Quasi-cylindrical approximation to the swirling flow in an atomizer chamber

A quasi-cylindrical approximation is used to analyse the axisymmetric swirling flow of a liquid with a hollow air core in the chamber of a pressure swirl atomizer. The liquid is injected into the chamber with an azimuthal velocity component through a number of slots at the periphery of one end of the chamber, and flows out as an annular sheet through a central orifice at the other end, following a conical convergence of the chamber wall. An effective inlet condition is used to model the effects of the slots and the boundary layer that develops at the nearby endwall of the chamber. An analysis is presented of the structure of the liquid sheet at the end of the exit orifice, where the flow becomes critical in the sense that upstream propagation of long-wave perturbations ceases to be possible. This analysis leads to a boundary condition at the end of the orifice that is an extension of the condition of maximum flux used with irrotational models of the flow. As is well known, the radial pressure gradient induced by the swirling flow in the bulk of the chamber causes the overpressure that drives the liquid towards the exit orifice, and also leads to Ekman pumping in the boundary layers of reduced azimuthal velocity at the convergent wall of the chamber and at the wall opposite to the exit orifice. The numerical results confirm the important role played by the boundary layers. They make the thickness of the liquid sheet at the end of the orifice larger than predicted by irrotational models, and at the same time tend to decrease the overpressure required to pass a given flow rate through the chamber, because the large axial velocity in the boundary layers takes care of part of the flow rate. The thickness of the boundary layers increases when the atomizer constant (the inverse of a swirl number, proportional to the flow rate scaled with the radius of the exit orifice and the circulation around the air core) decreases. A minimum value of this parameter is found below which the layer of reduced azimuthal velocity around the air core prevents the pressure from increasing and steadily driving the flow through the exit orifice. The effects of other parameters not accounted for by irrotational models are also analysed in terms of their influence on the boundary layers.