative using Richardson extrapolation is calculated as:

The accuracy of the above approximation is of order and the uncertainty associated with the expression is