Effect of Eddy Viscosity and Turbulent Schmidt Number on Suspended Sediment Concentration Profiles: interest of the Fractional Advection-Diffusion Equation

The aim of this study is to find an improved model for concentration profiles in turbulent flows. We investigated three parameters to improve concentration profiles: the inverse of the Schmidt number (beta), eddy viscosity and the use of a fractional derivative order 0<alpha<=1 in the diffusion term instead of the traditional advection Diffusion Equation (ADE). We introduce a numerical scheme based on the L1 method to solve the fractional advection-diffusion equation (fADE) and make the following assessment: (1) beta using the Rouse equation and (2) eddy viscosity profiles using the ADE and fADE. The results show that beta depends on the flow and sediment characteristics while the assumption of equality between the eddy viscosity and the sediment diffusivity is inaccurate. For the eddy viscosity, similar results are obtained from the parabolic and exponential formulations, indicating a weak contribution of the vertical fluid momentum on sediment concentration profiles. Knowing the best values for the parameters, we resolved the fADE by numerically optimizing the value for alpha. Our results show an improvement provided by the fractional calculus (alpha<1) with respect to experimental data and demonstrated the non-local behavior of sediment distribution in turbulent flow.