Linear Dynamics of Turbulent Structures in the Log Layer

The long streamwise-velocity $u-$structures of the log layer are analyzed using the linearized Navier-Stokes equation for a logarithmic mean velocity profile and an appropriate eddy viscosity. A concentrated wall-normal $v$ velocity diffuses into a $v$-puff which leaves upstream a $u$ ``log layer streak" with an energy maximum in the wall region. The lifetime of $v$ is short and the $u$-streak grows even after $v$ decays, eventually becoming self-similar. These results compare well with the conditionally-averaged structures obtained from turbulent channels by del \'Alamo {\em et al} (2005), except that here there is no wake downstream of the puff. This suggests that the puffs are created at the tails of the streaks, leading to long $u$-structures containing several puffs. This is essentially the same vortex-streak cycle known to be responsible for buffer layer streaks, but acting in the log-layer with larger self-similarly growing structures. The short life of $v$ implies that this process does not always originate at the wall. Indeed, using rough-wall profiles, the wall component weakens but the log-layer one is not affected.