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.