Modelling of inherently non-linear Kelvin-Helmholtz instability

The Kelvin-Helmholtz (KH) instability is one of the most elementary and widespread models of fluid dynamics. The model is successful in predicting several physical situations, although in its basic formulation it does not consider the finite thickness of the shear layer and the non-linear saturation of the exponential amplification of disturbances. However, there are flow configurations, such as those of fuel injection systems, where the basic KH linear model fails because the non linear effects play a major role in the process of transition from stratified to slug flow, and therefore the finite-amplitude of the instability wave has to be taken into account starting from the early instants. In the case of two-phase gas-liquid flow confined in a channel, which is the subject of the present contribution, the shear between parallel flow streams is produced by merging them from two separate channels divided by a rigid boundary, that leads to the formation of a mixing layer having a continuous velocity profile, hence introducing a finite length scale. Previous contributions of literature are due to Orazzo et al., for non-parallel channel shear flow, and to Hoepffner et al., for unconfined flows and disturbance wave produced by a localized impulse force. In both cases, the relevant characteristic of the inherently non linear instability is the emergence of a single travelling wave, whose characteristic velocity is different from the one of the classic linear KH theory. The present work aims to provide an in-depth modelling analysis of the essential features of the non linear single-wave, as well as to highlight the different features that distinguish the single-wave scenario from the classical train of KH linear waves. The model is based on the streamwise Bernoulli’s relationship imposed on both gas and liquid phases; contrary to the linear KH model, the suction velocity within the liquid bump (wave) under the action of the normal-to-flow pressure head, is obtained via the appropriate vertical Bernoulli’s equation, including gravity. The model takes into account also the pressure jump due to the surface tension. The condition of vanishing suction velocity yields the emergence onset of the single wave. The time amplification of the wave amplitude and its propagation velocity are predicted as well. Theoretical results are compared to direct numerical simulations of Navier-Stokes equations performed by means of the interFoam solver, included within the open-source package OpenFOAM, and widely validated for the flows of interest.