Fractional Derivative Modeling For Suspended Sediment In Unsteady Flows

Suspended sediment transport is a frontier problem in river dynamics.The research on the movement law of suspended sediment is the prerequisite to guide the construction of various water conservancy projects,including coastal dike construction,port and waterway construction,estuary regulation,resource development,etc.,and it is also the key to solve all kinds of flood control problems,reservoir sedimentation,estuary sedimentation,and maintenance of built water conservancy projects.Suspended sediment is the main part of sediment transport process.Suspended sediment moves under the action of water flow,and it also has an effect on water body.The interaction between sediment and water and the interaction between sediment particles complicate the sediment movement,which is nonlinear and belongs to anomalous diffusion behavior.Therefore,the study of unsteady flow suspended sediment movement has important engineering and theoretical significance.In this paper,the suspended sediment under unsteady flow is studied by using continuous time random walk theory and fractional calculus Therefore,it is of great theoretical and engineering significance to study the suspended sediment transport in unsteady flow.The main research contents and conclusions of this paper can be summarized as the following three aspects:(1)Fractional order modeling and numerical solution of suspended sediment movement in unsteady flow.In order to better describe the anomalous diffuseness of suspended sediment movement under unsteady flow and its nonlinear characteristics,the space-time fractional model is used to describe the suspended sediment movement.Firstly,based on the continuous time random walk theory,the power-law function is selected as the probability density function of waiting time and jumping distance.Through Laplace Fourier transform,a fractional order model describing the suspended sediment movement in unsteady flow is established.The time fractional derivative and space fractional derivative in the model correspond to the time and space fractional derivative in the process of suspended sediment movement respectively Secondly,the time fractional term and space fractional term are discretized by L₁ algorithm and G algorithm respectively,and the implicit difference scheme is established for the established model.Finally,the stability and convergence of the established difference scheme are analyzed,and the results show that the established difference scheme is unconditionally stable and convergent.(2)Analysis and validation of fractional model for unsteady suspended sediment flow.In order to verify the validity of the spatiotemporal fractional order model and explore the influence of the parameters of the model on the calculation results,this paper uses the experimental data to analyze and verify the model,and carries on the sensitivity analysis for the parameters.Firstly,the time-space fractional order model is used to analyze the influence of time-space fractional order and Peclet number on the temporal and spatial vertical distribution of suspended sediment concentration;Secondly,the validity of the model is verified by comparing with the experimental data from the time-varying and spatial vertical distribution of concentration;Finally,the sensitivity of concentration to temporal and spatial fractional derivative was analyzed.The results show that,compared with the classical convection diffusion model,the fractional order model can better describe the abnormal diffusion phenomenon in the suspended sediment movement,reflecting the non local characteristics of its movement in time and space.In comparison with the experiment,it is found that the time fractional order is related to the average sediment concentration,and the smaller the average sediment concentration is,the smaller the time fractional order is.(3)Distributed fractional order modeling and analysis of suspended sediment movement in unsteady flow.In order to study the movement of non-uniform suspended load,considering the sediment gradation,different fractional order is selected,and a distributed fractional order model is established to fit the reality.In this paper,the distributed fractional order model is used to simulate the motion of non-uniform suspended load.Firstly,on the basis of the above research,considering the problem of sediment gradation,the waiting time probability density distribution is selected as the cumulative distribution in the continuous time random walk model,so that the fractional order can adapt to the overall sediment size distribution.Secondly,the finite difference method is used to establish the difference scheme for the distributed fractional order model.Finally,through numerical simulation,the influence of distributed fractional order and time coefficient on the spatial and temporal distribution of suspended load concentration is analyzed.The results show that by changing the weight coefficient and fractional order of the time derivative term,the distributed fractional order model can better simulate the spatial and temporal concentration distribution of suspended sediment under different particle size distribution,and describe the abnormal diffusion phenomenon in time and space.