High-order linear and non-linear residual distribution schemes for turbulent compressible flows

A high-order Residual Distribution scheme for the solution of the compressible RANS equations is developed. The one-equation Spalart-Allmaras turbulence model is solved in a fully coupled approach; the mean flow equations as well as the turbulence equation are solved with high-order of accuracy. A continuous approximation of the solution is adopted and standard Lagrangian basis functions are used to construct the discrete space. Since viscous terms involve the gradient of the numerical solution which has a discontinuous normal component across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same basis functions used for the solution. The non-linear system of equations resulting from the numerical discretization is solved with the non-linear LU-SGS method. Several numerical experiments are performed to verify the accuracy of the numerical method, these are based on both subsonic and transonic flows in two and three spatial dimensions.

[1]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[2]  John A. Ekaterinaris,et al.  High-order accurate, low numerical diffusion methods for aerodynamics , 2005 .

[3]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[4]  Patrick J. Roache,et al.  Verification and Validation in Computational Science and Engineering , 1998 .

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

[6]  Rémi Abgrall,et al.  Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes , 2011, J. Comput. Phys..

[7]  Heinrich Lüdeke,et al.  Numerical investigations on the VFE-2 65-degree rounded leading edge delta wing using the unstructured DLR TAU-Code , 2013 .

[8]  Arild Bertelrud,et al.  Prediction of high-lift flows using turbulent closure models , 1997 .

[9]  Eli Turkel,et al.  An implicit high-order spectral difference approach for large eddy simulation , 2010, J. Comput. Phys..

[10]  Thomas J. R. Hughes,et al.  Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations†‡ , 1987 .

[11]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[12]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[13]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[14]  D. Mavriplis,et al.  High-order Discontinuous Galerkin Methods for Turbulent High-lift Flows , 2012 .

[15]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[16]  Moulay D. Tidriri,et al.  Preconditioning Techniques for the Newton-Krylov Solution of Compressible Flows , 1997 .

[17]  R. Abgrall,et al.  Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier-Stokes equations , 2015, J. Comput. Phys..

[18]  P. Spalart.  A One-Equation Turbulence Model for Aerodynamic Flows , 1992 .

[19]  Jaime Peraire,et al.  Navier-Stokes Solution Using Hybridizable Discontinuous Galerkin methods , 2011 .

[20]  Vincent Mousseau,et al.  A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids , 2010, J. Comput. Phys..

[21]  Wei Song,et al.  Finite-Element Method , 2012 .

[22]  R. Abgrall,et al.  Numerical approximation of parabolic problems by residual distribution schemes , 2013 .

[23]  Rémi Abgrall,et al.  Essentially non-oscillatory Residual Distribution schemes for hyperbolic problems , 2006, J. Comput. Phys..

[24]  Rémi Abgrall,et al.  High-Order Preserving Residual Distribution Schemes for Advection-Diffusion Scalar Problems on Arbitrary Grids , 2014, SIAM J. Sci. Comput..

[25]  Leonard R. Herrmann.  Interpretation of Finite Element Procedure as Stress Error Minimization Procedure , 1972 .

[26]  Bram Van Leer,et al.  Discontinuous Galerkin for Diffusion , 2005 .

[27]  K Wieghardt,et al.  On the turbulent friction layer for rising pressure , 1951 .

[28]  V. Selmin,et al.  Finite element methods for nonlinear advection , 1985 .

[29]  Graham F. Carey,et al.  Superconvergent derivatives: A Taylor series analysis , 1989 .

[30]  Takanori Haga,et al.  An Implicit LU-SGS Scheme for the Spectral Volume Method on Unstructured Tetrahedral Grids , 2009 .

[31]  Todd A. Oliver.  A High-Order, Adaptive, Discontinuous Galerkin Finite Element Method for the Reynolds-Averaged Navier-Stokes Equations , 2008 .

[32]  Rémi Abgrall,et al.  Numerical approximation of parabolic problems by means of residual distribution schemes , 2011 .

[33]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

[34]  Herman Deconinck,et al.  Residual Distribution Schemes: Foundations and Analysis , 2007 .

[35]  Rémi Abgrall,et al.  High Order Fluctuation Schemes on Triangular Meshes , 2003, J. Sci. Comput..

[36]  S. Ying,et al.  Prediction of High-Lift Flows Using Turbulent Closure Models , 1997 .

[37]  James M. Luckring,et al.  Experimental Surface Pressure Data Obtained on 65 deg Delta Wing Across Reynolds Number and Mach Number Ranges. Volume 1; Sharp Leading Edge; [conducted in the Langley National Transonic Facility (NTF)] , 1996 .