© 2023, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.Up to this day, the Computational Fluid Dynamics (CFD) field struggles to generate accurate and computationally viable turbulent flow simulations for aeronautical problems. The absence of a proper spatial resolution reduces the accuracy of simulations and may lead to nonphysical results and numerical instabilities. This problem may be addressed by increasing the number of degrees of freedom in the simulation. Since this also leads to higher computational costs, this process must be performed parsimoniously and focus on where it is the most efficient. However, the process of identification and refinement of those regions can be far from trivial. The current work is an initial step to investigate the performance of adaptation drivers that can be used to make industrial simulations more viable. The drivers are based on a jump indicator for high-order spectral/hp schemes. It takes the difference between averaged values on overlapping borders of two different elements as a measurement of error. The chosen adaptation method is a p-adaptation framework that increases the polynomial order of 10% of the mesh elements. The governing equations employed in the study are the two-dimensional Navier-Stokes equations, and the simulated test case is one of a tilted flat plate.

© 2023, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.Recently, waviness applied on leading edge of airfoils has been extensively researched. As a biomimetic solution, the also called tubercle has brought up many insights on passive flow control mechanisms and inspired other studies. Therefore, the present study aims to investigate the potential of waviness now applied on the trailing edge of airfoils. For this purpose, experimental tests in wind tunnel is carried out in order to understand the effects of different wavy geometries on the flow. A NACA 0020 airfoil is selected as a baseline profile and three configurations of waviness are tested: A = 0.11c, λ = 0.40c; A = 0.03c, λ = 0.40c; and A = 0.03c, λ = 0.11c. The phenomena are evaluated measuring forces at a single Reynolds numbers of 250,000, and correlating it with a flow topology analysis provided by an oil flow visualization technique. Main results show that the wavy model with parameters A = 0.11c, λ = 0.40c presents the best aerodynamic efficiency, with similar lift values compared to the baseline profile, but with reduced drag coefficients, also briefly delaying stall separation. Flow visualization shows that this case has larger regions of attached flow.

© 2022 Elsevier Inc.In recent years, different dispersion-diffusion (eigen)analyses have been developed and used to assess various spectral element methods (SEMs) with regards to accuracy and stability, both of which are very important aspects for under-resolved computations of transitional and turbulent flows. Not surprisingly, eigenanalysis has been used recurrently to probe the inner-workings of SEM-based implicit LES approaches, where numerical dissipation acts alone in lieu of a subgrid model. In this study we present and discuss an intriguing linear mechanism that causes energy transfer across Fourier modes as seen in the energy spectrum of SEM computations. Despite its linear nature, this mechanism has not been considered in eigenanalyses so far, possibly due to its connection to the often overlooked multiple eigencurves feature of periodic eigenanalysis. As we unveil the mechanism in the simplified context of linear advection, we point out how its effects might take place in actual turbulence simulations. In particular, we highlight how taking it into account in eigenanalysis can improve dissipation estimates in wavenumber space, potentially allowing for a superior correlation between dissipation estimates and energy spectra measured in SEM-based eddy-resolving turbulence computations.

© 2022 by the authors.A numerical investigation is proposed to explore the flow past a novel wavy circular cylinder as a passive flow control, whose shape is determined by a sinusoidal function applied to its leading edge line, similar to studies with wavy leading-edge airfoils. The latter are motivated by the wavy-shaped tubercles found in the flippers of humpback whales, which are believed to improve their maneuverability. Our attempt is, therefore, to assess the effects of leading-edge waviness now on a simpler and canonical geometry: circular cylinders. The present work relies on iLES simulations conducted with Nektar++ at a Reynolds number of 3900. Besides the straight cylinder, two wavy geometries are assessed, which are determined by a single wavelength of 37.5% for two amplitudes, 3% and 11%, based on the mean diameter of the wavy cylinder. Our results showed that, contrary to what is usually the case with traditional wavy cylinders at similar Reynolds numbers, waviness caused a reduction in the near-wake recirculation length and an increase in the mean near-wake turbulent kinetic energy compared to the straight cylinder. This was followed by a reduction in base pressure (up to about 36%) leading to a rise in lift oscillations and also to a significant increase in the mean drag coefficient of up to about 28%. An attempt to detail the flow phenomena is provided, evidencing the emergence of counter-rotating pairs of streamwise vortices between peaks. It is argued that the differences observed in recirculation length, turbulent kinetic energy, and force coefficients start even prior to the formation of these coherent structures and end up with interactions with the near wake.

© 2022, American Institute of Aeronautics and Astronautics Inc, AIAA. All rights reserved.Flow over a humpback whale flippers bio-inspired wavy cylinder is experimentally investigated. Besides the smooth model, a selection of a set of 4 wave combinations (2 amplitudes x 2 wavelengths) is compared using pressure distribution and aerodynamic forces. The study is performed in a wind tunnel at Reynolds numbers ranging from 3.9 × 104 to 2 × 105, within the sub-critical regime. The most notable results show that, for the 12% wavelength, the 3% and 11% amplitudes have opposite effects, with a drag coefficient reduction of up to 25%, and an increase of up to 25%, respectively. Flow visualizations shows the formation of three-dimensional laminar separation bubbles and the action of counter-rotating vortex pairs, with a shift in the separation line and a change in the base pressure, which suggest the mechanisms behind drag variation.

© 2021 Elsevier B.V.One of the strengths of the discontinuous Galerkin (DG) method has been its balance between accuracy and robustness, which stems from DG's intrinsic (upwind) dissipation being biased towards high frequencies/wavenumbers. This is particularly useful in high Reynolds-number flow simulations where limitations on mesh resolution typically lead to potentially unstable under-resolved scales. In continuous Galerkin (CG) discretisations, similar properties are achievable through the addition of artificial diffusion such as spectral vanishing viscosity (SVV). However although SVV is recognised as very useful in CG-based high-fidelity turbulence simulations, this approach has been observed to be sub-optimal when compared to DG at intermediate polynomials orders (P≈3). In this paper we explore an alternative stabilisation approach through the introduction of a continuous interior penalty on the gradient discontinuity at elemental boundaries, which we refer to as a gradient jump penalisation (GJP). Analogous to DG methods, this introduces a penalisation at the elemental interfaces as opposed to the interior element stabilisation of SVV. Detailed eigenanalysis of the GJP approach shows its potential as equivalent (sometimes superior) to DG dissipation and hence superior to previous SVV approaches. Through eigenanalysis, a judicious choice of GJP's P-dependent scaling parameter is made and found to be consistent with previous a-priori error analysis. The favourable properties of the GJP stabilisation approach are also supported by turbulent flow simulations of the incompressible Navier–Stokes equation, as we achieve higher quality flow solutions at P=3 using GJP, whereas SVV performs marginally worse at P=5 with twice as many degrees of freedom in total.

\bigcirc c 2021 Society for Industrial and Applied MathematicsWe 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 high-order accurate transient LES of an entire real car geometry. Moreover, it constitutes a key milestone toward considerably expanding the computational design envelope currently allowed in industry, where steady-state modeling remains the standard. 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 translating 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++.

© 2019 Elsevier Inc.This study considers the spatial eigensolution analysis of spectral/hp continuous Galerkin (CG) schemes, complementing a recent work by Moura et al. (2016) [15] which addressed CG's temporal analysis. While the latter assumes periodic boundary conditions, the spatial approach presumes inflow/outflow type conditions and therefore provides insights for a different class of problems. The linear advection-diffusion problem is considered for a wide range of Péclet numbers, allowing for viscous effects at different intensities. The inviscid (linear advection) case receives particular attention owing to the manifestation of peculiar characteristics previously observed for discontinuous Galerkin (DG) schemes in the limit of strong over-upwinding. These effects are discussed in detail due to their potential to negatively affect solution quality and numerical stability of under-resolved simulations at high Reynolds numbers. The spectral vanishing viscosity (SVV) technique is subsequently considered as a natural stabilization strategy, in the context of linear advection. An optimization procedure is employed to match SVV diffusion levels to those of DG at appropriate polynomial orders. The resulting CG-SVV discretisations are tested against under-resolved computations of spatially developing vortex-dominated flows and display excellent robustness at high Reynolds numbers along with superior eddy-resolving characteristics at higher polynomial orders. This highlights the importance of appropriate stabilization techniques to improve the potential of spectral/hp CG methods for high-fidelity simulations of transitional and turbulent flows, including implicit LES / under-resolved DNS approaches.

© 2020, The Author(s).Erosion wear is the loss of mass on the surface of a solid body in contact with a moving fluid. When the fluid under study has solid particulate material or sediments, the erosion could be significant (Kulu and Kleis, Solid particle erosion: occurrence, prediction and control. Springer, London, 2008; Karimi and Schmid, Wear 156(1):33–47, 1992). Several studies (Karimi and Schmid, Wear 156(1):33–47, 1992; Neopane and Cervantes, Global J Res Eng Mech Mech Eng 11:17–26, 2011; Zeng et al., Proc IMechE 231(3):182–196, 2017; Padhy and Saini, Energy 39(1):286–293, 2012; Bajracharya et al., Wear 264(3–4):177–184, 2008; Kumar and Saini, Renew Sust Energ Rev 14(1):374–383, 2010; Xiao et al., J Hydrodyn B 19(3):356–364, 2007; Zhu et al., Nucl Eng Des 273:396–411, 2014; Chongji et al., IOP Conf Ser Earth Environ Sci 22(5):052019, 2014; Finnie and Kabil, Wear 8(1):60–69, 1965; Finnie, Wear 3(2):87–103, 1960; Finnie, Wear 19(1): 81–90, 1972) have been developed to investigate this phenomenon. Nevertheless, few attention has been paid to influence of small eddies during the erosion process, limiting the accuracy of the results in the simulation. The use of spectral element methods can allow increasing accuracy in the simulation due to the potential to consider these smaller scales.To the best of the authors’ knowledge, there is no work that uses high order methods to evaluate erosion wear rate. This research aim is to assess the impact of higher resolution methods on the prediction of erosion wear rate and distribution. Nektar+ + is a framework to solve partial differential equations using spectral/hp element method (Cantwell et al., Comput Phys Commun 192:205–219, 2015). This work presents a methodology to model erosion wear in Nektar+ +, with the creation of two modules. The first one is for the particle tracking evaluation and the record of impact on the walls of interest. The second one implements the Hutchings and Finnie models (Finnie, Wear 3(2):87–103, 1960; Finnie, Wear 19(1):81–90, 1972; Padhy and Saini, Renew Sust Energ Rev 12(7):1974–1987, 2008; Dutta, Bulle-effect and its implications for morphodynamics of river diversions. Ph.D. Thesis. University of Illinois at Urbana-Champaign, 2017) to predict the material removal rate as a function of the velocity and the angle of impact of each particle, the particle diameter and density, and the hardness relationship between particles and the wall. At the end, the particles trajectories and erosion distribution on walls are obtained. To assess effectiveness of the methodology a Backward Facing Step was investigate and the erosion patterns were found.

© 2020, The Author(s).We present the first eigenanalysis of hybridisable discontinuous Galerkin (HDG) schemes for the advection-diffusion equation in one dimension. This study is also one of the first to include viscous diffusion effects in the eigenanalysis of discontinuous spectral element methods. The interplay between upwind dissipation and viscous diffusion is discussed and preliminary insights deemed relevant to (under-resolved) turbulence computation approaches are presented.

© Springer Nature Switzerland AG 2020.This study assesses the suitability of spectral/hp continuous Galerkin (CG) schemes [1] for model-free under-resolved simulations of a non-trivial turbulent boundary layer flow. We consider a model problem proposed by Spalart in [2] that features a rotating free-stream velocity and admits an asymptotic solution with significant crossflow effects. Note this test case is substantially more complex than typical turbulent boundary layer canonical problems owing to its unsteadiness and enhanced small-scale anisotropy. Reported LES-based solutions to this problem are known to require sophisticated modelling and relatively fine grids to achieve meaningful results, with traditional models exhibiting poor performance. The model-free CG-based approach advocated, on the other hand, yields surprisingly good results with considerably less degrees of freedom for higher order discretisations. Usefully accurate results for the mean flow quantities could even be obtained with half as many degrees of freedom per direction (in comparison to reference LES solutions). Usage of high-order spectral element methods (CG in particular) is therefore strongly motivated for wall-bounded turbulence simulations via under-resolved DNS (uDNS), sometimes called implicit LES (iLES), approaches.

© 2018 Elsevier B.V.High-order spectral element methods (SEM) for large-eddy simulation (LES) are still very limited in industry. One of the main reasons behind this is the lack of robustness of SEM for under-resolved simulations, which can lead to the failure of the computation or to inaccurate results, aspects that are critical in an industrial setting. To help address this issue, we introduce a non-modal analysis technique that characterizes the numerical diffusion properties of spectral element methods for linear convection–diffusion problems, including the scales affected by numerical diffusion and the relationship between the amount of numerical diffusion and the level of under-resolution in the simulation. This framework differs from traditional eigenanalysis techniques in that all eigenmodes are taken into account with no need to differentiate them as physical or unphysical. While strictly speaking only valid for linear problems, the non-modal analysis is devised so that it can give critical insights for under-resolved nonlinear problems. For example, why do SEM sometimes suffer from numerical stability issues in LES? And, why do they at other times be robust and successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these questions in turn provides crucial guidelines to construct more robust and accurate schemes for LES. For illustration purposes, the non-modal analysis is applied to the hybridized discontinuous Galerkin methods as representatives of SEM. The effects of the polynomial order, the upwinding parameter and the Péclet number on the so-called short-term diffusion of the scheme are investigated. From a non-modal analysis point of view, and for the particular case of hybridized discontinuous Galerkin methods, polynomial orders between 2 and 4 with standard upwinding are found to be well suited for under-resolved turbulence simulations. For lower polynomial orders, diffusion is introduced in scales that are much larger than the grid resolution. For higher polynomial orders, as well as for strong under/over-upwinding, robustness issues can be expected due to low and non-monotonic numerical diffusion. The non-modal analysis results are tested against under-resolved turbulence simulations of the Burgers, Euler and Navier–Stokes equations. While devised in the linear setting, non-modal analysis successfully predicts the behavior of the scheme in the nonlinear problems considered. Although the focus of this paper is on LES, the non-modal analysis can be applied to other simulation fields characterized by under-resolved scales.

© Springer Nature Switzerland AG 2019.We consider the suitability of implicit large-eddy simulation (iLES) approaches via discontinuous Galerkin (DG) schemes. These are model-free eddy-resolving approaches which solve the governing equations in unfiltered form and rely on numerical stabilization techniques to account for the missing scales. In DG, upwind dissipation from the Riemann solver provides the baseline mechanism for regularization. DG-based iLES approaches are currently under rapid dissemination due to their success in predicting complex transitional and turbulent flows at moderate Reynolds numbers (Uranga et al, Int J Numer Meth Eng 87(1–5):232–261, 2011, [1], Gassner and Beck, Theor Comput Fluid Dyn 27(3–4):221–237, 2013, [2], Beck et al, Int J Numer Methods Fluids 76(8):522–548, 2014, [3], Wiart et al Int J Numer Methods Fluids 78:335–354, 2015, [4]). However, at higher Reynolds number, accuracy and stability issues can arise due the highly under-resolved character of the computations and the suppression of stabilizing viscous effects.

© 2018 The Author(s)This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor–Green vortex (TGV) flow to analyse the implicit large eddy simulation (iLES) capabilities of DG methods at very high Reynolds numbers. The governing equations are discretised in two ways in order to suppress aliasing errors introduced into the discrete variational forms due to the under-integration of non-linear terms. The first, more straightforward way relies on consistent/over-integration, where quadrature accuracy is improved by using a larger number of integration points, consistent with the degree of the non-linearities. The second strategy, originally applied in the high-order finite difference community, relies on a split (or skew-symmetric) form of the governing equations. Different split forms are available depending on how the variables in the non-linear terms are grouped. The desired split form is then built by averaging conservative and non-conservative forms of the governing equations, although conservativity of the DG scheme is fully preserved. A preliminary analysis based on Burgers’ turbulence in one spatial dimension is conducted and shows the potential of split forms in keeping the energy of higher-order polynomial modes close to the expected levels. This indicates that the favourable dealiasing properties observed from split-form approaches in more classical schemes seem to hold for DG. The remainder of the study considers a comprehensive set of (under-resolved) computations of the inviscid TGV flow and compares the accuracy and robustness of consistent/over-integration and split form discretisations based on the local Lax–Friedrichs and Roe-type Riemann solvers. Recent works showed that relevant split forms can stabilize higher-order inviscid TGV test cases otherwise unstable even with consistent integration. Here we show that stable high-order cases achievable with both strategies have comparable accuracy, further supporting the good dealiasing properties of split form DG. The higher-order cases achieved only with split form schemes also displayed all the main features expected from consistent/over-integration. Among test cases with the same number of degrees of freedom, best solution quality is obtained with Roe-type fluxes at moderately high orders (around sixth order). Solutions obtained with very high polynomial orders displayed spurious features attributed to a sharper dissipation in wavenumber space. Accuracy differences between the two dealiasing strategies considered were, however, observed for the low-order cases, which also yielded reduced solution quality compared to high-order results.

© 2017 The AuthorsThe study focusses on the dispersion and diffusion characteristics of discontinuous spectral element methods - specifically discontinuous Galerkin (DG) - via the spatial eigensolution analysis framework built around a one-dimensional linear problem, namely the linear advection equation. Dispersion and diffusion characteristics are of critical importance when dealing with under-resolved computations, as they affect both the numerical stability of the simulation and the solution accuracy. The spatial eigensolution analysis carried out in this paper complements previous analyses based on the temporal approach, which are more commonly found in the literature. While the latter assumes periodic boundary conditions, the spatial approach assumes inflow/outflow type boundary conditions and is therefore better suited for the investigation of open flows typical of aerodynamic problems, including transitional and fully turbulent flows and aeroacoustics. The influence of spurious/reflected eigenmodes is assessed with regard to the presence of upwind dissipation, naturally present in DG methods. This provides insights into the accuracy and robustness of these schemes for under-resolved computations, including under-resolved direct numerical simulation (uDNS) and implicit large-eddy simulation (iLES). The results estimated from the spatial eigensolution analysis are verified using the one-dimensional linear advection equation and successively by performing two-dimensional compressible Euler simulations that mimic (spatially developing) grid turbulence.

© 2017 The Author(s)This study focuses on the dispersion and diffusion characteristics of high-order energy-stable flux reconstruction (ESFR) schemes via the spatial eigensolution analysis framework proposed in [1]. The analysis is performed for five ESFR schemes, where the parameter ‘c’ dictating the properties of the specific scheme recovered is chosen such that it spans the entire class of ESFR methods, also referred to as VCJH schemes, proposed in [2]. In particular, we used five values of ‘c’ two that correspond to its lower and upper bounds and the others that identify three schemes that are linked to common high-order methods, namely the ESFR recovering two versions of discontinuous Galerkin methods and one recovering the spectral difference scheme. The performance of each scheme is assessed when using different numerical intercell fluxes (e.g. different levels of upwinding), ranging from “under-” to “over-upwinding”. In contrast to the more common temporal analysis, the spatial eigensolution analysis framework adopted here allows one to grasp crucial insights into the diffusion and dispersion properties of FR schemes for problems involving non-periodic boundary conditions, typically found in open-flow problems, including turbulence, unsteady aerodynamics and aeroacoustics.

© 2016 The AuthorsWe present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods. The ‘1% rule’ based on linear dispersion–diffusion analysis introduced by Moura et al. (2015) [10] is here adapted for 3D energy spectra and validated through the inviscid Taylor–Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra. As the original rule relies on standard upwinding, different Riemann solvers are tested. Very good agreement is found for solvers which treat the different physical waves in a consistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spurious features attributed to the inconsistent treatment of different physical waves. It is argued that, in the limit of vanishing viscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.

We present a study on the suitability of under-resolved DNS (uDNS)-also called implicit LES (iLES)-approaches based on spectral element methods (SEM), with emphasis on high-order continuous and discontinuous Galerkin (i.e. CG and DG) schemes. Broadly speaking, these are model-free eddy-resolving approaches to turbulence which solve the governing equations in unfiltered form and rely on numerical stabilization techniques for small-scale regularization. Model problems in 1D, 2D and 3D are used in the assessment of solution quality and numerical stability. A rationale for the excellent potential of these methods for transitional and turbulent flows is offered on the basis of linear dispersion-diffusion analysis.

© 2017, Springer International Publishing AG.We suggest a new interpretation of implicit large eddy simulation (iLES) approaches based on discontinuous Galerkin (DG) methods by analogy with the LES-PLB framework (Pope, Fluid mechanics and the environment: dynamical approaches. Springer, Berlin, 2001), where PLB stands for ‘projection onto local basis functions’. Within this framework, the DG discretization of the unfiltered compressible Navier-Stokes equations can be recognized as a Galerkin solution of a PLB-based (and hence filtered) version of the equations with extra terms originating from DG’s implicit subgrid-scale modelling. It is shown that for under-resolved simulations of isotropic turbulence at very high Reynolds numbers, energy dissipation is primarily determined by the property-jump term of the Riemann flux employed. Additionally, in order to assess how this dissipation is distributed in Fourier space, we compare energy spectra obtained from inviscid simulations of the Taylor-Green vortex with different Riemann solvers and polynomial orders. An explanation is proposed for the spectral ‘energy bump’ observed when the Lax-Friedrichs flux is employed.

© 2016 Elsevier Inc.This paper proposes two important improvements to shock-capturing strategies using a discontinuous Galerkin scheme, namely, accurate shock identification via finite-time Lyapunov exponent (FTLE) operators and efficient shock treatment through a point-implicit discretization of a PDE-based artificial viscosity technique. The advocated approach is based on the FTLE operator, originally developed in the context of dynamical systems theory to identify certain types of coherent structures in a flow. We propose the application of FTLEs in the detection of shock waves and demonstrate the operator's ability to identify strong and weak shocks equally well. The detection algorithm is coupled with a mesh refinement procedure and applied to transonic and supersonic flows. While the proposed strategy can be used potentially with any numerical method, a high-order discontinuous Galerkin solver is used in this study. In this context, two artificial viscosity approaches are employed to regularize the solution near shocks: an element-wise constant viscosity technique and a PDE-based smooth viscosity model. As the latter approach is more sophisticated and preferable for complex problems, a point-implicit discretization in time is proposed to reduce the extra stiffness introduced by the PDE-based technique, making it more competitive in terms of computational cost.

© 2015 The Authors.This study addresses linear dispersion-diffusion analysis for the spectral/hp continuous Galerkin (CG) formulation in one dimension. First, numerical dispersion and diffusion curves are obtained for the advection-diffusion problem and the role of multiple eigencurves peculiar to spectral/hp methods is discussed. From the eigencurves' behaviour, we observe that CG might feature potentially undesirable non-smooth dispersion/diffusion characteristics for under-resolved simulations of problems strongly dominated by either convection or diffusion. Subsequently, the linear advection equation augmented with spectral vanishing viscosity (SVV) is analysed. Dispersion and diffusion characteristics of CG with SVV-based stabilization are verified to display similar non-smooth features in flow regions where convection is much stronger than dissipation or vice-versa, owing to a dependency of the standard SVV operator on a local Péclet number. First a modification is proposed to the traditional SVV scaling that enforces a globally constant Péclet number so as to avoid the previous issues. In addition, a new SVV kernel function is suggested and shown to provide a more regular behaviour for the eigencurves along with a consistent increase in resolution power for higher-order discretizations, as measured by the extent of the wavenumber range where numerical errors are negligible. The dissipation characteristics of CG with the SVV modifications suggested are then verified to be broadly equivalent to those obtained through upwinding in the discontinuous Galerkin (DG) scheme. Nevertheless, for the kernel function proposed, the full upwind DG scheme is found to have a slightly higher resolution power for the same dissipation levels. These results show that improved CG-SVV characteristics can be pursued via different kernel functions with the aid of optimization algorithms.

© 2015John Wiley & Sons, Ltd.When dealing with high-order numerical methods, an adequate treatment of curved surfaces is required not only to guarantee that the expected high-order is maintained in the vicinity of surfaces but also to avoid steady-state convergence issues. Among the variety of high-order surface treatment techniques that have been proposed, the ones employing NURBS (non-uniform rational B-splines) to describe curved surfaces can be considered superior both in terms of accuracy and compatibility with computer-aided design softwares. The current study describes in detail the integration of NURBS-based geometry description in a high-order solver based on the discontinuous Galerkin formulation. Particularly, this work also discusses how and why NURBS curves of very high order can be employed within standard NURBS-based boundary treatment techniques to yield reduced implementation complexity and computational overhead. Theoretical estimates are provided along with numerical experiments in order to support the proposed approach. Minding engineering applications in the context of compressible aerodynamics, additional simulations are addressed as numerical examples to illustrate the advantages of using higher-order NURBS in practical situations.

© 2015.We investigate the potential of linear dispersion-diffusion analysis in providing direct guidelines for turbulence simulations through the under-resolved DNS (sometimes called implicit LES) approach via spectral/. hp methods. The discontinuous Galerkin (DG) formulation is assessed in particular as a representative of these methods. We revisit the eigensolutions technique as applied to linear advection and suggest a new perspective to the role of multiple numerical modes, peculiar to spectral/. hp methods. From this new perspective, "secondary" eigenmodes are seen to replicate the propagation behaviour of a "primary" mode, so that DG's propagation characteristics can be obtained directly from the dispersion-diffusion curves of the primary mode. Numerical dissipation is then appraised from these primary eigencurves and its effect over poorly-resolved scales is quantified. Within this scenario, a simple criterion is proposed to estimate DG's effective resolution in terms of the largest wavenumber it can accurately resolve in a given hp approximation space, also allowing us to present points per wavelength estimates typically used in spectral and finite difference methods. Although strictly valid for linear advection, the devised criterion is tested against (1D) Burgers turbulence and found to predict with good accuracy the beginning of the dissipation range on the energy spectra of under-resolved simulations. The analysis of these test cases through the proposed methodology clarifies why and how the DG formulation can be used for under-resolved turbulence simulations without explicit subgrid-scale modelling. In particular, when dealing with communication limited hardware which forces one to consider the performance for a fixed number of degrees of freedom, the use of higher polynomial orders along with moderately coarser meshes is shown to be the best way to translate available degrees of freedom into resolution power.

© Springer International Publishing Switzerland 2015.In this paper we present an assessment of the discontinuous Galerkin (DG) formulation through modified equation analysis (MEA). When applied to linear advection, MEA can help to clarify wave-propagation properties previously observed in DG. In particular, a connection between MEA and dispersion-diffusion (eigensolution) analysis is highlighted. To the authors’ knowledge this is the first application of MEA to DG schemes, and as such this study focuses only on element-wise constant and linear discretizations in one dimension. For the linear discretization, we found that the physical mode’s accuracy can be increased via upwinding. MEA’s application to higher order solutions and non-linear problems is also briefly discussed. In special, we point out that MEA’s applicability in the analysis of DG-based implicit large eddy simulations seems infeasible due to convergence issues.