We are considering a steady-state turbulent Reynolds-Averaged Navier-Stokes (RA\-NS) model that couples the equation for the velocity-pressure mean field with the equation for the turbulent kinetic energy. Eddy viscosities vanish at the boundary, characterized by terms like $d(x, \Gamma)^\eta$ and $d(x, \Gamma)^\beta$, where $0 < \eta, \beta < 1$. We determine critical values $\eta_c$ and $\beta_c$ for which the system has a weak solution. This solution is obtained as the limit of viscous regularizations for $0 < \eta < \eta_c$ and $0 < \beta < \beta_c$.