Chemical vapour deposition on a Cu substrate is becoming a very important approach to obtain high quality graphene samples. Previous studies of graphene growth on Cu mainly focus on surface processes. However, recent experiments suggest that gas-phase dynamics also plays an important role in graphene growth. In this article, gas-phase processes are systematically studied using computational fluid dynamics. Our simulations clearly show that graphene growth is limited by mass transport under ambient pressures while it is limited by surface reactions under low pressures. The carbon deposition rate at different positions in the tube furnace and the concentration of different gas phase species are calculated. Our results confirm that the previously realized graphene thickness control by changing the position of the Cu foil is a result of gas-phase methane decomposition reactions.