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Mean flow of turbulent–laminar patterns in plane Couette flow

Published online by Cambridge University Press:  28 March 2007

DWIGHT BARKLEY  and
LAURETTE S. TUCKERMAN
DWIGHT BARKLEY
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, [email protected]
LAURETTE S. TUCKERMAN
Affiliation:
LIMSI-CNRS, BP 133, 91403 Orsay, [email protected]
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Abstract

A turbulent–laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the variation of the mean flow along the pattern wavevector is found to be almost exactly harmonic: the flow can be represented via only three cross-channel profiles as U(x, y, z) ≈ U0(y) + Uc(y) cos(kz) + Us(y) sin(kz). A model is formulated which relates the cross-channel profiles of the mean flow and of the Reynolds stress. Regimes computed for a full range of angle and Reynolds number in a tilted rectangular periodic computational domain are presented. Observations of regular turbulent–laminar patterns in other shear flows – Taylor–Couette, rotor–stator, and plane Poiseuille – are compared.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 576 , 10 April 2007 , pp. 109 - 137
Copyright
Copyright © Cambridge University Press 2007

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