The present study extends the range of the Rayleigh numbers to systematically investigate the flow transition during the melting process in an isothermally heated horizontal cylinder, emphasizing the complicated multi-cell flow pattern and the thermal instability. The enthalpy-porosity formulation is employed with appropriate source terms to account for the phase-change. At the low Rayleigh number, the flow is in a stable state because a single-cell base how is developed at the initial melting stage. At the high Rayleigh number, the Benard-type convection is found to develop within a narrow liquid gap between the unmelted solid surface and the cylinder bottom wall. At the intermediate Rayleigh number, the on-set of the Benard convection depends on the strength of the base flow under development and a delicate interaction between the two flows determines the flow pattern afterwards. The effective Rayleigh number and the corresponding wave number are in excellent agreement with those from the linear stability theory.