A note on optimal transient growth in turbulent channel flows

We compute the optimal transient growth of perturbations sustained by a turbulent channel flow following the same approach recently used by del Alamo and Jimenez [J. Fluid Mech. 559, 205 (2006)]. Contrary to this previous analysis, we use generalized Orr-Sommerfeld and Squire operators consistent with previous investigations of mean flows with variable viscosity. The optimal perturbations are streamwise vortices evolving into streamwise streaks. In accordance with del Alamo and Jimenez, it is found that for very elongated structures and for sufficiently large Reynolds numbers, the optimal energy growth presents a primary peak in the spanwise wavelength, scaling in outer units, and a secondary peak scaling in inner units and corresponding to lambda(+)(z)approximate to 100. Contrary to the previous results, however, it is found that the maximum energy growth associated with the primary peak increases with the Reynolds number. This growth, in a first approximation, scales linearly with an effective Reynolds number based on the centerline velocity, the channel half width and the maximum eddy viscosity associated. The optimal streaks associated with the primary peak have an optimal spacing of lambda(z)=4h and scale in outer units in the outer region and in wall units in the near wall region, where they still have up to 50% of their maximum amplitude near y(+)=10. (C) 2009 American Institute of Physics. [DOI: 10.1063/1.3068760]