Development of Finite Analytic Algorithms for Solute Transport Modeling and Non-Negative Solutions of Advection-Diffusion-Reaction Equations
Numerical simulations have been widely used in providing solutions to various engineering problems, including those related to the fluid flows and transport of concerned constituents. In this study, the finite analytic (FA) based numerical models are developed to solve the convection-diffusion transport equations. Different from other numerical approaches, the FA method adopts the locally derived analytic solutions to form a system of discretized equations with unknowns being functions of the values of the neighboring points and associated weighting coefficients. One of the developed two-dimensional FA models with anisotropic diffusion property is firstly applied to investigate the solution property of the transport equations in terms of the maximum principle (or the non-negative solutions). Cases considering different scenarios of the source concentrations were simulated. The computed concentrations inside the model domain with use of different grid sizes are presented to illustrate the solution values and the required grid size for satisfying the non-negative solution condition without imposing any limiters, which is fundamentally different from the conclusions required by other numerical methods with needed additional constraints. The results are compared with other published solutions to demonstrate the robust of the FA model in producing the non-negative solutions. Additionally, an unsteady FA transport model is developed to simulate the process of solute transport in a domain with either a layer of porous medium or a fluid-porous medium two-layer system under the action of standing waves. In the case of a layer of porous medium, the simulated results agree nicely with published finite-difference solutions. Depending on the ratio of fluid velocity versus diffusion coefficient, different transport processes of the concentration are analyzed. The FA model, with the automatic adjustment of the upwind effect, is proved to be able to obtain accurate concentration distributions without using any limiters or other numerically imposed convective controls. The modeling capability is also extended to simulate a two-layer system, which includes an upper fluid body and a lower porous medium. A vertically stretched coordinate system is applied in the fluid domain. The simulated time varying concentrations under various flow and porous medium conditions are presented. The effect of particle size contained in the porous medium on the transient concentration distributions are analyzed in detail. Generally, the concentration level increases when the pore size of the porous medium become larger. The phase changes of the variation trend in velocity and concentration between the fluid and porous medium layers are also examined and discussed.