The effect of confinement on the stability of two-dimensional shear flows

Abstract

It has been shown recently that the instability of a two-dimensional wake increases when it is confined in the transverse direction by two flat plates. Confinement causes the transition from convective to absolute instability to occur at lower values of shear. This paper examines this effect comprehensively and concludes that it is due to the constructive interaction of modes with zero group velocity in the wake (or jet) and in the surrounding flow. Maximum instability occurs when the wavenumber of the fundamental mode in the wake (or jet) matches that of the fundamental mode in the surrounding flow. Other regions of high instability occur when the harmonics of one mode interact with the fundamental of the other. This effect is examined at density ratios from 0.001 to 1000. At each density ratio, the confinement which causes maximum absolute instability can be predicted. This study also shows that it is vital to examine the wavenumber of absolutely unstable modes in order to avoid over-predicting the absolute instability. In some situations this wavenumber is vanishingly small and the mode must be discounted on physical grounds.

Stability analysis and breakup length calculations for steady planar liquid jets

“…We investigated the effect of thickening the liquid shear layer on the absolute instability of the jet and found that the finite shear layer produces an absolute instability in jets with smaller values of the density parameter than was found in the work of Juniper (2006) who considered only infinitely thin shear layers. Juniper (2006) found that absolute instability occurs for q & 1.25, while we have shown that an absolute instability occurs for at least q > 0.5.…”

“…It is known that the density ratio has a large effect on the behaviour of absolute instabilities, in particular that low-density jets (q > 1) are almost always absolutely unstable (Sreenivasan, Raghu & Kyle 1989;Yu & Monkewitz 1990;Juniper 2006). In this work we are concerned with jets which have q < 1, but we find that absolute instabilities can occur at these density ratios for particular velocity profiles.…”

“…In a steady parallel jet, an absolute instability is significant because the jet will eventually break up at the nozzle as long as enough time is allowed to pass. In this paper we examine the fluid response to the forcing in frames of reference moving at various speeds downstream (or possibly upstream) of the source of disturbances as in the problems considered by Healey (2006) and Juniper (2006).…”

“…The density ratio q = 1/500 is considered because it corresponds to cold liquid diesel liquid fuel being injected into compressed air at about 5 atmospheres pressure, which is similar to the smallest density ratio in the experiments in figure 1. In this paper we adopt the notation of Juniper (2006Juniper ( , 2007 for the labelling of the saddle points. We denote the saddle whose position is controlled by the thickness of the shear layer and surface tension (Rees & Juniper 2009) as s 1 .…”

“…The destabilizing effect of surface tension for absolute instabilities has been studied by Lin & Lian (1989), and we look at its effect on our jet profiles in figures 11 and 12. The major part of this study up to now has focused on the sinuous modes of the jet, but these modes are more stable than the varicose modes when it comes to absolute instabilities (Juniper 2006). Therefore, for the remainder of this section we focus our attention on varicose modes and investigate the conditions when they produce absolute instabilities in the jet.…”