A procedure for computing pulsatile flows in an arbitrary geometry is presented. The procedure is developed in a boundary-fitted coordinate system with a nonstaggered grid arrangement. The covariant velocity components are chosen as the dependent variables. To obtain the covariant velocities needed on the control surfaces, the physical contravanant velocities at the main grid are interpolated first and then convened to their covariant counterparts. A direct interpolation of the nonphysical covariant velocity components may induce a virtual velocity that leads to an erroneous solution. The procedure is employed to solve a moving indentation problem and a pulsatile backward-facing step flow.