a1 Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA
a2 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
a3 Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
Streamwise and quasi-streamwise elongated structures have been shown to play a significant role in turbulent shear flows. We model the mean behaviour 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 direct numerical simulation (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 systems-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.
(Received October 21 2009)
(Revised July 14 2010)
(Accepted July 14 2010)
(Online publication October 19 2010)