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Published 2010 | Published + Submitted
Journal Article Open

A streamwise constant model of turbulence in plane Couette flow


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.

Additional Information

© 2010 Cambridge University Press. Received 21 October 2009; revised 14 July 2010; accepted 14 July 2010; first published online 19 October 2010. The authors would like to thank H. Kawamura and T. Tsukahara for providing us with their DNS data. This research is sponsored in part through a grant from the Boeing Corporation. B.J.M. gratefully acknowledges support from NSF-CAREER award number 0747672 (programme managers W. W. Schultz and H. H. Winter).

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Published - Gayme2010p12460J_Fluid_Mech.pdf

Submitted - 1001.0782.pdf


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August 19, 2023
October 23, 2023