A non-planar or a bilateral mixing-layer is studied by means of a series of direct numerical simulations (DNSs). This mixing-layer forms at the interface of two co-current plane Couette flows of different Reynolds numbers. The current DNS study determined the conditions for the onset of shear-layer instability at the interface. The influence of different Reynolds number (of the co-current plane Couette flows) and their Reynolds number ratio on the mixing-layer is studied. A critical Reynolds number of about 500 (or more particularly one of the co-current plane Couette flows must be turbulent) and a Reynolds number ratio greater than 2 is required for the genesis of this bilateral shear-layer instability. Independent of the Reynolds number and the Reynolds number ratio, the temporal evolution of the shear-layer instability followed the same pattern. In addition, the oscillation frequency of the instability was found to increase with increasing Reynolds number and increasing Reynolds number ratio. Further, influence of instability on the local skin friction and the two-point correlation is elaborated on. © 2021 Elsevier Inc.