A unified model of suspension concentration distribution in sediment mixed turbulent flows using generalized fractional advection-diffusion equation

The fractional operator in a space fractional advection-diffusion equation (FADE) plays a significant role in the mixing and vertical movement of sediment particles in a sediment-laden turbulent flow under non-local effects. Turbulent flow exhibits non-local mixing properties, which leads to the non-Fickian diffusion process that cannot be captured by the traditional diffusion equation. In this work, we present a generalized FADE that includes the generalized fractional differential operator in the Caputo sense. The full analytical solution is proposed utilizing the general Laplace transformation method. This generalized solution contains weight and scale functions and includes the effects of non-locality. It has been shown that several existing famous models of suspension concentration distribution for sediment particles (including both type-I and type-II distributions) in turbulent flows can be obtained from the proposed generalized solution with proper choices of the scale and weight functions in particular. Here a total of fourteen different types of concentration distribution equations including type-I and type-II profiles are derived from the general solution. Further possible generalizations of the model are also discussed which are more useful for practical applications. It is found that the several existing sediment distribution models are equivalent up to choices of weight and scale functions. Further, we found that the scale function could be physically related to the characteristic Lagrangian length of sediment mixing. The choice of the scale and weight function for both the type-I and type-II profiles are discussed and analyzed. Finally, the model is validated with experimental data as well as field data from the Missouri River, Mississippi River, and Rio Grande conveyance channels, and in each case, satisfactory agreements are obtained. These suggest the broader applicability of the present study.