  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.
