Bounded oscillatory flows occur in many situations. The Lattice-Boltzmann model (LBM) is used to simulate oscillatory flows into the intermittently turbulent regime, in both two- and three-dimensions. The result of one of the three-dimensional simulations is also compared to Particle Image Velocimetry (PIV) measurements of an equivalent flow. After a review of some theory of oscillatory flow and also the methods used in the LBM simulations and PIV measurements, a two-dimensional LBM is used to simulate oscillatory flow in an infinite two-dimensional channel. The development of turbulence over the oscillatory cycle is observed and presented. A three-dimensional LBM simulation of oscillatory flow in an infinite three-dimensional channel is performed and the result presented and compared to results from the infinite two-dimensional channel. A further simulation is performed with the three-dimensional LBM applied to an infinite square duct. The development of turbulence is again observed and compared to the LBM simulation in an infinite three-dimensional channel. The results of PIV measurements of a flow equivalent to that in the LBM simulation in an infinite square duct are then presented and compared to the results of the three-dimensional infinite square duct simulation. The results of this work clearly show that the LBM is a useful tool for simulating bounded oscillatory flows and also provides a clear insight into the transition to turbulence in this type of flow.