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 k, and where K(k, Re) is a function of k and the Reynolds number Re. Moreover, for k → ∞ or k1 → 0, K → 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 k1 = 0 using the rough boundary layer experiments by Vanderwel and Ganapathisubramani (J. Fluid Mech. 774, R2, 2015) and are in good agreement for ℓ k /δ ≤ 0.5, with δ the boundary-layer thickness and ℓ k = 2π/k.

Effects of heterogeneous surface geometry on secondary flows in turbulent boundary layers

“…Therefore, the presence of a peak instead of a classical plateau in τ xy is considered to be a consequence of the increased lateral motions which enhance the secondary flows and lead to higher momentum transport. These results are also in agreement with recent studies that assessed the contributions of the dispersive stresses (Chan et al 2018;Meyers et al 2019;Vanderwel et al 2019). They found that these form-induced stresses are prominent within the canopy but can markedly increase spatially depending on the surface heterogeneity length scale.…”

“…This decomposition allows us to quantify the amount of stresses arising from the mean flow heterogeneity, which contribute to the total stresses as forminduced (dispersive) stresses. In the case of strong spatial heterogeneities, these dispersive stresses can play a major role in the transport of momentum flux, which can impact the organisation of the turbulent boundary layer structure (Türk et al 2014;Meyers et al 2019).…”

“…This decomposition allows us to quantify the amount of stresses arising from the mean flow heterogeneity, which contribute to the total stresses as form-induced (dispersive) stresses. In the case of strong spatial heterogeneities, Turbulent boundary layers over heterogeneous surfaces 886 A31-17 these dispersive stresses can play a major role in the transport of momentum flux, which can have an impact on the organisation of the turbulent boundary layer structure (Türk et al 2014;Meyers, Ganapathisubramani & Cal 2019). Comparative maps of the turbulent, dispersive and total shear stresses denoted uv, uṽ and τ xy , respectively, are presented in figure 8 (in columns), for different cases (in rows).…”

Turbulent boundary-layer flow over regular multiscale roughness

“…This method is usually employed in flows where spatial heterogeneity is prevalent, in order to quantify the magnitude of the stresses originating from the mean flow inhomogeneity. In this scenario, the dispersive stresses can have substantial contributions to the total stresses and play a major role in the transport of momentum fluxes (Meyers, Ganapathisubramani & Cal 2019;Nikora et al 2019). In this scheme, the streamwise velocity field can be written as u i (y, z, t) where u i is the instantaneous velocity field measured at a fixed streamwise location.…”

Secondary motions above a staggered multi-scale rough wall

“…2014; Anderson et al. 2015; Vanderwel & Ganapathisubramani 2015; Meyers, Ganapathisubramani & Cal 2019). Furthermore, the decay of dispersive stresses for such streamwise roughness elements has been found to scale with the spanwise spacing (Chan et al.…”