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

TL;DR

This paper develops a theoretical framework to analyze how dispersive motions in rough-wall boundary layers decay in the outer region, providing analytical solutions and validating them with experimental data.

Contribution

It introduces a linearized Fourier-space model for dispersive motion decay and derives explicit analytical solutions, enhancing understanding of rough-wall boundary layer dynamics.

Findings

Dispersive motions decay exponentially with wall distance.

Analytical solutions match experimental data for certain scales.

Decay behavior is independent of Reynolds number in specific limits.

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 $K(\mathbf{k},\textit{Re})$ is a function of $\mathbf{k}$ and the Reynolds number $\textit{Re}$. Moreover, for $k\rightarrow \infty$ or $k_1\rightarrow 0$, $K\rightarrow k$ 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 $\ell_k/\delta \leq 0.5$, with $\delta$ the boundary-layer thickness and $\ell_k = 2\pi/k$.