Computational Fluid Dynamics (CFD) is a powerful tool for early-stage product development and optimization of biomedical devices such as blood pumps [1]. Despite the growing popularity of CFD in assessing design and performances also in predictive manner, the complex physics in addition to the components motion pose several challenges to CFD investigation, particularly with regards to mesh generation. In order to achieve results of adequate accuracy, high quality grids are necessary, leading to often expensive generation process, further complicated by the transient nature of certain phenomena, resulting in a bottleneck for the design process. In view of addressing these challenges, the present work proposes an alternative approach, which employs an autonomous mesh generation based on a modified Cartesian cut-cell methodology [3]. This allows the boundary motion to be directly imposed and enables on-the-fly adaptive mesh refinement (AMR) based on local velocity gradients, together with a dynamic y*-driven near-wall refinement. The proposed approach is applied to the United States Food and Drug Administration (FDA) Blood Pump benchmark for a Newtonian fluid, comparing the simulation results with the extensive experimental data available [1,2]. Two different methodologies are tested to address the impeller rotation: (i) fully moving geometry and (ii) multiple reference frame (MRF), highlighting the benefits of both and showing a good agreement with the experimental data. The pressure jumps due to the impeller blade passage on the outlet nozzle and velocity profiles at various sections are finally discussed.