Axisymmetric free jet of Re → 0 from a round tube

Abstract

An axisymmetric free jet of Reynolds number Re → 0 from a round tube is investigated in a theoretical way in order to elucidate pressure and velocity fields in the downstream of the tube exit. The Stokes approximation is performed to the Navier–Stokes equations expressed by the spherical coordinate system. The whole flow field is divided into two regions, a closed region consisting of both the tube exit and the hemisphere with the tube radius, i.e. inner region, and a region outside of the hemisphere, i.e. outer region. The pressure and velocity components in the inner region are expanded by a series of unknown functions of the angle θ, multiplied by powers of the radial distance r, whilst those in the outer region by a series of unknown functions of θ, multiplied by inverse powers of r, both of which satisfy the respective boundary conditions at the tube exit and infinity. The inner-region and outer-region pressures are patched together well at an arbitrary point over the hemisphere common to both regions, and the velocity components are done at the same point likewise. The parabolic velocity distribution at the exit is derived from the velocity in the inner region. The jet-ejection pressure and the pressure gradient coefficient are related by one parameter σ and the role of the parameter in the patching is clarified. Stream lines show a typical flow characteristic of the free jet flow for Re → 0 in the outer region.