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 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 $\mathbf{k}$, and where $K(\mathbf{k},\text{Re})$ is a function of $\mathbf{k}$ and the Reynolds number $\text{Re}$. Moreover, for $k\to \infty$ or $k_1\to 0$, $K\to k$ is found, in which case solutions consist of a linear combination of $\exp (-kz)$ and $z\exp (-kz)$, and are Reynolds number independent. These analytical relations are verified in the limit of $k_1=0$ using the rough boundary layer experiments by Vanderwel and Ganapathisubramani (J. Fluid Mech. 774, R2, 2015) and are in good agreement for ${\ell }_k/\delta \le 0.5$, with $\delta$ the boundary-layer thickness and ${\ell }_k=2\pi /k$.