Oil-water core-annular flow has the potential to significantly reduce the pumping power required for heavy oil transportation. We study axisymmetric, laminar-laminar oil-water core-annular flow in a pipe (radius 14.2mm) under three different channel conditions: vertically upward flow, vertically downward flow, and without considering the gravitational effect. For the no-gravity' case, a smooth oil-water interface is obtained and the results correspond to the analytical solution for a fully-developed' annular flow. For the vertically upward flow, saw-tooth' waves are observed at the interface. No perturbations are introduced to the flow initially; the interfacial waves take shape from the interplay between buoyancy, viscous force, and interfacial tension. CFD simulations are compared with published experimental observations, and found in qualitative agreement with each other. Both the CFD simulations and experiments show the wave amplitude reducing with increasing oil flow rate while keeping the water flow rate constant for a vertically upward flow. CFD simulation for a vertically downward flow shows the presence of very short waves of small amplitude, and the mean interfacial radius is larger than for the no-gravity case. Moreover, the maximum velocity in the annular water film occurs between the interface and the wall, unlike the upward and no-gravity cases where the maximum velocity in the water film occurs at the interface.