Effects of liquid and air swirls on the three-dimensional instability of an annular viscous liquid sheet

A three-dimensional temporal linear instability model is proposed for an annular viscous liquid sheet moving between two inviscid gas streams. The model is numerically solved by Chebyshev collocation method. Effects of liquid swirl, inner and outer gas swirls, liquid viscosity, radius ratio, and density ratio on the liquid sheet instability are systematically investigated. The energy budget is calculated to provide insight into the instability driving mechanism. Results show that without gas swirl the liquid swirl has a dual effect on the sheet instability mainly through the axisymmetric mode. For a non-swirling liquid sheet, inner gas swirl induces the maximum wave growth rates of non-axisymmetric modes about 2–3 times higher than those induced by outer gas swirl, suggesting that inner gas swirl is more effective than outer one in promoting the sheet instability. For a swirling liquid sheet, simultaneous inner and outer gas swirls have the greatest destabilizing effect; outer gas swirl shows a slightly greater destabilizing effect than the inner one. Regarding the relative rotating direction, the combination of both co-rotating inner and outer gases is more favored to producing finer droplets. Liquid viscosity is found to have a evident stabilizing effect when Reynolds number is smaller than 500, it dampens the disturbance by dissipating the disturbance energy. Increasing radius ratio decreases the maximum wave growth rates of modes with n=0 and 5⩽n⩽7 but increases those of modes with 1⩽n⩽4. Maximum wave growth rate and optimal wave number are found to first decrease and then increase with increasing density ratio.