A class of lower―upper symmetric-Gauss―Seidel, implicit, antidiffusive, weighted essentially nonoscillatory schemes for solving the two- and three-dimensional compressible Navier―Stokes equations with the Spalart― Allmaras one-equation turbulence model is presented. A weighted essentially nonoscillatory spatial operator with and without antidiffusive flux is employed for inviscid fluxes and central differencing for viscous fluxes. A numerical flux of weighted essentially nonoscillatory scheme in flux-limiter form is adopted, which consists of first-order and high-order fluxes and allows for a more flexible choice of first-order dissipative methods. The computations are performed for the two-dimensional turbulent flows over NACA 0012 and RAE 2822 airfoils and the three-dimensional turbulent flow over an ONERA M6 wing. By using the weighted essentially nonoscillatory scheme with antidiffusive flux corrections, the present solutions indicate that better resolution of contact discontinuities related flow structures and the convergence rate for steady-state computation are good, as compared with that using the original fifth-order weighted essentially nonoscillatory scheme combined with the Roe scheme on the same meshes. The present solutions are also compared with experimental data and other computational results and exhibit good agreement.