Capturing multiscale interactions in fluid flow via Lagrangian coherent structures and modal analysis

TL;DR

This work develops and applies a mode-sensitivity framework that ties Eulerian modal decompositions to Lagrangian coherent structures (LCSs) via the model-sensitivity approach of Kaszás and Haller. By decomposing flows into a baseline $(\tilde{\mathbf{u}})$ and perturbations $(\mathbf{u}')$, the authors derive a mode-sensitivity upper bound $\mathscr{MS}$ and a manifold response $\zeta$ that quantify how specific modal components influence FTLE-based LCSs. Through cylinder wake, oscillating-foil wake, and turbulent channel flow examples, they show that $\mathscr{MS}$ captures regions where modal content strongly affects particle transport, while $\zeta$ highlights localized perturbation-induced adjustments to LCS structure. The framework is flexible across modal representations (POD, DMD, SPOD, etc.) and offers a pathway to interpret and predict multi-scale transport phenomena in complex flows, with potential extensions to three-dimensional and geophysical contexts.

Abstract

We consider the relationship between Eulerian modal decompositions and Lagrangian coherent structures (LCSs). The model sensitivity framework developed by Kaszás and Haller (2020) is used to express data-driven modal representations of fluid flow in a Lagrangian space. The method, based on the computation of the finite-time Lyapunov exponent, computes the amplitude perturbations experienced by fluid particles due to specific modal components of the flow. Demonstrations of the method are presented for both periodic and turbulent flows, including experimental data from the wake past an oscillating foil, numerical data of the classical cylinder wake flow, and a direct numerical simulation (DNS) of a turbulent channel flow. This method provides a way to understand how Eulerian mode structures interact dynamically with features of the Lagrangian coherent structure across scales, offering additional physical insight into modal decompositions.

Figures (17)

• Figure 1: Illustration of perturbed (black) and unperturbed (gray) trajectories for a single fluid particle.
• Figure 2: Forward FTLE field showing LCS structure of the unperturbed kinematic system with $\epsilon=0$.
• Figure 3: LCS structure of the perturbed kinematic system $\mathbf{f}(\mathbf{x},t)+\epsilon \mathbf{g}(\mathbf{x},t)$ for various initial times $t_0$ with $\epsilon=0.03$.
• Figure 4: Scaled upper-bound uncertainty fields for the perturbed kinematic system for various initial times $t_0$ with $\epsilon=0.03$. The green points are particle positions referred to in figure \ref{['fig:currentbounds']}
• Figure 5: Trajectory uncertainty (black) described by $|\mathbf{x}^\epsilon - \mathbf{x}^0|^2$ and upper-bound estimate (blue) of trajectory uncertainty represented by $\Delta^2_\infty \mathrm{MS}$. The initial conditions ($x,y$) chosen for each of the curves are (a) ($0.48, 0.63$), (b) ($1.13, 0.92$) and (c) ($1.50, 0.75$). These positions are marked with green points in figure \ref{['fig:currentcse']}.