Hybrid Spectral Difference/Embedded Finite Volume Method for Conservation Laws

A novel hybrid spectral difference/embedded finite volume method is introduced in order to apply a discontinuous high-order method for large scale engineering applications involving discontinuities in the flows with complex geometries. In the proposed hybrid approach, the finite volume (FV) element, consisting of structured FV subcells, is embedded in the base hexahedral element containing discontinuity, and an FV based high-order shock-capturing scheme is employed to overcome the Gibbs phenomena. Thus, a discontinuity is captured at the resolution of FV subcells within an embedded FV element. In the smooth flow region, the SD element is used in the base hexahedral element. Then, the governing equations are solved by the SD method. The SD method is chosen for its low numerical dissipation and computational efficiency preserving high-order accurate solutions. The coupling between the SD element and the FV element is achieved by the globally conserved mortar method. In this paper, the 5th-order WENO scheme with the characteristic decomposition is employed as the shock-capturing scheme in the embedded FV element, and the 5th-order SD method is used in the smooth flow field. The order of accuracy study and various 1D and 2D test cases are carried out, which involve the discontinuities and vortex flows. Overall, it is shown that the proposed hybrid method results in comparable or better simulation results compared with the standalone WENO scheme when the same number of solution DOF is considered in both SD and FV elements.Comment: 27 pages, 17 figures, 2 tables, Accepted for publication in the Journal of Computational Physics, April 201

A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, $\mathrm{M}\approx 0.1$, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to a genuinely incompressible formulation. Our contributions to the state-of-the-art are twofold: Firstly, we present a high-performance discontinuous Galerkin solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures. The performance results presented in this work focus on the node-level performance and our results suggest that there is great potential for further performance improvements for current state-of-the-art discontinuous Galerkin implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows

Non-modal analysis of spectral element methods: Towards accurate and robust large-eddy simulations

We introduce a \textit{non-modal} analysis technique that characterizes the diffusion properties of spectral element methods for linear convection-diffusion systems. While strictly speaking only valid for linear problems, the analysis is devised so that it can give critical insights on two questions: (i) Why do spectral element methods suffer from stability issues in under-resolved computations of nonlinear problems? And, (ii) why do they successfully predict under-resolved turbulent flows even without a subgrid-scale model? The answer to these two questions can in turn provide crucial guidelines to construct more robust and accurate schemes for complex under-resolved flows, commonly found in industrial applications. For illustration purposes, this analysis technique is applied to the hybridized discontinuous Galerkin methods as representatives of spectral element methods. The effect of the polynomial order, the upwinding parameter and the P\'eclet number on the so-called \textit{short-term diffusion} of the scheme are investigated. From a purely non-modal analysis point of view, polynomial orders between $2$ and $4$ with standard upwinding are 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. The non-modal analysis results are then tested against under-resolved turbulence simulations of the Burgers, Euler and Navier-Stokes equations. While devised in the linear setting, our non-modal analysis succeeds to predict the behavior of the scheme in the nonlinear problems considered

On the Properties of Energy Stable Flux Reconstruction Schemes for Implicit Large Eddy Simulation

We begin by investigating the stability, order of accuracy, and dispersion and dissipation characteristics of the extended range of energy stable flux reconstruction (E-ESFR) schemes in the context of implicit large eddy simulation (ILES). We proceed to demonstrate that subsets of the E-ESFR schemes are more stable than collocation nodal discontinuous Galerkin methods recovered with the flux reconstruction approach (FRDG) for marginally-resolved ILES simulations of the Taylor-Green vortex. These schemes are shown to have reduced dissipation and dispersion errors relative to FRDG schemes of the same polynomial degree and, simultaneously, have increased CourantFriedrichs-Lewy (CFL) limits. Finally, we simulate turbulent flow over an SD7003 aerofoil using two of the most stable E-ESFR schemes identified by the aforementioned Taylor-Green vortex experiments. Results demonstrate that subsets of E-ESFR schemes appear more stable than the commonly used FRDG method, have increased CFL limits, and are suitable for ILES of complex turbulent flows on unstructured grids

On the use of spectral element methods for under-resolved simulations of transitional and turbulent flows

The present thesis comprises a sequence of studies that investigate the suitability of spectral element methods for model-free under-resolved computations of transitional and turbulent flows. More specifically, the continuous and the discontinuous Galerkin (i.e. CG and DG) methods have their performance assessed for under-resolved direct numerical simulations (uDNS) / implicit large eddy simulations (iLES). In these approaches, the governing equations of fluid motion are solved in unfiltered form, as in a typical direct numerical simulation, but the degrees of freedom employed are insufficient to capture all the turbulent scales. Numerical dissipation introduced by appropriate stabilisation techniques complements molecular viscosity in providing small-scale regularisation at very large Reynolds numbers. Added spectral vanishing viscosity (SVV) is considered for CG, while upwind dissipation is relied upon for DG-based computations. In both cases, the use of polynomial dealiasing strategies is assumed. Focus is given to the so-called eigensolution analysis framework, where numerical dispersion and diffusion errors are appraised in wavenumber/frequency space for simplified model problems, such as the one-dimensional linear advection equation. In the assessment of CG and DG, both temporal and spatial eigenanalyses are considered. While the former assumes periodic boundary conditions and is better suited for temporally evolving problems, the latter considers inflow / outflow type boundaries and should be favoured for spatially developing flows. Despite the simplicity of linear eigensolution analyses, surprisingly useful insights can be obtained from them and verified in actual turbulence problems. In fact, one of the most important contributions of this thesis is to highlight how linear eigenanalysis can be helpful in explaining why and how to use spectral element methods (particularly CG and DG) in uDNS/iLES approaches. Various aspects of solution quality and numerical stability are discussed by connecting observations from eigensolution analyses and under-resolved turbulence computations. First, DG’s temporal eigenanalysis is revisited and a simple criterion named "the 1% rule" is devised to estimate DG’s effective resolution power in spectral space. This criterion is shown to pinpoint the wavenumber beyond which a numerically induced dissipation range appears in the energy spectra of Burgers turbulence simulations in one dimension. Next, the temporal eigenanalysis of CG is discussed with and without SVV. A modified SVV operator based on DG’s upwind dissipation is proposed to enhance CG’s accuracy and robustness for uDNS / iLES. In the sequence, an extensive set of DG computations of the inviscid Taylor-Green vortex model problem is considered. These are used for the validation of the 1% rule in actual three-dimensional transitional / turbulent flows. The performance of various Riemann solvers is also discussed in this infinite Reynolds number scenario, with high quality solutions being achieved. Subsequently, the capabilities of CG for uDNS/iLES are tested through a complex turbulent boundary layer (periodic) test problem. While LES results of this test case are known to require sophisticated modelling and relatively fine grids, high-order CG approaches are shown to deliver surprisingly good quality with significantly less degrees of freedom, even without SVV. Finally, spatial eigenanalyses are conducted for DG and CG. Differences caused by upwinding levels and Riemann solvers are explored in the DG case, while robust SVV design is considered for CG, again by reference to DG’s upwind dissipation. These aspects are then tested in a two-dimensional test problem that mimics spatially developing grid turbulence. In summary, a point is made that uDNS/iLES approaches based on high-order spectral element methods, when properly stabilised, are very powerful tools for the computation of practically all types of transitional and turbulent flows. This capability is argued to stem essentially from their superior resolution power per degree of freedom and the absence of (often restrictive) modelling assumptions. Conscientious usage is however necessary as solution quality and numerical robustness may depend strongly on discretisation variables such as polynomial order, appropriate mesh spacing, Riemann solver, SVV parameters, dealiasing strategy and alternative stabilisation techniques.Open Acces

Subgrid-scale modeling and implicit numerical dissipation in DG-based Large-Eddy Simulation

Over the past few years, high-order discontinuous Galerkin (DG) methods for Large-Eddy Simulation (LES) have emerged as a promising approach to solve complex turbulent flows. However, despite the significant research investment, the relation between the discretization scheme, the subgrid-scale (SGS) model and the resulting LES solver remains unclear. This paper aims to shed some light on this matter. To that end, we investigate the role of the Riemann solver, the SGS model, the time resolution, and the accuracy order in the ability to predict a variety of flow regimes, including transition to turbulence, wall-free turbulence, wall-bounded turbulence, and turbulence decay. The transitional flow over the Eppler 387 wing, the TaylorGreen vortex problem and the turbulent channel flow are considered to this end. The focus is placed on post-processing the LES results and providing with a rationale for the performance of the various approaches.United States. Air Force. Office of Scientific Research (FA9550-16-1-0214