Theoretical interpretation of coherent structures in near-wall turbulence

In the past 20 years significant progress has been made in understanding the dynamics of near-wall coherent structures, i.e. low-velocity streaks and streamwise vortices. Here we review outcomes obtained from recent studies on the theoretical descriptions of generations of coherent structures and of their regeneration cycle in near-wall turbulence, with emphasis being placed on the roles of coherent structures in characterizations of near-wall turbulent flows. The low-velocity streak is generated by cross-streamwise advection of mean-shear vorticity lines induced by streamwise vortices, while streamwise vortices arise from the instability of the streak. These mutual generations lead to a closed fully nonlinear regeneration cycle of streaks and vortices. This regeneration cycle can be described theoretically in terms of nonlinear equilibrium or temporally periodic solutions, with wall-parallel periodicities, to the incompressible Navier-Stokes equation. The confinement of solutions to the range of the spanwise period around 100 wall units provides us with theoretical interpretation of the observed preferential spanwise spacing of the low-velocity streaks. Not only the whole cyclic dynamics but also turbulence statistics in the buffer region are shown to be represented by the periodic solution. The role of coherent structures in the appearance of the mean secondary flow of the Prandtl second kind is also discussed in marginally turbulent square-duct flow. (invited review)