Model for transitional turbulence in a planar shear flow

TL;DR

This paper develops a simplified mathematical model for transitional turbulence in planar shear flows, capturing key phenomena like turbulent patterns and large-scale flows, thus advancing theoretical understanding of turbulence onset.

Contribution

The authors derive a minimal model from Navier-Stokes equations for planar shear flow turbulence, enabling analysis of pattern formation and turbulence transition mechanisms.

Findings

Model reproduces turbulent-laminar patterns and large-scale flows.

Patterns emerge via linear instability as Reynolds number decreases.

Provides a criterion for pattern orientation at turbulence onset.

Abstract

A central obstacle to understanding the route to turbulence in wall-bounded flows is that the flows are composed of complex, highly fluctuating, and strongly nonlinear states. In the case of pipe flow, models have deepened our understanding of turbulent onset by providing valuable theory to complement experiments and simulations. In planar cases, the large-scale flows associated with transitional turbulence are considerably more complex than for pipes, limiting our ability to develop models and provide theoretical analyses for these cases. We address this challenge here by deriving from the Navier-Stokes equations a simplified model for transitional turbulence in a planar setting. The Reynolds-averaged and turbulent-kinetic-energy equations are projected onto a minimal set of wall-normal modes and justified model closures are used for the Reynolds stresses and turbulent dissipation and transport. The model reproduces phenomena found at the onset of turbulence in planar shear flows, such as turbulent-laminar patterns (turbulent bands) oriented obliquely to the streamwise direction and large-scale flows associated with both stationary patterns and growing turbulent spots. We demonstrate the model's utility by showing that patterns arise with decreasing Reynolds number via a linear instability of uniform turbulence and by deriving a selection criterion for the pattern orientation at onset.