Numerical study of natural convection in a horizontal annular channel

About the authors:

Pavel T. Zubkov, Dr. Sci. (Phys.-Math.), Professor, Department of Fundamental Mathematics and Mechanics, School of Computer Sciences, University of Tyumen, Tyumen, Russia

Eduard I. Narygin, Postgraduate Student, Department of Fundamental Mathematics and Mechanics, University of Tyumen; e.i.narygin@yandex.ru

Abstract:

The authors of this article study the natural convection of a viscous incompressible fluid that completely fills a horizontal annular channel, on the outer boundary of which a constant temperature differential is maintained. The inner cylinder can rotate around its axis. The movement of fluid in the annular cavity due to viscous friction will cause the rotation of the inner cylinder, which can be used to perform mechanical work. This system can be considered as a stationary heat engine, operating in the presence of a gravitational field, where the work is done through an irreversible process — viscous friction.

Two extreme cases were considered: when the inner cylinder is heat-insulated and when the inner cylinder is made of a material having a very high thermal conductivity.

This paper analyzes the amount of kinetic energy of the rotating cylinder depending on the inner radius and the size of the area where a constant temperature is maintained. The results show that the kinetic energy of the cylinder essentially depends on both the thermal conductivity and the radius. For both types of the inner cylinder, the authors have found the values of the inner radius, at which the maximum kinetic energy of the cylinder is reached. They have also established that this radius does not depend on the size of the region on which the constant temperature is maintained. The Boussinesq approximation was chosen as the mathematical model. To solve the problem, the control volume method and the SIMPLER algorithm were used. The calculations were carried out at Pr = 1, 10⁴ ≤ Gr ≤ 2∙10⁴, 0 < 2α ≤ π, 0 < R_inside < 1.

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