The instability of a liquid jet in a compressible airstream

The instability of a liquid jet in a coflowing compressible airstream has been studied, motivated by the experimentally observed surface wave phenomenon and breakup of a liquid jet injected transversely into a compressible gas flow. The linear stability analysis of a two-dimensional plane liquid jet in a compressible airstream is formulated first to correlate the instability characteristics with the compressibility of the airstream. The small-amplitude disturbances superimposed on the mean flow inside and outside the jet are governed by Laplace's equation and the convective wave equation, respectively, along with suitable boundary conditions at the interfaces. The conditions for a valid linearization of the convective wave equation are obtained. It is established that the transient motion of the gas can dominate so that the nonlinear term due to accumulation of disturbances in the basic flow can be negligible even as the Mach number approaches unity. The dispersion relation has been developed and studied for both temporally and spatially growing modes. A physical understanding of how compressibility affects the growth rate has been made clear by applying a 1D compressible continuity equation at the interface region and studying the convection and accumulation of vorticity. Furthermore, the kinetic energy of the disturbed liquid jet is examined to see how surface tension and airstream pressure transfer energy between the liquid jet and the compressible airstream. A similar linear study using a circular liquid jet model is then formulated in order to further investigate the behavior of the axisymmetric and non-axisymmetric unstable modes. The empirically obtained sonic point criterion for predicting jet breakup has been partially explained by studying both the maximum growth rates and the transition from convective instability to absolute instability near M approx. 1 as obtained through these linear instability analyses. Furthermore, numerical simulations of the interfacial instability of a 2D plane liquid jet and of an axisymmetrical circular liquid jet in a compressible airstream are performed to investigate the nonlinear aspects of these problems. A high resolution scheme which has second order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with the instability characteristics of the jet flow established by the linear theory. The process of the jet disruption in compressible flow involves the formation of spikes and the stretching and detachment of the liquid main center core.