Laminar-turbulent patterning in wall-bounded shear flows: a Galerkin model

TL;DR

This paper develops a Galerkin model for plane Couette flow to understand laminar-turbulent pattern formation, successfully reproducing observed patterns and offering insights into the flow's spatiotemporal dynamics.

Contribution

It introduces a Galerkin modeling approach with functional basis expansions that accurately capture wall-normal dependence, improving pattern prediction over simpler models.

Findings

Successfully reproduces laminar-turbulent pattern formation

Demonstrates the importance of wall-normal resolution in modeling

Provides a framework for analyzing flow pattern dynamics

Abstract

On its way to turbulence, plane Couette flow - the flow between counter-translating parallel plates - displays a puzzling steady oblique laminar-turbulent pattern. We approach this problem via Galerkin modelling of the Navier-Stokes equations. The wall-normal dependence of the hydrodynamic field is treated by means of expansions on functional bases fitting the boundary conditions exactly. This yields a set of partial differential equations for the spatiotemporal dynamics in the plane of the flow. Truncating this set beyond lowest nontrivial order is numerically shown to produce the expected pattern, therefore improving over what was obtained at cruder effective wall-normal resolution. Perspectives opened by the approach are discussed.