The displacement effect of a sphere in a two-dimensional shear flow

Abstract

This paper contains a theoretical investigation of the displacement effect of a pitot tube in a shear flow. Viscosity is neglected throughout so that the vorticity field alone is considered. It is first shown that a two-dimensional approach does not produce a large enough displacement effect because it does not include the stretching of vortex tubes that takes place around a three-dimensional pitot tube. Then the three-dimensional problem is considered. A solution is obtained in the plane of symmetry for a sphere in a shear flow. This solution is found by making an assumption about the rate of stetching of vortex tubes perpendicular to the plane of symmetry and then considering the shear flow as a small perturbation of a uniform flow. A solution in the plane of symmetry is sufficient to obtain the displacement effect, which is found to be of the same order as the experimental result obtained by Young & Maas (1936) for a conventional pitot tube. The sphere may be considered to represent a conventional pitot tube (of slightly smaller diameter), so it is concluded that a large part of the displacement effect of a pitot tube may be accounted for without the inclusion of viscosity, i.e. by consideration of the vorticity field alone. To a first approximation, the vorticity in the plane of symmetry is found to depend only on the distance from the centre of the sphere. An outline of shear flows past some two-dimensional bodies is given in an appendix. The bodies considered are a circular cylinder and a two-dimensional ‘pitot-tube’ consisting of two parallel semi-infinite plates.