Noise-driven wave transitions on a vertically falling film

Abstract

We decipher the rich and complex two-dimensional wave transition and evolution dynamics on a falling film both theoretically and numerically. Small-amplitude white noise at the inlet is filtered by the classical linear instability into a narrow Gaussian band of primary frequency harmonics centred about ω_m. Weakly nonlinear zero-mode excitation and a secondary modulation instability then introduce a distinct characteristic modulation frequency Δ [Lt ] ω_m. The primary wave field evolves into trains of solitary pulses with an average wave period of 2π/ω_m. Abnormally large ‘excited’ pulses appear within this train at a relative frequency of Δ/ω_m due to the modulation. The excited pulses travel faster than the equilibrium ones and eliminate them via coalescence to coarsen the pulse field downstream. The linear coarsening of wave period downstream is a universal (0.015/〈u〉) s cm⁻¹ and the final wave frequency is the modulation frequency Δ for 0.1 < δ < 0.4 where 〈u〉 is the flat-film average Nusselt velocity, δ = (3R²/W)^1/3/15 is a normalized Reynolds number, R is the flat-film Reynolds number and W the Weber number.