Flow in an agitated vessel with a paddle impeller is described with the stream functions which satisfy the boundary conditions in order to understand the nature of the large scale and low frequency fluctuations of the flow. Differential equations for the stream functions are derived from the Navier-Stokes equations, which are solved numerically with a periodically changed external stream function. Non-periodic and long time fluctuations of the velocity field and the quantity related to the torque (equivalent torque) are found by frequency analysis of the time series of the velocities and the equivalent torque, the frequencies of which are clearly different from the frequency of the external input (about twentieth as low as the input frequency). The peak positions of the power spectra agree excellently with some reports of experimental analysis concerning the turbulent-flow in an agitated vessel, which shows that the results obtained by this numerical analysis are valid.