In order to understand the nature ofbuoyant-thermocapillary convection in an annular pool with the outer heatedcontainer of radius ro=40 mm and the inner cooled cylinder ofri=20 mm, and an adjustable depth d=3~17 mm, we conducted a series ofunsteady three-dimensional numerical simulations with the finite differencemethod. The pool was filled with the 0.65cSt silicone oil (Prandtl numberPr=6.7). Results show that a small temperature difference in the radialdirection generates steady roll-cell buoyant-thermocapillary flow. Withlarge temperature difference, the simulations can predict two types of flowinstability. In a shallow pool (d=3 mm), the hydrothermal wave characterizedby curved spokes traveling in the azimuthal direction is dominant while indeep pools (d≥6 mm), the three-dimensional stationary instabilityappears. The critical conditions for the onset of the instability flows andthe free surface temperature fluctuation are determined and compared withthe experimental results.