High-order methods, such as the discontinuous Galerkin method, have gained increasing attention in the computational fluid dynamics community because they are expected to deliver high-accuracy results on complex flows and geometries. However, in realistic high Reynolds number flows, the significant computational cost and lack of robustness are unavoidable concerns when compared to low-order methods. The present paper develops a detached eddy simulation model based on the $k\text{−}g$ equations, where the near-wall scaling of the new transport variables suits the polynomial nature of the discontinuous Galerkin method. The hybrid modeling framework of “${\ell }^2\times \frac{1}{T}$” mimics traditional Smagorinsky and dynamic subgrid models in the eddy-resolving regions, and provides seamless hybrid modeling of the unresolved scales in coarse-mesh simulations of wall turbulence and separated flows. Robustness is achieved by both proper near-wall scaling of turbulence variables and the scale-dependent hybrid modeling of unresolved turbulence. For the latter, no extra cutoff filtering, overintegration, or additional viscosity is needed regardless of the mesh resolution. Our model can perform both traditional delayed detached eddy simulation and wall-modeled large eddy simulation for high Reynolds number turbulent flows like its predecessor, the adaptive ${\ell }^2\text{−}\omega$ model [Yin and Durbin, Int. J. Heat Fluid Flow 62, 499 (2016)]. By implementing the method in a spectral element discontinuous Galerkin solver, the proposed model is tested in a transonic airfoil with shock-turbulent boundary layer interaction.

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• Turbulence modeling
• Large eddy simulations
• Numerical techniques