This paper is devoted to the numerical study of time-dependent flows that develop in a Taylor-Couette-like configuration (the outer cylinder exhibiting a side cavity) with a superposed axial through-flow, which is relevant to turbomachinery applications. Depending on the differential rotation rate between the inner cylinder and the cavity walls, the base flow can be either convectively or absolutely unstable, and then wave packets consisting of several small toroidal vortices appear near the inner cylinder and travel downstream. The influence of the different control parameters and of the inlet condition on the occurence of the patterns is analyzed. The flow simulations were obtained by using a spectral approximation coupled with a domain decomposition method for the solution of the axisymmetric time-dependent Navier-Stokes equations in the vorticity-streamfunction formulation.