Dealiasing techniques for high-order spectral element methods on regular and irregular grids

High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations

On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence

AbstractWe 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

Dealiasing techniques for high-order spectral element methods on regular and irregular grids

This is the final version of the article. Available from Elsevier via the DOI in this record.High-order methods are becoming increasingly attractive in both academia and industry, especially in the context of computational fluid dynamics. However, before they can be more widely adopted, issues such as lack of robustness in terms of numerical stability need to be addressed, particularly when treating industrial-type problems where challenging geometries and a wide range of physical scales, typically due to high Reynolds numbers, need to be taken into account. One source of instability is aliasing effects which arise from the nonlinearity of the underlying problem. In this work we detail two dealiasing strategies based on the concept of consistent integration. The first uses a localised approach, which is useful when the nonlinearities only arise in parts of the problem. The second is based on the more traditional approach of using a higher quadrature. The main goal of both dealiasing techniques is to improve the robustness of high order spectral element methods, thereby reducing aliasing-driven instabilities. We demonstrate how these two strategies can be effectively applied to both continuous and discontinuous discretisations, where, in the latter, both volumetric and interface approximations must be considered. We show the key features of each dealiasing technique applied to the scalar conservation law with numerical examples and we highlight the main differences in terms of implementation between continuous and discontinuous spatial discretisations.This work was supported by the Laminar Flow Control Centre funded by Airbus/EADS and EPSRC under grant EP/I037946. We thank Dr. Colin Cotter for helpful discussions and Jean-Eloi Lombard for his assistance in the generation of results and figures for the NACA 0012 simulation. PV acknowledges the Engineering and Physical Sciences Research Council for their support via an Early Career Fellowship (EP/K027379/1). SJS additionally acknowledges Royal Academy of Engineering support under their research chair scheme. Data supporting this publication can be obtained on request from [email protected]

Deterministic and stochastic chaos characterize laboratory earthquakes

We analyze frictional motion for a laboratory fault as it passes through the stability transition from stable sliding to unstable motion. We study frictional stick-slip events, which are the lab equivalent of earthquakes, via dynamical system tools in order to retrieve information on the underlying dynamics and to assess whether there are dynamical changes associated with the transition from stable to unstable motion. We find that the seismic cycle exhibits characteristics of a low-dimensional system with average dimension similar to that of natural slow earthquakes (<5). We also investigate local properties of the attractor and find maximum instantaneous dimension ≳10, indicating that some regions of the phase space require a high number of degrees of freedom (dofs). Our analysis does not preclude deterministic chaos, but the lab seismic cycle is best explained by a random attractor based on rate- and state-dependent friction whose dynamics is stochastically perturbed. We find that minimal variations of 0.05% of the shear and normal stresses applied to the experimental fault influence the large-scale dynamics and the recurrence time of labquakes. While complicated motion including period doubling is observed near the stability transition, even in the fully unstable regime we do not observe truly periodic behavior. Friction's nonlinear nature amplifies small scale perturbations, reducing the predictability of the otherwise periodic macroscopic dynamics. As applied to tectonic faults, our results imply that even small stress field fluctuations (≲150 kPa) can induce coefficient of variations in earthquake repeat time of a few percent. Moreover, these perturbations can drive an otherwise fast-slipping fault, close to the critical stability condition, into a mixed behavior involving slow and fast ruptures

Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

The 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

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