This paper presents a novel computational method for the flow simulations in the automotive oil pumps. The objective of this effort is to develop an advanced Computational Fluid Dynamics (CFD) tool to improve oil pump designs and efficiency by detailed analysis of unsteady fluid flow patterns inside stationary and rotating passages of an oil pump. To achieve this goal, several state-of-the-art computational technologies have been implemented into a general purpose unstructured grid code to handle numerical difficulties posed by complex geometry and moving parts of oil pumps. Most challenging numerical issues resolved in this paper include moving/deforming cells inside pump pockets, arbitrary sliding interface to connect moving and stationary parts and large grid distortions due to the great volume change of the pump pockets etc. A practical validation case, a vane oil pump, is studied using the presented method. The numerical results are compared with available experiment data.