Industry-Relevant Implicit Large-Eddy Simulation of a High-Performance Road Car via Spectral/hp Element Methods

We present a successful deployment of high-fidelity Large-Eddy Simulation (LES) technologies based on spectral/hp element methods to industrial flow problems, which are characterized by high Reynolds numbers and complex geometries. In particular, we describe the numerical methods, software development and steps that were required to perform the implicit LES of a real automotive car, namely the Elemental Rp1 model. To the best of the authors' knowledge, this simulation represents the first fifth-order accurate transient LES of an entire real car geometry. Moreover, this constitutes a key milestone towards considerably expanding the computational design envelope currently allowed in industry, where steady-state modelling remains the standard. To this end, a number of novel developments had to be made in order to overcome obstacles in mesh generation and solver technology to achieve this simulation, which we detail in this paper. The main objective is to present to the industrial and applied mathematics community, a viable pathway to translate academic developments into industrial tools, that can substantially advance the analysis and design capabilities of high-end engineering stakeholders. The novel developments and results were achieved using the academic-driven open-source framework Nektar++.

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

[2]  N. Kuiper,et al.  On C1-isometric imbeddings. II , 1955 .

[3]  Jon Louis Bentley,et al.  Multidimensional binary search trees used for associative searching , 1975, CACM.

[4]  E. Tadmor,et al.  Convergence of spectral methods for nonlinear conservation laws. Final report , 1989 .

[5]  S. Orszag,et al.  High-order splitting methods for the incompressible Navier-Stokes equations , 1991 .

[6]  George Em Karniadakis,et al.  Nodes, modes and flow codes , 1993 .

[7]  Paul Fischer,et al.  PROJECTION TECHNIQUES FOR ITERATIVE SOLUTION OF Ax = b WITH SUCCESSIVE RIGHT-HAND SIDES , 1993 .

[8]  Gregory A. Blaisdell,et al.  The effect of the formulation of nonlinear terms on aliasing errors in spectral methods , 1996 .

[9]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[10]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[11]  George Em Karniadakis,et al.  A Spectral Vanishing Viscosity Method for Large-Eddy Simulations , 2000 .

[12]  Paul F. Fischer,et al.  Fast Parallel Direct Solvers for Coarse Grid Problems , 2001, J. Parallel Distributed Comput..

[13]  D. Gottlieb,et al.  Spectral methods for hyperbolic problems , 2001 .

[14]  Robert M. O'Bara,et al.  Towards curvilinear meshing in 3D: the case of quadratic simplices , 2001, Comput. Aided Des..

[15]  Julia S. Mullen,et al.  Filter-based stabilization of spectral element methods , 2001 .

[16]  M. A. Casarin,et al.  Low-energy basis preconditioning for elliptic substructured solvers based on unstructured spectral/ hp element discretization , 2001 .

[17]  L. Margolin,et al.  A rationale for implicit turbulence modelling , 2001 .

[18]  Michael J. Pratt,et al.  Introduction to ISO 10303 - the STEP Standard for Product Data Exchange. pp , 2001, J. Comput. Inf. Sci. Eng..

[19]  S. Sherwin,et al.  Mesh generation in curvilinear domains using high‐order elements , 2002 .

[20]  Francis Loth,et al.  Spectral Element Methods for Transitional Flows in Complex Geometries , 2002, J. Sci. Comput..

[21]  Jie Shen,et al.  Velocity-Correction Projection Methods for Incompressible Flows , 2003, SIAM J. Numer. Anal..

[22]  Robert M. O'Bara,et al.  Automatic p-version mesh generation for curved domains , 2004, Engineering with Computers.

[23]  S. Pope.  Ten questions concerning the large-eddy simulation of turbulent flows , 2004 .

[24]  G. Karniadakis,et al.  Spectral/hp Element Methods for Computational Fluid Dynamics , 2005 .

[25]  Robert M. O'Bara,et al.  Adaptive mesh generation for curved domains , 2005 .

[26]  S. Sherwin,et al.  STABILISATION OF SPECTRAL/HP ELEMENT METHODS THROUGH SPECTRAL VANISHING VISCOSITY: APPLICATION TO FLUID MECHANICS MODELLING , 2006 .

[27]  Robert Michael Kirby,et al.  Filtering in Legendre spectral methods , 2008, Math. Comput..

[28]  Per-Olof Persson,et al.  Curved mesh generation and mesh refinement using Lagrangian solid mechanics , 2008 .

[29]  Spencer J. Sherwin,et al.  Parallel performance of the coarse space linear vertex solver and low energy basis preconditioner for spectral/hp elements , 2009, Parallel Comput..

[30]  A. U.S.,et al.  Curved Mesh Generation and Mesh Refinement using Lagrangian Solid Mechanics , 2009 .

[31]  M. Shephard,et al.  Curved boundary layer meshing for adaptive viscous flow simulations , 2010 .

[32]  Spencer J. Sherwin,et al.  A generic framework for time-stepping partial differential equations (PDEs): general linear methods, object-oriented implementation and application to fluid problems , 2011 .

[33]  Jörg Stiller,et al.  Adapting the spectral vanishing viscosity method for large-eddy simulations in cylindrical configurations , 2012, J. Comput. Phys..

[34]  Mariano Vázquez,et al.  Variational multiscale stabilization of high-order spectral elements for the advection-diffusion equation , 2012, J. Comput. Phys..

[35]  Christophe Geuzaine,et al.  Robust untangling of curvilinear meshes , 2013, J. Comput. Phys..

[36]  Gregor Gassner,et al.  A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods , 2013, SIAM J. Sci. Comput..

[37]  Bruno Lévy,et al.  Particle-based anisotropic surface meshing , 2013, ACM Trans. Graph..

[38]  K. Morgan,et al.  The generation of arbitrary order curved meshes for 3D finite element analysis , 2013 .

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

[40]  Robert Michael Kirby,et al.  High-order spectral/hp element discretisation for reaction–diffusion problems on surfaces: Application to cardiac electrophysiology , 2014, J. Comput. Phys..

[41]  David Moxey,et al.  23rd International Meshing Roundtable (IMR23) A thermo-elastic analogy for high-order curvilinear meshing with control of mesh validity and quality , 2014 .

[42]  J. Avrin,et al.  Convergence results for a class of spectrally hyperviscous models of 3-D turbulent flow , 2014 .

[43]  William Gropp,et al.  CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences , 2014 .

[44]  Suchuan Dong,et al.  A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains , 2014, J. Comput. Phys..

[45]  Franco Dassi,et al.  Curvature-adapted Remeshing of CAD Surfaces☆ , 2014 .

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

[47]  Spencer J. Sherwin,et al.  Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods , 2015, J. Comput. Phys..

[48]  Spencer J. Sherwin,et al.  Dealiasing techniques for high-order spectral element methods on regular and irregular grids , 2015, J. Comput. Phys..

[49]  Robert Michael Kirby,et al.  Nektar++: An open-source spectral/hp element framework , 2015, Comput. Phys. Commun..

[50]  Xevi Roca,et al.  Optimization of a regularized distortion measure to generate curved high‐order unstructured tetrahedral meshes , 2015 .

[51]  Giacomo Castiglioni,et al.  A numerical dissipation rate and viscosity in flow simulations with realistic geometry using low-order compressible Navier–Stokes solvers , 2015 .

[52]  Spencer J. Sherwin,et al.  Modified Equation Analysis for the Discontinuous Galerkin Formulation , 2015 .

[53]  S. Sherwin,et al.  An isoparametric approach to high-order curvilinear boundary-layer meshing , 2015 .

[54]  Spencer J. Sherwin,et al.  Implicit Large-Eddy Simulation of a Wingtip Vortex , 2016 .

[55]  Per-Olof Persson,et al.  High-order unstructured curved mesh generation using the Winslow equations , 2016, J. Comput. Phys..

[56]  Antony Jameson,et al.  A study on the numerical dissipation of the Spectral Difference method for freely decaying and wall-bounded turbulence , 2016 .

[57]  Spencer J. Sherwin,et al.  Optimising the performance of the spectral/hp element method with collective linear algebra operations , 2016 .

[58]  Ralf Hartmann,et al.  Generation of unstructured curvilinear grids and high‐order discontinuous Galerkin discretization applied to a 3D high‐lift configuration , 2016 .

[59]  Spencer J. Sherwin,et al.  Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: Insights into spectral vanishing viscosity , 2016, J. Comput. Phys..

[60]  A. J. Gil,et al.  A unified approach for a posteriori high-order curved mesh generation using solid mechanics , 2016 .

[61]  Paul G. Tucker,et al.  Implicit large eddy simulation using the high‐order correction procedure via reconstruction scheme , 2016 .

[62]  Spencer J. Sherwin,et al.  Direct numerical simulations of the flow around wings with spanwise waviness , 2017, Journal of Fluid Mechanics.

[63]  John H. Bucklow,et al.  An automated workflow for high quality CFD meshing using the 3D medial object , 2017 .

[64]  Michael Turner,et al.  High-order mesh generation for CFD solvers , 2017 .

[65]  Spencer J. Sherwin,et al.  On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence , 2017, J. Comput. Phys..

[66]  Sylvain Laizet,et al.  Numerical dissipation vs. subgrid-scale modelling for large eddy simulation , 2017, J. Comput. Phys..

[67]  Spencer J. Sherwin,et al.  An LES Setting for DG-Based Implicit LES with Insights on Dissipation and Robustness , 2017 .

[68]  R. Moura.  On the use of spectral element methods for under-resolved simulations of transitional and turbulent flows , 2017 .

[69]  Spencer J. Sherwin,et al.  A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations , 2017, J. Comput. Phys..

[70]  Spencer J. Sherwin,et al.  Spatial eigensolution analysis of energy-stable flux reconstruction schemes and influence of the numerical flux on accuracy and robustness , 2018, J. Comput. Phys..

[71]  Spencer J. Sherwin,et al.  Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES , 2017, Computers & Fluids.

[72]  David Moxey,et al.  Curvilinear mesh generation using a variational framework , 2017, Comput. Aided Des..

[73]  Mesbah Uddin,et al.  Turbulence modeling effects on the CFD predictions of flow over a NASCAR Gen 6 racecar , 2018 .

[74]  Robert H. Bush,et al.  Recommendations for Future Efforts in RANS Modeling and Simulation , 2019, AIAA Scitech 2019 Forum.

[75]  Spencer J. Sherwin,et al.  Implicit LES Approaches via Discontinuous Galerkin Methods at Very Large Reynolds , 2019, Direct and Large-Eddy Simulation XI.

[76]  Mesbah Uddin,et al.  Turbulence Modeling Effects on the CFD Predictions of Flow over a Detailed Full-Scale Sedan Vehicle , 2019, Fluids.

[77]  Gianmarco Mengaldo,et al.  Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations , 2018, Computer Methods in Applied Mechanics and Engineering.

[78]  Spencer J. Sherwin,et al.  Spatial eigenanalysis of spectral/hp continuous Galerkin schemes and their stabilisation via DG-mimicking spectral vanishing viscosity for high Reynolds number flows , 2020, J. Comput. Phys..

[79]  Robert Michael Kirby,et al.  Nektar++: enhancing the capability and application of high-fidelity spectral/hp element methods , 2019, Comput. Phys. Commun..

[80]  Danna Zhou,et al.  d. , 1840, Microbial pathogenesis.

[81]  Nodes , 2021, Understanding Energy Innovation.

[82]  P. Alam,et al.  R , 1823, The Herodotus Encyclopedia.