  In this study, we introduce a robust central Gradient-Based Reconstruction
(GBR) scheme for the compressible Navier-Stokes equations. The method leverages
transformation to characteristic space, allowing selective treatment of waves
from the compressible Euler equations. By averaging left- and right-biased
state interpolations, a central scheme is achieved for all but the acoustic
waves, which require upwinding for stability. Distinct differences were
observed between transformations using either primitive or conservative
variables. We evaluated the method's robustness and superiority using benchmark
problems, including the two-dimensional shock entropy problem, two-dimensional
viscous shock tube, and three-dimensional inviscid Taylor-Green vortex.
Subsequently, we assessed the method in the context of Wall Modelled Large Eddy
Simulations (WMLES), where coarse grids are used to reduce computational cost
but also introduce substantial numerical dissipation. Using WMLES, we simulated
oblique shock impingement on a Mach 6 disturbed boundary layer and a Mach 7.7
flow over a $15^{\circ}$ compression ramp. Our findings reveal that: 1)
transformation to characteristic space using conservative variables leads to
more accurate results; 2) minimizing numerical dissipation through centralized
interpolation is crucial. In the compression ramp case, boundary layer
separation was shifted slightly upstream, and there was an over-prediction of
wall heating, likely attributable to the equilibrium-assuming wall model.
Overall, this work showcases the method's potential in accurately capturing
complex flow dynamics with reduced numerical dissipation.
