This study investigates the flow through round orifices with cross flow at the inlet. Emphasis is placed on the change in tangential velocity component for orifices with low L/D. A definition of velocity pick-up is developed based on the orifice exit to cross-flow tangential velocity ratio. Steady, incompressible, and 3D computational fluid dynamics models with the SST k - turbulent model are employed to calculate orifice flows with different geometrical and flow conditions. Stationary orifices and axial orifices in a rotating disk are considered. Computational fluid dynamics solutions are compared with experimental results and published correlations for discharge coefficients, and good agreement is generally demonstrated. It is found that the non-dimensional velocity pick-up depends strongly on the ratio of characteristic times for flow to travel across the orifice in the tangential direction and that for the flow to pass through the orifice. A correlation of velocity pick-up as a function of this ratio is given.