Ginzburg--Landau description of laminar-turbulent oblique band formation in transitional plane Couette flow

TL;DR

This paper models the formation and disappearance of oblique turbulent-laminar bands in transitional plane Couette flow using a Ginzburg--Landau framework, supported by numerical simulations and experimental comparisons.

Contribution

It introduces a Ginzburg--Landau model for laminar-turbulent band patterns in plane Couette flow, linking numerical results with experimental data.

Findings

The model accurately describes the pattern formation and transition thresholds.

Coefficients of the model match experimental values from cylindrical Couette flow.

Numerical simulations support the continuous transition scenario.

Abstract

Plane Couette flow, the flow between two parallel planes moving in opposite directions, is an example of wall-bounded flow experiencing a transition to turbulence with an ordered coexistence of turbulent and laminar domains in some range of Reynolds numbers [R_g,R_t]. When the aspect-ratio is sufficiently large, this coexistence occurs in the form of alternately turbulent and laminar oblique bands. As R goes up trough the upper threshold R_t, the bands disappear progressively to leave room to a uniform regime of featureless turbulence. This continuous transition is studied here by means of under-resolved numerical simulations understood as a modelling approach adapted to the long time, large aspect-ratio limit. The state of the system is quantitatively characterised using standard observables (turbulent fraction and turbulence intensity inside the bands). A pair of complex order parameters is defined for the pattern which is further analysed within a standard Ginzburg--Landau formalism. Coefficients of the model turn out to be comparable to those experimentally determined for cylindrical Couette flow.