Assessment of high-order finite volume methods on unstructured meshes for RANS solutions of aeronautical configurations

This paper is concerned with the application of k-exact finite volume methods for compressible Reynolds-Averaged Navier-Stokes computations of flows around aeronautical configurations including the NACA0012, RAE2822, MDA30P30N, ONERA-M6, CRM and DLR-F11. High-order spatial discretisation is obtained with the Weighted Essentially Non-Oscillatory and the Monotone-Upstream Central Scheme for Conservation Laws methods on hybrid unstructured grids in two- and three- dimensions. Schemes of fifth, third and second order comprise the foundation of the analysis, with main findings suggesting that enhanced accuracy can be obtained with at least a third-order scheme. Steady state solutions are achieved with the implicit approximately factored Lower-Upper Symmetric Gauss-Seidel time advancing technique, convergence properties of each scheme are discussed. The Spalart-Allmaras turbulence model is employed where its discretisation with respect to the high-order framework is assessed. A low-Mach number treatment technique is studied, where recovery of accuracy in low speed regions is exemplified. Results are compared with referenced data and discussed in terms of accuracy, grid dependence and computational budget.

[1]  Krzysztof J. Fidkowski,et al.  A Robust Adaptive Solution Strategy for High-Order Implicit CFD Solvers , 2011 .

[2]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[3]  Dimitris Drikakis,et al.  Implicit Large Eddy Simulation of weakly-compressible turbulent channel flow , 2015 .

[4]  Jue Yan,et al.  THE DIRECT DISCONTINUOUS GALERKIN (DDG) METHOD FOR DIFFUSION WITH INTERFACE CORRECTIONS , 2010 .

[5]  Juan-Chen Huang,et al.  Implicit preconditioned WENO scheme for steady viscous flow computation , 2009, J. Comput. Phys..

[6]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[7]  Jeffrey A. Housman,et al.  Preconditioned methods for simulations of low speed compressible flows , 2009 .

[8]  F. Menter.  Two-equation eddy-viscosity turbulence models for engineering applications , 1994 .

[9]  Ruo Li,et al.  A Robust WENO Type Finite Volume Solver for Steady Euler Equations on Unstructured Grids , 2011 .

[10]  Lisa J. Cowles,et al.  High Reynolds number , 1987 .

[11]  Eli Turkel,et al.  Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier-Stokes equations , 2010, J. Comput. Phys..

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

[13]  Antonis F. Antoniadis,et al.  Numerical Accuracy in RANS Computations of High-Lift Multi-element Airfoil and Aicraft Configurations , 2015 .

[14]  Timothy J. Barth,et al.  The design and application of upwind schemes on unstructured meshes , 1989 .

[15]  Z. J. Wang,et al.  Efficient Implicit Non-linear LU-SGS Approach for Compressible Flow Computation Using High-Order Spectral Difference Method , 2008 .

[16]  Eleuterio F. Toro,et al.  ADER schemes for three-dimensional non-linear hyperbolic systems , 2005 .

[17]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[18]  Antony Jameson,et al.  Lower-upper implicit schemes with multiple grids for the Euler equations , 1987 .

[19]  Peter Eliasson,et al.  Investigation of a Half-Model High-Lift Configuration in a Wind Tunnel , 2008 .

[20]  Boris Diskin,et al.  Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations , 2013 .

[21]  Zhi J. Wang,et al.  Fast, Block Lower-Upper Symmetric Gauss-Seidel Scheme for Arbitrary Grids , 2000 .

[22]  Claus-Dieter Munz,et al.  A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes , 2007, J. Comput. Phys..

[23]  Boris Diskin,et al.  Comparison of Node-Centered and Cell-Centered Unstructured Finite-Volume Discretizations: Inviscid Fluxes , 2011 .

[24]  Christopher L. Rumsey,et al.  Overview and Summary of the Second AIAA High-Lift Prediction Workshop , 2014 .

[25]  Edward N. Tinoco,et al.  Summary of the Fourth AIAA Computational Fluid Dynamics Drag Prediction Workshop , 2014 .

[26]  D. Drikakis,et al.  Large eddy simulation using high-resolution and high-order methods , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[28]  Hailiang Liu,et al.  The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems , 2008, SIAM J. Numer. Anal..

[29]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[30]  Ben Thornber,et al.  Steady Turbulent Flow Computations Using a Low Mach Fully Compressible Scheme , 2014 .

[31]  Elizabeth M. Lee-Rausch,et al.  Three-dimensional effects in multi-element high lift computations , 2002 .

[32]  Rainald Löhner,et al.  A fast, matrix-free implicit method for compressible flows on unstructured grids , 1998 .

[33]  Clinton P. T. Groth,et al.  High-Order Solution-Adaptive Central Essentially Non-Oscillatory (CENO) Method for Viscous Flows , 2011 .

[34]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[35]  Panagiotis Tsoutsanis,et al.  Azure: an advanced CFD software suite based on high-resolution and high-order methods , 2015 .

[36]  D. Drikakis,et al.  Comparison of structured- and unstructured-grid, compressible and incompressible methods using the vortex pairing problem , 2015 .

[37]  Y. Mukaigawa,et al.  Large Deviations Estimates for Some Non-local Equations I. Fast Decaying Kernels and Explicit Bounds , 2022 .

[38]  O. Friedrich,et al.  Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids , 1998 .

[39]  Dimitri J. Mavriplis,et al.  hp-Adaptive Discontinuous Galerkin Solver for the Navier-Stokes Equations , 2012 .

[40]  W. K. Anderson,et al.  Navier-Stokes Computations and Experimental Comparisons for Multielement Airfoil Configurations , 1993 .

[41]  D. Drikakis,et al.  Uniformly high-order schemes on arbitrary unstructured meshes for advection–diffusion equations , 2011 .

[42]  Eli Turkel,et al.  Assessment of Preconditioning Methods for Multidimensional Aerodynamics , 1997 .

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

[44]  Dimitris Drikakis,et al.  WENO schemes on arbitrary unstructured meshes for laminar, transitional and turbulent flows , 2014, J. Comput. Phys..

[45]  David L. Darmofal,et al.  Impact of Turbulence Model Irregularity on High-Order Discretizations , 2009 .

[46]  Ching-Hua Wang,et al.  Implicit Weighted Essentially Nonoscillatory Schemes with Antidiffusive Flux for Compressible Viscous Flows , 2009 .

[47]  Stuart E. Rogers,et al.  A comparison of turbulence models in computing multi-element airfoil flows , 1994 .

[48]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[49]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

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

[51]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[52]  Michael A. Leschziner,et al.  Average-State Jacobians and Implicit Methods for Compressible Viscous and Turbulent Flows , 1997 .

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

[54]  Kazuhiro Nakahashi,et al.  Applications of unstructured hybrid grid method to high‐Reynolds number viscous flows , 1999 .

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

[56]  J. Mixter.  Fast , 2012 .

[57]  Panagiotis Tsoutsanis,et al.  High-Order Methods for Hypersonic Shock Wave Turbulent Boundary Layer Interaction Flow , 2015 .

[58]  Michael Dumbser,et al.  Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier–Stokes equations , 2010 .

[59]  Shia-Hui Peng,et al.  Results from the Second AIAA CFD High-Lift Prediction Workshop Using Edge , 2015 .

[60]  D. Hill,et al.  Unstructured-Grid Third-Order Finite Volume Discretization Using a Multistep Quadratic Data-Reconstruction Method , 2010 .

[61]  Frank W. Spaid.  High Reynolds Number, Multielement Airfoil Flowfield Measurements , 2000 .

[62]  Per-Olof Persson,et al.  Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier--Stokes Equations , 2008, SIAM J. Sci. Comput..

[63]  R. J. R. Williams,et al.  An improved reconstruction method for compressible flows with low Mach number features , 2008, J. Comput. Phys..

[64]  P. Frederickson,et al.  Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction , 1990 .

[65]  Yan Chao,et al.  A Spalart–allmaras Turbulence Model Implementation for High-order Discontinuous Galerkin Solution of the Reynolds-averaged Navier-stokes Equations , 2016 .

[66]  João Luiz F. Azevedo,et al.  High-Order Unstructured Essentially Nonoscillatory and Weighted Essentially Nonoscillatory Schemes for Aerodynamic Flows , 2006 .

[67]  Michael Dumbser,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[68]  C. Rumsey,et al.  Grid-Adapted FUN3D Computations for the Second High Lift Prediction Workshop , 2015 .

[69]  Ralf Rudnik,et al.  DLR Contribution to the 2nd High Lift Prediction Workshop , 2014 .

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

[71]  Panagiotis Tsoutsanis,et al.  High-order schemes on mixed-element unstructured grids for aerodynamic flows , 2012 .

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

[73]  Vladimir A. Titarev,et al.  WENO schemes on arbitrary mixed-element unstructured meshes in three space dimensions , 2011, J. Comput. Phys..

[74]  Michael Dumbser,et al.  A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes , 2004, J. Comput. Phys..

[75]  Fermín Navarrina,et al.  Finite volume solvers and movingleast-squares approximations for thecompressible Navier-Stokes equations onunstructured grids , 2007 .

[76]  Dimitri J. Mavriplis,et al.  Multigrid solution of compressible turbulent flow on unstructured meshes using a two-equation model , 1991 .

[77]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[78]  V. Schmitt,et al.  Pressure distributions on the ONERA M6 wing at transonic Mach numbers , 1979 .

[79]  D. Kwak,et al.  Three-dimensional incompressible Navier-Stokes solver using lower-upper symmetric-Gauss-Seidel algorithm , 1991 .

[80]  H. T. Huynh,et al.  High-Order Methods for Computational Fluid Dynamics: A Brief Review of Compact Differential Formulations on Unstructured Grids , 2013 .

[81]  M. Dumbser,et al.  High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows , 2013, 1304.4816.

[82]  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..

[83]  C. Ollivier-Gooch,et al.  A high-order-accurate unstructured mesh finite-volume scheme for the advection-diffusion equation , 2002 .

[84]  A. Jameson,et al.  Multi-Element High-Lift Configuration Design Optimization Using Viscous Continuous Adjoint Method , 2004 .

[85]  Michael Dumbser,et al.  Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems , 2007, J. Comput. Phys..

[86]  Edward N. Tinoco,et al.  Summary of Data from the Fifth AIAA CFD Drag Prediction Workshop , 2013 .

[87]  Rainald Löhner,et al.  A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids , 2007, J. Comput. Phys..