Energy growth of initial perturbations in two-dimensional gravitational jets

The potential for transient growth of small perturbations to plane liquid jets falling under gravity in a still gas atmosphere is studied extending concepts thus far exclusively applied to one phase shear flows. The relevant initial value problem in time is formulated keeping into account the effect of the liquid viscosity, besides those of surface tension and gravity. The numerical counterpart of the differential operator, obtained by means of a spectral discretisation, is examined focusing attention on both eigenvalues and pseudospectra with the aim of highlighting its non-normal character. The trend of an opportunely defined energy of perturbations is computed as a function of time in different flow regimes. The far field behaviour is always recovered according to the classical temporal modes analysis, whereas non-normality is shown to amplify perturbations energy over short time scales even in situations where the spectrum is entirely confined to the stable half-plane.