Traveling waves on vertical films: Numerical analysis using the finite element method

Abstract

Finite-amplitude waves propagating at constant speed down an inclined fluid layer are computed by finite element analysis of the Navier–Stokes equations written in a reference frame translating at the wave speed. The velocity and pressure fields, free-surface shape and wave speed are computed simultaneously as functions of the Reynolds number Re and the wave number μ. The finite element results are compared with predictions of long-wave, asymptotic theories and boundary-layer approximations for the form and nonlinear transitions of finite-amplitude waves that evolve from the flat film state. Comparisons between the finite element calculations and the long-wave predictions for fixed μ and increasing Re agree well for small-amplitude waves. However, for larger-amplitude waves the long-wave results diverge qualitatively from the finite element predictions; the long-wave theories predict limit points in the solution families that do not exist in the finite element solutions. Comparisons between the finite element predictions, previous numerical simulations and experimental results for the shape and speed of periodic and solitary-like waves are in good agreement. Nonlinear interactions are demonstrated between multiple waves in a periodic wave train that cause secondary bifurcations to families of waves that differ from those that evolve from the neutral stability curve. These predictions for fixed Re and decreasing μ are in quantitative agreement with the results of long-wave approximations for small-amplitude waves. Comparisons with the predictions of boundary-layer approximations show sensitivity of the solution structure to the value of the Weber number We.

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

“…Chang (1994) already evocated this condition speaking about constant-average thickness or constant-flux formulation. Many authors, as for instance (Joo et al 1991;Salamon et al 1994;Oron & Gottlieb 2002;Scheid et al 2002), implicitly prescribed the closed flow condition. This is due to the fact that in time-dependent simulations, using periodic boundary conditions, the amount of liquid leaving the domain downstream is reinjected upstream.…”

“…For instance, an increase of the phase speed leads to an underestimation of the Reynolds number q N < q ξ . Salamon et al (1994) computed travelling-wave solutions of the Navier-Stokes equations for a film falling on a vertical (S = 1) and isothermal (Ma = 0) wall varying Re at fixed We = 1000 and ε L = 0.04/2π. They imposed the closed flow condition (2.20).…”

“…The γ 1 and γ 2 families can bifurcate either in respective Hopf and period doubling bifurcations, or vice versa. Actually, the bifurcations of the families may be reversed if the dispersion of the waves is modified (Salamon et al 1994). The Kawahara equation that contains such dispersion terms has been studied by Chang et al (1993).…”

“…‡ Some authors we refer to (Salamon et al 1994;Oron & Gottlieb 2002) based the Reynolds number on the surface velocity rather than on the mean velocity, which yields R = 3Re/2. Therefore, Re = 2.0667 corresponds to R = 3.1 and Ka = 3375 to We = 1000 with hN = 1.8371. ingly one-hump and faster as k decreases.…”

Wave patterns in film flows: modelling and three-dimensional waves

“…Their characteristics are then compared to available DNS results (Salamon et al 1994;Ramaswamy et al 1996;Malamataris et al 2002) and laboratory experiments (Liu & Gollub 1994). For stationary waves, the mass conservation equation (3.1) can be integrated once to give…”