A planar duct flow configuration with a cross-flow injected from a longitudinal slit close to the upper wall of the duct is studied by using a direct numerical simulation approach to explore the underlying flow mechanism in relation to the tip-leakage vortex (TLV), which is one of the most important flow phenomena in turbomachinery. Major characteristics of TLV in a rotor of turbomachinery are identified in the current flow model. The analysis of mean and instantaneous flow fields reveals that the interaction between the main (axial) flow and jet (cross) flow is the primary source of the generation of the TLV. The evolution of the TLV is then investigated, and a vortex breakup phenomenon is identified. The evolution of TLV can be divided into three phases, i.e. the formation phase, the break-up phase, and the diffusion phase. Mean streamlines and turbulence kinetic energy (TKE) budgets are analysed, showing that the high TKE central spot in the formation phase is due to the interaction between highly swirling vortex filaments and mean velocity gradient. In the outer part of the TLV, the TKE is mainly produced in the shear-layer and transported towards the centre by the turbulence transport.