SUMMARYThis paper focuses on the assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number. The Taylor–Green vortex at Re = 1600 is considered. The results are compared with those obtained using a pseudo‐spectral solver, converged on a 5123 grid and taken as the reference. The temporal evolution of the dissipation rate, visualisations of the vortical structures and the kinetic energy spectrum at the instant of maximal dissipation are compared to assess the results. At an effective resolution of 2883, the fourth‐order accurate discontinuous Galerkin method (DGM) solution (p = 3) is already very close to the pseudo‐spectral reference; the error on the dissipation rate is then essentially less than a percent, and the vorticity contours at times around the dissipation peak overlap everywhere. At a resolution of 3843, the solutions are indistinguishable. Then, an order convergence study is performed on the slightly under‐resolved grid (resolution of 1923). From the fourth order, the decrease of the error is no longer significant when going to a higher order. The fourth‐order DGM is also compared with an energy conserving fourth‐order finite difference method (FD4). The results show that, for the same number of DOF and the same order of accuracy, the errors of the DGM computation are significantly smaller. In particular, it takes 7683 DOF to converge the FD4 solution. Finally, the method is also successfully applied on unstructured high quality meshes. It is found that the dissipation rate captured is not significantly impacted by the element type. However, the element type impacts the energy spectrum in the large wavenumber range and thus the small vortical structures. In particular, at the same resolution, the results obtained using a tetrahedral mesh are much noisier than those obtained using a hexahedral mesh. Those obtained using a prismatic mesh are already much better, yet still slightly noisier. Copyright © 2013 John Wiley & Sons, Ltd.

An implicit time integration, high-order discontinuous Galerkin method is assessed on the DNS of the flow in the T106C cascade at low Reynolds number. This code, aimed at providing high orders of accuracy on unstructured meshes for DNS and LES simulations on industrial geometries, was previously successfully assessed on fundamental, academic test cases. The computational results are compared to the experimental values and literature, and the obtained flow field characteristics are discussed. Although adequate resolution is supposed to be attained, discrepancies with respect to the experiment are found. These differences were furthermore consistently found by all authors in the workshop on high-order methods for CFD. The origins are therefore conjectured to result from insufficient adequation between computational setup and experiments, as no modeling is assumed. A plan for further investigation is proposed.

between the flexibility of industrial finite volume methods (FVMs) and the accuracy of academic solvers, such as high-order finite difference (FDM) or pseudo-spectral (PSM) methods. Because of their computational compacity, most of these methods-in particular, those with discontinuous interpolation-also provide an excellent serial and parallel computational efficiencies. In view of these advantages, it is mainly in the field of scale-resolving simulations of industrial turbulent flows, that is direct numerical simulation (DNS) and large eddy simulation (LES), that these methods offer the best perspectives. Indeed, as DNS and LES require a nearly flawless representation of the (resolved) turbulent scales, current industrial solvers require huge computational resources to provide sufficient accuracy, and hence, up to date, most computations appear to be under-resolved (see Tucker [6,7] for a recent review in turbomachinery).In this paper, we focus on the DGM combined with a symmetric interior penalty (SIP) stabilisation.For the past few years, DGM has been assessed for compressible and incompressible DNS of simple and complex flow configurations [8][9][10]. Those investigations have highlighted the advantages of DGM for these kind of problems. Indeed, the very low dispersion of the method, typical for Galerkin approaches, combined to a dissipation targeted on the smallest scales, offers an accuracy similar to spectral methods (e.g. [11]). These properties also indicate the potential of the method to perform accurate implicit LES (ILES), that is where the dissipation given by the discretisation acts like a subgrid-scale (SGS) model. Several recent publications [12,13], where the method is applied to under-resolved flows, seem to corroborate the accuracy of ILES/DGM. Nevertheless, those studies only validate basic flow statistics (integral values, velocity profiles, etc.) without true reference results and concern transitional rather than fully turbulent conditions. We therefore believe that a validation on more canonical, fully turbulent cases is therefore required to assess whether the ILES/DGM can really compete with state-of-the-art SGS models and academic high-accuracy solvers.This study presents the validation of the compressible version of Argo, the DGM solver developed at Cenaero, for the ILES of equilibrium turbulent flows. The solver has already been intensively assessed for DNS of canonical flows [11] as well as on more industrial cases [14,15], partly during the European FP7 research project IDIHOM. The first sections of the paper describe the numerical method and discuss the ILES strategy. Then, the method is investigated for free turbulent flows on the simulation of homogeneous isotropic turbulence (HIT) at very high Reynolds number. Very few studies on HIT using unstructured high-order methods can be found in literature and the conclusions are not directly applicable to the discretisation and LES approach under study here. To our knowledge, only one publication is dedicated to the assessment of LES of ...

Wall-resolved Large-Eddy Simulations (LES) are still limited to moderate Reynolds number flows due to the high computational cost required to capture the inner part of the boundary layer. Wall-modeled LES (WMLES) provide more affordable LES by modeling the near-wall layer. Wall function-based WMLES solve LES equations up to the wall, where the coarse mesh resolution essentially renders the calculation under-resolved. This makes the accuracy of WMLES very sensitive to the behavior of the numerical method. Therefore, best practice rules regarding the use and implementation of WMLES cannot be directly transferred from one methodology to another regardless of the type of discretization approach. Whilst numerous studies present guidelines on the use of WMLES, there is a lack of knowledge for discontinuous finite-element-like high-order methods. Incidentally, these methods are increasingly used on the account of their high accuracy on unstructured meshes and their strong computational efficiency. The present paper proposes best practice guidelines for the use of WMLES in these methods. The study is based on sensitivity analyses of turbulent channel flow simulations by means of a Discontinuous Galerkin approach. It appears that good results can be obtained without the use of a spatial or temporal averaging. The study confirms the importance of the wall function input data location and suggests to take it at the bottom of the second off-wall element. These data being available through the ghost element, the suggested method prevents the loss of computational scalability experienced in unstructured WMLES. The study also highlights the influence of the polynomial degree used in the wall-adjacent element. It should preferably be of even degree as using polynomials of degree two in the first off-wall element provides, surprisingly, better results than using polynomials of degree three.

This paper follows a previous one that was dealing with high-quality surface remeshing using harmonic maps (Int. J. Numer. Meth. Engng 2010; 83:403-425). In (Int. J. Numer. Meth. Engng 2010; 83:403-425), it has been demonstrated that harmonic parametrizations can be used as input for surface meshers to produce high-quality triangulations. However, two important limitations were pointed out, namely surfaces with high genus and/or of large aspect ratio. This paper addresses those two issues. We first develop a multiscale version of the harmonic parametrization of (Int. J. Numer. Meth. Engng 2010; 83:403-425) and then combine it with a multilevel partitioning algorithm to come up with an automatic remeshing algorithm that overcomes the above-mentioned limitations of harmonic maps. The overall procedure is implemented in the open-source mesh generator Gmsh (Int. E. MARCHANDISE ET AL. local checks to guarantee its bijectivity: we called that approach hybrid [1]. Non-linear algorithms are considered to be slow and will not be considered here.The concept of discrete harmonic map has been described first by Eck et al. [2]. Floater [3] introduced a concept of convex combination map that guarantees that the discrete mapping is one-to-one. Sheffer and Strurler [4] presented a constrained minimization approach, the so-called angle-based flattening (ABF), such that the variation between the set of angles of an original mesh and one of the 2D flattened version is minimized. In order to obtain a valid and flippingfree parametrization, several additional constrained algorithms are developed. More recently, they improved the performance of the ABF technique by using an advanced numerical approach and a hierarchical technique [5]. Much research has also been incorporated within the theory of differential geometry. For example, Levy et al. [6] apply the Cauchy Riemann equations to compute a least-square conformal map (LSCM). This approach is quite similar to that of Desbrun and co-workers [7] that minimize a combination of Dirichlet and distortion energy to compute a conformal map for interactive geometry remeshing. More recently, Mullen et al. [8] have also presented spectral computation of the conformal map [8].Recently, discrete parametrization methods have received some attention from non-CGspecialists and in particular, in the domain of finite element mesh generation. Here, the target application is clearly surface meshing, especially for surfaces that are defined by a triangulation only or for cross-patch meshing. In [1], we have demonstrated the use of such mappings as parametrizations allowed to generate quality finite element meshes. Yet, two important issues have not been completely addressed: reparameterization techniques fail when the surface has a large aspect ratio and/or when it has a high genus. Those issues are critical in the domain of mesh generation, maybe more than in CG. This paper aims at addressing those two issues.In [1], we have demonstrated that parametric coordinates computed using a discrete harmonic ma...

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The design of a modular multi-physics high-order space-time finite-element framework is presented together with its extension to allow monolithic coupling of different physics. One of the main objectives of the framework is to perform efficient high-fidelity simulations of capsule/parachute systems. This problem requires simulating multiple physics including, but not limited to, the compressible Navier-Stokes equations, the dynamics of a moving body with mesh deformations and adaptation, the linear shell equations, non-reflective boundary conditions and wall modeling. The solver is based on high-order space-time finite element methods. Continuous, discontinuous and C 1 -discontinuous Galerkin methods are implemented, allowing one to discretize various physical models. Tangent and adjoint sensitivity analysis are also targeted in order to conduct gradient-based optimization, error estimation, mesh adaptation, and flow control, adding another layer of complexity to the framework. The decisions made to tackle these challenges are presented. The discussion focuses first on the "single-physics" solver and later on its extension to the monolithic coupling of different physics. The implementation of different physics modules, relevant to the capsule/parachute system, are also presented. Finally, examples of coupled computations are presented, paving the way to the simulation of the full capsule/parachute system. * USRA/NASA Postdoctoral Program Fellow. † Science and Technology Corp. ‡ NASA ARC. AIAA Member.

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