Direct numerical simulation of gravity-capillary waves breaking with surfactant

Surfactant at fluid interfaces plays a crucial role in fluid dynamics by modifying surface tension. This study investigates its effect on waves by integrating an insoluble surfactant transport equation at the interface and incorporating Marangoni forces into an incompressible two-phase Navier–Stokes framework, using the Basilisk solver. Surface tension is modeled via a nonlinear equation of state, following previous experimental observations. We conduct simulations of two-dimensional gravity-capillary waves and analyze the impacts of control parameters on surface roughness, wave regime transitions, and energy budget. Our results reveal a non-monotonic relationship between surfactant concentration and wave roughness: roughness initially decreases and then increases with surfactant concentration as parasitic capillary waves are either suppressed or enhanced, while the energy budget displays the opposite trend. The presence of surfactants also induces earlier transitions between wave regimes. We further quantify the non-monotonic behavior in roughness and energy by the effective Marangoni number based on surface tension gradient, while transitions between wave regimes are governed by the effective Bond number.