Reynolds-averaged equations for free-surface flows with application to high-Froude-number jet spreading

Abstract

The goals of this study were to develop a set of Reynolds-averaged governing equations for turbulent free-surface flow, and to use the resulting equations to determine the origin of the surface current in high-Froude-number jet flows. To develop the Reynolds-averaged equations, free-surface turbulent flow is treated as a two-fluid flow separated by an interface. It is shown that the general Navier–Stokes equations written for variable property flow embody the field equations applicable to each fluid, as well as the boundary conditions for the interface and, therefore, can be applied across the entire fluid domain, including the interface. With this as a starting point, a formulation of the Reynolds-averaged governing equations for turbulent free-surface flows can be developed rigorously. The resulting Reynolds-averaged equations are written in terms of density-weighted averages, their derivatives, and the probability density function for the free-surface position. These equations are similar to the conventional Reynolds-averaged equations, but include additional terms which represent the average effect of the forces acting instantaneously on the free surface, forces normally associated with the boundary conditions. These averaged equations are applied to the interaction of a turbulent jet with the free surface in order to establish, for arbitrary-Froude-number flows, the origin of the surface current, the large outward velocity which occurs in a thin layer adjacent to the surface. It is shown via an order-of-magnitude analysis that the outward acceleration associated with the surface current results from a combination of the Reynolds-stress anisotropy and the free-surface fluctuations. For low Froude number, the surface current is mainly driven by the Reynolds stress anisotropy, consistent with the results of Walker (1997); when the Froude number is large, the Reynolds-stress anisotropy is smaller and the free-surface fluctuations make a significant contribution.