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Wavy liquid films in interaction with a confined laminar gas flow

Published online by Cambridge University Press:  28 March 2013

Georg F. Dietze  and
Christian Ruyer-Quil
Georg F. Dietze*
Affiliation:
UPMC Universitaire Paris 06, Universitaire Paris-Sud, CNRS, Laboratoire FAST, Bâtiment 502, Campus Universitaire d’Orsay, Orsay, F-91405, France
Christian Ruyer-Quil
Affiliation:
UPMC Universitaire Paris 06, Universitaire Paris-Sud, CNRS, Laboratoire FAST, Bâtiment 502, Campus Universitaire d’Orsay, Orsay, F-91405, France
*
†Email address for correspondence: dietze@fast.u-psud.fr
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Abstract

A low-dimensional model capturing the fully coupled dynamics of a wavy liquid film in interaction with a strongly confined laminar gas flow is introduced. It is based on the weighted residual integral boundary layer approach of Ruyer-Quil & Manneville (Eur. Phys. J. B, vol. 15, 2000, pp. 357–369) and accounts for viscous diffusion up to second order in the film parameter. The model is applied to study two scenarios: a horizontal pressure-driven water film/air flow and a gravity-driven liquid film interacting with a co- or counter-current air flow. In the horizontal case, interfacial viscous drag is weak and interfacial waves are primarily driven by pressure variations induced by the velocity difference between the two layers. This produces an extremely thin interfacial shear layer which is pinched at the main and capillary wave humps, creating several elongated vortices in the wave-fixed reference frame. In the capillary wave region preceding a large wave hump, flow separation occurs in the liquid in the form of a vortex transcending the liquid–gas interface. For the gravity-driven film, a twin vortex (in the wave-fixed reference frame) linked to the occurrence of rolling waves has been identified. It consists of the known liquid-side vortex within the wave hump and a previously unknown counter-rotating gas-side vortex, which are connected by the same interfacial stagnation points. At large counter-current gas velocities, interfacial waves on the falling liquid film are amplified and cause flooding of the channel in a noise-driven scenario, while this can be delayed by forcing regular waves at the most amplified linear wave frequency. Our model is shown to exactly capture the long-wave linear stability threshold for the general case of two-phase channel flow. Further, for the two considered scenarios, it predicts growth rates and celerity of linear waves in convincing agreement with Orr–Sommerfeld calculations. Finally, model calculations of nonlinear interfacial waves are in good agreement with film thickness and velocity measurements as well as streamline patterns in both phases obtained from direct numerical simulations.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Nonlinear Dynamical Systems: Low-dimensional models Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 348 - 393
Copyright
©2013 Cambridge University Press

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