Direct Numerical Simulation of a Shallow Free-surface Turbulent Flow םver a Rough Bed

A lot of studies on ocean waves conducted in last few decades give many findings such as their statistical properties and spectrum characteristics. On the other hand, there are few studies regarding free-surface patterns in turbulent open-channel flows and the free-surface feature is not fully understood yet. In actual rivers, the free-surface patterns are caused by various factors, e.g. sub-surface turbulence, rough bed or wind. Further understanding of the relation between the free-surface patterns and the factors would contribute to the development of remote measurement techniques for actual river flows.

This paper provides results from direct numerical simulations of a shallow free-surface turbulent flow over a rough bed. The flow considered in this study is spatially periodic and uniform. The rough bed is composed of spherical roughness elements that are packed in a staggered arrangement. The free surface is captured by a single-phase volume-of-fluid method that needs dynamic boundary conditions at the free surface to neglect air motion. The spheres as the roughness elements are represented by an immersed boundary method in an orthogonal coordinate system. The frictional Reynolds number is 400, corresponding to the bulk Reynolds number of 2400, and the Froude number ranges from 0.3 to 0.9. The resulting roughness Reynolds number is approximately 120 in which the flow is in a fully-rough regime. To ensure that typical turbulence properties are resolved, grid convergence test is also conducted with different grid resolutions.

The free-surface patterns obtained by the simulations has a similar feature to that observed in a laboratory experiment. The wavenumber–frequency spectra of the free-surface show two interesting features related to turbulence and waves, indicating that the free surface is advected downstream with the sub-surface turbulence while it propagates to the surroundings as waves. The energy decomposition of the free-surface indicates that the energy associated with the waves propagating downstream becomes more dominant as the Froude number increases. Furthermore, although the distributions of the directional wave spectrum mainly spread in the downstream direction, some peaks appear also in the upstream direction at the low-Froude number.