Falling liquid films are thin liquid layers flowing down rigid vertical or inclined walls driven by gravity. In industry falling films are used in various applications such as refrigeration, cooling of mechanical or electronic systems, chemical processing, desalination and food processing. They generally exhibit higher heat and mass transfer rates compared to flows which are strictly aligned to the rigid wall. This increase in heat and mass transfer is caused by the wavy topology, which is attended by different types of vortices in the liquid and the gaseous phase. The characteristic vortices in the liquid phase are a circulating eddy in the main wave hump (also called circulating wave) and a vortex in the first capillary minimum associated with the phenomenon of flow reversal. The existence and the physical mechanism involved in those vortices are well-known. However, the flow conditions under which they occur have still been unclear.
In addition to films flowing down the upper side of an inclined wall, films flowing down the underside of an inclined plate have been considered in recent studies. In the latter case, the gravitational force acts destabilizing and, owing to the negative value of the inclination number, this condition is called “negative gravity”. Besides the two vortices which can occur in both configurations, a further phenomenon that occurs only in hanging film flows is dripping.
In my doctoral thesis, two model approaches have been utilized, namely the weighted integral boundary layer (WIBL) model and direct numerical simulations (DNS) in order to identify the conditions for the onset of circulating waves, flow reversal, and gravitational dripping. For investigating electrostatically induced spraying of a dielectric fluid, the WIBL model and the numerical code of the DNS have been extended by the electrostatic surface force. From the modeling point of view, the developed equations of the WIBL model revealed that the electrostatic force is similar to the gravitational body force with an additional non-linear contribution.
From the knowledge obtained by the simulations, analytical criteria for the onset of circulating waves and flow reversal based on the wave celerity, the average film thickness and the maximum and minimum film thickness have been approximated using self-similar parabolic velocity profiles. This approximation has been validated by second-order WIBL and direct numerical simulations. It is shown that the onset of circulating waves in the phase diagram for homoclinic solutions (waves of infinite wavelength) is strongly dependent on the inclination, but independent of the streamwise viscous dissipation effect. On the contrary, the onset of flow reversal shows a clear dependence on the viscous dissipation. Furthermore, simulation results for limit cycles (finite wavelength) reveal a strong increase of the corresponding critical Reynolds number with the excitation frequency.
The phenomenon of gravitational dripping is associated with a pressure gradient in crosswise direction. A criterion for the onset of dripping is derived based on a force balance between gravitational forces of the liquid in the main wave hump and surface tension. This force balance is found to be violated in the simulation results of the WIBL model. The model does not account for a crosswise force balance, owing to the integration of the streamwise boundary layer momentum equation across the depth of the film. As a consequence, the model is found to be not well-suited for the prediction of dripping. However, DNS and WIBL results agree well before dripping occurs.
The influence of the electrostatic surface force is found to be well-captured by the WIBL model in a specific range of parameters. Due to a quasi-constant pressure in crosswise direction of the film, the physical mechanisms of electrostatically induced spraying differ from the mechanisms involved in dripping. An explanation for the occurrence of spraying is the non-linear growth of the force with an increase in film thickness. Above a critical value of the electric potential, the increase in the destabilizing electrostatic pressure exceeds the increase of the stabilizing surface tension force. Consequently, small perturbations grow infinitely up to spraying, which finally ruptures the film surface.
Benoit Scheid – Université Libre de Bruxelles (Belgium)
Georg Dietze – Laboratoire FAST, Orsay Cedex (France)
Oleg Tsvelodub – Novosibirsk State University (Russia)
Herman Haustein – Tel Aviv University (Israel)
Rietz, M., Scheid, B., Gallaire, F., Kofman, N., Kneer, R., Rohlfs, W.: Dynamics of falling films on the outside of a vertical rotating cylinder: waves, rivulets and dripping transitions. J. of Fluid Mechanics, 832, 180-211. (Providing front cover image of JFM Volume 832)
Rohlfs, W., Pischke, P., Scheid, B.: Hydrodynamic waves in films flowing under an inclined plane, Physical Review Fluids, 2, 4, 044003, 2017.
Scheid, B., Kofman, N., Rohlfs, W.: Critical inclination for Absolute/Convective instability transition in inverted falling films, Physics of Fluids, 28, 044107, 2016.
Seel, K., Reddemann, M., Verbaere, V. Mathieu. F., Cárdenas, M., Rohlfs, W., Haustein, H. D., Kneer, R.: A spin coating device for the investigation of spray-film interactions under engine relevant conditions. Atomization and Sprays, 26, 11, 1111-1125, 2016.
Rohlfs, W., Scheid, B.: Phase diagram for the onset of circulating waves and flow reversal in inclined falling films, J. of Fluid Mechanics, 763, 322-351, 2015.
Rietz, M., Rohlfs, W., Kneer, R., Scheid, B.: Experimental investigation of thermal structures in regular three-dimensional falling films, European Physics J. Special Topic, 224, 355-368, 2015.
Dietze, G. F., Rohlfs, W., Nährich, K., Kneer, R., Scheid, B.: Three-dimensional flow structures in laminar falling liquid films. J. of Fluid Mechanics, 743, 75-123, 2014.
Tsvelodub, O., Rohlfs, W., Kneer, R.: Modeling of wave modes on a vertical film of a viscous ferromagnetic fluid flowing down a cylindrical electric conductor, Physics of Fluids, 25, 092101, 2013.
Rohlfs, W., Dietze, G., F., Haustein, D., Kneer, R.: Experimental investigation of 3-dimensional wavy liquid films under the coupled influence of thermo-capillary and electrostatic forces European Physics J. Special Topic, 219, 111-119, 2013.
Rohlfs, W., Dietze, G. F., Haustein, H. D., Tsvelodub, O., Kneer, R.: Experimental investigation of 3-dimensional wavy liquid films under the influence of electrostatic forces. Experiments in Fluids, 53 (4), 1045-1056, 2012.
Rohlfs, W., Haustein, H. D., Dietze, G. F., Kneer, R.: Experimental and numerical investigation of falling liquid films influenced by the presence of electric fields, Proceedings of the 6th International Berlin Workshop – IBW6 on Transport Phenomena with Moving Boundaries, November 24th-25th, 2011, Berlin, Germany, Published VDI Fortschritt-Bericht VDI Reihe 3, Verfahrenstechnik 929, Düsseldorf VDI-Verlag, 2012.
Rohlfs, W., Kneer, R., Scheid, B.: Influence of gravity and electrostatic surface forces on the onset of rolling waves and flow reversal in falling liquid films, Proceedings of the 7th International Berlin Workshop – IBW7 on Transport Phenomena with Moving Boundaries, October 30th-31st, 2014, Berlin, Germany, Published VDI Fortschritt-Bericht VDI Reihe 3, Verfahrenstechnik 947, Düsseldorf VDI-Verlag, 2015.