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In this article, we present a discontinuous Galerkin (DG) method designed to improve the accuracy and efficiency of steady solutions of the compressible fully coupled Reynolds-averaged Navier–Stokes and k − ω turbulence model equations for solving all-speed flows. The system of equations is iterated to steady state by means of an implicit scheme. The DG solution is extended to the incompressible limit by implementing a low Mach number preconditioning technique. A full preconditioning approach is adopted, which modifies both the unsteady terms of the governing equations and the dissipative term of the numerical flux function by means of a new preconditioner, on the basis of a modified version of Turkel's preconditioning matrix. At sonic speed the preconditioner reduces to the identity matrix thus recovering the non-preconditioned DG discretization. An artificial viscosity term is added to the DG discretized equations to stabilize the solution in the presence of shocks when piecewise approximations of order of accuracy higher than 1 are used. Moreover, several rescaling techniques are implemented in order to overcome ill-conditioning problems that, in addition to the low Mach number stiffness, can limit the performance of the flow solver. These approaches, through a proper manipulation of the governing equations, reduce unbalances between residuals as a result of the dependence on the size of elements in the computational mesh and because of the inherent differences between turbulent and mean-flow variables, influencing both the evolution of the Courant Friedrichs Lewy (CFL) number and the inexact solution of the linear systems. The performance of the method is demonstrated by solving three turbulent aerodynamic test cases: the flat plate, the L1T2 high-lift configuration and the RAE2822 airfoil (Case 9). The computations are performed at different Mach numbers using various degrees of polynomial approximations to analyze the influence of the proposed numerical strategies on the accuracy, efficiency and robustness of a high-order DG solver at different flow regimes.

We present an eigen-decomposition of the quasi-linear convective flux formulation of the completely coupled Reynolds-averaged Navier–Stokes and kω turbulence model equations. Based on these results, we formulate different approximate Riemann solvers that can be used as numerical flux functions in a DG discretization. The effect of the different strategies on the solution accuracy is investigated with numerical examples. The actual computations are performed using a p-multigrid algorithm. To this end, we formulate a framework with a backward-Euler smoother in which the linear systems are solved with a general preconditioned Krylov method. We present matrix-free implementations and memory-lean line-Jacobi preconditioners and compare the effects of some parameter choices. In particular, p-multigrid is found to be less efficient than might be expected from recent findings by other authors. This might be due to the consideration of turbulent flow. Copyright © 2012 John Wiley & Sons, Ltd.

In this chapter we collect results obtained within the IDIHOM project on the development of Discontinuous Galerkin (DG) methods and their application to aerodynamic flows. In particular, we present an application of multigrid algorithms to a higher order DG discretization of the Reynolds-averaged Navier-Stokes (RANS) equations in combination with the Spalart-Allmaras as well as the Wilcox-kω turbulence model. Based on either lower order discretizations or agglomerated coarse meshes the resulting solver algorithms are characterized as p- or h-multigrid, respectively. Linear and nonlinear multigrid algorithms are applied to IDIHOM test cases, namely theL1T2 high lift configuration and the deltawing of the second Vortex Flow Experiment (VFE-2) with rounded leading edge. All presented algorithms are compared to a strongly implicit single grid solver in terms of number of nonlinear iterations and computing time. Furthermore, higher order DG methods are combined with adaptive mesh refinement, in particular, with residual-based and adjoint-based mesh refinement. These adaptive methods are applied to a subsonic and transonic flow around the VFE-2 delta wing.

A comparison of nonlinear and linear p- and h-multigrid algorithms will be presented with respect to both algorithmic convergence properties and run-time behavior. In addition to that a comparison with a single-level solver, namely a Backward--Euler method, will be presented as well. The algorithms will be used to solve the RANS-equations in combination with a k-w turbulence models for CFD applications in the area of compressible aerodynamic fows with high Reynolds numbers. The h-multigrid algorithms are formulated on agglomerated unstructured meshes. Results will be presented on both structured and unstructred meshes.

We present an application of multigrid algorithms to a Discontinuous Galerkin discretization of the 3D Reynolds-averaged Navier–Stokes equations in combination with a kω turbulence model. Based on either lower order discretizations or agglomerated coarse meshes the resulting algorithms can be characterized as p- or h-multigrid, respectively. Linear and nonlinear multigrid algorithms are applied to a 3D numerical test case, namely the VFE-2 delta-wing with rounded leading edge. All presented algorithms are compared to a strongly implicit single grid solver in terms of run time behavior and nonlinear iterations.