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We present a comprehensive study of the dispersion of capillary waves with finite amplitude, based on direct numerical simulations. The presented results show an increase of viscous attenuation and, consequently, a smaller frequency of capillary waves with increasing initial wave amplitude. Interestingly, however, the critical wavenumber as well as the wavenumber at which the maximum frequency is observed remain the same for a given two-phase system, irrespective of the wave amplitude. By devising an empirical correlation that describes the effect of the wave amplitude on the viscous attenuation, the dispersion of capillary waves with finite initial amplitude is shown to be, in very good approximation, self-similar throughout the entire underdamped regime and independent of the fluid properties. The results also shown that analytical solutions for capillary waves with infinitesimal amplitude are applicable with reasonable accuracy for capillary waves with moderate amplitude. [ABSTRACT FROM AUTHOR]

The propagation of capillary waves on material interfaces between two fluids imposes a strict constraint on the numerical time-step applied to solve the equations governing this problem and is directly associated with the stability of interfacial flow simulations. The explicit implementation of surface tension is the generally accepted reason for the restrictions on the temporal resolution caused by capillary waves. In this article, a fully-coupled numerical framework with an implicit treatment of surface tension is proposed and applied, demonstrating that the capillary time-step constraint is in fact a constraint imposed by the temporal sampling of capillary waves, irrespective of the type of implementation. The presented results show that the capillary time-step constraint can be exceeded by several orders of magnitude, with the explicit as well as the implicit treatment of surface tension, if capillary waves are absent. Furthermore, a revised capillary time-step constraint is derived by studying the temporal resolution of capillary waves based on numerical stability and signal processing theory, including the Doppler shift caused by an underlying fluid motion. The revised capillary time-step constraint assures a robust, aliasing-free result, as demonstrated by representative numerical experiments, and is in the static case less restrictive than previously proposed time-step limits associated with capillary waves. [ABSTRACT FROM AUTHOR]

We derive a general integro-differential equation for the transient behavior of small-amplitude capillary waves on the planar surface of a viscous fluid in the presence of the Marangoni effect. The equation is solved for an insoluble surfactant solution in concentration below the critical micelle concentration undergoing convective-diffusive surface transport. The special case of a diffusion-driven surfactant is considered near the the critical damping wavelength. The Marangoni effect is shown to contribute to the overall damping mechanism, and a first-order term correction to the critical wavelength with respect to the surfactant concentration difference and the Schmidt number is proposed. [ABSTRACT FROM AUTHOR]

This work presents a detailed study of the dispersion of capillary waves with small amplitude in viscous fluids using an analytically derived solution to the initial value problem of a small-amplitude capillary wave as well as direct numerical simulation. A rational parametrization for the dispersion of capillary waves in the underdamped regime is proposed, including predictions for the wave number of critical damping based on a harmonic-oscillator model. The scaling resulting from this parametrization leads to a self-similar solution of the frequency dispersion of capillary waves that covers the entire underdamped regime, which allows an accurate evaluation of the frequency at a given wave number, irrespective of the fluid properties. This similarity also reveals characteristic features of capillary waves, for instance that critical damping occurs when the characteristic time scales of dispersive and dissipative mechanisms are balanced. In addition, the presented results suggest that the widely adopted hydrodynamic theory for damped capillary waves does not accurately predict the dispersion when viscous damping is significant, and an alternative definition of the damping rate, which provides consistent accuracy in the underdamped regime, is presented. [ABSTRACT FROM AUTHOR]