Lagrangian approach for modal analysis of fluid flows
arXiv:2104.01308 · doi:10.1017/jfm.2021.834
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.
References in corpus (16)
- Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
- Particles and fields in fluid turbulence
- Mixing structures in the Mediterranean Sea from Finite-Size Lyapunov Exponents
- An Introduction to Adjoint Problems
- Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
- On coherent structure in wall turbulence
- Two-dimensional compressible viscous flow around a circular cylinder
- Finite-time Lagrangian transport analysis: Stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents
- On non-normality and classification of amplification mechanisms in stability and resolvent analysis
- Lagrangian Dynamic Mode Decomposition for Construction of Reduced-Order Models of Advection-Dominated Phenomena
- Dynamic mode decomposition based analysis of flow over a sinusoidally pitching airfoil
- Modal decomposition of fluid-structure interaction with application to flag flapping
- A Robust Approach for Stability Analysis of Complex Flows Using Navier-Stokes Solvers
- Finite-time Lyapunov exponents in the instantaneous limit and material transport
- Geometrical Constraints on Finite-time Lyapunov Exponents in Two and Three Dimensions
- Lagrangian Data-Driven Reduced Order Modeling of Finite Time Lyapunov Exponents