The purpose of the present numerical method is to simulate the flow in between two concentric cylinders of finite axial length. The numerical integration of the transient three-dimensional incompressible Navier-Stokes equations is performed by a Legendre spectral element method in space while the time marching scheme is fully explicit. By taking full advantage of the tensor product bases, it is possible to set up fast diagonalization techniques in order to speed up the solver. Numerical results are compared favorably with experimental data. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.