Stationary rotary force waves on the liquid–air core interface of a swirl atomizer

Abstract

A one-dimensional wave equation, applicable to the waves on the surface of the air-core of a swirl atomizer is derived analytically, by analogy to the similar one-dimensional wave equation derivation for shallow-water gravity waves. In addition an analogy to the flow of water over a weir is used to produce an analytical derivation of the flow over the lip of the outlet of a swirl atomizer using the principle of maximum flow. The principle of maximum flow is substantiated by reference to continuity of the discharge in the direction of streaming. For shallow-water gravity waves, the phase velocity is the same expression as for the critical velocity over the weir. Similarly, in the present work, the wave phase velocity on the surface of the air-core is shown to be the same expression as for the critical velocity for the flow at the outlet. In addition, this wave phase velocity is shown to be the square root of the product of the radial acceleration and the liquid thickness, as analogous with the wave phase velocity for shallow water gravity waves, which is the square root of the product of the acceleration due to gravity and the water depth. The work revisits the weirs and flumes work of Binnie et al. but using a different methodology. The results corroborate with the work of Binnie. High speed video, Laser Doppler Anemometry and deflected laser beam experimental work has been carried out on an oversize Perspex (Plexiglas) swirl atomizer. Three distinctive types of waves were detected: helical striations, low amplitude random ripples and low frequency stationary waves. It is the latter wave type that is considered further in this article. The experimentally observed waves appear to be stationary upon the axially moving flow. The mathematical analysis allows for the possibility of a negative value for the phase velocity expression. Therefore the critical velocity and the wave phase velocity do indeed lead to stationary waves in the atomizer. A quantitative comparison between the analytically derived wave phase velocity and that measured experimentally, for this stationary pulsating wave, show very good agreement within a few percent.