Full numerical simulations of the Navier–Stokes equations for four cases of vertically falling liquid films with three-dimensional surface waves have been performed. Flow conditions are based on several previous experimental studies where the streamwise and spanwise wavelengths were imposed, which we exploit by simulating periodic wave segments. The considered flows are laminar but approach conditions at which intermittent wave-induced turbulence has been observed elsewhere. Working liquids range from water to silicone oil and cover a large interval of the Kapitza number ( ), which relates capillary to viscous forces. Simulations were performed on a supercomputer, using a finite-volume code and the volume of fluid and continuum surface force methods to account for the multiphase nature of the flow. Our results show that surface waves, consisting of large horseshoe-shaped wave humps concentrating most of the liquid and preceded by capillary ripples on a thin residual film, segregate the flow field into two regions: an inertia-dominated one in the large humps, where the local Reynolds number is up to five times larger than its mean value, and a visco-capillary region, where capillary and/or viscous forces dominate. In the inertial region, an intricate structure of different-scale vortices arises, which is more complicated than film thickness variations there suggest. Conversely, the flow in the visco-capillary region of large- fluids is entirely governed by the local free-surface curvature through the action of capillary forces, which impose the pressure distribution in the liquid film. This results in flow separation zones underneath the capillary troughs and a spanwise cellular flow pattern in the region of capillary wave interference. In some cases, capillary waves bridge the large horseshoe humps in the spanwise direction, coupling the two aforementioned regions and leading the flow to oscillate between three- and two-dimensional wave patterns. This persists over long times, as we show by simulations with the low-dimensional model of Scheid et al. (J. Fluid Mech., vol. 562, 2006, pp. 183–222) after satisfactory comparison with our direct simulations at short times. The governing mechanism is connected to the bridging capillary waves, which drain liquid from the horseshoe humps, decreasing their amplitude and wave speed and causing them to retract in the streamwise direction. Overall, it is observed that spanwise flow structures (not accounted for in two-dimensional investigations) are particularly complex due to the absence of gravity in this direction.

Three-dimensional flow structures in laminar falling liquid films

Interfacial waves on the surface of a falling liquid film are known to modify heat and mass transfer. Under non-isothermal conditions, the wave topology is strongly influenced by the presence of thermocapillary (Marangoni) forces at the interface which leads to a destabilization of the film flow and potentially to critical film thinning. In this context, the present study investigates the evolution of the surface topology and the evolution of the surface temperature for the case of regularly excited solitary-type waves on a falling liquid film under the influence of a wall-side heat flux. Combining film thickness (chromatic confocal imaging) and surface temperature information (infrared thermography), interactions between hydrodynamics and thermocapillary forces are revealed. These include the formation of rivulets, film thinning and wave number doubling in spanwise direction. Distinct thermal structures on the films’ surface can be associated to characteristics of the surface topology.

Experimental investigation of thermal structures in regular three-dimensional falling films

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 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 the present 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 quasiconstant 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.

Wave characteristics of falling liquid films under the influence of positive and negative inclination or electrostatic forces

Here a review is presented about the stability problem of thin liquid films flowing down heated walls. The period reviewed starts from 2013, after the publication of the extensive review paper by Dávalos-Orozco [Interfacial Phenom. Heat Transfer, vol. 1, pp. 93–138 (2013)]. Emphasis is given to flows where the film is affected by thermocapillarity under different mechanical and thermal boundary conditions. The subjects under discussion include thin films flowing down heated flat and cylindrical walls. Besides, the results on non-uniformly heated films are considered for flows under three different boundary situations: (1) temperature gradient along the wall, (2) local heating (hot plates), and (3) wall topography. Finally, pure experimental research papers are also taken into account in this review. The formation and separation of rivulets are examined along with their local increase of maximum thickness with distance down a hot plate.

Thin liquid films flowing down heated walls: a review of recent results

Arrays of impinging jets can cool large areas with good thermal uniformity and are often used in industrial processes, such as drying and electronics cooling. However, due to cross-flow of the spent liquid and interference between adjacent jets, a significant amount of the available cooling performance is lost. Under free-surface jet impingement the area beyond the hydraulic jump is associated with significantly reduced heat transfer, and locally increased temperatures, therefore the hydrodynamics in this area must be better understood. Specifically, the interaction of jets in the vicinity of this location is expected to shed light on improving multi-jet array cooling uniformity and performance. Beyond this, it has been shown that micro-scale (sub-millimeter) jets tend to behave somewhat differently from larger jets, due to the increased significance of surface tension, pumping noise and edge-effects (such as small recirculation zones, and jet widening due to contact-angle at the nozzle exit, noise and nozzle imperfections due to manufacturing). These become much more dominant at the micro-scale. These effects cannot usually be accounted for by traditional scaling laws or numerical simulations, and are preferably investigated experimentally. Moreover, at these scales a micro-machined fixed-geometry array of jets is typically used, leaving no possibility for geometric variation, optimization and limited observation.

Influence of micro-scale aspects and jet-to-jet interaction on free-surface liquid jet impingement for micro-jet array cooling

Three-dimensional interfacial waves that develop on the free surface of falling liquid films are known to intensify heat and mass transfer. In this context, the present paper studies the effect of electrostatic forces applied to a falling film of dielectric liquid on its three-dimensional nonlinear wave dynamics. Therefore, measurements of the local film thickness using a confocal chromatic imaging method were taken, and the complex wave topology was characterized through photography. The experiments show a complex interaction between the electric field and the hydrodynamics of the falling film, whereby electrostatic forces were found to both increase and decrease wave peak height in different regions of the wave. Additionally, an electrically induced breakup of the three-dimensional wave fronts, which leads to a locally doubled frequency in streamwise direction, is found. The ability to influence the wave topology demonstrated here opens the possibility to optimize heat transfer processes in falling liquid films.

Oleg Yu. Tsvelodub

Papers

Experimental investigation into three-dimensional wavy liquid films under the influence of electrostatic forces