Global eigenmodes of thin liquid sheets by means of Volume-of-Fluid simulations

The unsteady dynamics of planar liquid sheet flows, interacting with unconfined gaseous environments located on both sides of the liquid phase, is numerically investigated by means of the Volume-Of-Fluid (VOF) technique for supercritical regimes. The global behavior of the non-parallel flow is analyzed by perturbing the initial steady configuration by means of a Gaussian bump in the transverse velocity component of relatively small amplitude, thereby exciting sinuous modes. To gain more physical insights on the fluid system, a theoretical linear one-dimensional model is also developed. A physical interpretation of this model relates the sheet dynamics to transverse vibrations of tensional string forced by terms containing the lateral velocity, and subjected to a total damping coefficient which can assume negative values. The VOF simulation satisfactorily confirms that the velocity impulse perturbation splits into two wave fronts traveling downstream with the theoretical wave velocities. A good agreement is found in comparing the crossing times over the entire domain length of such waves with the almost constant spacing between the frequencies of the eigenvalues spectrum. The surface tension plays a stabilizing role, and for relatively high values of density ratio (r/rho) of gaseous-to-liquid phases the sheet becomes unstable. It is argued that the distribution of transverse velocity component of the gaseous phase represents the forcing term which leads the system towards the instability when, for relatively high r/rho, the total damping becomes negative. An analogy seems to exist between the global unstable behavior exhibited by the liquid sheet as r/rho increases, and the shear-induced global instability found by Tammisola et al. (JFM, 2012) in the presence of surface tension. However, for the gravitational sheet the surface tension is stabilizing.