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

Turbulent flows over canopies of rigid filaments with different densities are studied using direct numerical simulations at moderate Reynolds numbers Re-tau approximate to 550-1000. The canopies have heights h(+ )approximate to 110-220, 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 K & aacute;rm & aacute;n 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 approximate to 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 K & aacute;rm & aacute;n constant, kappa approximate to 0.34-0.36. 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.