An analytical solution is presented for the flow of an adiabatic turbulent boundary layer on a uniformly rough surface over a two-dimensional hump with small curvature, e.g. a low hill. The theory is valid in the limit L/y0 → ∞ when h/L < 1/8(y0/L)0.1 and δ/L > 2k2/ln(δ/y0) where L and h are the characteristic length and height of the hump, y0 the roughness length of the surface and δ the thickness of the boundary layer. For rural terrain, taking δ ∼ 600m these conditions imply that 102 < L < 104m and h/L < 0·05. Considerations of the turbulent energy balance suggest that the eddy viscosity distribution for equilibrium flow near a wall may still be used to a good approximation to determine the changes in Reynolds stress. This result is only required in a thin layer adjacent to the surface - in the main part of the boundary layer the perturbation stresses are shown to be negligible and the disturbance to be almost irrotational. The theory shows that for a log-profile upwind the increase in wind speed near the surface of the hill is O((h/L)u0(L)) where u0(L) is the velocity of the incident wind at a height L. Thus the increase in surface winds can be considerably greater than is predicted by potential flow theory based on an upwind velocity u0(h). It is also found that, at the point above the top of a low hill at which the increase in velocity is a maximum, the velocity is approximately equal to the velocity at the same elevation above level ground upwind of the hill. The surface stress is highly sensitive to changes in the surface elevation, being doubled by a slope as small as one in five. The turning of the wind in the Ekman layer may induce a change in direction of the wind above the hill.
The main object of this analysis is to show how the changes in wind speed and shear stress are related to the size and shape of the hill and to the roughness of the surface. Some comparisons are made with measurements of the natural wind and wind tunnel flows. These suggest that the theory may be useful in giving rough estimates of the effect of hills on the wind. The theory and the quoted measurements suggest that the present design recommendation for the increase in wind speeds over hills to be used in wind loading calculations may be an underestimate. It is to be hoped that this analysis will encourage more detailed measurements to be made of the wind over hills.