Eigensolution analysis of spectral/hp continuous Galerkin approximations to advection-diffusion problems: insights into spectral vanishing viscosity

AbstractThis 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

Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods

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

Gradient jump penalty stabilisation of spectral/hp element discretisation for under-resolved turbulence simulations

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). The latter, although recognised as very useful in CG-based high-fidelity turbulence simulations, 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 by 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 apriori 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 high-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

Brazilian Bidens pilosa Linne´ yields fraction containing quercetin-derived flavonoid with free radical scavenger activity and hepatoprotective effects

Bidens pilosa is a plant used by Amazonian and Asian folks for some hepatopathies. The hydroethanol crude extract and three fractions were assessed for antioxidant and hepatoprotective effects. Higher levels of scavenger activity on the 1,1-diphenyl-2-picrylhydrazyl radical, inhibition of deoxyribose oxidation and lipid peroxidation in vitro were detected for the ethyl acetate fraction (IC50∼4.3-32.3 mg/ml) followed by the crude extract (IC50∼14.2-98.0 mg/ml). The ethyl acetate fraction, again followed by the crude extract, showed high contents of total soluble polyphenols (3.6±0.2 and 2.1±0.2 GAE/mg, respectively) and presence of a quercetin-derived flavonoid identified as quercetin 3,3′-dimethyl ether 7-Ο-β-D-glycopyranoside. Both products were assayed for hepatoprotector effects against CCl4-induced liver injury in mice. Markers of oxidative stress and hepatic injury were evaluated. The results showed that the 10-day pretreatments (15 mg/kg, p.o.) protected the livers against injury by blocking CCl4-induced lipid peroxidation and protein carbonylation and the DNA fragmentation was decreased (∼60%). The pretreatments avoided the loss of the plasma ferric reducing/antioxidant power and the elevation of serum transaminases and lactate dehydrogenase activities. The results suggest that the main constituents responsible for the hepatoprotective effects with free radical scavenger power associated are well extracted by performing fractionation with ethyl acetate. The findings support the Brazilian traditional use of this plant and justify further evaluations for the therapeutic efficacy and safety of the constituents of the ethyl acetate fraction to treat some liver diseases.Keywords: Bidens pilosa L.; hydroethanol maceration; ethyl acetate fractionation; free radical scavenger; hepatoprotection; CCl4 toxicit

High-order propagation of jet noise on a tetrahedral mesh using large eddy simulation sources

Jet noise is an important area of research in commercial aviation due to its high contribution to the overall noise generated by an aircraft. Conventionally, CFD combined with surface integral methods is used to study jet noise because of its low cost. However, it is not always trivial to define integration surfaces around complex geometries. This study employs a different two-step approach that can handle complex geometries. It combines a large-eddy simulation (LES) to obtain the acoustic sources from the flow field, and an acoustic perturbation equations (APE) solver to propagate the sound to the far field. The LES is performed with an industrial 2nd-order finite volume solver. The APE code is a high-order discontinuous Galerkin (DG) spectral/hp solver of the Nektar+ + framework. The APE solver is validated on a canonical test case. A study on different polynomial expansion orders and meshes is further performed to estimate the mesh size for noise propagation in the high-order spectral/hp DG context. Finally, a three-dimensional jet noise case (Re = 10, 000 and Mach = 0.9) is simulated using unstructured tetrahedral mesh for the APE solver and improved noise results for high frequencies are obtained. The results demonstrate that the present approach is capable of predicting noise in complex geometry scenarios, such as installed jets under the aircraft wings

Industry-relevant implicit large-eddy simulation of a high-performance road car via spectral/HP element methods

This is the author accepted manuscript. The final version is available from the Society for Industrial and Applied Mathematics via the DOI in this recordWe 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, 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++.Engineering and Physical Sciences Research Council (EPSRC)European Union Horizon 2020Imperial College High Performance Computing ServiceNational University of SingaporeAirbu

Stringy effects in black hole decay

We compute the low energy decay rates of near-extremal three(four) charge black holes in five(four) dimensional N=4 string theory to sub-leading order in the large charge approximation. This involves studying stringy corrections to scattering amplitudes of a scalar field off a black hole. We adapt and use recently developed techniques to compute such amplitudes as near-horizon quantities. We then compare this with the corresponding calculation in the microscopic configuration carrying the same charges as the black hole. We find perfect agreement between the microscopic and macroscopic calculations; in the cases we study, the zero energy limit of the scattering cross section is equal to four times the Wald entropy of the black hole.Comment: 32 page