Discretization of differential equations of convection and diffusion based on control volume method

The object of research is a mathematical model of convective heat and mass transfer. This model is based on the equations of hydrodynamics. Taking into account that the equations governing heat transfer, mass transfer and fluid flow are similar in many respects, and the dependent variables of our interest satisfy a generalized conservation law, we apply numerical methods to a differential equation of a generalized variable. The research objective is construction of a discrete analogue of the generalized differential equation describing the proposed mathematical model of convection in a viscous incompressible fluid.
To implement the mathematical model, it is suggested to use the control volume method. This method has a number of significant advantages in comparison to the method of finite differences. It provides strict compliance with the laws of mass, momentum, energy conservation for any group of cells, and therefore, in the whole computational domain. The method itself and its application are sufficiently described in the book by S. Patankar [5]. The computational domain is divided into a multiplicity of control volumes. The differential equation is integrated over the control volumes. In this case, the assumption about the form of the function describing the change in the variable between two adjacent nodes is made. To represent the profiles of functions between the nodes of the grid, we use templates that approximate the exact solution obtained analytically.
As a rule, researchers use discrete analogues obtained for rectangular grids in numerical simulation. Use of such templates for curvilinear coordinate systems leads to large computational errors. Many applied problems are easier to solve in the curvilinear coordinates, for example, in cylindrical coordinates. Further, to obtain the exact templates, we need different solutions that take into account specific features of curvilinear grids. Modern computers allow performance of complex mathematical calculations. It also makes possible the use of more accurate functional dependencies for approximations. This all led to the desire of authors to get a more efficient tool—the calculated schemes and algorithms that use an exponential scheme for the cylindrical coordinate system as a basis.
The task is to obtain the best approximations of the generalized variable profiles in a variety of directions. For this purpose, we find the exact solutions of the equations of conservation separately for each coordinate. The article investigates the meaning of exact solutions made in order to ensure their correctness.
Using the obtained analytical solutions, we have built the discrete analogue of the generalized differential equation. We obtained a more general formulation for the discrete analogue based on the research of S. Patankar. The resulting discrete analogue of the generalized differential equation written in cylindrical coordinates can be used for numerical simulations of convective heat and mass transfer under conditions of axisymmetric laminar and turbulent flows of viscous incompressible fluid and fluid flows in channels and pipes. Using discrete analogues constructed on the basis of exact transfer equations may be more advantageous taking into account the present level of computer technology.