Two-Parameter Mittag-Leffler Solution of Space Fractional Advection-Diffusion Equation for Sediment Suspension in Turbulent Flows

Abstract

In this paper, a simple derivation of the space fractional advection-diffusion equation (fADE) under the steady condition is proposed from a generalized mixing length concept. The obtained equation is a fractional Fickian-type diffusion equation in the steady-state condition. The space fADE is solved and an analytical solution is obtained in terms of the generalized two-parameter Mittag-Leffler function. The model has been applied to investigate the suspension distribution of sediment particles in open-channel turbulent flows over erodible sediment beds. The obtained results show that the fADE is able to predict the Type II profile (where the maximum concentration appears at some distance above the bottom of suspended-load layer) of the suspension concentration distribution in such flows, unlike the standard advection-diffusion equation. Apart from this, the model also describes the Type I profile (where the maximum concentration appears at the bottom of the suspended-load layer) of the suspension concentration distribution of particles well. The outcome of this study indicates that underlying diffusion of particles can be anomalous and exchange of momentum can happen within nonadjacent layers in the Type II distribution. The model is validated with experimental data available in the literature, and the results are satisfactory. Validation results show that the parameter β in the Type II profile depends on specific gravity, diameter, and settling velocity of particles. A nonlinear regression analysis is carried out to express the dependency and practical application of the model.