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

Turbulent flows over canopies of rigid filaments with different densities, $\lambda_f$, are studied using direct simulations at Reynolds numbers $Re_\tau\approx550-1000$. The canopies have heights $h^+\approx110-220$, and are an instance of obstructing substrate. We show that conventional methods used to determine the zero-plane displacement 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. We also investigate if the zero-plane displacement and the friction velocity can be set independently, but find that outer-layer similarity is more consistently recovered when they are coupled. Our results suggest a modified outer-layer similarity, where the K\'arm\'an constant, $\kappa$, 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 $\kappa\approx0.39$ and origin essentially at the tip plane. For intermediate densities, the overlying flow perceives a deeper zero-plane displacement, in agreement with previous studies, but exhibits a lower K\'arm\'an constant, $\kappa\approx0.34-0.36$. 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.