Statistical Methods and Modal Decompositions for Gridded and Scattered Data: Meshless Statistics and Meshless Data Driven Modal Analysis

TL;DR

This paper introduces meshless statistical and modal analysis methods for processing scattered and gridded flow data, enabling detailed turbulence analysis through advanced, data-driven decompositions like POD and DMD.

Contribution

It presents a novel meshless framework using RBF regression for turbulence statistics and modal decompositions from scattered data, enhancing traditional grid-based methods.

Findings

Meshless RBF regression effectively computes turbulent statistics from scattered data.

The framework enables data-driven modal decompositions like POD and DMD without a mesh.

Codes for the methods are publicly available for implementation and further research.

Abstract

Statistical tools are crucial for studying and modeling turbulent flows, where chaotic velocity fluctuations span a wide range of spatial and temporal scales. Advances in image velocimetry, especially in tracking-based methods, now allow for high-speed, high-density particle image processing, enabling the collection of detailed 3D flow fields. This lecture provides a set of tutorials on processing such datasets to extract essential quantities like averages, second-order moments (turbulent stresses) and coherent patterns using modal decompositions such as the Proper Orthogonal Decomposition (POD). After a brief review of the fundamentals of statistical and modal analysis, the lecture delves into the challenges of processing scattered data from tracking velocimetry, comparing it to traditional gridded-data approaches. It also covers research topics, including physics-based Radial Basis Function (RBF) regression for meshless computation of turbulent statistics and the definition of an RBF inner product, which enables meshless computation of data-driven decompositions. These include traditional methods like Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD), as well as advanced variants such as Spectral POD (SPOD) and Multiscale POD (mPOD). We refer to this approach as the "Meshless Data-Driven Decomposition" framework. All codes are available at https://github.com/mendezVKI/PIV_LS_2024_Signal