The virtual origin as a first-order correction for the far-field description of laminar jets

Abstract

The far-field velocity and composition fields of a submerged laminar jet are known to approach a self-similar solution corresponding to the flow induced by a point source of momentum and scalar. Previous efforts to improve this far-field description have introduced a virtual origin for the streamwise coordinate to remedy the singular behavior of the self-similar solution near the jet origin. The purpose of this note is to show, by means of a perturbative analysis of the point-source solution, that this virtual origin is in fact the first-order correction to the leading-order description. The perturbative analysis, which uses the ratio x of the streamwise distance to the length of jet development as an asymptotically large quantity, also indicates that the displaced point source provides the description in the far field with small relative errors of order x-3 for the round jet and of order x-10/3 for the plane jet. The values of the virtual origin are obtained by numerical integration of the boundary-layer equations in the region of jet development, giving values that depend on the shape of the jet velocity profile at the exit.