Forced Solutions of Streamwise Constant Plane Couette Flow

A two-dimensional, three-velocity component ($2D/3C$) model simulated under small-amplitude Gaussian forcing has been shown to capture salient features of turbulent plane Couette flow (Gayme et. al 2010). Periodic spanwise/wall-normal plane stream functions are used as input to develop forced $2D/3C$ streamwise velocities. The resulting steady-state solutions are qualitatively similar to a fully turbulent spatial field of DNS. Our analysis indicates that the momentum transfer which produces a `turbulent-like' mean profile requires a nonlinear streamwise velocity equation. A system theoretic approach is used to study amplification mechanisms which arise through this $2D/3C$ nonlinear coupling. The forcing required to produce each input is used to define an induced norm. The associated input-output response determines the energy optimal spanwise wavelength over a range of Reynolds numbers. We identify an important tradeoff between the linear amplification mechanism and the nonlinearity required to develop an appropriately shaped turbulent velocity profile. \textbf{Acknowledgements:} The research is supported by Boeing and AFOSR. B.J.M. gratefully acknowledges NSF-CAREER award no. 0747672 (program managers W. W. Schultz \& H. H. Winter).