On the Decay of Dispersive Motions in the Outer Region of Rough-wall Boundary Layers

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Journal of Fluid Mechanics

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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 and , with the wall distance, the magnitude of the horizontal wavevector , and where is a function of and the Reynolds number . Moreover, for or (with the stream-wise wavenumber), is found, in which case solutions consist of a linear combination of and , and are independent of the Reynolds number. These analytical relations are compared in the limit of to the rough boundary layer experiments by Vanderwel & Ganapathisubramani (J. Fluid Mech., vol. 774, 2015, R2) and are in reasonable agreement for 𝛿 , with 𝛿 the boundary-layer thickness and π .


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