Linear energy amplification in turbulent channels.

*(English)*Zbl 1095.76021Summary: We study the temporal stability of Orr-Sommerfeld and Squire equations in channels with turbulent mean velocity profiles and turbulent eddy viscosities. Friction Reynolds numbers up to \(\text{Re}_{\tau} = 2 \times 10^4\) 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.