Abstract: In this study, the SST k-ω turbulence model is applied within the framework of the internal penalty discontinuous Galerkin (IPDG) method. The applicability of the method is verified by numerical simulations of the flow fields and the turbulence characteristics under subsonic, transonic and supersonic conditions. Based on the present method, the numerical characteristics of four different flux schemes (AUSM, Lax F, HLL and Roe) for the convection term in the weak form of the RANS equations with IPDG are systematically analyzed. The results show that, the AUSM scheme presents an obvious folding phenomenon on the detached shock surface under supersonic condition. The Lax-F and HLL schemes have large numerical dissipation and perform poorly in the shock wave analysis. Moreover, flow separation is induced after the shock surface in the Lax-F scheme. The Roe scheme has the applicability for a wide range of velocity, and presents a high calculation accuracy that can precisely resolve the shock structure.