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

Abstract

In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow solutions existthat lead to so-called dispersive velocity components and dispersive stresses. They play asignificant role in the mean-flow momentum balance near the wall, but typically disappearin the outer layer. A theoretical framework is presented to study the decay of dispersivemotions 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 isderived by linearizing the Reynolds-averaged Navier–Stokes equations around a constantbackground velocity. With further simplifications, analytically tractable solutions arefound consisting of linear combinations of exp(−kz) and exp(−Kz), with z the walldistance, k the magnitude of the horizontal wavevector k, and where K(k, Re) is afunction of k and the Reynolds number Re. Moreover, for k → ∞ or k1 → 0 (withk1 the stream-wise wavenumber), K → k is found, in which case solutions consist of alinear combination of exp(−kz) and z exp(−kz), and are Reynolds number independent.These analytical relations are compared in the limit of k1 = 0 to the rough boundarylayer experiments by Vanderwel and Ganapathisubramani (J. Fluid Mech. 774, R2, 2015)and are in reasonable agreement for `k/δ 6 0.5, with δ the boundary-layer thickness and 'k = 2π/k.

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ORCID for Bharathram Ganapathisubramani: orcid.org/0000-0001-9817-0486

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