Turbulent shear flows over low hills

TLDR: In this paper, a general analysis is developed for turbulent shear flows over two and three-dimensional hills with low-slopes which allows for a wide range of upwind velocity profiles, such as those caused by wakes of up-wind hills, roughness changes, or changes in stratification.

Abstract: A general analysis is developed for turbulent shear flows over two- and three-dimensional hills with low-slopes which allows for a wide range of upwind velocity profiles, such as those caused by wakes of upwind hills, roughness changes, or changes in stratification. In this paper the atmosphere is assumed to be neutrally stable and the half-lengths of the hills, L, are large compared with their heights, H, which are very large compared with the roughness length zo. The general structure of the solution is defined by dividing the flow into two regions, each of which is divided into two sublayers: an inviscid outer region composed of an upper layer in which there is potential flow when the atmosphere is neutrally stable, and a middle layer in which the wind shear dominates; and an inner region of thickness l ≤ L in which the effects of perturbation shear stresses are confined. The latter region is divided into two: a shear stress layer where the shear stresses, although weak, determine that the maximum of the perturbation velocity is located in this layer; and an inner surface layer of thickness ls where the shear stress gradient varies rapidly and the perturbation velocity tends to zero. The details of the middle layer are given here for different kinds of upwind profiles, including logarithmic, ‘power law’ and linear profiles. It is shown that the analysis can be extended to allow for nonlinear inertial effects in the middle layer. Analytical solutions are derived for the inner region as asymptotic expansions in δ = [ln(l/zo)]−1, which is assumed to be small, and this shows that ls ∼ zo(l/zo)1/2. The results of the analytical model are compared with our own and with previously published numerical computations of the full equations (applying the same assumptions used for calculating the turbulent shear stresses as used in the analytical work), which have largely been validated against full-scale measurements. These results confirm that the relative increase of surface stress is significantly greater than the increase of wind speed near the surface except when there is no upwind shear (as for example in a logarithmic boundary layer when the roughness length tends to zero). Finally, the paper shows that the outer regions of laminar (or constant eddy viscosity) and of turbulent flows over hills are broadly similar, but that the effects of the flow in the inner region on the outer regions are much smaller in the latter case.