SUMMARYAfter several years of planning, the 1st International Workshop on High‐Order CFD Methods was successfully held in Nashville, Tennessee, on January 7–8, 2012, just before the 50th Aerospace Sciences Meeting. The American Institute of Aeronautics and Astronautics, the Air Force Office of Scientific Research, and the German Aerospace Center provided much needed support, financial and moral. Over 70 participants from all over the world across the research spectrum of academia, government labs, and private industry attended the workshop. Many exciting results were presented. In this review article, the main motivation and major findings from the workshop are described. Pacing items requiring further effort are presented. Copyright © 2013 John Wiley & Sons, Ltd.

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 ...

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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.

A numerical procedure to predict the impeller/volute interaction in a single-stage centrifugal compressor is presented. The method couples a three-dimensional unsteady flow calculation in the impeller with a three-dimensional time-averaged flow calculation in the volute through an iterative updating of the boundary conditions on the interface of both calculation domains. The method has been used to calculate the flow in a compressor with an external volute at off-design operation. Computed circumferential variations of flow angles, total temperature and pressure are shown and compared with measurements. The good agreement between the predictions and measurements confirms the validity of the approach.

At high Reynolds numbers the use of explicit in time compressible flow simulations with spectral/hp element discretization can become significantly limited by time step. To alleviate this limitation we extend the capability of the spectral/hp element open-source software framework, Nektar++, to include an implicit discontinuous Galerkin compressible flow solver. The integration in time is carried out by a singly diagonally implicit Runge-Kutta method. The non-linear system arising from the implicit time integration is iteratively solved by the Jacobian-free Newton Krylov (JFNK) method. A favorable feature of the JFNK approach is its extensive use of the explicit operators available from the previous explicit in time implementation. The functionalities of different building blocks of the implicit solver are analyzed from the point of view of software design and placed in appropriate hierarchical levels in the C++ libraries. In the detailed implementation, the contributions of different parts of the solver to computational cost, memory consumption and programming complexity are also analyzed. A combination of analytical and numerical methods is adopted to simplify the programming complexity in forming the preconditioning matrix. The solver is verified and tested using cases such as manufactured compressible tex, turbulent flow over a circular cylinder at Re = 3900 and shock wave boundary-layer interaction. The results show that the implicit solver can speed-up the simulations while maintaining good simulation accuracy.