Lagrangian approach for modal analysis of fluid flows

TL;DR

This work introduces Lagrangian modal analysis (LMA) to apply POD and DMD in moving/deforming domains by transforming Eulerian flow fields into Lagrangian flow maps via a diffeomorphic transformation. It develops LPOD and LDMD, and links their dominant modes to finite-time Lyapunov exponents (FTLE), enabling interpretation of flow stability and coherent structures in a Lagrangian frame. The authors validate LMA on two canonical 2D compressible flows (lid-driven cavity and flow past a cylinder), including a deforming mesh scenario and a geophysically inspired double-gyre model, and they establish forward and adjoint LMA procedures to capture both upstream and downstream coherent features. The results show LMA yields Lagrangian coherent structures that parallel FTLE fields, separates deformation-induced modes from primary flow modes, and reveals post-bifurcation dynamics inaccessible to traditional Eulerian POD/DMD analyses. This framework thus provides a robust, frame-invariant approach to modal analysis in fluid-structure interaction and other deforming-domain problems, with practical implications for flow control and transport studies.

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.

Figures (16)

• Figure 1: Schematic representation of a fluid element in Lagrangian frame of reference, where the deforming flow trajectories lead to Lyapunov exponents and a data matrix for LMA over a finite time.
• Figure 2: Flow recirculation patterns inside the lid-driven cavity at Mach number $M_\infty=0.5$ and pre- and post-critical Reynolds numbers. (a) Steady flow velocity $|\pmb{u}|$ at $Re_L=7{,}000$ and (b) time-averaged flow velocity $|\overline{\pmb{u}}|$ at $Re_L=15{,}000$. Streamlines display the flow recirculation patterns.
• Figure 3: Lid-driven cavity flow at $M_\infty=0.5$ and $Re_L=15{,}000$ with forced bottom surface/mesh deformation. (a) Instantaneous flow velocity magnitude with contours on a deformed domain. The inset shows a closer view of the deformed domain ($\pmb{\chi}$). (b) Skin-friction coefficient on the bottom surface with/without deformation.
• Figure 4: Compressible flow past a cylinder, in terms of the streamwise velocity $u_1$, at Mach number $M_\infty=0.5$ and Reynolds numbers $Re_D=40$ and $Re_D=100$. (a) Steady flow at $Re_D=40$. (b) The pressure coefficient $C_p$ is compared with the DNS profile of canuto2015two (dashed line.) (c) Unsteady instantaneous flow at $Re_D=100$. (d) Time-mean flow at $Re_D=100$.
• Figure 5: Leading POD and DMD modes of the streamwise velocity for the lid-driven cavity at $Re_L=15000$ in Eulerian frame of reference. (a) POD modes $\Phi_2^{u_1}$, $\Phi_3^{u_1}$, $\Phi_4^{u_1}$, and $\Phi_5^{u_1}$. (b) DMD modes $\phi_1^{u_1}$, $\phi_2^{u_1}$, and $\phi_3^{u_1}$. (c) POD modal energies and (d) DMD eigenvalues corresponding to the spatial modes.