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We present Large-Eddy Simulations (LES) of the turbulent compressible flow in curved ducts of square cross section. The aim is to investigate the influence of the curvature radius Rc on the flow and the heat transfer. We consider three different curvature radii : 4Dh, 7Dh and 11Dh (Dh hydraulic diameter). We observe a rise in the number of streamwise vortices of Görtler type on the unstable concave wall when the curvature radius decreases. The main effect is a strong intensification of the secondary flows with the reduction of Rc: a rise of 100 % of the intensity between the smaller and the higher case of the curvature radius. We determine the influence of Rc on heat transfer by considering the case of convex wall heating. Due to the modification of the secondary flows, we observe an enhancement of the heat flux for the smaller value of the curvature radius, specially close to the sidewalls.