A high-order Discontinuous Galerkin solver for unsteady incompressible turbulent flows

In this work we investigate the use of adaptive linearly implicit Rosenbrock-type Runge–Kutta and Explicit Singly Diagonally Implicit Runge–Kutta schemes to integrate in time high-order Discontinuous Galerkin space discretizations of the incompressible Navier–Stokes (INS) and Reynolds Averaged Navier–Stokes (URANS) equations. The objective of this activity is to assess the efficiency and accuracy of the considered schemes coupled with a time-step adaptation technique for incompressible URANS simulations. The schemes have been first investigated for the computation of the laminar travelling waves and of the turbulent flow around a circular cylinder at a Reynolds number Re=5×104, verifying the convergence order, a simple relation to set the system tolerance starting from the tolerance of the adaptation strategy, and their computational efficiency. Finally, the best scheme resulting from our analysis has been applied to the URANS simulation of the flow through a vertical axis wind turbine, comparing the results with CFD and experimental data available in literature.

[1]  H. Kredel,et al.  Integrated Performance Analysis of Computer Systems (IPACS). Benchmarks for Distributed Computer Systems , 2005, Prax. Inf.verarb. Kommun..

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

[3]  Daniele A. Di Pietro,et al.  A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure , 2011, J. Comput. Phys..

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

[5]  Jeff Cash,et al.  The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae , 1983 .

[6]  Alessandro Colombo,et al.  On the development of an implicit high-order Discontinuous Galerkin method for DNS and implicit LES of turbulent flows , 2016 .

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

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

[9]  Gustaf Söderlind,et al.  Digital filters in adaptive time-stepping , 2003, TOMS.

[10]  Joachim Rang,et al.  A New Stiffly Accurate Rosenbrock-Wanner Method for Solving the Incompressible Navier-Stokes Equations , 2013 .

[11]  Esteban Ferrer,et al.  Blade–wake interactions in cross-flow turbines , 2015 .

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

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

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

[15]  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..

[16]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[17]  Jeff Cash,et al.  A stability result for general linear methods with characteristic function having real poles only , 1998 .

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

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

[20]  Jens Lang,et al.  Towards a Fully Space-Time Adaptive FEM for Magnetoquasistatics , 2008, IEEE Transactions on Magnetics.

[21]  Per-Olof Persson,et al.  Validation of a High-Order Large-Eddy Simulation Solver Using a Vertical-Axis Wind Turbine , 2016 .

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

[23]  David I. Gottlieb,et al.  The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error , 1995, SIAM J. Sci. Comput..

[24]  Jan S. Hesthaven,et al.  Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes , 2007, J. Comput. Phys..

[25]  Gustaf Söderlind,et al.  Adaptive Time-Stepping and Computational Stability , 2006 .

[26]  C F Curtiss,et al.  Integration of Stiff Equations. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

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

[28]  A. Montlaur,et al.  Numerical Study of 2D Vertical Axis Wind and Tidal Turbines with a Degree-Adaptive Hybridizable Discontinuous Galerkin Method , 2015 .

[29]  Alessandro Colombo,et al.  Discontinuous Galerkin for Turbulent Flows , 2011 .

[30]  V. Selmin,et al.  3D anisotropic unstructured grid generation , 2006 .

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

[32]  Spencer J. Sherwin,et al.  Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations , 2014 .

[33]  Ernesto Benini,et al.  The Darrieus wind turbine: Proposal for a new performance prediction model based on CFD , 2011 .

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

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

[36]  Hester Bijl,et al.  Fourth-Order Runge–Kutta Schemes for Fluid Mechanics Applications , 2005, J. Sci. Comput..

[37]  Esteban Ferrer,et al.  A high order Discontinuous Galerkin - Fourier incompressible 3D Navier-Stokes solver with rotating sliding meshes , 2012, J. Comput. Phys..

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

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

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

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