Extraction of Lagrangian Coherent Structures in the framework of the Lagrangian–Eulerian Stabilized Collocation Method (LESCM)

Abstract

In recent years, the investigations on Lagrangian Coherent Structures (LCS) in complex flows have attracted increasing attention due to their ability to accurately describe flow field details. In this study, we propose a novel and accurate numerical technique based on the Lagrangian–Eulerian Stabilized Collocation Method (LESCM) for computing the Finite Time Lyapunov Exponents (FTLEs), which is essential for extracting LCSs in viscous incompressible flows. LESCM, previously employed for simulating fluid–structure interaction problems, was developed based on the Material Point Method (MPM). The hybrid Lagrangian–Eulerian​ description in LESCM enables explicit tracking of fluid flow trajectories throughout the simulation and direct evaluation of the FTLE field. Furthermore. The errors in FTLEs caused by the Particle Shifting Technique (PST) are completely avoided due to the Eulerian characteristic of LESCM as the deformation gradient is calculated based on fixed Eulerian nodes rather than fluid particles with unphysical shifting. Consequently, the novel technique based on LESCM surpasses the accuracy of pure Lagrangian particle methods and provides an accurate way of detecting complex LCSs in flow fields. Additionally, By harnessing the remarkable efficiency of LESCM, MATLAB can now handle up to 16 million particles with ease, eliminating the need for parallel computation techniques. Several numerical examples, such as 2D Taylor-Green​ vortices, flow passing a circular cylinder and a 3-dimensional shear driven cavity problem, have been tested to demonstrate the effectiveness of this novel approach under various conditions.