Lagrangian approach for modal analysis of fluid flows

TL;DR

This paper introduces a Lagrangian modal analysis (LMA) framework for fluid flows, transforming Eulerian flow data into Lagrangian coordinates to better analyze flows with deforming domains, and demonstrates its effectiveness through DNS of canonical flows.

Contribution

The paper develops a novel Lagrangian modal analysis approach that extends traditional Eulerian methods to deforming and complex flows, providing new insights into flow stability and structures.

Findings

LMA effectively captures flow stability and post-bifurcation dynamics.

LMA reveals Lagrangian coherent structures and links with FTLE.

Application to DNS validates the approach for various flow configurations.

Abstract

Common modal decomposition techniques for flowfield analysis, data-driven modeling and flow control, such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are usually performed in an Eulerian (fixed) frame of reference with snapshots from measurements or evolution equations. The Eulerian description poses some difficulties, however, when the domain or the mesh deforms with time as, for example, in fluid-structure interactions. For such cases, we first formulate a Lagrangian modal analysis (LMA) ansatz by a posteriori transforming the Eulerian flow fields into Lagrangian flow maps through an orientation and measure-preserving domain diffeomorphism. The development is then verified for Lagrangian variants of POD and DMD using direct numerical simulations (DNS) of two canonical flow configurations at Mach 0.5, the lid-driven cavity and flow past a cylinder, representing internal and external flows, respectively, at pre- and post-bifurcation Reynolds numbers. The LMA is demonstrated for several situations encompassing unsteady flow without and with boundary and mesh deformation as well as non-uniform base flows that are steady in Eulerian but not in Lagrangian frames. We show that LMA application to steady nonuniform base flow yields insights into flow stability and post-bifurcation dynamics. LMA naturally leads to Lagrangian coherent flow structures and connections with finite-time Lyapunov exponents (FTLE). We examine the mathematical link between FTLE and LMA by considering a double-gyre flow pattern. Dynamically important flow features in the Lagrangian sense are recovered by performing LMA with forward and backward (adjoint) time procedures.