J. zyxwvutsrq Fluid zyxwvutsrq Mech. zyxwvutsrqp (1993), vol. 251, pp. 149-172 Copyright zyxwvutsrq Q 1993 Cambridge University Press zyxwvut 149 The influence of swirl and confinement on the stability of counterflowing streams By J. D. GODDARD, A. K. DIDWANIA AND C.-Y. WU Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla CA 92093-0310, USA (Received 10 July 1992 and in revised form 5 December 1992) A linear stability analysis of laterally confined swirling flow is given, of the type described by Long’s equation in the inviscid limit or by the von Kdrmdn similarity equations in the absence of lateral confinement. The flow of interest involves identical counterflowing fluid streams injected with equal velocity W, through opposing porous disks, rotating with angular velocities SZ and 552, respectively, about a common normal axis. By means of mass transfer experiments on an aqueous system of this type we have detected an apparent hydrodynamic instability having the appearance of an inviscid supercritical bifurcation at a certain (521 > 0. zyxw As an attempt to elucidate this phenomenon, linear stability analyses are performed on several idealized flows, by means of a numerical Galerkin technique. An analysis of high-Reynolds-number similarity flow predicts oscillatory instability for all non-zero Q. The spatial structure of the most unstable modes suggests that finite container geometry, as represented by the confining cylindrical sidewalls, may have a strong influence on flow stability. This is borne out by an inviscid stability analysis of a confined flow described by Long’s equation. This analysis suggests a novel bifurcation of the inviscid variety, which serves qualitatively to explain the results of our mass transfer experiments. 1. Introduction The present work stems in part from the design of a laboratory device capable of generating a uniformly accessible surface for convective heat or mass transport between fluid streams. In previous works on the subject (Goddard, Melville & Zhang 1987; Zhang & Goddard 1989) it has been shown, by yet another variant of the Karman similarity equation, that the requisite flow field can in principle be realized by means of counterflowing rotating fluid streams, such as might be created by uniform injection through porous rotating disks. A similar flow configuration has also been the subject of recent theoretical and experimental investigations of the effects of swirl on gaseous flames (Kim, Libby & Williams 1992). For the purposes of the present discussion and for later reference, we recall that for steady similarity flows of the von Karman type, the surface-normal velocity depends spatially only on surface-normal distance zyxw z with, say, u, = zyx W(z). Hence, the mass (or heat) transfer coefficient at a planar stagnation surface z = 0 between counterflowing rotating streams is given asymptotically for large PCclet numbers by the non- dimensional surface-normal velocity gradient W’(0) as (Zhang & Goddard 1989) where Nu and Pe denote the appropriate Nusselt and Piclet numbers and the primes