High-order (HO) methods are of academic and industrial interest owing to greater accuracy per degree- of-freedom, favorable parallel scalability and quasi-mesh-independence. Their application to turbulence modeling using Reynolds-averaged Navier-Stokes (RANS) equations and hybrid RANS-LES (Large-Eddy Simulation) techniques is of particular interest to industry, given the fact that pure LES and Direct Numerical Simulation (DNS) still remain infeasible in an industrial context. Convergence acceleration is a major area of research in this context for steady-state problems as well as unsteady problems modeled using pseudo- time-stepping. This paper analyzes the performance of a combination of h-multigrid and p-multigrid as applied to steady- state RANS-based turbulent flows. The one-equation Spalart-Allmaras model with negative-correction is used to account for turbulence and is verified through the use of realistic near-wall manufactured solutions. Static p-adaptation is used to attain appropriate near-wall resolution and to reduce the computational cost by limiting the degrees-of-freedom. Through numerical experiments on turbulent flow over a flat-plate at Reynolds number 5 million, we show that the combination of hp-multigrid and p-adaptation significantly enhances convergence when compared to simple p-multigrid. p-adaptation achieves the same accuracy as uniform polynomial-orders at a much lower number of degrees-of-freedom. Using even a single additional h-level reduces the number of iterations by ∼ 60%.