On the accuracy of high-order discretizations for underresolved turbulence simulations
01 Jun 2013-Theoretical and Computational Fluid Dynamics (Springer-Verlag)-Vol. 27, Iss: 3, pp 221-237
TL;DR: It is shown that a low-order approximation exhibits unacceptable numerical discretization errors, whereas a naive application of high-order discretizations in those situations is often unstable due to aliasing, so proper stabilization is necessary for a successful computation of underresolved turbulence.
Abstract: In this paper, we investigate the accuracy of a high-order discontinuous Galerkin discretization for the coarse resolution simulation of turbulent flow. We show that a low-order approximation exhibits unacceptable numerical discretization errors, whereas a naive application of high-order discretizations in those situations is often unstable due to aliasing. Thus, for high-order simulations of underresolved turbulence, proper stabilization is necessary for a successful computation. Two different mechanisms are chosen, and their impact on the accuracy of underresolved high-order computations of turbulent flows is investigated. Results of these approximations for the Taylor–Green Vortex problem are compared to direct numerical simulation results from literature. Our findings show that the superior discretization properties of high-order approximations are retained even for these coarsely resolved computations.
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TL;DR: It is shown that besides the construction of entropy stable high-order schemes, a careful choice of subcell finite volume fluxes generates split formulations of quadratic or cubic terms, which are able to generate, in a systematic way, all common split forms of the compressible Euler advection terms.
278 citations
Cites background or methods from "On the accuracy of high-order discr..."
...However, recent investigations show that it is possible to achieve quite accurate results with (very) high-order DG schemes, even with under-resolution as long as proper de-aliasing mechanisms are augmented [15]....
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...For solutions of the incompressible viscous Taylor-Green vortex, the dissipation rate is directly linked to the enstrophy via the physical viscosity μ [15] − d κ dt = 2μσ....
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...In the discontinuous Galerkin community, stabilising an approximation is frequently done by polynomial de-aliasing through over-integration [15, 23, 30]....
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TL;DR: In this paper, a data-based approach to turbulence modeling by artificial neural networks is presented, which can generalize from the data and learn approximations with a cross correlation of up to 47% and even 73% for the inner elements, demonstrating that the perfect closure can indeed be learned from provided coarse grid data.
216 citations
TL;DR: In this paper, the authors investigated the accuracy and efficiency of discontinuous Galerkin spectral method simulations of under-resolved transitional and turbulent flows at moderate Reynolds numbers, where the accurate prediction of closely coupled laminar regions, transition and developed turbulence presents a great challenge to large eddy simulation modelling.
Abstract: SUMMARY
In this paper, we investigate the accuracy and efficiency of discontinuous Galerkin spectral method simulations of under-resolved transitional and turbulent flows at moderate Reynolds numbers, where the accurate prediction of closely coupled laminar regions, transition and developed turbulence presents a great challenge to large eddy simulation modelling. We take full advantage of the low numerical errors and associated superior scale resolving capabilities of high-order spectral methods by using high-order ansatz functions up to 12th order. We employ polynomial de-aliasing techniques to prevent instabilities arising from inexact quadrature of nonlinearities. Without the need for any additional filtering, explicit or implicit modelling, or artificial dissipation, our high-order schemes capture the turbulent flow at the considered Reynolds number range very well. Three classical large eddy simulation benchmark problems are considered: a circular cylinder flow at ReD=3900, a confined periodic hill flow at Reh=2800 and the transitional flow over a SD7003 airfoil at Rec=60,000. For all computations, the total number of degrees of freedom used for the discontinuous Galerkin spectral method simulations is chosen to be equal or considerably less than the reported data in literature. In all three cases, we achieve an equal or better match to direct numerical simulation results, compared with other schemes of lower order with explicitly or implicitly added subgrid scale models. Copyright © 2014 John Wiley & Sons, Ltd.
194 citations
Cites background or methods from "On the accuracy of high-order discr..."
...These efforts were accompanied by a number of researchers focusing on alternative approximations for the viscous fluxes [10–14], shock capturing *Correspondence to: Andrea D. Beck, Institute for Aerodynamics and Gasdynamics, University of Stuttgart, Pfaffenwaldring 21, 70550 Stuttgart, Germany....
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...Beck AD, Gassner GJ, Munz CD....
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...In this paper, we continue and extend the work from [32] by investigating the applicability of high-order DG schemes for transitional and turbulent flows....
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...For higher orders (N D 7:::15), Gassner and Beck [32] showed that stabilized DG schemes applied to under-resolved simulations of isotropic turbulence compare very well with low-order finite volume schemes with explicit or implicit subgrid scale models....
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...Gassner GJ, Beck AD....
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TL;DR: Why and how the discontinuous Galerkin (DG) formulation can be used for under-resolved turbulence simulations without explicit subgrid-scale modelling is clarified and 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.
142 citations
Cites background or result from "On the accuracy of high-order discr..."
...Regarding under-resolved simulations, a detailed study carried in [18] showed that, for a given number of DOFs, increasing the polynomial order is more effective than refining the mesh in order to improve the accuracy....
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...The advantages of this strategy (as compared to mesh refinement) were first pointed out in [18], but on a more qualitative basis....
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...This feature has also been recognized in previous works [17, 18]....
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TL;DR: Estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods are presented and are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.
137 citations
Cites result from "On the accuracy of high-order discr..."
...However, for example when DG-based uDNS test cases of the same DOFs are compared, higher-order solutions (on coarser meshes), can outperform low-order ones (on finer grids) and follow much more closely reference DNS results [11]....
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References
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TL;DR: In this article, the authors propose a definition of vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor, which captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers.
Abstract: Considerable confusion surrounds the longstanding question of what constitutes a vortex, especially in a turbulent flow. This question, frequently misunderstood as academic, has recently acquired particular significance since coherent structures (CS) in turbulent flows are now commonly regarded as vortices. An objective definition of a vortex should permit the use of vortex dynamics concepts to educe CS, to explain formation and evolutionary dynamics of CS, to explore the role of CS in turbulence phenomena, and to develop viable turbulence models and control strategies for turbulence phenomena. We propose a definition of a vortex in an incompressible flow in terms of the eigenvalues of the symmetric tensor ${\bm {\cal S}}^2 + {\bm \Omega}^2$ are respectively the symmetric and antisymmetric parts of the velocity gradient tensor ${\bm \Delta}{\bm u}$. This definition captures the pressure minimum in a plane perpendicular to the vortex axis at high Reynolds numbers, and also accurately defines vortex cores at low Reynolds numbers, unlike a pressure-minimum criterion. We compare our definition with prior schemes/definitions using exact and numerical solutions of the Euler and Navier–Stokes equations for a variety of laminar and turbulent flows. In contrast to definitions based on the positive second invariant of ${\bm \Delta}{\bm u}$ or the complex eigenvalues of ${\bm \Delta}{\bm u}$, our definition accurately identifies the vortex core in flows where the vortex geometry is intuitively clear.
5,837 citations
Additional excerpts
...It shows a visualization of the vortex structure defined by the λ2 criterion [24] of an O(2) and a stabilized O(16) computation with the same number of degrees of freedom alongside a DNS reference solution....
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Book•
18 Dec 2007
TL;DR: The text offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations.
Abstract: The text offers an introduction to the key ideas, basic analysis, and efficient implementation of discontinuous Galerkin finite element methods (DG-FEM) for the solution of partial differential equations. All key theoretical results are either derived or discussed, including an overview of relevant results from approximation theory, convergence theory for numerical PDEs, orthogonal polynomials etc. Through embedded Matlab codes, the algorithms are discussed and implemented for a number of classic systems of PDEs, e.g., Maxwells equations, Euler equations, incompressible Navier-Stokes equations, and Poisson- and Helmholtz equations. These developments are done in detail inone and two dimensions on general unstructured grids with high-order elements and all essential routines for 3D extensions are also included and discussed briefly. The three appendices contain an overview of orthogonal polynomials and associated library routines used throughout, a brief introduction to grid generation, and an overview of the associated software (where to get it, list of variables etc). A variety of exercises are included at the end of most chapters.
2,014 citations
Additional excerpts
...the book of Hesthaven and Warburton [20]....
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...[20]....
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TL;DR: This paper extends a discontinuous finite element discretization originally considered for hyperbolic systems such as the Euler equations to the case of the Navier?Stokes equations by treating the viscous terms with a mixed formulation, and finds the method is ideally suited to compute high-order accurate solution of theNavier?
1,750 citations
"On the accuracy of high-order discr..." refers methods in this paper
...Instead, the modified gradients Q ≈∇xU introduced by Bassi and Rebay [3,2] are used ∫...
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TL;DR: In this article, the dynamics of inviscid and viscous Taylor-Green (TG) vortex flows are investigated by both direct spectral numerical solution of the Navier-Stokes equations and by power-series analysis in time.
Abstract: The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 256 3 modes) and by power-series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance $\hat{\delta}(t)$ of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that $\hat{\delta}(t)$ decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity ( $\hat{\delta}(t) = 0$ ) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t 44 to order t 80 . Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t max at which the maximum energy dissipation is achieved. Early-time, high- R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t max the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near t max when R = 3000 and R λ = 110:
a k − n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times); th energy dissipation has considerable spatial intermittency; its spectrum has a k −1+μ inertial range with the codimension μ ≈ 0.3−0.7. Skewness and flatness results are also presented.
684 citations
"On the accuracy of high-order discr..." refers background or methods in this paper
...DNS data from Brachet [7,8] (Re = 800, 1600) and Fauconnier et al....
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...involved physics, the TGV is well suited as a benchmark problem for code validation and turbulence model evaluation and has been studied extensively in literature ([15],[7],[21],[8])....
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TL;DR: In this article, a drag reduction mechanism by riblets with small spacings was proposed to reduce viscous drag by restricting the location of the streamwise vortices above the wetted surface.
Abstract: Direct numerical simulations of turbulent flows over riblet-mounted surfaces are performed to educe the mechanism of drag reduction by riblets. The computed drag on the riblet surfaces is in good agreement with the existing experimental data. The mean-velocity profiles show upward and downward shifts in the log–law for drag-decreasing and drag-increasing cases, respectively. Turbulence statistics above the riblets are computed and compared with those above a flat plate. Differences in the mean-velocity profile and turbulence quantities are found to be limited to the inner region of the boundary layer. Velocity and vorticity fluctuations as well as the Reynolds shear stresses above the riblets are reduced in drag-reducing configurations. Quadrant analysis indicates that riblets mitigate the positive Reynolds-shear-stress-producing events in drag-reducing configurations. From examination of the instantaneous flow fields, a drag reduction mechanism by riblets is proposed: riblets with small spacings reduce viscous drag by restricting the location of the streamwise vortices above the wetted surface such that only a limited area of the riblets is exposed to the downwash of high-speed fluid that the vortices induce.
657 citations