Free-surface turbulent flows: Governing equations and surface elevation measurements

The goals of this study were to develop a set of Reynolds-averaged governing equations for turbulent free-surface flows, to characterize the surface fluctuations in free-surface jets, and to determine the origin of jet spreading in high-Froude-number jet flows. To develop the Reynolds-averaged equations, a free-surface turbulent flow was treated as a two-fluid flow separated by an interface. It was shown that the general Navier-Stokes equations written for variable property flows 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. From there, a formulation of the Reynolds-averaged governing equations for turbulent free-surface flows was 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. They are similar to the conventional Reynolds-averaged equations, but include additional terms representing the average effect of the forces acting instantaneously on the free surface. The turbulent kinetic energy equation was also derived in a similar way. Comparisons to the terms in the instantaneous free surface kinematic boundary condition were made to identify terms representing the energy transfer between sub-surface turbulence and surface waves. Measurements of the surface elevation for free-surface turbulent jet flows at two Froude numbers were presented and discussed. The r.m.s. surface fluctuations in the low-Froude-number flow are an order-of-magnitude smaller than that in the high-Froude-number flow. For the high-Froude-number flow the probability density functions for surface positions are nearly Gaussian. The measured apparent-frequency spectra are consistent with the Doppler effect caused by the velocity field. The averaged equations were applied to the free-surface turbulent jet flows in order to establish, for arbitrary-Froude-number flows, the origin of the jet spreading, the large outward velocity which occurs in a thin layer adjacent to the surface. It was shown via an order-of-magnitude analysis that the outward acceleration associated with the jet spreading results from a combination of the Reynolds-stress anisotropy and the free-surface fluctuations.