A priori assessment of a wall model for turbulent boundary layers over cubical roughness

Surface roughness is ubiquitous in numerous types of flows, such as those in relevant engineering applications and surface flows in environmental scenarios. Both natural and artificial bodies present a rough surface, which in first approximation can be considered aerodynamically smooth, but roughness effects cannot be neglected in refined aerodynamic modeling. In this context, the aim of this work is to model the effects of cubical roughness elements on a turbulent boundary layer. This is important to evaluate the mechanical loads on vehicle surfaces and urban canopies. The presence of roughness elements induces a sheltering mechanism, changing momentum exchange in the viscous sublayer. Thus, a distinction is required between the near-wall drag producing height and the outer turbulent flow height, that is the region from the wall to the height of the roughness elements and the region from this height up to the edge of the boundary layer. The velocity profile follows an exponential law in the first region and a logarithmic law for the second region (Yang et al., 2016). The mean velocity profile is described assuming a shape function, following von Kármán-Pohlhausen method. The flow parameters are obtained on momentum conservation. A wake term is included based on Coles (1956), which accounts for outer layer turbulence and momentum transport, enabling realistic modeling in turbulent boundary layers. The model adopts a wake parameter expressed as a function of the Reynolds number, as proposed by Hasan et al. (2024). The results are compared with other data from high fidelity simulations and experiments. The predictions, in terms of total stress at the wall, can be improved using a different model for the displacement height for low frontal solidities. The base model relation, which uses the the drag force as in Jackson (1981), is replaced by the empirical expression in Coceal et al. (2004) for low frontal solidities. Another model is proposed for further improvement, based on outer layer universality, reflecting the universal behavior of the flow above the roughness elements. Following this assumption, this additional model establishes that the region above the roughness elements is divided into a first logarithmic part and a second parabolic part (Pirozzoli et al., 2023). The models are applied to aligned or staggered cube arrays and for different heights and different Reynolds numbers, thus covering a large parameter space. Further improvements can be achieved in the future by accounting for a wider set of roughness geometries.