Covariant approach to relativistic large-eddy simulations: Lagrangian filtering

TL;DR

This paper introduces a covariant filtering scheme for relativistic fluid turbulence that is observer-dependent but spacetime-independent, enabling more accurate large-eddy simulations with effective sub-grid models.

Contribution

It presents the first covariant filtering method for relativistic turbulence, applicable in arbitrary spacetimes, and demonstrates how to extract sub-grid closure models from the filtered equations.

Findings

Filtering residuals can be modeled as a non-ideal fluid with effective viscosities.

Most non-ideal terms correlate with thermodynamic and flow invariants.

A power-law model fits the non-ideal coefficients for sub-grid closure.

Abstract

We present a proof-of-principle implementation of the first fully covariant filtering scheme applied to relativistic fluid turbulence. The filtering is performed with respect to special observers, identified dynamically as moving with the "bulk of the flow". This means that filtering does not depend on foliations of spacetime but rather on the intrinsic fibration traced out by the observers. The covariance of the approach means that the results may be lifted into an arbitrary, curved spacetime. This practical step follows theoretical work showing that the residuals introduced by filtering a fine-scale ideal fluid can be represented by a non-ideal fluid prescription at the coarse scale. We interpret such non-ideal terms using a simple first-order gradient model, which allows us to extract effective turbulent viscosities and conductivity. A statistical regression on these terms shows that the majority of their variation may be explained by the thermodynamic properties of the filtered fluid and invariants of its flow, such as the shear and vorticity. This serves as a validation of the method and enables us to fit a functional, power-law form for the non-ideal coefficients -- an approach that may be used practically to give a sub-grid closure model in large-eddy simulations.