Airborne wind energy is an emerging technology that aims at harvesting wind power at high altitudes. In the present work, we propose a framework combining optimal control and large-eddy simulation to investigate the wake characteristics of large-scale airborne wind energy systems. We consider systems operating in pumping mode which alternate between power-generating and -consuming phases. We investigate the downstream wake development in non-turbulent and turbulent sheared inflow conditions. The optimal system operation leads to a non-uniform radial wake development and results show that the maximal wake velocity deficit for the current system is half the deficit predicted by Betz limit for a conventional wind turbine.