The boundary condition at the air-water interface plays a major role in the stability of a rotating flow with a free surface. We consider here a generic configuration to investigate such effects both experimentally and numerically. For the flow driven by a rotating disc in a fixed cylindrical vessel partially filled with water, the standard free-slip condition in numerical simulations does not predict the instability threshold found experimentally. The unavoidable surface contamination changes the stresses at the interface and has a strong impact on the velocity field, at least when H, the fluid height, is small compared to R, the disc radius. The possible effect of unidentified pollutants at the interface can be modelled using an advection-diffusion equation and a closure equation linking the surface tension to their concentration. This modelling has been proposed in Faugaret, et al. J. Fluid Mech., 900 (2020), A42. A even simpler numerical model without superficial transport of the surfactants, adaptable into any code for single-phase flows has been proposed in Faugaret, et al. J. Fluid Mech., 935 (2022), A2. The model does not possess any free parameter and is independent on the closure model for surfactants. For the stationary axisymmetric base flow, the radial velocity at the interface is set to zero whereas the usual stress-free boundary conditions are retained for the perturbations. For a geometrical aspect ratio G = H/R equal to 1/4, known to display ambiguous behaviour regarding stability thresholds, the modal selection as well as a nonlinear stability island found in the experiments are well reproduced by the model, both qualitatively and quantitatively. We present experimental and numerical results in a systematic study for G ranging from 0.1 to 2. By varying the aspect ratio, we describe the modal selection of the primary instability. We show that the simple model is robust over the entire range studied.