Numerical Study of Unsteady Natural Convection in a Horizontal Annular Channel

The problem of natural convection of a viscous incompressible fluid between two concentric cylinders is investigated. It is assumed that the inner cylinder can rotate around its axis without friction. Convection in the area is created due to the constant temperature difference at the outer boundary. The movement of fluid in the annular cavity, due to viscous friction, will cause the rotation of the inner cylinder. This rotation 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. In the work, two limiting cases were considered: the inner cylinder is thermally insulated, the inner cylinder is made of a material having a very high thermal conductivity. The work analyzed the amount of kinetic energy of the rotating cylinder depending on the inner radius, the size of the area where the temperature is maintained and the location of these areas. It has been found that the kinetic energy of the cylinder depends strongly on the thermal conductivity and the radius. For both types of the inner cylinder, the values of the inner radius were established at which the maximum kinetic energy of the cylinder is reached. It has also been found that this radius does not depend on the size of the area over which the temperature is maintained, or on the location of these areas. The Boussinesq approximation was chosen as the mathematical model. To solve this 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 ≤ γ ≤ 2π, 0 < Ri < 1.