On the Steady Flow of a Fluid of Variable Density Past an Obstacle

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Abstract
A model of steady airflow past a three-dimensional mountain is considered. The fluid is inviscid and incompressible. The flow is under the influence of gravity, but for most of the work the earth's rotation is neglected. Far upstream the flow is parallel and horizontal, the velocity U(z) and the density ƍ(z) varying with height z. By use of conservation of density and head along streamlines, and thence of the dependence of density and head on one variable z0 only, the equations of motion are simplified, z0 (r) being the height far upstream of the streamline through the point with position vector r. The flow is considered first with small and then with large density variations far upstream. In the case when these variations are small, results of the known method of simple-shear secondary flows are rederived and extended. The chief consideration of this paper is the buoyancy effects of large variations of density in the oncoming stream. For simplicity, the shear far upstream is neglected. The primary and secondary velocities are found for large Richardson number (proportional to gdƍ/ƍdz0). The method is time-symmetric and analogous to taking the first two terns of the Janzen-Rayleigh expansion, the inverse of the Richardson number being the analogue of the square of the Mach number. The primary velocity, for infinite buoyancy forces, is horizontal, the flow in each horizontal plane being the same as the potential flow about the section of the obstacle in that plane. In general there is shear between horizontal planes, with horizontal primary vorticity. The secondary flow is found explicitly for a circular cylinder with vertical axis and for a hemisphere resting in a horizontal plane. The secondary approximation is not uniformly valid near the horizontal plane at the height of the top (if any) of the obstacle. Near that plane there is an inviscid shear layer, and the velocity gradient cannot be neglected, however large the Richardson number.

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