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

Abstract 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 R e D = 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 R e c = 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.

[1]  Francesco Bassi,et al.  A High Order Discontinuous Galerkin Method for Compressible Turbulent Flows , 2000 .

[2]  Andrea Crivellini,et al.  An artificial compressibility flux for the discontinuous Galerkin solution of the incompressible Navier-Stokes equations , 2006, J. Comput. Phys..

[3]  David L. Darmofal,et al.  An unsteady adaptation algorithm for discontinuous Galerkin discretizations of the RANS equations , 2007 .

[4]  E. Lamballais,et al.  Evaluation of a high-order discontinuous Galerkin method for the DNS of turbulent flows , 2014 .

[5]  C. L. Merkle,et al.  The application of preconditioning in viscous flows , 1993 .

[6]  Valerio D’Alessandro,et al.  Assessment of a high-order discontinuous Galerkin method for incompressible three-dimensional Navier–Stokes equations: Benchmark results for the flow past a sphere up to Re = 500 , 2013 .

[7]  Andrea Crivellini,et al.  An implicit matrix-free Discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations , 2011 .

[8]  S. Scott Collis,et al.  Discontinuous Galerkin Methods for Turbulence Simulation , 2002 .

[9]  H. Sakamoto,et al.  A STUDY ON VORTEX SHEDDING FROM SPHERES IN A UNIFORM FLOW , 1990 .

[10]  Luca Bonaventura,et al.  Dynamic models for Large Eddy Simulation of compressible flows with a high order DG method , 2014, Computers & Fluids.

[11]  Alessandro Colombo,et al.  Very High-Order Accurate Discontinuous Galerkin Computation of Transonic Turbulent Flows on Aeronautical Configurations , 2010 .

[12]  Claus-Dieter Munz,et al.  High-Order Discontinuous Galerkin Schemes for Large-Eddy Simulations of Moderate Reynolds Number Flows , 2015 .

[13]  Wayne A. Smith,et al.  Preconditioning Applied to Variable and Constant Density Flows , 1995 .

[14]  L. Bricteux,et al.  Implicit LES of free and wall‐bounded turbulent flows based on the discontinuous Galerkin/symmetric interior penalty method , 2015 .

[15]  Marco Luciano Savini,et al.  Discontinuous Galerkin solution of the Reynolds-averaged Navier–Stokes and k–ω turbulence model equations , 2005 .

[16]  R. Tognaccini,et al.  RANS analysis of the low-Reynolds number flow around the SD7003 airfoil , 2011 .

[17]  Alessandro Colombo,et al.  Investigation of high-order temporal schemes for the discontinuous Galerkin solution of the navier-stokes equations , 2014 .

[18]  David W. Zingg,et al.  Time-accurate flow simulations using an efficient Newton-Krylov-Schur approach with high-order temporal and spatial discretization , 2013 .

[19]  Gerard M. Faeth,et al.  Sphere wakes in still surroundings at intermediate Reynolds numbers , 1993 .

[20]  Claus-Dieter Munz,et al.  A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions , 2008, J. Sci. Comput..

[21]  Per-Olof Persson,et al.  RANS Solutions Using High Order Discontinuous Galerkin Methods , 2007 .

[22]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[23]  Rémi Abgrall,et al.  High‐order CFD methods: current status and perspective , 2013 .

[24]  Bertil Gustafsson,et al.  Numerical Methods for Differential Equations , 2011 .

[25]  Francesco Bassi,et al.  Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier-Stokes equations , 2014, J. Comput. Phys..

[26]  Claus-Dieter Munz,et al.  High‐order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations , 2014 .

[27]  Ralf Hartmann,et al.  Adjoint-based error estimation and adaptive mesh refinement for the RANS and k-ω turbulence model equations , 2011, J. Comput. Phys..

[28]  Claus-Dieter Munz,et al.  Explicit Discontinuous Galerkin methods for unsteady problems , 2012 .

[29]  Valerio D'Alessandro,et al.  A Spalart-Allmaras turbulence model implementation in a discontinuous Galerkin solver for incompressible flows , 2013, J. Comput. Phys..

[30]  Alessandro Colombo,et al.  Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows , 2015 .

[31]  Guillermo Hauke,et al.  a Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation of the K - Model. , 1994 .

[32]  S. Orszag,et al.  Numerical investigation of transitional and weak turbulent flow past a sphere , 2000, Journal of Fluid Mechanics.

[33]  Lorenzo Botti,et al.  Influence of Reference-to-Physical Frame Mappings on Approximation Properties of Discontinuous Piecewise Polynomial Spaces , 2012, J. Sci. Comput..

[34]  Clinton P. T. Groth,et al.  Assessment of Riemann solvers for unsteady one-dimensional inviscid flows for perfect gases , 1988 .

[35]  Marco Luciano Savini,et al.  A high-order Discontinuous Galerkin solver for the incompressible RANS and k–ω turbulence model equations , 2014 .

[36]  Francesco Bassi,et al.  Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows , 2014 .

[37]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[38]  P. Tesini,et al.  On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations , 2012, J. Comput. Phys..

[39]  Koen Hillewaert,et al.  IDIHOM: Industrialization of high-order methods - a top-down approach : results of a collaborative research project funded by the European Union, 2010 - 2014 , 2015 .

[40]  Ralf Hartmann,et al.  Discontinuous Galerkin discretization of the Reynolds-averaged Navier-Stokes equations with the shear-stress transport model , 2014, J. Comput. Phys..

[41]  Ralf Hartmann,et al.  A discontinuous Galerkin method for inviscid low Mach number flows , 2009, J. Comput. Phys..

[42]  Michael S. Warren,et al.  Vortex Methods for Direct Numerical Simulation of Three-Dimensional Bluff Body Flows , 2002 .

[43]  A. J. Baker,et al.  A stiffly-stable implicit Runge-Kutta algorithm for CFD applications , 1988 .

[44]  S. Scott Collis,et al.  The DG/VMS Method for Unified Turbulence Simulation , 2002 .

[45]  Koen Hillewaert,et al.  Assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number , 2014 .

[46]  Ewald Krämer,et al.  A parallel, high-order discontinuous Galerkin code for laminar and turbulent flows , 2008 .

[47]  D. Mavriplis,et al.  Robust Computation of Turbulent Flows Using a Discontinuous Galerkin Method , 2012 .

[48]  R. Schwane,et al.  ON THE ACCURACY OF UPWIND SCHEMES FOR THE SOLUTION OF THE NAVIER-STOKES EQUATIONS , 1987 .

[49]  Jens Lang,et al.  ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems , 2000 .

[50]  A. Beck,et al.  On the accuracy of high-order discretizations for underresolved turbulence simulations , 2013 .

[51]  S. Rebay,et al.  An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows , 2007 .

[52]  Andrew Pollard,et al.  Direct numerical simulation of compressible turbulent channel flows using the discontinuous Galerkin method , 2011 .

[53]  F. Brezzi,et al.  Discontinuous Galerkin approximations for elliptic problems , 2000 .

[54]  Per-Olof Persson,et al.  Implicit Large Eddy Simulation of transition to turbulence at low Reynolds numbers using a Discontinuous Galerkin method , 2011 .

[55]  Alessandro Colombo,et al.  Time Integration in the Discontinuous Galerkin Code MIGALE - Unsteady Problems , 2015 .

[56]  M Giacobello,et al.  Numerical studies of High Reynolds Number flow past a stationary and rotating sphere , 2009 .