On the development of an implicit high-order Discontinuous Galerkin method for DNS and implicit LES of turbulent flows

In recent years Discontinuous Galerkin (DG) methods have emerged as one of the most promising high-order discretization techniques for CFD. DG methods have been successfully applied to the simulation of turbulent flows by solving the Reynolds averaged Navier–Stokes (RANS) equations with first-moment closures. More recently, due to their favorable dispersion and dissipation properties, DG discretizations have also been found very well suited for the Direct Numerical Simulation (DNS) and Implicit Large Eddy Simulation (ILES) of turbulent flows. The growing interest in the implementation of DG methods for DNS and ILES is motivated by their attractive features. In particular, these methods can easily achieve high-order accuracy on arbitrarily shaped elements and are perfectly suited to hp-adaptation techniques. Moreover, their compact stencil is independent of the degree of polynomial approximation and is thus well suited for implicit time discretization and for massively parallel implementations. In this paper we focus on recent developments and applications of an implicit high-order DG method for the DNS and ILES of both compressible and incompressible flows. High-order spatial and temporal accuracy has been achieved using the same numerical technology in both cases. Numerical inviscid flux formulations are based on the exact solution of Riemann problems (suitably perturbed in the incompressible case), and viscous flux discretizations rely on the BR2 scheme. Several types of high-order (up to order six) implicit schemes, suited also for DAEs, can be employed for accurate time integration. In particular, linearly implicit Rosenbrock-type Runge–Kutta schemes have been used for all the simulations presented in this work. The massively separated incompressible flow past a sphere at ReD=1000, with transition to turbulence in the wake region, is considered as a DNS test case, while the potential of the ILES is demonstrated by computing the compressible transitional flow at Rec=60 000, M∞=0.1 and α=8°, around the Selig–Donovan 7003 airfoil. The computed solutions are compared with experimental data and numerical results available in the literature, showing good agreement.