Session G22: Falling and Heated Films

Modelling a transitional falling liquid films

Liquid film plate exchangers are state-of-the-art technical solutions for the mass or heat transfer between a liquid and a gas whenever pressure drop in the gas phase is critical, for instance, evaporators. Operating conditions generally correspond to the wavy regime of the film with a Reynolds number in the range 100 to 1000, in which case turbulent spots may be observed at the front of the most prominent waves (see Adomeit et al. Int. J. Multiphase Flow (2000)). In this paper, we present a new model of falling liquid film accounting for the possible presence of turbulent spots. The proposed approach is based on the zero-equation model of turbulence (van Driest hypothesis) and the weighted residual technique. The obtained model is consistent up to second order with respect to the film parameter for diffusion terms. The model enables to reproduce convincingly most of the features of the roll waves observed by Brock (J.Hydraulics Div. 1970) at large Reynolds number. The influence of the turbulence-induced diffusion on the wave characteristics is discussed.   

Intensification of heat transfer across falling liquid films

The wavy motion of a liquid film is well known to intensify heat or mass transfers. Yet, if film thinning and wave merging are generally invoked, the physical mechanisms which enable this intensification are still unclear. We propose a systematic investigation of the impact of wavy motions on the heat transfer across 2D falling films on hot plates as a function of the inlet frequency and flow parameters. Computations over extended domains and for sufficient durations to achieve statistically established flows have been made possible by low-dimensional modeling and the development of a fast temporal solver based on graph optimizations. Heat transfer has been modeled using the weighted residual technique as a set of two evolution equations for the free-surface temperature and the wall heat flux. This new model solves the shortcomings of previous attempts, namely their inability to capture the onset of thermal boundary layers in large-amplitude waves and their limitation to low Prandtl numbers. Our study reveals that heat transfer is enhanced at the crests of the waves and that heat transfer intensification is maximum at the maximum of density of wave crests, which does not correspond to the natural wavy regime (no inlet forcing).   

Statistical Characteristics of Falling-Film Flows: A Synergistic Experimental-Computational Approach

We undertake an extensive statistical study of the hydrodynamics of gravity-driven falling film-flows based on carefully conducted experiments and advanced direct numerical simulations (DNSs). Specifically, we measure the instantaneous and local film-heights and velocity fields of harmonically excited falling-film flows for a wide range of flow conditions by simultaneous application of planar laser-induced fluorescence and particle tracking velocimetry. A Reynolds decomposition of the time-varying flow-rate into steady and unsteady terms seeks to provide novel statistical relations linking the film-height and bulk-velocity statistics. We observe that the covariance of the film-height and bulk-velocity fluctuations varies near-linearly with the product of the coefficients of variation of the film-height and bulk-velocity while the ratio between the Nusselt height and mean film-height varies linearly with both aforementioned quantities, decreasing as either of the two increases. We also show that the bulk-velocity statistics can be predicted using these relations together with information on the film-height statistics with only moderate errors ($\leq 5$\% for the DNSs and $\leq 10$\% for the experiments).   

Bifurcation analysis of liquid films flowing under an inclined plane

Consider a liquid film flowing under an inclined plane. This flow is analyzed using both a long-wave model and the Stokes equations for zero Reynolds number. The solution space is investigated by constructing bifurcation diagrams using a novel continuation method allowing for the computation of travelling waves, including solitary pulses and their bound states, as well as spatially varying time-periodic solutions. As the inclination angle decreases the amplitude and speed of pulses grow and become infinite at a critical value of the angle. Asymptotics for this limit are developed. The effect of an electric field is also considered, and it is found that it can be used to alter the dynamics from a chaotic regime to a regularized one described by interacting large-amplitude pulses with recirculation zones in the humps. A coherent-structure theory for such pulses is developed and good agreement is found with numerical simulations.\\ $[1]$ T.S. Lin, M. Pradas, S. Kalliadasis, D.T. Papageorgiou, D. Tseluiko, Coherent structures in nonlocal dispersive active-dissipative systems. SIAM J Appl Math 75, 538 (2015)\\ $[2]$ M.G. Blyth, D. Tseluiko, T.-S. Lin, S. Kalliadasis, Coherent-structure theory and the formation of bound states on electrified falling films. Submitted   

Three dimensional massively-parallel simulation of falling liquid films

We present results on the numerical study of falling liquid films using direct numerical simulations. Falling films due to their rich dynamics have been a subject of many interesting studies over the past decades. However, the majority of the research in the literature has focused only on the two-dimensional case due to the complexity of three-dimensional studies. In this work, we solve the full Navier-Stokes equations using a massively-parallelised numerical code – {\it Blue}. The code utilises a domain-decomposition strategy for parallelization with MPI, and an hybrid front-tracking/level set method is designed to handle the deforming interface. Parallel GMRES and Multigrid iterative solvers are then employed to appropriately handle the linear system arising from the implicit solution for the fluid velocities and pressure in the presence of strong density and viscosity discontinuities across the fluid phases. Our result show many interesting dynamics, which cannot be observed in the two-dimensional studies.   

Consistent three-equation model for thin films.

Numerical simulations of thin films of newtonian fluids down an inclined plane use reduced models for computational cost reasons. These models are usually derived by averaging over the fluid depth the physical equations of fluid mechanics with an asymptotic method in the long-wave limit. Two-equation models are based on the mass conservation equation and either on the momentum balance equation or on the work-energy theorem. We show that there is no two-equation model that is both consistent and theoretically coherent and that a third variable and a three-equation model are required to solve all theoretical contradictions. The linear and nonlinear properties of two and three-equation models are tested on various practical problems. We present a new consistent three-equation model with a simple mathematical structure which allows an easy and reliable numerical resolution. The numerical calculations agree fairly well with experimental measurements or with direct numerical resolutions for neutral stability curves, speed of kinematic waves and of solitary waves and depth profiles of wavy films. The model can also predict the flow reversal at the first capillary trough ahead of the main wave hump.   

Instability of thin liquid films in strongly confined channels

The flow of a falling liquid film in contact with a gas within a very narrow inclined channel may occur in several chemical engineering devices, e.g. within structured packings in distillation columns. Surface waves on the liquid film are known to greatly intensify inter-phase heat/mass transfer. It is also known that a counter-current gas flow may destabilize the non-linear surface waves, possibly leading to the flooding of the channel. Conversely, we show in our current study that the confinement can strongly stabilize the film when the gas velocity is quite low. In particular, we find that the critical Reynolds number can be increased by up to 30\% at moderate relative confinement. This effect depends on the inclination angle of the channel due to a competition between lubrication- versus acceleration-induced pressure variations in the gas. We show this by way of linear stability analysis based on the Orr-Sommerfeld equation as well as experimental comparisons. In addition, simulations with an integral boundary layer model and direct numerical simulations show that the opposing bounding wall causes a flattening of the crests of large-amplitude non-linear surface waves. This effect may be important in understanding the onset of flooding in strongly-confined geometries.   

Kelvin-Helmholtz instability of a thin liquid sheet: Effect of the gas-boundary layer

It is well known that when a thin liquid sheet moves with respect to a surrounding gas phase, the liquid sheet is susceptible to the Kelvin-Helmholtz instability. Here, flow in both the liquid and the gas phases are assumed to be inviscid. In this work, we include exactly via a perturbation analysis, the influence of the growing boundary layer in the gas phase in the base flow and show that both temporal and spatial growth rates obtained from the linear stability analysis are significantly reduced due to the presence of the boundary layer. These results are in line with the simulation results of Lozano et al [1] and Tammisola et al[2]. We conclude with the implication of these results on the break-up of radially expanding liquid sheets.\\ \\1. A. Lozano, F. Barreras, G. Hauke, and C. Dopazo, “Longitudinal instabilities in an air-blasted liquid sheet,” J. Fluid Mech. 437, 143 (2001).\newline 2. O. Tammisola, A. Sasaki, F. Lundell, M. Matsubara, and L. Söderberg, “Stabilizing effect of surrounding gas flow on a plane liquid sheet,” J. Fluid Mech. 672, 5 (2011).   

Instability of a liquid film non locally heated from below.

By invoking the long-wave approximation to study thin liquid films, one typically derives a single nonlinear PDE for the evolution of the local film thickness. Without advection, linear analysis of these equations predict that perturbations that grow/decay monotonically because the evolution equation is first order in time. If, however, the film evolution equation is coupled to a second process with its own characteristic time scale, it is common to encounter linear operators that are not self-adjoint, and therefore one must consider the possibility of oscillatory dynamics.this talk, present oscillatory regimes that arise for long-wavelength, thermocapillary destabilization of a liquid film that is heated from the bottom of a solid substrate. For thick substrates, the film evolution equation is nonlocally coupled to the full substrate heat equation, and linear analysis leads to a transcendental, implicit dispersion relation between the perturbation growth rate and wavenumber. Towards highlighting the underlying physical mechanisms, we present analytical results for various asymptotic limits of the input parameters. Finally, conditions that lead to oscillatory dynamics in real film-substrate systems will be predicted.   

Stability of film boiling on inclined plates and spheres

In film boiling, a continuous sub-millimeter vapor film forms between a liquid and a heated surface, insulating the two from each other. While quite accurate steady state solutions are readily obtained, the intermediate Reynolds numbers can make transient analysis challenging. The present work is a theoretical study of film boiling instabilities. We study the formation of travelling waves that are a combination of Kelvin--Helmholtz and the Rayleigh--Taylor instabilities. In particular, we study how the nature of this process depends on the Reynolds number, the Bond number, and the inclination of the submerged heated plate. In addition we extend the analysis to the case of a submerged heated sphere. Modelling of the transient dynamics of such films is important for answering practical questions such as how instabilities affect the overall heat transfer, and whether they can lead to complete film boiling collapse (Leidenfrost point).