Efficient Lagrangian averaging with exponential filters

TL;DR

This paper introduces a PDE-based method using exponential filters to efficiently compute Lagrangian averages in fluid dynamics, enabling real-time analysis of multiscale turbulent flows.

Contribution

It develops a novel exponential filter approach that allows for single-pass, on-the-fly Lagrangian averaging via PDEs, improving computational efficiency over traditional methods.

Findings

Effective filtering of Poincaré waves in shallow-water simulations

Retains key features of slow turbulent flows

Enables real-time Lagrangian averaging during simulations

Abstract

Lagrangian averaging is a valuable tool for the analysis and modelling of multiscale processes in fluid dynamics. The numerical computation of Lagrangian (time) averages from simulation data is challenging, however. It can be carried out by tracking a large number of particles or, following a recent approach, by solving a dedicated set of partial differential equations (PDEs). Both approaches are computationally demanding because they require an entirely new computation for each time at which the Lagrangian mean fields are desired. We overcome this drawback by developing a PDE-based method that delivers Lagrangian mean fields for all times through the single solution of evolutionary PDEs. This allows for an on-the-fly implementation, in which Lagrangian averages are computed along with the dynamical variables. This is made possible by the use of a special class of temporal filters whose kernels are sums of exponential functions. We focus on two specific kernels involving one and two exponential functions. We implement these in the rotating shallow-water model and demonstrate their effectiveness at filtering out large-amplitude Poincar\'e waves while retaining the salient features of an underlying slowly evolving turbulent flow.