High-order Discontinuous Galerkin (DG) schemes offer an highly accurate and efficient numerical framework to simulate fluid dynamics problems. % In this abstract we propose a modal DG solver adopting a projection-correction approach to embed a constraint of physical compatibility on the discrete entropy evolution. Particularly, an $L_2-$projection is exploited to construct a polynomial approximation of the entropy variables for the computation of the DG space operators, while the explicit entropy correction introduced by Abgrall~\cite{abgrall} is used to account for the inexact integration of the Navier-Stokes non-linear terms. % The proposed strategy ensures an entropy conserving/stable (EC/ES) discretization of the convective terms upon appropriate choice of the numerical flux function, without relying on the construction of Summation-By-Parts operators or flux splitting techniques. % The robustness improvement ensured by the projection-correction can be effectively leveraged to tackle highly challenging flow scenarios, see~\cite{Alberti} for examples in the inviscid limit, which would be otherwise unfeasible to simulate with the standard baseline solver. % We show that highly accurate solutions of implicit Large Eddy Simulations may be derived for severely under-resolved discretizations, even in presence of strongly compressible phenomena. As low-Mach number channel flows are considered, accurate solutions are derived by implementing a low-dissipation framework, obtained coupling an EC convective flux and minimizing solution jumps penalization. On the contrary, as high-Mach number channel flows are considered, we note that an adequate modulation of the introduced numerical dissipation can be effectively leveraged to improve the accuracy in condition of extremely coarse space resolutions.