The thermal management of variable-geometry supersonic exhaust nozzles is critical for structural integrity and infrared signature reduction [1]. Early NASA tests [2] on variable two-dimensional convergent–divergent nozzles revealed that, while cooling flows provide necessary thermal protection in exhaust nozzles, the integration of cooling mechanisms often results in thrust penalties. To mitigate such losses, we analyze a leakage cooling model that replicates a cooling strategy used in modern aircraft engines. The primary objective is to computationally investigate complex coolant-induced flow physics, including flow separation and shock wave-shear layer interaction. The computational domain models the internal flow path of a 12-sided faceted nozzle based on contemporary aircraft engines. To simulate realistic operating conditions, coolant air at 500 K was injected through leakage slits located at the convergent flap and the throat hinge under a Nozzle Pressure Ratio (NPR) of 3.5. The results from preliminary steady RANS simulations demonstrate that throat leakage injection effectively forms a film cooling layer on the faceted divergent flaps. Notably, the flap wall temperature was significantly reduced by over 200 K. However, the cooling flow was observed to intensify the shock wave-shear layer interaction, particularly around the nozzle corners at the exit, suggesting the formation of anisotropic turbulence structures. To further analyze these phenomena, Detached Eddy Simulation (DES) is to be performed and the detailed findings are to be discussed.