A particle-tracking approach for accurate material derivative measurements with tomographic PIV

Abstract

The evaluation of the instantaneous 3D pressure field from tomographic PIV data relies on the accurate estimate of the fluid velocity material derivative, i.e., the velocity time rate of change following a given fluid element. To date, techniques that reconstruct the fluid parcel trajectory from a time sequence of 3D velocity fields obtained with Tomo-PIV have already been introduced. However, an accurate evaluation of the fluid element acceleration requires trajectory reconstruction over a relatively long observation time, which reduces random errors. On the other hand, simple integration and finite difference techniques suffer from increasing truncation errors when complex trajectories need to be reconstructed over a long time interval. In principle, particle-tracking velocimetry techniques (3D-PTV) enable the accurate reconstruction of single particle trajectories over a long observation time. Nevertheless, PTV can be reliably performed only at limited particle image number density due to errors caused by overlapping particles. The particle image density can be substantially increased by use of tomographic PIV. In the present study, a technique to combine the higher information density of tomographic PIV and the accurate trajectory reconstruction of PTV is proposed (Tomo-3D-PTV). The particle-tracking algorithm is applied to the tracers detected in the 3D domain obtained by tomographic reconstruction. The 3D particle information is highly sparse and intersection of trajectories is virtually impossible. As a result, ambiguities in the particle path identification over subsequent recordings are easily avoided. Polynomial fitting functions are introduced that describe the particle position in time with sequences based on several recordings, leading to the reduction in truncation errors for complex trajectories. Moreover, the polynomial regression approach provides a reduction in the random errors due to the particle position measurement. Finally, the acceleration can be evaluated analytically, which greatly reduces the truncation errors due to finite differences. The approach is first assessed using computer-generated data of an advecting vortex ring. Precision errors in the material derivative can be reduced with a factor 2–3. This is achieved when a long sequence is considered (e.g. 15–20 recordings). Similarly, truncation errors typically associated with direct integration and finite differences from the PIV-based technique are almost eliminated. It is shown that the material derivative information obtained at the scattered locations in the 3D domain can be reduced to a uniform Cartesian grid by means of a second-order spatial regression with no significant artefact. The technique is applied to a benchmark Tomo-PIV experiment of a transitional jet in water. The results confirm the conclusions obtained with the numerical study. Moreover, it is shown that the evaluation of the instantaneous 3D pressure field can be retrieved with significant reduction in artefacts associated with random and truncation errors.