On displacement-thickness, wall-layer and mid-flow scales in turbulent boundary layers, and slugs of vorticity in channel and pipe flows

Abstract

This theoretical study is motivated by the experimental observations ( a ) on the thickening of a turbulent boundary layer compared with its laminar counterpart, ( b ) on the erupting tongue of fluid that forms the leading edge of a turbulent spot in a boundary layer, ( c ) on the wall-layer and mid-flow scales, and ( d ) on the slugs of vorticity that occur in the middle of turbulent channel and pipe flows. It appears that no previous rational explanation has been put forward for these experimental observations. The present tentative suggestions for ( a ), ( b ) and ( d ) centre on the existence of small-deficit fast-travelling zones of concentrated vorticity governed by the nonlinear Euler equations to leading order at high Reynolds numbers Re but crucially influenced by viscosity nevertheless. In the boundary-layer case these zones travel outside the original boundary layer and hence act to increase the effective boundary-layer thickness. The structure of such zones and their scales, governing equations and amplitude dependence are discussed for assumed planar boundary layers and channel flows and for three-dimensional pipe flows in turn. Allied with this, the theory addresses the closure of the amplitude-dependent neutral curve at high Reynolds numbers, the connection with other Euler-type flows and the possibility of delay in sublayer bursting, as well as aiming to give some guidance on nonlinear aspects of unsteady two- and three-dimensional computations for Euler and related flows. The aspects in ( c ) above, concerning the turbulent scales both of the thin wall layer ( O ( Re -1 In Re ), from a renormalizing and scale-cascade argument) and of the thicker mid-flow zone (containing the Kolmogorov microscale O ( Re -3/4 )) which lies between that layer and the extensive small-deficit outer zone, are also discussed tentatively in terms of their dynamics, leading to apparently good agreement with turbulent-flow experiments and empirical models, for those scales. Other qualitative comparisons are presented.