A countercurrent vortex of a gas centrifuge is simulated. The mathematical model of the motion of a viscous heat-conducting gas includes an equation for density, velocities and specific energy in cylindrical geometry. After the introduction of the grid, the partial derivatives over the space are replaced by finite differences, and the problem is reduced to a system of ordinary differential equations (ODES). This technique is called the Lines Method. Since the flow is supersonic, and the design area includes thin boundary layers, the ODE system is stiff due to the presence of different-time scales and a decay. In the language of mathematics, this means a significant difference between the eigenvalues of the Jacobi matrix and the negative real parts. Therefore, to solve the problem, it is useful to use the implicit Geer method for the ODE system without splitting the problem into physical processes and directions. An effective method for solving the Jacobin matrix inversion is the use of the cyclic reduction method in the matrix variant. As an example, the countercurrent flow arising due to the temperature gradient is demonstrated.
Аксенов А.Г. Двумерная модель и метод расчета противотока во вращающемся вязком теплопроводном газе. Математическое моделирование и численные методы, 2023, № 4, с. 3–14.