Downstream evolution of unconfined vortices: mechanical and thermal aspects

Abstract

We present a numerical study of the downstream evolution (mechanical and thermal) of vortex-jet cores whose velocity and temperature fields far from the axis match a family of inviscid and non-conducting vortices. The far-velocity field is rotational, except for a particular case which corresponds to the well-known Long's vortex. The evolution of the vortex core depends on both the conditions at a certain upstream station, characterized by the dimensionless value of the velocity at the axis, and a dimensionless swirling parameter L defined as the ratio of the values of the azimuthal and axial velocities outside the vortex core. This numerical study, based on the quasi-cylindrical approximation (QC) of the Navier–Stokes equations, determines the conditions under which the vortex evolution proceeds smoothly, eventually reaching an asymptotic self-similar behaviour as described in the literature (Fernández-Feria, Fernández de la Mora & Barrero 1995; Herrada, Pérez-Saborid & Barrero 1999), or breaks in a non-slender solution (vortex breakdown). In particular, the critical value L = Lb(a) beyond which vortex breakdown occurs downstream is a function of a dimensionless parameter a characterizing the axial momentum of the vortex jet at an initial upstream station. It is found numerically that for very large values of a this vortex breakdown criterion tends to an asymptote which is precisely the value L = L* predicted by the self-similar analysis, and beyond which a self-similar structure of the vortex core does not exist. In addition, the computation of the total temperature field provides useful information on the physical mechanisms responsible for the thermal separation phenomenon observed in Ranque–Hilsch tubes and other swirling jet devices. In particular, the mechanical work of viscous forces which gives rise to an intense loss of kinetic energy during the initial stages of the evolution has been identified as the physical mechanism responsible for thermal separation.