Perron-Frobenius analysis of burst regeneration in wall turbulence*

Intermittent bursting in wall-bounded turbulence can be modelled by the transient amplification of the wall-normal velocity as it is tilted by the shear, and can be represented as a clockwise path along the top of a roughly triangular pdf of flow states in an inclination-amplitude space. How bursts are regenerated is less well understood. The Perron-Frobenius operator describes probability transfer in time, and allows the identification of precursors and effects of particular states. When applied to an Re_τ ≈10³ small-box channel, it shows that the precursors of regeneration are at the lower left corner of the triangle. It also provides an estimate of the regeneration time and identifies the state-space trajectories involved. Conditional averaging along these trajectories shows that the most likely regenerative state contains a low-shear region near the wall and a high-shear layer above it. The new burst is generated from the latter. This is confirmed by numerical experiments in which the effect of the bursts on the mean shear is modified.