Dynamics and instabilities occurring in the near-wake of wind turbines have a crucial role for the wake downstream evolution, and for the onset of far-wake instabilities. Furthermore, wake dynamics significantly affect the intra-wind farm wake flow, wake interactions and potential power losses. Therefore, the physical understanding and predictability of wind turbine wake instabilities become a nodal point for prediction of wind power harvesting and optimization of wind farm layout. This study is focused on the prediction of the hub vortex instability encountered within wind turbine wakes under different operational conditions of the wind turbine. Linear stability analysis of the wake flow is performed by means of a novel approach that enables to take effects of turbulence on wake instabilities into account. Stability analysis is performed by using as base flow the time-averaged wake velocity field at a specific downstream location. The latter is modeled through Carton-McWilliams velocity profiles by mimicking the presence of the hub vortex and helicoidal tip vortices, and matching the wind turbine thrust coefficient predicted through the actuator disc model. The results show that hub vortex instability is promoted by increasing the turbine thrust coefficient. Indeed, a larger aerodynamic load produces an enhanced wake velocity deficit and axial shear, which are considered the main sources for the wake instability. Nonetheless, wake swirl also promotes hub vortex instability, and it can also affect the azimuthal wavenumber of the most unstable mode.