A combined VOF-RANS approach for studying the evolution of incipient two-dimensional wind-driven waves over a viscous liquid

Abstract

Recent laboratory experiments have revealed that important insights into the physical processes involved in the wind-driven generation of surface waves may be obtained by varying the viscosity of the carrying liquid over several orders of magnitude. The present paper reports on the development of a companion approach aimed at studying similar phenomena through numerical simulation, a way expected to remove some of the experimental limitations, especially in the near-interface region, and to allow the relative influence of several physical processes to be assessed by disregarding or inactivating arbitrarily some of them. After reviewing available options, we select and approach based on the combination of a volume of fluid technique to track the evolution of the air–liquid interface, and a two-dimensional Reynolds-averaged version of the Navier–Stokes equations supplemented with a turbulence model to predict the velocity and pressure fields in both fluids. We examine the formal and physical frameworks in which such a time-dependent two-dimensional formulation is meaningful, and close the governing momentum equations with the one-equation Spalart–Allmaras model which directly solves a transport equation for the eddy viscosity. For this purpose, we assume the interface to behave as a rigid wall with respect to turbulent fluctuations in the air, and implement a versatile algorithm to compute the local distance to the interface whatever its shape. We first assess the performance of this model in unseparated and separated single-phase flows over a wavy rigid wall, which are of specific relevance with respect to wind-wave generation. Then, we discuss the initialization protocol used in two-phase simulations, which involves an impulse disturbance with a white noise distribution applied to the interface position. We finally present some examples of interface evolutions obtained at several wind speeds with liquids of various viscosities, and discuss the underlying physics revealed by the associated statistics of interface disturbances, streamline patterns and energy spectra.