Analytical and numerical modeling of a stationary boundary value problem of diffusion-convection-decay for a homogeneous layer based on the equations of turbulent diffusion

Abstract

For comparison and testing purposes, an analytical and numerical method has been developed for solving the boundary value problem of convection-diffusion-decay for a homogeneous layer. A model stationary boundary value problem of the third kind is described by the equations of turbulent diffusion. In the analytical model, the Fourier symbol of the Green's function for the boundary value problem is constructed, the calculation of the Fourier integral is based on the theory of residues. In the numerical model, the Fourier integral is calculated using an algorithm based on the Gauss-Kronrod formulas.
In the two-dimensional case, comparative calculations for the near and middle zones showed good agreement between the results. The accuracy of calculations within each model can be changed by several orders of magnitude. Calculations of the substance concentration function for two plane problems of convection-diffusion-decay are given as examples. The implementation of the analytical method in the flat case is relatively straightforward. Numerical calculation is much simpler and more convenient for engineering and serial scientific calculations.
Numerical integration is realized just as easily in the spatial case as in the two-dimensional case, but the counting time increases significantly. Therefore, a balance is required between the required accuracy and the calculation time. For the spatial case, an analytical approach based on the theory of residues was fundamentally developed in the works of other authors. However, the method is rather cumbersome and better suited for theoretical research.