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The technique of identifying Lagrangian Coherent Structures (LCSs) has emerged as a powerful tool for studying incompressible flows. Yet, the discrepancies arising from incompressibility assumptions often compromise the accuracy of LCSs constructed using Lagrangian particle methods. In this study, we introduce a modified framework to compute Finite-Time Lyapunov Exponents (FTLEs), addressing the misalignment between the fully incompressible assumption of LCS theory and the inherent incompressibility loss in simulations of particle methods. We begin by examining the correlation between the minimum and maximum FTLEs. By incorporating the deformation gradient and Cauchy-Green strain tensor which account for the time-advancing errors of incompressibility based on continuum theory, we enhance the computational accuracy of FTLEs. Moreover, we introduce the modified FTLE algorithm to the incompressible particle-in-cell (PIC) method for resolving free surface flows and fluid-structure interaction problems. Finally, numerical examples including the Tayler-Green vortices, water sloshing with baffles, an eccentric box sinking in water, and three-dimensional shear-driven cavity problems with high Reynolds numbers are tested to validate the effectiveness of the modified FTLE algorithm and the improved LCSs. These results demonstrate that the proposed modification scheme adeptly counteracts the errors caused by incompressibility loss, enabling accurate computation of FTLEs and detection of LCSs. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]