On the frequency of hydrodynamic perturbations. From the early transient through the intermediate term to the asymptotic state

We present recent findings concerning the frequency in the transient evolution of threedimensional perturbations in sheared flows. Sheared incompressible flows are usually considered non-dispersive media, as a consequence, the frequency evolution in transients has received much less attention than the wave energy density or growth factor. By carrying out a large number of long term transient simulations, we observe frequency jumps which appear quite far along within the time interval needed to reach the asymptotic state. We interpret these jumps as the end of the early transient and the beginning of the intermediate transient, where the process of the establishment of the final wave characteristics takes place. In the presence of a wall, where the no-slip condition applies, regardless of the symmetry of the initial condition, the frequency of non orthogonal waves jumps to values that can be 30 − 40% higher than those observed in the early transient. In the case of the wake, which is the free flow considered in this paper, the situation is similar, but the jumps are generally lower. These are accompanied by oscillations that begin in the early transient, may last throughout the intermediate term and disappear when the asymptotic state is reached. The early transient and the intermediate term durations are of the same order in the wake and in the channel solely in the case of long waves. Medium-short waves in the channel flow have an intermediate evolution one order of magnitude longer than the early transient. In both cases, the frequency and the phase speed fall to zero when φ, the obliquity angle between the perturbation and basic flow plane, approaches π/2. As a consequence, orthogonal waves are standing waves, a result that can be explained in terms of the symmetry of the system and agrees with laboratory findings concerning the unstable-turbulent spots observed in wall flows during the transition to turbulence.

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