Analytical study of turbulent wind flow past a low hill

摘要

The mechanisms and characteristics of turbulent air flow past a low hill are studied analytically. The objective of the analysis is to study how the wind profile changes as it approaches and flows past the hill, and to understand how wind speed and shear stresses are related to the hill topography and roughness. The wind, idealized by an adiabatic turbulent boundary layer, is set to flow past a uniformly rough surface of a two-dimensional symmetric low hill of mild slope. Considerations of the turbulent energy suggest that the eddy viscosity distribution for stable flow is a good approximation to determine the variations of Reynolds stresses within the turbulent boundary layer. The calculations for both regions are thus based on the full Navier-Stokes equations with closures by the use of eddy viscosity. As in typical boundary layer problems, the entire flow field is divided into two regions: the inner region where viscous effect is significant and the outer region where the flow is almost irrotational. In each respective region, the horizontal and vertical velocity components and the pressure are obtained in closed forms using perturbation techniques and Fourier transform analyses based on the theory developed by Jackson & Hunt (1975). Assuming sufficiently high Rossby and Reynolds numbers, the greater part of the boundary-layer profile can be described by a logarithmic upwind profile. From the analytical results, it is observed that the increase in wind speed near the surface is of the order of h/l u{sub}0 [h], where h and l is the characteristic height and length of the hill, and u{sub}0 [h] is the velocity of the incident wind at height h. It is also found that as the wind approaches the hill, the flow accelerates until it reaches the top, where the velocity is approximately equal to the velocity at the same elevation above level ground. On the downward side, the increase in wind speed over the surface of the hill, as in typical high Reynolds number flow past bluff bodies. It is found that the velocity increases along the upwind side of the hill until a maximum is reached at the summit of the hill, and then decelerates along the downhill side.