Geometrical and Rayleigh number effects in the transient laminar free convection between two vertically eccentric spheres

The purpose of this paper is to study the transient natural convection of a Newtonian fluid which develops in a closed spherical annulus delimited by two vertically eccentric spheres by using a bispherical coordinates system. The inner sphere is heated by a heat flux of constant density and the outer one is maintained isothermal.

The transfer equations are written by using a bispherical coordinates system. The Navier‐Stokes equations are solved and coupled with the energy equation by using the alternating direction implicit (ADI) and the successive over relaxation (SOR) methods.

The study of the stream function and the Nusselt number shows that the convection motion is reinforced for the geometries characterized by positive values of the eccentricity with heat exchange increasing. The Nusselt number increases with the modified Rayleigh number. The heat exchange increases with the radius ratio. The results show that the steady state is reached faster when the modified Rayleigh number increases and the influence of the eccentricity is very low on the establishment of the steady state. The fluids flow depends strongly on the eccentricity and the modified Rayleigh number.

Simulations are performed for modified Rayleigh numbers ranging from 10³ to 10⁶, for eccentricities varying between –0.6 and +0.6 and for radius ratio between 1.5 and 2.

The results of eccentricity and modified Rayleigh number effects in transient natural convection between vertically eccentric spheres have been displayed.