The `Multifractal Model' of Turbulence and {\em A Priori} Estimates in Large-Eddy Simulation, I. Subgrid Flux and Locality of Energy Transfer

TL;DR

This paper rigorously analyzes subgrid stress and flux in Large-Eddy Simulation of turbulence, demonstrating that local energy transfer dominates under realistic conditions, except for certain filters like the sharp cutoff.

Contribution

It provides the first rigorous a priori estimates linking LES subgrid flux to the multifractal turbulence model, emphasizing the dominance of local energy cascade in realistic filtering schemes.

Findings

Local triads dominate energy transfer in LES with most filters.

Sharp cutoff filter leads to nonlocal energy transfer at high Reynolds numbers.

Energy flux estimates are derived as consequences of Navier-Stokes exact formulas.

Abstract

We establish and discuss {\em a priori} estimates on subgrid stress and subgrid flux for filtering schemes used in the turbulence modelling method of Large-Eddy Simulation (LES). Our estimates are derived as rigorous consequences of the exact subgrid stress formulae from Navier-Stokes equations under realistic conditions for inertial-range velocity fields, those conjectured in the Parisi-Frisch ``multifractal model.'' The estimates are shown to be an expression of ``local energy cascade,'' i.e. the dominance of local wavevector triads in the energy transfer. We prove that for nearly any reasonable filter function the LES method defines an energy flux in which local triads dominate in individual realizations, due to cancellation of distant triadic contributions by detailed conservation. A somewhat similar observation of Leslie and Quarini on graded filters in the EDQNM closure is shown to be unrelated to the cancellation we establish in Navier-Stokes solutions. The sharp Fourier cutoff filter is one example which does not satisfy the modest conditions of our proof and, in fact, we show that with that filter the energy transfer in individual realizations at arbitrarily high Reynolds number will be dominated by nonlocal, convective sweeping.