Isothermal Case: Three-Dimensional Flow

Abstract

The three-dimensional wave regime on film flows down an inclined plane has a very rich phenomenology. In particular, isolated synchronous depressions, rugged-modulated waves as well as horseshoe-like three-dimensional solitary waves and oblique solitary waves are observable. This phenomenology is far from being fully understood. However, recent well-controlled experiments in the inclined plane geometry by Gollub and coworkers have enabled us to understand the transition from two-dimensional to three-dimensional flows in terms of two different secondary instability mechanisms, leading to in-phase span-wise/synchronous modulations or to herringbone patterns, and ultimately, to modulated or horseshoe-like solitary waves. To scrutinize the transition from two-dimensional to three-dimensional flows, the two-dimensional averaged models derived in Chap. 6 have been extended to include the span-wise dependence of the flow. The stability of two-dimensional periodic waves based on a Floquet analysis shows that the secondary instability is not selective, which makes the resulting three-dimensional instability strongly dependent on the initial conditions. Provided that initial conditions are appropriately tuned, the experimental results reported by Gollub and coworkers are recovered by numerical simulations. The widespread observation of the synchronous instability in the above experiments is likely to be related to the span-wise nonuniformities at the inlet, favoring in-phase modulations of the wave fronts. In some cases, the three-dimensional patterns emerge from a two-dimensional oscillatory mode rather than from saturated traveling waves, as also observed in DNS. The competition between the growing two-dimensional modulation and the secondary three-dimensional instability makes the evolution of the film even more sensitive to initial conditions. The agreement of the simulations of the three-dimensional low-dimensional models obtained from the weighted residuals methods, to the available experimental data is encouraging. The regularized and the full second-order model are able to recover the synchronous scenario of transition from two-dimensional to three-dimensional wave patterns observed in the experiments by Gollub and coworkers, whereas simulations based on the simplified model systematically show a subharmonic transition scenario (herringbone pattern).