Computing Lagrangian means

TL;DR

This paper introduces a PDE-based method for computing Lagrangian means that reduces memory use and improves wave-vortex separation in fluid simulations, advancing multiscale flow analysis.

Contribution

A novel PDE approach for calculating Lagrangian means directly, avoiding particle tracking and enhancing efficiency in fluid data analysis.

Findings

PDE-based Lagrangian mean computation reduces memory demands.

Method improves wave-vortex separation in turbulent flows.

Demonstrated effectiveness with rotating shallow-water model simulations.

Abstract

Lagrangian averaging plays an important role in the analysis of wave--mean-flow interactions and other multiscale fluid phenomena. The numerical computation of Lagrangian means, e.g. from simulation data, is however challenging. Typical implementations require tracking a large number of particles to construct Lagrangian time series which are then averaged using a low-pass filter. This has drawbacks that include large memory demands, particle clustering and complications of parallelisation. We develop a novel approach in which the Lagrangian means of various fields (including particle positions) are computed by solving partial differential equations (PDEs) that are integrated over successive averaging time intervals. We propose two strategies, distinguished by their spatial independent variables. The first, which generalises the algorithm of Kafiabad (2022, J. Fluid Mech. 940, A2), uses end-of-interval particle positions; the second directly uses the Lagrangian mean positions. The PDEs can be discretised in a variety of ways, e.g. using the same discretisation as that employed for the governing dynamical equations, and solved on-the-fly to minimise the memory footprint. We illustrate the new approach with a pseudospectral implementation for the rotating shallow-water model. Two applications to flows that combine vortical turbulence and Poincare waves demonstrate the superiority of Lagrangian averaging over Eulerian averaging for wave--vortex separation.