Improved Lagrangian coherent structures with modified finite-time Lyapunov exponents in the PIC framework

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.