Analytical Eddy Viscosity Model for Turbulent Wave Boundary Layers: Application to Suspended Sediment Concentrations over Wave Ripples

Abstract

Turbulence related to flow oscillations near the seabed, in the wave bottom boundary layer (WBBL), is the phenomenon responsible for the suspension and transport of sediments. The vertical distribution of turbulent eddy viscosity within the WBBL is a key parameter that determines the vertical distribution of suspended sediments. For practical coastal engineering applications, the most used method to parameterize turbulence consists in specifying the shape of the one-dimensional-vertical (1DV) profile of eddy viscosity. Different empirical models have been proposed for the vertical variation of eddy viscosity in the WBBL. In this study, we consider the exponential-type profile, which was validated and calibrated by direct numerical simulation (DNS) and experimental data for turbulent channel and open-channel flows, respectively. This model is generalized to the WBBL, and the period-averaged eddy viscosity is calibrated by a two-equation baseline (BSL) k-ω model for different conditions. This model, together with a β-function (where β is the inverse of the turbulent Schmidt number), is used in modeling suspended sediment concentration (SSC) profiles over wave ripples, where field and laboratory measurements of SSC show two kinds of concentration profiles depending on grain particles size. Our study shows that the convection–diffusion equation, for SSC in WBBLs over sand ripples with an upward convection term, reverts to the classical advection–diffusion equation (ADE) with an “apparent” sediment diffusivity εs*=α εs related to the sediment diffusivity εs by an additional parameter α associated with the convective sediment entrainment process over sand ripples, which is defined by two equations. In the first, α depends on the relative importance of upward convection related to coherent vortex shedding and downward settling of sediments. When the convective transfer is very small, above low-steepness ripples, α≈1. In the second, α depends on the relative importance of coherent vortex shedding and random turbulence. When random turbulence is more important than coherent vortex shedding, α≈1, and “apparent” sediment diffusivity reverts to the classical sediment diffusivity εs*≈ εs. Comparisons with experimental data show that the proposed method allows a good description of both SSC for fine and coarse sand and “apparent” sediment diffusivity εs*.

Keywords

• analytical model
• eddy viscosity
• turbulence
• wave bottom boundary layer (WBBL)
• suspended sediments
• concentrations