Design of a non-linear hp-multigrid scheme with static near-wall p-adaptation for RANS simulations in a high-order flux reconstruction framework

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%.