Minimum requirements for outer layer similarity and roughness function in wall turbulence over obstructing surfaces

Turbulent flows over rough and complex surfaces are generally believed to be essentially smooth-wall-like sufficiently far above the surface, exhibiting outer-layer similarity. The only difference with smooth-wall flows is then a constant shift in the mean velocity profile, also known as the roughness function, △U⁺. However, experimental and numerical studies have reported the loss of outer-layer similarity over certain obstructing layouts, such as porous substrates and canopies, and values for the Karman constant, κ, different from the smooth-wall value by factors of up to one half. Motivated by this, we have conducted an extensive set of direct simulations for flows over surfaces with tall elongated obstructions (canopies), aiming to disrupt the overlying flow as much as possible, varying the element height and spacing, friction Reynolds number, Re_τ, and ratio of boundary-layer to canopy height, δ/h. The objective is to delimit under which conditions outer-layer similarity can be meaningfully defined, and when not. We find that spurious values of κ can result easily from trying to match the velocity profile to a log, when not even a smooth-wall flow at the same Re_τ follows a log exactly, or from trying to extend that match into the roughness sublayer or above the log layer. We also suggest that outer-layer similarity implies a match not only of the log layer, but also of the wake, to a corresponding smooth wall flow, and that this should be taken into account. When this is done, outer-layer similarity can be recovered provided there is a sufficient extension of flow above the roughness sublayer. For δ/h≥20, a sufficient depth of log layer can develop and canonical values of κ can be observed. For δ/h∽10, however, the log layer is partially disrupted, and small decreases of order 10-15% in κ can be observed, although not as large as previously reported. At those low δ/h values, outer-layer similarity is only partially recovered, but the flow thickness is nevertheless sufficient to estimate △U⁺ accurately.