Grain-size distribution in suspension through open channel turbulent flow using space-fractional ADE

The main aim of the present study is to develop a mathematical formulation that can describe the grain-size distribution (GSD) of non-uniform sediments in suspension over erodible sediment beds in a wide open channel under non-equilibrium conditions. The two important characteristics of sediment particles, such as non-local mixing and the hiding-hindering effect during settling, has been considered in the current study. The traditional advection–diffusion equation (ADE) is taken in the modified form as fractional ADE which is capable of capturing non-local movement of particles unlike the traditional diffusion theory where particles jump within a restricted distance along the vertical direction over a small time interval. The equation is further modified for any kth class of non-uniform sediment and the effect of non-local movement is included in the expression of depth averaged sediment diffusivity also. The settling velocity of a particle is taken in a form that contains the effect of hiding and hindering due to the presence of non-uniform particles in the flow. The space-fractional derivative is taken in the Caputo sense where the order α of the fractional derivative varies from 1<α≤2. The non-local effect is considered in the bottom and top boundary conditions also and the governing equation with boundary and initial condition have been solved by Chebyshev collocation method along with Euler’s backward method. The temporal variation of concentration profiles for different size particles is studied by varying α and it is observed that the magnitude of concentration decreases for each size as α increases. Non-local effect on bottom concentration for different size particle is also studied and it is seen that overshooting of bottom concentration gradually decreases as α increases for a particular size. Sensitivity analysis of α on GSD at different heights also has been done. The model has been validated for GSD under equilibrium conditions with available experimental data and it is found that the model aligns well when the non-local mixing effect is taken into account. At the end, an error analysis has been conducted to confirm the accuracy of the presented model.