Proper Orthogonal Decomposition Analysis of Particle Image Velocimetry Data at the Inlet of a Centrifugal Compressor
Authors: Banerjee Deb , Selamet Ahmet , Dehner Rick 
Published: February 2023

Abstract
Stereoscopic Particle Image Velocimetry (SPIV) measurements are carried out at the inlet of a turbocharger compressor at various rotational speeds from 80,000 to 140,000 rpm and mass flow rates spanning the entire compressor flow range. The data obtained is then used to perform a Proper Orthogonal Decomposition (POD) analysis of the flow field. POD is a modal decomposition technique that allows for the study of coherent structures (robust vortical structures that retain their identity for many turnover times) in turbulent flow fields. In the present work, the POD analysis is focused at a fixed rotational speed of 80,000 rpm and two mass flow rates: choke (maximum flow rate) and mild surge (minimum flow rate before encountering deep surge) at the specified speed. The POD analysis is performed using the singular value decomposition algorithm. The results at these two operating conditions illustrate the differences in the overall description of the POD modes for a fully developed turbulent flow field (choke), and a highly three-dimensional, swirling, wall-bounded shear flow (mild surge) at the centrifugal compressor inlet. At mild surge, all three velocity components were separately analyzed using POD, while only the axial velocity component was used for the analysis at choke as the radial and tangential velocities were nearly negligible. The POD analysis at mild surge revealed the presence of travelling structures through certain mode pairs. Although the axial, radial, and tangential velocities have significantly different magnitudes and radial profiles, the distribution of singular values (a quantity associated with each POD mode reflecting its energy content) for the three velocity components are similar. As expected, the magnitudes of the singular values decrease progressively with mode number illustrating that the contribution of the lower order modes is much higher. At mild surge, the cumulative energy distribution showed that 98% of the total energy was resolved using the first 300 out of a total number of 430 modes (for each velocity component). Reduced order reconstruction of a sample velocity field revealed that the large scale vortical structures could be recovered by just using the first 50 modes, whereas using a larger number of modes (about 300) ensured that even the small-scale structures are properly captured. A statistical measure of ‘L2Norm’ of difference in modal values is employed to characterize the similarity (or difference) between any specific mode number of the three velocity components. The first 100 POD modes of the three velocity components which cumulatively capture roughly 90% of the energy are shown to exhibit shapes which are fairly distinct from each other, whereas the higher order modes (mode number above 100) of the different velocity components are quite similar. At choke, the singular value as well as the cumulative energy distribution were qualitatively similar to that at mild surge, with the first 300 modes capturing 96.6% of the total energy. At this operating point, the most dominant POD modes for axial velocity showed zones of strong correlation only near the periphery of the duct walls (within the boundary layer), while at mild surge, these regions extended over the entire PIV investigation domain.