Application of Higher Order Multigrid Algorithms for turbulent flows
Marcel Wallraff, Tobias Leicht · elib (German Aerospace Center) · 2013
Computational Fluid Dynamics (CFD) methods have advanced substantially in the past decades.Moreover, CFD tools have become essential in the design process and analysis of modern aircraft design.The last decade has seen an interest in high order numerical methods, in particular in the discontinuous Galerkin (DG) Finite Element method.The analysis of turbulent flows employing steady-state computations based on Reynolds-averaged Navier-Stokes (RANS) equations and a turbulence model might be considered as the work-horse in this field.Nevertheless, DG results are relatively rare for this particular application.One of the reasons for this might be the stiffness introduced by both the turbulence model equations and the highly stretched meshes typically used for an efficient resolution of turbulent boundary layers.In order to solve the RANS equations in combination with a turbulence model several authors suggested strongly implicit schemes that are close to Newton's method.A Backward-Euler method in combination with an iterative linear solver can be considered as standard approach to solve a nonlinear set of equations for DG [2].Here, we focus on a combination of nonlinear multigrid algorithms using strongly implicit schemes as smoothers and linear multigrid algorithms to exploit hierarchies of coarse level problems in solver algorithms [3].Based on either lower order discretizations or agglomerated coarse meshes the resulting algorithms can be characterized as either p-or h-multigrid, respectively.The only difference between these multigrid algorithms is the use of different coarse level DG discretizations and, therefore, transfer operators.All other ingredients like smoothers, timestep control, usage of a Galerkin-transfer [3], startup strategy, etc. will stay the same for both kinds of multigrid algorithms.The proposed algorithms will then be applied to the DG discretizations of the steadystate RANS equations in combination with two different turbulence models: the Wilcoxkω two equation model [4] and the negativ Spalart-Allmaras one equation turbulence model [1].Results based on various combinations of multigrid algorithms are shown in comparison to a strongly implicit single grid solver.As a test case we consider the MDA 30P30N configuration which is a 2D high-lift three element airfoil and was recently considered as a test case for the Second International Workshop on High-Order CFD Methods in Cologne on May 2013.