A 3-D analytical model has been developed to predict the instability of a swirling annular liquid sheet sandwiched by swirling air streams. The model incorporates essential features of the prefilming and subsequent atomization process inside pressure-swirl airblast atomizer. Effects of relative axial velocity, liquid swirl, inner and outer air swirl on the growth of various disturbance modes were examined. Effect of relative swirl orientation of the air swirl with respect to liquid swirl has also been investigated. It was found that increasing the relative axial velocity between the liquid and the gas phases significantly improves the fuel atomization. A combination of the inner and outer air leads to higher growth rates and larger most unstable wave numbers than a single air stream. Swirl has a significant effect on the instability of the liquid sheet. It not only increases the maximum growth rate and the most unstable wave number, but also switches the dominant mode from the axisymmetric mode to a helical mode. A combination of inner and outer air swirl leads to better atomization than a single air swirl while the inner air swirl is more effective than the outer air swirl in enhancing the instability of the liquid sheet. Liquid swirl is far more effective than air swirl in promoting the breakup of the liquid sheet. In regard to the effect of swirl orientation on atomization, co-swirl leads to the highest growth rate, but outer coand inner counterswirl leads to the largest most unstable wave number thus finest spray. Inner and outer counter-swirl results in worst atomization quality. NOMENCLATURES A Vortex strength m/s h Ratio of inner and outer radii /„ n* order modified Bessel function of first kind Kn n^order modified Bessel function of second kind k Axial wave number i/m n Azimuthal wave number * Member, AIAA Copyright ©2001. American Institute of Aeronautics and Astronautics, Inc. All rights reserved. P Mean pressure N/m p Disturbance pressure N/m Ra Inner diameter of liquid sheet m Rh Outer diameter of liquid sheet m r Radial coordinate m t Time s U Mean axial velocity m/s V Mean radial velocity m/s W Mean tangential velocity m/s u Disturbance axial velocity m/s v Disturbance radial velocity m/s w Disturbance tangential velocity m/s We Weber number (We = pUR/a) X Axial coordinate m Greek letters T| Displacement disturbance m 9 Azimuthal angle radian p Fluid density kg/m n Angular velocity 1/s co Temporal frequency 1/s Subscripts / Inner gas / Liquid phase o Outer gas s Based on swirling component INTRODUCTION In the past years, extensive research efforts have been made to optimize fuel atomization process and combustor aerodynamics aiming at improving combustion efficiency and reduce pollutant emissions. It was realized that it is crucial to achieve rapid and uniform fuel-air mixing. As pressure-swirl airblast atomizer combines desirable attributes of simplex and airblast atomizers, it leads to remarkable improvements in combustor aerodynamic and fuel/air mixing. In addition, it has such advantages as larger flow turndown ratio, lower pollutant emissions, and better patternation. As such, it is widely used in aircraft engine combustors and power generation furnaces. Inside a pressure-swirl airblast atomizer as shown in Fig. 1, liquid fuel emanates from a pressureswirl or simplex atomizer in the form of a conical 1 American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. liquid sheet and impinges with the swirier cup. Due to strong shear action of the fast moving inner air, a thin annular liquid film is formed on the cup surface. The kinetic energy of the high-speed swirling air streams and the pressure energy of the liquid fuel act together to breakup the liquid sheet. As a result, pressure-swirl airblast atomizer has improved spray quality compared to simplex atomizer. However, our knowledge on its internal prefilming process and the subsequent breakup of the liquid sheet is very limited. Understanding factors that influence the instability and breakup of the liquid sheet will not only provide guidance for atomizer design but also improve spray model for engine combustion simulations. Pressure swirl atomizer Outer venturi Inner Swirler Outer Swirler Figure 1 Schematic of a pressure-swirl airblast atomizer It is well known that disintegration of liquid sheet is due to growth of unstable waves at the interface between the gas and the liquid sheet. Among all unstable waves, there is a so-called most unstable frequency (or wavelength) as illustrated in Fig. 2. This wave has the maximum growth rate and dominates the breakup of the liquid sheet. Its wavelength and growth rate are related to the mean size of droplets formed and the breakup length. Figure 2 A dominant frequency exists among unstable waves at liquid/air interface There are numerous studies" on the instability of two-dimensional liquid sheets. Flow conditions and liquid film geometry considered are nevertheless over simplified and can not be applied to atomization process in practical devices. These two-dimensional models were later extended to annular liquid sheets" 16 to simulate liquid breakup. However, flow conditions in these studies ~ were in the low speed regime and disturbances are assumed to be twodimensional. More importantly, swirl effect on the instability of the annular liquid sheet had not been considered. Theoretical, experimental, and computational efforts have been made to study internal flow field, liquid filming process, instability and breakup of liquid sheet, and resultant spray characteristics such as droplet size". Recently, we developed a theoretical model considering both inner and outer air swirl and found that air swirl has a destabilizing effect on the annular liquid sheet. Based on our previous model, in a more recent study, we developed a 3-D viscous instability model for airlast atomization. It was found that liquid viscosity not only reduces the growth rates of unstable waves, but also shifts the dispersion diagram towards long waves. However, the model does not consider the effect of liquid swirl. As such, the effectiveness of liquid and air swirl in enhancing the instability of the liquid sheet remains unclear. Furthermore, effects of the relative air swirl orientation with respect to liquid swirl are not understood. The objective of present paper is to develop a 3-D instability model for pressure-swirl airblast atomization by incorporating features such as three-dimensional disturbance, liquid swirl, inner and outer air swirl, and an annular geometry with a finite film thickness. The flow conditions and liquid film geometry considered here are very similar to their practical counterparts. Therefore, it is expected that this model is capable of providing insights into the underlying physics of airblast atomization and can be applied to predict drop sizes at the nozzle orifice. MATHEMATICAL MODEL We consider an inviscid swirling annular liquid sheet surrounded by coaxial swirling air streams as shown in Fig. 3. In the instability model, the gas phase is assumed to be inviscid and incompressible. Mean flows of the liquid, the inner and the outer air are assumed to be (£/,, 0, A/r), (Uit 0, Or} and (U0, 0, A/r), respectively. This combination of solid body rotation and free vortex profile is quite similar to the practical tangential velocity profile inside gas turbine combustors. Due to the swirl effect, to maintain an annular shape of the liquid surface, a constraint on the American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. mean pressure of the inner and outer air streams must be imposed by Eq. (1).