Turbulent flows through concentric annuli with the oscillating inner-wall around the axis were investigated by means of direct numerical simulation (DNS). In the inner-wall oscillation with the small oscillating period, the drag reduction occurs and the flow is stabilized like the outer-wall oscillation case, while in the inner-wall oscillation with the large one, the flow becomes unstable. The variation of the phase mean quantities is strongly dependent on the magnitude of the oscillating period.