A quadratic Reynolds stress development for the turbulent Kolmogorov flow

TL;DR

This study investigates the turbulent Kolmogorov flow using direct numerical simulations, proposing a nonlinear quadratic Reynolds stress model that extends Boussinesq's linear approximation, and examines effects of different forcing shapes.

Contribution

A new nonlinear quadratic Reynolds stress development model is proposed and validated for turbulent Kolmogorov flow, extending traditional linear models.

Findings

Boussinesq's linear relation holds in half the flow volume.

The quadratic model parameters vary with Reynolds number.

Differences between sinusoidal and Gaussian forcing are highlighted.

Abstract

We study the three-dimensional turbulent Kolmogorov flow, i.e. the Navier-Stokes equations forced by a low-single-wave-number sinusoidal force in a periodic domain, by means of direct numerical simulations. This classical model system is a realization of anisotropic and non-homogeneous hydrodynamic turbulence. Boussinesq's eddy viscosity linear relation is checked and found to be approximately valid over half of the system volume. A more general nonlinear quadratic Reynolds stress development is proposed and its parameters estimated at varying the Taylor scale-based Reynolds number in the flow up to the value 200. The case of a forcing with a different shape, here chosen Gaussian, is considered and the differences with the sinusoidal forcing are emphasized.