How close are shell models to the 3D Navier-Stokes equations?

TL;DR

This paper rigorously compares shell models to the 3D Navier-Stokes equations, revealing similarities in energy dissipation but weaker Reynolds number dependence in shell models, supported by numerical simulations.

Contribution

It provides a rigorous analysis of the relationship between shell models and the 3D Navier-Stokes equations, highlighting their similarities and differences.

Findings

Mean energy dissipation rates are comparable in both systems.

Velocity derivatives show weaker Reynolds dependence in shell models.

Numerical simulations confirm theoretical estimates.

Abstract

Shell models have found wide application in the study of hydrodynamic turbulence because they are easily solved numerically even at very large Reynolds numbers. Although bereft of spatial variation, they accurately reproduce the main statistical properties of fully-developed homogeneous and isotropic turbulence. Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3D) Navier-Stokes equations (NSEs). The goal of this study is to make a rigorous comparison between shell models and the NSEs. It turns out that only the estimate of the mean energy dissipation rate is the same in both systems. The estimates of the velocity and its higher-order derivatives display a weaker Reynolds number dependence for shell models than for the 3D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those corresponding to a velocity gradient averaged version of the 3D Navier-Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over a wide range of Reynolds numbers confirm the estimates for shell models.