Linear energy amplification in turbulent channels

Abstract

We study the temporal stability of the Orr–Sommerfeld and Squire equations in channels with turbulent mean velocity profiles and turbulent eddy viscosities. Friction Reynolds numbers up to $Re_\tau \,{=}\, 2\,{\times}\, 10^4$ are considered. All the eigensolutions of the problem are damped, but initial perturbations with wavelengths $\lambda_x \,{>}\, \lambda_z$ can grow temporarily before decaying. The most amplified solutions reproduce the organization of turbulent structures in actual channels, including their self-similar spreading in the logarithmic region. The typical widths of the near-wall streaks and of the large-scale structures of the outer layer, $\lambda_z^+ \,{=}\, 100$ and $\lambda_z/h \,{=}\, 3$, are predicted well. The dynamics of the most amplified solutions is roughly the same regardless of the wavelength of the perturbations and of the Reynolds number. They start with a wall-normal $v$ event which does not grow but which forces streamwise velocity fluctuations by stirring the mean shear ($uv\,{<}\,0$). The resulting $u$ fluctuations grow significantly and last longer than the $v$ ones, and contain nearly all the kinetic energy at the instant of maximum amplification.