Vortex–wave interaction arrays: a sustaining mechanism for the log layer?

Abstract

Vortex–wave interaction theory is used to describe new kinds of localised and distributed exact coherent structures. Starting with a localised vortex–wave interaction state driven by a single inviscid wave, regular arrays of interacting vortex–wave states are investigated. In the first instance the arrays described are operational in an infinite uniform shear flow; we refer to them as ‘uniform shear vortex–wave arrays’. The basic form of the interaction remains identical to the canonical one found by Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666) and subsequently used to describe exact coherent structures by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). Thus in each cell of a vortex–wave array a roll stress jump is induced across the critical layer of an inviscid wave riding on the streak part of the flow. The theory is extended to arbitrary shear flows using a nonlinear Wentzel–Kramers–Brillouin–Jeffreys or ray theory approach with the wave–roll–streak field operating on a shorter length scale than the mean flow. The evolution equation governing the slow dynamics of the interaction turns out to be a modified form of the well-known mean equation for a turbulent flow, and its particular form can be interpreted as a ‘closure’ between the small and large scales of the flow. If the array structure is taken to be universal, in the sense that it applies to arbitrary shear flows, then the array takes on a form which supports a logarithmic mean velocity profile trapped between what can be identified with the ‘wake region’ and a ‘buffer layer’ well known in the context of wall-bounded turbulent flows. The many similarities between the distributed structures described and wall-bounded turbulence suggest that vortex–wave arrays might be involved in the self-sustaining process supporting the log layer. The modification of the mean profile within each cell of the array leads to ‘staircase’-like streamwise velocity profiles similar to those observed experimentally in turbulent flows. The wave field supporting the ‘staircase’ is concentrated in critical layers which can be associated with the shear layer structures that have been attributed by experimentalists to be the mechanism supporting the uniform-momentum zones of the staircase.