This paper reports the main findings of a numerical investigation aimed at characterizing the flow field and the wall heat transfer resulting from the interaction of a swirling flow provided by lean-burn injectors and a slot cooling system, which generates film cooling in the first part of the combustor liner. In order to overcome some well-known limitations of Reynolds-averaged Navier–Stokes (RANS) approach, e.g., the underestimation of mixing, the simulations were performed with hybrid RANS–large eddy simulation (LES) models, namely, scale-adaptive simulation (SAS)–shear stress transport (SST) and detached eddy simulation (DES)–SST, which are proving to be a viable approach to resolve the main structures of the flow field. The numerical results were compared to experimental data obtained on a nonreactive three-sector planar rig developed in the context of the EU project LEMCOTEC. The analysis of the flow field has highlighted a generally good agreement against particle image velocimetry (PIV) measurements, especially for the SAS–SST model, whereas DES–SST returns some discrepancies in the opening angle of the swirling flow, altering the location of the corner vortex. Also the assessment in terms of Nu/Nu0 distribution confirms the overall accuracy of SAS–SST, where a constant overprediction in the magnitude of the heat transfer is shown by DES–SST, even though potential improvements with mesh refinement are pointed out.