Linear instability analysis of an inviscid coaxial swirling jet is carried out by deriving an analytical dispersion relation of perturbation growth. The azimuthal Rankine vortex and the axial discontinuous velocity distribution are utilized as the jet basic flow. Due to the existence of double interfaces, the instability mechanisms of the coaxial swirling jet are much more complex than those of the single-layered swirling jet. The effects of control parameters (including the swirling ratio, the Weber number, the jet radius ratio, the velocity ratios between different fluids, and the azimuthal velocity jump at the inner interface) on the temporal instability of coaxial swirling jet with different azimuthal modes are studied. By comparing the growth rate of different azimuthal modes, the predominant mode that determines the jet breakup is identified. The results indicate that an increase in the swirling ratio, the Weber number, and the radius ratio can lead to predominant mode transition to larger azimuthal wavenumbers. The velocity ratio between the inner jet and the annular jet and that between the surrounding fluid and the annular jet mainly affect the axial Kelvin–Helmholtz (KH) instability. An enhancement of the KH instability leads to the jet breakup with smaller azimuthal wavenumbers. The azimuthal velocity jump affects the azimuthal KH instability, the centrifugal instability, and the Coriolis instability simultaneously, thus leading to a multiple influence on modes transition. The phase-diagram of the predominant modes is further given, showing that the relative importance between the centrifugal force and the interfacial tension plays a significant role on the transition of predominant modes.
