J. C. R. Hunt

Abstract 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/y 0 → ∞ when h/L < 1/8( y0/L ) 0.1 and δ/ L > 2 k 2 /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 10 2 < L < 10 4 m 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 ) u 0 (L)) where u 0 (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 u 0 ( 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.

In flows around three-dimensional surface obstacles in laminar or turbulent streamsthere are a number of points where the shear stress or where two or more component,s of the mean velocity are zero. In the first part of this paper we summarize and extend the kinematical theory for the flow near these points, particularly by emphasizing the topological classification of these points as nodes or saddles. We show that the zero-shear-stress points on the surface and on the obstacle must be such that the sum of the nodes Σ N and the sum of the saddles Σ s satisfy \[ \Sigma_N -\Sigma_S = 0. \] If the obstacle has a hole through it, such as a passageway under a building, \[ \Sigma_N -\Sigma_S =-2. \] If the surface is a junction between two pipes, \[ \Sigma_N -\Sigma_S =-1. \] We also consider, in two-dimensional plane sections of the flow, the points where the components of the mean velocity parallel to the planes are zero, both in the flow and near surfaces cutting the sections. The latter points are half-nodes N′ or half-saddles S′. We find that \[ (\Sigma_N +{\textstyle\frac{1}{2}}\Sigma_{N^{\prime}}-(\Sigma_{S^{\prime}}+{\textstyle\frac{1}{2}}\Sigma_{S^{\prime}}) = 1-n, \] where n is the connectivity of the section of the flow considered. In the second part new flow-visualization studies of laminar and turbulent flows around cuboids and axisymmetric humps (i.e. model hills) are reported. A new method of obtaining a high resolution of the surface shear-stress lines was used. These studies show how enumerating the nodes and saddle points acts as a check on the inferred flow pattern. Two specific conclusions drawn from these studies are that: for all the flows we observed, there are no closed surfaces of mean streamlines around the separated flows behind three-dimensional surface obstacles, which con-tradicts most of the previous suggestions for such flows (e.g. Halitsky 1968); the separation streamline on the centre-line of a three-dimensional bluff obstacle does not, in general, reattach to the surface.

A general expression is derived for the fluid force on a body of simple shape moving with a velocity v through inviscid fluid in which there is an unsteady non-uniform rotational velocity field u 0 ( x , t ) in two or three dimensions. It is assumed that the radius is small compared with the scale over which the strain rate changes, though for the sphere it is also assumed that the changes in the ambient velocity field over the scale of the sphere are small compared with the velocity of the body relative to the flow. Given these approximations it is shown that the effects of the rate of change of the vorticity of the ambient flow is of second order and can be neglected. However the rate of change of the irrotational straining motion is included in the analysis. It is shown that the inertial forces derived by many authors for irrotational flow can be simply added to a generalization of the lift force derived by Auton (1987) in a companion paper. It is shown how this lift force is made up of a rotational and an inertial or added-mass component. For three-dimensional bluff bodies the latter is generally larger (by a factor of three for a sphere), and can be simply calculated from the added-mass coefficient. For illustration, the general expression is used to derive formulae for (i) the motion of a spherical bubble in a steady non-uniform flow to contrast with the motion in an unsteady flow, and (ii) the motion of rigid volumes of neutral density across an inviscid shear flow. These results show how added-mass (and lift) forces lead to different motions for a sphere and a cylinder. The general expression is useful in two-phase flow calculations, and for indicating the forces and motions of ‘lumps of fluid’ in turbulent flows.

The paper presents an analysis of laminar motion of a conducting liquid in a rectangular duct under a uniform transverse magnetic field. The effects of the duct having conducting walls are investigated. Exact solutions are obtained for two cases, (i) perfectly conducting walls perpendicular to the field and thin walls of arbitrary conductivity parallel to the field, and (ii) non-conducting walls parallel to the field and thin walls of arbitrary conductivity perpendicular to the field. The boundary layers on the walls parallel to the field are studied in case (i) and it is found that at high Hartmann number ( M ), large positive and negative velocities of order MV c are induced, where V c is the velocity of the core. It is suggested that contrary to previous assumptions the magnetic field may in some cases have a destabilizing effect on flow in ducts.

This paper describes the flow structure observed over a bell-shaped hill with height h (the profile of which is the reciprocal of a fourth-order polynomial) when it was placed first in a large towing tank containing stratified saline solutions with uniform stable density gradients and second in an unstratified wind tunnel. (A similarly shaped model hill was also studied in a small towing tank.) Observations were made at values of the Froude number F (≃ U / Nh ) in the range 0·1 to 1·7 and at F = ∞, where U is the towing speed and N is the Brunt-Väisälä frequency, and at values of the Reynolds number from 400 to 275000. For F ≲ 0·4, the observations verify Drazin's (1961) theory for low-Froude-number flow over three-dimensional obstacles and establish limits of applicability. For Froude numbers of the order of unity, it is found that a classification of the lee-wave patterns and separated-flow regions observed in two-dimensional flows also appears to apply to three-dimensional hills. Flow-visualization techniques were used extensively in obtaining both qualitative and quantitative information on the flow structure around the hill. Representative photographs of dye tracers, potassium permanganate dye streaks, shadowgraphs, surface dye smears, and hydrogen-bubble patterns are included here. While emphasis is centred on obtaining a basic understanding of the flow around three-dimensional hills, the results are applicable to the estimation of air pollutant dispersion around hills.