Numeric Metrics for Capturing Variations in Flow Fields: An Improvement Towards a Robust Comparison of Vector Fields

Quantifying the similarity of velocity vector fields is a critical task across numerous applications within fluid mechanics research, such as computational fluid dynamics validation and quantifying the levels of variability in a flow field. However, this task remains challenging for widely used vector comparison metrics at present. Traditional metrics include the Relevance Index (RI) and Magnitude Similarity Index (MSI) as well as their local versions, Local Structural Index (LSI) and Local Magnitude Index (LMI). These metrics, however, are often sensitive to low-velocity magnitude areas, which can distort the results. To address this, improved metrics like the Weighted Relevance Index (WRI), the Weighted Magnitude Index (WMI), and their amalgamated Combined Magnitude And Relevance Index (CMRI), have been introduced in the literature. Despite having reduced sensitivity to low-velocity areas, CMRI in its original form does not equally consider the significance of WRI and WMI, and introduces a degree of subjectivity. In the present work, we propose two enhanced metrics to address this problem: the modified CMRI for one-by-one flow field comparison, and the ensemble CMRI for comparing collections of vector fields. We compare their properties to the previously developed CMRI and spatially averaged CMRI, and investigate their usage in an applied example for quantifying cyclic variations in a flow from a combustion engine cylinder. The newly proposed metrics were found to more robustly isolate the effects of discrepant vector magnitudes and directions, leading to improved diagnostics of in-cylinder flow fields. In particular, the modified CMRI, which ensures equal treatment of WMI and WRI, can serve as a baseline for flow field comparison, providing a more objective target for quantifying flow similarity.