This paper presents a study of the order of grid convergence of the Hybrid convection discretization scheme that blends first-order upwind with the second-order central-difference scheme. The aim of this study is to demonstrate that the relation between the order of grid convergence and the blending parameter is not linear. Three manufactured solutions that mimic statistically steady, two-dimensional, incompressible, turbulent, near-wall viscous flows were selected to enable the evaluation of discretization errors. To avoid any possible disturbances of the solution of turbulence quantities transport equations on the asymptotic order of grid convergence of mean flow quantities, we use a manufactured eddy-viscosity field in the RANS equations. Grid refinement studies were performed with the RANS solver ReFRESCO in geometrically similar stretched Cartesian grids, which ensure that the discretization schemes of all remaining terms of the RANS equations are second-order accurate. For flows dominated by convection, the Hybrid scheme remains first-order accurate up to values of the blending parameter very close to 1. On the other hand, the decrease of the error level with the blending parameter is close to linear.