Amplification and nonlinear mechanisms in plane Couette flow
We study the input-output response of a streamwise constant projection of the Navier-Stokes equations for plane Couette flow, the so-called 2D/3C model. Study of a streamwise constant model is motivated by numerical and experimental observations that suggest the prevalence and importance of streamwise and quasi-streamwise elongated structures. Periodic spanwise/wall-normal (z–y) plane stream functions are used as input to develop a forced 2D/3C streamwise velocity field that is qualitatively similar to a fully turbulent spatial field of direct numerical simulation data. The input-output response associated with the 2D/3C nonlinear coupling is used to estimate the energy optimal spanwise wavelength over a range of Reynolds numbers. The results of the input-output analysis agree with previous studies of the linearized Navier-Stokes equations. The optimal energy corresponds to minimal nonlinear coupling. On the other hand, the nature of the forced 2D/3C streamwise velocity field provides evidence that the nonlinear coupling in the 2D/3C model is responsible for creating the well known characteristic "S" shaped turbulent velocity profile. This indicates that there is an important tradeoff between energy amplification, which is primarily linear, and the seemingly nonlinear momentum transfer mechanism that produces a turbulent-like mean profile.
© 2011 American Institute of Physics. Received 18 November 2010; accepted 17 May 2011; published online 17 June 2011. The authors would like to thank H. Kawamura and T. Tsukahara for providing them with their DNS data. This research is partially supported by AFOSR (FA9550-08-1-0043). B.J.M. gratefully acknowledges support from NSFCAREER Award No. 0747672 (program managers W. W. Schultz and H. H. Winter).
Submitted - 1011.5675.pdf
Published - Gayme2011p15144Phys_Fluids.pdf