Finite element (FE) method has been widely used for gaining insights into the mechanical response of brain tissue during impacts. In this study, a coupled Eulerian-Lagrangian (CEL) formulation is implemented in impact simulations of a head system to overcome the mesh distortion difficulties due to large deformation in the cerebrospinal fluid (CSF) region and provide a biofidelic model of the interaction between the brain and skull. The head system used in our FE model is constructed from the transverse section of the human brain, with CSF modeled by Eulerian elements. Spring connectors are applied to represent the pia-arachnoid connection between the brain and skull. Validations of the CEL formulation and the FE model are performed using experimental results. The dynamic response of brain tissue under non-contact impacts and the brain regions susceptible to injury are evaluated based on the intracranial pressure (ICP), maximum principal strain (MPS), and von Mises stress. While tracking the critical MPS location on the brain, higher likelihood of contrecoup injury than coup injury is found when sudden brain-skull motion takes place. The accumulation effect of CSF in the ventricle system, under large relative brain-skull motion, is also identified. The FE results show that adding relative angular velocities, to the translational impact model, not only causes a diffuse high strain area , but also cause the temporal lobes to be susceptible to cerebral contusions since the protecting CSF is prone to be squeezed away at the temporal sites due to the head rotations.