A Streamwise Constant Model of Turbulence in Plane Couette Flow

TL;DR

This paper introduces a simplified 2D/3C model of turbulent plane Couette flow that captures key features of turbulence at low Reynolds numbers, highlighting the role of nonlinear coupling and input-output amplification.

Contribution

It presents a novel streamwise constant projection of Navier-Stokes equations that models turbulence, demonstrating its effectiveness through simulations and system theoretic analysis.

Findings

The 2D/3C model reproduces salient turbulence features.

Nonlinear coupling shapes the turbulent velocity profile.

Large input-output amplification enables turbulence generation.

Abstract

Streamwise and quasi-streamwise elongated structures have been shown to play a significant role in turbulent shear flows. We model the mean behavior of fully turbulent plane Couette flow using a streamwise constant projection of the Navier Stokes equations. This results in a two-dimensional, three velocity component ($2D/3C$) model. We first use a steady state version of the model to demonstrate that its nonlinear coupling provides the mathematical mechanism that shapes the turbulent velocity profile. Simulations of the $2D/3C$ model under small amplitude Gaussian forcing of the cross-stream components are compared to DNS data. The results indicate that a streamwise constant projection of the Navier Stokes equations captures salient features of fully turbulent plane Couette flow at low Reynolds numbers. A system theoretic approach is used to demonstrate the presence of large input-output amplification through the forced $2D/3C$ model. It is this amplification coupled with the appropriate nonlinearity that enables the $2D/3C$ model to generate turbulent behaviour under the small amplitude forcing employed in this study.