Optimal coordinates for squire's jet

Abstract

HE role of the coordinate system in fluid mechanics problems has long intrigued workers in the field. Specifically, one often attempts to find coordinate systems in which a boundary-layer solution is uniformly valid throughout the entire flowfield. Such coordinates, following Kaplun,l are said to be optimal. In his classic paper, which deals with steady, planar, incompressible flows, rules for constructing optimal coordinate systems are given. For three-dimensional flows, the general problem concerning the existence of optimal coordinates is still unsolved. In this paper, the problem of the laminar round (without swirl), issuing into an infinite fluid at rest, is considered. The flow here is assumed to be steady and incompressible. An exact solution for the entire flowfield exists, and is discussed at length by Batchelor 2 and Yih.3 The solution was first obtained by Landau in 1944, but his results were never published, and, independently, by Squire 4 in 1951. More recently, the same results were derived by Coles 5 using the method of matched asymptotic expansions. In the above analyses, the governing equations were posed in spherical coordinates. Simple geometric considerations show that the outer (potential flow) stream surfaces are paraboloids. Thus, the choice of a paraboloidal coordinate system at least appears to be a natural one, and here, the method of inner and outer expansions is applied to the Navier-Stokes equations written in this system. It is shown that the solution obtained describes an optimal solution; more surprisingly, the optimal coordinate system obtained here coincides with the stream-surface coordinate system of the outer entrained flow. Also, as expected, the solution in paraboloidal coordinates takes on a particularly simplified form. For the momentum equations written in spherical coordinates, Squire has shown that the stream function (1) describes the full solution to the laminar round jet. It is noted that r measures the distance from the source; 6 measures the width or jet half-angle (taking the source point as vertex); and, measures the angular position in planes perpendicular to the axis (i.e., swirl coordinate). Here, the velocity components in the er, ee, and e^ directions are, respectively,