HP-Multigrid for RANS-Modeled Turbulent Flows in a High-Order Flux Reconstruction Framework

Abstract

High-order (HO) methods are of concerted academic and industrial interest in recent years due to their improved accuracy and their capability to deal with complex geometries [1]. Of particular note is the flux reconstruction method [2], which unifies several existing HO schemes into a simpler and computationally efficient approach that has been shown to work on all element types (including simplices) in two and three dimensions. There is considerable interest to apply HO methods to industrially relevant problems. At the same time, accurate and robust turbulence modeling techniques are essential for reliable results. As outlined in the National Aeronautics and Space Administration's CFD vision 2030 study, Large Eddy Simulation (LES) still remains impractical for industrial cases therefore, Reynolds-Averaged Navier Stokes (RANS) and hybrid RANS-LES methods hold high significance in the near future [4]. Achieving fastest convergence to steady-state is important in the context of RANS simulations, for which several convergence acceleration techniques are being investigated. Multigrid methods are an industry standard in Finite Volume (FV) type schemes and are increasingly being applied to HO methods in the form of p-multigrid [22]. They exploit the polynomial hierarchy of the solution space to represent errors on a coarser resolution. A natural extension of this idea is hp-multigrid, where we can augument the classical h-multigrid to the polynomial hierarchy [23]. In this paper we illustrate the application of high-order flux reconstruction methods to simulate compressible, turbulent flows on body-fitted meshes. The case in point is the turbulent flow over a flat plate [24]. Turbulence is modeled through the RANS approach using the one-equation Spalart-Allmaras model. Grid-coarsening for the h-levels is performed by removing every other line in each direction from the original mesh. The system is driven to a steady-state solution using hp-multigrid convergence acceleration with local time-stepping using an explicit Runge-Kutta time-marcher. We show that the augumented h-multigrid is highly effective with a 10X to 24X drop in convergence time.