Nonlinear evolution of waves on falling films at high Reynolds numbers

Abstract

A new model describing the dynamics of large-amplitude waves on laminar falling wavy films at high Reynolds numbers (Re≳300) is presented. The model is based on second-order boundary layer theory and includes the pressure variation across the film as well as higher-order viscous terms. The consistency and accuracy of the model is verified by comparing the linear stability results with Kapitza’s classical boundary layer model and Orr–Sommerfeld studies of the two-dimensional Navier–Stokes equations. Numerical integration of a traveling wave simplification of the model predicts the existence of chaotic large-amplitude, nonperiodic waves, as observed in the experiments. The computed wave statistics such as wave celerities, root-mean-square (RMS) values of film thickness, probability density function (PDF), and film thickness power spectrum using the present model are in reasonable agreement with those measured on naturally excited fully developed flows at Re≳300. The present model also overcomes the main deficiency of the classical boundary layer model (namely, negative wall shear stress) predicts large-amplitude waves (with peak to substrate ratios of 3 to 4) and gives better agreement with data.

Flow structure underneath the large amplitude waves of a vertically falling film

“…Miyara 10 also computed a circulation of flow in a moving coordinate system for smaller Reynolds numbers (Re 5 100). Many other studies 4,6,11,12 have also shown this region of circulation underneath the hump of a large amplitude wave when viewed in a coordinate system moving with the wave.…”

Modeling film flows down inclined planes

“…4 and 5). We follow the same general strategy as Yu et al [9] and use BL equations as a starting point, which has the interest of focusing on the appropriate long wavelength properties of the flow right from the beginning. We also use polynomials to expand the velocity field but, to stay closer to the physics of the problem, instead of choosing some general systematic expressions of increasing degree, we prefer to take the specific polynomials that appear in Benney's gradient expansion and to introduce combinations of coefficients of the lowest order terms that may be given an immediate physical interpretation.…”

“…Better approximations of the flow have to be developed in order to get more realistic results. This requisite has lead to the derivation of improved models by weighted residual methods [9,10], by expanding the hydrodynamic fields on a functional basis of the cross-stream variable y, finding relations between the coefficients of the (truncated) expansion from the NS or BL equations by some specific projection rule, and further applying the resulting set of equations to concrete problems such as the structure of solitary waves. In the absence of clear physical meaning for the coefficients appearing in the expansion, the interpretation of such studies is not straightforward and one is often confined to a comparison of the obtained output with that of concurrent models and numerical solutions of BL or NS equations, or with the results of laboratory experiments.…”

“…Starting from this long-wave expansion, a certain number of models have then been derived since the pioneering work of Kapitza [7], e.g. [8][9][10] for the most recent ones, see the review by Demekhin et al [11] for earlier attempts. The simplest useful result obtained in this way is a partial differential equation called Benney's equation [12] to be written explicitly later (36).…”

Inertial effects on thin-film wave structures with imposed surface shear on an inclined plane

“…It should also be noted that Malamataris et al comment on a parabolic profile, which is the exact solution of uniform film flow, as making the problem analytically tractable. Studies by Prokopiou et al [16] and Yu et al [25] extend the Shkadov model in order to allow for higher order description of film profiles relevant at higher Reynolds numbers. Referred to as the second-order boundary layer model, this retains terms of O(" 2 ), and hence the modified model includes additional viscous terms, tangential and normal stress conditions and pressure variations across the film.…”