Natural heat convection within an annular annulus bounded by two horizontal vertically eccentric long cylinders has been investigated. The annulus inner wall has been heated and maintained at either constant wall temperature CWT or constant heat flux CHF while the outer wall is cooled and maintained at constant temperature. The induced buoyancy driven flow and the associated heat convection are predicted through solving numerically the full conservation equations for mass, momentum, and energy using Fourier spectral method. Beside Rayleigh and Prandtl numbers, the heat convection process in the annulus depends on the annulus radius ratio and eccentricity (normalized by the radius difference). The study considered a moderate range of Rayleigh numbers up to 105 while Prandtl number is fixed at 0.7. The radius ratio is considered up to 3.2 while the eccentricity is varied between − 0.65 and + 0.65. The study has revealed that at certain radius ratio for a given Rayleigh number and eccentricity, the heat transfer is minimum in case of CWT and the mean inner wall temperature is maximum in case of CHF. The study has also shown, in the range considered for controlling parameters, that multiple convection cells only exist in case of CWT and only for positive eccentricity. Moreover, the study has shown that the present numerical solution of the pure conduction problem is almost identical with the newly presented analytical solution which confirms the high accuracy of the numerical solution.