Examination of outer-layer similarity in wall turbulence over obstructing surfaces

Abstract

Turbulent flows over canopies of rigid filaments with different densities are studied using direct numerical simulations at moderate Reynolds numbers $Re_\tau \approx 550\unicode{x2013}1000$ . The canopies have heights $h^+\approx 110\unicode{x2013}220$ , and are used as an instance of obstructing substrate for the assessment of outer-layer similarity. We show that conventional methods used to determine the zero-plane displacement $\Delta y$ can be at odds with proper outer-layer similarity and may not be applicable for flows at moderate $Re_\tau$ . Instead, we determine $\Delta y$ and the length and velocity scales that recover outer-layer similarity by minimising the difference between the smooth-wall and canopy diagnostic function everywhere above the roughness sublayer, not just in the logarithmic layer. In addition, we explore the possibility of the zero-plane displacement and the friction velocity being set independently, but find that outer-layer similarity is recovered more consistently when they are coupled. We observe that although the Kármán constant, $\kappa$ may not have smooth-wall-like values, the flow statistics are smooth-wall-like in the logarithmic layer and above if the surface effect is limited within the near-wall region. This suggests a modified outer-layer similarity, where $\kappa$ is not 0.39, but turbulence is otherwise smooth-wall-like. When the canopy is dense, the flow above the tips is essentially smooth-wall-like, with smooth-wall-like $\kappa \approx 0.39$ and origin essentially at the tip plane. For canopies with intermediate density, the overlying flow perceives a deeper zero-plane displacement into the canopy, which is consistent with observations reported by previous studies, but exhibits a lower Kármán constant, $\kappa \approx 0.34\unicode{x2013}0.36$ . For sparse canopies, $\kappa$ tends back to its smooth-wall value, and the zero-plane displacement height is at the canopy bed. For all canopies studied, the decrease in $\kappa$ never exceeds $15\,\%$ , which is significantly less than that obtained in some previous works using conventional methods to assess outer-layer similarity.