On the decay of dispersive motions in the outer region of rough-wall boundary layers

In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow solutions exist that lead to so-called dispersive velocity components and dispersive stresses. They play a significant role in the mean-flow momentum balance near the wall, but typically disappear in the outer layer. A theoretical framework is presented to study the decay of dispersive motions in the outer layer. To this end, the problem is formulated in Fourier space, and a set of governing ordinary differential equations per mode in wavenumber space is derived by linearizing the Reynolds-averaged Navier-Stokes equations around a constant background velocity. With further simplifications, analytically tractable solutions are found consisting of linear combinations of exp (kz) and exp (-kz), with z the wall distance, k the magnitude of the horizontal wavevector k, and where K. k; Re /is a function of k and the Reynolds number Re. Moreover, for k -> infinity or k(1) -> 0 (with k(1) the stream-wise wavenumber), K -> k is found, in which case solutions consist of a linear combination of exp (-kz) and z exp (-kz) and are independent of the Reynolds number. These analytical relations are compared in the limit of k(1) = 0 to the rough boundary layer experiments by Vanderwel & Ganapathisubramani (J. Fluid Mech., vol. 774, 2015, R2) and are in reasonable agreement for l(k) /delta <= 0.5, with delta the boundary-layer thickness and l(k) = 2 pi/k.