Core-annular flow is an efficient way of transporting viscous oil through a pipeline. A sharp increase in the pressure drop will occur when the oil waves at the water-oil interface touch the pipe wall. Depending on the oil and pipe material physical properties, the oil may adhere to the wall leading to fouling. Therefore, a necessary requirement for the onset of oil fouling of the pipe wall is that the flow hydrodynamics allow the oil to reach and touch the wall. With respect to the problem statement, this study deals with finding the hydrodynamic conditions under which core-annular flow becomes unstable and the oil waves touch the pipe wall. The method that is followed is to resolve the first-principle set of equations that describe the hydrodynamics: the Reynolds-Averaged Navier-Stokes (RANS) equations are solved using Computational Fluid Dynamics (CFD) in the opensource package OpenFOAM. Simulations were carried out for the horizontal pipe with two diameters (10.5 and 21 mm), at a range of imposed pressure drops and water holdup fractions (giving the mixture velocity and watercut as output). Most simulations were carried out for an oil to water viscosity ratio of 1040 (but also a variation of this was considered). For each value of the pressure drop (or mixture velocity) there is a critical value of the watercut below which the oil reaches the pipe wall. This critical value of the watercut is lower for the larger pipe diameter of 21 mm, namely about 9.6%, than for the smaller pipe diameter of 10.5 mm, namely about 14% (for a viscosity ratio m = 1040). Wall touching occurs when the mixture velocity is too low, but this lower limit is significantly higher for the larger pipe diameter of 21 mm, namely about 1.1 m/s, than for the smaller pipe diameter, namely about 0.3 m/s (for a viscosity ratio m = 1040). The main conclusion is that a state-of-art CFD approach is capable of simulating the growth of waves at the oil-water interface until they touch the pipe wall, which is a necessary condition for the onset of fouling.