The effect of density gradient and shear on the flow over a hemisphere

Abstract

When an inviscid fluid with a slight vertical variation of stagnation pressure and density flows over an obstacle lying on a horizontal plane a vorticity component in the direction of flow appears. An expression for this secondary vorticity is obtained on the assumption that velocity and density gradients and gravitational effects are small enough for the flow to be regarded as a small perturbation of the potential flow about the obstacle. Calculations made for flow over a hemispherical hump show that, if the density is constant and the velocity in the approaching stream increases with height, there is a vorticity component which produces a downward flow on the surface of the obstacle; when the density alone varies the effect is found to depend on the Richardson number associated with the flow, and for an unstable density gradient, i.e. one in which the density increases with height, or, for example, in the atmosphere when there is a non-adiabatic lapse rate, the secondary vorticity produced causes an upward flow on the surface of the hemisphere. Experiments with smoke in a low-velocity wind tunnel with a heated bottom wall show qualitatively that vorticity components due to both effects may exist together, and that their relative importance depends on the sign and magnitude of the Richardson number.