Abstract
Convection-diffusion problems, due to its fundamental nature, are found in various science and engineering applications. In this research, the importance of the relationship between grid structure and flow parameters in such problems is emphasized. In particular, we propose a systematic technique in the selection of the grid expansion factor based on its logarithmic relationship with low Peclet number. Such linear mathematical connection between the two non-dimensional parameters serves as a guideline for more structured decision-making and improves the heuristic process in the determination of the computational domain grid for the numerical solution of convection-diffusion equations especially in the prediction of the concentration of the scalar. Results confirm the effectiveness of the new approach.
Highlights
In Cartesian coordinates and tensor notation following the Einstein convention, the generic conservation equation in the partial differential form is ( )+ − − = 0where is the density, is the conserved property, are velocity components of the fluid in the axes directions at the point (, ) at time, is the diffusivity of, and is the source or sink of
We investigate the relationship between the flow parameter of interest in convection-diffusion equation (CDE) and the expansion factor based grid structure in finite difference www.etasr.com
CONVECTION DIFFUSION PROBLEMS Various numerical methods for solving CDE are well formulated and many useful schemes can be found such as finite differences, finite elements, spectral procedures, and the method of lines [1,2,3,4,5,6,7,8,9,10,11,12]
Summary
In Cartesian coordinates and tensor notation following the Einstein convention, the generic conservation equation in the partial differential form is ( )+. We consider the steady one-dimensional convection-diffusion problem where (4) reduces to ( ) − ( ) = 0, involving the scalar whose concentration is denoted by. Such scalar is carried along with the moving fluid element (convection) and spreads due to diffusion. Numerical scheme, and formulate the mathematical relationship between and which is necessary in achieving numerical accuracy in the solution of the equation, unify the deduction of heuristic selections of grid expansion factor for solving the contaminated fluids problem that leads to less precomputation time. Note that the relatively smaller grid expansion factor does not unconditionally lead to higher numerical accuracy
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