•RANS solutions with a medium size mesh are close to the experimental data;•The surface film effectiveness of the cylindrical hole converges well, though the 3D spatial variables are challenging;•Mesh convergence of fan-shaped hole is more difficult than that of a cylindrical counterpart;•Mesh convergence at high blowing ratios provides a greater challenge than at lower blowing ratios. Accurate prediction of film cooling flow is a necessity in the design of high-pressure turbine components given the temperature of hot gas exceeds the melting point of the component material; temperatures that are necessary to achieve high thermal efficiency. Nowadays, Computational Fluid Dynamics (CFD) methods are becoming increasingly popular to predict film cooling flows. When compared to large eddy simulation (LES) and direct numerical simulation (DNS), the Reynolds Averaged Navier-Stokes (RANS) method is most frequently employed due to its much lower computational cost, though it struggles to accurately solve the highly three-dimensional and anisotropic cooling flows. However, few publications have focused on the mesh sensitivity problem of RANS simulations. And yet mesh generation is fundamental to CFD simulations and directly affects the accuracy of the calculated solution. Two typical film cooling hole geometries, namely cylindrical and fan-shaped, are utilised to investigate the mesh sensitivity of RANS simulations with low, medium, and high blowing ratios. Seven mesh sizes ranging from super coarse to super fine (4000 times larger) are employed for analysis. In general, the comparisons show that the computational results are close to the experimentally observed film features and averaged effectiveness, while also demonstrating that mesh convergence becomes challenging for RANS modelling, especially for the fan-shaped hole.