On the emergence of the Navier-Stokes-$\alpha$ model for turbulent channel flows

TL;DR

This paper demonstrates how wall roughness in turbulent channel flows naturally leads to the emergence of the Navier-Stokes-$oldsymbol{ extalpha}$ model through a specific Reynolds averaging process, linking turbulence modeling to physical wall effects.

Contribution

It introduces a simple Reynolds averaging method that, due to wall roughness, transforms the Navier-Stokes equations into the Navier-Stokes-$oldsymbol{ extalpha}$ model, providing a physical basis for this turbulence model.

Findings

Wall roughness induces the Navier-Stokes-$\alpha$ model from Navier-Stokes equations.

Reynolds averaging linked to wall effects explains turbulence modeling.

Supports the Navier-Stokes-$\alpha$ model as a Reynolds-averaged turbulence model.

Abstract

In a series of papers (see \cite{CDT02} and the pertinent references therein) the 3D Navier-Stokes-$\alpha$ model were shown to be a useful complement to the 3D Navier-Stokes equations; and in particular, to be a good Reynolds version of the latter equations. In this work, we introduce a simple Reynolds averaging which, due to the wall roughness, transforms the Navier-Stokes equations into the Navier-Stokes-$\alpha$ model.