A Modified Logarithmic Law for Neutrally Stratified Flow over Low-Sloped Hills

The study of the atmospheric boundary layer flow over two-dimensional low-sloped hills under a neutral atmosphere finds numerous applications in meteorology and engineering, such as the development of large-scale atmospheric models, the siting of wind turbines, and the estimation of wind loads on transmission towers and antennas. In this paper, the intermediate variable technique is applied to the momentum equations in streamline coordinates to divide the flow into regions, with each characterized by the dominance of different terms. Using a simple mixing-length turbulence closure, a simplified form of the x momentum equation is solved for the fully turbulent region, resulting in a modified logarithmic law. The solution is expressed as a power series correction to the classical logarithmic law that is valid for flat terrain. A new parameter appears: the effective radius of curvature of the hill. The modified logarithmic law is used to obtain new equations for the speedup, the relative speedup, the maximum speedup, and the height at which it occurs. A new speedup ratio is proposed to calculate the relative speedup at specific heights. The results are in very good agreement with the Askervein and Black Mountain field data.