Compressible flow models are widely used in many industrial sectors. In the aerospace industry, Reynolds-Averaged Navier–Stokes (RANS) modeling is one of the main tools for aircraft design, flow analysis, and related applications. High-order (HO) numerical methods are particularly promising for these models, as they can drastically reduce the computational cost required to resolve smooth regions of the solution. However, the stability of HO methods for compressible turbulent flows remains a major challenge, especially when the numerical scheme is expected to respect invariant properties of the underlying continuous problem. Violations of these invariants may prevent the numerical scheme from capturing the physical solution. The von Neumann stability condition associated with the discretization of the viscous terms makes the use of explicit time integration impractical. For this reason, we focus on the study of time-implicit discretizations. In this work, we first analyze the Discontinuous Galerkin Spectral Element Method (DGSEM) applied to the discretization of compressible turbulent flows and investigate its fundamental properties. We then introduce a low-order scheme and examine its stability and admissibility properties, with particular attention to the entropy behavior and the preservation of invariants. We will in particular consider the flux-corrected transport (FCT) limiting, within the DGSEM framework. The theoretical results are assessed through numerical experiments on standard benchmark test cases.