Three regularization models of the Navier-Stokes equations

TL;DR

This paper compares three regularization models of the Navier-Stokes equations, analyzing their effectiveness in turbulence simulation and their ability to replicate DNS results, with Clark-alpha emerging as the most accurate SGS model.

Contribution

The study provides a detailed comparison of Clark-alpha, LANS-alpha, and Leray-alpha models, deriving their properties and assessing their suitability as SGS models for turbulence.

Findings

Clark-alpha accurately reproduces DNS energy spectrum and intermittency.

Leray-alpha is inadequate as a SGS model at studied parameters.

LANS-alpha reduces intermittency but struggles to match DNS large-scale spectra.

Abstract

We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier-Stokes. The consequences on the applicability of the regularizations as SGS models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-alpha model are compared to two previously employed regularizations, LANS-alpha and Leray-alpha (at Re ~ 3300, Taylor Re ~ 790) and to a DNS. We derive the Karman-Howarth equation for both the Clark-alpha and Leray-alpha models. We confirm one of two possible scalings resulting from this equation for Clark as well as its associated k^(-1) energy spectrum. At sub-filter scales, Clark-alpha possesses similar total dissipation and characteristic time to reach a statistical turbulent steady-state as Navier-Stokes, but exhibits greater intermittency. As a SGS model, Clark reproduces the energy spectrum and intermittency properties of the DNS. For the Leray model, increasing the filter width decreases the nonlinearity and the effective Re is substantially decreased. Even for the smallest value of alpha studied, Leray-alpha was inadequate as a SGS model. The LANS energy spectrum k^1, consistent with its so-called "rigid bodies," precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in resolution. However, that this same feature reduces its intermittency compared to Clark-alpha (which shares a similar Karman-Howarth equation). Clark is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than alpha, whereas high-order intermittency properties for larger values of alpha are best reproduced by LANS-alpha.