Lagrangian approach for modal analysis of fluid flows

Abstract

Common modal decomposition techniques for flow-field analysis, data-driven modelling and flow control, such as proper orthogonal decomposition and dynamic mode decomposition, 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 proper orthogonal decomposition and dynamic mode decomposition using direct numerical simulations of two canonical flow configurations at Mach 0.5, i.e. 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 application of LMA to steady non-uniform 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. We examine the mathematical link between finite-time Lyapunov exponents 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.

Identification and reconstruction of high-frequency fluctuations evolving on a low-frequency periodic limit cycle: application to turbulent cylinder flow

“…This example is intimately connected to a broader category of problems, which are fluid–structure interactions, where mean-flow analyses do not make much sense (in those cases, a change in the reference frame may be needed; see e.g. Shinde & Gaitonde (2021)). Also, flows around bluff bodies typically present a strong VS motion where the small-scale turbulent structures evolving on top of them may exhibit different amplitudes, frequencies and spatial locations depending on the phase and thus the position of the large-scale vortices.…”

A Measurement-Aided Control System for Stabilization of the Real-Life Stewart Platform

“…Observe that following a notion of Refs. [ 22 , 23 ], the commonly known six degrees of freedom-originated Stewart platform system derived from the Lagrangian paradigm can be described in the comprehensive manner. Thus, from the Lagrangian method, we obtain for the chosen linearized operating point where and stand for the kinetic and potential energies, respectively, whereas the symbols m , v , I , , g and h denote the corresponding mass, velocity, inertia moment, angular velocity, gravity and altitude.…”

Investigation on stall characteristics of marine centrifugal pump considering transition effect

“…Stall phenomena usually contain multiple frequencies, and it is a challenge to identify the frequencies when the stall occurs (Murata et al, 2020;Shinde and Gaitonde, 2021). The proper orthogonal decomposition (POD) method was first proposed by Lumley (2007) and Sirovich (1987) and used in engineering.…”