Temporal decay of secondary motions in turbulent channel flows

We perform direct numerical simulations of turbulent channel flows. Secondary motions are produced by applying a streamwise-homogeneous, spanwise-heterogeneous roughness pattern of spanwise period $\Lambda _s$ to the walls of the channel; their time evolution is observed. Notice that, owing to the geometry, the secondary motions are streamwise-invariant at any instant of time, so that no spatial development is seen. Once the secondary motions reach a statistically steady state, the roughness pattern is suddenly removed, so that the secondary motions decay. The time needed for the secondary motions to vanish is then measured; in doing so, we distinguish between the streamwise-momentum pathways and the cross-sectional circulatory motions that compose the secondary motions. Larger values of $\Lambda _s$ are generally associated with a longer time scale for the decay of the momentum pathways, although this might not hold true for $\Lambda _s/h\gt 4$ (where $h$ is the channel half-height). The value of such a time scale for the circulatory motions, instead, saturates for $\Lambda _s/h \geqslant 2$ ; this may be related to the observed spatial confinement of said circulatory motions. For specific values of $\Lambda _s$ ( $2 \leqslant \Lambda _s/h \leqslant 4$ ), the volume-averaged energy associated with the momentum pathways undergoes an unexpected transient growth with respect to its value at the beginning of the decay. This might indicate that structures of such a specific size are able to self-sustain as postulated by Townsend ( The Structure of Turbulent Shear Flow , 2 nd edition, 1976, ch. 7.19); the evidence we gather in this respect is however inconclusive. Finally, the present data suggest that most of the energy of the momentum pathways is produced by the circulatory motions transporting the mean (spanwise-averaged) velocity.